LMFDB - Embedded newform 51.7.b.a.35.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [51,7,Mod(35,51)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(51, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0]))

N = Newforms(chi, 7, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("51.35");

S:= CuspForms(chi, 7);

N := Newforms(S);

Level:	$ N $	$=$	$ 51 = 3 \cdot 17 $
Weight:	$ k $	$=$	$ 7 $
Character orbit:	$[\chi]$	$=$	51.b (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$11.7327582646$
Analytic rank:	$0$
Dimension:	$32$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			35.3
Character	$\chi$	$=$	51.35
Dual form			51.7.b.a.35.30

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-13.9722i q^{2} +(23.6370 + 13.0496i) q^{3} -131.222 q^{4} +76.5124i q^{5} +(182.331 - 330.261i) q^{6} -548.925 q^{7} +939.239i q^{8} +(388.417 + 616.906i) q^{9} +1069.05 q^{10} +1542.64i q^{11} +(-3101.70 - 1712.39i) q^{12} -3442.11 q^{13} +7669.68i q^{14} +(-998.455 + 1808.53i) q^{15} +4725.01 q^{16} +1191.58i q^{17} +(8619.53 - 5427.03i) q^{18} +3758.59 q^{19} -10040.1i q^{20} +(-12974.9 - 7163.24i) q^{21} +21554.0 q^{22} -15746.1i q^{23} +(-12256.7 + 22200.8i) q^{24} +9770.85 q^{25} +48093.8i q^{26} +(1130.64 + 19650.5i) q^{27} +72031.0 q^{28} -13783.9i q^{29} +(25269.1 + 13950.6i) q^{30} -43753.9 q^{31} -5907.44i q^{32} +(-20130.8 + 36463.4i) q^{33} +16648.9 q^{34} -41999.6i q^{35} +(-50968.8 - 80951.7i) q^{36} -11046.9 q^{37} -52515.8i q^{38} +(-81361.2 - 44918.1i) q^{39} -71863.5 q^{40} +29854.9i q^{41} +(-100086. + 181288. i) q^{42} -11876.0 q^{43} -202428. i q^{44} +(-47201.0 + 29718.7i) q^{45} -220007. q^{46} +24476.0i q^{47} +(111685. + 61659.4i) q^{48} +183669. q^{49} -136520. i q^{50} +(-15549.6 + 28165.3i) q^{51} +451681. q^{52} +164876. i q^{53} +(274560. - 15797.6i) q^{54} -118031. q^{55} -515571. i q^{56} +(88841.9 + 49048.1i) q^{57} -192591. q^{58} +81554.6i q^{59} +(131019. - 237318. i) q^{60} -430458. q^{61} +611338. i q^{62} +(-213212. - 338635. i) q^{63} +219861. q^{64} -263364. i q^{65} +(509473. + 281271. i) q^{66} -6416.60 q^{67} -156361. i q^{68} +(205479. - 372190. i) q^{69} -586826. q^{70} +374466. i q^{71} +(-579422. + 364816. i) q^{72} +468151. q^{73} +154349. i q^{74} +(230954. + 127505. i) q^{75} -493210. q^{76} -846792. i q^{77} +(-627604. + 1.13679e6i) q^{78} -72781.8 q^{79} +361522. i q^{80} +(-229706. + 479234. i) q^{81} +417138. q^{82} -821984. i q^{83} +(1.70260e6 + 939975. i) q^{84} -91170.5 q^{85} +165933. i q^{86} +(179874. - 325810. i) q^{87} -1.44891e6 q^{88} +467567. i q^{89} +(415236. + 659501. i) q^{90} +1.88946e6 q^{91} +2.06623e6i q^{92} +(-1.03421e6 - 570971. i) q^{93} +341983. q^{94} +287579. i q^{95} +(77089.6 - 139634. i) q^{96} -931439. q^{97} -2.56626e6i q^{98} +(-951663. + 599186. i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q + 32 q^{3} - 1024 q^{4} + 286 q^{6} + 568 q^{7} - 912 q^{9} - 744 q^{10} + 194 q^{12} - 2312 q^{13} - 6240 q^{15} + 13208 q^{16} + 2936 q^{18} + 7936 q^{19} - 21688 q^{21} + 13176 q^{22} + 18282 q^{24}+ \cdots + 1619864 q^{99}+O(q^{100}) $ 32 * q + 32 * q^3 - 1024 * q^4 + 286 * q^6 + 568 * q^7 - 912 * q^9 - 744 * q^10 + 194 * q^12 - 2312 * q^13 - 6240 * q^15 + 13208 * q^16 + 2936 * q^18 + 7936 * q^19 - 21688 * q^21 + 13176 * q^22 + 18282 * q^24 - 76792 * q^25 + 87056 * q^27 + 45712 * q^28 - 45620 * q^30 - 36728 * q^31 + 73456 * q^33 - 52272 * q^36 + 133336 * q^37 - 48720 * q^39 - 81084 * q^40 - 305352 * q^42 - 412832 * q^43 - 9936 * q^45 + 283044 * q^46 + 233266 * q^48 + 342960 * q^49 + 50152 * q^52 + 196798 * q^54 + 7104 * q^55 - 22576 * q^57 + 167484 * q^58 - 58564 * q^60 - 235496 * q^61 - 331312 * q^63 + 299888 * q^64 + 485816 * q^66 - 774272 * q^67 + 553800 * q^69 - 1318656 * q^70 + 447312 * q^72 - 33152 * q^73 + 86872 * q^75 + 1045000 * q^76 - 8652 * q^78 + 1789576 * q^79 - 2391192 * q^81 - 178380 * q^82 - 751024 * q^84 + 235824 * q^85 - 155232 * q^87 - 3091500 * q^88 - 728056 * q^90 + 2080448 * q^91 + 1012816 * q^93 + 3173160 * q^94 + 422058 * q^96 - 1502960 * q^97 + 1619864 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/51\mathbb{Z}\right)^\times$.

$n$	$35$	$37$
$\chi(n)$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		−	13.9722i		−	1.74652i	−0.487251	−	0.873262i	$-0.662000\pi$
$2$		−	13.9722i		−	1.74652i	0.487251	−	0.873262i	$-0.338000\pi$
$3$	23.6370	+	13.0496i	0.875445	+	0.483318i
$3$	23.6370	+	13.0496i	0.875445	+	0.483318i
$4$	−131.222			−2.05034
$5$			76.5124i			0.612100i	0.952016	+	0.306050i	$0.0990074\pi$
$5$			76.5124i			0.612100i	−0.952016	+	0.306050i	$0.900993\pi$
$6$	182.331	−	330.261i	0.844126	−	1.52899i
$7$	−548.925			−1.60036			−0.800182	−	0.599757i	$-0.795264\pi$
$7$	−548.925			−1.60036			−0.800182	+	0.599757i	$0.795264\pi$
$8$			939.239i			1.83445i
$9$	388.417	+	616.906i	0.532808	+	0.846236i
$10$	1069.05			1.06905
$11$			1542.64i			1.15901i	0.814970	+	0.579503i	$0.196753\pi$
$11$			1542.64i			1.15901i	−0.814970	+	0.579503i	$0.803247\pi$
$12$	−3101.70	−	1712.39i	−1.79496	−	0.990968i
$13$	−3442.11			−1.56673			−0.783366	−	0.621560i	$-0.786499\pi$
$13$	−3442.11			−1.56673			−0.783366	+	0.621560i	$0.786499\pi$
$14$			7669.68i			2.79507i
$15$	−998.455	+	1808.53i	−0.295839	+	0.535859i
$16$	4725.01			1.15357
$17$			1191.58i			0.242536i
$17$			1191.58i			0.242536i
$18$	8619.53	−	5427.03i	1.47797	−	0.930561i
$19$	3758.59			0.547980			0.273990	−	0.961733i	$-0.411657\pi$
$19$	3758.59			0.547980			0.273990	+	0.961733i	$0.411657\pi$
$20$		−	10040.1i		−	1.25501i
$21$	−12974.9	−	7163.24i	−1.40103	−	0.773484i
$22$	21554.0			2.02423
$23$		−	15746.1i		−	1.29416i	−0.762422	−	0.647080i	$-0.775990\pi$
$23$		−	15746.1i		−	1.29416i	0.762422	−	0.647080i	$-0.224010\pi$
$24$	−12256.7	+	22200.8i	−0.886623	+	1.60596i
$25$	9770.85			0.625334
$26$			48093.8i			2.73633i
$27$	1130.64	+	19650.5i	0.0574427	+	0.998349i
$28$	72031.0			3.28130
$29$		−	13783.9i		−	0.565169i	−0.959242	−	0.282584i	$-0.908808\pi$
$29$		−	13783.9i		−	0.565169i	0.959242	−	0.282584i	$-0.0911917\pi$
$30$	25269.1	+	13950.6i	0.935891	+	0.516689i
$31$	−43753.9			−1.46870			−0.734348	−	0.678773i	$-0.762512\pi$
$31$	−43753.9			−1.46870			−0.734348	+	0.678773i	$0.762512\pi$
$32$		−	5907.44i		−	0.180281i
$33$	−20130.8	+	36463.4i	−0.560169	+	1.01465i
$34$	16648.9			0.423594
$35$		−	41999.6i		−	0.979582i
$36$	−50968.8	−	80951.7i	−1.09244	−	1.73508i
$37$	−11046.9			−0.218089			−0.109044	−	0.994037i	$-0.534779\pi$
$37$	−11046.9			−0.218089			−0.109044	+	0.994037i	$0.534779\pi$
$38$		−	52515.8i		−	0.957060i
$39$	−81361.2	−	44918.1i	−1.37159	−	0.757230i
$40$	−71863.5			−1.12287
$41$			29854.9i			0.433175i	0.976263	+	0.216588i	$0.0694927\pi$
$41$			29854.9i			0.433175i	−0.976263	+	0.216588i	$0.930507\pi$
$42$	−100086.	+	181288.i	−1.35091	+	2.44693i
$43$	−11876.0			−0.149370			−0.0746850	−	0.997207i	$-0.523795\pi$
$43$	−11876.0			−0.149370			−0.0746850	+	0.997207i	$0.523795\pi$
$44$		−	202428.i		−	2.37636i
$45$	−47201.0	+	29718.7i	−0.517981	+	0.326131i
$46$	−220007.			−2.26028
$47$			24476.0i			0.235747i	0.993029	+	0.117873i	$0.0376077\pi$
$47$			24476.0i			0.235747i	−0.993029	+	0.117873i	$0.962392\pi$
$48$	111685.	+	61659.4i	1.00988	+	0.557539i
$49$	183669.			1.56116
$50$		−	136520.i		−	1.09216i
$51$	−15549.6	+	28165.3i	−0.117222	+	0.212327i
$52$	451681.			3.21234
$53$			164876.i			1.10746i	0.832695	+	0.553732i	$0.186797\pi$
$53$			164876.i			1.10746i	−0.832695	+	0.553732i	$0.813203\pi$
$54$	274560.	−	15797.6i	1.74364	−	0.100325i
$55$	−118031.			−0.709427
$56$		−	515571.i		−	2.93579i
$57$	88841.9	+	49048.1i	0.479726	+	0.264848i
$58$	−192591.			−0.987080
$59$			81554.6i			0.397093i	0.980091	+	0.198547i	$0.0636221\pi$
$59$			81554.6i			0.397093i	−0.980091	+	0.198547i	$0.936378\pi$
$60$	131019.	−	237318.i	0.606571	−	1.09870i
$61$	−430458.			−1.89645			−0.948225	−	0.317600i	$-0.897123\pi$
$61$	−430458.			−1.89645			−0.948225	+	0.317600i	$0.897123\pi$
$62$			611338.i			2.56511i
$63$	−213212.	−	338635.i	−0.852686	−	1.35429i
$64$	219861.			0.838702
$65$		−	263364.i		−	0.958996i
$66$	509473.	+	281271.i	1.77210	+	0.978348i
$67$	−6416.60			−0.0213344			−0.0106672	−	0.999943i	$-0.503396\pi$
$67$	−6416.60			−0.0213344			−0.0106672	+	0.999943i	$0.503396\pi$
$68$		−	156361.i		−	0.497281i
$69$	205479.	−	372190.i	0.625491	−	1.13297i
$70$	−586826.			−1.71086
$71$			374466.i			1.04626i	0.852254	+	0.523128i	$0.175235\pi$
$71$			374466.i			1.04626i	−0.852254	+	0.523128i	$0.824765\pi$
$72$	−579422.	+	364816.i	−1.55238	+	0.977409i
$73$	468151.			1.20342			0.601710	−	0.798715i	$-0.294486\pi$
$73$	468151.			1.20342			0.601710	+	0.798715i	$0.294486\pi$
$74$			154349.i			0.380897i
$75$	230954.	+	127505.i	0.547446	+	0.302235i
$76$	−493210.			−1.12355
$77$		−	846792.i		−	1.85483i
$78$	−627604.	+	1.13679e6i	−1.32252	+	2.39551i
$79$	−72781.8			−0.147619			−0.0738093	−	0.997272i	$-0.523516\pi$
$79$	−72781.8			−0.147619			−0.0738093	+	0.997272i	$0.523516\pi$
$80$			361522.i			0.706098i
$81$	−229706.	+	479234.i	−0.432232	+	0.901762i
$82$	417138.			0.756551
$83$		−	821984.i		−	1.43757i	−0.695233	−	0.718785i	$-0.744699\pi$
$83$		−	821984.i		−	1.43757i	0.695233	−	0.718785i	$-0.255301\pi$
$84$	1.70260e6	+	939975.i	2.87259	+	1.58591i
$85$	−91170.5			−0.148456
$86$			165933.i			0.260878i
$87$	179874.	−	325810.i	0.273156	−	0.494774i
$88$	−1.44891e6			−2.12614
$89$			467567.i			0.663244i	0.943412	+	0.331622i	$0.107596\pi$
$89$			467567.i			0.663244i	−0.943412	+	0.331622i	$0.892404\pi$
$90$	415236.	+	659501.i	0.569596	+	0.904666i
$91$	1.88946e6			2.50734
$92$			2.06623e6i			2.65347i
$93$	−1.03421e6	−	570971.i	−1.28576	−	0.709847i
$94$	341983.			0.411738
$95$			287579.i			0.335418i
$96$	77089.6	−	139634.i	0.0871329	−	0.157826i
$97$	−931439.			−1.02056			−0.510281	−	0.860008i	$-0.670458\pi$
$97$	−931439.			−1.02056			−0.510281	+	0.860008i	$0.670458\pi$
$98$		−	2.56626e6i		−	2.72661i
$99$	−951663.	+	599186.i	−0.980794	+	0.617528i
$100$	−1.28215e6			−1.28215
$101$		−	1.47373e6i		−	1.43038i	−0.698928	−	0.715192i	$-0.746339\pi$
$101$		−	1.47373e6i		−	1.43038i	0.698928	−	0.715192i	$-0.253661\pi$
$102$	393531.	+	217262.i	0.370833	+	0.204731i
$103$	1.16204e6			1.06344			0.531718	−	0.846922i	$-0.321547\pi$
$103$	1.16204e6			1.06344			0.531718	+	0.846922i	$0.321547\pi$
$104$		−	3.23296e6i		−	2.87409i
$105$	548077.	−	992745.i	0.473449	−	0.857570i
$106$	2.30368e6			1.93421
$107$			558307.i			0.455745i	0.973691	+	0.227872i	$0.0731769\pi$
$107$			558307.i			0.455745i	−0.973691	+	0.227872i	$0.926823\pi$
$108$	−148365.	−	2.57858e6i	−0.117777	−	2.04696i
$109$	259325.			0.200246			0.100123	−	0.994975i	$-0.468076\pi$
$109$	259325.			0.200246			0.100123	+	0.994975i	$0.468076\pi$
$110$			1.64915e6i			1.23903i
$111$	−261115.	−	144157.i	−0.190925	−	0.105406i
$112$	−2.59367e6			−1.84613
$113$			130617.i			0.0905241i	0.998975	+	0.0452620i	$0.0144123\pi$
$113$			130617.i			0.0905241i	−0.998975	+	0.0452620i	$0.985588\pi$
$114$	685309.	−	1.24132e6i	0.462564	−	0.837853i
$115$	1.20477e6			0.792155
$116$			1.80875e6i			1.15879i
$117$	−1.33697e6	−	2.12346e6i	−0.834767	−	1.32583i
$118$	1.13950e6			0.693532
$119$		−	654086.i		−	0.388145i
$120$	−1.69864e6	−	937788.i	−0.983008	−	0.542701i
$121$	−608170.			−0.343296
$122$			6.01444e6i			3.31219i
$123$	−389594.	+	705680.i	−0.209361	+	0.379221i
$124$	5.74148e6			3.01133
$125$			1.94310e6i			0.994866i
$126$	−4.73147e6	+	2.97903e6i	−2.36529	+	1.48924i
$127$	442172.			0.215864			0.107932	−	0.994158i	$-0.465577\pi$
$127$	442172.			0.215864			0.107932	+	0.994158i	$0.465577\pi$
$128$		−	3.45001e6i		−	1.64509i
$129$	−280712.	−	154976.i	−0.130765	−	0.0721932i
$130$	−3.67978e6			−1.67491
$131$			2.15450e6i			0.958367i	0.877715	+	0.479183i	$0.159067\pi$
$131$			2.15450e6i			0.958367i	−0.877715	+	0.479183i	$0.840933\pi$
$132$	2.64160e6	−	4.78479e6i	1.14854	−	2.08037i
$133$	−2.06319e6			−0.876967
$134$			89653.9i			0.0372610i
$135$	−1.50351e6	+	86508.4i	−0.611089	+	0.0351606i
$136$	−1.11918e6			−0.444920
$137$		−	2.02372e6i		−	0.787026i	−0.919319	−	0.393513i	$-0.871260\pi$
$137$		−	2.02372e6i		−	0.787026i	0.919319	−	0.393513i	$-0.128740\pi$
$138$	−5.20030e6	−	2.87100e6i	−1.97875	−	1.09243i
$139$	−2.54076e6			−0.946062			−0.473031	−	0.881046i	$-0.656840\pi$
$139$	−2.54076e6			−0.946062			−0.473031	+	0.881046i	$0.656840\pi$
$140$			5.51127e6i			2.00848i
$141$	−319401.	+	578539.i	−0.113941	+	0.206383i
$142$	5.23211e6			1.82731
$143$		−	5.30993e6i		−	1.81585i
$144$	1.83527e6	+	2.91489e6i	0.614629	+	0.976190i
$145$	1.05464e6			0.345939
$146$		−	6.54109e6i		−	2.10180i
$147$	4.34140e6	+	2.39681e6i	1.36671	+	0.754539i
$148$	1.44959e6			0.447157
$149$			6.24567e6i			1.88808i	0.329833	+	0.944039i	$0.393008\pi$
$149$			6.24567e6i			1.88808i	−0.329833	+	0.944039i	$0.606992\pi$
$150$	1.78153e6	−	3.22693e6i	0.527861	−	0.956127i
$151$	−294031.			−0.0854009			−0.0427005	−	0.999088i	$-0.513596\pi$
$151$	−294031.			−0.0854009			−0.0427005	+	0.999088i	$0.513596\pi$
$152$			3.53022e6i			1.00524i
$153$	−735092.	+	462829.i	−0.205242	+	0.129225i
$154$	−1.18315e7			−3.23951
$155$		−	3.34772e6i		−	0.898988i
$156$	1.06764e7	+	5.89425e6i	2.81223	+	1.55258i
$157$	4.98082e6			1.28707			0.643534	−	0.765417i	$-0.277467\pi$
$157$	4.98082e6			1.28707			0.643534	+	0.765417i	$0.277467\pi$
$158$			1.01692e6i			0.257819i
$159$	−2.15156e6	+	3.89718e6i	−0.535257	+	0.969524i
$160$	451993.			0.110350
$161$			8.64340e6i			2.07113i
$162$	6.69594e6	+	3.20949e6i	1.57495	+	0.754903i
$163$	2.24420e6			0.518202			0.259101	−	0.965850i	$-0.416574\pi$
$163$	2.24420e6			0.518202			0.259101	+	0.965850i	$0.416574\pi$
$164$		−	3.91762e6i		−	0.888158i
$165$	−2.78990e6	−	1.54026e6i	−0.621065	−	0.342879i
$166$	−1.14849e7			−2.51075
$167$		−	4.33896e6i		−	0.931615i	−0.884886	−	0.465807i	$-0.845764\pi$
$167$		−	4.33896e6i		−	0.931615i	0.884886	−	0.465807i	$-0.154236\pi$
$168$	6.72799e6	−	1.21866e7i	1.41892	−	2.57012i
$169$	7.02132e6			1.45465
$170$			1.27385e6i			0.259282i
$171$	1.45990e6	+	2.31870e6i	0.291968	+	0.463720i
$172$	1.55839e6			0.306260
$173$			9.55679e6i			1.84575i	0.385094	+	0.922877i	$0.374169\pi$
$173$			9.55679e6i			1.84575i	−0.385094	+	0.922877i	$0.625831\pi$
$174$	−4.55228e6	−	2.51323e6i	−0.864134	−	0.477073i
$175$	−5.36346e6			−1.00076
$176$			7.28898e6i			1.33699i
$177$	−1.06425e6	+	1.92771e6i	−0.191922	+	0.347633i
$178$	6.53293e6			1.15837
$179$			2.01726e6i			0.351725i	0.984415	+	0.175862i	$0.0562714\pi$
$179$			2.01726e6i			0.351725i	−0.984415	+	0.175862i	$0.943729\pi$
$180$	6.19381e6	−	3.89975e6i	1.06204	−	0.668682i
$181$	6.73662e6			1.13607			0.568037	−	0.823003i	$-0.307703\pi$
$181$	6.73662e6			1.13607			0.568037	+	0.823003i	$0.307703\pi$
$182$		−	2.63999e7i		−	4.37913i
$183$	−1.01747e7	−	5.61730e6i	−1.66024	−	0.916588i
$184$	1.47893e7			2.37407
$185$		−	845222.i		−	0.133492i
$186$	−7.97771e6	+	1.44502e7i	−1.23976	+	2.24561i
$187$	−1.83817e6			−0.281100
$188$		−	3.21178e6i		−	0.483362i
$189$	−620639.	−	1.07866e7i	−0.0919292	−	1.59772i
$190$	4.01811e6			0.585816
$191$			7.36801e6i			1.05743i	0.848800	+	0.528713i	$0.177325\pi$
$191$			7.36801e6i			1.05743i	−0.848800	+	0.528713i	$0.822675\pi$
$192$	5.19685e6	+	2.86909e6i	0.734238	+	0.405360i
$193$	−1.28611e7			−1.78898			−0.894492	−	0.447085i	$-0.852462\pi$
$193$	−1.28611e7			−1.78898			−0.894492	+	0.447085i	$0.852462\pi$
$194$			1.30142e7i			1.78243i
$195$	3.43679e6	−	6.22515e6i	0.463500	−	0.839548i
$196$	−2.41015e7			−3.20092
$197$			1.14789e6i			0.150142i	0.997178	+	0.0750711i	$0.0239184\pi$
$197$			1.14789e6i			0.150142i	−0.997178	+	0.0750711i	$0.976082\pi$
$198$	8.37195e6	+	1.32968e7i	1.07853	+	1.71298i
$199$	−6.00763e6			−0.762332			−0.381166	−	0.924507i	$-0.624477\pi$
$199$	−6.00763e6			−0.762332			−0.381166	+	0.924507i	$0.624477\pi$
$200$			9.17716e6i			1.14714i
$201$	−151669.	−	83733.9i	−0.0186771	−	0.0103113i
$202$	−2.05912e7			−2.49820
$203$			7.56632e6i			0.904475i
$204$	2.04045e6	−	3.69591e6i	0.240345	−	0.435343i
$205$	−2.28427e6			−0.265146
$206$		−	1.62363e7i		−	1.85731i
$207$	9.71384e6	−	6.11603e6i	1.09517	−	0.689539i
$208$	−1.62640e7			−1.80733
$209$			5.79815e6i			0.635112i
$210$	−1.38708e7	−	7.65783e6i	−1.49777	−	0.826891i
$211$	6.01757e6			0.640581			0.320290	−	0.947319i	$-0.396220\pi$
$211$	6.01757e6			0.640581			0.320290	+	0.947319i	$0.396220\pi$
$212$		−	2.16354e7i		−	2.27068i
$213$	−4.88663e6	+	8.85126e6i	−0.505674	+	0.915939i
$214$	7.80077e6			0.795969
$215$		−	908658.i		−	0.0914293i
$216$	−1.84565e7	+	1.06194e6i	−1.83142	+	0.105376i
$217$	2.40176e7			2.35045
$218$		−	3.62333e6i		−	0.349735i
$219$	1.10657e7	+	6.10917e6i	1.05353	+	0.581634i
$220$	1.54883e7			1.45457
$221$		−	4.10154e6i		−	0.379988i
$222$	−2.01419e6	+	3.64834e6i	−0.184095	+	0.333455i
$223$	−5.02279e6			−0.452930			−0.226465	−	0.974019i	$-0.572717\pi$
$223$	−5.02279e6			−0.452930			−0.226465	+	0.974019i	$0.572717\pi$
$224$			3.24274e6i			0.288515i
$225$	3.79516e6	+	6.02770e6i	0.333183	+	0.529180i
$226$	1.82500e6			0.158102
$227$		−	1.68620e7i		−	1.44156i	−0.693166	−	0.720778i	$-0.743785\pi$
$227$		−	1.68620e7i		−	1.44156i	0.693166	−	0.720778i	$-0.256215\pi$
$228$	−1.16580e7	−	6.43619e6i	−0.983604	−	0.543030i
$229$	1.50996e7			1.25736			0.628678	−	0.777666i	$-0.283596\pi$
$229$	1.50996e7			1.25736			0.628678	+	0.777666i	$0.283596\pi$
$230$		−	1.68333e7i		−	1.38352i
$231$	1.10503e7	−	2.00156e7i	0.896474	−	1.62380i
$232$	1.29464e7			1.03677
$233$			5.33260e6i			0.421571i	0.977532	+	0.210786i	$0.0676022\pi$
$233$			5.33260e6i			0.421571i	−0.977532	+	0.210786i	$0.932398\pi$
$234$	−2.96694e7	+	1.86804e7i	−2.31559	+	1.45794i
$235$	−1.87272e6			−0.144301
$236$		−	1.07018e7i		−	0.814177i
$237$	−1.72034e6	−	949772.i	−0.129232	−	0.0713467i
$238$	−9.13902e6			−0.677905
$239$			1.14958e6i			0.0842062i	0.999113	+	0.0421031i	$0.0134058\pi$
$239$			1.14958e6i			0.0842062i	−0.999113	+	0.0421031i	$0.986594\pi$
$240$	−4.71771e6	+	8.54530e6i	−0.341270	+	0.618150i
$241$	−2.38751e7			−1.70566			−0.852832	−	0.522185i	$-0.825117\pi$
$241$	−2.38751e7			−1.70566			−0.852832	+	0.522185i	$0.825117\pi$
$242$			8.49747e6i			0.599575i
$243$	−1.16834e7	+	8.33009e6i	−0.814233	+	0.580538i
$244$	5.64856e7			3.88837
$245$			1.40530e7i			0.955588i
$246$	9.85989e6	+	5.44347e6i	0.662318	+	0.365654i
$247$	−1.29375e7			−0.858538
$248$		−	4.10954e7i		−	2.69425i
$249$	1.07265e7	−	1.94292e7i	0.694803	−	1.25851i
$250$	2.71493e7			1.73756
$251$		−	1.62523e7i		−	1.02776i	−0.857861	−	0.513881i	$-0.828207\pi$
$251$		−	1.62523e7i		−	1.02776i	0.857861	−	0.513881i	$-0.171793\pi$
$252$	2.79781e7	+	4.44364e7i	1.74830	+	2.77675i
$253$	2.42905e7			1.49994
$254$		−	6.17811e6i		−	0.377012i
$255$	−2.15500e6	−	1.18974e6i	−0.129965	−	0.0717514i
$256$	−3.41331e7			−2.03449
$257$			5.62767e6i			0.331535i	0.986165	+	0.165767i	$0.0530101\pi$
$257$			5.62767e6i			0.331535i	−0.986165	+	0.165767i	$0.946990\pi$
$258$	−2.16536e6	+	3.92216e6i	−0.126087	+	0.228384i
$259$	6.06389e6			0.349022
$260$			3.45592e7i			1.96627i
$261$	8.50337e6	−	5.35390e6i	0.478266	−	0.301126i
$262$	3.01030e7			1.67381
$263$		−	7.99601e6i		−	0.439548i	−0.975551	−	0.219774i	$-0.929468\pi$
$263$		−	7.99601e6i		−	0.439548i	0.975551	−	0.219774i	$-0.0705319\pi$
$264$	−3.42478e7	−	1.89076e7i	−1.86132	−	1.02760i
$265$	−1.26151e7			−0.677878
$266$			2.88272e7i			1.53164i
$267$	−6.10155e6	+	1.10519e7i	−0.320558	+	0.580634i
$268$	841999.			0.0437429
$269$			3.82451e6i			0.196480i	0.995163	+	0.0982402i	$0.0313213\pi$
$269$			3.82451e6i			0.196480i	−0.995163	+	0.0982402i	$0.968679\pi$
$270$	1.20871e6	+	2.10073e7i	0.0614089	+	1.06728i
$271$	−3.24302e7			−1.62945			−0.814726	−	0.579846i	$-0.803113\pi$
$271$	−3.24302e7			−1.62945			−0.814726	+	0.579846i	$0.803113\pi$
$272$			5.63021e6i			0.279781i
$273$	4.46612e7	+	2.46567e7i	2.19504	+	1.21184i
$274$	−2.82758e7			−1.37456
$275$			1.50729e7i			0.724766i
$276$	−2.69634e7	+	4.88395e7i	−1.28247	+	2.32297i
$277$	1.89457e7			0.891396			0.445698	−	0.895183i	$-0.352955\pi$
$277$	1.89457e7			0.891396			0.445698	+	0.895183i	$0.352955\pi$
$278$			3.55000e7i			1.65232i
$279$	−1.69948e7	−	2.69921e7i	−0.782533	−	1.24286i
$280$	3.94476e7			1.79699
$281$			5.35271e6i			0.241243i	0.992699	+	0.120622i	$0.0384888\pi$
$281$			5.35271e6i			0.241243i	−0.992699	+	0.120622i	$0.961511\pi$
$282$	8.08345e6	+	4.46273e6i	0.360454	+	0.199000i
$283$	−1.07877e7			−0.475961			−0.237980	−	0.971270i	$-0.576485\pi$
$283$	−1.07877e7			−0.475961			−0.237980	+	0.971270i	$0.576485\pi$
$284$		−	4.91382e7i		−	2.14518i
$285$	−3.75279e6	+	6.79751e6i	−0.162114	+	0.293640i
$286$	−7.41913e7			−3.17143
$287$		−	1.63881e7i		−	0.693238i
$288$	3.64434e6	−	2.29455e6i	0.152560	−	0.0960550i
$289$	−1.41986e6			−0.0588235
$290$		−	1.47356e7i		−	0.604191i
$291$	−2.20164e7	−	1.21549e7i	−0.893445	−	0.493255i
$292$	−6.14317e7			−2.46742
$293$		−	1.26877e7i		−	0.504407i	−0.967674	−	0.252204i	$-0.918845\pi$
$293$		−	1.26877e7i		−	0.504407i	0.967674	−	0.252204i	$-0.0811553\pi$
$294$	3.34887e7	−	6.06588e7i	1.31782	−	2.38700i
$295$	−6.23994e6			−0.243060
$296$		−	1.03756e7i		−	0.400073i
$297$	−3.03136e7	+	1.74417e6i	−1.15709	+	0.0665765i
$298$	8.72656e7			3.29757
$299$			5.41997e7i			2.02760i
$300$	−3.03062e7	−	1.67315e7i	−1.12245	−	0.619686i
$301$	6.51901e6			0.239046
$302$			4.10826e6i			0.149155i
$303$	1.92315e7	−	3.48345e7i	0.691330	−	1.25222i
$304$	1.77594e7			0.632131
$305$		−	3.29354e7i		−	1.16082i
$306$	6.46673e6	+	1.02708e7i	0.225694	+	0.358461i
$307$	−3.39493e7			−1.17332			−0.586659	−	0.809834i	$-0.699557\pi$
$307$	−3.39493e7			−1.17332			−0.586659	+	0.809834i	$0.699557\pi$
$308$			1.11118e8i			3.80304i
$309$	2.74673e7	+	1.51642e7i	0.930979	+	0.513977i
$310$	−4.67750e7			−1.57010
$311$		−	1.64551e7i		−	0.547041i	−0.961866	−	0.273520i	$-0.911812\pi$
$311$		−	1.64551e7i		−	0.547041i	0.961866	−	0.273520i	$-0.0881881\pi$
$312$	4.21888e7	−	7.64176e7i	1.38910	−	2.51611i
$313$	−8.44685e6			−0.275462			−0.137731	−	0.990470i	$-0.543981\pi$
$313$	−8.44685e6			−0.275462			−0.137731	+	0.990470i	$0.543981\pi$
$314$		−	6.95929e7i		−	2.24789i
$315$	2.59098e7	−	1.63133e7i	0.828958	−	0.521929i
$316$	9.55057e6			0.302669
$317$			5.80618e7i			1.82269i	0.411644	+	0.911345i	$0.364955\pi$
$317$			5.80618e7i			1.82269i	−0.411644	+	0.911345i	$0.635045\pi$
$318$	5.44521e7	+	3.00620e7i	1.69330	+	0.934839i
$319$	2.12636e7			0.655034
$320$			1.68221e7i			0.513369i
$321$	−7.28567e6	+	1.31967e7i	−0.220270	+	0.398979i
$322$	1.20767e8			3.61727
$323$			4.47866e6i			0.132905i
$324$	3.01425e7	−	6.28860e7i	0.886224	−	1.84892i
$325$	−3.36323e7			−0.979731
$326$		−	3.13564e7i		−	0.905052i
$327$	6.12966e6	+	3.38408e6i	0.175305	+	0.0967826i
$328$	−2.80408e7			−0.794638
$329$		−	1.34355e7i		−	0.377281i
$330$	−2.15207e7	+	3.89810e7i	−0.598846	+	1.08470i
$331$	−2.14944e7			−0.592710			−0.296355	−	0.955078i	$-0.595771\pi$
$331$	−2.14944e7			−0.592710			−0.296355	+	0.955078i	$0.595771\pi$
$332$			1.07862e8i			2.94751i
$333$	−4.29079e6	−	6.81488e6i	−0.116199	−	0.184555i
$334$	−6.06248e7			−1.62709
$335$		−	490950.i		−	0.0130588i
$336$	−6.13067e7	−	3.38464e7i	−1.61618	−	0.892266i
$337$	−6.30235e7			−1.64669			−0.823347	−	0.567538i	$-0.807896\pi$
$337$	−6.30235e7			−1.64669			−0.823347	+	0.567538i	$0.807896\pi$
$338$		−	9.81032e7i		−	2.54058i
$339$	−1.70450e6	+	3.08739e6i	−0.0437519	+	0.0792488i
$340$	1.19636e7			0.304386
$341$		−	6.74965e7i		−	1.70223i
$342$	3.23973e7	−	2.03980e7i	0.809899	−	0.509929i
$343$	−3.62402e7			−0.898067
$344$		−	1.11544e7i		−	0.274012i
$345$	2.84771e7	+	1.57217e7i	0.693488	+	0.382863i
$346$	1.33529e8			3.22365
$347$		−	3.94698e7i		−	0.944661i	−0.881422	−	0.472331i	$-0.843413\pi$
$347$		−	3.94698e7i		−	0.944661i	0.881422	−	0.472331i	$-0.156587\pi$
$348$	−2.36034e7	+	4.27535e7i	−0.560064	+	1.01446i
$349$	1.49778e7			0.352347			0.176174	−	0.984359i	$-0.443628\pi$
$349$	1.49778e7			0.352347			0.176174	+	0.984359i	$0.443628\pi$
$350$			7.49393e7i			1.74785i
$351$	−3.89180e6	−	6.76392e7i	−0.0899973	−	1.56415i
$352$	9.11304e6			0.208947
$353$			3.92245e7i			0.891730i	0.895100	+	0.445865i	$0.147104\pi$
$353$			3.92245e7i			0.891730i	−0.895100	+	0.445865i	$0.852896\pi$
$354$	2.69343e7	+	1.48699e7i	0.607149	+	0.335197i
$355$	−2.86513e7			−0.640412
$356$		−	6.13550e7i		−	1.35988i
$357$	8.53555e6	−	1.54606e7i	0.187598	−	0.339800i
$358$	2.81855e7			0.614295
$359$			6.33820e7i			1.36988i	0.728600	+	0.684940i	$0.240172\pi$
$359$			6.33820e7i			1.36988i	−0.728600	+	0.684940i	$0.759828\pi$
$360$	−2.79130e7	−	4.43330e7i	−0.598272	−	0.950210i
$361$	−3.29189e7			−0.699718
$362$		−	9.41253e7i		−	1.98418i
$363$	−1.43753e7	−	7.93637e6i	−0.300537	−	0.165921i
$364$	−2.47939e8			−5.14091
$365$			3.58194e7i			0.736613i
$366$	−7.84859e7	+	1.42163e8i	−1.60084	+	2.89964i
$367$	−7.82479e6			−0.158298			−0.0791489	−	0.996863i	$-0.525220\pi$
$367$	−7.82479e6			−0.158298			−0.0791489	+	0.996863i	$0.525220\pi$
$368$		−	7.44003e7i		−	1.49290i
$369$	−1.84177e7	+	1.15961e7i	−0.366569	+	0.230799i
$370$	−1.18096e7			−0.233147
$371$		−	9.05045e7i		−	1.77235i
$372$	1.35711e8	+	7.49239e7i	2.63626	+	1.45543i
$373$	−7.37682e6			−0.142149			−0.0710744	−	0.997471i	$-0.522643\pi$
$373$	−7.37682e6			−0.142149			−0.0710744	+	0.997471i	$0.522643\pi$
$374$			2.56833e7i			0.490948i
$375$	−2.53566e7	+	4.59290e7i	−0.480837	+	0.870951i
$376$	−2.29888e7			−0.432466
$377$			4.74457e7i			0.885468i
$378$	−1.50713e8	+	8.67168e6i	−2.79046	+	0.160556i
$379$	2.88715e6			0.0530336			0.0265168	−	0.999648i	$-0.491558\pi$
$379$	2.88715e6			0.0530336			0.0265168	+	0.999648i	$0.491558\pi$
$380$		−	3.77367e7i		−	0.687723i
$381$	1.04516e7	+	5.77016e6i	0.188977	+	0.104331i
$382$	1.02947e8			1.84682
$383$		−	8.15610e7i		−	1.45173i	−0.687837	−	0.725865i	$-0.741440\pi$
$383$		−	8.15610e7i		−	1.45173i	0.687837	−	0.725865i	$-0.258560\pi$
$384$	4.50212e7	−	8.15480e7i	0.795103	−	1.44019i
$385$	6.47901e7			1.13534
$386$			1.79698e8i			3.12450i
$387$	−4.61282e6	−	7.32635e6i	−0.0795855	−	0.126402i
$388$	1.22225e8			2.09250
$389$			8.91326e7i			1.51422i	0.653290	+	0.757108i	$0.273388\pi$
$389$			8.91326e7i			1.51422i	−0.653290	+	0.757108i	$0.726612\pi$
$390$	−8.69789e7	−	4.80195e7i	−1.46629	−	0.809514i
$391$	1.87626e7			0.313880
$392$			1.72509e8i			2.86388i
$393$	−2.81153e7	+	5.09258e7i	−0.463196	+	0.838997i
$394$	1.60386e7			0.262227
$395$		−	5.56871e6i		−	0.0903573i
$396$	1.24879e8	−	7.86265e7i	2.01096	−	1.26614i
$397$	6.47825e7			1.03535			0.517674	−	0.855578i	$-0.326798\pi$
$397$	6.47825e7			1.03535			0.517674	+	0.855578i	$0.326798\pi$
$398$			8.39398e7i			1.33143i
$399$	−4.87675e7	−	2.69237e7i	−0.767736	−	0.423854i
$400$	4.61673e7			0.721365
$401$			6.19346e7i			0.960507i	0.877130	+	0.480254i	$0.159455\pi$
$401$			6.19346e7i			0.960507i	−0.877130	+	0.480254i	$0.840545\pi$
$402$	−1.16995e6	+	2.11915e6i	−0.0180089	+	0.0326200i
$403$	1.50606e8			2.30105
$404$			1.93385e8i			2.93278i
$405$	−3.66673e7	−	1.75753e7i	−0.551968	−	0.264569i
$406$	1.05718e8			1.57969
$407$		−	1.70413e7i		−	0.252766i
$408$	−2.64540e7	−	1.46048e7i	−0.389503	−	0.215038i
$409$	6.80218e7			0.994210			0.497105	−	0.867690i	$-0.334396\pi$
$409$	6.80218e7			0.994210			0.497105	+	0.867690i	$0.334396\pi$
$410$			3.19162e7i			0.463084i
$411$	2.64087e7	−	4.78347e7i	0.380384	−	0.688998i
$412$	−1.52486e8			−2.18041
$413$		−	4.47673e7i		−	0.635493i
$414$	−8.54543e7	−	1.35724e8i	−1.20430	−	1.91273i
$415$	6.28920e7			0.879936
$416$			2.03341e7i			0.282452i
$417$	−6.00560e7	−	3.31559e7i	−0.828225	−	0.457249i
$418$	8.10128e7			1.10924
$419$		−	8.89863e7i		−	1.20971i	−0.796336	−	0.604855i	$-0.793231\pi$
$419$		−	8.89863e7i		−	1.20971i	0.796336	−	0.604855i	$-0.206769\pi$
$420$	−7.19198e7	+	1.30270e8i	−0.970734	+	1.75831i
$421$	−4.87978e7			−0.653964			−0.326982	−	0.945030i	$-0.606032\pi$
$421$	−4.87978e7			−0.653964			−0.326982	+	0.945030i	$0.606032\pi$
$422$		−	8.40786e7i		−	1.11879i
$423$	−1.50994e7	+	9.50687e6i	−0.199498	+	0.125608i
$424$	−1.54858e8			−2.03159
$425$			1.16427e7i			0.151666i
$426$	1.23672e8	+	6.82769e7i	1.59971	+	0.883171i
$427$	2.36289e8			3.03501
$428$		−	7.32621e7i		−	0.934433i
$429$	6.92924e7	−	1.25511e8i	0.877634	−	1.58968i
$430$	−1.26959e7			−0.159683
$431$			4.20458e7i			0.525159i	0.964910	+	0.262580i	$0.0845733\pi$
$431$			4.20458e7i			0.525159i	−0.964910	+	0.262580i	$0.915427\pi$
$432$	5.34231e6	+	9.28488e7i	0.0662640	+	1.15166i
$433$	−5.84441e7			−0.719908			−0.359954	−	0.932970i	$-0.617208\pi$
$433$	−5.84441e7			−0.719908			−0.359954	+	0.932970i	$0.617208\pi$
$434$		−	3.35579e8i		−	4.10511i
$435$	2.49285e7	+	1.37626e7i	0.302851	+	0.167199i
$436$	−3.40291e7			−0.410574
$437$		−	5.91830e7i		−	0.709174i
$438$	8.53585e7	−	1.54612e8i	1.01584	−	1.84001i
$439$	−1.03339e7			−0.122144			−0.0610718	−	0.998133i	$-0.519452\pi$
$439$	−1.03339e7			−0.122144			−0.0610718	+	0.998133i	$0.519452\pi$
$440$		−	1.10859e8i		−	1.30141i
$441$	7.13403e7	+	1.13307e8i	0.831800	+	1.32111i
$442$	−5.73075e7			−0.663659
$443$			1.70511e7i			0.196129i	0.995180	+	0.0980645i	$0.0312651\pi$
$443$			1.70511e7i			0.196129i	−0.995180	+	0.0980645i	$0.968735\pi$
$444$	3.42640e7	+	1.89166e7i	0.391462	+	0.216119i
$445$	−3.57747e7			−0.405971
$446$			7.01794e7i			0.791052i
$447$	−8.15034e7	+	1.47629e8i	−0.912542	+	1.65291i
$448$	−1.20687e8			−1.34223
$449$			3.33949e7i			0.368927i	0.982839	+	0.184464i	$0.0590548\pi$
$449$			3.33949e7i			0.368927i	−0.982839	+	0.184464i	$0.940945\pi$
$450$	8.42201e7	−	5.30267e7i	0.924226	−	0.581912i
$451$	−4.60552e7			−0.502053
$452$		−	1.71398e7i		−	0.185605i
$453$	−6.95002e6	−	3.83698e6i	−0.0747638	−	0.0412758i
$454$	−2.35599e8			−2.51771
$455$			1.44567e8i			1.53474i
$456$	−4.60679e7	+	8.34438e7i	−0.485851	+	0.880034i
$457$	9.61317e7			1.00721			0.503603	−	0.863935i	$-0.332008\pi$
$457$	9.61317e7			1.00721			0.503603	+	0.863935i	$0.332008\pi$
$458$		−	2.10974e8i		−	2.19600i
$459$	−2.34151e7	+	1.34725e6i	−0.242135	+	0.0139319i
$460$	−1.58092e8			−1.62419
$461$			1.15458e8i			1.17848i	0.807959	+	0.589238i	$0.200572\pi$
$461$			1.15458e8i			1.17848i	−0.807959	+	0.589238i	$0.799428\pi$
$462$	−2.79662e8	−	1.54397e8i	−2.83601	−	1.56571i
$463$	−3.79753e7			−0.382612			−0.191306	−	0.981530i	$-0.561272\pi$
$463$	−3.79753e7			−0.382612			−0.191306	+	0.981530i	$0.561272\pi$
$464$		−	6.51290e7i		−	0.651960i
$465$	4.36864e7	−	7.91301e7i	0.434497	−	0.787015i
$466$	7.45081e7			0.736284
$467$		−	6.42028e7i		−	0.630382i	−0.949028	−	0.315191i	$-0.897931\pi$
$467$		−	6.42028e7i		−	0.630382i	0.949028	−	0.315191i	$-0.102069\pi$
$468$	1.75440e8	+	2.78645e8i	1.71156	+	2.71840i
$469$	3.52223e6			0.0341428
$470$			2.61659e7i			0.252024i
$471$	1.17732e8	+	6.49976e7i	1.12676	+	0.622063i
$472$	−7.65992e7			−0.728448
$473$		−	1.83203e7i		−	0.173121i
$474$	−1.32704e7	+	2.40370e7i	−0.124609	+	0.225707i
$475$	3.67246e7			0.342670
$476$			8.58305e7i			0.795831i
$477$	−1.01713e8	+	6.40406e7i	−0.937177	+	0.590066i
$478$	1.60621e7			0.147068
$479$		−	3.33892e7i		−	0.303808i	−0.988395	−	0.151904i	$-0.951460\pi$
$479$		−	3.33892e7i		−	0.303808i	0.988395	−	0.151904i	$-0.0485405\pi$
$480$	1.06838e7	+	5.89832e6i	0.0966052	+	0.0533340i
$481$	3.80245e7			0.341687
$482$			3.33587e8i			2.97898i
$483$	−1.12793e8	+	2.04304e8i	−1.00101	+	1.81316i
$484$	7.98053e7			0.703876
$485$		−	7.12666e7i		−	0.624685i
$486$	1.16390e8	+	1.63242e8i	1.01392	+	1.42208i
$487$	1.18889e7			0.102933			0.0514666	−	0.998675i	$-0.483610\pi$
$487$	1.18889e7			0.102933			0.0514666	+	0.998675i	$0.483610\pi$
$488$		−	4.04303e8i		−	3.47894i
$489$	5.30463e7	+	2.92859e7i	0.453658	+	0.250456i
$490$	1.96351e8			1.66896
$491$		−	1.14678e8i		−	0.968801i	−0.874846	−	0.484400i	$-0.839038\pi$
$491$		−	1.14678e8i		−	0.968801i	0.874846	−	0.484400i	$-0.160962\pi$
$492$	5.11233e7	−	9.26007e7i	0.429263	−	0.777534i
$493$	1.64246e7			0.137074
$494$			1.80765e8i			1.49946i
$495$	−4.58452e7	−	7.28141e7i	−0.377988	−	0.600343i
$496$	−2.06738e8			−1.69424
$497$		−	2.05554e8i		−	1.67439i
$498$	−2.71469e8	−	1.49873e8i	−2.19802	−	1.21349i
$499$	−1.68425e8			−1.35552			−0.677759	−	0.735284i	$-0.737049\pi$
$499$	−1.68425e8			−1.35552			−0.677759	+	0.735284i	$0.737049\pi$
$500$		−	2.54977e8i		−	2.03982i
$501$	5.66216e7	−	1.02560e8i	0.450266	−	0.815577i
$502$	−2.27080e8			−1.79501
$503$		−	8.97159e7i		−	0.704961i	−0.935819	−	0.352481i	$-0.885338\pi$
$503$		−	8.97159e7i		−	0.704961i	0.935819	−	0.352481i	$-0.114662\pi$
$504$	3.18059e8	−	2.00257e8i	2.48437	−	1.56421i
$505$	1.12758e8			0.875537
$506$		−	3.39391e8i		−	2.61968i
$507$	1.65963e8	+	9.16253e7i	1.27347	+	0.703058i
$508$	−5.80227e7			−0.442596
$509$			1.85400e8i			1.40590i	0.711238	+	0.702951i	$0.248135\pi$
$509$			1.85400e8i			1.40590i	−0.711238	+	0.702951i	$0.751865\pi$
$510$	−1.66232e7	+	3.01100e7i	−0.125316	+	0.226987i
$511$	−2.56980e8			−1.92591
$512$			2.56114e8i			1.90820i
$513$	4.24963e6	+	7.38582e7i	0.0314774	+	0.547075i
$514$	7.86309e7			0.579033
$515$			8.89109e7i			0.650928i
$516$	3.68356e7	+	2.03363e7i	0.268114	+	0.148021i
$517$	−3.77575e7			−0.273232
$518$		−	8.47259e7i		−	0.609574i
$519$	−1.24712e8	+	2.25894e8i	−0.892086	+	1.61586i
$520$	2.47362e8			1.75923
$521$		−	7.29135e6i		−	0.0515579i	−0.999668	−	0.0257789i	$-0.991793\pi$
$521$		−	7.29135e6i		−	0.0515579i	0.999668	−	0.0257789i	$-0.00820660\pi$
$522$	−7.48057e7	−	1.18811e8i	−0.525924	−	0.835303i
$523$	−1.02948e8			−0.719634			−0.359817	−	0.933023i	$-0.617161\pi$
$523$	−1.02948e8			−0.719634			−0.359817	+	0.933023i	$0.617161\pi$
$524$		−	2.82717e8i		−	1.96498i
$525$	−1.26776e8	−	6.99909e7i	−0.876112	−	0.483686i
$526$	−1.11722e8			−0.767680
$527$		−	5.21362e7i		−	0.356211i
$528$	−9.51181e7	+	1.72290e8i	−0.646192	+	1.17046i
$529$	−9.99023e7			−0.674852
$530$			1.76260e8i			1.18393i
$531$	−5.03115e7	+	3.16772e7i	−0.336035	+	0.211574i
$532$	2.70735e8			1.79808
$533$		−	1.02764e8i		−	0.678670i
$534$	1.54419e8	+	8.52520e7i	1.01409	+	0.559862i
$535$	−4.27174e7			−0.278961
$536$		−	6.02672e6i		−	0.0391369i
$537$	−2.63244e7	+	4.76820e7i	−0.169995	+	0.307916i
$538$	5.34368e7			0.343158
$539$			2.83335e8i			1.80940i
$540$	1.97293e8	−	1.13518e7i	1.25294	−	0.0720914i
$541$	5.04741e7			0.318769			0.159385	−	0.987217i	$-0.449049\pi$
$541$	5.04741e7			0.318769			0.159385	+	0.987217i	$0.449049\pi$
$542$			4.53121e8i			2.84588i
$543$	1.59234e8	+	8.79101e7i	0.994570	+	0.549084i
$544$	7.03917e6			0.0437245
$545$			1.98416e7i			0.122571i
$546$	3.44508e8	−	6.24015e8i	2.11651	−	3.83369i
$547$	2.51788e8			1.53841			0.769207	−	0.639000i	$-0.220652\pi$
$547$	2.51788e8			1.53841			0.769207	+	0.639000i	$0.220652\pi$
$548$			2.65557e8i			1.61367i
$549$	−1.67197e8	−	2.65552e8i	−1.01044	−	1.60484i
$550$	2.10601e8			1.26582
$551$		−	5.18081e7i		−	0.309701i
$552$	3.49575e8	+	1.92994e8i	2.07837	+	1.14743i
$553$	3.99517e7			0.236244
$554$		−	2.64713e8i		−	1.55684i
$555$	1.10298e7	−	1.99785e7i	0.0645191	−	0.116865i
$556$	3.33404e8			1.93975
$557$			2.50076e8i			1.44712i	0.690259	+	0.723562i	$0.257496\pi$
$557$			2.50076e8i			1.44712i	−0.690259	+	0.723562i	$0.742504\pi$
$558$	−3.77138e8	+	2.37454e8i	−2.17069	+	1.36671i
$559$	4.08784e7			0.234023
$560$		−	1.98448e8i		−	1.13001i
$561$	−4.34489e7	−	2.39874e7i	−0.246088	−	0.135861i
$562$	7.47891e7			0.421337
$563$		−	3.02110e8i		−	1.69293i	−0.532443	−	0.846466i	$-0.678726\pi$
$563$		−	3.02110e8i		−	1.69293i	0.532443	−	0.846466i	$-0.321274\pi$
$564$	4.19124e7	−	7.59170e7i	0.233618	−	0.423157i
$565$	−9.99382e6			−0.0554097
$566$			1.50728e8i			0.831276i
$567$	1.26091e8	−	2.63063e8i	0.691728	−	1.44315i
$568$	−3.51713e8			−1.91930
$569$		−	1.03975e7i		−	0.0564408i	−0.999602	−	0.0282204i	$-0.991016\pi$
$569$		−	1.03975e7i		−	0.0564408i	0.999602	−	0.0282204i	$-0.00898402\pi$
$570$	9.49761e7	+	5.24347e7i	0.512850	+	0.283135i
$571$	1.02221e8			0.549077			0.274538	−	0.961576i	$-0.411475\pi$
$571$	1.02221e8			0.549077			0.274538	+	0.961576i	$0.411475\pi$
$572$			6.96780e8i			3.72312i
$573$	−9.61495e7	+	1.74158e8i	−0.511073	+	0.925719i
$574$	−2.28977e8			−1.21076
$575$		−	1.53852e8i		−	0.809283i
$576$	8.53976e7	+	1.35633e8i	0.446867	+	0.709740i
$577$	1.06043e8			0.552018			0.276009	−	0.961155i	$-0.410988\pi$
$577$	1.06043e8			0.552018			0.276009	+	0.961155i	$0.410988\pi$
$578$			1.98385e7i			0.102737i
$579$	−3.03998e8	−	1.67832e8i	−1.56616	−	0.864648i
$580$	−1.38392e8			−0.709295
$581$			4.51207e8i			2.30063i
$582$	−1.69830e8	+	3.07618e8i	−0.861482	+	1.56042i
$583$	−2.54344e8			−1.28356
$584$			4.39705e8i			2.20761i
$585$	1.62471e8	−	1.02295e8i	0.811537	−	0.510961i
$586$	−1.77275e8			−0.880959
$587$			6.29843e7i			0.311400i	0.987804	+	0.155700i	$0.0497633\pi$
$587$			6.29843e7i			0.311400i	−0.987804	+	0.155700i	$0.950237\pi$
$588$	−5.69687e8	−	3.14514e8i	−2.80223	−	1.54706i
$589$	−1.64453e8			−0.804816
$590$			8.71856e7i			0.424511i
$591$	−1.49795e7	+	2.71328e7i	−0.0725664	+	0.131441i
$592$	−5.21965e7			−0.251580
$593$		−	1.28440e8i		−	0.615935i	−0.951397	−	0.307968i	$-0.900351\pi$
$593$		−	1.28440e8i		−	0.615935i	0.951397	−	0.307968i	$-0.0996488\pi$
$594$	2.43699e7	+	4.23547e8i	0.116277	+	2.02089i
$595$	5.00458e7			0.237584
$596$		−	8.19569e8i		−	3.87121i
$597$	−1.42002e8	−	7.83971e7i	−0.667380	−	0.368449i
$598$	7.57288e8			3.54126
$599$		−	2.23285e8i		−	1.03891i	−0.854497	−	0.519456i	$-0.826135\pi$
$599$		−	2.23285e8i		−	1.03891i	0.854497	−	0.519456i	$-0.173865\pi$
$600$	−1.19758e8	+	2.16921e8i	−0.554435	+	1.00426i
$601$	−7.23276e7			−0.333181			−0.166591	−	0.986026i	$-0.553276\pi$
$601$	−7.23276e7			−0.333181			−0.166591	+	0.986026i	$0.553276\pi$
$602$		−	9.10848e7i		−	0.417500i
$603$	−2.49232e6	−	3.95844e6i	−0.0113671	−	0.0180539i
$604$	3.85834e7			0.175101
$605$		−	4.65326e7i		−	0.210132i
$606$	−4.86714e8	−	2.68706e8i	−2.18703	−	1.20742i
$607$	1.95537e8			0.874306			0.437153	−	0.899387i	$-0.355987\pi$
$607$	1.95537e8			0.874306			0.437153	+	0.899387i	$0.355987\pi$
$608$		−	2.22037e7i		−	0.0987902i
$609$	−9.87373e7	+	1.78845e8i	−0.437149	+	0.791818i
$610$	−4.60180e8			−2.02739
$611$		−	8.42490e7i		−	0.369352i
$612$	9.64602e7	−	6.07333e7i	0.420818	−	0.264955i
$613$	3.05389e8			1.32578			0.662891	−	0.748715i	$-0.269329\pi$
$613$	3.05389e8			1.32578			0.662891	+	0.748715i	$0.269329\pi$
$614$			4.74346e8i			2.04923i
$615$	−5.39933e7	−	2.98088e7i	−0.232121	−	0.128150i
$616$	7.95340e8			3.40260
$617$			3.37059e7i			0.143500i	0.997423	+	0.0717498i	$0.0228583\pi$
$617$			3.37059e7i			0.143500i	−0.997423	+	0.0717498i	$0.977142\pi$
$618$	2.11877e8	−	3.83778e8i	0.897673	−	1.62598i
$619$	7.18540e7			0.302955			0.151478	−	0.988461i	$-0.451597\pi$
$619$	7.18540e7			0.302955			0.151478	+	0.988461i	$0.451597\pi$
$620$			4.39295e8i			1.84324i
$621$	3.09418e8	−	1.78032e7i	1.29202	−	0.0743401i
$622$	−2.29914e8			−0.955419
$623$		−	2.56659e8i		−	1.06143i
$624$	−3.84433e8	−	2.12238e8i	−1.58222	−	0.873515i
$625$	3.99824e6			0.0163768
$626$			1.18021e8i			0.481101i
$627$	−7.56634e7	+	1.37051e8i	−0.306961	+	0.556006i
$628$	−6.53593e8			−2.63893
$629$		−	1.31632e7i		−	0.0528943i
$630$	−2.27933e8	−	3.62017e8i	−0.911561	−	1.44779i
$631$	−2.30047e8			−0.915649			−0.457825	−	0.889042i	$-0.651371\pi$
$631$	−2.30047e8			−0.915649			−0.457825	+	0.889042i	$0.651371\pi$
$632$		−	6.83594e7i		−	0.270799i
$633$	1.42237e8	+	7.85268e7i	0.560793	+	0.309604i
$634$	8.11250e8			3.18337
$635$			3.38317e7i			0.132130i
$636$	2.82332e8	−	5.11395e8i	1.09746	−	1.98786i
$637$	−6.32211e8			−2.44593
$638$		−	2.97098e8i		−	1.14403i
$639$	−2.31011e8	+	1.45449e8i	−0.885379	+	0.557453i
$640$	2.63969e8			1.00696
$641$		−	2.00468e8i		−	0.761151i	−0.924750	−	0.380576i	$-0.875726\pi$
$641$		−	2.00468e8i		−	0.761151i	0.924750	−	0.380576i	$-0.124274\pi$
$642$	1.84387e8	+	1.01797e8i	0.696827	+	0.384706i
$643$	7.38859e7			0.277926			0.138963	−	0.990298i	$-0.455623\pi$
$643$	7.38859e7			0.277926			0.138963	+	0.990298i	$0.455623\pi$
$644$		−	1.13420e9i		−	4.24653i
$645$	1.18576e7	−	2.14780e7i	0.0441894	−	0.0800413i
$646$	6.25766e7			0.232121
$647$			1.33382e8i			0.492476i	0.969209	+	0.246238i	$0.0791944\pi$
$647$			1.33382e8i			0.492476i	−0.969209	+	0.246238i	$0.920806\pi$
$648$	−4.50115e8	−	2.15749e8i	−1.65424	−	0.792908i
$649$	−1.25809e8			−0.460233
$650$			4.69917e8i			1.71112i
$651$	5.67705e8	+	3.13420e8i	2.05769	+	1.13601i
$652$	−2.94489e8			−1.06249
$653$			1.29106e8i			0.463667i	0.972755	+	0.231834i	$0.0744725\pi$
$653$			1.29106e8i			0.463667i	−0.972755	+	0.231834i	$0.925528\pi$
$654$	4.72830e7	−	8.56448e7i	0.169033	−	0.306174i
$655$	−1.64846e8			−0.586616
$656$			1.41065e8i			0.499696i
$657$	1.81838e8	+	2.88805e8i	0.641191	+	1.01838i
$658$	−1.87723e8			−0.658930
$659$		−	2.07847e8i		−	0.726252i	−0.931740	−	0.363126i	$-0.881710\pi$
$659$		−	2.07847e8i		−	0.726252i	0.931740	−	0.363126i	$-0.118290\pi$
$660$	3.66096e8	+	2.02115e8i	1.27340	+	0.703020i
$661$	4.35233e8			1.50701			0.753507	−	0.657440i	$-0.228360\pi$
$661$	4.35233e8			1.50701			0.753507	+	0.657440i	$0.228360\pi$
$662$			3.00324e8i			1.03518i
$663$	5.35234e7	−	9.69482e7i	0.183655	−	0.332659i
$664$	7.72039e8			2.63715
$665$		−	1.57859e8i		−	0.536791i
$666$	−9.52187e7	+	5.99517e7i	−0.322329	+	0.202945i
$667$	−2.17042e8			−0.731419
$668$			5.69367e8i			1.91013i
$669$	−1.18724e8	−	6.55453e7i	−0.396515	−	0.218909i
$670$	−6.85964e6			−0.0228075
$671$		−	6.64041e8i		−	2.19800i
$672$	−4.23164e7	+	7.66487e7i	−0.139444	+	0.252579i
$673$	4.51544e8			1.48134			0.740670	−	0.671869i	$-0.234508\pi$
$673$	4.51544e8			1.48134			0.740670	+	0.671869i	$0.234508\pi$
$674$			8.80577e8i			2.87599i
$675$	1.10474e7	+	1.92002e8i	0.0359209	+	0.624302i
$676$	−9.21352e8			−2.98253
$677$		−	5.23061e8i		−	1.68572i	−0.538130	−	0.842862i	$-0.680869\pi$
$677$		−	5.23061e8i		−	1.68572i	0.538130	−	0.842862i	$-0.319131\pi$
$678$	4.31376e7	+	2.38155e7i	0.138410	+	0.0764137i
$679$	5.11290e8			1.63327
$680$		−	8.56309e7i		−	0.272335i
$681$	2.20042e8	−	3.98567e8i	0.696730	−	1.26200i
$682$	−9.43073e8			−2.97298
$683$			2.01564e8i			0.632631i	0.948654	+	0.316315i	$0.102446\pi$
$683$			2.01564e8i			0.632631i	−0.948654	+	0.316315i	$0.897554\pi$
$684$	−1.91571e8	−	3.04265e8i	−0.598635	−	0.950787i
$685$	1.54840e8			0.481738
$686$			5.06355e8i			1.56850i
$687$	3.56909e8	+	1.97043e8i	1.10075	+	0.607703i
$688$	−5.61140e7			−0.172308
$689$		−	5.67521e8i		−	1.73510i
$690$	2.19667e8	−	3.97888e8i	0.668679	−	1.21119i
$691$	−2.97598e8			−0.901978			−0.450989	−	0.892530i	$-0.648929\pi$
$691$	−2.97598e8			−0.901978			−0.450989	+	0.892530i	$0.648929\pi$
$692$		−	1.25406e9i		−	3.78443i
$693$	5.22391e8	−	3.28908e8i	1.56963	−	0.988269i
$694$	−5.51479e8			−1.64987
$695$		−	1.94400e8i		−	0.579084i
$696$	3.06013e8	+	1.68945e8i	0.907638	+	0.501091i
$697$	−3.55744e7			−0.105060
$698$		−	2.09272e8i		−	0.615383i
$699$	−6.95882e7	+	1.26047e8i	−0.203753	+	0.369063i
$700$	7.03804e8			2.05191
$701$		−	4.47494e8i		−	1.29907i	−0.760331	−	0.649535i	$-0.774963\pi$
$701$		−	4.47494e8i		−	1.29907i	0.760331	−	0.649535i	$-0.225037\pi$
$702$	−9.45068e8	+	5.43770e7i	−2.73182	+	0.157182i
$703$	−4.15207e7			−0.119508
$704$			3.39165e8i			0.972061i
$705$	−4.42654e7	−	2.44382e7i	−0.126327	−	0.0697431i
$706$	5.48052e8			1.55743
$707$			8.08964e8i			2.28913i
$708$	1.39653e8	−	2.52958e8i	0.393506	−	0.712767i
$709$	−6.10696e8			−1.71351			−0.856755	−	0.515724i	$-0.827523\pi$
$709$	−6.10696e8			−1.71351			−0.856755	+	0.515724i	$0.827523\pi$
$710$			4.00322e8i			1.11850i
$711$	−2.82697e7	−	4.48995e7i	−0.0786524	−	0.124920i
$712$	−4.39157e8			−1.21669
$713$			6.88952e8i			1.90073i
$714$	−2.16019e8	−	1.19260e8i	−0.593468	−	0.327643i
$715$	4.06276e8			1.11148
$716$		−	2.64709e8i		−	0.721157i
$717$	−1.50015e7	+	2.71726e7i	−0.0406984	+	0.0737179i
$718$	8.85585e8			2.39253
$719$			6.74433e8i			1.81448i	0.420614	+	0.907240i	$0.361815\pi$
$719$			6.74433e8i			1.81448i	−0.420614	+	0.907240i	$0.638185\pi$
$720$	−2.23025e8	+	1.40421e8i	−0.597526	+	0.376214i
$721$	−6.37875e8			−1.70188
$722$			4.59948e8i			1.22207i
$723$	−5.64335e8	−	3.11560e8i	−1.49322	−	0.824378i
$724$	−8.83993e8			−2.32934
$725$		−	1.34680e8i		−	0.353419i
$726$	−1.10888e8	+	2.00855e8i	−0.289785	+	0.524895i
$727$	4.49333e7			0.116941			0.0584703	−	0.998289i	$-0.481378\pi$
$727$	4.49333e7			0.116941			0.0584703	+	0.998289i	$0.481378\pi$
$728$			1.77465e9i			4.59959i
$729$	−3.84864e8	+	4.44355e7i	−0.993401	+	0.114696i
$730$	5.00475e8			1.28651
$731$		−	1.41511e7i		−	0.0362275i
$732$	1.33515e9	+	7.37113e8i	3.40406	+	1.87932i
$733$	2.94605e8			0.748045			0.374022	−	0.927420i	$-0.377978\pi$
$733$	2.94605e8			0.748045			0.374022	+	0.927420i	$0.377978\pi$
$734$			1.09329e8i			0.276471i
$735$	−1.83386e8	+	3.32171e8i	−0.461853	+	0.836565i
$736$	−9.30189e7			−0.233312
$737$		−	9.89849e6i		−	0.0247267i
$738$	1.62023e8	+	2.57335e8i	0.403096	+	0.640221i
$739$	2.17628e7			0.0539240			0.0269620	−	0.999636i	$-0.491417\pi$
$739$	2.17628e7			0.0539240			0.0269620	+	0.999636i	$0.491417\pi$
$740$			1.10912e8i			0.273705i
$741$	−3.05804e8	−	1.68829e8i	−0.751603	−	0.414947i
$742$	−1.26455e9			−3.09544
$743$		−	9.30103e7i		−	0.226759i	−0.993552	−	0.113379i	$-0.963832\pi$
$743$		−	9.30103e7i		−	0.226759i	0.993552	−	0.113379i	$-0.0361676\pi$
$744$	5.36278e8	−	9.71372e8i	1.30218	−	2.35867i
$745$	−4.77871e8			−1.15569
$746$			1.03070e8i			0.248266i
$747$	5.07087e8	−	3.19272e8i	1.21652	−	0.765948i
$748$	2.41209e8			0.576353
$749$		−	3.06468e8i		−	0.729357i
$750$	6.41729e8	+	3.54287e8i	1.52114	+	0.839792i
$751$	−1.10996e8			−0.262053			−0.131027	−	0.991379i	$-0.541827\pi$
$751$	−1.10996e8			−0.262053			−0.131027	+	0.991379i	$0.541827\pi$
$752$			1.15649e8i			0.271950i
$753$	2.12085e8	−	3.84155e8i	0.496736	−	0.899749i
$754$	6.62920e8			1.54649
$755$		−	2.24971e7i		−	0.0522739i
$756$	8.14415e7	+	1.41545e9i	0.188486	+	3.27588i
$757$	−7.58602e8			−1.74874			−0.874372	−	0.485257i	$-0.838726\pi$
$757$	−7.58602e8			−1.74874			−0.874372	+	0.485257i	$0.838726\pi$
$758$		−	4.03397e7i		−	0.0926244i
$759$	5.74154e8	+	3.16980e8i	1.31312	+	0.724948i
$760$	−2.70106e8			−0.615308
$761$			5.43052e8i			1.23222i	0.787661	+	0.616109i	$0.211292\pi$
$761$			5.43052e8i			1.23222i	−0.787661	+	0.616109i	$0.788708\pi$
$762$	8.06218e7	−	1.46032e8i	0.182216	−	0.330053i
$763$	−1.42350e8			−0.320467
$764$		−	9.66846e8i		−	2.16809i
$765$	−3.54122e7	−	5.62437e7i	−0.0790985	−	0.125629i
$766$	−1.13959e9			−2.53548
$767$		−	2.80720e8i		−	0.622139i
$768$	−8.06805e8	−	4.45423e8i	−1.78109	−	0.983307i
$769$	−1.95292e8			−0.429444			−0.214722	−	0.976675i	$-0.568885\pi$
$769$	−1.95292e8			−0.429444			−0.214722	+	0.976675i	$0.568885\pi$
$770$		−	9.05260e8i		−	1.98290i
$771$	−7.34387e7	+	1.33021e8i	−0.160237	+	0.290241i
$772$	1.68766e9			3.66803
$773$		−	4.49070e8i		−	0.972245i	−0.873891	−	0.486123i	$-0.838411\pi$
$773$		−	4.49070e8i		−	0.972245i	0.873891	−	0.486123i	$-0.161589\pi$
$774$	−1.02365e8	+	6.44512e7i	−0.220765	+	0.138998i
$775$	−4.27513e8			−0.918426
$776$		−	8.74843e8i		−	1.87217i
$777$	1.43332e8	+	7.91313e7i	0.305549	+	0.168688i
$778$	1.24538e9			2.64461
$779$			1.12212e8i			0.237371i
$780$	−4.50983e8	+	8.16876e8i	−0.950334	+	1.72136i
$781$	−5.77666e8			−1.21262
$782$		−	2.62155e8i		−	0.548199i
$783$	2.70860e8	−	1.55847e7i	0.564235	−	0.0324648i
$784$	8.67840e8			1.80091
$785$			3.81094e8i			0.787814i
$786$	7.11545e8	+	3.92832e8i	1.46533	+	0.808982i
$787$	−3.95231e8			−0.810824			−0.405412	−	0.914134i	$-0.632872\pi$
$787$	−3.95231e8			−0.810824			−0.405412	+	0.914134i	$0.632872\pi$
$788$		−	1.50629e8i		−	0.307843i
$789$	1.04345e8	−	1.89002e8i	0.212441	−	0.384800i
$790$	−7.78071e7			−0.157811
$791$		−	7.16988e7i		−	0.144871i
$792$	−5.62779e8	−	8.93839e8i	−1.13282	−	1.79922i
$793$	1.48168e9			2.97123
$794$		−	9.05153e8i		−	1.80826i
$795$	−2.98182e8	−	1.64621e8i	−0.593445	−	0.327631i
$796$	7.88334e8			1.56304
$797$			2.08748e8i			0.412333i	0.978517	+	0.206166i	$0.0660988\pi$
$797$			2.08748e8i			0.412333i	−0.978517	+	0.206166i	$0.933901\pi$
$798$	−3.76183e8	+	6.81389e8i	−0.740271	+	1.34087i
$799$	−2.91650e7			−0.0571770
$800$		−	5.77207e7i		−	0.112736i
$801$	−2.88445e8	+	1.81611e8i	−0.561261	+	0.353382i
$802$	8.65362e8			1.67755
$803$			7.22187e8i			1.39477i
$804$	1.99023e7	+	1.09877e7i	0.0382945	+	0.0211417i
$805$	−6.61328e8			−1.26774
$806$		−	2.10429e9i		−	4.01885i
$807$	−4.99083e7	+	9.04001e7i	−0.0949625	+	0.172008i
$808$	1.38418e9			2.62397
$809$			1.09068e8i			0.205992i	0.994682	+	0.102996i	$0.0328429\pi$
$809$			1.09068e8i			0.205992i	−0.994682	+	0.102996i	$0.967157\pi$
$810$	−2.45566e8	+	5.12323e8i	−0.462076	+	0.964026i
$811$	5.71123e8			1.07070			0.535349	−	0.844631i	$-0.320180\pi$
$811$	5.71123e8			1.07070			0.535349	+	0.844631i	$0.320180\pi$
$812$		−	9.92868e8i		−	1.85449i
$813$	−7.66553e8	−	4.23200e8i	−1.42650	−	0.787543i
$814$	−2.38104e8			−0.441463
$815$			1.71709e8i			0.317191i
$816$	−7.34719e7	+	1.33081e8i	−0.135223	+	0.244933i
$817$	−4.46369e7			−0.0818517
$818$		−	9.50413e8i		−	1.73641i
$819$	7.33898e8	+	1.16562e9i	1.33593	+	2.12180i
$820$	2.99746e8			0.543641
$821$			4.06859e8i			0.735215i	0.929981	+	0.367607i	$0.119823\pi$
$821$			4.06859e8i			0.735215i	−0.929981	+	0.367607i	$0.880177\pi$
$822$	−6.68356e8	−	3.68988e8i	−1.20335	−	0.664349i
$823$	4.10677e7			0.0736717			0.0368358	−	0.999321i	$-0.488272\pi$
$823$	4.10677e7			0.0736717			0.0368358	+	0.999321i	$0.488272\pi$
$824$			1.09144e9i			1.95082i
$825$	−1.96695e8	+	3.56278e8i	−0.350293	+	0.634493i
$826$	−6.25497e8			−1.10990
$827$			5.95170e8i			1.05226i	0.850403	+	0.526132i	$0.176358\pi$
$827$			5.95170e8i			1.05226i	−0.850403	+	0.526132i	$0.823642\pi$
$828$	−1.27467e9	+	8.02558e8i	−2.24547	+	1.41379i
$829$	5.98747e8			1.05095			0.525473	−	0.850811i	$-0.323889\pi$
$829$	5.98747e8			1.05095			0.525473	+	0.850811i	$0.323889\pi$
$830$		−	8.78738e8i		−	1.53683i
$831$	4.47819e8	+	2.47233e8i	0.780369	+	0.430828i
$832$	−7.56785e8			−1.31402
$833$			2.18856e8i			0.378638i
$834$	−4.63260e8	+	8.39114e8i	−0.798595	+	1.44651i
$835$	3.31984e8			0.570241
$836$		−	7.60845e8i		−	1.30220i
$837$	−4.94701e7	−	8.59787e8i	−0.0843659	−	1.46627i
$838$	−1.24333e9			−2.11279
$839$			1.20600e8i			0.204203i	0.994774	+	0.102102i	$0.0325567\pi$
$839$			1.20600e8i			0.204203i	−0.994774	+	0.102102i	$0.967443\pi$
$840$	9.32424e8	+	5.14775e8i	1.57317	+	0.868520i
$841$	4.04828e8			0.680585
$842$			6.81812e8i			1.14216i
$843$	−6.98507e7	+	1.26522e8i	−0.116597	+	0.211195i
$844$	−7.89638e8			−1.31341
$845$			5.37218e8i			0.890391i
$846$	1.32832e8	+	2.10971e8i	0.219377	+	0.348427i
$847$	3.33840e8			0.549399
$848$			7.79040e8i			1.27753i
$849$	−2.54990e8	−	1.40775e8i	−0.416677	−	0.230040i
$850$	1.62674e8			0.264888
$851$			1.73944e8i			0.282242i
$852$	6.41233e8	−	1.16148e9i	1.03681	−	1.87799i
$853$	−3.40592e8			−0.548766			−0.274383	−	0.961621i	$-0.588474\pi$
$853$	−3.40592e8			−0.548766			−0.274383	+	0.961621i	$0.588474\pi$
$854$		−	3.30148e9i		−	5.30071i
$855$	−1.77409e8	+	1.11701e8i	−0.283843	+	0.178713i
$856$	−5.24383e8			−0.836041
$857$			3.90623e8i			0.620606i	0.950638	+	0.310303i	$0.100430\pi$
$857$			3.90623e8i			0.620606i	−0.950638	+	0.310303i	$0.899570\pi$
$858$	−1.75366e9	−	9.68166e8i	−2.77641	−	1.53281i
$859$	1.76674e8			0.278737			0.139368	−	0.990241i	$-0.455493\pi$
$859$	1.76674e8			0.278737			0.139368	+	0.990241i	$0.455493\pi$
$860$			1.19236e8i			0.187461i
$861$	2.13858e8	−	3.87365e8i	0.335054	−	0.606892i
$862$	5.87472e8			0.917203
$863$			1.07362e9i			1.67039i	0.549952	+	0.835196i	$0.314646\pi$
$863$			1.07362e9i			1.67039i	−0.549952	+	0.835196i	$0.685354\pi$
$864$	1.16084e8	−	6.67921e6i	0.179983	−	0.0103558i
$865$	−7.31214e8			−1.12979
$866$			8.16592e8i			1.25734i
$867$	−3.35612e7	−	1.85285e7i	−0.0514968	−	0.0284305i
$868$	−3.15164e9			−4.81923
$869$		−	1.12276e8i		−	0.171091i
$870$	1.92294e8	−	3.48306e8i	0.292016	−	0.528936i
$871$	2.20866e7			0.0334253
$872$			2.43568e8i			0.367342i
$873$	−3.61786e8	−	5.74610e8i	−0.543763	−	0.863636i
$874$	−8.26916e8			−1.23859
$875$		−	1.06661e9i		−	1.59215i
$876$	−1.45206e9	−	8.01658e8i	−2.16009	−	1.19255i
$877$	1.27042e9			1.88342			0.941711	−	0.336423i	$-0.109217\pi$
$877$	1.27042e9			1.88342			0.941711	+	0.336423i	$0.109217\pi$
$878$			1.44387e8i			0.213327i
$879$	1.65570e8	−	2.99900e8i	0.243789	−	0.441581i
$880$	−5.57698e8			−0.818372
$881$		−	7.13915e8i		−	1.04404i	−0.852932	−	0.522022i	$-0.825178\pi$
$881$		−	7.13915e8i		−	1.04404i	0.852932	−	0.522022i	$-0.174822\pi$
$882$	1.58314e9	−	9.96780e8i	2.30736	−	1.45276i
$883$	7.16412e8			1.04059			0.520296	−	0.853986i	$-0.325821\pi$
$883$	7.16412e8			1.04059			0.520296	+	0.853986i	$0.325821\pi$
$884$			5.38213e8i			0.779107i
$885$	−1.47494e8	−	8.14286e7i	−0.212786	−	0.117475i
$886$	2.38241e8			0.342544
$887$		−	1.12421e9i		−	1.61093i	−0.592642	−	0.805466i	$-0.701915\pi$
$887$		−	1.12421e9i		−	1.61093i	0.592642	−	0.805466i	$-0.298085\pi$
$888$	1.35398e8	−	2.45249e8i	0.193363	−	0.350242i
$889$	−2.42719e8			−0.345461
$890$			4.99850e8i			0.709039i
$891$	−7.39284e8	−	3.54353e8i	−1.04515	−	0.500960i
$892$	6.59101e8			0.928661
$893$			9.19952e7i			0.129185i
$894$	2.06270e9	+	1.13878e9i	2.88684	+	1.59378i
$895$	−1.54346e8			−0.215291
$896$			1.89380e9i			2.63275i
$897$	−7.07283e8	+	1.28112e9i	−0.979977	+	1.77506i
$898$	4.66599e8			0.644340
$899$			6.03100e8i			0.830061i
$900$	−4.98009e8	−	7.90966e8i	−0.683139	−	1.08500i
$901$	−1.96462e8			−0.268600
$902$			6.43493e8i			0.876847i
$903$	1.54090e8	+	8.50703e7i	0.209272	+	0.115535i
$904$	−1.22680e8			−0.166062
$905$			5.15435e8i			0.695390i
$906$	−5.36111e7	+	9.71070e7i	−0.0720891	+	0.130577i
$907$	−1.43794e8			−0.192717			−0.0963583	−	0.995347i	$-0.530719\pi$
$907$	−1.43794e8			−0.192717			−0.0963583	+	0.995347i	$0.530719\pi$
$908$			2.21267e9i			2.95568i
$909$	9.09150e8	−	5.72420e8i	1.21044	−	0.762119i
$910$	2.01992e9			2.68046
$911$			9.56775e8i			1.26548i	0.774365	+	0.632739i	$0.218069\pi$
$911$			9.56775e8i			1.26548i	−0.774365	+	0.632739i	$0.781931\pi$
$912$	4.19779e8	+	2.31753e8i	0.553396	+	0.305520i
$913$	1.26802e9			1.66615
$914$		−	1.34317e9i		−	1.75911i
$915$	4.29793e8	−	7.78494e8i	0.561043	−	1.01623i
$916$	−1.98140e9			−2.57801
$917$		−	1.18266e9i		−	1.53374i
$918$	1.88240e7	+	3.27160e8i	0.0243324	+	0.422895i
$919$	2.96256e8			0.381699			0.190850	−	0.981619i	$-0.438876\pi$
$919$	2.96256e8			0.381699			0.190850	+	0.981619i	$0.438876\pi$
$920$			1.13157e9i			1.45317i
$921$	−8.02460e8	−	4.43024e8i	−1.02717	−	0.567085i
$922$	1.61320e9			2.05824
$923$		−	1.28895e9i		−	1.63920i
$924$	−1.45004e9	+	2.62649e9i	−1.83808	+	3.32936i
$925$	−1.07937e8			−0.136378
$926$			5.30598e8i			0.668241i
$927$	4.51358e8	+	7.16873e8i	0.566607	+	0.899918i
$928$	−8.14275e7			−0.101889
$929$		−	6.82768e8i		−	0.851581i	−0.904822	−	0.425790i	$-0.859996\pi$
$929$		−	6.82768e8i		−	0.851581i	0.904822	−	0.425790i	$-0.140004\pi$
$930$	−1.10562e9	−	6.10394e8i	−1.37454	−	0.758860i
$931$	6.90339e8			0.855487
$932$		−	6.99754e8i		−	0.864366i
$933$	2.14732e8	−	3.88950e8i	0.264395	−	0.478904i
$934$	−8.97054e8			−1.10098
$935$		−	1.40643e8i		−	0.172061i
$936$	1.99444e9	−	1.25574e9i	2.43216	−	1.53134i
$937$	8.52089e7			0.103578			0.0517889	−	0.998658i	$-0.483508\pi$
$937$	8.52089e7			0.103578			0.0517889	+	0.998658i	$0.483508\pi$
$938$		−	4.92133e7i		−	0.0596312i
$939$	−1.99658e8	−	1.10228e8i	−0.241152	−	0.133136i
$940$	2.45742e8			0.295866
$941$			1.41772e9i			1.70146i	0.525607	+	0.850728i	$0.323838\pi$
$941$			1.41772e9i			1.70146i	−0.525607	+	0.850728i	$0.676162\pi$
$942$	9.08158e8	−	1.64497e9i	1.08645	−	1.96791i
$943$	4.70096e8			0.560598
$944$			3.85346e8i			0.458073i
$945$	8.25313e8	−	4.74866e7i	0.977965	−	0.0562698i
$946$	−2.55975e8			−0.302359
$947$			9.97674e8i			1.17473i	0.809321	+	0.587366i	$0.199835\pi$
$947$			9.97674e8i			1.17473i	−0.809321	+	0.587366i	$0.800165\pi$
$948$	2.25747e8	+	1.24631e8i	0.264970	+	0.146285i
$949$	−1.61143e9			−1.88544
$950$		−	5.13124e8i		−	0.598482i
$951$	−7.57682e8	+	1.37241e9i	−0.880938	+	1.59566i
$952$	6.14343e8			0.712033
$953$		−	1.21868e8i		−	0.140803i	−0.997519	−	0.0704013i	$-0.977572\pi$
$953$		−	1.21868e8i		−	0.140803i	0.997519	−	0.0704013i	$-0.0224280\pi$
$954$	8.94787e8	+	1.42115e9i	1.03056	+	1.63680i
$955$	−5.63745e8			−0.647250
$956$		−	1.50850e8i		−	0.172652i
$957$	5.02607e8	+	2.77481e8i	0.573446	+	0.316590i
$958$	−4.66520e8			−0.530608
$959$			1.11087e9i			1.25953i
$960$	−2.19521e8	+	3.97624e8i	−0.248121	+	0.449426i
$961$	1.02690e9			1.15707
$962$		−	5.31286e8i		−	0.596764i
$963$	−3.44423e8	+	2.16856e8i	−0.385668	+	0.242824i
$964$	3.13294e9			3.49720
$965$		−	9.84035e8i		−	1.09504i
$966$	2.85458e9	+	1.57596e9i	3.16672	+	1.74829i
$967$	−5.77633e7			−0.0638811			−0.0319405	−	0.999490i	$-0.510169\pi$
$967$	−5.77633e7			−0.0638811			−0.0319405	+	0.999490i	$0.510169\pi$
$968$		−	5.71217e8i		−	0.629760i
$969$	−5.84446e7	+	1.05862e8i	−0.0642352	+	0.116351i
$970$	−9.95751e8			−1.09103
$971$		−	1.07632e9i		−	1.17566i	−0.808984	−	0.587831i	$-0.799982\pi$
$971$		−	1.07632e9i		−	1.17566i	0.808984	−	0.587831i	$-0.200018\pi$
$972$	1.53311e9	−	1.09309e9i	1.66946	−	1.19030i
$973$	1.39469e9			1.51404
$974$		−	1.66114e8i		−	0.179775i
$975$	−7.94968e8	−	4.38888e8i	−0.857701	−	0.473522i
$976$	−2.03392e9			−2.18768
$977$		−	2.67412e7i		−	0.0286746i	−0.999897	−	0.0143373i	$-0.995436\pi$
$977$		−	2.67412e7i		−	0.0286746i	0.999897	−	0.0143373i	$-0.00456386\pi$
$978$	4.09188e8	−	7.41172e8i	0.437428	−	0.792324i
$979$	−7.21286e8			−0.768704
$980$		−	1.84406e9i		−	1.95928i
$981$	1.00726e8	+	1.59979e8i	0.106693	+	0.169456i
$982$	−1.60230e9			−1.69203
$983$		−	6.25782e8i		−	0.658813i	−0.944188	−	0.329407i	$-0.893151\pi$
$983$		−	6.25782e8i		−	0.658813i	0.944188	−	0.329407i	$-0.106849\pi$
$984$	−6.62802e8	−	3.65921e8i	−0.695662	−	0.384063i
$985$	−8.78281e7			−0.0919020
$986$		−	2.29487e8i		−	0.239402i
$987$	1.75327e8	−	3.17574e8i	0.182347	−	0.330289i
$988$	1.69768e9			1.76030
$989$			1.86999e8i			0.193309i
$990$	−1.01737e9	+	6.40558e8i	−1.04851	+	0.660166i
$991$	4.79795e8			0.492986			0.246493	−	0.969145i	$-0.420722\pi$
$991$	4.79795e8			0.492986			0.246493	+	0.969145i	$0.420722\pi$
$992$			2.58474e8i			0.264778i
$993$	−5.08064e8	−	2.80494e8i	−0.518885	−	0.286467i
$994$	−2.87204e9			−2.92436
$995$		−	4.59659e8i		−	0.466623i
$996$	−1.40756e9	+	2.54954e9i	−1.42459	+	2.58038i
$997$	4.30412e8			0.434309			0.217155	−	0.976137i	$-0.430322\pi$
$997$	4.30412e8			0.434309			0.217155	+	0.976137i	$0.430322\pi$
$998$			2.35327e9i			2.36744i
$999$	−1.24901e7	−	2.17076e8i	−0.0125276	−	0.217729i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	51.7.b.a.35.3	✓	32
3.2	odd	2	inner	51.7.b.a.35.30	yes	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
51.7.b.a.35.3	✓	32	1.1	even	1	trivial
51.7.b.a.35.30	yes	32	3.2	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 51 = 3 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	51.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-13.9722i q^{2} +(23.6370 + 13.0496i) q^{3} -131.222 q^{4} +76.5124i q^{5} +(182.331 - 330.261i) q^{6} -548.925 q^{7} +939.239i q^{8} +(388.417 + 616.906i) q^{9} +1069.05 q^{10} +1542.64i q^{11} +(-3101.70 - 1712.39i) q^{12} -3442.11 q^{13} +7669.68i q^{14} +(-998.455 + 1808.53i) q^{15} +4725.01 q^{16} +1191.58i q^{17} +(8619.53 - 5427.03i) q^{18} +3758.59 q^{19} -10040.1i q^{20} +(-12974.9 - 7163.24i) q^{21} +21554.0 q^{22} -15746.1i q^{23} +(-12256.7 + 22200.8i) q^{24} +9770.85 q^{25} +48093.8i q^{26} +(1130.64 + 19650.5i) q^{27} +72031.0 q^{28} -13783.9i q^{29} +(25269.1 + 13950.6i) q^{30} -43753.9 q^{31} -5907.44i q^{32} +(-20130.8 + 36463.4i) q^{33} +16648.9 q^{34} -41999.6i q^{35} +(-50968.8 - 80951.7i) q^{36} -11046.9 q^{37} -52515.8i q^{38} +(-81361.2 - 44918.1i) q^{39} -71863.5 q^{40} +29854.9i q^{41} +(-100086. + 181288. i) q^{42} -11876.0 q^{43} -202428. i q^{44} +(-47201.0 + 29718.7i) q^{45} -220007. q^{46} +24476.0i q^{47} +(111685. + 61659.4i) q^{48} +183669. q^{49} -136520. i q^{50} +(-15549.6 + 28165.3i) q^{51} +451681. q^{52} +164876. i q^{53} +(274560. - 15797.6i) q^{54} -118031. q^{55} -515571. i q^{56} +(88841.9 + 49048.1i) q^{57} -192591. q^{58} +81554.6i q^{59} +(131019. - 237318. i) q^{60} -430458. q^{61} +611338. i q^{62} +(-213212. - 338635. i) q^{63} +219861. q^{64} -263364. i q^{65} +(509473. + 281271. i) q^{66} -6416.60 q^{67} -156361. i q^{68} +(205479. - 372190. i) q^{69} -586826. q^{70} +374466. i q^{71} +(-579422. + 364816. i) q^{72} +468151. q^{73} +154349. i q^{74} +(230954. + 127505. i) q^{75} -493210. q^{76} -846792. i q^{77} +(-627604. + 1.13679e6i) q^{78} -72781.8 q^{79} +361522. i q^{80} +(-229706. + 479234. i) q^{81} +417138. q^{82} -821984. i q^{83} +(1.70260e6 + 939975. i) q^{84} -91170.5 q^{85} +165933. i q^{86} +(179874. - 325810. i) q^{87} -1.44891e6 q^{88} +467567. i q^{89} +(415236. + 659501. i) q^{90} +1.88946e6 q^{91} +2.06623e6i q^{92} +(-1.03421e6 - 570971. i) q^{93} +341983. q^{94} +287579. i q^{95} +(77089.6 - 139634. i) q^{96} -931439. q^{97} -2.56626e6i q^{98} +(-951663. + 599186. i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 32 q^{3} - 1024 q^{4} + 286 q^{6} + 568 q^{7} - 912 q^{9} - 744 q^{10} + 194 q^{12} - 2312 q^{13} - 6240 q^{15} + 13208 q^{16} + 2936 q^{18} + 7936 q^{19} - 21688 q^{21} + 13176 q^{22} + 18282 q^{24}+ \cdots + 1619864 q^{99}+O(q^{100}) \) 32 * q + 32 * q^3 - 1024 * q^4 + 286 * q^6 + 568 * q^7 - 912 * q^9 - 744 * q^10 + 194 * q^12 - 2312 * q^13 - 6240 * q^15 + 13208 * q^16 + 2936 * q^18 + 7936 * q^19 - 21688 * q^21 + 13176 * q^22 + 18282 * q^24 - 76792 * q^25 + 87056 * q^27 + 45712 * q^28 - 45620 * q^30 - 36728 * q^31 + 73456 * q^33 - 52272 * q^36 + 133336 * q^37 - 48720 * q^39 - 81084 * q^40 - 305352 * q^42 - 412832 * q^43 - 9936 * q^45 + 283044 * q^46 + 233266 * q^48 + 342960 * q^49 + 50152 * q^52 + 196798 * q^54 + 7104 * q^55 - 22576 * q^57 + 167484 * q^58 - 58564 * q^60 - 235496 * q^61 - 331312 * q^63 + 299888 * q^64 + 485816 * q^66 - 774272 * q^67 + 553800 * q^69 - 1318656 * q^70 + 447312 * q^72 - 33152 * q^73 + 86872 * q^75 + 1045000 * q^76 - 8652 * q^78 + 1789576 * q^79 - 2391192 * q^81 - 178380 * q^82 - 751024 * q^84 + 235824 * q^85 - 155232 * q^87 - 3091500 * q^88 - 728056 * q^90 + 2080448 * q^91 + 1012816 * q^93 + 3173160 * q^94 + 422058 * q^96 - 1502960 * q^97 + 1619864 * q^99

Introduction

L-functions

Modular forms

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Database

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