LMFDB - Embedded newform 30.8.c.b.19.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [30,8,Mod(19,30)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(30, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("30.19");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 30 = 2 \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	30.c (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:	$9.37155076452$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{2641})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1321x^{2} + 435600 $ x^4 + 1321*x^2 + 435600
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2}\cdot 5 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			19.2
Root			$25.1953i$ of defining polynomial
Character	$\chi$	$=$	30.19
Dual form			30.8.c.b.19.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-8.00000i q^{2} -27.0000i q^{3} -64.0000 q^{4} +(271.563 - 66.1720i) q^{5} -216.000 q^{6} -1309.81i q^{7} +512.000i q^{8} -729.000 q^{9} +(-529.376 - 2172.50i) q^{10} -5457.35 q^{11} +1728.00i q^{12} +5237.72i q^{13} -10478.5 q^{14} +(-1786.64 - 7332.19i) q^{15} +4096.00 q^{16} -4870.92i q^{17} +5832.00i q^{18} -32781.2 q^{19} +(-17380.0 + 4235.01i) q^{20} -35365.0 q^{21} +43658.8i q^{22} -83486.0i q^{23} +13824.0 q^{24} +(69367.5 - 35939.7i) q^{25} +41901.8 q^{26} +19683.0i q^{27} +83828.0i q^{28} +73409.7 q^{29} +(-58657.5 + 14293.1i) q^{30} +244719. q^{31} -32768.0i q^{32} +147348. i q^{33} -38967.3 q^{34} +(-86672.9 - 355696. i) q^{35} +46656.0 q^{36} -107537. i q^{37} +262250. i q^{38} +141418. q^{39} +(33880.1 + 139040. i) q^{40} +396774. q^{41} +282920. i q^{42} +725839. i q^{43} +349270. q^{44} +(-197969. + 48239.4i) q^{45} -667888. q^{46} -801478. i q^{47} -110592. i q^{48} -892068. q^{49} +(-287517. - 554940. i) q^{50} -131515. q^{51} -335214. i q^{52} -1.22395e6i q^{53} +157464. q^{54} +(-1.48201e6 + 361123. i) q^{55} +670624. q^{56} +885092. i q^{57} -587278. i q^{58} -2.32931e6 q^{59} +(114345. + 469260. i) q^{60} +2.63553e6 q^{61} -1.95775e6i q^{62} +954854. i q^{63} -262144. q^{64} +(346590. + 1.42237e6i) q^{65} +1.17879e6 q^{66} +209887. i q^{67} +311739. i q^{68} -2.25412e6 q^{69} +(-2.84557e6 + 693383. i) q^{70} +3.76578e6 q^{71} -373248. i q^{72} +2.81943e6i q^{73} -860298. q^{74} +(-970371. - 1.87292e6i) q^{75} +2.09800e6 q^{76} +7.14810e6i q^{77} -1.13135e6i q^{78} +8.68583e6 q^{79} +(1.11232e6 - 271040. i) q^{80} +531441. q^{81} -3.17419e6i q^{82} -5.77317e6i q^{83} +2.26336e6 q^{84} +(-322318. - 1.32276e6i) q^{85} +5.80672e6 q^{86} -1.98206e6i q^{87} -2.79416e6i q^{88} -1.02714e7 q^{89} +(385915. + 1.58375e6i) q^{90} +6.86043e6 q^{91} +5.34310e6i q^{92} -6.60742e6i q^{93} -6.41183e6 q^{94} +(-8.90215e6 + 2.16920e6i) q^{95} -884736. q^{96} +5.19683e6i q^{97} +7.13654e6i q^{98} +3.97841e6 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 256 q^{4} + 264 q^{5} - 864 q^{6} - 2916 q^{9} + 2816 q^{10} - 7440 q^{11} - 9024 q^{14} + 9504 q^{15} + 16384 q^{16} - 69456 q^{19} - 16896 q^{20} - 30456 q^{21} + 55296 q^{24} + 60396 q^{25} + 118272 q^{26}+ \cdots + 5423760 q^{99}+O(q^{100}) $ 4 * q - 256 * q^4 + 264 * q^5 - 864 * q^6 - 2916 * q^9 + 2816 * q^10 - 7440 * q^11 - 9024 * q^14 + 9504 * q^15 + 16384 * q^16 - 69456 * q^19 - 16896 * q^20 - 30456 * q^21 + 55296 * q^24 + 60396 * q^25 + 118272 * q^26 - 372384 * q^29 - 57024 * q^30 + 436192 * q^31 + 386816 * q^34 - 534576 * q^35 + 186624 * q^36 + 399168 * q^39 - 180224 * q^40 + 608616 * q^41 + 476160 * q^44 - 192456 * q^45 - 204800 * q^46 - 1249524 * q^49 - 1656576 * q^50 + 1305504 * q^51 + 629856 * q^54 - 3448960 * q^55 + 577536 * q^56 - 9117840 * q^59 - 608256 * q^60 + 11972840 * q^61 - 1048576 * q^64 - 350232 * q^65 + 1607040 * q^66 - 691200 * q^69 - 7356544 * q^70 + 18158880 * q^71 - 6121728 * q^74 - 5590944 * q^75 + 4445184 * q^76 + 3958272 * q^79 + 1081344 * q^80 + 2125764 * q^81 + 1949184 * q^84 - 14713336 * q^85 + 16780416 * q^86 - 15587448 * q^89 - 2052864 * q^90 + 10507488 * q^91 - 5617280 * q^94 - 17260896 * q^95 - 3538944 * q^96 + 5423760 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/30\mathbb{Z}\right)^\times$.

$n$	$7$	$11$
$\chi(n)$	$-1$	$1$

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 $

$2$		−	8.00000i		−	0.707107i
$2$		−	8.00000i		−	0.707107i
$3$		−	27.0000i		−	0.577350i
$3$		−	27.0000i		−	0.577350i
$4$	−64.0000			−0.500000
$5$	271.563	−	66.1720i	0.971572	−	0.236744i
$5$	271.563	−	66.1720i	0.971572	−	0.236744i
$6$	−216.000			−0.408248
$7$		−	1309.81i		−	1.44333i	−0.692241	−	0.721666i	$-0.743377\pi$
$7$		−	1309.81i		−	1.44333i	0.692241	−	0.721666i	$-0.256623\pi$
$8$			512.000i			0.353553i
$9$	−729.000			−0.333333
$10$	−529.376	−	2172.50i	−0.167403	−	0.687005i
$11$	−5457.35			−1.23625			−0.618126	−	0.786079i	$-0.712108\pi$
$11$	−5457.35			−1.23625			−0.618126	+	0.786079i	$0.712108\pi$
$12$			1728.00i			0.288675i
$13$			5237.72i			0.661212i	0.943769	+	0.330606i	$0.107253\pi$
$13$			5237.72i			0.661212i	−0.943769	+	0.330606i	$0.892747\pi$
$14$	−10478.5			−1.02059
$15$	−1786.64	−	7332.19i	−0.136684	−	0.560937i
$16$	4096.00			0.250000
$17$		−	4870.92i		−	0.240458i	−0.992746	−	0.120229i	$-0.961637\pi$
$17$		−	4870.92i		−	0.240458i	0.992746	−	0.120229i	$-0.0383629\pi$
$18$			5832.00i			0.235702i
$19$	−32781.2			−1.09645			−0.548223	−	0.836332i	$-0.684695\pi$
$19$	−32781.2			−1.09645			−0.548223	+	0.836332i	$0.684695\pi$
$20$	−17380.0	+	4235.01i	−0.485786	+	0.118372i
$21$	−35365.0			−0.833308
$22$			43658.8i			0.874162i
$23$		−	83486.0i		−	1.43076i	−0.698737	−	0.715379i	$-0.746254\pi$
$23$		−	83486.0i		−	1.43076i	0.698737	−	0.715379i	$-0.253746\pi$
$24$	13824.0			0.204124
$25$	69367.5	−	35939.7i	0.887904	−	0.460028i
$26$	41901.8			0.467547
$27$			19683.0i			0.192450i
$28$			83828.0i			0.721666i
$29$	73409.7			0.558934			0.279467	−	0.960155i	$-0.409842\pi$
$29$	73409.7			0.558934			0.279467	+	0.960155i	$0.409842\pi$
$30$	−58657.5	+	14293.1i	−0.396643	+	0.0966504i
$31$	244719.			1.47537			0.737687	−	0.675142i	$-0.235918\pi$
$31$	244719.			1.47537			0.737687	+	0.675142i	$0.235918\pi$
$32$		−	32768.0i		−	0.176777i
$33$			147348.i			0.713751i
$34$	−38967.3			−0.170030
$35$	−86672.9	−	355696.i	−0.341700	−	1.40230i
$36$	46656.0			0.166667
$37$		−	107537.i		−	0.349022i	−0.984655	−	0.174511i	$-0.944166\pi$
$37$		−	107537.i		−	0.349022i	0.984655	−	0.174511i	$-0.0558345\pi$
$38$			262250.i			0.775304i
$39$	141418.			0.381751
$40$	33880.1	+	139040.i	0.0837017	+	0.343503i
$41$	396774.			0.899082			0.449541	−	0.893260i	$-0.351588\pi$
$41$	396774.			0.899082			0.449541	+	0.893260i	$0.351588\pi$
$42$			282920.i			0.589238i
$43$			725839.i			1.39220i	0.717946	+	0.696099i	$0.245082\pi$
$43$			725839.i			1.39220i	−0.717946	+	0.696099i	$0.754918\pi$
$44$	349270.			0.618126
$45$	−197969.	+	48239.4i	−0.323857	+	0.0789147i
$46$	−667888.			−1.01170
$47$		−	801478.i		−	1.12603i	−0.826447	−	0.563014i	$-0.809642\pi$
$47$		−	801478.i		−	1.12603i	0.826447	−	0.563014i	$-0.190358\pi$
$48$		−	110592.i		−	0.144338i
$49$	−892068.			−1.08321
$50$	−287517.	−	554940.i	−0.325289	−	0.627843i
$51$	−131515.			−0.138829
$52$		−	335214.i		−	0.330606i
$53$		−	1.22395e6i		−	1.12927i	−0.825340	−	0.564636i	$-0.809017\pi$
$53$		−	1.22395e6i		−	1.12927i	0.825340	−	0.564636i	$-0.190983\pi$
$54$	157464.			0.136083
$55$	−1.48201e6	+	361123.i	−1.20111	+	0.292675i
$56$	670624.			0.510295
$57$			885092.i			0.633033i
$58$		−	587278.i		−	0.395226i
$59$	−2.32931e6			−1.47654			−0.738270	−	0.674506i	$-0.764357\pi$
$59$	−2.32931e6			−1.47654			−0.738270	+	0.674506i	$0.764357\pi$
$60$	114345.	+	469260.i	0.0683421	+	0.280469i
$61$	2.63553e6			1.48667			0.743334	−	0.668921i	$-0.233244\pi$
$61$	2.63553e6			1.48667			0.743334	+	0.668921i	$0.233244\pi$
$62$		−	1.95775e6i		−	1.04325i
$63$			954854.i			0.481111i
$64$	−262144.			−0.125000
$65$	346590.	+	1.42237e6i	0.156538	+	0.642415i
$66$	1.17879e6			0.504698
$67$			209887.i			0.0852557i	0.999091	+	0.0426279i	$0.0135730\pi$
$67$			209887.i			0.0852557i	−0.999091	+	0.0426279i	$0.986427\pi$
$68$			311739.i			0.120229i
$69$	−2.25412e6			−0.826049
$70$	−2.84557e6	+	693383.i	−0.991577	+	0.241619i
$71$	3.76578e6			1.24868			0.624339	−	0.781154i	$-0.285368\pi$
$71$	3.76578e6			1.24868			0.624339	+	0.781154i	$0.285368\pi$
$72$		−	373248.i		−	0.117851i
$73$			2.81943e6i			0.848265i	0.905600	+	0.424132i	$0.139421\pi$
$73$			2.81943e6i			0.848265i	−0.905600	+	0.424132i	$0.860579\pi$
$74$	−860298.			−0.246796
$75$	−970371.	−	1.87292e6i	−0.265597	−	0.512632i
$76$	2.09800e6			0.548223
$77$			7.14810e6i			1.78432i
$78$		−	1.13135e6i		−	0.269939i
$79$	8.68583e6			1.98206			0.991030	−	0.133642i	$-0.0426671\pi$
$79$	8.68583e6			1.98206			0.991030	+	0.133642i	$0.0426671\pi$
$80$	1.11232e6	−	271040.i	0.242893	−	0.0591860i
$81$	531441.			0.111111
$82$		−	3.17419e6i		−	0.635747i
$83$		−	5.77317e6i		−	1.10826i	−0.832431	−	0.554129i	$-0.813051\pi$
$83$		−	5.77317e6i		−	1.10826i	0.832431	−	0.554129i	$-0.186949\pi$
$84$	2.26336e6			0.416654
$85$	−322318.	−	1.32276e6i	−0.0569271	−	0.233622i
$86$	5.80672e6			0.984432
$87$		−	1.98206e6i		−	0.322701i
$88$		−	2.79416e6i		−	0.437081i
$89$	−1.02714e7			−1.54441			−0.772206	−	0.635373i	$-0.780846\pi$
$89$	−1.02714e7			−1.54441			−0.772206	+	0.635373i	$0.780846\pi$
$90$	385915.	+	1.58375e6i	0.0558011	+	0.229002i
$91$	6.86043e6			0.954348
$92$			5.34310e6i			0.715379i
$93$		−	6.60742e6i		−	0.851808i
$94$	−6.41183e6			−0.796222
$95$	−8.90215e6	+	2.16920e6i	−1.06528	+	0.259577i
$96$	−884736.			−0.102062
$97$			5.19683e6i			0.578146i	0.957307	+	0.289073i	$0.0933470\pi$
$97$			5.19683e6i			0.578146i	−0.957307	+	0.289073i	$0.906653\pi$
$98$			7.13654e6i			0.765943i
$99$	3.97841e6			0.412084
$100$	−4.43952e6	+	2.30014e6i	−0.443952	+	0.230014i
$101$	2.84866e6			0.275116			0.137558	−	0.990494i	$-0.456075\pi$
$101$	2.84866e6			0.275116			0.137558	+	0.990494i	$0.456075\pi$
$102$			1.05212e6i			0.0981666i
$103$		−	1.49365e7i		−	1.34685i	−0.739257	−	0.673423i	$-0.764823\pi$
$103$		−	1.49365e7i		−	1.34685i	0.739257	−	0.673423i	$-0.235177\pi$
$104$	−2.68171e6			−0.233774
$105$	−9.60380e6	+	2.34017e6i	−0.809619	+	0.197281i
$106$	−9.79161e6			−0.798516
$107$			1.04317e6i			0.0823213i	0.999153	+	0.0411606i	$0.0131055\pi$
$107$			1.04317e6i			0.0823213i	−0.999153	+	0.0411606i	$0.986894\pi$
$108$		−	1.25971e6i		−	0.0962250i
$109$	−5.23084e6			−0.386882			−0.193441	−	0.981112i	$-0.561965\pi$
$109$	−5.23084e6			−0.386882			−0.193441	+	0.981112i	$0.561965\pi$
$110$	2.88899e6	+	1.18561e7i	0.206953	+	0.849312i
$111$	−2.90351e6			−0.201508
$112$		−	5.36499e6i		−	0.360833i
$113$			1.58182e7i			1.03130i	0.856800	+	0.515649i	$0.172449\pi$
$113$			1.58182e7i			1.03130i	−0.856800	+	0.515649i	$0.827551\pi$
$114$	7.08074e6			0.447622
$115$	−5.52443e6	−	2.26717e7i	−0.338723	−	1.39008i
$116$	−4.69822e6			−0.279467
$117$		−	3.81830e6i		−	0.220404i
$118$			1.86345e7i			1.04407i
$119$	−6.37999e6			−0.347061
$120$	3.75408e6	−	914761.i	0.198321	−	0.0483252i
$121$	1.02955e7			0.528320
$122$		−	2.10842e7i		−	1.05123i
$123$		−	1.07129e7i		−	0.519085i
$124$	−1.56620e7			−0.737687
$125$	1.64594e7	−	1.43501e7i	0.753754	−	0.657156i
$126$	7.63883e6			0.340197
$127$		−	4.24970e6i		−	0.184096i	−0.995755	−	0.0920481i	$-0.970659\pi$
$127$		−	4.24970e6i		−	0.184096i	0.995755	−	0.0920481i	$-0.0293414\pi$
$128$			2.09715e6i			0.0883883i
$129$	1.95977e7			0.803786
$130$	1.13790e7	−	2.77272e6i	0.454256	−	0.110689i
$131$	5.59155e6			0.217312			0.108656	−	0.994079i	$-0.465345\pi$
$131$	5.59155e6			0.217312			0.108656	+	0.994079i	$0.465345\pi$
$132$		−	9.43029e6i		−	0.356875i
$133$			4.29372e7i			1.58254i
$134$	1.67910e6			0.0602849
$135$	1.30246e6	+	5.34517e6i	0.0455614	+	0.186979i
$136$	2.49391e6			0.0850148
$137$			2.21174e7i			0.734874i	0.930048	+	0.367437i	$0.119765\pi$
$137$			2.21174e7i			0.734874i	−0.930048	+	0.367437i	$0.880235\pi$
$138$			1.80330e7i			0.584105i
$139$	2.55140e7			0.805799			0.402900	−	0.915244i	$-0.368002\pi$
$139$	2.55140e7			0.805799			0.402900	+	0.915244i	$0.368002\pi$
$140$	5.54707e6	+	2.27646e7i	0.170850	+	0.701150i
$141$	−2.16399e7			−0.650113
$142$		−	3.01262e7i		−	0.882948i
$143$		−	2.85841e7i		−	0.817424i
$144$	−2.98598e6			−0.0833333
$145$	1.99353e7	−	4.85767e6i	0.543045	−	0.132324i
$146$	2.25554e7			0.599814
$147$			2.40858e7i			0.625390i
$148$			6.88238e6i			0.174511i
$149$	1.64001e6			0.0406158			0.0203079	−	0.999794i	$-0.493535\pi$
$149$	1.64001e6			0.0406158			0.0203079	+	0.999794i	$0.493535\pi$
$150$	−1.49834e7	+	7.76297e6i	−0.362485	+	0.187806i
$151$	−4.65795e7			−1.10097			−0.550485	−	0.834845i	$-0.685557\pi$
$151$	−4.65795e7			−1.10097			−0.550485	+	0.834845i	$0.685557\pi$
$152$		−	1.67840e7i		−	0.387652i
$153$			3.55090e6i			0.0801527i
$154$	5.71848e7			1.26171
$155$	6.64566e7	−	1.61936e7i	1.43343	−	0.349286i
$156$	−9.05078e6			−0.190875
$157$			6.21860e7i			1.28246i	0.767349	+	0.641230i	$0.221575\pi$
$157$			6.21860e7i			1.28246i	−0.767349	+	0.641230i	$0.778425\pi$
$158$		−	6.94867e7i		−	1.40153i
$159$	−3.30467e7			−0.651985
$160$	−2.16832e6	−	8.89856e6i	−0.0418508	−	0.171751i
$161$	−1.09351e8			−2.06506
$162$		−	4.25153e6i		−	0.0785674i
$163$			5.86864e7i			1.06140i	0.847558	+	0.530702i	$0.178072\pi$
$163$			5.86864e7i			1.06140i	−0.847558	+	0.530702i	$0.821928\pi$
$164$	−2.53935e7			−0.449541
$165$	9.75033e6	+	4.00143e7i	0.168976	+	0.693460i
$166$	−4.61854e7			−0.783657
$167$		−	2.74875e7i		−	0.456696i	−0.973580	−	0.228348i	$-0.926668\pi$
$167$		−	2.74875e7i		−	0.456696i	0.973580	−	0.228348i	$-0.0733325\pi$
$168$		−	1.81069e7i		−	0.294619i
$169$	3.53148e7			0.562799
$170$	−1.05821e7	+	2.57855e6i	−0.165196	+	0.0402535i
$171$	2.38975e7			0.365482
$172$		−	4.64537e7i		−	0.696099i
$173$		−	6.19491e7i		−	0.909649i	−0.890581	−	0.454825i	$-0.849702\pi$
$173$		−	6.19491e7i		−	0.909649i	0.890581	−	0.454825i	$-0.150298\pi$
$174$	−1.58565e7			−0.228184
$175$	−4.70743e7	−	9.08585e7i	−0.663973	−	1.28154i
$176$	−2.23533e7			−0.309063
$177$			6.28913e7i			0.852480i
$178$			8.21709e7i			1.09206i
$179$	−1.04816e7			−0.136598			−0.0682988	−	0.997665i	$-0.521757\pi$
$179$	−1.04816e7			−0.136598			−0.0682988	+	0.997665i	$0.521757\pi$
$180$	1.26700e7	−	3.08732e6i	0.161929	−	0.0394573i
$181$	−9.64546e7			−1.20906			−0.604530	−	0.796582i	$-0.706639\pi$
$181$	−9.64546e7			−1.20906			−0.604530	+	0.796582i	$0.706639\pi$
$182$		−	5.48835e7i		−	0.674826i
$183$		−	7.11593e7i		−	0.858328i
$184$	4.27448e7			0.505849
$185$	−7.11595e6	−	2.92031e7i	−0.0826289	−	0.339100i
$186$	−5.28594e7			−0.602319
$187$			2.65823e7i			0.297267i
$188$			5.12946e7i			0.563014i
$189$	2.57811e7			0.277769
$190$	1.73536e7	+	7.12172e7i	0.183549	+	0.753264i
$191$	9.71449e7			1.00880			0.504398	−	0.863471i	$-0.331714\pi$
$191$	9.71449e7			1.00880			0.504398	+	0.863471i	$0.331714\pi$
$192$			7.07789e6i			0.0721688i
$193$		−	5.17514e7i		−	0.518169i	−0.965855	−	0.259085i	$-0.916579\pi$
$193$		−	5.17514e7i		−	0.518169i	0.965855	−	0.259085i	$-0.0834208\pi$
$194$	4.15746e7			0.408811
$195$	3.84040e7	−	9.35794e6i	0.370898	−	0.0903772i
$196$	5.70923e7			0.541604
$197$			1.55737e8i			1.45131i	0.688057	+	0.725657i	$0.258464\pi$
$197$			1.55737e8i			1.45131i	−0.688057	+	0.725657i	$0.741536\pi$
$198$		−	3.18272e7i		−	0.291387i
$199$	−1.49558e8			−1.34532			−0.672658	−	0.739954i	$-0.734847\pi$
$199$	−1.49558e8			−1.34532			−0.672658	+	0.739954i	$0.734847\pi$
$200$	1.84011e7	+	3.55162e7i	0.162644	+	0.313922i
$201$	5.66695e6			0.0492224
$202$		−	2.27892e7i		−	0.194536i
$203$		−	9.61530e7i		−	0.806728i
$204$	8.41695e6			0.0694143
$205$	1.07749e8	−	2.62553e7i	0.873523	−	0.212852i
$206$	−1.19492e8			−0.952364
$207$			6.08613e7i			0.476919i
$208$			2.14537e7i			0.165303i
$209$	1.78898e8			1.35548
$210$	1.87214e7	+	7.68304e7i	0.139499	+	0.572487i
$211$	−4.83513e6			−0.0354340			−0.0177170	−	0.999843i	$-0.505640\pi$
$211$	−4.83513e6			−0.0354340			−0.0177170	+	0.999843i	$0.505640\pi$
$212$			7.83329e7i			0.564636i
$213$		−	1.01676e8i		−	0.720924i
$214$	8.34536e6			0.0582099
$215$	4.80302e7	+	1.97111e8i	0.329595	+	1.35262i
$216$	−1.00777e7			−0.0680414
$217$		−	3.20537e8i		−	2.12946i
$218$			4.18467e7i			0.273567i
$219$	7.61246e7			0.489746
$220$	9.48487e7	−	2.31119e7i	0.600554	−	0.146338i
$221$	2.55125e7			0.158994
$222$			2.32280e7i			0.142488i
$223$			2.61843e8i			1.58116i	0.612361	+	0.790578i	$0.290220\pi$
$223$			2.61843e8i			1.58116i	−0.612361	+	0.790578i	$0.709780\pi$
$224$	−4.29200e7			−0.255147
$225$	−5.05689e7	+	2.62000e7i	−0.295968	+	0.153343i
$226$	1.26546e8			0.729237
$227$			1.28619e8i			0.729820i	0.931043	+	0.364910i	$0.118900\pi$
$227$			1.28619e8i			0.729820i	−0.931043	+	0.364910i	$0.881100\pi$
$228$		−	5.66459e7i		−	0.316517i
$229$	−1.01465e8			−0.558330			−0.279165	−	0.960243i	$-0.590058\pi$
$229$	−1.01465e8			−0.558330			−0.279165	+	0.960243i	$0.590058\pi$
$230$	−1.81373e8	+	4.41955e7i	−0.982938	+	0.239514i
$231$	1.92999e8			1.03018
$232$			3.75858e7i			0.197613i
$233$		−	2.74718e8i		−	1.42279i	−0.702791	−	0.711397i	$-0.748063\pi$
$233$		−	2.74718e8i		−	1.42279i	0.702791	−	0.711397i	$-0.251937\pi$
$234$	−3.05464e7			−0.155849
$235$	−5.30354e7	−	2.17652e8i	−0.266580	−	1.09402i
$236$	1.49076e8			0.738270
$237$		−	2.34518e8i		−	1.14434i
$238$			5.10399e7i			0.245409i
$239$	3.77861e7			0.179035			0.0895177	−	0.995985i	$-0.471467\pi$
$239$	3.77861e7			0.179035			0.0895177	+	0.995985i	$0.471467\pi$
$240$	−7.31809e6	−	3.00327e7i	−0.0341711	−	0.140234i
$241$	1.70494e8			0.784600			0.392300	−	0.919837i	$-0.371680\pi$
$241$	1.70494e8			0.784600			0.392300	+	0.919837i	$0.371680\pi$
$242$		−	8.23637e7i		−	0.373578i
$243$		−	1.43489e7i		−	0.0641500i
$244$	−1.68674e8			−0.743334
$245$	−2.42252e8	+	5.90299e7i	−1.05241	+	0.256443i
$246$	−8.57031e7			−0.367049
$247$		−	1.71699e8i		−	0.724983i
$248$			1.25296e8i			0.521624i
$249$	−1.55876e8			−0.639854
$250$	−1.14800e8	−	1.31675e8i	−0.464680	−	0.532985i
$251$	−3.41080e8			−1.36144			−0.680719	−	0.732544i	$-0.738333\pi$
$251$	−3.41080e8			−1.36144			−0.680719	+	0.732544i	$0.738333\pi$
$252$		−	6.11106e7i		−	0.240555i
$253$			4.55612e8i			1.76878i
$254$	−3.39976e7			−0.130176
$255$	−3.57145e7	+	8.70259e6i	−0.134882	+	0.0328669i
$256$	1.67772e7			0.0625000
$257$			1.94713e8i			0.715530i	0.933812	+	0.357765i	$0.116461\pi$
$257$			1.94713e8i			0.715530i	−0.933812	+	0.357765i	$0.883539\pi$
$258$		−	1.56781e8i		−	0.568362i
$259$	−1.40854e8			−0.503755
$260$	−2.21818e7	−	9.10316e7i	−0.0782690	−	0.321207i
$261$	−5.35157e7			−0.186311
$262$		−	4.47324e7i		−	0.153663i
$263$			4.85601e8i			1.64602i	0.568029	+	0.823009i	$0.307706\pi$
$263$			4.85601e8i			1.64602i	−0.568029	+	0.823009i	$0.692294\pi$
$264$	−7.54424e7			−0.252349
$265$	−8.09913e7	−	3.32379e8i	−0.267348	−	1.09717i
$266$	3.43498e8			1.11902
$267$			2.77327e8i			0.891666i
$268$		−	1.34328e7i		−	0.0426279i
$269$	2.89744e7			0.0907572			0.0453786	−	0.998970i	$-0.485551\pi$
$269$	2.89744e7			0.0907572			0.0453786	+	0.998970i	$0.485551\pi$
$270$	4.27613e7	−	1.04197e7i	0.132214	−	0.0322168i
$271$	3.02460e8			0.923156			0.461578	−	0.887100i	$-0.347283\pi$
$271$	3.02460e8			0.923156			0.461578	+	0.887100i	$0.347283\pi$
$272$		−	1.99513e7i		−	0.0601146i
$273$		−	1.85232e8i		−	0.550993i
$274$	1.76939e8			0.519634
$275$	−3.78563e8	+	1.96135e8i	−1.09767	+	0.568710i
$276$	1.44264e8			0.413024
$277$			1.78252e8i			0.503912i	0.967739	+	0.251956i	$0.0810739\pi$
$277$			1.78252e8i			0.503912i	−0.967739	+	0.251956i	$0.918926\pi$
$278$		−	2.04112e8i		−	0.569786i
$279$	−1.78400e8			−0.491792
$280$	1.82117e8	−	4.43765e7i	0.495788	−	0.120809i
$281$	2.21487e7			0.0595492			0.0297746	−	0.999557i	$-0.490521\pi$
$281$	2.21487e7			0.0595492			0.0297746	+	0.999557i	$0.490521\pi$
$282$			1.73119e8i			0.459699i
$283$			1.08099e8i			0.283511i	0.989902	+	0.141756i	$0.0452747\pi$
$283$			1.08099e8i			0.283511i	−0.989902	+	0.141756i	$0.954725\pi$
$284$	−2.41010e8			−0.624339
$285$	5.85683e7	+	2.40358e8i	0.149867	+	0.615038i
$286$	−2.28672e8			−0.578006
$287$		−	5.19699e8i		−	1.29767i
$288$			2.38879e7i			0.0589256i
$289$	3.86613e8			0.942180
$290$	−3.88613e7	−	1.59483e8i	−0.0935675	−	0.383991i
$291$	1.40314e8			0.333793
$292$		−	1.80444e8i		−	0.424132i
$293$		−	5.08345e8i		−	1.18065i	−0.807165	−	0.590326i	$-0.798999\pi$
$293$		−	5.08345e8i		−	1.18065i	0.807165	−	0.590326i	$-0.201001\pi$
$294$	1.92687e8			0.442217
$295$	−6.32553e8	+	1.54135e8i	−1.43456	+	0.349562i
$296$	5.50591e7			0.123398
$297$		−	1.07417e8i		−	0.237917i
$298$		−	1.31201e7i		−	0.0287197i
$299$	4.37276e8			0.946034
$300$	6.21038e7	+	1.19867e8i	0.132799	+	0.256316i
$301$	9.50714e8			2.00940
$302$			3.72636e8i			0.778503i
$303$		−	7.69137e7i		−	0.158838i
$304$	−1.34272e8			−0.274111
$305$	7.15712e8	−	1.74398e8i	1.44440	−	0.351960i
$306$	2.84072e7			0.0566765
$307$			4.60852e8i			0.909027i	0.890740	+	0.454514i	$0.150187\pi$
$307$			4.60852e8i			0.909027i	−0.890740	+	0.454514i	$0.849813\pi$
$308$		−	4.57479e8i		−	0.892161i
$309$	−4.03285e8			−0.777602
$310$	−1.29549e8	−	5.31653e8i	−0.246983	−	1.01359i
$311$	7.63906e8			1.44005			0.720027	−	0.693946i	$-0.244129\pi$
$311$	7.63906e8			1.44005			0.720027	+	0.693946i	$0.244129\pi$
$312$			7.24062e7i			0.134969i
$313$			4.96971e8i			0.916064i	0.888936	+	0.458032i	$0.151446\pi$
$313$			4.96971e8i			0.916064i	−0.888936	+	0.458032i	$0.848554\pi$
$314$	4.97488e8			0.906836
$315$	6.31846e7	+	2.59303e8i	0.113900	+	0.467434i
$316$	−5.55893e8			−0.991030
$317$		−	4.18865e8i		−	0.738528i	−0.929325	−	0.369264i	$-0.879610\pi$
$317$		−	4.18865e8i		−	0.738528i	0.929325	−	0.369264i	$-0.120390\pi$
$318$			2.64373e8i			0.461023i
$319$	−4.00622e8			−0.690984
$320$	−7.11885e7	+	1.73466e7i	−0.121447	+	0.0295930i
$321$	2.81656e7			0.0475282
$322$			8.74808e8i			1.46022i
$323$			1.59675e8i			0.263649i
$324$	−3.40122e7			−0.0555556
$325$	1.88242e8	+	3.63328e8i	0.304176	+	0.587093i
$326$	4.69491e8			0.750527
$327$			1.41233e8i			0.223366i
$328$			2.03148e8i			0.317873i
$329$	−1.04979e9			−1.62523
$330$	3.20114e8	−	7.80027e7i	0.490350	−	0.119484i
$331$	−8.00262e8			−1.21293			−0.606463	−	0.795112i	$-0.707412\pi$
$331$	−8.00262e8			−1.21293			−0.606463	+	0.795112i	$0.707412\pi$
$332$			3.69483e8i			0.554129i
$333$			7.83946e7i			0.116341i
$334$	−2.19900e8			−0.322933
$335$	1.38886e7	+	5.69975e7i	0.0201838	+	0.0828321i
$336$	−1.44855e8			−0.208327
$337$		−	1.38535e9i		−	1.97177i	−0.167437	−	0.985883i	$-0.553549\pi$
$337$		−	1.38535e9i		−	1.97177i	0.167437	−	0.985883i	$-0.446451\pi$
$338$		−	2.82518e8i		−	0.397959i
$339$	4.27093e8			0.595420
$340$	2.06284e7	+	8.46566e7i	0.0284635	+	0.116811i
$341$	−1.33552e9			−1.82394
$342$		−	1.91180e8i		−	0.258435i
$343$			8.97545e7i			0.120096i
$344$	−3.71630e8			−0.492216
$345$	−6.12135e8	+	1.49160e8i	−0.802566	+	0.195562i
$346$	−4.95593e8			−0.643219
$347$		−	9.62881e8i		−	1.23714i	−0.785729	−	0.618571i	$-0.787712\pi$
$347$		−	9.62881e8i		−	1.23714i	0.785729	−	0.618571i	$-0.212288\pi$
$348$			1.26852e8i			0.161350i
$349$	5.40052e8			0.680058			0.340029	−	0.940415i	$-0.389563\pi$
$349$	5.40052e8			0.680058			0.340029	+	0.940415i	$0.389563\pi$
$350$	−7.26868e8	+	3.76594e8i	−0.906186	+	0.469500i
$351$	−1.03094e8			−0.127250
$352$			1.78826e8i			0.218541i
$353$		−	1.02599e9i		−	1.24145i	−0.784027	−	0.620727i	$-0.786837\pi$
$353$		−	1.02599e9i		−	1.24145i	0.784027	−	0.620727i	$-0.213163\pi$
$354$	5.03131e8			0.602795
$355$	1.02264e9	−	2.49189e8i	1.21318	−	0.295617i
$356$	6.57367e8			0.772206
$357$			1.72260e8i			0.200376i
$358$			8.38530e7i			0.0965891i
$359$	7.47383e8			0.852536			0.426268	−	0.904597i	$-0.359828\pi$
$359$	7.47383e8			0.852536			0.426268	+	0.904597i	$0.359828\pi$
$360$	−2.46986e7	−	1.01360e8i	−0.0279006	−	0.114501i
$361$	1.80735e8			0.202194
$362$			7.71637e8i			0.854935i
$363$		−	2.77977e8i		−	0.305026i
$364$	−4.39068e8			−0.477174
$365$	1.86567e8	+	7.65652e8i	0.200822	+	0.824150i
$366$	−5.69275e8			−0.606929
$367$		−	4.77968e8i		−	0.504740i	−0.967631	−	0.252370i	$-0.918790\pi$
$367$		−	4.77968e8i		−	0.504740i	0.967631	−	0.252370i	$-0.0812100\pi$
$368$		−	3.41959e8i		−	0.357689i
$369$	−2.89248e8			−0.299694
$370$	−2.33625e8	+	5.69276e7i	−0.239780	+	0.0584274i
$371$	−1.60315e9			−1.62991
$372$			4.22875e8i			0.425904i
$373$		−	1.41621e9i		−	1.41302i	−0.707704	−	0.706509i	$-0.750269\pi$
$373$		−	1.41621e9i		−	1.41302i	0.707704	−	0.706509i	$-0.249731\pi$
$374$	2.12658e8			0.210200
$375$	−3.87452e8	−	4.44405e8i	−0.379409	−	0.435180i
$376$	4.10357e8			0.398111
$377$			3.84500e8i			0.369574i
$378$		−	2.06248e8i		−	0.196413i
$379$	−4.23809e8			−0.399883			−0.199941	−	0.979808i	$-0.564075\pi$
$379$	−4.23809e8			−0.399883			−0.199941	+	0.979808i	$0.564075\pi$
$380$	5.69738e8	−	1.38829e8i	0.532638	−	0.129789i
$381$	−1.14742e8			−0.106288
$382$		−	7.77159e8i		−	0.713327i
$383$			8.56505e8i			0.778994i	0.921028	+	0.389497i	$0.127351\pi$
$383$			8.56505e8i			0.778994i	−0.921028	+	0.389497i	$0.872649\pi$
$384$	5.66231e7			0.0510310
$385$	4.73004e8	+	1.94116e9i	0.422428	+	1.73360i
$386$	−4.14011e8			−0.366401
$387$		−	5.29137e8i		−	0.464066i
$388$		−	3.32597e8i		−	0.289073i
$389$	1.08932e8			0.0938282			0.0469141	−	0.998899i	$-0.485061\pi$
$389$	1.08932e8			0.0938282			0.0469141	+	0.998899i	$0.485061\pi$
$390$	−7.48635e7	−	3.07232e8i	−0.0639064	−	0.262265i
$391$	−4.06653e8			−0.344037
$392$		−	4.56739e8i		−	0.382972i
$393$		−	1.50972e8i		−	0.125465i
$394$	1.24590e9			1.02623
$395$	2.35875e9	−	5.74759e8i	1.92571	−	0.469241i
$396$	−2.54618e8			−0.206042
$397$			2.13997e9i			1.71649i	0.513239	+	0.858245i	$0.328445\pi$
$397$			2.13997e9i			1.71649i	−0.513239	+	0.858245i	$0.671555\pi$
$398$			1.19646e9i			0.951281i
$399$	1.15931e9			0.913677
$400$	2.84129e8	−	1.47209e8i	0.221976	−	0.115007i
$401$	−9.00462e8			−0.697365			−0.348682	−	0.937241i	$-0.613371\pi$
$401$	−9.00462e8			−0.697365			−0.348682	+	0.937241i	$0.613371\pi$
$402$		−	4.53356e7i		−	0.0348055i
$403$			1.28177e9i			0.975535i
$404$	−1.82314e8			−0.137558
$405$	1.44320e8	−	3.51665e7i	0.107952	−	0.0263049i
$406$	−7.69224e8			−0.570443
$407$			5.86868e8i			0.431479i
$408$		−	6.73356e7i		−	0.0490833i
$409$	2.15143e9			1.55487			0.777436	−	0.628962i	$-0.216520\pi$
$409$	2.15143e9			1.55487			0.777436	+	0.628962i	$0.216520\pi$
$410$	−2.10042e8	−	8.61991e8i	−0.150509	−	0.617674i
$411$	5.97171e8			0.424279
$412$			9.55935e8i			0.673423i
$413$			3.05096e9i			2.13114i
$414$	4.86890e8			0.337233
$415$	−3.82022e8	−	1.56778e9i	−0.262374	−	1.07675i
$416$	1.71630e8			0.116887
$417$		−	6.88878e8i		−	0.465228i
$418$		−	1.43119e9i		−	0.958472i
$419$	−7.96072e7			−0.0528693			−0.0264346	−	0.999651i	$-0.508415\pi$
$419$	−7.96072e7			−0.0528693			−0.0264346	+	0.999651i	$0.508415\pi$
$420$	6.14643e8	−	1.49771e8i	0.404809	−	0.0986404i
$421$	3.29700e8			0.215344			0.107672	−	0.994186i	$-0.465660\pi$
$421$	3.29700e8			0.215344			0.107672	+	0.994186i	$0.465660\pi$
$422$			3.86810e7i			0.0250556i
$423$			5.84278e8i			0.375343i
$424$	6.26663e8			0.399258
$425$	−1.75059e8	−	3.37884e8i	−0.110617	−	0.213504i
$426$	−8.13408e8			−0.509771
$427$		−	3.45205e9i		−	2.14575i
$428$		−	6.67629e7i		−	0.0411606i
$429$	−7.71769e8			−0.471940
$430$	1.57689e9	−	3.84242e8i	0.956447	−	0.233059i
$431$	5.57393e8			0.335345			0.167672	−	0.985843i	$-0.446375\pi$
$431$	5.57393e8			0.335345			0.167672	+	0.985843i	$0.446375\pi$
$432$			8.06216e7i			0.0481125i
$433$		−	1.00562e9i		−	0.595287i	−0.954677	−	0.297644i	$-0.903799\pi$
$433$		−	1.00562e9i		−	0.595287i	0.954677	−	0.297644i	$-0.0962007\pi$
$434$	−2.56429e9			−1.50575
$435$	−1.31157e8	−	5.38254e8i	−0.0763975	−	0.313527i
$436$	3.34774e8			0.193441
$437$			2.73677e9i			1.56875i
$438$		−	6.08997e8i		−	0.346303i
$439$	−2.43906e9			−1.37593			−0.687967	−	0.725742i	$-0.741496\pi$
$439$	−2.43906e9			−1.37593			−0.687967	+	0.725742i	$0.741496\pi$
$440$	−1.84895e8	−	7.58790e8i	−0.103476	−	0.424656i
$441$	6.50317e8			0.361069
$442$		−	2.04100e8i		−	0.112426i
$443$			1.15960e9i			0.633715i	0.948473	+	0.316858i	$0.102628\pi$
$443$			1.15960e9i			0.633715i	−0.948473	+	0.316858i	$0.897372\pi$
$444$	1.85824e8			0.100754
$445$	−2.78932e9	+	6.79676e8i	−1.50051	+	0.365630i
$446$	2.09475e9			1.11805
$447$		−	4.42804e7i		−	0.0234496i
$448$			3.43360e8i			0.180416i
$449$	−6.32037e8			−0.329519			−0.164759	−	0.986334i	$-0.552685\pi$
$449$	−6.32037e8			−0.329519			−0.164759	+	0.986334i	$0.552685\pi$
$450$	2.09600e8	+	4.04551e8i	0.108430	+	0.209281i
$451$	−2.16533e9			−1.11149
$452$		−	1.01237e9i		−	0.515649i
$453$			1.25765e9i			0.635645i
$454$	1.02895e9			0.516061
$455$	1.86304e9	−	4.53969e8i	0.927218	−	0.225936i
$456$	−4.53167e8			−0.223811
$457$			1.78534e9i			0.875015i	0.899215	+	0.437507i	$0.144139\pi$
$457$			1.78534e9i			0.875015i	−0.899215	+	0.437507i	$0.855861\pi$
$458$			8.11718e8i			0.394799i
$459$	9.58743e7			0.0462762
$460$	3.53564e8	+	1.45099e9i	0.169362	+	0.695042i
$461$	−1.23239e9			−0.585862			−0.292931	−	0.956134i	$-0.594631\pi$
$461$	−1.23239e9			−0.585862			−0.292931	+	0.956134i	$0.594631\pi$
$462$		−	1.54399e9i		−	0.728447i
$463$		−	3.27184e9i		−	1.53200i	−0.642841	−	0.766000i	$-0.722244\pi$
$463$		−	3.27184e9i		−	1.53200i	0.642841	−	0.766000i	$-0.277756\pi$
$464$	3.00686e8			0.139734
$465$	−4.37226e8	−	1.79433e9i	−0.201661	−	0.827593i
$466$	−2.19775e9			−1.00607
$467$		−	4.20893e8i		−	0.191233i	−0.995418	−	0.0956165i	$-0.969518\pi$
$467$		−	4.20893e8i		−	0.191233i	0.995418	−	0.0956165i	$-0.0304822\pi$
$468$			2.44371e8i			0.110202i
$469$	2.74913e8			0.123052
$470$	−1.74121e9	+	4.24283e8i	−0.773587	+	0.188501i
$471$	1.67902e9			0.740428
$472$		−	1.19261e9i		−	0.522036i
$473$		−	3.96116e9i		−	1.72111i
$474$	−1.87614e9			−0.809172
$475$	−2.27395e9	+	1.17815e9i	−0.973539	+	0.504396i
$476$	4.08320e8			0.173531
$477$			8.92260e8i			0.376424i
$478$		−	3.02288e8i		−	0.126597i
$479$	2.91500e7			0.0121189			0.00605946	−	0.999982i	$-0.498071\pi$
$479$	2.91500e7			0.0121189			0.00605946	+	0.999982i	$0.498071\pi$
$480$	−2.40261e8	+	5.85447e7i	−0.0991607	+	0.0241626i
$481$	5.63250e8			0.230777
$482$		−	1.36395e9i		−	0.554796i
$483$			2.95248e9i			1.19226i
$484$	−6.58909e8			−0.264160
$485$	3.43885e8	+	1.41126e9i	0.136873	+	0.561710i
$486$	−1.14791e8			−0.0453609
$487$			2.58798e9i			1.01534i	0.861553	+	0.507668i	$0.169492\pi$
$487$			2.58798e9i			1.01534i	−0.861553	+	0.507668i	$0.830508\pi$
$488$			1.34939e9i			0.525616i
$489$	1.58453e9			0.612802
$490$	4.72239e8	+	1.93802e9i	0.181332	+	0.744169i
$491$	−2.01157e9			−0.766918			−0.383459	−	0.923558i	$-0.625267\pi$
$491$	−2.01157e9			−0.766918			−0.383459	+	0.923558i	$0.625267\pi$
$492$			6.85625e8i			0.259543i
$493$		−	3.57573e8i		−	0.134400i
$494$	−1.37359e9			−0.512640
$495$	1.08039e9	−	2.63259e8i	0.400369	−	0.0975585i
$496$	1.00237e9			0.368844
$497$		−	4.93246e9i		−	1.80226i
$498$			1.24701e9i			0.452445i
$499$	6.31261e8			0.227435			0.113717	−	0.993513i	$-0.463724\pi$
$499$	6.31261e8			0.227435			0.113717	+	0.993513i	$0.463724\pi$
$500$	−1.05340e9	+	9.18404e8i	−0.376877	+	0.328578i
$501$	−7.42162e8			−0.263674
$502$			2.72864e9i			0.962682i
$503$		−	1.57252e9i		−	0.550944i	−0.961309	−	0.275472i	$-0.911166\pi$
$503$		−	1.57252e9i		−	0.550944i	0.961309	−	0.275472i	$-0.0888342\pi$
$504$	−4.88885e8			−0.170098
$505$	7.73589e8	−	1.88501e8i	0.267295	−	0.0651320i
$506$	3.64490e9			1.25071
$507$		−	9.53500e8i		−	0.324932i
$508$			2.71981e8i			0.0920481i
$509$	2.18665e9			0.734965			0.367482	−	0.930031i	$-0.380220\pi$
$509$	2.18665e9			0.734965			0.367482	+	0.930031i	$0.380220\pi$
$510$	6.96208e7	+	2.85716e8i	0.0232404	+	0.0953760i
$511$	3.69293e9			1.22433
$512$		−	1.34218e8i		−	0.0441942i
$513$		−	6.45232e8i		−	0.211011i
$514$	1.55770e9			0.505956
$515$	−9.88376e8	−	4.05619e9i	−0.318858	−	1.30856i
$516$	−1.25425e9			−0.401893
$517$			4.37394e9i			1.39205i
$518$			1.12683e9i			0.356208i
$519$	−1.67263e9			−0.525186
$520$	−7.28253e8	+	1.77454e8i	−0.227128	+	0.0553445i
$521$	5.81389e9			1.80109			0.900543	−	0.434766i	$-0.143169\pi$
$521$	5.81389e9			1.80109			0.900543	+	0.434766i	$0.143169\pi$
$522$			4.28126e8i			0.131742i
$523$		−	2.36475e9i		−	0.722819i	−0.932407	−	0.361409i	$-0.882296\pi$
$523$		−	2.36475e9i		−	0.722819i	0.932407	−	0.361409i	$-0.117704\pi$
$524$	−3.57859e8			−0.108656
$525$	−2.45318e9	+	1.27101e9i	−0.739898	+	0.383345i
$526$	3.88481e9			1.16391
$527$		−	1.19201e9i		−	0.354766i
$528$			6.03539e8i			0.178438i
$529$	−3.56509e9			−1.04707
$530$	−2.65903e9	+	6.47930e8i	−0.775816	+	0.189044i
$531$	1.69807e9			0.492180
$532$		−	2.74798e9i		−	0.791268i
$533$			2.07819e9i			0.594483i
$534$	2.21861e9			0.630503
$535$	6.90286e7	+	2.83286e8i	0.0194891	+	0.0799810i
$536$	−1.07462e8			−0.0301425
$537$			2.83004e8i			0.0788647i
$538$		−	2.31795e8i		−	0.0641751i
$539$	4.86832e9			1.33912
$540$	−8.33576e7	−	3.42091e8i	−0.0227807	−	0.0934896i
$541$	5.64040e9			1.53151			0.765754	−	0.643134i	$-0.222366\pi$
$541$	5.64040e9			1.53151			0.765754	+	0.643134i	$0.222366\pi$
$542$		−	2.41968e9i		−	0.652770i
$543$			2.60428e9i			0.698051i
$544$	−1.59610e8			−0.0425074
$545$	−1.42050e9	+	3.46135e8i	−0.375884	+	0.0915920i
$546$	−1.48185e9			−0.389611
$547$			1.78063e9i			0.465177i	0.972575	+	0.232589i	$0.0747195\pi$
$547$			1.78063e9i			0.465177i	−0.972575	+	0.232589i	$0.925280\pi$
$548$		−	1.41552e9i		−	0.367437i
$549$	−1.92130e9			−0.495556
$550$	1.56908e9	+	3.02850e9i	0.402139	+	0.776173i
$551$	−2.40646e9			−0.612841
$552$		−	1.15411e9i		−	0.292052i
$553$		−	1.13768e10i		−	2.86077i
$554$	1.42602e9			0.356320
$555$	−7.88483e8	+	1.92131e8i	−0.195779	+	0.0477058i
$556$	−1.63290e9			−0.402900
$557$			1.56532e9i			0.383805i	0.981414	+	0.191902i	$0.0614657\pi$
$557$			1.56532e9i			0.383805i	−0.981414	+	0.191902i	$0.938534\pi$
$558$			1.42720e9i			0.347749i
$559$	−3.80174e9			−0.920537
$560$	−3.55012e8	−	1.45693e9i	−0.0854251	−	0.350575i
$561$	7.17722e8			0.171627
$562$		−	1.77190e8i		−	0.0421077i
$563$			2.73932e9i			0.646939i	0.946239	+	0.323470i	$0.104849\pi$
$563$			2.73932e9i			0.646939i	−0.946239	+	0.323470i	$0.895151\pi$
$564$	1.38495e9			0.325056
$565$	1.04672e9	+	4.29564e9i	0.244153	+	1.00198i
$566$	8.64794e8			0.200473
$567$		−	6.96088e8i		−	0.160370i
$568$			1.92808e9i			0.441474i
$569$	−7.38630e8			−0.168087			−0.0840435	−	0.996462i	$-0.526783\pi$
$569$	−7.38630e8			−0.168087			−0.0840435	+	0.996462i	$0.526783\pi$
$570$	1.92286e9	−	4.68547e8i	0.434897	−	0.105972i
$571$	3.23235e9			0.726594			0.363297	−	0.931673i	$-0.381651\pi$
$571$	3.23235e9			0.726594			0.363297	+	0.931673i	$0.381651\pi$
$572$			1.82938e9i			0.408712i
$573$		−	2.62291e9i		−	0.582429i
$574$	−4.15759e9			−0.917594
$575$	−3.00046e9	−	5.79122e9i	−0.658189	−	1.27038i
$576$	1.91103e8			0.0416667
$577$			2.65831e9i			0.576090i	0.957617	+	0.288045i	$0.0930053\pi$
$577$			2.65831e9i			0.576090i	−0.957617	+	0.288045i	$0.906995\pi$
$578$		−	3.09290e9i		−	0.666222i
$579$	−1.39729e9			−0.299165
$580$	−1.27586e9	+	3.10891e8i	−0.271522	+	0.0661622i
$581$	−7.56178e9			−1.59959
$582$		−	1.12252e9i		−	0.236027i
$583$			6.67952e9i			1.39606i
$584$	−1.44355e9			−0.299907
$585$	−2.52664e8	−	1.03691e9i	−0.0521793	−	0.214138i
$586$	−4.06676e9			−0.834848
$587$			9.42375e9i			1.92305i	0.274720	+	0.961524i	$0.411415\pi$
$587$			9.42375e9i			1.92305i	−0.274720	+	0.961524i	$0.588585\pi$
$588$		−	1.54149e9i		−	0.312695i
$589$	−8.02219e9			−1.61767
$590$	1.23308e9	+	5.06043e9i	0.247178	+	1.01439i
$591$	4.20491e9			0.837916
$592$		−	4.40472e8i		−	0.0872555i
$593$		−	2.34902e9i		−	0.462589i	−0.972884	−	0.231294i	$-0.925704\pi$
$593$		−	2.34902e9i		−	0.462589i	0.972884	−	0.231294i	$-0.0742960\pi$
$594$	−8.59336e8			−0.168233
$595$	−1.73257e9	+	4.22177e8i	−0.337195	+	0.0821646i
$596$	−1.04961e8			−0.0203079
$597$			4.03807e9i			0.776718i
$598$		−	3.49821e9i		−	0.668947i
$599$	−2.66914e9			−0.507431			−0.253715	−	0.967279i	$-0.581653\pi$
$599$	−2.66914e9			−0.507431			−0.253715	+	0.967279i	$0.581653\pi$
$600$	9.58937e8	−	4.96830e8i	0.181243	−	0.0939028i
$601$	−1.24294e9			−0.233555			−0.116777	−	0.993158i	$-0.537256\pi$
$601$	−1.24294e9			−0.233555			−0.116777	+	0.993158i	$0.537256\pi$
$602$		−	7.60571e9i		−	1.42086i
$603$		−	1.53008e8i		−	0.0284186i
$604$	2.98109e9			0.550485
$605$	2.79586e9	−	6.81271e8i	0.513301	−	0.125077i
$606$	−6.15310e8			−0.112315
$607$			1.29515e9i			0.235050i	0.993070	+	0.117525i	$0.0374960\pi$
$607$			1.29515e9i			0.235050i	−0.993070	+	0.117525i	$0.962504\pi$
$608$			1.07417e9i			0.193826i
$609$	−2.59613e9			−0.465764
$610$	−1.39519e9	−	5.72569e9i	−0.248873	−	1.02135i
$611$	4.19792e9			0.744543
$612$		−	2.27258e8i		−	0.0400764i
$613$		−	5.13495e9i		−	0.900377i	−0.892934	−	0.450188i	$-0.851357\pi$
$613$		−	5.13495e9i		−	0.900377i	0.892934	−	0.450188i	$-0.148643\pi$
$614$	3.68681e9			0.642779
$615$	−7.08893e8	−	2.90922e9i	−0.122890	−	0.504329i
$616$	−3.65983e9			−0.630853
$617$		−	1.20368e9i		−	0.206306i	−0.994665	−	0.103153i	$-0.967107\pi$
$617$		−	1.20368e9i		−	0.206306i	0.994665	−	0.103153i	$-0.0328932\pi$
$618$			3.22628e9i			0.549848i
$619$	9.74534e9			1.65150			0.825752	−	0.564034i	$-0.190751\pi$
$619$	9.74534e9			1.65150			0.825752	+	0.564034i	$0.190751\pi$
$620$	−4.25322e9	+	1.03639e9i	−0.716716	+	0.174643i
$621$	1.64325e9			0.275350
$622$		−	6.11125e9i		−	1.01827i
$623$			1.34536e10i			2.22910i
$624$	5.79250e8			0.0954377
$625$	3.52019e9	−	4.98609e9i	0.576749	−	0.816922i
$626$	3.97577e9			0.647755
$627$		−	4.83026e9i		−	0.782589i
$628$		−	3.97990e9i		−	0.641230i
$629$	−5.23805e8			−0.0839252
$630$	2.07442e9	−	5.05477e8i	0.330526	−	0.0805395i
$631$	−9.82029e9			−1.55604			−0.778021	−	0.628238i	$-0.783776\pi$
$631$	−9.82029e9			−1.55604			−0.778021	+	0.628238i	$0.783776\pi$
$632$			4.44715e9i			0.700764i
$633$			1.30549e8i			0.0204578i
$634$	−3.35092e9			−0.522218
$635$	−2.81211e8	−	1.15406e9i	−0.0435837	−	0.178863i
$636$	2.11499e9			0.325993
$637$		−	4.67240e9i		−	0.716229i
$638$			3.20498e9i			0.488599i
$639$	−2.74525e9			−0.416226
$640$	1.38773e8	+	5.69508e8i	0.0209254	+	0.0858756i
$641$	−5.05603e9			−0.758240			−0.379120	−	0.925347i	$-0.623773\pi$
$641$	−5.05603e9			−0.758240			−0.379120	+	0.925347i	$0.623773\pi$
$642$		−	2.25325e8i		−	0.0336075i
$643$			1.13650e10i			1.68590i	0.537990	+	0.842951i	$0.319184\pi$
$643$			1.13650e10i			1.68590i	−0.537990	+	0.842951i	$0.680816\pi$
$644$	6.99847e9			1.03253
$645$	5.32199e9	−	1.29682e9i	0.780936	−	0.190292i
$646$	1.27740e9			0.186428
$647$			1.08126e8i			0.0156952i	0.999969	+	0.00784758i	$0.00249799\pi$
$647$			1.08126e8i			0.0156952i	−0.999969	+	0.00784758i	$0.997502\pi$
$648$			2.72098e8i			0.0392837i
$649$	1.27118e10			1.82538
$650$	2.90662e9	−	1.50594e9i	0.415137	−	0.215085i
$651$	−8.65449e9			−1.22944
$652$		−	3.75593e9i		−	0.530702i
$653$		−	3.37806e9i		−	0.474757i	−0.971417	−	0.237378i	$-0.923712\pi$
$653$		−	3.37806e9i		−	0.474757i	0.971417	−	0.237378i	$-0.0762882\pi$
$654$	1.12986e9			0.157944
$655$	1.51846e9	−	3.70004e8i	0.211134	−	0.0514472i
$656$	1.62518e9			0.224770
$657$		−	2.05536e9i		−	0.282755i
$658$			8.39829e9i			1.14921i
$659$	2.43494e9			0.331428			0.165714	−	0.986174i	$-0.447007\pi$
$659$	2.43494e9			0.331428			0.165714	+	0.986174i	$0.447007\pi$
$660$	−6.24021e8	−	2.56092e9i	−0.0844881	−	0.346730i
$661$	6.61695e9			0.891154			0.445577	−	0.895244i	$-0.352999\pi$
$661$	6.61695e9			0.891154			0.445577	+	0.895244i	$0.352999\pi$
$662$			6.40210e9i			0.857668i
$663$		−	6.88838e8i		−	0.0917951i
$664$	2.95586e9			0.391829
$665$	2.84124e9	+	1.16602e10i	0.374656	+	1.53755i
$666$	6.27157e8			0.0822653
$667$		−	6.12868e9i		−	0.799700i
$668$			1.75920e9i			0.228348i
$669$	7.06977e9			0.912881
$670$	4.55980e8	−	1.11109e8i	0.0585711	−	0.0142721i
$671$	−1.43830e10			−1.83790
$672$			1.15884e9i			0.147309i
$673$		−	4.76258e9i		−	0.602267i	−0.953582	−	0.301134i	$-0.902635\pi$
$673$		−	4.76258e9i		−	0.602267i	0.953582	−	0.301134i	$-0.0973650\pi$
$674$	−1.10828e10			−1.39425
$675$	7.07401e8	+	1.36536e9i	0.0885324	+	0.170877i
$676$	−2.26015e9			−0.281400
$677$		−	2.85988e9i		−	0.354232i	−0.984190	−	0.177116i	$-0.943323\pi$
$677$		−	2.85988e9i		−	0.354232i	0.984190	−	0.177116i	$-0.0566768\pi$
$678$		−	3.41674e9i		−	0.421025i
$679$	6.80688e9			0.834456
$680$	6.77253e8	−	1.65027e8i	0.0825980	−	0.0201268i
$681$	3.47272e9			0.421362
$682$			1.06841e10i			1.28972i
$683$			5.11045e9i			0.613744i	0.951751	+	0.306872i	$0.0992823\pi$
$683$			5.11045e9i			0.613744i	−0.951751	+	0.306872i	$0.900718\pi$
$684$	−1.52944e9			−0.182741
$685$	1.46355e9	+	6.00627e9i	0.173977	+	0.713983i
$686$	7.18036e8			0.0849204
$687$			2.73955e9i			0.322352i
$688$			2.97304e9i			0.348049i
$689$	6.41071e9			0.746688
$690$	1.19328e9	+	4.89708e9i	0.138283	+	0.567500i
$691$	9.23300e9			1.06456			0.532279	−	0.846569i	$-0.321336\pi$
$691$	9.23300e9			1.06456			0.532279	+	0.846569i	$0.321336\pi$
$692$			3.96475e9i			0.454825i
$693$		−	5.21097e9i		−	0.594774i
$694$	−7.70304e9			−0.874791
$695$	6.92865e9	−	1.68831e9i	0.782892	−	0.190768i
$696$	1.01482e9			0.114092
$697$		−	1.93265e9i		−	0.216192i
$698$		−	4.32041e9i		−	0.480874i
$699$	−7.41739e9			−0.821450
$700$	3.01275e9	+	5.81495e9i	0.331986	+	0.640770i
$701$	−4.27775e9			−0.469031			−0.234516	−	0.972112i	$-0.575350\pi$
$701$	−4.27775e9			−0.469031			−0.234516	+	0.972112i	$0.575350\pi$
$702$			8.24752e8i			0.0899795i
$703$			3.52520e9i			0.382684i
$704$	1.43061e9			0.154532
$705$	−5.87659e9	+	1.43196e9i	−0.631631	+	0.153910i
$706$	−8.20790e9			−0.877841
$707$		−	3.73121e9i		−	0.397083i
$708$		−	4.02505e9i		−	0.426240i
$709$	−6.00248e9			−0.632512			−0.316256	−	0.948674i	$-0.602426\pi$
$709$	−6.00248e9			−0.632512			−0.316256	+	0.948674i	$0.602426\pi$
$710$	−1.99351e9	−	8.18115e9i	−0.209033	−	0.857848i
$711$	−6.33197e9			−0.660686
$712$		−	5.25894e9i		−	0.546032i
$713$		−	2.04306e10i		−	2.11090i
$714$	1.37808e9			0.141687
$715$	−1.89146e9	−	7.76236e9i	−0.193520	−	0.794187i
$716$	6.70824e8			0.0682988
$717$		−	1.02022e9i		−	0.103366i
$718$		−	5.97907e9i		−	0.602834i
$719$	−1.01189e9			−0.101527			−0.0507634	−	0.998711i	$-0.516165\pi$
$719$	−1.01189e9			−0.101527			−0.0507634	+	0.998711i	$0.516165\pi$
$720$	−8.10882e8	+	1.97588e8i	−0.0809643	+	0.0197287i
$721$	−1.95640e10			−1.94395
$722$		−	1.44588e9i		−	0.142973i
$723$		−	4.60333e9i		−	0.452989i
$724$	6.17310e9			0.604530
$725$	5.09225e9	−	2.63832e9i	0.496280	−	0.257125i
$726$	−2.22382e9			−0.215686
$727$		−	1.27391e10i		−	1.22961i	−0.788678	−	0.614807i	$-0.789234\pi$
$727$		−	1.27391e10i		−	1.22961i	0.788678	−	0.614807i	$-0.210766\pi$
$728$			3.51254e9i			0.337413i
$729$	−3.87420e8			−0.0370370
$730$	6.12521e9	−	1.49254e9i	0.582762	−	0.142002i
$731$	3.53550e9			0.334765
$732$			4.55420e9i			0.429164i
$733$			1.02714e10i			0.963314i	0.876360	+	0.481657i	$0.159965\pi$
$733$			1.02714e10i			0.963314i	−0.876360	+	0.481657i	$0.840035\pi$
$734$	−3.82374e9			−0.356905
$735$	1.59381e9	+	6.54081e9i	0.148057	+	0.607611i
$736$	−2.73567e9			−0.252925
$737$		−	1.14543e9i		−	0.105398i
$738$			2.31398e9i			0.211916i
$739$	−7.46828e9			−0.680714			−0.340357	−	0.940296i	$-0.610548\pi$
$739$	−7.46828e9			−0.680714			−0.340357	+	0.940296i	$0.610548\pi$
$740$	4.55421e8	+	1.86900e9i	0.0413144	+	0.169550i
$741$	−4.63587e9			−0.418569
$742$			1.28252e10i			1.15252i
$743$		−	1.13615e10i		−	1.01619i	−0.861302	−	0.508093i	$-0.830351\pi$
$743$		−	1.13615e10i		−	1.01619i	0.861302	−	0.508093i	$-0.169649\pi$
$744$	3.38300e9			0.301160
$745$	4.45366e8	−	1.08523e8i	0.0394612	−	0.00961556i
$746$	−1.13297e10			−0.999155
$747$			4.20864e9i			0.369420i
$748$		−	1.70127e9i		−	0.148634i
$749$	1.36636e9			0.118817
$750$	−3.55524e9	+	3.09961e9i	−0.307719	+	0.268283i
$751$	9.08552e9			0.782726			0.391363	−	0.920236i	$-0.372004\pi$
$751$	9.08552e9			0.782726			0.391363	+	0.920236i	$0.372004\pi$
$752$		−	3.28285e9i		−	0.281507i
$753$			9.20916e9i			0.786027i
$754$	3.07600e9			0.261328
$755$	−1.26492e10	+	3.08226e9i	−1.06967	+	0.260648i
$756$	−1.64999e9			−0.138885
$757$			1.00537e10i			0.842348i	0.906980	+	0.421174i	$0.138382\pi$
$757$			1.00537e10i			0.842348i	−0.906980	+	0.421174i	$0.861618\pi$
$758$			3.39047e9i			0.282760i
$759$	1.23015e10			1.02120
$760$	−1.11063e9	−	4.55790e9i	−0.0917744	−	0.376632i
$761$	1.04527e9			0.0859768			0.0429884	−	0.999076i	$-0.486312\pi$
$761$	1.04527e9			0.0859768			0.0429884	+	0.999076i	$0.486312\pi$
$762$			9.17934e8i			0.0751570i
$763$			6.85142e9i			0.558399i
$764$	−6.21728e9			−0.504398
$765$	2.34970e8	+	9.64292e8i	0.0189757	+	0.0778742i
$766$	6.85204e9			0.550832
$767$		−	1.22003e10i		−	0.976305i
$768$		−	4.52985e8i		−	0.0360844i
$769$	−1.24461e10			−0.986939			−0.493470	−	0.869763i	$-0.664272\pi$
$769$	−1.24461e10			−0.986939			−0.493470	+	0.869763i	$0.664272\pi$
$770$	1.55293e10	−	3.78403e9i	1.22584	−	0.298702i
$771$	5.25724e9			0.413112
$772$			3.31209e9i			0.259085i
$773$		−	2.86975e9i		−	0.223469i	−0.993738	−	0.111734i	$-0.964359\pi$
$773$		−	2.86975e9i		−	0.223469i	0.993738	−	0.111734i	$-0.0356405\pi$
$774$	−4.23310e9			−0.328144
$775$	1.69756e10	−	8.79513e9i	1.30999	−	0.678714i
$776$	−2.66078e9			−0.204405
$777$			3.80305e9i			0.290843i
$778$		−	8.71459e8i		−	0.0663465i
$779$	−1.30067e10			−0.985795
$780$	−2.45785e9	+	5.98908e8i	−0.185449	+	0.0451886i
$781$	−2.05511e10			−1.54368
$782$			3.25323e9i			0.243271i
$783$			1.44492e9i			0.107567i
$784$	−3.65391e9			−0.270802
$785$	4.11497e9	+	1.68874e10i	0.303615	+	1.24600i
$786$	−1.20778e9			−0.0887171
$787$			1.10693e10i			0.809487i	0.914430	+	0.404743i	$0.132639\pi$
$787$			1.10693e10i			0.809487i	−0.914430	+	0.404743i	$0.867361\pi$
$788$		−	9.96719e9i		−	0.725657i
$789$	1.31112e10			0.950329
$790$	−4.59807e9	−	1.88700e10i	−0.331803	−	1.36169i
$791$	2.07189e10			1.48850
$792$			2.03694e9i			0.145694i
$793$			1.38042e10i			0.983002i
$794$	1.71198e10			1.21374
$795$	−8.97424e9	+	2.18676e9i	−0.633451	+	0.154354i
$796$	9.57172e9			0.672658
$797$			9.16190e9i			0.641034i	0.947243	+	0.320517i	$0.103857\pi$
$797$			9.16190e9i			0.641034i	−0.947243	+	0.320517i	$0.896143\pi$
$798$		−	9.27445e9i		−	0.646067i
$799$	−3.90393e9			−0.270763
$800$	−1.17767e9	−	2.27304e9i	−0.0813222	−	0.156961i
$801$	7.48782e9			0.514804
$802$			7.20369e9i			0.493111i
$803$		−	1.53866e10i		−	1.04867i
$804$	−3.62685e8			−0.0246112
$805$	−2.96957e10	+	7.23598e9i	−2.00635	+	0.488890i
$806$	1.02542e10			0.689807
$807$		−	7.82308e8i		−	0.0523987i
$808$			1.45851e9i			0.0972681i
$809$	7.65555e9			0.508343			0.254171	−	0.967159i	$-0.418197\pi$
$809$	7.65555e9			0.508343			0.254171	+	0.967159i	$0.418197\pi$
$810$	−2.81332e8	−	1.15456e9i	−0.0186004	−	0.0763339i
$811$	1.37797e10			0.907121			0.453561	−	0.891225i	$-0.350154\pi$
$811$	1.37797e10			0.907121			0.453561	+	0.891225i	$0.350154\pi$
$812$			6.15380e9i			0.403364i
$813$		−	8.16641e9i		−	0.532984i
$814$	4.69494e9			0.305102
$815$	3.88340e9	+	1.59370e10i	0.251281	+	1.03123i
$816$	−5.38685e8			−0.0347072
$817$		−	2.37939e10i		−	1.52647i
$818$		−	1.72114e10i		−	1.09946i
$819$	−5.00126e9			−0.318116
$820$	−6.89593e9	+	1.68034e9i	−0.436761	+	0.106426i
$821$	3.07675e10			1.94040			0.970200	−	0.242304i	$-0.0779032\pi$
$821$	3.07675e10			1.94040			0.970200	+	0.242304i	$0.0779032\pi$
$822$		−	4.77736e9i		−	0.300011i
$823$			2.78795e10i			1.74335i	0.490081	+	0.871677i	$0.336967\pi$
$823$			2.78795e10i			1.74335i	−0.490081	+	0.871677i	$0.663033\pi$
$824$	7.64748e9			0.476182
$825$	5.29565e9	+	1.02212e10i	0.328345	+	0.633742i
$826$	2.44077e10			1.50694
$827$		−	1.30484e10i		−	0.802211i	−0.916032	−	0.401105i	$-0.868626\pi$
$827$		−	1.30484e10i		−	0.802211i	0.916032	−	0.401105i	$-0.131374\pi$
$828$		−	3.89512e9i		−	0.238460i
$829$	2.37741e10			1.44932			0.724659	−	0.689108i	$-0.241997\pi$
$829$	2.37741e10			1.44932			0.724659	+	0.689108i	$0.241997\pi$
$830$	−1.25422e10	+	3.05618e9i	−0.761380	+	0.185526i
$831$	4.81280e9			0.290934
$832$		−	1.37304e9i		−	0.0826515i
$833$			4.34519e9i			0.260466i
$834$	−5.51103e9			−0.328966
$835$	−1.81890e9	−	7.46458e9i	−0.108120	−	0.443713i
$836$	−1.14495e10			−0.677742
$837$			4.81681e9i			0.283936i
$838$			6.36858e8i			0.0373842i
$839$	−1.02523e10			−0.599315			−0.299658	−	0.954047i	$-0.596872\pi$
$839$	−1.02523e10			−0.599315			−0.299658	+	0.954047i	$0.596872\pi$
$840$	−1.19817e9	−	4.91715e9i	−0.0697493	−	0.286243i
$841$	−1.18609e10			−0.687593
$842$		−	2.63760e9i		−	0.152271i
$843$		−	5.98015e8i		−	0.0343808i
$844$	3.09448e8			0.0177170
$845$	9.59018e9	−	2.33685e9i	0.546800	−	0.133239i
$846$	4.67422e9			0.265407
$847$		−	1.34851e10i		−	0.762541i
$848$		−	5.01330e9i		−	0.282318i
$849$	2.91868e9			0.163685
$850$	−2.70307e9	+	1.40047e9i	−0.150970	+	0.0782184i
$851$	−8.97785e9			−0.499366
$852$			6.50726e9i			0.360462i
$853$		−	1.30152e10i		−	0.718008i	−0.933336	−	0.359004i	$-0.883116\pi$
$853$		−	1.30152e10i		−	0.718008i	0.933336	−	0.359004i	$-0.116884\pi$
$854$	−2.76164e10			−1.51728
$855$	6.48967e9	−	1.58134e9i	0.355092	−	0.0865257i
$856$	−5.34103e8			−0.0291050
$857$			3.47955e10i			1.88838i	0.329396	+	0.944192i	$0.393155\pi$
$857$			3.47955e10i			1.88838i	−0.329396	+	0.944192i	$0.606845\pi$
$858$			6.17415e9i			0.333712i
$859$	2.22903e9			0.119989			0.0599943	−	0.998199i	$-0.480892\pi$
$859$	2.22903e9			0.119989			0.0599943	+	0.998199i	$0.480892\pi$
$860$	−3.07393e9	−	1.26151e10i	−0.164797	−	0.676310i
$861$	−1.40319e10			−0.749212
$862$		−	4.45915e9i		−	0.237124i
$863$			2.28044e10i			1.20776i	0.797075	+	0.603880i	$0.206379\pi$
$863$			2.28044e10i			1.20776i	−0.797075	+	0.603880i	$0.793621\pi$
$864$	6.44973e8			0.0340207
$865$	−4.09930e9	−	1.68231e10i	−0.215354	−	0.883790i
$866$	−8.04497e9			−0.420932
$867$		−	1.04385e10i		−	0.543968i
$868$			2.05143e10i			1.06473i
$869$	−4.74016e10			−2.45033
$870$	−4.30603e9	+	1.04926e9i	−0.221697	+	0.0540212i
$871$	−1.09933e9			−0.0563721
$872$		−	2.67819e9i		−	0.136783i
$873$		−	3.78849e9i		−	0.192715i
$874$	2.18942e10			1.10927
$875$	−1.87959e10	−	2.15588e10i	−0.948495	−	1.08792i
$876$	−4.87198e9			−0.244873
$877$		−	4.21151e9i		−	0.210833i	−0.994428	−	0.105417i	$-0.966382\pi$
$877$		−	4.21151e9i		−	0.210833i	0.994428	−	0.105417i	$-0.0336176\pi$
$878$			1.95125e10i			0.972932i
$879$	−1.37253e10			−0.681650
$880$	−6.07032e9	+	1.47916e9i	−0.300277	+	0.0731689i
$881$	2.60856e10			1.28524			0.642622	−	0.766184i	$-0.277847\pi$
$881$	2.60856e10			1.28524			0.642622	+	0.766184i	$0.277847\pi$
$882$		−	5.20254e9i		−	0.255314i
$883$			1.24090e9i			0.0606560i	0.999540	+	0.0303280i	$0.00965519\pi$
$883$			1.24090e9i			0.0606560i	−0.999540	+	0.0303280i	$0.990345\pi$
$884$	−1.63280e9			−0.0794969
$885$	4.16164e9	+	1.70789e10i	0.201820	+	0.828246i
$886$	9.27678e9			0.448104
$887$		−	2.74043e10i		−	1.31852i	−0.751915	−	0.659260i	$-0.770870\pi$
$887$		−	2.74043e10i		−	1.31852i	0.751915	−	0.659260i	$-0.229130\pi$
$888$		−	1.48659e9i		−	0.0712438i
$889$	−5.56631e9			−0.265712
$890$	5.43741e9	+	2.23145e10i	0.258540	+	1.06102i
$891$	−2.90026e9			−0.137361
$892$		−	1.67580e10i		−	0.790578i
$893$			2.62734e10i			1.23463i
$894$	−3.54243e8			−0.0165813
$895$	−2.84642e9	+	6.93590e8i	−0.132714	+	0.0323387i
$896$	2.74688e9			0.127574
$897$		−	1.18065e10i		−	0.546193i
$898$			5.05629e9i			0.233005i
$899$	1.79648e10			0.824637
$900$	3.23641e9	−	1.67680e9i	0.147984	−	0.0766713i
$901$	−5.96176e9			−0.271543
$902$			1.73226e10i			0.785943i
$903$		−	2.56693e10i		−	1.16013i
$904$	−8.09894e9			−0.364619
$905$	−2.61935e10	+	6.38259e9i	−1.17469	+	0.286238i
$906$	1.00612e10			0.449469
$907$		−	1.54071e10i		−	0.685638i	−0.939402	−	0.342819i	$-0.888618\pi$
$907$		−	1.54071e10i		−	0.685638i	0.939402	−	0.342819i	$-0.111382\pi$
$908$		−	8.23163e9i		−	0.364910i
$909$	−2.07667e9			−0.0917052
$910$	−3.63175e9	−	1.49043e10i	−0.159761	−	0.655642i
$911$	1.20985e9			0.0530171			0.0265085	−	0.999649i	$-0.491561\pi$
$911$	1.20985e9			0.0530171			0.0265085	+	0.999649i	$0.491561\pi$
$912$			3.62534e9i			0.158258i
$913$			3.15062e10i			1.37009i
$914$	1.42828e10			0.618729
$915$	−4.70875e9	−	1.93242e10i	−0.203204	−	0.833927i
$916$	6.49374e9			0.279165
$917$		−	7.32389e9i		−	0.313653i
$918$		−	7.66994e8i		−	0.0327222i
$919$	−1.95758e10			−0.831986			−0.415993	−	0.909368i	$-0.636566\pi$
$919$	−1.95758e10			−0.831986			−0.415993	+	0.909368i	$0.636566\pi$
$920$	1.16079e10	−	2.82851e9i	0.491469	−	0.119757i
$921$	1.24430e10			0.524827
$922$			9.85913e9i			0.414267i
$923$			1.97241e10i			0.825640i
$924$	−1.23519e10			−0.515090
$925$	−3.86485e9	−	7.45959e9i	−0.160560	−	0.309898i
$926$	−2.61747e10			−1.08329
$927$			1.08887e10i			0.448949i
$928$		−	2.40549e9i		−	0.0988065i
$929$	−2.43152e10			−0.994999			−0.497500	−	0.867464i	$-0.665748\pi$
$929$	−2.43152e10			−0.994999			−0.497500	+	0.867464i	$0.665748\pi$
$930$	−1.43546e10	+	3.49781e9i	−0.585197	+	0.142596i
$931$	2.92430e10			1.18768
$932$			1.75820e10i			0.711397i
$933$		−	2.06255e10i		−	0.831416i
$934$	−3.36714e9			−0.135222
$935$	1.75900e9	+	7.21876e9i	0.0703762	+	0.288816i
$936$	1.95497e9			0.0779245
$937$		−	2.87271e9i		−	0.114078i	−0.998372	−	0.0570391i	$-0.981834\pi$
$937$		−	2.87271e9i		−	0.114078i	0.998372	−	0.0570391i	$-0.0181660\pi$
$938$		−	2.19930e9i		−	0.0870111i
$939$	1.34182e10			0.528890
$940$	3.39427e9	+	1.39297e10i	0.133290	+	0.547009i
$941$	3.33166e10			1.30346			0.651729	−	0.758452i	$-0.274044\pi$
$941$	3.33166e10			1.30346			0.651729	+	0.758452i	$0.274044\pi$
$942$		−	1.34322e10i		−	0.523562i
$943$		−	3.31250e10i		−	1.28637i
$944$	−9.54085e9			−0.369135
$945$	7.00117e9	−	1.70598e9i	0.269873	−	0.0657603i
$946$	−3.16893e10			−1.21701
$947$			5.07496e9i			0.194182i	0.995276	+	0.0970908i	$0.0309537\pi$
$947$			5.07496e9i			0.194182i	−0.995276	+	0.0970908i	$0.969046\pi$
$948$			1.50091e10i			0.572171i
$949$	−1.47674e10			−0.560882
$950$	9.42517e9	+	1.81916e10i	0.356662	+	0.688396i
$951$	−1.13094e10			−0.426389
$952$		−	3.26656e9i		−	0.122705i
$953$		−	2.69161e10i		−	1.00737i	−0.863889	−	0.503683i	$-0.831978\pi$
$953$		−	2.69161e10i		−	1.00737i	0.863889	−	0.503683i	$-0.168022\pi$
$954$	7.13808e9			0.266172
$955$	2.63809e10	−	6.42827e9i	0.980118	−	0.238826i
$956$	−2.41831e9			−0.0895177
$957$			1.08168e10i			0.398940i
$958$		−	2.33200e8i		−	0.00856937i
$959$	2.89697e10			1.06067
$960$	4.68358e8	+	1.92209e9i	0.0170855	+	0.0701172i
$961$	3.23749e10			1.17673
$962$		−	4.50600e9i		−	0.163184i
$963$		−	7.60471e8i		−	0.0274404i
$964$	−1.09116e10			−0.392300
$965$	−3.42449e9	−	1.40537e10i	−0.122673	−	0.503439i
$966$	2.36198e10			0.843057
$967$			1.62053e10i			0.576320i	0.957582	+	0.288160i	$0.0930434\pi$
$967$			1.62053e10i			0.576320i	−0.957582	+	0.288160i	$0.906957\pi$
$968$			5.27127e9i			0.186789i
$969$	4.31121e9			0.152218
$970$	1.12901e10	−	2.75108e9i	0.397189	−	0.0967836i
$971$	−1.58919e10			−0.557068			−0.278534	−	0.960426i	$-0.589848\pi$
$971$	−1.58919e10			−0.557068			−0.278534	+	0.960426i	$0.589848\pi$
$972$			9.18330e8i			0.0320750i
$973$		−	3.34186e10i		−	1.16304i
$974$	2.07039e10			0.717951
$975$	9.80985e9	−	5.08253e9i	0.338958	−	0.175616i
$976$	1.07951e10			0.371667
$977$			2.98228e10i			1.02310i	0.859254	+	0.511549i	$0.170928\pi$
$977$			2.98228e10i			1.02310i	−0.859254	+	0.511549i	$0.829072\pi$
$978$		−	1.26763e10i		−	0.433317i
$979$	5.60544e10			1.90928
$980$	1.55041e10	−	3.77791e9i	0.526207	−	0.128221i
$981$	3.81328e9			0.128961
$982$			1.60925e10i			0.542293i
$983$		−	3.04567e10i		−	1.02270i	−0.859374	−	0.511348i	$-0.829146\pi$
$983$		−	3.04567e10i		−	1.02270i	0.859374	−	0.511348i	$-0.170854\pi$
$984$	5.48500e9			0.183524
$985$	1.03055e10	+	4.22925e10i	0.343590	+	1.41006i
$986$	−2.86058e9			−0.0950354
$987$			2.83442e10i			0.938328i
$988$			1.09887e10i			0.362491i
$989$	6.05974e10			1.99190
$990$	−2.10607e9	−	8.64309e9i	−0.0689843	−	0.283104i
$991$	−4.03364e10			−1.31656			−0.658278	−	0.752775i	$-0.728715\pi$
$991$	−4.03364e10			−1.31656			−0.658278	+	0.752775i	$0.728715\pi$
$992$		−	8.01896e9i		−	0.260812i
$993$			2.16071e10i			0.700283i
$994$	−3.94597e10			−1.27439
$995$	−4.06144e10	+	9.89655e9i	−1.30707	+	0.318495i
$996$	9.97604e9			0.319927
$997$		−	1.00858e10i		−	0.322312i	−0.986929	−	0.161156i	$-0.948478\pi$
$997$		−	1.00858e10i		−	0.322312i	0.986929	−	0.161156i	$-0.0515221\pi$
$998$		−	5.05009e9i		−	0.160821i
$999$	2.11666e9			0.0671693

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	30.8.c.b.19.2	✓	4
3.2	odd	2		90.8.c.b.19.3		4
4.3	odd	2		240.8.f.d.49.4		4
5.2	odd	4		150.8.a.s.1.2		2
5.3	odd	4		150.8.a.r.1.1		2
5.4	even	2	inner	30.8.c.b.19.4	yes	4
15.2	even	4		450.8.a.be.1.2		2
15.8	even	4		450.8.a.bh.1.1		2
15.14	odd	2		90.8.c.b.19.1		4
20.19	odd	2		240.8.f.d.49.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
30.8.c.b.19.2	✓	4	1.1	even	1	trivial
30.8.c.b.19.4	yes	4	5.4	even	2	inner
90.8.c.b.19.1		4	15.14	odd	2
90.8.c.b.19.3		4	3.2	odd	2
150.8.a.r.1.1		2	5.3	odd	4
150.8.a.s.1.2		2	5.2	odd	4
240.8.f.d.49.2		4	20.19	odd	2
240.8.f.d.49.4		4	4.3	odd	2
450.8.a.be.1.2		2	15.2	even	4
450.8.a.bh.1.1		2	15.8	even	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:	\( N \)	\(=\)	\( 30 = 2 \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	30.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-8.00000i q^{2} -27.0000i q^{3} -64.0000 q^{4} +(271.563 - 66.1720i) q^{5} -216.000 q^{6} -1309.81i q^{7} +512.000i q^{8} -729.000 q^{9} +(-529.376 - 2172.50i) q^{10} -5457.35 q^{11} +1728.00i q^{12} +5237.72i q^{13} -10478.5 q^{14} +(-1786.64 - 7332.19i) q^{15} +4096.00 q^{16} -4870.92i q^{17} +5832.00i q^{18} -32781.2 q^{19} +(-17380.0 + 4235.01i) q^{20} -35365.0 q^{21} +43658.8i q^{22} -83486.0i q^{23} +13824.0 q^{24} +(69367.5 - 35939.7i) q^{25} +41901.8 q^{26} +19683.0i q^{27} +83828.0i q^{28} +73409.7 q^{29} +(-58657.5 + 14293.1i) q^{30} +244719. q^{31} -32768.0i q^{32} +147348. i q^{33} -38967.3 q^{34} +(-86672.9 - 355696. i) q^{35} +46656.0 q^{36} -107537. i q^{37} +262250. i q^{38} +141418. q^{39} +(33880.1 + 139040. i) q^{40} +396774. q^{41} +282920. i q^{42} +725839. i q^{43} +349270. q^{44} +(-197969. + 48239.4i) q^{45} -667888. q^{46} -801478. i q^{47} -110592. i q^{48} -892068. q^{49} +(-287517. - 554940. i) q^{50} -131515. q^{51} -335214. i q^{52} -1.22395e6i q^{53} +157464. q^{54} +(-1.48201e6 + 361123. i) q^{55} +670624. q^{56} +885092. i q^{57} -587278. i q^{58} -2.32931e6 q^{59} +(114345. + 469260. i) q^{60} +2.63553e6 q^{61} -1.95775e6i q^{62} +954854. i q^{63} -262144. q^{64} +(346590. + 1.42237e6i) q^{65} +1.17879e6 q^{66} +209887. i q^{67} +311739. i q^{68} -2.25412e6 q^{69} +(-2.84557e6 + 693383. i) q^{70} +3.76578e6 q^{71} -373248. i q^{72} +2.81943e6i q^{73} -860298. q^{74} +(-970371. - 1.87292e6i) q^{75} +2.09800e6 q^{76} +7.14810e6i q^{77} -1.13135e6i q^{78} +8.68583e6 q^{79} +(1.11232e6 - 271040. i) q^{80} +531441. q^{81} -3.17419e6i q^{82} -5.77317e6i q^{83} +2.26336e6 q^{84} +(-322318. - 1.32276e6i) q^{85} +5.80672e6 q^{86} -1.98206e6i q^{87} -2.79416e6i q^{88} -1.02714e7 q^{89} +(385915. + 1.58375e6i) q^{90} +6.86043e6 q^{91} +5.34310e6i q^{92} -6.60742e6i q^{93} -6.41183e6 q^{94} +(-8.90215e6 + 2.16920e6i) q^{95} -884736. q^{96} +5.19683e6i q^{97} +7.13654e6i q^{98} +3.97841e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 256 q^{4} + 264 q^{5} - 864 q^{6} - 2916 q^{9} + 2816 q^{10} - 7440 q^{11} - 9024 q^{14} + 9504 q^{15} + 16384 q^{16} - 69456 q^{19} - 16896 q^{20} - 30456 q^{21} + 55296 q^{24} + 60396 q^{25} + 118272 q^{26}+ \cdots + 5423760 q^{99}+O(q^{100}) \) 4 * q - 256 * q^4 + 264 * q^5 - 864 * q^6 - 2916 * q^9 + 2816 * q^10 - 7440 * q^11 - 9024 * q^14 + 9504 * q^15 + 16384 * q^16 - 69456 * q^19 - 16896 * q^20 - 30456 * q^21 + 55296 * q^24 + 60396 * q^25 + 118272 * q^26 - 372384 * q^29 - 57024 * q^30 + 436192 * q^31 + 386816 * q^34 - 534576 * q^35 + 186624 * q^36 + 399168 * q^39 - 180224 * q^40 + 608616 * q^41 + 476160 * q^44 - 192456 * q^45 - 204800 * q^46 - 1249524 * q^49 - 1656576 * q^50 + 1305504 * q^51 + 629856 * q^54 - 3448960 * q^55 + 577536 * q^56 - 9117840 * q^59 - 608256 * q^60 + 11972840 * q^61 - 1048576 * q^64 - 350232 * q^65 + 1607040 * q^66 - 691200 * q^69 - 7356544 * q^70 + 18158880 * q^71 - 6121728 * q^74 - 5590944 * q^75 + 4445184 * q^76 + 3958272 * q^79 + 1081344 * q^80 + 2125764 * q^81 + 1949184 * q^84 - 14713336 * q^85 + 16780416 * q^86 - 15587448 * q^89 - 2052864 * q^90 + 10507488 * q^91 - 5617280 * q^94 - 17260896 * q^95 - 3538944 * q^96 + 5423760 * q^99

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