LMFDB - Embedded newform 115.6.b.a.24.36

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [115,6,Mod(24,115)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(115, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("115.24");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 115 = 5 \cdot 23 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	115.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:	$18.4441392785$
Analytic rank:	$0$
Dimension:	$54$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			24.36
Character	$\chi$	$=$	115.24
Dual form			115.6.b.a.24.19

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.13410i q^{2} +27.5349i q^{3} +22.1774 q^{4} +(23.6107 + 50.6709i) q^{5} -86.2973 q^{6} -98.9863i q^{7} +169.798i q^{8} -515.172 q^{9} +O(q^{10})$	\(q+3.13410i q^{2} +27.5349i q^{3} +22.1774 q^{4} +(23.6107 + 50.6709i) q^{5} -86.2973 q^{6} -98.9863i q^{7} +169.798i q^{8} -515.172 q^{9} +(-158.808 + 73.9983i) q^{10} -172.976 q^{11} +610.653i q^{12} +499.623i q^{13} +310.233 q^{14} +(-1395.22 + 650.119i) q^{15} +177.514 q^{16} -185.294i q^{17} -1614.60i q^{18} +457.866 q^{19} +(523.624 + 1123.75i) q^{20} +2725.58 q^{21} -542.125i q^{22} -529.000i q^{23} -4675.36 q^{24} +(-2010.07 + 2392.75i) q^{25} -1565.87 q^{26} -7494.24i q^{27} -2195.26i q^{28} -4657.20 q^{29} +(-2037.54 - 4372.76i) q^{30} +9047.26 q^{31} +5989.87i q^{32} -4762.89i q^{33} +580.731 q^{34} +(5015.72 - 2337.14i) q^{35} -11425.2 q^{36} -5379.45i q^{37} +1435.00i q^{38} -13757.1 q^{39} +(-8603.79 + 4009.04i) q^{40} +957.099 q^{41} +8542.25i q^{42} -19714.5i q^{43} -3836.16 q^{44} +(-12163.6 - 26104.2i) q^{45} +1657.94 q^{46} +22583.7i q^{47} +4887.83i q^{48} +7008.71 q^{49} +(-7499.12 - 6299.77i) q^{50} +5102.06 q^{51} +11080.3i q^{52} +15608.8i q^{53} +23487.7 q^{54} +(-4084.09 - 8764.85i) q^{55} +16807.6 q^{56} +12607.3i q^{57} -14596.1i q^{58} -506.470 q^{59} +(-30942.3 + 14417.9i) q^{60} +31327.9 q^{61} +28355.1i q^{62} +50995.0i q^{63} -13092.4 q^{64} +(-25316.3 + 11796.4i) q^{65} +14927.4 q^{66} -43689.5i q^{67} -4109.34i q^{68} +14566.0 q^{69} +(7324.82 + 15719.8i) q^{70} -18754.4 q^{71} -87475.0i q^{72} +39706.2i q^{73} +16859.8 q^{74} +(-65884.1 - 55347.1i) q^{75} +10154.3 q^{76} +17122.3i q^{77} -43116.1i q^{78} +21075.0 q^{79} +(4191.23 + 8994.78i) q^{80} +81166.6 q^{81} +2999.65i q^{82} +59205.2i q^{83} +60446.3 q^{84} +(9389.01 - 4374.92i) q^{85} +61787.3 q^{86} -128236. i q^{87} -29370.9i q^{88} +74791.3 q^{89} +(81813.3 - 38121.9i) q^{90} +49455.8 q^{91} -11731.8i q^{92} +249116. i q^{93} -70779.5 q^{94} +(10810.5 + 23200.5i) q^{95} -164931. q^{96} +46828.3i q^{97} +21966.0i q^{98} +89112.5 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 54 q - 832 q^{4} + 98 q^{5} - 144 q^{6} - 4270 q^{9}+O(q^{10}) $ 54 * q - 832 * q^4 + 98 * q^5 - 144 * q^6 - 4270 * q^9	$ 54 q - 832 q^{4} + 98 q^{5} - 144 q^{6} - 4270 q^{9} + 942 q^{10} - 380 q^{11} - 3988 q^{14} + 690 q^{15} + 16632 q^{16} - 2440 q^{19} - 2436 q^{20} + 4768 q^{21} - 17940 q^{24} - 9150 q^{25} - 15144 q^{26} + 22958 q^{29} + 23034 q^{30} - 33970 q^{31} + 26120 q^{34} - 18694 q^{35} + 71560 q^{36} + 13416 q^{39} - 22230 q^{40} - 33842 q^{41} + 33908 q^{44} + 35260 q^{45} - 8464 q^{46} - 153772 q^{49} - 91728 q^{50} + 96576 q^{51} - 90500 q^{54} + 37344 q^{55} + 201796 q^{56} + 110106 q^{59} - 92484 q^{60} + 132924 q^{61} - 262108 q^{64} + 45090 q^{65} - 123288 q^{66} + 38088 q^{69} + 249664 q^{70} - 310814 q^{71} + 220660 q^{74} + 235776 q^{75} + 42220 q^{76} - 389348 q^{79} - 245138 q^{80} + 364422 q^{81} - 35836 q^{84} - 134118 q^{85} + 111980 q^{86} - 353788 q^{89} - 805800 q^{90} - 226632 q^{91} + 289804 q^{94} - 42676 q^{95} + 294416 q^{96} - 14088 q^{99}+O(q^{100}) $ 54 * q - 832 * q^4 + 98 * q^5 - 144 * q^6 - 4270 * q^9 + 942 * q^10 - 380 * q^11 - 3988 * q^14 + 690 * q^15 + 16632 * q^16 - 2440 * q^19 - 2436 * q^20 + 4768 * q^21 - 17940 * q^24 - 9150 * q^25 - 15144 * q^26 + 22958 * q^29 + 23034 * q^30 - 33970 * q^31 + 26120 * q^34 - 18694 * q^35 + 71560 * q^36 + 13416 * q^39 - 22230 * q^40 - 33842 * q^41 + 33908 * q^44 + 35260 * q^45 - 8464 * q^46 - 153772 * q^49 - 91728 * q^50 + 96576 * q^51 - 90500 * q^54 + 37344 * q^55 + 201796 * q^56 + 110106 * q^59 - 92484 * q^60 + 132924 * q^61 - 262108 * q^64 + 45090 * q^65 - 123288 * q^66 + 38088 * q^69 + 249664 * q^70 - 310814 * q^71 + 220660 * q^74 + 235776 * q^75 + 42220 * q^76 - 389348 * q^79 - 245138 * q^80 + 364422 * q^81 - 35836 * q^84 - 134118 * q^85 + 111980 * q^86 - 353788 * q^89 - 805800 * q^90 - 226632 * q^91 + 289804 * q^94 - 42676 * q^95 + 294416 * q^96 - 14088 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/115\mathbb{Z}\right)^\times$.

$n$	$47$	$51$
$\chi(n)$	$-1$	$1$

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 $

$2$			3.13410i			0.554036i	0.960865	+	0.277018i	$0.0893462\pi$
$2$			3.13410i			0.554036i	−0.960865	+	0.277018i	$0.910654\pi$
$3$			27.5349i			1.76637i	0.469028	+	0.883183i	$0.344604\pi$
$3$			27.5349i			1.76637i	−0.469028	+	0.883183i	$0.655396\pi$
$4$	22.1774			0.693044
$5$	23.6107	+	50.6709i	0.422361	+	0.906428i
$5$	23.6107	+	50.6709i	0.422361	+	0.906428i
$6$	−86.2973			−0.978631
$7$		−	98.9863i		−	0.763537i	−0.924258	−	0.381768i	$-0.875315\pi$
$7$		−	98.9863i		−	0.763537i	0.924258	−	0.381768i	$-0.124685\pi$
$8$			169.798i			0.938008i
$9$	−515.172			−2.12005
$10$	−158.808	+	73.9983i	−0.502194	+	0.234003i
$11$	−172.976			−0.431027			−0.215514	−	0.976501i	$-0.569143\pi$
$11$	−172.976			−0.431027			−0.215514	+	0.976501i	$0.569143\pi$
$12$			610.653i			1.22417i
$13$			499.623i			0.819944i	0.912098	+	0.409972i	$0.134462\pi$
$13$			499.623i			0.819944i	−0.912098	+	0.409972i	$0.865538\pi$
$14$	310.233			0.423027
$15$	−1395.22	+	650.119i	−1.60108	+	0.746044i
$16$	177.514			0.173353
$17$		−	185.294i		−	0.155503i	−0.996973	−	0.0777516i	$-0.975226\pi$
$17$		−	185.294i		−	0.155503i	0.996973	−	0.0777516i	$-0.0247741\pi$
$18$		−	1614.60i		−	1.17458i
$19$	457.866			0.290974			0.145487	−	0.989360i	$-0.453525\pi$
$19$	457.866			0.290974			0.145487	+	0.989360i	$0.453525\pi$
$20$	523.624	+	1123.75i	0.292715	+	0.628194i
$21$	2725.58			1.34869
$22$		−	542.125i		−	0.238805i
$23$		−	529.000i		−	0.208514i
$23$		−	529.000i		−	0.208514i
$24$	−4675.36			−1.65687
$25$	−2010.07	+	2392.75i	−0.643222	+	0.765679i
$26$	−1565.87			−0.454279
$27$		−	7494.24i		−	1.97842i
$28$		−	2195.26i		−	0.529164i
$29$	−4657.20			−1.02832			−0.514161	−	0.857693i	$-0.671897\pi$
$29$	−4657.20			−1.02832			−0.514161	+	0.857693i	$0.671897\pi$
$30$	−2037.54	−	4372.76i	−0.413336	−	0.887058i
$31$	9047.26			1.69088			0.845440	−	0.534070i	$-0.179338\pi$
$31$	9047.26			1.69088			0.845440	+	0.534070i	$0.179338\pi$
$32$			5989.87i			1.03405i
$33$		−	4762.89i		−	0.761352i
$34$	580.731			0.0861545
$35$	5015.72	−	2337.14i	0.692091	−	0.322488i
$36$	−11425.2			−1.46929
$37$		−	5379.45i		−	0.646002i	−0.946399	−	0.323001i	$-0.895308\pi$
$37$		−	5379.45i		−	0.646002i	0.946399	−	0.323001i	$-0.104692\pi$
$38$			1435.00i			0.161210i
$39$	−13757.1			−1.44832
$40$	−8603.79	+	4009.04i	−0.850236	+	0.396178i
$41$	957.099			0.0889195			0.0444598	−	0.999011i	$-0.485843\pi$
$41$	957.099			0.0889195			0.0444598	+	0.999011i	$0.485843\pi$
$42$			8542.25i			0.747221i
$43$		−	19714.5i		−	1.62598i	−0.582279	−	0.812989i	$-0.697839\pi$
$43$		−	19714.5i		−	1.62598i	0.582279	−	0.812989i	$-0.302161\pi$
$44$	−3836.16			−0.298721
$45$	−12163.6	−	26104.2i	−0.895426	−	1.92167i
$46$	1657.94			0.115525
$47$			22583.7i			1.49125i	0.666368	+	0.745623i	$0.267848\pi$
$47$			22583.7i			1.49125i	−0.666368	+	0.745623i	$0.732152\pi$
$48$			4887.83i			0.306206i
$49$	7008.71			0.417012
$50$	−7499.12	−	6299.77i	−0.424214	−	0.356369i
$51$	5102.06			0.274676
$52$			11080.3i			0.568257i
$53$			15608.8i			0.763271i	0.924313	+	0.381635i	$0.124639\pi$
$53$			15608.8i			0.763271i	−0.924313	+	0.381635i	$0.875361\pi$
$54$	23487.7			1.09612
$55$	−4084.09	−	8764.85i	−0.182049	−	0.390695i
$56$	16807.6			0.716203
$57$			12607.3i			0.513967i
$58$		−	14596.1i		−	0.569728i
$59$	−506.470			−0.0189419			−0.00947096	−	0.999955i	$-0.503015\pi$
$59$	−506.470			−0.0189419			−0.00947096	+	0.999955i	$0.503015\pi$
$60$	−30942.3	+	14417.9i	−1.10962	+	0.517041i
$61$	31327.9			1.07797			0.538985	−	0.842316i	$-0.318808\pi$
$61$	31327.9			1.07797			0.538985	+	0.842316i	$0.318808\pi$
$62$			28355.1i			0.936809i
$63$			50995.0i			1.61874i
$64$	−13092.4			−0.399549
$65$	−25316.3	+	11796.4i	−0.743220	+	0.346312i
$66$	14927.4			0.421817
$67$		−	43689.5i		−	1.18902i	−0.804087	−	0.594511i	$-0.797346\pi$
$67$		−	43689.5i		−	1.18902i	0.804087	−	0.594511i	$-0.202654\pi$
$68$		−	4109.34i		−	0.107771i
$69$	14566.0			0.368313
$70$	7324.82	+	15719.8i	0.178670	+	0.383444i
$71$	−18754.4			−0.441526			−0.220763	−	0.975327i	$-0.570855\pi$
$71$	−18754.4			−0.441526			−0.220763	+	0.975327i	$0.570855\pi$
$72$		−	87475.0i		−	1.98862i
$73$			39706.2i			0.872070i	0.899930	+	0.436035i	$0.143618\pi$
$73$			39706.2i			0.872070i	−0.899930	+	0.436035i	$0.856382\pi$
$74$	16859.8			0.357909
$75$	−65884.1	−	55347.1i	−1.35247	−	1.13617i
$76$	10154.3			0.201658
$77$			17122.3i			0.329105i
$78$		−	43116.1i		−	0.802423i
$79$	21075.0			0.379927			0.189964	−	0.981791i	$-0.439163\pi$
$79$	21075.0			0.379927			0.189964	+	0.981791i	$0.439163\pi$
$80$	4191.23	+	8994.78i	0.0732177	+	0.157132i
$81$	81166.6			1.37456
$82$			2999.65i			0.0492646i
$83$			59205.2i			0.943333i	0.881777	+	0.471666i	$0.156347\pi$
$83$			59205.2i			0.943333i	−0.881777	+	0.471666i	$0.843653\pi$
$84$	60446.3			0.934698
$85$	9389.01	−	4374.92i	0.140952	−	0.0656785i
$86$	61787.3			0.900851
$87$		−	128236.i		−	1.81639i
$88$		−	29370.9i		−	0.404307i
$89$	74791.3			1.00087			0.500433	−	0.865775i	$-0.333174\pi$
$89$	74791.3			1.00087			0.500433	+	0.865775i	$0.333174\pi$
$90$	81813.3	−	38121.9i	1.06468	−	0.496099i
$91$	49455.8			0.626057
$92$		−	11731.8i		−	0.144510i
$93$			249116.i			2.98672i
$94$	−70779.5			−0.826205
$95$	10810.5	+	23200.5i	0.122896	+	0.263747i
$96$	−164931.			−1.82651
$97$			46828.3i			0.505335i	0.967553	+	0.252668i	$0.0813079\pi$
$97$			46828.3i			0.505335i	−0.967553	+	0.252668i	$0.918692\pi$
$98$			21966.0i			0.231040i
$99$	89112.5			0.913799
$100$	−44578.1	+	53064.9i	−0.445781	+	0.530649i
$101$	124353.			1.21297			0.606487	−	0.795094i	$-0.292578\pi$
$101$	124353.			1.21297			0.606487	+	0.795094i	$0.292578\pi$
$102$			15990.4i			0.152180i
$103$			165230.i			1.53460i	0.641290	+	0.767299i	$0.278400\pi$
$103$			165230.i			1.53460i	−0.641290	+	0.767299i	$0.721600\pi$
$104$	−84834.8			−0.769114
$105$	64352.8	+	138107.i	0.569632	+	1.22249i
$106$	−48919.5			−0.422880
$107$			136352.i			1.15133i	0.817684	+	0.575667i	$0.195257\pi$
$107$			136352.i			1.15133i	−0.817684	+	0.575667i	$0.804743\pi$
$108$		−	166203.i		−	1.37113i
$109$	−223278.			−1.80003			−0.900016	−	0.435858i	$-0.856445\pi$
$109$	−223278.			−1.80003			−0.900016	+	0.435858i	$0.856445\pi$
$110$	27469.9	−	12800.0i	0.216459	−	0.100862i
$111$	148123.			1.14108
$112$		−	17571.4i		−	0.132362i
$113$		−	106206.i		−	0.782443i	−0.920297	−	0.391222i	$-0.872053\pi$
$113$		−	106206.i		−	0.782443i	0.920297	−	0.391222i	$-0.127947\pi$
$114$	−39512.6			−0.284756
$115$	26804.9	−	12490.1i	0.189003	−	0.0880683i
$116$	−103284.			−0.712673
$117$		−	257392.i		−	1.73832i
$118$		−	1587.33i		−	0.0104945i
$119$	−18341.6			−0.118732
$120$	−110389.	−	236905.i	−0.699795	−	1.50183i
$121$	−131130.			−0.814216
$122$			98184.8i			0.597234i
$123$			26353.6i			0.157064i
$124$	200645.			1.17185
$125$	−168702.	−	45357.5i	−0.965705	−	0.259642i
$126$	−159824.			−0.896839
$127$			295410.i			1.62524i	0.582797	+	0.812618i	$0.301958\pi$
$127$			295410.i			1.62524i	−0.582797	+	0.812618i	$0.698042\pi$
$128$			150643.i			0.812687i
$129$	542838.			2.87207
$130$	−36971.3	−	79344.0i	−0.191870	−	0.411771i
$131$	−75570.6			−0.384746			−0.192373	−	0.981322i	$-0.561618\pi$
$131$	−75570.6			−0.384746			−0.192373	+	0.981322i	$0.561618\pi$
$132$		−	105628.i		−	0.527650i
$133$		−	45322.5i		−	0.222170i
$134$	136927.			0.658762
$135$	379740.	−	176944.i	1.79329	−	0.835607i
$136$	31462.5			0.145863
$137$		−	400299.i		−	1.82215i	−0.412245	−	0.911073i	$-0.635255\pi$
$137$		−	400299.i		−	1.82215i	0.412245	−	0.911073i	$-0.364745\pi$
$138$			45651.3i			0.204059i
$139$	−66371.9			−0.291371			−0.145686	−	0.989331i	$-0.546539\pi$
$139$	−66371.9			−0.291371			−0.145686	+	0.989331i	$0.546539\pi$
$140$	111236.	−	51831.6i	0.479649	−	0.223498i
$141$	−621839.			−2.63409
$142$		−	58778.1i		−	0.244622i
$143$		−	86422.9i		−	0.353418i
$144$	−91450.2			−0.367518
$145$	−109960.	−	235984.i	−0.434323	−	0.932100i
$146$	−124443.			−0.483158
$147$			192984.i			0.736595i
$148$		−	119302.i		−	0.447708i
$149$	−181608.			−0.670147			−0.335073	−	0.942192i	$-0.608761\pi$
$149$	−181608.			−0.670147			−0.335073	+	0.942192i	$0.608761\pi$
$150$	173464.	−	206488.i	0.629478	−	0.749318i
$151$	−179415.			−0.640350			−0.320175	−	0.947358i	$-0.603742\pi$
$151$	−179415.			−0.640350			−0.320175	+	0.947358i	$0.603742\pi$
$152$			77744.5i			0.272936i
$153$			95458.4i			0.329675i
$154$	−53663.0			−0.182336
$155$	213612.	+	458433.i	0.714162	+	1.53266i
$156$	−305096.			−1.00375
$157$		−	574225.i		−	1.85923i	−0.368531	−	0.929615i	$-0.620139\pi$
$157$		−	574225.i		−	1.85923i	0.368531	−	0.929615i	$-0.379861\pi$
$158$			66051.3i			0.210494i
$159$	−429786.			−1.34822
$160$	−303512.	+	141425.i	−0.937293	+	0.436743i
$161$	−52363.8			−0.159208
$162$			254384.i			0.761558i
$163$		−	157294.i		−	0.463705i	−0.972751	−	0.231853i	$-0.925521\pi$
$163$		−	157294.i		−	0.463705i	0.972751	−	0.231853i	$-0.0744787\pi$
$164$	21226.0			0.0616251
$165$	241340.	−	112455.i	0.690111	−	0.321565i
$166$	−185555.			−0.522640
$167$			150221.i			0.416811i	0.978042	+	0.208406i	$0.0668274\pi$
$167$			150221.i			0.416811i	−0.978042	+	0.208406i	$0.933173\pi$
$168$			462797.i			1.26508i
$169$	121670.			0.327692
$170$	13711.5	+	29426.1i	0.0363883	+	0.0780928i
$171$	−235880.			−0.616880
$172$		−	437217.i		−	1.12687i
$173$			408467.i			1.03763i	0.854887	+	0.518813i	$0.173626\pi$
$173$			408467.i			1.03763i	−0.854887	+	0.518813i	$0.826374\pi$
$174$	401903.			1.00635
$175$	236849.	+	198969.i	0.584624	+	0.491124i
$176$	−30705.7			−0.0747200
$177$		−	13945.6i		−	0.0334584i
$178$			234404.i			0.554517i
$179$	−83182.2			−0.194043			−0.0970214	−	0.995282i	$-0.530932\pi$
$179$	−83182.2			−0.194043			−0.0970214	+	0.995282i	$0.530932\pi$
$180$	−269756.	−	578924.i	−0.620570	−	1.33180i
$181$	383086.			0.869160			0.434580	−	0.900633i	$-0.356897\pi$
$181$	383086.			0.869160			0.434580	+	0.900633i	$0.356897\pi$
$182$			155000.i			0.346859i
$183$			862611.i			1.90409i
$184$	89822.9			0.195588
$185$	272581.	−	127013.i	0.585554	−	0.272846i
$186$	−780754.			−1.65475
$187$			32051.5i			0.0670261i
$188$			500847.i			1.03350i
$189$	−741827.			−1.51060
$190$	−72712.6	+	33881.3i	−0.146125	+	0.0680889i
$191$	906396.			1.79777			0.898886	−	0.438182i	$-0.144377\pi$
$191$	906396.			1.79777			0.898886	+	0.438182i	$0.144377\pi$
$192$		−	360499.i		−	0.705750i
$193$			609358.i			1.17755i	0.808297	+	0.588775i	$0.200390\pi$
$193$			609358.i			1.17755i	−0.808297	+	0.588775i	$0.799610\pi$
$194$	−146765.			−0.279974
$195$	−324814.	−	697083.i	−0.611714	−	1.31280i
$196$	155435.			0.289007
$197$		−	690338.i		−	1.26735i	−0.773600	−	0.633675i	$-0.781546\pi$
$197$		−	690338.i		−	1.26735i	0.773600	−	0.633675i	$-0.218454\pi$
$198$			279288.i			0.506278i
$199$	118685.			0.212452			0.106226	−	0.994342i	$-0.466123\pi$
$199$	118685.			0.212452			0.106226	+	0.994342i	$0.466123\pi$
$200$	−406283.	−	341305.i	−0.718213	−	0.603348i
$201$	1.20299e6			2.10025
$202$			389734.i			0.672031i
$203$			460999.i			0.785162i
$204$	113150.			0.190362
$205$	22597.8	+	48497.0i	0.0375561	+	0.0805991i
$206$	−517846.			−0.850223
$207$			272526.i			0.442061i
$208$			88690.0i			0.142140i
$209$	−79199.9			−0.125418
$210$	−432843.	+	201688.i	−0.677302	+	0.315597i
$211$	1.18355e6			1.83012			0.915062	−	0.403314i	$-0.132142\pi$
$211$	1.18355e6			1.83012			0.915062	+	0.403314i	$0.132142\pi$
$212$			346162.i			0.528980i
$213$		−	516400.i		−	0.779897i
$214$	−427340.			−0.637881
$215$	998951.	−	465473.i	1.47383	−	0.686750i
$216$	1.27250e6			1.85577
$217$		−	895555.i		−	1.29105i
$218$		−	699777.i		−	0.997283i
$219$	−1.09331e6			−1.54039
$220$	−90574.5	−	194382.i	−0.126168	−	0.270769i
$221$	92577.3			0.127504
$222$			464232.i			0.632198i
$223$		−	1.37756e6i		−	1.85501i	−0.373806	−	0.927507i	$-0.621948\pi$
$223$		−	1.37756e6i		−	1.85501i	0.373806	−	0.927507i	$-0.378052\pi$
$224$	592915.			0.789537
$225$	1.03553e6	−	1.23268e6i	1.36366	−	1.62328i
$226$	332860.			0.433502
$227$			1.14830e6i			1.47908i	0.673114	+	0.739539i	$0.264957\pi$
$227$			1.14830e6i			1.47908i	−0.673114	+	0.739539i	$0.735043\pi$
$228$			279597.i			0.356202i
$229$	−171814.			−0.216506			−0.108253	−	0.994123i	$-0.534526\pi$
$229$	−171814.			−0.216506			−0.108253	+	0.994123i	$0.534526\pi$
$230$	39145.1	+	84009.2i	0.0487931	+	0.104715i
$231$	−471461.			−0.581320
$232$		−	790780.i		−	0.964575i
$233$		−	651670.i		−	0.786390i	−0.919455	−	0.393195i	$-0.871370\pi$
$233$		−	651670.i		−	0.786390i	0.919455	−	0.393195i	$-0.128630\pi$
$234$	806693.			0.963094
$235$	−1.14433e6	+	533216.i	−1.35171	+	0.629844i
$236$	−11232.2			−0.0131276
$237$			580300.i			0.671091i
$238$		−	57484.4i		−	0.0657821i
$239$	1.53723e6			1.74078			0.870389	−	0.492365i	$-0.163867\pi$
$239$	1.53723e6			1.74078			0.870389	+	0.492365i	$0.163867\pi$
$240$	−247671.	+	115405.i	−0.277553	+	0.129329i
$241$	913293.			1.01290			0.506451	−	0.862269i	$-0.330957\pi$
$241$	913293.			1.01290			0.506451	+	0.862269i	$0.330957\pi$
$242$		−	410976.i		−	0.451105i
$243$			413814.i			0.449562i
$244$	694771.			0.747080
$245$	165481.	+	355137.i	0.176129	+	0.377991i
$246$	−82595.0			−0.0870194
$247$			228760.i			0.238583i
$248$			1.53620e6i			1.58606i
$249$	−1.63021e6			−1.66627
$250$	142155.	−	528729.i	0.143851	−	0.535036i
$251$	432806.			0.433620			0.216810	−	0.976214i	$-0.430435\pi$
$251$	432806.			0.433620			0.216810	+	0.976214i	$0.430435\pi$
$252$			1.13094e6i			1.12186i
$253$			91504.4i			0.0898754i
$254$	−925846.			−0.900440
$255$	120463.	+	258526.i	0.116012	+	0.248974i
$256$	−891087.			−0.849807
$257$		−	1.43836e6i		−	1.35842i	−0.733944	−	0.679210i	$-0.762323\pi$
$257$		−	1.43836e6i		−	1.35842i	0.733944	−	0.679210i	$-0.237677\pi$
$258$			1.70131e6i			1.59123i
$259$	−532492.			−0.493246
$260$	−561450.	+	261615.i	−0.515084	+	0.240010i
$261$	2.39926e6			2.18010
$262$		−	236846.i		−	0.213163i
$263$			172554.i			0.153828i	0.997038	+	0.0769142i	$0.0245068\pi$
$263$			172554.i			0.153828i	−0.997038	+	0.0769142i	$0.975493\pi$
$264$	808727.			0.714154
$265$	−790909.	+	368534.i	−0.691850	+	0.322376i
$266$	142045.			0.123090
$267$			2.05937e6i			1.76790i
$268$		−	968920.i		−	0.824045i
$269$	243007.			0.204757			0.102378	−	0.994746i	$-0.467355\pi$
$269$	243007.			0.204757			0.102378	+	0.994746i	$0.467355\pi$
$270$	554561.	+	1.19014e6i	0.462957	+	0.993550i
$271$	−2.16968e6			−1.79462			−0.897310	−	0.441400i	$-0.854482\pi$
$271$	−2.16968e6			−1.79462			−0.897310	+	0.441400i	$0.854482\pi$
$272$		−	32892.3i		−	0.0269570i
$273$			1.36176e6i			1.10585i
$274$	1.25458e6			1.00954
$275$	347694.	−	413889.i	0.277246	−	0.330029i
$276$	323035.			0.255257
$277$		−	858654.i		−	0.672386i	−0.941793	−	0.336193i	$-0.890860\pi$
$277$		−	858654.i		−	0.672386i	0.941793	−	0.336193i	$-0.109140\pi$
$278$		−	208016.i		−	0.161430i
$279$	−4.66090e6			−3.58475
$280$	396840.	+	851657.i	0.302496	+	0.649187i
$281$	2.29812e6			1.73623			0.868115	−	0.496363i	$-0.165331\pi$
$281$	2.29812e6			1.73623			0.868115	+	0.496363i	$0.165331\pi$
$282$		−	1.94891e6i		−	1.45938i
$283$			382115.i			0.283614i	0.989894	+	0.141807i	$0.0452912\pi$
$283$			382115.i			0.283614i	−0.989894	+	0.141807i	$0.954709\pi$
$284$	−415923.			−0.305997
$285$	−638823.	+	297667.i	−0.465874	+	0.217080i
$286$	270858.			0.195807
$287$		−	94739.6i		−	0.0678933i
$288$		−	3.08581e6i		−	2.19224i
$289$	1.38552e6			0.975819
$290$	739598.	−	344625.i	0.516417	−	0.240631i
$291$	−1.28942e6			−0.892607
$292$			880580.i			0.604382i
$293$		−	4228.48i		−	0.00287750i	−0.999999	−	0.00143875i	$-0.999542\pi$
$293$		−	4228.48i		−	0.00287750i	0.999999	−	0.00143875i	$-0.000457969\pi$
$294$	−604833.			−0.408100
$295$	−11958.1	−	25663.3i	−0.00800032	−	0.0171695i
$296$	913418.			0.605955
$297$			1.29633e6i			0.852752i
$298$		−	569179.i		−	0.371286i
$299$	264301.			0.170970
$300$	−1.46114e6	−	1.22746e6i	−0.937321	−	0.787413i
$301$	−1.95147e6			−1.24149
$302$		−	562306.i		−	0.354777i
$303$			3.42404e6i			2.14255i
$304$	81277.6			0.0504414
$305$	739673.	+	1.58741e6i	0.455292	+	0.977101i
$306$	−299176.			−0.182652
$307$			454799.i			0.275406i	0.990474	+	0.137703i	$0.0439719\pi$
$307$			454799.i			0.275406i	−0.990474	+	0.137703i	$0.956028\pi$
$308$			379728.i			0.228084i
$309$	−4.54958e6			−2.71066
$310$	−1.43677e6	+	669483.i	−0.849150	+	0.395672i
$311$	−2.90140e6			−1.70101			−0.850506	−	0.525966i	$-0.823704\pi$
$311$	−2.90140e6			−1.70101			−0.850506	+	0.525966i	$0.823704\pi$
$312$		−	2.33592e6i		−	1.35854i
$313$		−	2.58180e6i		−	1.48957i	−0.667304	−	0.744785i	$-0.732552\pi$
$313$		−	2.58180e6i		−	1.48957i	0.667304	−	0.744785i	$-0.267448\pi$
$314$	1.79968e6			1.03008
$315$	−2.58396e6	+	1.20403e6i	−1.46727	+	0.683691i
$316$	467389.			0.263306
$317$		−	2.07352e6i		−	1.15894i	−0.814995	−	0.579468i	$-0.803260\pi$
$317$		−	2.07352e6i		−	1.15894i	0.814995	−	0.579468i	$-0.196740\pi$
$318$		−	1.34699e6i		−	0.746961i
$319$	805584.			0.443235
$320$	−309121.	−	663404.i	−0.168754	−	0.362162i
$321$	−3.75443e6			−2.03368
$322$		−	164113.i		−	0.0882073i
$323$		−	84839.9i		−	0.0452474i
$324$	1.80006e6			0.952632
$325$	−1.19547e6	−	1.00428e6i	−0.627814	−	0.527406i
$326$	492974.			0.256909
$327$		−	6.14795e6i		−	3.17951i
$328$			162513.i			0.0834072i
$329$	2.23547e6			1.13862
$330$	352446.	+	756383.i	0.178159	+	0.382346i
$331$	3.30721e6			1.65917			0.829587	−	0.558377i	$-0.188576\pi$
$331$	3.30721e6			1.65917			0.829587	+	0.558377i	$0.188576\pi$
$332$			1.31302e6i			0.653771i
$333$			2.77134e6i			1.36956i
$334$	−470808.			−0.230929
$335$	2.21378e6	−	1.03154e6i	1.07776	−	0.502197i
$336$	483828.			0.233799
$337$			1.83038e6i			0.877945i	0.898500	+	0.438973i	$0.144658\pi$
$337$			1.83038e6i			0.877945i	−0.898500	+	0.438973i	$0.855342\pi$
$338$			381325.i			0.181553i
$339$	2.92437e6			1.38208
$340$	208224.	−	97024.5i	0.0976862	−	0.0455181i
$341$	−1.56496e6			−0.728816
$342$		−	739272.i		−	0.341774i
$343$		−	2.35743e6i		−	1.08194i
$344$	3.34747e6			1.52518
$345$	343913.	+	738070.i	0.155561	+	0.333849i
$346$	−1.28018e6			−0.574883
$347$			1.81015e6i			0.807030i	0.914973	+	0.403515i	$0.132212\pi$
$347$			1.81015e6i			0.807030i	−0.914973	+	0.403515i	$0.867788\pi$
$348$		−	2.84393e6i		−	1.25884i
$349$	−1.63329e6			−0.717793			−0.358896	−	0.933377i	$-0.616847\pi$
$349$	−1.63329e6			−0.717793			−0.358896	+	0.933377i	$0.616847\pi$
$350$	−623591.	+	742310.i	−0.272101	+	0.323903i
$351$	3.74430e6			1.62219
$352$		−	1.03610e6i		−	0.445704i
$353$		−	833243.i		−	0.355906i	−0.984039	−	0.177953i	$-0.943053\pi$
$353$		−	833243.i		−	0.355906i	0.984039	−	0.177953i	$-0.0569475\pi$
$354$	43707.0			0.0185371
$355$	−442804.	−	950299.i	−0.186483	−	0.400212i
$356$	1.65868e6			0.693644
$357$		−	505034.i		−	0.209725i
$358$		−	260701.i		−	0.107507i
$359$	636036.			0.260463			0.130231	−	0.991484i	$-0.458428\pi$
$359$	636036.			0.260463			0.130231	+	0.991484i	$0.458428\pi$
$360$	4.43243e6	−	2.06534e6i	1.80254	−	0.839917i
$361$	−2.26646e6			−0.915334
$362$			1.20063e6i			0.481546i
$363$		−	3.61066e6i		−	1.43820i
$364$	1.09680e6			0.433885
$365$	−2.01195e6	+	937491.i	−0.790468	+	0.368328i
$366$	−2.70351e6			−1.05493
$367$		−	372220.i		−	0.144256i	−0.997395	−	0.0721282i	$-0.977021\pi$
$367$		−	372220.i		−	0.144256i	0.997395	−	0.0721282i	$-0.0229791\pi$
$368$		−	93904.8i		−	0.0361467i
$369$	−493071.			−0.188514
$370$	398071.	+	854298.i	0.151167	+	0.324418i
$371$	1.54505e6			0.582785
$372$			5.52474e6i			2.06992i
$373$		−	18520.2i		−	0.00689245i	−0.999994	−	0.00344622i	$-0.998903\pi$
$373$		−	18520.2i		−	0.00689245i	0.999994	−	0.00344622i	$-0.00109697\pi$
$374$	−100453.			−0.0371349
$375$	1.24892e6	−	4.64519e6i	0.458622	−	1.70579i
$376$	−3.83465e6			−1.39880
$377$		−	2.32684e6i		−	0.843167i
$378$		−	2.32496e6i		−	0.836925i
$379$	−4.32941e6			−1.54821			−0.774106	−	0.633056i	$-0.781800\pi$
$379$	−4.32941e6			−1.54821			−0.774106	+	0.633056i	$0.781800\pi$
$380$	239750.	+	514526.i	0.0851724	+	0.182788i
$381$	−8.13410e6			−2.87076
$382$			2.84074e6i			0.996031i
$383$			3.16493e6i			1.10247i	0.834350	+	0.551235i	$0.185843\pi$
$383$			3.16493e6i			1.10247i	−0.834350	+	0.551235i	$0.814157\pi$
$384$	−4.14794e6			−1.43550
$385$	−867600.	+	404269.i	−0.298310	+	0.139001i
$386$	−1.90979e6			−0.652406
$387$			1.01564e7i			3.44716i
$388$			1.03853e6i			0.350219i
$389$	2.47737e6			0.830074			0.415037	−	0.909804i	$-0.363769\pi$
$389$	2.47737e6			0.830074			0.415037	+	0.909804i	$0.363769\pi$
$390$	2.18473e6	−	1.01800e6i	0.727338	−	0.338912i
$391$	−98020.6			−0.0324247
$392$			1.19006e6i			0.391160i
$393$		−	2.08083e6i		−	0.679603i
$394$	2.16359e6			0.702158
$395$	497596.	+	1.06789e6i	0.160466	+	0.344377i
$396$	1.97628e6			0.633303
$397$		−	1.98642e6i		−	0.632550i	−0.948667	−	0.316275i	$-0.897568\pi$
$397$		−	1.98642e6i		−	0.632550i	0.948667	−	0.316275i	$-0.102432\pi$
$398$			371970.i			0.117706i
$399$	1.24795e6			0.392433
$400$	−356815.	+	424746.i	−0.111505	+	0.132733i
$401$	−3.05157e6			−0.947681			−0.473840	−	0.880611i	$-0.657133\pi$
$401$	−3.05157e6			−0.947681			−0.473840	+	0.880611i	$0.657133\pi$
$402$			3.77029e6i			1.16361i
$403$			4.52022e6i			1.38643i
$404$	2.75782e6			0.840643
$405$	1.91640e6	+	4.11278e6i	0.580562	+	1.24594i
$406$	−1.44482e6			−0.435008
$407$			930517.i			0.278444i
$408$			866318.i			0.257648i
$409$	1.54349e6			0.456243			0.228122	−	0.973633i	$-0.426742\pi$
$409$	1.54349e6			0.456243			0.228122	+	0.973633i	$0.426742\pi$
$410$	−151995.	+	70823.7i	−0.0446548	+	0.0208075i
$411$	1.10222e7			3.21858
$412$			3.66436e6i			1.06354i
$413$			50133.6i			0.0144628i
$414$	−854125.			−0.244918
$415$	−2.99998e6	+	1.39788e6i	−0.855063	+	0.398427i
$416$	−2.99268e6			−0.847865
$417$		−	1.82754e6i		−	0.514669i
$418$		−	248221.i		−	0.0694860i
$419$	−782997.			−0.217884			−0.108942	−	0.994048i	$-0.534746\pi$
$419$	−782997.			−0.217884			−0.108942	+	0.994048i	$0.534746\pi$
$420$	1.42718e6	+	3.06286e6i	0.394780	+	0.847236i
$421$	−1.89001e6			−0.519707			−0.259854	−	0.965648i	$-0.583674\pi$
$421$	−1.89001e6			−0.519707			−0.259854	+	0.965648i	$0.583674\pi$
$422$			3.70937e6i			1.01395i
$423$		−	1.16345e7i		−	3.16152i
$424$	−2.65033e6			−0.715954
$425$	443362.	+	372454.i	0.119066	+	0.100023i
$426$	1.61845e6			0.432091
$427$		−	3.10103e6i		−	0.823069i
$428$			3.02393e6i			0.797925i
$429$	2.37965e6			0.624266
$430$	1.45884e6	+	3.13081e6i	0.380484	+	0.816556i
$431$	1.47121e6			0.381489			0.190745	−	0.981640i	$-0.438910\pi$
$431$	1.47121e6			0.381489			0.190745	+	0.981640i	$0.438910\pi$
$432$		−	1.33033e6i		−	0.342966i
$433$			1638.32i			0.000419931i	1.00000		0.000209966i	$6.68341e-5\pi$
$433$			1638.32i			0.000419931i	−1.00000		0.000209966i	$0.999933\pi$
$434$	2.80676e6			0.715289
$435$	6.49780e6	−	3.02773e6i	1.64643	−	0.767174i
$436$	−4.95173e6			−1.24750
$437$		−	242211.i		−	0.0606723i
$438$		−	3.42654e6i		−	0.853435i
$439$	6.97689e6			1.72783			0.863914	−	0.503640i	$-0.168006\pi$
$439$	6.97689e6			1.72783			0.863914	+	0.503640i	$0.168006\pi$
$440$	1.48825e6	−	693468.i	0.366475	−	0.170763i
$441$	−3.61069e6			−0.884085
$442$			290147.i			0.0706418i
$443$		−	3.49744e6i		−	0.846723i	−0.905961	−	0.423361i	$-0.860850\pi$
$443$		−	3.49744e6i		−	0.846723i	0.905961	−	0.423361i	$-0.139150\pi$
$444$	3.28498e6			0.790816
$445$	1.76588e6	+	3.78974e6i	0.422727	+	0.907213i
$446$	4.31740e6			1.02774
$447$		−	5.00057e6i		−	1.18372i
$448$			1.29597e6i			0.305070i
$449$	4.00137e6			0.936684			0.468342	−	0.883547i	$-0.344851\pi$
$449$	4.00137e6			0.936684			0.468342	+	0.883547i	$0.344851\pi$
$450$	3.86334e6	+	3.24546e6i	0.899355	+	0.755519i
$451$	−165555.			−0.0383267
$452$		−	2.35537e6i		−	0.542267i
$453$		−	4.94019e6i		−	1.13109i
$454$	−3.59889e6			−0.819463
$455$	1.16769e6	+	2.50597e6i	0.264422	+	0.567476i
$456$	−2.14069e6			−0.482105
$457$			4.37124e6i			0.979072i	0.871983	+	0.489536i	$0.162834\pi$
$457$			4.37124e6i			0.979072i	−0.871983	+	0.489536i	$0.837166\pi$
$458$		−	538482.i		−	0.119952i
$459$	−1.38864e6			−0.307651
$460$	594463.	−	276997.i	0.130988	−	0.0610352i
$461$	1.70022e6			0.372610			0.186305	−	0.982492i	$-0.440349\pi$
$461$	1.70022e6			0.372610			0.186305	+	0.982492i	$0.440349\pi$
$462$		−	1.47761e6i		−	0.322073i
$463$		−	5.77097e6i		−	1.25111i	−0.780179	−	0.625557i	$-0.784872\pi$
$463$		−	5.77097e6i		−	1.25111i	0.780179	−	0.625557i	$-0.215128\pi$
$464$	−826717.			−0.178263
$465$	−1.26229e7	+	5.88180e6i	−2.70724	+	1.26147i
$466$	2.04240e6			0.435689
$467$		−	3.25780e6i		−	0.691246i	−0.938373	−	0.345623i	$-0.887668\pi$
$467$		−	3.25780e6i		−	0.691246i	0.938373	−	0.345623i	$-0.112332\pi$
$468$		−	5.70828e6i		−	1.20473i
$469$	−4.32466e6			−0.907863
$470$	−1.67115e6	−	3.58646e6i	−0.348957	−	0.748895i
$471$	1.58113e7			3.28408
$472$		−	85997.4i		−	0.0177677i
$473$			3.41014e6i			0.700841i
$474$	−1.81872e6			−0.371809
$475$	−920343.	+	1.09556e6i	−0.187161	+	0.222793i
$476$	−406769.			−0.0822868
$477$		−	8.04120e6i		−	1.61817i
$478$			4.81783e6i			0.964454i
$479$	5.35349e6			1.06610			0.533051	−	0.846083i	$-0.321045\pi$
$479$	5.35349e6			1.06610			0.533051	+	0.846083i	$0.321045\pi$
$480$	−3.89413e6	−	8.35717e6i	−0.771448	−	1.65560i
$481$	2.68770e6			0.529685
$482$			2.86236e6i			0.561185i
$483$		−	1.44183e6i		−	0.281220i
$484$	−2.90813e6			−0.564287
$485$	−2.37283e6	+	1.10565e6i	−0.458050	+	0.213434i
$486$	−1.29694e6			−0.249074
$487$		−	3.94606e6i		−	0.753948i	−0.926224	−	0.376974i	$-0.876965\pi$
$487$		−	3.94606e6i		−	0.753948i	0.926224	−	0.376974i	$-0.123035\pi$
$488$			5.31940e6i			1.01114i
$489$	4.33107e6			0.819073
$490$	−1.11304e6	+	518633.i	−0.209421	+	0.0975821i
$491$	−4.46612e6			−0.836038			−0.418019	−	0.908438i	$-0.637276\pi$
$491$	−4.46612e6			−0.836038			−0.418019	+	0.908438i	$0.637276\pi$
$492$			584455.i			0.108853i
$493$			862951.i			0.159908i
$494$	−716959.			−0.132183
$495$	2.10401e6	+	4.51541e6i	0.385953	+	0.828293i
$496$	1.60601e6			0.293120
$497$			1.85642e6i			0.337121i
$498$		−	5.10925e6i		−	0.923175i
$499$	−421495.			−0.0757777			−0.0378889	−	0.999282i	$-0.512063\pi$
$499$	−421495.			−0.0757777			−0.0378889	+	0.999282i	$0.512063\pi$
$500$	−3.74137e6	−	1.00591e6i	−0.669276	−	0.179943i
$501$	−4.13633e6			−0.736242
$502$			1.35646e6i			0.240241i
$503$		−	165744.i		−	0.0292090i	−0.999893	−	0.0146045i	$-0.995351\pi$
$503$		−	165744.i		−	0.0292090i	0.999893	−	0.0146045i	$-0.00464892\pi$
$504$	−8.65882e6			−1.51839
$505$	2.93605e6	+	6.30105e6i	0.512312	+	1.09947i
$506$	−286784.			−0.0497942
$507$			3.35017e6i			0.578824i
$508$			6.55143e6i			1.12636i
$509$	2.54136e6			0.434782			0.217391	−	0.976085i	$-0.430245\pi$
$509$	2.54136e6			0.434782			0.217391	+	0.976085i	$0.430245\pi$
$510$	−810246.	+	377544.i	−0.137940	+	0.0642750i
$511$	3.93037e6			0.665857
$512$			2.02781e6i			0.341863i
$513$		−	3.43136e6i		−	0.575669i
$514$	4.50796e6			0.752614
$515$	−8.37232e6	+	3.90118e6i	−1.39100	+	0.648154i
$516$	1.20387e7			1.99047
$517$		−	3.90643e6i		−	0.642768i
$518$		−	1.66889e6i		−	0.273276i
$519$	−1.12471e7			−1.83283
$520$	−2.00301e6	−	4.29865e6i	−0.324844	−	0.697146i
$521$	−1.00488e7			−1.62189			−0.810944	−	0.585124i	$-0.801046\pi$
$521$	−1.00488e7			−1.62189			−0.810944	+	0.585124i	$0.801046\pi$
$522$			7.51952e6i			1.20785i
$523$		−	5.25311e6i		−	0.839774i	−0.907576	−	0.419887i	$-0.862070\pi$
$523$		−	5.25311e6i		−	0.839774i	0.907576	−	0.419887i	$-0.137930\pi$
$524$	−1.67596e6			−0.266646
$525$	−5.47861e6	+	6.52163e6i	−0.867505	+	1.03266i
$526$	−540803.			−0.0852265
$527$		−	1.67641e6i		−	0.262937i
$528$		−	845479.i		−	0.131983i
$529$	−279841.			−0.0434783
$530$	−1.15502e6	−	2.47879e6i	−0.178608	−	0.383310i
$531$	260919.			0.0401578
$532$		−	1.00513e6i		−	0.153973i
$533$			478189.i			0.0729090i
$534$	−6.45429e6			−0.979479
$535$	−6.90906e6	+	3.21936e6i	−1.04360	+	0.486278i
$536$	7.41837e6			1.11531
$537$		−	2.29042e6i		−	0.342751i
$538$			761609.i			0.113443i
$539$	−1.21234e6			−0.179743
$540$	8.42164e6	−	3.92416e6i	1.24283	−	0.579112i
$541$	4.45549e6			0.654489			0.327244	−	0.944940i	$-0.393880\pi$
$541$	4.45549e6			0.654489			0.327244	+	0.944940i	$0.393880\pi$
$542$		−	6.80000e6i		−	0.994285i
$543$			1.05482e7i			1.53526i
$544$	1.10989e6			0.160798
$545$	−5.27175e6	−	1.13137e7i	−0.760263	−	1.63160i
$546$	−4.26791e6			−0.612679
$547$			2.77840e6i			0.397033i	0.980098	+	0.198516i	$0.0636123\pi$
$547$			2.77840e6i			0.397033i	−0.980098	+	0.198516i	$0.936388\pi$
$548$		−	8.87759e6i		−	1.26283i
$549$	−1.61393e7			−2.28535
$550$	1.29717e6	+	1.08971e6i	0.182848	+	0.153605i
$551$	−2.13237e6			−0.299215
$552$			2.47327e6i			0.345480i
$553$		−	2.08614e6i		−	0.290089i
$554$	2.69111e6			0.372527
$555$	3.49728e6	+	7.50551e6i	0.481946	+	1.03430i
$556$	−1.47196e6			−0.201933
$557$			1.26464e7i			1.72715i	0.504220	+	0.863575i	$0.331780\pi$
$557$			1.26464e7i			1.72715i	−0.504220	+	0.863575i	$0.668220\pi$
$558$		−	1.46077e7i		−	1.98608i
$559$	9.84982e6			1.33321
$560$	890360.	−	414874.i	0.119976	−	0.0559044i
$561$	−882535.			−0.118393
$562$			7.20255e6i			0.961935i
$563$		−	3.99645e6i		−	0.531377i	−0.964059	−	0.265689i	$-0.914401\pi$
$563$		−	3.99645e6i		−	0.531377i	0.964059	−	0.265689i	$-0.0855993\pi$
$564$	−1.37908e7			−1.82554
$565$	5.38155e6	−	2.50760e6i	0.709228	−	0.330473i
$566$	−1.19759e6			−0.157132
$567$		−	8.03438e6i		−	1.04953i
$568$		−	3.18444e6i		−	0.414155i
$569$	1.09077e6			0.141238			0.0706191	−	0.997503i	$-0.477503\pi$
$569$	1.09077e6			0.141238			0.0706191	+	0.997503i	$0.477503\pi$
$570$	−932920.	−	2.00214e6i	−0.120270	−	0.258111i
$571$	226191.			0.0290325			0.0145162	−	0.999895i	$-0.495379\pi$
$571$	226191.			0.0290325			0.0145162	+	0.999895i	$0.495379\pi$
$572$		−	1.91664e6i		−	0.244934i
$573$			2.49576e7i			3.17553i
$574$	296924.			0.0376154
$575$	1.26576e6	+	1.06333e6i	0.159655	+	0.134121i
$576$	6.74485e6			0.847064
$577$		−	3.28573e6i		−	0.410858i	−0.978672	−	0.205429i	$-0.934141\pi$
$577$		−	3.28573e6i		−	0.410858i	0.978672	−	0.205429i	$-0.0658590\pi$
$578$			4.34237e6i			0.540639i
$579$	−1.67786e7			−2.07999
$580$	−2.43862e6	−	5.23351e6i	−0.301005	−	0.645986i
$581$	5.86051e6			0.720269
$582$		−	4.04116e6i		−	0.494537i
$583$		−	2.69994e6i		−	0.328991i
$584$	−6.74202e6			−0.818008
$585$	1.30423e7	−	6.07720e6i	1.57566	−	0.734199i
$586$	13252.5			0.00159424
$587$			7.43016e6i			0.890027i	0.895524	+	0.445013i	$0.146801\pi$
$587$			7.43016e6i			0.890027i	−0.895524	+	0.445013i	$0.853199\pi$
$588$			4.27989e6i			0.510493i
$589$	4.14244e6			0.492003
$590$	80431.3	−	37478.0i	0.00951251	−	0.00443247i
$591$	1.90084e7			2.23860
$592$		−	954928.i		−	0.111987i
$593$		−	8.97683e6i		−	1.04830i	−0.851626	−	0.524151i	$-0.824383\pi$
$593$		−	8.97683e6i		−	1.04830i	0.851626	−	0.524151i	$-0.175617\pi$
$594$	−4.06282e6			−0.472456
$595$	−433058.	−	929384.i	−0.0501480	−	0.107622i
$596$	−4.02760e6			−0.464441
$597$			3.26797e6i			0.375269i
$598$			828345.i			0.0947237i
$599$	1.40377e7			1.59856			0.799280	−	0.600959i	$-0.205215\pi$
$599$	1.40377e7			1.59856			0.799280	+	0.600959i	$0.205215\pi$
$600$	9.39781e6	−	1.11870e7i	1.06573	−	1.26863i
$601$	9.15574e6			1.03397			0.516984	−	0.855995i	$-0.327055\pi$
$601$	9.15574e6			1.03397			0.516984	+	0.855995i	$0.327055\pi$
$602$		−	6.11610e6i		−	0.687833i
$603$			2.25076e7i			2.52079i
$604$	−3.97897e6			−0.443790
$605$	−3.09608e6	−	6.64448e6i	−0.343893	−	0.738028i
$606$	−1.07313e7			−1.18705
$607$		−	7.53570e6i		−	0.830141i	−0.909789	−	0.415070i	$-0.863757\pi$
$607$		−	7.53570e6i		−	0.830141i	0.909789	−	0.415070i	$-0.136243\pi$
$608$			2.74256e6i			0.300882i
$609$	−1.26936e7			−1.38688
$610$	−4.97511e6	+	2.31821e6i	−0.541350	+	0.252248i
$611$	−1.12833e7			−1.22274
$612$			2.11702e6i			0.228479i
$613$		−	9.98554e6i		−	1.07330i	−0.843805	−	0.536649i	$-0.819690\pi$
$613$		−	9.98554e6i		−	1.07330i	0.843805	−	0.536649i	$-0.180310\pi$
$614$	−1.42539e6			−0.152585
$615$	−1.33536e6	+	622228.i	−0.142368	+	0.0663379i
$616$	−2.90732e6			−0.308703
$617$		−	6.82945e6i		−	0.722226i	−0.932522	−	0.361113i	$-0.882397\pi$
$617$		−	6.82945e6i		−	0.722226i	0.932522	−	0.361113i	$-0.117603\pi$
$618$		−	1.42589e7i		−	1.50181i
$619$	9.08935e6			0.953468			0.476734	−	0.879047i	$-0.341820\pi$
$619$	9.08935e6			0.953468			0.476734	+	0.879047i	$0.341820\pi$
$620$	4.73736e6	+	1.01668e7i	0.494946	+	1.06220i
$621$	−3.96445e6			−0.412529
$622$		−	9.09330e6i		−	0.942422i
$623$		−	7.40332e6i		−	0.764199i
$624$	−2.44207e6			−0.251071
$625$	−1.68486e6	−	9.61918e6i	−0.172530	−	0.985004i
$626$	8.09161e6			0.825276
$627$		−	2.18076e6i		−	0.221534i
$628$		−	1.27348e7i		−	1.28853i
$629$	−996781.			−0.100455
$630$	−3.77354e6	−	8.09839e6i	−0.378790	−	0.812920i
$631$	5.93201e6			0.593100			0.296550	−	0.955017i	$-0.404164\pi$
$631$	5.93201e6			0.593100			0.296550	+	0.955017i	$0.404164\pi$
$632$			3.57849e6i			0.356375i
$633$			3.25890e7i			3.23267i
$634$	6.49862e6			0.642093
$635$	−1.49687e7	+	6.97484e6i	−1.47316	+	0.686436i
$636$	−9.53154e6			−0.934373
$637$			3.50172e6i			0.341926i
$638$			2.52478e6i			0.245568i
$639$	9.66173e6			0.936058
$640$	−7.63320e6	+	3.55678e6i	−0.736642	+	0.343247i
$641$	−1.20892e7			−1.16212			−0.581062	−	0.813859i	$-0.697363\pi$
$641$	−1.20892e7			−1.16212			−0.581062	+	0.813859i	$0.697363\pi$
$642$		−	1.17668e7i		−	1.12673i
$643$			1.09118e7i			1.04081i	0.853920	+	0.520404i	$0.174219\pi$
$643$			1.09118e7i			1.04081i	−0.853920	+	0.520404i	$0.825781\pi$
$644$	−1.16129e6			−0.110338
$645$	1.28168e7	+	2.75060e7i	1.21305	+	2.60333i
$646$	265897.			0.0250687
$647$			8.06391e6i			0.757330i	0.925534	+	0.378665i	$0.123617\pi$
$647$			8.06391e6i			0.757330i	−0.925534	+	0.378665i	$0.876383\pi$
$648$			1.37819e7i			1.28935i
$649$	87607.3			0.00816448
$650$	3.14751e6	−	3.74673e6i	0.292202	−	0.347832i
$651$	2.46590e7			2.28047
$652$		−	3.48836e6i		−	0.321368i
$653$			897296.i			0.0823479i	0.999152	+	0.0411740i	$0.0131098\pi$
$653$			897296.i			0.0823479i	−0.999152	+	0.0411740i	$0.986890\pi$
$654$	1.92683e7			1.76157
$655$	−1.78427e6	−	3.82923e6i	−0.162502	−	0.348745i
$656$	169898.			0.0154145
$657$		−	2.04555e7i		−	1.84883i
$658$			7.00620e6i			0.630838i
$659$	−1.32258e6			−0.118634			−0.0593168	−	0.998239i	$-0.518892\pi$
$659$	−1.32258e6			−0.118634			−0.0593168	+	0.998239i	$0.518892\pi$
$660$	5.35228e6	−	2.49396e6i	0.478277	−	0.222859i
$661$	−1.38914e7			−1.23663			−0.618317	−	0.785929i	$-0.712185\pi$
$661$	−1.38914e7			−1.23663			−0.618317	+	0.785929i	$0.712185\pi$
$662$			1.03651e7i			0.919243i
$663$			2.54911e6i			0.225219i
$664$	−1.00529e7			−0.884853
$665$	2.29653e6	−	1.07009e6i	0.201381	−	0.0938357i
$666$	−8.68568e6			−0.758784
$667$			2.46366e6i			0.214420i
$668$			3.33151e6i			0.288869i
$669$	3.79309e7			3.27663
$670$	3.23295e6	+	6.93823e6i	0.278235	+	0.597120i
$671$	−5.41898e6			−0.464634
$672$			1.63259e7i			1.39461i
$673$		−	1.38217e7i		−	1.17632i	−0.808745	−	0.588159i	$-0.799853\pi$
$673$		−	1.38217e7i		−	1.17632i	0.808745	−	0.588159i	$-0.200147\pi$
$674$	−5.73661e6			−0.486414
$675$	1.79318e7	+	1.50640e7i	1.51483	+	1.27256i
$676$	2.69832e6			0.227105
$677$		−	4.26668e6i		−	0.357782i	−0.983869	−	0.178891i	$-0.942749\pi$
$677$		−	4.26668e6i		−	0.357782i	0.983869	−	0.178891i	$-0.0572509\pi$
$678$			9.16529e6i			0.765723i
$679$	4.63536e6			0.385842
$680$	742851.	+	1.59423e6i	0.0616069	+	0.132215i
$681$	−3.16184e7			−2.61259
$682$		−	4.90475e6i		−	0.403790i
$683$		−	1.34750e6i		−	0.110529i	−0.998472	−	0.0552646i	$-0.982400\pi$
$683$		−	1.34750e6i		−	0.110529i	0.998472	−	0.0552646i	$-0.0176002\pi$
$684$	−5.23120e6			−0.427525
$685$	2.02835e7	−	9.45134e6i	1.65164	−	0.769603i
$686$	7.38843e6			0.599434
$687$		−	4.73088e6i		−	0.382429i
$688$		−	3.49960e6i		−	0.281869i
$689$	−7.79850e6			−0.625839
$690$	−2.31319e6	+	1.07786e6i	−0.184964	+	0.0861864i
$691$	−1.29881e7			−1.03478			−0.517392	−	0.855748i	$-0.673097\pi$
$691$	−1.29881e7			−1.03478			−0.517392	+	0.855748i	$0.673097\pi$
$692$			9.05873e6i			0.719121i
$693$		−	8.82092e6i		−	0.697719i
$694$	−5.67318e6			−0.447124
$695$	−1.56709e6	−	3.36312e6i	−0.123064	−	0.264107i
$696$	2.17741e7			1.70379
$697$		−	177345.i		−	0.0138273i
$698$		−	5.11889e6i		−	0.397683i
$699$	1.79437e7			1.38905
$700$	5.25270e6	+	4.41262e6i	0.405170	+	0.340370i
$701$	−1.78237e7			−1.36994			−0.684970	−	0.728571i	$-0.740185\pi$
$701$	−1.78237e7			−1.36994			−0.684970	+	0.728571i	$0.740185\pi$
$702$			1.17350e7i			0.898754i
$703$		−	2.46307e6i		−	0.187970i
$704$	2.26468e6			0.172216
$705$	−1.46821e7	−	3.15091e7i	−1.11254	−	2.38761i
$706$	2.61147e6			0.197185
$707$		−	1.23092e7i		−	0.926150i
$708$		−	309278.i		−	0.0231881i
$709$	1.38491e7			1.03468			0.517341	−	0.855779i	$-0.326922\pi$
$709$	1.38491e7			1.03468			0.517341	+	0.855779i	$0.326922\pi$
$710$	2.97834e6	−	1.38779e6i	0.221732	−	0.103319i
$711$	−1.08573e7			−0.805465
$712$			1.26994e7i			0.938821i
$713$		−	4.78600e6i		−	0.352573i
$714$	1.58283e6			0.116195
$715$	4.37912e6	−	2.04051e6i	0.320348	−	0.149270i
$716$	−1.84476e6			−0.134480
$717$			4.23274e7i			3.07485i
$718$			1.99340e6i			0.144306i
$719$	6.39208e6			0.461126			0.230563	−	0.973057i	$-0.425943\pi$
$719$	6.39208e6			0.461126			0.230563	+	0.973057i	$0.425943\pi$
$720$	−2.15920e6	−	4.63386e6i	−0.155225	−	0.333128i
$721$	1.63555e7			1.17172
$722$		−	7.10331e6i		−	0.507128i
$723$			2.51475e7i			1.78916i
$724$	8.49585e6			0.602366
$725$	9.36129e6	−	1.11435e7i	0.661440	−	0.787366i
$726$	1.13162e7			0.796817
$727$		−	9.15468e6i		−	0.642403i	−0.947011	−	0.321201i	$-0.895913\pi$
$727$		−	9.15468e6i		−	0.642403i	0.947011	−	0.321201i	$-0.104087\pi$
$728$			8.39748e6i			0.587247i
$729$	8.32912e6			0.580471
$730$	−2.93819e6	−	6.30565e6i	−0.204067	−	0.437948i
$731$	−3.65298e6			−0.252845
$732$			1.91305e7i			1.31962i
$733$		−	1.29905e7i		−	0.893028i	−0.894777	−	0.446514i	$-0.852665\pi$
$733$		−	1.29905e7i		−	0.893028i	0.894777	−	0.446514i	$-0.147335\pi$
$734$	1.16658e6			0.0799232
$735$	−9.77868e6	+	4.55650e6i	−0.667670	+	0.311109i
$736$	3.16864e6			0.215615
$737$			7.55725e6i			0.512501i
$738$		−	1.54533e6i		−	0.104444i
$739$	3.98014e6			0.268094			0.134047	−	0.990975i	$-0.457203\pi$
$739$	3.98014e6			0.268094			0.134047	+	0.990975i	$0.457203\pi$
$740$	6.04515e6	−	2.81681e6i	0.405815	−	0.189094i
$741$	−6.29890e6			−0.421424
$742$			4.84236e6i			0.322884i
$743$		−	1.96581e6i		−	0.130638i	−0.997864	−	0.0653192i	$-0.979193\pi$
$743$		−	1.96581e6i		−	0.130638i	0.997864	−	0.0653192i	$-0.0208065\pi$
$744$	−4.22992e7			−2.80156
$745$	−4.28790e6	−	9.20224e6i	−0.283044	−	0.607439i
$746$	58044.2			0.00381867
$747$		−	3.05009e7i		−	1.99991i
$748$			710819.i			0.0464520i
$749$	1.34970e7			0.879086
$750$	1.45585e7	+	3.91423e6i	0.945069	+	0.254093i
$751$	3.00617e6			0.194497			0.0972487	−	0.995260i	$-0.468996\pi$
$751$	3.00617e6			0.194497			0.0972487	+	0.995260i	$0.468996\pi$
$752$			4.00891e6i			0.258513i
$753$			1.19173e7i			0.765931i
$754$	7.29256e6			0.467145
$755$	−4.23612e6	−	9.09113e6i	−0.270459	−	0.580431i
$756$	−1.64518e7			−1.04691
$757$		−	8.54036e6i		−	0.541672i	−0.962626	−	0.270836i	$-0.912700\pi$
$757$		−	8.54036e6i		−	0.541672i	0.962626	−	0.270836i	$-0.0873001\pi$
$758$		−	1.35688e7i		−	0.857766i
$759$	−2.51957e6			−0.158753
$760$	−3.93938e6	+	1.83560e6i	−0.247397	+	0.115278i
$761$	−4.00401e6			−0.250630			−0.125315	−	0.992117i	$-0.539994\pi$
$761$	−4.00401e6			−0.250630			−0.125315	+	0.992117i	$0.539994\pi$
$762$		−	2.54931e7i		−	1.59051i
$763$			2.21015e7i			1.37439i
$764$	2.01015e7			1.24594
$765$	−4.83696e6	+	2.25384e6i	−0.298826	+	0.139242i
$766$	−9.91921e6			−0.610809
$767$		−	253044.i		−	0.0155313i
$768$		−	2.45360e7i		−	1.50107i
$769$	3.19689e6			0.194945			0.0974725	−	0.995238i	$-0.468924\pi$
$769$	3.19689e6			0.194945			0.0974725	+	0.995238i	$0.468924\pi$
$770$	−1.26702e6	−	2.71915e6i	−0.0770117	−	0.165275i
$771$	3.96051e7			2.39947
$772$			1.35140e7i			0.816094i
$773$			1.56246e7i			0.940503i	0.882532	+	0.470251i	$0.155837\pi$
$773$			1.56246e7i			0.940503i	−0.882532	+	0.470251i	$0.844163\pi$
$774$	−3.18311e7			−1.90985
$775$	−1.81856e7	+	2.16478e7i	−1.08761	+	1.29467i
$776$	−7.95134e6			−0.474008
$777$		−	1.46621e7i		−	0.871254i
$778$			7.76433e6i			0.459891i
$779$	438223.			0.0258733
$780$	−7.20354e6	−	1.54595e7i	−0.423945	−	0.909827i
$781$	3.24406e6			0.190310
$782$		−	307207.i		−	0.0179644i
$783$			3.49021e7i			2.03445i
$784$	1.24414e6			0.0722904
$785$	2.90965e7	−	1.35579e7i	1.68526	−	0.785266i
$786$	6.52154e6			0.376525
$787$		−	2.27061e7i		−	1.30679i	−0.757018	−	0.653395i	$-0.773344\pi$
$787$		−	2.27061e7i		−	1.30679i	0.757018	−	0.653395i	$-0.226656\pi$
$788$		−	1.53099e7i		−	0.878329i
$789$	−4.75127e6			−0.271717
$790$	−3.34688e6	+	1.55952e6i	−0.190797	+	0.0889043i
$791$	−1.05129e7			−0.597424
$792$			1.51311e7i			0.857151i
$793$			1.56521e7i			0.883874i
$794$	6.22565e6			0.350456
$795$	−1.01475e7	−	2.17776e7i	−0.569434	−	1.22206i
$796$	2.63212e6			0.147239
$797$		−	2.22676e7i		−	1.24173i	−0.783916	−	0.620866i	$-0.786781\pi$
$797$		−	2.22676e7i		−	1.24173i	0.783916	−	0.620866i	$-0.213219\pi$
$798$			3.91121e6i			0.217422i
$799$	4.18462e6			0.231894
$800$	−1.43322e7	−	1.20401e7i	−0.791752	−	0.665125i
$801$	−3.85304e7			−2.12189
$802$		−	9.56393e6i		−	0.525050i
$803$		−	6.86823e6i		−	0.375886i
$804$	2.66791e7			1.45557
$805$	−1.23634e6	−	2.65332e6i	−0.0672434	−	0.144311i
$806$	−1.41668e7			−0.768131
$807$			6.69118e6i			0.361676i
$808$			2.11148e7i			1.13778i
$809$	−3.12578e7			−1.67914			−0.839570	−	0.543251i	$-0.817193\pi$
$809$	−3.12578e7			−1.67914			−0.839570	+	0.543251i	$0.817193\pi$
$810$	−1.28899e7	+	6.00619e6i	−0.690297	+	0.321652i
$811$	−5.17398e6			−0.276231			−0.138116	−	0.990416i	$-0.544105\pi$
$811$	−5.17398e6			−0.276231			−0.138116	+	0.990416i	$0.544105\pi$
$812$			1.02237e7i			0.544152i
$813$		−	5.97420e7i		−	3.16996i
$814$	−2.91634e6			−0.154268
$815$	7.97020e6	−	3.71381e6i	0.420315	−	0.195851i
$816$	905687.			0.0476160
$817$		−	9.02660e6i		−	0.473118i
$818$			4.83747e6i			0.252775i
$819$	−2.54783e7			−1.32727
$820$	501160.	+	1.07554e6i	0.0260280	+	0.0558587i
$821$	−2.51567e7			−1.30255			−0.651277	−	0.758840i	$-0.725766\pi$
$821$	−2.51567e7			−1.30255			−0.651277	+	0.758840i	$0.725766\pi$
$822$			3.45447e7i			1.78321i
$823$			2.17605e7i			1.11988i	0.828535	+	0.559938i	$0.189175\pi$
$823$			2.17605e7i			1.11988i	−0.828535	+	0.559938i	$0.810825\pi$
$824$	−2.80556e7			−1.43946
$825$	1.13964e7	+	9.57374e6i	0.582952	+	0.489719i
$826$	−157124.			−0.00801294
$827$			2.25721e7i			1.14765i	0.818979	+	0.573823i	$0.194540\pi$
$827$			2.25721e7i			1.14765i	−0.818979	+	0.573823i	$0.805460\pi$
$828$			6.04392e6i			0.306368i
$829$	−1.17900e6			−0.0595836			−0.0297918	−	0.999556i	$-0.509484\pi$
$829$	−1.17900e6			−0.0595836			−0.0297918	+	0.999556i	$0.509484\pi$
$830$	−4.38109e6	−	9.40224e6i	−0.220743	−	0.473736i
$831$	2.36430e7			1.18768
$832$		−	6.54127e6i		−	0.327608i
$833$		−	1.29867e6i		−	0.0648467i
$834$	5.72771e6			0.285145
$835$	−7.61183e6	+	3.54682e6i	−0.377809	+	0.176045i
$836$	−1.75645e6			−0.0869200
$837$		−	6.78024e7i		−	3.34527i
$838$		−	2.45399e6i		−	0.120716i
$839$	−5.57632e6			−0.273491			−0.136746	−	0.990606i	$-0.543664\pi$
$839$	−5.57632e6			−0.273491			−0.136746	+	0.990606i	$0.543664\pi$
$840$	−2.34503e7	+	1.09270e7i	−1.14670	+	0.534319i
$841$	1.17832e6			0.0574477
$842$		−	5.92349e6i		−	0.287937i
$843$			6.32786e7i			3.06682i
$844$	2.62481e7			1.26836
$845$	2.87271e6	+	6.16511e6i	0.138404	+	0.297029i
$846$	3.64636e7			1.75160
$847$			1.29801e7i			0.621684i
$848$			2.77077e6i			0.132316i
$849$	−1.05215e7			−0.500966
$850$	−1.16731e6	+	1.38954e6i	−0.0554165	+	0.0659667i
$851$	−2.84573e6			−0.134701
$852$		−	1.14524e7i		−	0.540503i
$853$			2.78507e7i			1.31058i	0.755378	+	0.655289i	$0.227453\pi$
$853$			2.78507e7i			1.31058i	−0.755378	+	0.655289i	$0.772547\pi$
$854$	9.71895e6			0.456010
$855$	−5.56929e6	−	1.19522e7i	−0.260546	−	0.559157i
$856$	−2.31522e7			−1.07996
$857$		−	7.89202e6i		−	0.367059i	−0.983014	−	0.183530i	$-0.941248\pi$
$857$		−	7.89202e6i		−	0.367059i	0.983014	−	0.183530i	$-0.0587523\pi$
$858$			7.45806e6i			0.345866i
$859$	−1.81190e7			−0.837823			−0.418911	−	0.908027i	$-0.637588\pi$
$859$	−1.81190e7			−0.837823			−0.418911	+	0.908027i	$0.637588\pi$
$860$	2.21541e7	−	1.03230e7i	1.02143	−	0.475948i
$861$	2.60865e6			0.119924
$862$			4.61093e6i			0.211359i
$863$		−	272254.i		−	0.0124437i	−0.999981	−	0.00622183i	$-0.998020\pi$
$863$		−	272254.i		−	0.0124437i	0.999981	−	0.00622183i	$-0.00198048\pi$
$864$	4.48895e7			2.04579
$865$	−2.06973e7	+	9.64418e6i	−0.940534	+	0.438253i
$866$	−5134.65			−0.000232657
$867$			3.81503e7i			1.72365i
$868$		−	1.98611e7i		−	0.894754i
$869$	−3.64548e6			−0.163759
$870$	9.48922e6	+	2.03648e7i	0.425042	+	0.912182i
$871$	2.18283e7			0.974932
$872$		−	3.79121e7i		−	1.68844i
$873$		−	2.41247e7i		−	1.07134i
$874$	759115.			0.0336147
$875$	−4.48977e6	+	1.66992e7i	−0.198246	+	0.737351i
$876$	−2.42467e7			−1.06756
$877$		−	8.51694e6i		−	0.373925i	−0.982367	−	0.186962i	$-0.940136\pi$
$877$		−	8.51694e6i		−	0.373925i	0.982367	−	0.186962i	$-0.0598643\pi$
$878$			2.18663e7i			0.957279i
$879$	116431.			0.00508272
$880$	−724982.	−	1.55588e6i	−0.0315588	−	0.0677283i
$881$	−475215.			−0.0206277			−0.0103138	−	0.999947i	$-0.503283\pi$
$881$	−475215.			−0.0206277			−0.0103138	+	0.999947i	$0.503283\pi$
$882$		−	1.13163e7i		−	0.489815i
$883$			2.40143e7i			1.03650i	0.855230	+	0.518249i	$0.173416\pi$
$883$			2.40143e7i			1.03650i	−0.855230	+	0.518249i	$0.826584\pi$
$884$	2.05312e6			0.0883658
$885$	706636.	−	329266.i	0.0303276	−	0.0141315i
$886$	1.09613e7			0.469115
$887$		−	840273.i		−	0.0358601i	−0.999839	−	0.0179300i	$-0.994292\pi$
$887$		−	840273.i		−	0.0358601i	0.999839	−	0.0179300i	$-0.00570762\pi$
$888$			2.51509e7i			1.07034i
$889$	2.92416e7			1.24093
$890$	−1.18774e7	+	5.53443e6i	−0.502629	+	0.234206i
$891$	−1.40399e7			−0.592474
$892$		−	3.05506e7i		−	1.28561i
$893$			1.03403e7i			0.433914i
$894$	1.56723e7			0.655826
$895$	−1.96399e6	−	4.21491e6i	−0.0819561	−	0.175886i
$896$	1.49116e7			0.620517
$897$			7.27750e6i			0.301996i
$898$			1.25407e7i			0.518957i
$899$	−4.21349e7			−1.73877
$900$	2.29654e7	−	2.73376e7i	0.945079	−	1.12500i
$901$	2.89221e6			0.118691
$902$		−	518867.i		−	0.0212344i
$903$		−	5.37335e7i		−	2.19293i
$904$	1.80335e7			0.733938
$905$	9.04493e6	+	1.94113e7i	0.367099	+	0.787831i
$906$	1.54831e7			0.626666
$907$		−	2.64919e7i		−	1.06929i	−0.845078	−	0.534643i	$-0.820446\pi$
$907$		−	2.64919e7i		−	1.06929i	0.845078	−	0.534643i	$-0.179554\pi$
$908$			2.54663e7i			1.02507i
$909$	−6.40630e7			−2.57156
$910$	−7.85397e6	+	3.65965e6i	−0.314402	+	0.146500i
$911$	−1.82537e7			−0.728709			−0.364355	−	0.931260i	$-0.618710\pi$
$911$	−1.82537e7			−0.728709			−0.364355	+	0.931260i	$0.618710\pi$
$912$			2.23797e6i			0.0890979i
$913$		−	1.02411e7i		−	0.406602i
$914$	−1.36999e7			−0.542441
$915$	−4.37092e7	+	2.03668e7i	−1.72592	+	0.804213i
$916$	−3.81038e6			−0.150048
$917$			7.48045e6i			0.293768i
$918$		−	4.35214e6i		−	0.170450i
$919$	−3.48435e7			−1.36092			−0.680461	−	0.732784i	$-0.738220\pi$
$919$	−3.48435e7			−1.36092			−0.680461	+	0.732784i	$0.738220\pi$
$920$	2.12078e6	+	4.55140e6i	0.0826088	+	0.177287i
$921$	−1.25228e7			−0.486468
$922$			5.32868e6i			0.206439i
$923$		−	9.37011e6i		−	0.362027i
$924$	−1.04558e7			−0.402880
$925$	1.28717e7	+	1.08131e7i	0.494630	+	0.415523i
$926$	1.80868e7			0.693162
$927$		−	8.51217e7i		−	3.25342i
$928$		−	2.78960e7i		−	1.06334i
$929$	4.36870e6			0.166078			0.0830391	−	0.996546i	$-0.473537\pi$
$929$	4.36870e6			0.166078			0.0830391	+	0.996546i	$0.473537\pi$
$930$	−1.84342e7	−	3.95615e7i	−0.698901	−	1.49991i
$931$	3.20905e6			0.121340
$932$		−	1.44523e7i		−	0.545003i
$933$		−	7.98899e7i		−	3.00461i
$934$	1.02103e7			0.382975
$935$	−1.62408e6	+	756758.i	−0.0607544	+	0.0283092i
$936$	4.37045e7			1.63056
$937$			1.90567e7i			0.709087i	0.935040	+	0.354543i	$0.115364\pi$
$937$			1.90567e7i			0.709087i	−0.935040	+	0.354543i	$0.884636\pi$
$938$		−	1.35539e7i		−	0.502989i
$939$	7.10895e7			2.63113
$940$	−2.53783e7	+	1.18253e7i	−0.936792	+	0.436510i
$941$	−6.86363e6			−0.252685			−0.126343	−	0.991987i	$-0.540324\pi$
$941$	−6.86363e6			−0.252685			−0.126343	+	0.991987i	$0.540324\pi$
$942$			4.95541e7i			1.81950i
$943$		−	506305.i		−	0.0185410i
$944$	−89905.5			−0.00328364
$945$	−1.75151e7	−	3.75890e7i	−0.638017	−	1.36925i
$946$	−1.06877e7			−0.388291
$947$			2.04942e7i			0.742602i	0.928513	+	0.371301i	$0.121088\pi$
$947$			2.04942e7i			0.742602i	−0.928513	+	0.371301i	$0.878912\pi$
$948$			1.28695e7i			0.465095i
$949$	−1.98381e7			−0.715048
$950$	−3.43359e6	−	2.88445e6i	−0.123435	−	0.103694i
$951$	5.70942e7			2.04711
$952$		−	3.11436e6i		−	0.111372i
$953$		−	2.36728e6i		−	0.0844341i	−0.999108	−	0.0422170i	$-0.986558\pi$
$953$		−	2.36728e6i		−	0.0844341i	0.999108	−	0.0422170i	$-0.0134421\pi$
$954$	2.52019e7			0.896526
$955$	2.14007e7	+	4.59279e7i	0.759309	+	1.62955i
$956$	3.40917e7			1.20644
$957$			2.21817e7i			0.782916i
$958$			1.67784e7i			0.590659i
$959$	−3.96241e7			−1.39128
$960$	1.82668e7	−	8.51163e6i	0.639711	−	0.298081i
$961$	5.32238e7			1.85908
$962$			8.42353e6i			0.293465i
$963$		−	7.02446e7i		−	2.44089i
$964$	2.02545e7			0.701986
$965$	−3.08767e7	+	1.43874e7i	−1.06736	+	0.497351i
$966$	4.51885e6			0.155806
$967$			193505.i			0.00665465i	0.999994	+	0.00332732i	$0.00105912\pi$
$967$			193505.i			0.00665465i	−0.999994	+	0.00332732i	$0.998941\pi$
$968$		−	2.22656e7i		−	0.763740i
$969$	2.33606e6			0.0799236
$970$	−3.46522e6	−	7.43670e6i	−0.118250	−	0.253776i
$971$	1.18283e7			0.402602			0.201301	−	0.979529i	$-0.435483\pi$
$971$	1.18283e7			0.402602			0.201301	+	0.979529i	$0.435483\pi$
$972$			9.17733e6i			0.311566i
$973$			6.56990e6i			0.222473i
$974$	1.23674e7			0.417715
$975$	2.76527e7	−	3.29172e7i	0.931593	−	1.10895i
$976$	5.56113e6			0.186870
$977$			1.27220e7i			0.426400i	0.977009	+	0.213200i	$0.0683886\pi$
$977$			1.27220e7i			0.426400i	−0.977009	+	0.213200i	$0.931611\pi$
$978$			1.35740e7i			0.453796i
$979$	−1.29371e7			−0.431401
$980$	3.66993e6	+	7.87602e6i	0.122065	+	0.261964i
$981$	1.15027e8			3.81616
$982$		−	1.39973e7i		−	0.463196i
$983$		−	2.62540e7i		−	0.866588i	−0.901253	−	0.433294i	$-0.857351\pi$
$983$		−	2.62540e7i		−	0.866588i	0.901253	−	0.433294i	$-0.142649\pi$
$984$	−4.47478e6			−0.147328
$985$	3.49800e7	−	1.62994e7i	1.14876	−	0.535279i
$986$	−2.70458e6			−0.0885946
$987$			6.15536e7i			2.01122i
$988$			5.07331e6i			0.165348i
$989$	−1.04290e7			−0.339040
$990$	−1.41518e7	+	6.59418e6i	−0.458904	+	0.213832i
$991$	5.49258e7			1.77661			0.888306	−	0.459253i	$-0.151883\pi$
$991$	5.49258e7			1.77661			0.888306	+	0.459253i	$0.151883\pi$
$992$			5.41919e7i			1.74846i
$993$			9.10638e7i			2.93071i
$994$	−5.81823e6			−0.186778
$995$	2.80223e6	+	6.01385e6i	0.0897316	+	0.192573i
$996$	−3.61539e7			−1.15480
$997$			3.45837e7i			1.10188i	0.834545	+	0.550939i	$0.185730\pi$
$997$			3.45837e7i			1.10188i	−0.834545	+	0.550939i	$0.814270\pi$
$998$		−	1.32101e6i		−	0.0419836i
$999$	−4.03149e7			−1.27806

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	115.6.b.a.24.36	yes	54
5.2	odd	4		575.6.a.m.1.9		27
5.3	odd	4		575.6.a.l.1.19		27
5.4	even	2	inner	115.6.b.a.24.19	✓	54

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
115.6.b.a.24.19	✓	54	5.4	even	2	inner
115.6.b.a.24.36	yes	54	1.1	even	1	trivial
575.6.a.l.1.19		27	5.3	odd	4
575.6.a.m.1.9		27	5.2	odd	4

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:	\( N \)	\(=\)	\( 115 = 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	115.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+3.13410i q^{2} +27.5349i q^{3} +22.1774 q^{4} +(23.6107 + 50.6709i) q^{5} -86.2973 q^{6} -98.9863i q^{7} +169.798i q^{8} -515.172 q^{9} +O(q^{10})\)	\(q+3.13410i q^{2} +27.5349i q^{3} +22.1774 q^{4} +(23.6107 + 50.6709i) q^{5} -86.2973 q^{6} -98.9863i q^{7} +169.798i q^{8} -515.172 q^{9} +(-158.808 + 73.9983i) q^{10} -172.976 q^{11} +610.653i q^{12} +499.623i q^{13} +310.233 q^{14} +(-1395.22 + 650.119i) q^{15} +177.514 q^{16} -185.294i q^{17} -1614.60i q^{18} +457.866 q^{19} +(523.624 + 1123.75i) q^{20} +2725.58 q^{21} -542.125i q^{22} -529.000i q^{23} -4675.36 q^{24} +(-2010.07 + 2392.75i) q^{25} -1565.87 q^{26} -7494.24i q^{27} -2195.26i q^{28} -4657.20 q^{29} +(-2037.54 - 4372.76i) q^{30} +9047.26 q^{31} +5989.87i q^{32} -4762.89i q^{33} +580.731 q^{34} +(5015.72 - 2337.14i) q^{35} -11425.2 q^{36} -5379.45i q^{37} +1435.00i q^{38} -13757.1 q^{39} +(-8603.79 + 4009.04i) q^{40} +957.099 q^{41} +8542.25i q^{42} -19714.5i q^{43} -3836.16 q^{44} +(-12163.6 - 26104.2i) q^{45} +1657.94 q^{46} +22583.7i q^{47} +4887.83i q^{48} +7008.71 q^{49} +(-7499.12 - 6299.77i) q^{50} +5102.06 q^{51} +11080.3i q^{52} +15608.8i q^{53} +23487.7 q^{54} +(-4084.09 - 8764.85i) q^{55} +16807.6 q^{56} +12607.3i q^{57} -14596.1i q^{58} -506.470 q^{59} +(-30942.3 + 14417.9i) q^{60} +31327.9 q^{61} +28355.1i q^{62} +50995.0i q^{63} -13092.4 q^{64} +(-25316.3 + 11796.4i) q^{65} +14927.4 q^{66} -43689.5i q^{67} -4109.34i q^{68} +14566.0 q^{69} +(7324.82 + 15719.8i) q^{70} -18754.4 q^{71} -87475.0i q^{72} +39706.2i q^{73} +16859.8 q^{74} +(-65884.1 - 55347.1i) q^{75} +10154.3 q^{76} +17122.3i q^{77} -43116.1i q^{78} +21075.0 q^{79} +(4191.23 + 8994.78i) q^{80} +81166.6 q^{81} +2999.65i q^{82} +59205.2i q^{83} +60446.3 q^{84} +(9389.01 - 4374.92i) q^{85} +61787.3 q^{86} -128236. i q^{87} -29370.9i q^{88} +74791.3 q^{89} +(81813.3 - 38121.9i) q^{90} +49455.8 q^{91} -11731.8i q^{92} +249116. i q^{93} -70779.5 q^{94} +(10810.5 + 23200.5i) q^{95} -164931. q^{96} +46828.3i q^{97} +21966.0i q^{98} +89112.5 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 54 q - 832 q^{4} + 98 q^{5} - 144 q^{6} - 4270 q^{9}+O(q^{10}) \) 54 * q - 832 * q^4 + 98 * q^5 - 144 * q^6 - 4270 * q^9	\( 54 q - 832 q^{4} + 98 q^{5} - 144 q^{6} - 4270 q^{9} + 942 q^{10} - 380 q^{11} - 3988 q^{14} + 690 q^{15} + 16632 q^{16} - 2440 q^{19} - 2436 q^{20} + 4768 q^{21} - 17940 q^{24} - 9150 q^{25} - 15144 q^{26} + 22958 q^{29} + 23034 q^{30} - 33970 q^{31} + 26120 q^{34} - 18694 q^{35} + 71560 q^{36} + 13416 q^{39} - 22230 q^{40} - 33842 q^{41} + 33908 q^{44} + 35260 q^{45} - 8464 q^{46} - 153772 q^{49} - 91728 q^{50} + 96576 q^{51} - 90500 q^{54} + 37344 q^{55} + 201796 q^{56} + 110106 q^{59} - 92484 q^{60} + 132924 q^{61} - 262108 q^{64} + 45090 q^{65} - 123288 q^{66} + 38088 q^{69} + 249664 q^{70} - 310814 q^{71} + 220660 q^{74} + 235776 q^{75} + 42220 q^{76} - 389348 q^{79} - 245138 q^{80} + 364422 q^{81} - 35836 q^{84} - 134118 q^{85} + 111980 q^{86} - 353788 q^{89} - 805800 q^{90} - 226632 q^{91} + 289804 q^{94} - 42676 q^{95} + 294416 q^{96} - 14088 q^{99}+O(q^{100}) \) 54 * q - 832 * q^4 + 98 * q^5 - 144 * q^6 - 4270 * q^9 + 942 * q^10 - 380 * q^11 - 3988 * q^14 + 690 * q^15 + 16632 * q^16 - 2440 * q^19 - 2436 * q^20 + 4768 * q^21 - 17940 * q^24 - 9150 * q^25 - 15144 * q^26 + 22958 * q^29 + 23034 * q^30 - 33970 * q^31 + 26120 * q^34 - 18694 * q^35 + 71560 * q^36 + 13416 * q^39 - 22230 * q^40 - 33842 * q^41 + 33908 * q^44 + 35260 * q^45 - 8464 * q^46 - 153772 * q^49 - 91728 * q^50 + 96576 * q^51 - 90500 * q^54 + 37344 * q^55 + 201796 * q^56 + 110106 * q^59 - 92484 * q^60 + 132924 * q^61 - 262108 * q^64 + 45090 * q^65 - 123288 * q^66 + 38088 * q^69 + 249664 * q^70 - 310814 * q^71 + 220660 * q^74 + 235776 * q^75 + 42220 * q^76 - 389348 * q^79 - 245138 * q^80 + 364422 * q^81 - 35836 * q^84 - 134118 * q^85 + 111980 * q^86 - 353788 * q^89 - 805800 * q^90 - 226632 * q^91 + 289804 * q^94 - 42676 * q^95 + 294416 * q^96 - 14088 * q^99

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