LMFDB - Embedded newform 25.8.b.c.24.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [25,8,Mod(24,25)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(25, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("25.24");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 25 = 5^{2} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	25.b (of order $2$, degree $1$, not 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:	$7.80962563710$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{19})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 9x^{2} + 25 $ x^4 - 9*x^2 + 25
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6}\cdot 5^{2} $
Twist minimal:	no (minimal twist has level 5)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			24.4
Root			$2.17945 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	25.24
Dual form			25.8.b.c.24.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+18.7178i q^{2} +59.7424i q^{3} -222.356 q^{4} -1118.25 q^{6} +438.197i q^{7} -1766.14i q^{8} -1382.15 q^{9} +O(q^{10})$	\(q+18.7178i q^{2} +59.7424i q^{3} -222.356 q^{4} -1118.25 q^{6} +438.197i q^{7} -1766.14i q^{8} -1382.15 q^{9} +5759.12 q^{11} -13284.1i q^{12} +3530.42i q^{13} -8202.08 q^{14} +4596.61 q^{16} -23991.9i q^{17} -25870.8i q^{18} -16590.3 q^{19} -26178.9 q^{21} +107798. i q^{22} +65626.9i q^{23} +105513. q^{24} -66081.7 q^{26} +48083.5i q^{27} -97435.6i q^{28} -134041. q^{29} +129002. q^{31} -140027. i q^{32} +344064. i q^{33} +449075. q^{34} +307330. q^{36} +161108. i q^{37} -310534. i q^{38} -210916. q^{39} -362989. q^{41} -490012. i q^{42} -588189. i q^{43} -1.28057e6 q^{44} -1.22839e6 q^{46} +343895. i q^{47} +274612. i q^{48} +631527. q^{49} +1.43333e6 q^{51} -785010. i q^{52} +1.66139e6i q^{53} -900018. q^{54} +773915. q^{56} -991144. i q^{57} -2.50896e6i q^{58} +2.54214e6 q^{59} +2.52337e6 q^{61} +2.41464e6i q^{62} -605655. i q^{63} +3.20936e6 q^{64} -6.44011e6 q^{66} +1.56618e6i q^{67} +5.33474e6i q^{68} -3.92071e6 q^{69} -299354. q^{71} +2.44107e6i q^{72} -312494. i q^{73} -3.01558e6 q^{74} +3.68895e6 q^{76} +2.52363e6i q^{77} -3.94788e6i q^{78} +1.95247e6 q^{79} -5.89539e6 q^{81} -6.79435e6i q^{82} +621372. i q^{83} +5.82104e6 q^{84} +1.10096e7 q^{86} -8.00795e6i q^{87} -1.01714e7i q^{88} -5.78298e6 q^{89} -1.54702e6 q^{91} -1.45925e7i q^{92} +7.70690e6i q^{93} -6.43696e6 q^{94} +8.36554e6 q^{96} +7.20152e6i q^{97} +1.18208e7i q^{98} -7.95998e6 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 192 q^{4} - 2032 q^{6} - 11108 q^{9}+O(q^{10}) $ 4 * q - 192 * q^4 - 2032 * q^6 - 11108 * q^9	$ 4 q - 192 q^{4} - 2032 q^{6} - 11108 q^{9} + 9088 q^{11} - 15024 q^{14} + 40704 q^{16} - 77520 q^{19} - 138192 q^{21} + 263040 q^{24} - 114032 q^{26} + 144520 q^{29} + 613648 q^{31} + 906736 q^{34} - 439616 q^{36} - 1549456 q^{39} + 528728 q^{41} - 2868224 q^{44} - 2606832 q^{46} + 2330828 q^{49} + 2332688 q^{51} - 2205920 q^{54} + 1898880 q^{56} + 2240240 q^{59} + 4514088 q^{61} + 10293248 q^{64} - 13128704 q^{66} + 1490544 q^{69} + 1243568 q^{71} - 5302544 q^{74} + 1775360 q^{76} - 8666080 q^{79} - 4795756 q^{81} + 796416 q^{84} + 21596368 q^{86} - 12051240 q^{89} - 10704592 q^{91} - 11721584 q^{94} + 26610688 q^{96} - 5781376 q^{99}+O(q^{100}) $ 4 * q - 192 * q^4 - 2032 * q^6 - 11108 * q^9 + 9088 * q^11 - 15024 * q^14 + 40704 * q^16 - 77520 * q^19 - 138192 * q^21 + 263040 * q^24 - 114032 * q^26 + 144520 * q^29 + 613648 * q^31 + 906736 * q^34 - 439616 * q^36 - 1549456 * q^39 + 528728 * q^41 - 2868224 * q^44 - 2606832 * q^46 + 2330828 * q^49 + 2332688 * q^51 - 2205920 * q^54 + 1898880 * q^56 + 2240240 * q^59 + 4514088 * q^61 + 10293248 * q^64 - 13128704 * q^66 + 1490544 * q^69 + 1243568 * q^71 - 5302544 * q^74 + 1775360 * q^76 - 8666080 * q^79 - 4795756 * q^81 + 796416 * q^84 + 21596368 * q^86 - 12051240 * q^89 - 10704592 * q^91 - 11721584 * q^94 + 26610688 * q^96 - 5781376 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/25\mathbb{Z}\right)^\times$.

$n$	$2$
$\chi(n)$	$-1$

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 $

$2$	18.7178i	1.65444i	0.561882	+	0.827218i	$0.310078\pi$
$2$	18.7178i	1.65444i	−0.561882	+	0.827218i	$0.689922\pi$
$3$	59.7424i	1.27749i	0.769418	+	0.638746i	$0.220547\pi$
$3$	59.7424i	1.27749i	−0.769418	+	0.638746i	$0.779453\pi$
$4$	−222.356	−1.73716
$5$	0	0
$5$	0	0
$6$	−1118.25	−2.11353
$7$	438.197i	0.482865i	0.970418	+	0.241433i	$0.0776173\pi$
$7$	438.197i	0.482865i	−0.970418	+	0.241433i	$0.922383\pi$
$8$	− 1766.14i	− 1.21958i
$9$	−1382.15	−0.631986
$10$	0	0
$11$	5759.12	1.30461	0.652306	−	0.757955i	$-0.273802\pi$
$11$	5759.12	1.30461	0.652306	+	0.757955i	$0.273802\pi$
$12$	− 13284.1i	− 2.21920i
$13$	3530.42i	0.445682i	0.974855	+	0.222841i	$0.0715330\pi$
$13$	3530.42i	0.445682i	−0.974855	+	0.222841i	$0.928467\pi$
$14$	−8202.08	−0.798869
$15$	0	0
$16$	4596.61	0.280555
$17$	− 23991.9i	− 1.18439i	−0.805797	−	0.592193i	$-0.798262\pi$
$17$	− 23991.9i	− 1.18439i	0.805797	−	0.592193i	$-0.201738\pi$
$18$	− 25870.8i	− 1.04558i
$19$	−16590.3	−0.554903	−0.277451	−	0.960740i	$-0.589490\pi$
$19$	−16590.3	−0.554903	−0.277451	+	0.960740i	$0.589490\pi$
$20$	0	0
$21$	−26178.9	−0.616856
$22$	107798.i	2.15840i
$23$	65626.9i	1.12469i	0.826902	+	0.562347i	$0.190101\pi$
$23$	65626.9i	1.12469i	−0.826902	+	0.562347i	$0.809899\pi$
$24$	105513.	1.55800
$25$	0	0
$26$	−66081.7	−0.737351
$27$	48083.5i	0.470136i
$28$	− 97435.6i	− 0.838812i
$29$	−134041.	−1.02058	−0.510289	−	0.860003i	$-0.670461\pi$
$29$	−134041.	−1.02058	−0.510289	+	0.860003i	$0.670461\pi$
$30$	0	0
$31$	129002.	0.777734	0.388867	−	0.921294i	$-0.372867\pi$
$31$	129002.	0.777734	0.388867	+	0.921294i	$0.372867\pi$
$32$	− 140027.i	− 0.755417i
$33$	344064.i	1.66663i
$34$	449075.	1.95949
$35$	0	0
$36$	307330.	1.09786
$37$	161108.i	0.522890i	0.965218	+	0.261445i	$0.0841990\pi$
$37$	161108.i	0.522890i	−0.965218	+	0.261445i	$0.915801\pi$
$38$	− 310534.i	− 0.918050i
$39$	−210916.	−0.569355
$40$	0	0
$41$	−362989.	−0.822526	−0.411263	−	0.911517i	$-0.634912\pi$
$41$	−362989.	−0.822526	−0.411263	+	0.911517i	$0.634912\pi$
$42$	− 490012.i	− 1.02055i
$43$	− 588189.i	− 1.12818i	−0.825714	−	0.564089i	$-0.809228\pi$
$43$	− 588189.i	− 1.12818i	0.825714	−	0.564089i	$-0.190772\pi$
$44$	−1.28057e6	−2.26632
$45$	0	0
$46$	−1.22839e6	−1.86073
$47$	343895.i	0.483151i	0.970382	+	0.241576i	$0.0776642\pi$
$47$	343895.i	0.483151i	−0.970382	+	0.241576i	$0.922336\pi$
$48$	274612.i	0.358406i
$49$	631527.	0.766841
$50$	0	0
$51$	1.43333e6	1.51304
$52$	− 785010.i	− 0.774219i
$53$	1.66139e6i	1.53287i	0.642320	+	0.766436i	$0.277972\pi$
$53$	1.66139e6i	1.53287i	−0.642320	+	0.766436i	$0.722028\pi$
$54$	−900018.	−0.777809
$55$	0	0
$56$	773915.	0.588891
$57$	− 991144.i	− 0.708884i
$58$	− 2.50896e6i	− 1.68848i
$59$	2.54214e6	1.61145	0.805726	−	0.592289i	$-0.201776\pi$
$59$	2.54214e6	1.61145	0.805726	+	0.592289i	$0.201776\pi$
$60$	0	0
$61$	2.52337e6	1.42340	0.711699	−	0.702484i	$-0.247926\pi$
$61$	2.52337e6	1.42340	0.711699	+	0.702484i	$0.247926\pi$
$62$	2.41464e6i	1.28671i
$63$	− 605655.i	− 0.305164i
$64$	3.20936e6	1.53034
$65$	0	0
$66$	−6.44011e6	−2.75734
$67$	1.56618e6i	0.636178i	0.948061	+	0.318089i	$0.103041\pi$
$67$	1.56618e6i	0.636178i	−0.948061	+	0.318089i	$0.896959\pi$
$68$	5.33474e6i	2.05746i
$69$	−3.92071e6	−1.43679
$70$	0	0
$71$	−299354.	−0.0992615	−0.0496307	−	0.998768i	$-0.515804\pi$
$71$	−299354.	−0.0992615	−0.0496307	+	0.998768i	$0.515804\pi$
$72$	2.44107e6i	0.770755i
$73$	− 312494.i	− 0.0940183i	−0.998894	−	0.0470091i	$-0.985031\pi$
$73$	− 312494.i	− 0.0940183i	0.998894	−	0.0470091i	$-0.0149690\pi$
$74$	−3.01558e6	−0.865087
$75$	0	0
$76$	3.68895e6	0.963952
$77$	2.52363e6i	0.629952i
$78$	− 3.94788e6i	− 0.941961i
$79$	1.95247e6	0.445542	0.222771	−	0.974871i	$-0.428490\pi$
$79$	1.95247e6	0.445542	0.222771	+	0.974871i	$0.428490\pi$
$80$	0	0
$81$	−5.89539e6	−1.23258
$82$	− 6.79435e6i	− 1.36082i
$83$	621372.i	0.119283i	0.998220	+	0.0596414i	$0.0189957\pi$
$83$	621372.i	0.119283i	−0.998220	+	0.0596414i	$0.981004\pi$
$84$	5.82104e6	1.07158
$85$	0	0
$86$	1.10096e7	1.86650
$87$	− 8.00795e6i	− 1.30378i
$88$	− 1.01714e7i	− 1.59108i
$89$	−5.78298e6	−0.869534	−0.434767	−	0.900543i	$-0.643169\pi$
$89$	−5.78298e6	−0.869534	−0.434767	+	0.900543i	$0.643169\pi$
$90$	0	0
$91$	−1.54702e6	−0.215204
$92$	− 1.45925e7i	− 1.95377i
$93$	7.70690e6i	0.993549i
$94$	−6.43696e6	−0.799343
$95$	0	0
$96$	8.36554e6	0.965039
$97$	7.20152e6i	0.801167i	0.916260	+	0.400584i	$0.131193\pi$
$97$	7.20152e6i	0.801167i	−0.916260	+	0.400584i	$0.868807\pi$
$98$	1.18208e7i	1.26869i
$99$	−7.95998e6	−0.824496
$100$	0	0
$101$	−2.91989e6	−0.281995	−0.140997	−	0.990010i	$-0.545031\pi$
$101$	−2.91989e6	−0.281995	−0.140997	+	0.990010i	$0.545031\pi$
$102$	2.68288e7i	2.50323i
$103$	− 3.94639e6i	− 0.355852i	−0.984044	−	0.177926i	$-0.943061\pi$
$103$	− 3.94639e6i	− 0.355852i	0.984044	−	0.177926i	$-0.0569388\pi$
$104$	6.23520e6	0.543543
$105$	0	0
$106$	−3.10976e7	−2.53604
$107$	− 3.81991e6i	− 0.301446i	−0.988576	−	0.150723i	$-0.951840\pi$
$107$	− 3.81991e6i	− 0.301446i	0.988576	−	0.150723i	$-0.0481602\pi$
$108$	− 1.06917e7i	− 0.816699i
$109$	8.82259e6	0.652534	0.326267	−	0.945278i	$-0.394209\pi$
$109$	8.82259e6	0.652534	0.326267	+	0.945278i	$0.394209\pi$
$110$	0	0
$111$	−9.62496e6	−0.667988
$112$	2.01422e6i	0.135470i
$113$	− 2.12074e7i	− 1.38265i	−0.722545	−	0.691324i	$-0.757028\pi$
$113$	− 2.12074e7i	− 1.38265i	0.722545	−	0.691324i	$-0.242972\pi$
$114$	1.85520e7	1.17280
$115$	0	0
$116$	2.98049e7	1.77290
$117$	− 4.87958e6i	− 0.281664i
$118$	4.75832e7i	2.66604i
$119$	1.05132e7	0.571898
$120$	0	0
$121$	1.36803e7	0.702015
$122$	4.72319e7i	2.35492i
$123$	− 2.16858e7i	− 1.05077i
$124$	−2.86844e7	−1.35105
$125$	0	0
$126$	1.13365e7	0.504874
$127$	− 2.55822e7i	− 1.10822i	−0.832445	−	0.554108i	$-0.813059\pi$
$127$	− 2.55822e7i	− 1.10822i	0.832445	−	0.554108i	$-0.186941\pi$
$128$	4.21487e7i	1.77644i
$129$	3.51398e7	1.44124
$130$	0	0
$131$	1.30640e7	0.507722	0.253861	−	0.967241i	$-0.418299\pi$
$131$	1.30640e7	0.507722	0.253861	+	0.967241i	$0.418299\pi$
$132$	− 7.65046e7i	− 2.89520i
$133$	− 7.26982e6i	− 0.267943i
$134$	−2.93154e7	−1.05252
$135$	0	0
$136$	−4.23729e7	−1.44445
$137$	2.14021e7i	0.711106i	0.934656	+	0.355553i	$0.115707\pi$
$137$	2.14021e7i	0.711106i	−0.934656	+	0.355553i	$0.884293\pi$
$138$	− 7.33870e7i	− 2.37707i
$139$	−4.00656e7	−1.26538	−0.632688	−	0.774406i	$-0.718049\pi$
$139$	−4.00656e7	−1.26538	−0.632688	+	0.774406i	$0.718049\pi$
$140$	0	0
$141$	−2.05451e7	−0.617222
$142$	− 5.60324e6i	− 0.164222i
$143$	2.03321e7i	0.581442i
$144$	−6.35321e6	−0.177307
$145$	0	0
$146$	5.84921e6	0.155547
$147$	3.77289e7i	0.979633i
$148$	− 3.58233e7i	− 0.908341i
$149$	5.96142e7	1.47638	0.738190	−	0.674593i	$-0.235681\pi$
$149$	5.96142e7	1.47638	0.738190	+	0.674593i	$0.235681\pi$
$150$	0	0
$151$	−5.21166e6	−0.123185	−0.0615924	−	0.998101i	$-0.519618\pi$
$151$	−5.21166e6	−0.123185	−0.0615924	+	0.998101i	$0.519618\pi$
$152$	2.93007e7i	0.676746i
$153$	3.31604e7i	0.748514i
$154$	−4.72367e7	−1.04222
$155$	0	0
$156$	4.68984e7	0.989058
$157$	− 1.10197e7i	− 0.227259i	−0.993523	−	0.113630i	$-0.963752\pi$
$157$	− 1.10197e7i	− 0.227259i	0.993523	−	0.113630i	$-0.0362477\pi$
$158$	3.65458e7i	0.737120i
$159$	−9.92554e7	−1.95823
$160$	0	0
$161$	−2.87575e7	−0.543075
$162$	− 1.10349e8i	− 2.03922i
$163$	− 2.32415e7i	− 0.420346i	−0.977664	−	0.210173i	$-0.932597\pi$
$163$	− 2.32415e7i	− 0.420346i	0.977664	−	0.210173i	$-0.0674028\pi$
$164$	8.07127e7	1.42886
$165$	0	0
$166$	−1.16307e7	−0.197346
$167$	− 5.84152e7i	− 0.970550i	−0.874361	−	0.485275i	$-0.838719\pi$
$167$	− 5.84152e7i	− 0.970550i	0.874361	−	0.485275i	$-0.161281\pi$
$168$	4.62355e7i	0.752304i
$169$	5.02846e7	0.801368
$170$	0	0
$171$	2.29303e7	0.350690
$172$	1.30787e8i	1.95982i
$173$	− 1.18828e6i	− 0.0174485i	−0.999962	−	0.00872427i	$-0.997223\pi$
$173$	− 1.18828e6i	− 0.0174485i	0.999962	−	0.00872427i	$-0.00277706\pi$
$174$	1.49891e8	2.15702
$175$	0	0
$176$	2.64724e7	0.366015
$177$	1.51873e8i	2.05862i
$178$	− 1.08245e8i	− 1.43859i
$179$	−1.28635e8	−1.67638	−0.838191	−	0.545377i	$-0.816387\pi$
$179$	−1.28635e8	−1.67638	−0.838191	+	0.545377i	$0.816387\pi$
$180$	0	0
$181$	1.40320e8	1.75892	0.879458	−	0.475976i	$-0.157905\pi$
$181$	1.40320e8	1.75892	0.879458	+	0.475976i	$0.157905\pi$
$182$	− 2.89568e7i	− 0.356041i
$183$	1.50752e8i	1.81838i
$184$	1.15906e8	1.37165
$185$	0	0
$186$	−1.44256e8	−1.64376
$187$	− 1.38172e8i	− 1.54516i
$188$	− 7.64671e7i	− 0.839309i
$189$	−2.10700e7	−0.227012
$190$	0	0
$191$	−3.81784e7	−0.396461	−0.198231	−	0.980155i	$-0.563519\pi$
$191$	−3.81784e7	−0.396461	−0.198231	+	0.980155i	$0.563519\pi$
$192$	1.91735e8i	1.95500i
$193$	1.35915e8i	1.36087i	0.732807	+	0.680436i	$0.238210\pi$
$193$	1.35915e8i	1.36087i	−0.732807	+	0.680436i	$0.761790\pi$
$194$	−1.34797e8	−1.32548
$195$	0	0
$196$	−1.40424e8	−1.33212
$197$	6.16154e7i	0.574193i	0.957902	+	0.287096i	$0.0926900\pi$
$197$	6.16154e7i	0.574193i	−0.957902	+	0.287096i	$0.907310\pi$
$198$	− 1.48993e8i	− 1.36408i
$199$	1.84377e8	1.65852	0.829261	−	0.558862i	$-0.188762\pi$
$199$	1.84377e8	1.65852	0.829261	+	0.558862i	$0.188762\pi$
$200$	0	0
$201$	−9.35671e7	−0.812713
$202$	− 5.46539e7i	− 0.466542i
$203$	− 5.87365e7i	− 0.492801i
$204$	−3.18710e8	−2.62839
$205$	0	0
$206$	7.38677e7	0.588734
$207$	− 9.07063e7i	− 0.710790i
$208$	1.62280e7i	0.125038i
$209$	−9.55455e7	−0.723933
$210$	0	0
$211$	1.71174e8	1.25444	0.627218	−	0.778844i	$-0.284194\pi$
$211$	1.71174e8	1.25444	0.627218	+	0.778844i	$0.284194\pi$
$212$	− 3.69420e8i	− 2.66284i
$213$	− 1.78841e7i	− 0.126806i
$214$	7.15003e7	0.498723
$215$	0	0
$216$	8.49220e7	0.573366
$217$	5.65283e7i	0.375541i
$218$	1.65139e8i	1.07958i
$219$	1.86692e7	0.120108
$220$	0	0
$221$	8.47014e7	0.527859
$222$	− 1.80158e8i	− 1.10514i
$223$	− 2.67014e8i	− 1.61238i	−0.591657	−	0.806190i	$-0.701526\pi$
$223$	− 2.67014e8i	− 1.61238i	0.591657	−	0.806190i	$-0.298474\pi$
$224$	6.13594e7	0.364765
$225$	0	0
$226$	3.96955e8	2.28750
$227$	378641.i	0.00214851i	0.999999	+	0.00107425i	$0.000341946\pi$
$227$	378641.i	0.00214851i	−0.999999	+	0.00107425i	$0.999658\pi$
$228$	2.20387e8i	1.23144i
$229$	2.24429e8	1.23497	0.617483	−	0.786584i	$-0.288152\pi$
$229$	2.24429e8	1.23497	0.617483	+	0.786584i	$0.288152\pi$
$230$	0	0
$231$	−1.50767e8	−0.804759
$232$	2.36735e8i	1.24467i
$233$	− 1.55173e8i	− 0.803656i	−0.915715	−	0.401828i	$-0.868375\pi$
$233$	− 1.55173e8i	− 0.803656i	0.915715	−	0.401828i	$-0.131625\pi$
$234$	9.13350e7	0.465995
$235$	0	0
$236$	−5.65260e8	−2.79934
$237$	1.16645e8i	0.569176i
$238$	1.96783e8i	0.946169i
$239$	−4.07160e7	−0.192918	−0.0964588	−	0.995337i	$-0.530752\pi$
$239$	−4.07160e7	−0.192918	−0.0964588	+	0.995337i	$0.530752\pi$
$240$	0	0
$241$	−3.06501e8	−1.41050	−0.705249	−	0.708960i	$-0.749165\pi$
$241$	−3.06501e8	−1.41050	−0.705249	+	0.708960i	$0.749165\pi$
$242$	2.56065e8i	1.16144i
$243$	− 2.47046e8i	− 1.10448i
$244$	−5.61086e8	−2.47266
$245$	0	0
$246$	4.05911e8	1.73843
$247$	− 5.85708e7i	− 0.247310i
$248$	− 2.27835e8i	− 0.948506i
$249$	−3.71222e7	−0.152383
$250$	0	0
$251$	−1.30381e8	−0.520421	−0.260211	−	0.965552i	$-0.583792\pi$
$251$	−1.30381e8	−0.520421	−0.260211	+	0.965552i	$0.583792\pi$
$252$	1.34671e8i	0.530117i
$253$	3.77953e8i	1.46729i
$254$	4.78842e8	1.83347
$255$	0	0
$256$	−3.78133e8	−1.40866
$257$	3.23514e8i	1.18885i	0.804151	+	0.594426i	$0.202620\pi$
$257$	3.23514e8i	1.18885i	−0.804151	+	0.594426i	$0.797380\pi$
$258$	6.57740e8i	2.38443i
$259$	−7.05969e7	−0.252485
$260$	0	0
$261$	1.85266e8	0.644990
$262$	2.44529e8i	0.839994i
$263$	− 2.39895e7i	− 0.0813159i	−0.999173	−	0.0406579i	$-0.987055\pi$
$263$	− 2.39895e7i	− 0.0813159i	0.999173	−	0.0406579i	$-0.0129454\pi$
$264$	6.07663e8	2.03259
$265$	0	0
$266$	1.36075e8	0.443295
$267$	− 3.45489e8i	− 1.11082i
$268$	− 3.48249e8i	− 1.10514i
$269$	1.73612e8	0.543809	0.271905	−	0.962324i	$-0.412346\pi$
$269$	1.73612e8	0.543809	0.271905	+	0.962324i	$0.412346\pi$
$270$	0	0
$271$	5.08478e8	1.55196	0.775978	−	0.630760i	$-0.217257\pi$
$271$	5.08478e8	1.55196	0.775978	+	0.630760i	$0.217257\pi$
$272$	− 1.10281e8i	− 0.332285i
$273$	− 9.24226e7i	− 0.274922i
$274$	−4.00600e8	−1.17648
$275$	0	0
$276$	8.71792e8	2.49592
$277$	− 6.01050e8i	− 1.69915i	−0.527468	−	0.849575i	$-0.676859\pi$
$277$	− 6.01050e8i	− 1.69915i	0.527468	−	0.849575i	$-0.323141\pi$
$278$	− 7.49940e8i	− 2.09348i
$279$	−1.78301e8	−0.491517
$280$	0	0
$281$	−6.36212e8	−1.71053	−0.855264	−	0.518193i	$-0.826605\pi$
$281$	−6.36212e8	−1.71053	−0.855264	+	0.518193i	$0.826605\pi$
$282$	− 3.84559e8i	− 1.02115i
$283$	− 5.46181e7i	− 0.143247i	−0.997432	−	0.0716233i	$-0.977182\pi$
$283$	− 5.46181e7i	− 0.143247i	0.997432	−	0.0716233i	$-0.0228179\pi$
$284$	6.65631e7	0.172433
$285$	0	0
$286$	−3.80572e8	−0.961958
$287$	− 1.59061e8i	− 0.397169i
$288$	1.93539e8i	0.477413i
$289$	−1.65271e8	−0.402768
$290$	0	0
$291$	−4.30236e8	−1.02348
$292$	6.94850e7i	0.163324i
$293$	1.22481e8i	0.284467i	0.989833	+	0.142234i	$0.0454284\pi$
$293$	1.22481e8i	0.284467i	−0.989833	+	0.142234i	$0.954572\pi$
$294$	−7.06202e8	−1.62074
$295$	0	0
$296$	2.84538e8	0.637704
$297$	2.76919e8i	0.613345i
$298$	1.11585e9i	2.44257i
$299$	−2.31690e8	−0.501255
$300$	0	0
$301$	2.57743e8	0.544758
$302$	− 9.75509e7i	− 0.203801i
$303$	− 1.74441e8i	− 0.360246i
$304$	−7.62591e7	−0.155681
$305$	0	0
$306$	−6.20690e8	−1.23837
$307$	5.58187e8i	1.10102i	0.834829	+	0.550510i	$0.185567\pi$
$307$	5.58187e8i	1.10102i	−0.834829	+	0.550510i	$0.814433\pi$
$308$	− 5.61143e8i	− 1.09433i
$309$	2.35767e8	0.454598
$310$	0	0
$311$	8.94564e7	0.168636	0.0843180	−	0.996439i	$-0.473129\pi$
$311$	8.94564e7	0.168636	0.0843180	+	0.996439i	$0.473129\pi$
$312$	3.72506e8i	0.694372i
$313$	2.75895e8i	0.508556i	0.967131	+	0.254278i	$0.0818378\pi$
$313$	2.75895e8i	0.508556i	−0.967131	+	0.254278i	$0.918162\pi$
$314$	2.06265e8	0.375986
$315$	0	0
$316$	−4.34142e8	−0.773976
$317$	− 4.40449e8i	− 0.776584i	−0.921536	−	0.388292i	$-0.873065\pi$
$317$	− 4.40449e8i	− 0.776584i	0.921536	−	0.388292i	$-0.126935\pi$
$318$	− 1.85784e9i	− 3.23977i
$319$	−7.71960e8	−1.33146
$320$	0	0
$321$	2.28211e8	0.385095
$322$	− 5.38277e8i	− 0.898483i
$323$	3.98032e8i	0.657218i
$324$	1.31088e9	2.14118
$325$	0	0
$326$	4.35030e8	0.695436
$327$	5.27082e8i	0.833607i
$328$	6.41088e8i	1.00313i
$329$	−1.50694e8	−0.233297
$330$	0	0
$331$	1.68079e8	0.254751	0.127376	−	0.991855i	$-0.459345\pi$
$331$	1.68079e8	0.254751	0.127376	+	0.991855i	$0.459345\pi$
$332$	− 1.38166e8i	− 0.207213i
$333$	− 2.22675e8i	− 0.330459i
$334$	1.09340e9	1.60571
$335$	0	0
$336$	−1.20334e8	−0.173062
$337$	8.38651e8i	1.19365i	0.802372	+	0.596824i	$0.203571\pi$
$337$	8.38651e8i	1.19365i	−0.802372	+	0.596824i	$0.796429\pi$
$338$	9.41218e8i	1.32581i
$339$	1.26698e9	1.76632
$340$	0	0
$341$	7.42939e8	1.01464
$342$	4.29205e8i	0.580195i
$343$	6.37607e8i	0.853146i
$344$	−1.03882e9	−1.37590
$345$	0	0
$346$	2.22421e7	0.0288675
$347$	− 1.16128e9i	− 1.49205i	−0.665921	−	0.746023i	$-0.731961\pi$
$347$	− 1.16128e9i	− 1.49205i	0.665921	−	0.746023i	$-0.268039\pi$
$348$	1.78062e9i	2.26487i
$349$	8.37482e8	1.05460	0.527298	−	0.849680i	$-0.323205\pi$
$349$	8.37482e8	1.05460	0.527298	+	0.849680i	$0.323205\pi$
$350$	0	0
$351$	−1.69755e8	−0.209531
$352$	− 8.06432e8i	− 0.985527i
$353$	− 7.61561e8i	− 0.921496i	−0.887531	−	0.460748i	$-0.847581\pi$
$353$	− 7.61561e8i	− 0.921496i	0.887531	−	0.460748i	$-0.152419\pi$
$354$	−2.84274e9	−3.40585
$355$	0	0
$356$	1.28588e9	1.51052
$357$	6.28081e8i	0.730596i
$358$	− 2.40776e9i	− 2.77347i
$359$	−3.19728e8	−0.364712	−0.182356	−	0.983233i	$-0.558372\pi$
$359$	−3.19728e8	−0.364712	−0.182356	+	0.983233i	$0.558372\pi$
$360$	0	0
$361$	−6.18634e8	−0.692083
$362$	2.62649e9i	2.91001i
$363$	8.17293e8i	0.896818i
$364$	3.43989e8	0.373843
$365$	0	0
$366$	−2.82175e9	−3.00839
$367$	1.25415e8i	0.132440i	0.997805	+	0.0662199i	$0.0210939\pi$
$367$	1.25415e8i	0.132440i	−0.997805	+	0.0662199i	$0.978906\pi$
$368$	3.01661e8i	0.315538i
$369$	5.01706e8	0.519825
$370$	0	0
$371$	−7.28016e8	−0.740171
$372$	− 1.71367e9i	− 1.72595i
$373$	− 1.71505e9i	− 1.71118i	−0.517657	−	0.855588i	$-0.673196\pi$
$373$	− 1.71505e9i	− 1.71118i	0.517657	−	0.855588i	$-0.326804\pi$
$374$	2.58628e9	2.55637
$375$	0	0
$376$	6.07365e8	0.589240
$377$	− 4.73223e8i	− 0.454853i
$378$	− 3.94385e8i	− 0.375577i
$379$	1.07297e8	0.101239	0.0506196	−	0.998718i	$-0.483880\pi$
$379$	1.07297e8	0.101239	0.0506196	+	0.998718i	$0.483880\pi$
$380$	0	0
$381$	1.52834e9	1.41574
$382$	− 7.14615e8i	− 0.655919i
$383$	6.77468e8i	0.616160i	0.951361	+	0.308080i	$0.0996864\pi$
$383$	6.77468e8i	0.616160i	−0.951361	+	0.308080i	$0.900314\pi$
$384$	−2.51807e9	−2.26938
$385$	0	0
$386$	−2.54403e9	−2.25148
$387$	8.12967e8i	0.712992i
$388$	− 1.60130e9i	− 1.39175i
$389$	−1.94836e9	−1.67821	−0.839104	−	0.543971i	$-0.816920\pi$
$389$	−1.94836e9	−1.67821	−0.839104	+	0.543971i	$0.816920\pi$
$390$	0	0
$391$	1.57451e9	1.33207
$392$	− 1.11536e9i	− 0.935222i
$393$	7.80474e8i	0.648611i
$394$	−1.15331e9	−0.949965
$395$	0	0
$396$	1.76995e9	1.43228
$397$	− 1.11752e9i	− 0.896369i	−0.893941	−	0.448185i	$-0.852071\pi$
$397$	− 1.11752e9i	− 0.896369i	0.893941	−	0.448185i	$-0.147929\pi$
$398$	3.45113e9i	2.74392i
$399$	4.34316e8	0.342295
$400$	0	0
$401$	1.61315e9	1.24931	0.624655	−	0.780901i	$-0.285240\pi$
$401$	1.61315e9	1.24931	0.624655	+	0.780901i	$0.285240\pi$
$402$	− 1.75137e9i	− 1.34458i
$403$	4.55432e8i	0.346622i
$404$	6.49254e8	0.489869
$405$	0	0
$406$	1.09942e9	0.815308
$407$	9.27838e8i	0.682169i
$408$	− 2.53146e9i	− 1.84527i
$409$	−1.32866e7	−0.00960248	−0.00480124	−	0.999988i	$-0.501528\pi$
$409$	−1.32866e7	−0.00960248	−0.00480124	+	0.999988i	$0.501528\pi$
$410$	0	0
$411$	−1.27861e9	−0.908432
$412$	8.77503e8i	0.618170i
$413$	1.11396e9i	0.778114i
$414$	1.69782e9	1.17596
$415$	0	0
$416$	4.94354e8	0.336676
$417$	− 2.39361e9i	− 1.61651i
$418$	− 1.78840e9i	− 1.19770i
$419$	5.93249e7	0.0393992	0.0196996	−	0.999806i	$-0.493729\pi$
$419$	5.93249e7	0.0393992	0.0196996	+	0.999806i	$0.493729\pi$
$420$	0	0
$421$	−2.93348e9	−1.91600	−0.958000	−	0.286769i	$-0.907419\pi$
$421$	−2.93348e9	−1.91600	−0.958000	+	0.286769i	$0.907419\pi$
$422$	3.20399e9i	2.07538i
$423$	− 4.75315e8i	− 0.305345i
$424$	2.93424e9	1.86946
$425$	0	0
$426$	3.34751e8	0.209792
$427$	1.10573e9i	0.687310i
$428$	8.49380e8i	0.523659i
$429$	−1.21469e9	−0.742788
$430$	0	0
$431$	−1.96307e8	−0.118104	−0.0590520	−	0.998255i	$-0.518808\pi$
$431$	−1.96307e8	−0.118104	−0.0590520	+	0.998255i	$0.518808\pi$
$432$	2.21021e8i	0.131899i
$433$	7.68256e8i	0.454777i	0.973804	+	0.227389i	$0.0730187\pi$
$433$	7.68256e8i	0.454777i	−0.973804	+	0.227389i	$0.926981\pi$
$434$	−1.05809e9	−0.621308
$435$	0	0
$436$	−1.96175e9	−1.13355
$437$	− 1.08877e9i	− 0.624095i
$438$	3.49446e8i	0.198710i
$439$	1.35112e9	0.762197	0.381099	−	0.924534i	$-0.375546\pi$
$439$	1.35112e9	0.762197	0.381099	+	0.924534i	$0.375546\pi$
$440$	0	0
$441$	−8.72866e8	−0.484633
$442$	1.58542e9i	0.873308i
$443$	− 2.20956e9i	− 1.20752i	−0.797167	−	0.603759i	$-0.793669\pi$
$443$	− 2.20956e9i	− 1.20752i	0.797167	−	0.603759i	$-0.206331\pi$
$444$	2.14017e9	1.16040
$445$	0	0
$446$	4.99792e9	2.66758
$447$	3.56150e9i	1.88606i
$448$	1.40633e9i	0.738950i
$449$	−3.16264e9	−1.64887	−0.824436	−	0.565955i	$-0.808507\pi$
$449$	−3.16264e9	−1.64887	−0.824436	+	0.565955i	$0.808507\pi$
$450$	0	0
$451$	−2.09050e9	−1.07308
$452$	4.71558e9i	2.40188i
$453$	− 3.11357e8i	− 0.157368i
$454$	−7.08732e6	−0.00355457
$455$	0	0
$456$	−1.75050e9	−0.864538
$457$	5.97325e8i	0.292755i	0.989229	+	0.146377i	$0.0467614\pi$
$457$	5.97325e8i	0.292755i	−0.989229	+	0.146377i	$0.953239\pi$
$458$	4.20082e9i	2.04317i
$459$	1.15361e9	0.556821
$460$	0	0
$461$	−2.01803e9	−0.959346	−0.479673	−	0.877447i	$-0.659245\pi$
$461$	−2.01803e9	−0.959346	−0.479673	+	0.877447i	$0.659245\pi$
$462$	− 2.82204e9i	− 1.33142i
$463$	− 1.32264e8i	− 0.0619310i	−0.999520	−	0.0309655i	$-0.990142\pi$
$463$	− 1.32264e8i	− 0.0619310i	0.999520	−	0.0309655i	$-0.00985820\pi$
$464$	−6.16136e8	−0.286328
$465$	0	0
$466$	2.90449e9	1.32960
$467$	− 3.62018e9i	− 1.64483i	−0.568887	−	0.822416i	$-0.692626\pi$
$467$	− 3.62018e9i	− 1.64483i	0.568887	−	0.822416i	$-0.307374\pi$
$468$	1.08500e9i	0.489295i
$469$	−6.86293e8	−0.307188
$470$	0	0
$471$	6.58344e8	0.290322
$472$	− 4.48976e9i	− 1.96529i
$473$	− 3.38745e9i	− 1.47183i
$474$	−2.18334e9	−0.941665
$475$	0	0
$476$	−2.33766e9	−0.993477
$477$	− 2.29629e9i	− 0.968753i
$478$	− 7.62113e8i	− 0.319170i
$479$	2.77900e9	1.15535	0.577675	−	0.816267i	$-0.303960\pi$
$479$	2.77900e9	1.15535	0.577675	+	0.816267i	$0.303960\pi$
$480$	0	0
$481$	−5.68778e8	−0.233042
$482$	− 5.73703e9i	− 2.33358i
$483$	− 1.71804e9i	− 0.693774i
$484$	−3.04189e9	−1.21951
$485$	0	0
$486$	4.62416e9	1.82728
$487$	1.86684e9i	0.732414i	0.930533	+	0.366207i	$0.119344\pi$
$487$	1.86684e9i	0.732414i	−0.930533	+	0.366207i	$0.880656\pi$
$488$	− 4.45661e9i	− 1.73594i
$489$	1.38850e9	0.536989
$490$	0	0
$491$	5.06515e9	1.93111	0.965555	−	0.260198i	$-0.0837879\pi$
$491$	5.06515e9	1.93111	0.965555	+	0.260198i	$0.0837879\pi$
$492$	4.82197e9i	1.82535i
$493$	3.21590e9i	1.20876i
$494$	1.09632e9	0.409158
$495$	0	0
$496$	5.92973e8	0.218197
$497$	− 1.31176e8i	− 0.0479299i
$498$	− 6.94846e8i	− 0.252108i
$499$	5.56606e8	0.200538	0.100269	−	0.994960i	$-0.468030\pi$
$499$	5.56606e8	0.200538	0.100269	+	0.994960i	$0.468030\pi$
$500$	0	0
$501$	3.48986e9	1.23987
$502$	− 2.44044e9i	− 0.861004i
$503$	7.79533e8i	0.273116i	0.990632	+	0.136558i	$0.0436040\pi$
$503$	7.79533e8i	0.273116i	−0.990632	+	0.136558i	$0.956396\pi$
$504$	−1.06967e9	−0.372171
$505$	0	0
$506$	−7.07445e9	−2.42754
$507$	3.00412e9i	1.02374i
$508$	5.68835e9i	1.92515i
$509$	1.12351e9	0.377629	0.188814	−	0.982013i	$-0.439536\pi$
$509$	1.12351e9	0.377629	0.188814	+	0.982013i	$0.439536\pi$
$510$	0	0
$511$	1.36934e8	0.0453981
$512$	− 1.68278e9i	− 0.554094i
$513$	− 7.97720e8i	− 0.260879i
$514$	−6.05547e9	−1.96688
$515$	0	0
$516$	−7.81355e9	−2.50365
$517$	1.98053e9i	0.630326i
$518$	− 1.32142e9i	− 0.417721i
$519$	7.09909e7	0.0222904
$520$	0	0
$521$	−6.43550e8	−0.199366	−0.0996828	−	0.995019i	$-0.531783\pi$
$521$	−6.43550e8	−0.199366	−0.0996828	+	0.995019i	$0.531783\pi$
$522$	3.46777e9i	1.06709i
$523$	− 6.30863e9i	− 1.92832i	−0.265324	−	0.964159i	$-0.585479\pi$
$523$	− 6.30863e9i	− 1.92832i	0.265324	−	0.964159i	$-0.414521\pi$
$524$	−2.90486e9	−0.881993
$525$	0	0
$526$	4.49030e8	0.134532
$527$	− 3.09500e9i	− 0.921137i
$528$	1.58153e9i	0.467582i
$529$	−9.02060e8	−0.264936
$530$	0	0
$531$	−3.51362e9	−1.01841
$532$	1.61649e9i	0.465459i
$533$	− 1.28150e9i	− 0.366585i
$534$	6.46679e9	1.83778
$535$	0	0
$536$	2.76608e9	0.775868
$537$	− 7.68495e9i	− 2.14156i
$538$	3.24963e9i	0.899697i
$539$	3.63704e9	1.00043
$540$	0	0
$541$	−3.71277e9	−1.00811	−0.504055	−	0.863672i	$-0.668159\pi$
$541$	−3.71277e9	−1.00811	−0.504055	+	0.863672i	$0.668159\pi$
$542$	9.51758e9i	2.56761i
$543$	8.38306e9i	2.24700i
$544$	−3.35951e9	−0.894705
$545$	0	0
$546$	1.72995e9	0.454840
$547$	4.19241e9i	1.09524i	0.836728	+	0.547619i	$0.184466\pi$
$547$	4.19241e9i	1.09524i	−0.836728	+	0.547619i	$0.815534\pi$
$548$	− 4.75888e9i	− 1.23530i
$549$	−3.48768e9	−0.899567
$550$	0	0
$551$	2.22379e9	0.566321
$552$	6.92450e9i	1.75227i
$553$	8.55564e8i	0.215137i
$554$	1.12503e10	2.81113
$555$	0	0
$556$	8.90883e9	2.19816
$557$	2.52750e9i	0.619723i	0.950782	+	0.309861i	$0.100283\pi$
$557$	2.52750e9i	0.619723i	−0.950782	+	0.309861i	$0.899717\pi$
$558$	− 3.33740e9i	− 0.813182i
$559$	2.07656e9	0.502808
$560$	0	0
$561$	8.25473e9	1.97393
$562$	− 1.19085e10i	− 2.82996i
$563$	4.12649e9i	0.974543i	0.873250	+	0.487272i	$0.162008\pi$
$563$	4.12649e9i	0.974543i	−0.873250	+	0.487272i	$0.837992\pi$
$564$	4.56833e9	1.07221
$565$	0	0
$566$	1.02233e9	0.236992
$567$	− 2.58334e9i	− 0.595170i
$568$	5.28700e8i	0.121057i
$569$	−5.27044e8	−0.119937	−0.0599686	−	0.998200i	$-0.519100\pi$
$569$	−5.27044e8	−0.119937	−0.0599686	+	0.998200i	$0.519100\pi$
$570$	0	0
$571$	−3.18083e9	−0.715014	−0.357507	−	0.933911i	$-0.616373\pi$
$571$	−3.18083e9	−0.715014	−0.357507	+	0.933911i	$0.616373\pi$
$572$	− 4.52097e9i	− 1.01006i
$573$	− 2.28087e9i	− 0.506476i
$574$	2.97726e9	0.657091
$575$	0	0
$576$	−4.43583e9	−0.967155
$577$	4.88332e9i	1.05828i	0.848535	+	0.529139i	$0.177485\pi$
$577$	4.88332e9i	1.05828i	−0.848535	+	0.529139i	$0.822515\pi$
$578$	− 3.09351e9i	− 0.666354i
$579$	−8.11990e9	−1.73850
$580$	0	0
$581$	−2.72283e8	−0.0575976
$582$	− 8.05307e9i	− 1.69329i
$583$	9.56814e9i	1.99981i
$584$	−5.51907e8	−0.114662
$585$	0	0
$586$	−2.29257e9	−0.470632
$587$	8.66750e8i	0.176873i	0.996082	+	0.0884363i	$0.0281870\pi$
$587$	8.66750e8i	0.176873i	−0.996082	+	0.0884363i	$0.971813\pi$
$588$	− 8.38925e9i	− 1.70178i
$589$	−2.14019e9	−0.431567
$590$	0	0
$591$	−3.68105e9	−0.733527
$592$	7.40549e8i	0.146699i
$593$	1.43710e9i	0.283007i	0.989938	+	0.141503i	$0.0451936\pi$
$593$	1.43710e9i	0.283007i	−0.989938	+	0.141503i	$0.954806\pi$
$594$	−5.18331e9	−1.01474
$595$	0	0
$596$	−1.32556e10	−2.56470
$597$	1.10151e10i	2.11875i
$598$	− 4.33674e9i	− 0.829294i
$599$	−7.50069e9	−1.42596	−0.712980	−	0.701185i	$-0.752655\pi$
$599$	−7.50069e9	−1.42596	−0.712980	+	0.701185i	$0.752655\pi$
$600$	0	0
$601$	5.05436e9	0.949741	0.474870	−	0.880056i	$-0.342495\pi$
$601$	5.05436e9	0.949741	0.474870	+	0.880056i	$0.342495\pi$
$602$	4.82437e9i	0.901266i
$603$	− 2.16469e9i	− 0.402055i
$604$	1.15884e9	0.213991
$605$	0	0
$606$	3.26515e9	0.596004
$607$	− 9.94598e8i	− 0.180504i	−0.995919	−	0.0902521i	$-0.971233\pi$
$607$	− 9.94598e8i	− 0.180504i	0.995919	−	0.0902521i	$-0.0287673\pi$
$608$	2.32309e9i	0.419183i
$609$	3.50906e9	0.629550
$610$	0	0
$611$	−1.21409e9	−0.215332
$612$	− 7.37342e9i	− 1.30029i
$613$	− 1.58283e9i	− 0.277539i	−0.990325	−	0.138769i	$-0.955685\pi$
$613$	− 1.58283e9i	− 0.277539i	0.990325	−	0.138769i	$-0.0443147\pi$
$614$	−1.04480e10	−1.82157
$615$	0	0
$616$	4.45707e9	0.768275
$617$	− 8.35079e9i	− 1.43130i	−0.698461	−	0.715648i	$-0.746132\pi$
$617$	− 8.35079e9i	− 1.43130i	0.698461	−	0.715648i	$-0.253868\pi$
$618$	4.41303e9i	0.752103i
$619$	−1.12049e9	−0.189886	−0.0949428	−	0.995483i	$-0.530267\pi$
$619$	−1.12049e9	−0.189886	−0.0949428	+	0.995483i	$0.530267\pi$
$620$	0	0
$621$	−3.15557e9	−0.528758
$622$	1.67443e9i	0.278997i
$623$	− 2.53408e9i	− 0.419868i
$624$	−9.69497e8	−0.159735
$625$	0	0
$626$	−5.16415e9	−0.841372
$627$	− 5.70812e9i	− 0.924819i
$628$	2.45030e9i	0.394785i
$629$	3.86528e9	0.619303
$630$	0	0
$631$	3.72303e9	0.589921	0.294961	−	0.955509i	$-0.404693\pi$
$631$	3.72303e9	0.589921	0.294961	+	0.955509i	$0.404693\pi$
$632$	− 3.44832e9i	− 0.543372i
$633$	1.02263e10i	1.60253i
$634$	8.24424e9	1.28481
$635$	0	0
$636$	2.20700e10	3.40176
$637$	2.22956e9i	0.341767i
$638$	− 1.44494e10i	− 2.20281i
$639$	4.13753e8	0.0627318
$640$	0	0
$641$	−1.08733e9	−0.163065	−0.0815323	−	0.996671i	$-0.525981\pi$
$641$	−1.08733e9	−0.163065	−0.0815323	+	0.996671i	$0.525981\pi$
$642$	4.27160e9i	0.637115i
$643$	− 9.30287e9i	− 1.38000i	−0.723810	−	0.689999i	$-0.757611\pi$
$643$	− 9.30287e9i	− 1.38000i	0.723810	−	0.689999i	$-0.242389\pi$
$644$	6.39440e9	0.943407
$645$	0	0
$646$	−7.45029e9	−1.08733
$647$	− 3.75633e9i	− 0.545254i	−0.962120	−	0.272627i	$-0.912108\pi$
$647$	− 3.75633e9i	− 0.545254i	0.962120	−	0.272627i	$-0.0878925\pi$
$648$	1.04121e10i	1.50323i
$649$	1.46405e10	2.10232
$650$	0	0
$651$	−3.37714e9	−0.479750
$652$	5.16788e9i	0.730207i
$653$	− 6.47262e9i	− 0.909671i	−0.890576	−	0.454835i	$-0.849698\pi$
$653$	− 6.47262e9i	− 0.909671i	0.890576	−	0.454835i	$-0.150302\pi$
$654$	−9.86582e9	−1.37915
$655$	0	0
$656$	−1.66852e9	−0.230764
$657$	4.31915e8i	0.0594182i
$658$	− 2.82065e9i	− 0.385975i
$659$	−8.31257e9	−1.13145	−0.565726	−	0.824593i	$-0.691404\pi$
$659$	−8.31257e9	−1.13145	−0.565726	+	0.824593i	$0.691404\pi$
$660$	0	0
$661$	−1.42966e9	−0.192543	−0.0962714	−	0.995355i	$-0.530692\pi$
$661$	−1.42966e9	−0.192543	−0.0962714	+	0.995355i	$0.530692\pi$
$662$	3.14608e9i	0.421470i
$663$	5.06026e9i	0.674335i
$664$	1.09743e9	0.145475
$665$	0	0
$666$	4.16799e9	0.546723
$667$	− 8.79672e9i	− 1.14784i
$668$	1.29890e10i	1.68600i
$669$	1.59521e10	2.05980
$670$	0	0
$671$	1.45324e10	1.85698
$672$	3.66575e9i	0.465984i
$673$	− 7.03224e9i	− 0.889285i	−0.895708	−	0.444642i	$-0.853331\pi$
$673$	− 7.03224e9i	− 0.889285i	0.895708	−	0.444642i	$-0.146669\pi$
$674$	−1.56977e10	−1.97481
$675$	0	0
$676$	−1.11811e10	−1.39210
$677$	− 5.75322e8i	− 0.0712608i	−0.999365	−	0.0356304i	$-0.988656\pi$
$677$	− 5.75322e8i	− 0.0712608i	0.999365	−	0.0356304i	$-0.0113439\pi$
$678$	2.37150e10i	2.92227i
$679$	−3.15568e9	−0.386856
$680$	0	0
$681$	−2.26209e7	−0.00274470
$682$	1.39062e10i	1.67866i
$683$	− 7.67069e9i	− 0.921217i	−0.887603	−	0.460609i	$-0.847631\pi$
$683$	− 7.67069e9i	− 0.921217i	0.887603	−	0.460609i	$-0.152369\pi$
$684$	−5.09870e9	−0.609204
$685$	0	0
$686$	−1.19346e10	−1.41148
$687$	1.34079e10i	1.57766i
$688$	− 2.70368e9i	− 0.316516i
$689$	−5.86541e9	−0.683173
$690$	0	0
$691$	3.62855e9	0.418369	0.209185	−	0.977876i	$-0.432919\pi$
$691$	3.62855e9	0.418369	0.209185	+	0.977876i	$0.432919\pi$
$692$	2.64222e8i	0.0303108i
$693$	− 3.48804e9i	− 0.398121i
$694$	2.17365e10	2.46849
$695$	0	0
$696$	−1.41431e10	−1.59006
$697$	8.70878e9i	0.974188i
$698$	1.56758e10i	1.74476i
$699$	9.27040e9	1.02666
$700$	0	0
$701$	−1.71375e10	−1.87904	−0.939518	−	0.342499i	$-0.888727\pi$
$701$	−1.71375e10	−1.87904	−0.939518	+	0.342499i	$0.888727\pi$
$702$	− 3.17744e9i	− 0.346655i
$703$	− 2.67283e9i	− 0.290153i
$704$	1.84831e10	1.99651
$705$	0	0
$706$	1.42547e10	1.52456
$707$	− 1.27948e9i	− 0.136166i
$708$	− 3.37700e10i	− 3.57614i
$709$	−1.34055e10	−1.41260	−0.706302	−	0.707911i	$-0.749638\pi$
$709$	−1.34055e10	−1.41260	−0.706302	+	0.707911i	$0.749638\pi$
$710$	0	0
$711$	−2.69860e9	−0.281576
$712$	1.02135e10i	1.06046i
$713$	8.46601e9i	0.874712i
$714$	−1.17563e10	−1.20872
$715$	0	0
$716$	2.86027e10	2.91214
$717$	− 2.43247e9i	− 0.246451i
$718$	− 5.98460e9i	− 0.603392i
$719$	5.99982e9	0.601987	0.300994	−	0.953626i	$-0.402682\pi$
$719$	5.99982e9	0.601987	0.300994	+	0.953626i	$0.402682\pi$
$720$	0	0
$721$	1.72929e9	0.171829
$722$	− 1.15795e10i	− 1.14501i
$723$	− 1.83111e10i	− 1.80190i
$724$	−3.12010e10	−3.05551
$725$	0	0
$726$	−1.52979e10	−1.48373
$727$	− 1.84312e10i	− 1.77903i	−0.456909	−	0.889513i	$-0.651044\pi$
$727$	− 1.84312e10i	− 1.77903i	0.456909	−	0.889513i	$-0.348956\pi$
$728$	2.73225e9i	0.262458i
$729$	1.86590e9	0.178378
$730$	0	0
$731$	−1.41118e10	−1.33620
$732$	− 3.35206e10i	− 3.15881i
$733$	7.01069e9i	0.657502i	0.944417	+	0.328751i	$0.106628\pi$
$733$	7.01069e9i	0.657502i	−0.944417	+	0.328751i	$0.893372\pi$
$734$	−2.34749e9	−0.219113
$735$	0	0
$736$	9.18953e9	0.849613
$737$	9.01980e9i	0.829966i
$738$	9.39083e9i	0.860016i
$739$	−3.58081e9	−0.326382	−0.163191	−	0.986595i	$-0.552179\pi$
$739$	−3.58081e9	−0.326382	−0.163191	+	0.986595i	$0.552179\pi$
$740$	0	0
$741$	3.49916e9	0.315936
$742$	− 1.36268e10i	− 1.22456i
$743$	4.23347e9i	0.378648i	0.981915	+	0.189324i	$0.0606297\pi$
$743$	4.23347e9i	0.378648i	−0.981915	+	0.189324i	$0.939370\pi$
$744$	1.36114e10	1.21171
$745$	0	0
$746$	3.21019e10	2.83103
$747$	− 8.58830e8i	− 0.0753851i
$748$	3.07234e10i	2.68419i
$749$	1.67387e9	0.145558
$750$	0	0
$751$	3.94072e9	0.339497	0.169749	−	0.985487i	$-0.445704\pi$
$751$	3.94072e9	0.339497	0.169749	+	0.985487i	$0.445704\pi$
$752$	1.58075e9i	0.135550i
$753$	− 7.78925e9i	− 0.664834i
$754$	8.85769e9	0.752524
$755$	0	0
$756$	4.68505e9	0.394355
$757$	− 8.01225e8i	− 0.0671303i	−0.999437	−	0.0335652i	$-0.989314\pi$
$757$	− 8.01225e8i	− 0.0671303i	0.999437	−	0.0335652i	$-0.0106861\pi$
$758$	2.00836e9i	0.167494i
$759$	−2.25798e10	−1.87445
$760$	0	0
$761$	−3.84688e9	−0.316419	−0.158209	−	0.987406i	$-0.550572\pi$
$761$	−3.84688e9	−0.316419	−0.158209	+	0.987406i	$0.550572\pi$
$762$	2.86072e10i	2.34225i
$763$	3.86603e9i	0.315086i
$764$	8.48919e9	0.688715
$765$	0	0
$766$	−1.26807e10	−1.01940
$767$	8.97482e9i	0.718194i
$768$	− 2.25906e10i	− 1.79955i
$769$	−4.40578e9	−0.349366	−0.174683	−	0.984625i	$-0.555890\pi$
$769$	−4.40578e9	−0.349366	−0.174683	+	0.984625i	$0.555890\pi$
$770$	0	0
$771$	−1.93275e10	−1.51875
$772$	− 3.02216e10i	− 2.36405i
$773$	1.54451e10i	1.20272i	0.798979	+	0.601359i	$0.205374\pi$
$773$	1.54451e10i	1.20272i	−0.798979	+	0.601359i	$0.794626\pi$
$774$	−1.52170e10	−1.17960
$775$	0	0
$776$	1.27189e10	0.977085
$777$	− 4.21762e9i	− 0.322548i
$778$	− 3.64690e10i	− 2.77649i
$779$	6.02210e9	0.456422
$780$	0	0
$781$	−1.72401e9	−0.129498
$782$	2.94714e10i	2.20382i
$783$	− 6.44518e9i	− 0.479810i
$784$	2.90288e9	0.215141
$785$	0	0
$786$	−1.46088e10	−1.07309
$787$	− 2.40331e10i	− 1.75751i	−0.477274	−	0.878755i	$-0.658375\pi$
$787$	− 2.40331e10i	− 1.75751i	0.477274	−	0.878755i	$-0.341625\pi$
$788$	− 1.37006e10i	− 0.997463i
$789$	1.43319e9	0.103880
$790$	0	0
$791$	9.29299e9	0.667633
$792$	1.40584e10i	1.00554i
$793$	8.90856e9i	0.634383i
$794$	2.09174e10	1.48299
$795$	0	0
$796$	−4.09973e10	−2.88111
$797$	1.37624e10i	0.962918i	0.876469	+	0.481459i	$0.159893\pi$
$797$	1.37624e10i	0.962918i	−0.876469	+	0.481459i	$0.840107\pi$
$798$	8.12944e9i	0.566305i
$799$	8.25068e9	0.572237
$800$	0	0
$801$	7.99296e9	0.549533
$802$	3.01947e10i	2.06690i
$803$	− 1.79969e9i	− 0.122657i
$804$	2.08052e10	1.41181
$805$	0	0
$806$	−8.52468e9	−0.573463
$807$	1.03720e10i	0.694712i
$808$	5.15692e9i	0.343914i
$809$	−5.00163e9	−0.332118	−0.166059	−	0.986116i	$-0.553104\pi$
$809$	−5.00163e9	−0.332118	−0.166059	+	0.986116i	$0.553104\pi$
$810$	0	0
$811$	8.14966e9	0.536496	0.268248	−	0.963350i	$-0.413555\pi$
$811$	8.14966e9	0.536496	0.268248	+	0.963350i	$0.413555\pi$
$812$	1.30604e10i	0.856073i
$813$	3.03777e10i	1.98261i
$814$	−1.73671e10	−1.12860
$815$	0	0
$816$	6.58847e9	0.424491
$817$	9.75824e9i	0.626029i
$818$	− 2.48697e8i	− 0.0158867i
$819$	2.13822e9	0.136006
$820$	0	0
$821$	1.30820e10	0.825035	0.412518	−	0.910950i	$-0.364650\pi$
$821$	1.30820e10	0.825035	0.412518	+	0.910950i	$0.364650\pi$
$822$	− 2.39328e10i	− 1.50294i
$823$	3.12969e10i	1.95705i	0.206133	+	0.978524i	$0.433912\pi$
$823$	3.12969e10i	1.95705i	−0.206133	+	0.978524i	$0.566088\pi$
$824$	−6.96985e9	−0.433989
$825$	0	0
$826$	−2.08508e10	−1.28734
$827$	− 1.98645e10i	− 1.22126i	−0.791916	−	0.610630i	$-0.790916\pi$
$827$	− 1.98645e10i	− 1.22126i	0.791916	−	0.610630i	$-0.209084\pi$
$828$	2.01691e10i	1.23475i
$829$	−1.06183e10	−0.647310	−0.323655	−	0.946175i	$-0.604912\pi$
$829$	−1.06183e10	−0.647310	−0.323655	+	0.946175i	$0.604912\pi$
$830$	0	0
$831$	3.59082e10	2.17065
$832$	1.13304e10i	0.682046i
$833$	− 1.51515e10i	− 0.908235i
$834$	4.48032e10	2.67441
$835$	0	0
$836$	2.12451e10	1.25758
$837$	6.20288e9i	0.365640i
$838$	1.11043e9i	0.0651834i
$839$	−4.19640e9	−0.245307	−0.122654	−	0.992450i	$-0.539140\pi$
$839$	−4.19640e9	−0.245307	−0.122654	+	0.992450i	$0.539140\pi$
$840$	0	0
$841$	7.17225e8	0.0415785
$842$	− 5.49082e10i	− 3.16990i
$843$	− 3.80088e10i	− 2.18518i
$844$	−3.80615e10	−2.17915
$845$	0	0
$846$	8.89685e9	0.505173
$847$	5.99465e9i	0.338979i
$848$	7.63676e9i	0.430055i
$849$	3.26302e9	0.182996
$850$	0	0
$851$	−1.05730e10	−0.588091
$852$	3.97664e9i	0.220281i
$853$	− 2.52606e10i	− 1.39355i	−0.717289	−	0.696775i	$-0.754617\pi$
$853$	− 2.52606e10i	− 1.39355i	0.717289	−	0.696775i	$-0.245383\pi$
$854$	−2.06969e10	−1.13711
$855$	0	0
$856$	−6.74648e9	−0.367637
$857$	1.07248e10i	0.582046i	0.956716	+	0.291023i	$0.0939957\pi$
$857$	1.07248e10i	0.582046i	−0.956716	+	0.291023i	$0.906004\pi$
$858$	− 2.27363e10i	− 1.22889i
$859$	3.02643e10	1.62913	0.814563	−	0.580075i	$-0.196977\pi$
$859$	3.02643e10	1.62913	0.814563	+	0.580075i	$0.196977\pi$
$860$	0	0
$861$	9.50266e9	0.507381
$862$	− 3.67443e9i	− 0.195396i
$863$	1.39177e10i	0.737106i	0.929607	+	0.368553i	$0.120147\pi$
$863$	1.39177e10i	0.737106i	−0.929607	+	0.368553i	$0.879853\pi$
$864$	6.73299e9	0.355148
$865$	0	0
$866$	−1.43801e10	−0.752399
$867$	− 9.87370e9i	− 0.514533i
$868$	− 1.25694e10i	− 0.652373i
$869$	1.12445e10	0.581260
$870$	0	0
$871$	−5.52926e9	−0.283533
$872$	− 1.55819e10i	− 0.795815i
$873$	− 9.95360e9i	− 0.506326i
$874$	2.03794e10	1.03253
$875$	0	0
$876$	−4.15120e9	−0.208646
$877$	− 1.81702e10i	− 0.909622i	−0.890588	−	0.454811i	$-0.849707\pi$
$877$	− 1.81702e10i	− 0.909622i	0.890588	−	0.454811i	$-0.150293\pi$
$878$	2.52900e10i	1.26101i
$879$	−7.31731e9	−0.363404
$880$	0	0
$881$	−3.14352e10	−1.54882	−0.774409	−	0.632685i	$-0.781953\pi$
$881$	−3.14352e10	−1.54882	−0.774409	+	0.632685i	$0.781953\pi$
$882$	− 1.63381e10i	− 0.801793i
$883$	2.11722e10i	1.03491i	0.855709	+	0.517457i	$0.173121\pi$
$883$	2.11722e10i	1.03491i	−0.855709	+	0.517457i	$0.826879\pi$
$884$	−1.88339e10	−0.916973
$885$	0	0
$886$	4.13582e10	1.99776
$887$	3.72771e10i	1.79353i	0.442505	+	0.896766i	$0.354090\pi$
$887$	3.72771e10i	1.79353i	−0.442505	+	0.896766i	$0.645910\pi$
$888$	1.69990e10i	0.814662i
$889$	1.12100e10	0.535119
$890$	0	0
$891$	−3.39523e10	−1.60804
$892$	5.93722e10i	2.80095i
$893$	− 5.70532e9i	− 0.268102i
$894$	−6.66634e10	−3.12037
$895$	0	0
$896$	−1.84694e10	−0.857780
$897$	− 1.38417e10i	− 0.640350i
$898$	− 5.91976e10i	− 2.72795i
$899$	−1.72916e10	−0.793738
$900$	0	0
$901$	3.98599e10	1.81551
$902$	− 3.91295e10i	− 1.77534i
$903$	1.53982e10i	0.695923i
$904$	−3.74551e10	−1.68625
$905$	0	0
$906$	5.82792e9	0.260354
$907$	1.00189e10i	0.445854i	0.974835	+	0.222927i	$0.0715612\pi$
$907$	1.00189e10i	0.445854i	−0.974835	+	0.222927i	$0.928439\pi$
$908$	− 8.41930e7i	− 0.00373229i
$909$	4.03573e9	0.178217
$910$	0	0
$911$	3.16088e10	1.38514	0.692571	−	0.721350i	$-0.256478\pi$
$911$	3.16088e10	1.38514	0.692571	+	0.721350i	$0.256478\pi$
$912$	− 4.55590e9i	− 0.198881i
$913$	3.57855e9i	0.155618i
$914$	−1.11806e10	−0.484344
$915$	0	0
$916$	−4.99032e10	−2.14533
$917$	5.72460e9i	0.245161i
$918$	2.15931e10i	0.921225i
$919$	−7.18397e8	−0.0305324	−0.0152662	−	0.999883i	$-0.504860\pi$
$919$	−7.18397e8	−0.0305324	−0.0152662	+	0.999883i	$0.504860\pi$
$920$	0	0
$921$	−3.33474e10	−1.40654
$922$	− 3.77732e10i	− 1.58718i
$923$	− 1.05685e9i	− 0.0442390i
$924$	3.35240e10	1.39799
$925$	0	0
$926$	2.47569e9	0.102461
$927$	5.45451e9i	0.224893i
$928$	1.87694e10i	0.770962i
$929$	1.63954e10	0.670915	0.335458	−	0.942055i	$-0.391109\pi$
$929$	1.63954e10	0.670915	0.335458	+	0.942055i	$0.391109\pi$
$930$	0	0
$931$	−1.04772e10	−0.425522
$932$	3.45036e10i	1.39608i
$933$	5.34434e9i	0.215431i
$934$	6.77618e10	2.72127
$935$	0	0
$936$	−8.61800e9	−0.343511
$937$	− 9.79890e9i	− 0.389125i	−0.980890	−	0.194562i	$-0.937671\pi$
$937$	− 9.79890e9i	− 0.389125i	0.980890	−	0.194562i	$-0.0623286\pi$
$938$	− 1.28459e10i	− 0.508223i
$939$	−1.64826e10	−0.649676
$940$	0	0
$941$	1.68542e10	0.659394	0.329697	−	0.944087i	$-0.393053\pi$
$941$	1.68542e10	0.659394	0.329697	+	0.944087i	$0.393053\pi$
$942$	1.23227e10i	0.480319i
$943$	− 2.38218e10i	− 0.925090i
$944$	1.16852e10	0.452100
$945$	0	0
$946$	6.34056e10	2.43506
$947$	2.47225e10i	0.945948i	0.881076	+	0.472974i	$0.156820\pi$
$947$	2.47225e10i	0.945948i	−0.881076	+	0.472974i	$0.843180\pi$
$948$	− 2.59367e10i	− 0.988748i
$949$	1.10324e9	0.0419022
$950$	0	0
$951$	2.63135e10	0.992080
$952$	− 1.85677e10i	− 0.697474i
$953$	4.36738e9i	0.163454i	0.996655	+	0.0817271i	$0.0260436\pi$
$953$	4.36738e9i	0.163454i	−0.996655	+	0.0817271i	$0.973956\pi$
$954$	4.29816e10	1.60274
$955$	0	0
$956$	9.05343e9	0.335128
$957$	− 4.61188e10i	− 1.70093i
$958$	5.20167e10i	1.91145i
$959$	−9.37832e9	−0.343368
$960$	0	0
$961$	−1.08711e10	−0.395130
$962$	− 1.06463e10i	− 0.385554i
$963$	5.27970e9i	0.190510i
$964$	6.81523e10	2.45025
$965$	0	0
$966$	3.21579e10	1.14780
$967$	− 1.00818e10i	− 0.358547i	−0.983799	−	0.179274i	$-0.942625\pi$
$967$	− 1.00818e10i	− 0.358547i	0.983799	−	0.179274i	$-0.0573747\pi$
$968$	− 2.41612e10i	− 0.856161i
$969$	−2.37794e10	−0.839591
$970$	0	0
$971$	−1.19066e10	−0.417369	−0.208685	−	0.977983i	$-0.566918\pi$
$971$	−1.19066e10	−0.417369	−0.208685	+	0.977983i	$0.566918\pi$
$972$	5.49322e10i	1.91865i
$973$	− 1.75566e10i	− 0.611006i
$974$	−3.49432e10	−1.21173
$975$	0	0
$976$	1.15989e10	0.399341
$977$	1.99598e10i	0.684741i	0.939565	+	0.342370i	$0.111230\pi$
$977$	1.99598e10i	0.684741i	−0.939565	+	0.342370i	$0.888770\pi$
$978$	2.59897e10i	0.888414i
$979$	−3.33049e10	−1.13441
$980$	0	0
$981$	−1.21942e10	−0.412392
$982$	9.48085e10i	3.19490i
$983$	3.22193e10i	1.08188i	0.841061	+	0.540940i	$0.181931\pi$
$983$	3.22193e10i	1.08188i	−0.841061	+	0.540940i	$0.818069\pi$
$984$	−3.83001e10	−1.28150
$985$	0	0
$986$	−6.01947e10	−1.99981
$987$	− 9.00280e9i	− 0.298035i
$988$	1.30236e10i	0.429616i
$989$	3.86010e10	1.26885
$990$	0	0
$991$	−1.42552e10	−0.465281	−0.232640	−	0.972563i	$-0.574737\pi$
$991$	−1.42552e10	−0.465281	−0.232640	+	0.972563i	$0.574737\pi$
$992$	− 1.80638e10i	− 0.587513i
$993$	1.00415e10i	0.325443i
$994$	2.45532e9	0.0792969
$995$	0	0
$996$	8.25435e9	0.264713
$997$	− 2.73327e10i	− 0.873475i	−0.899589	−	0.436737i	$-0.856134\pi$
$997$	− 2.73327e10i	− 0.873475i	0.899589	−	0.436737i	$-0.143866\pi$
$998$	1.04184e10i	0.331776i
$999$	−7.74662e9	−0.245829

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	25.8.b.c.24.4		4
3.2	odd	2		225.8.b.m.199.1		4
4.3	odd	2		400.8.c.m.49.2		4
5.2	odd	4		25.8.a.b.1.1		2
5.3	odd	4		5.8.a.b.1.2	✓	2
5.4	even	2	inner	25.8.b.c.24.1		4
15.2	even	4		225.8.a.w.1.2		2
15.8	even	4		45.8.a.h.1.1		2
15.14	odd	2		225.8.b.m.199.4		4
20.3	even	4		80.8.a.g.1.2		2
20.7	even	4		400.8.a.bb.1.1		2
20.19	odd	2		400.8.c.m.49.3		4
35.13	even	4		245.8.a.c.1.2		2
40.3	even	4		320.8.a.u.1.1		2
40.13	odd	4		320.8.a.l.1.2		2
55.43	even	4		605.8.a.d.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5.8.a.b.1.2	✓	2	5.3	odd	4
25.8.a.b.1.1		2	5.2	odd	4
25.8.b.c.24.1		4	5.4	even	2	inner
25.8.b.c.24.4		4	1.1	even	1	trivial
45.8.a.h.1.1		2	15.8	even	4
80.8.a.g.1.2		2	20.3	even	4
225.8.a.w.1.2		2	15.2	even	4
225.8.b.m.199.1		4	3.2	odd	2
225.8.b.m.199.4		4	15.14	odd	2
245.8.a.c.1.2		2	35.13	even	4
320.8.a.l.1.2		2	40.13	odd	4
320.8.a.u.1.1		2	40.3	even	4
400.8.a.bb.1.1		2	20.7	even	4
400.8.c.m.49.2		4	4.3	odd	2
400.8.c.m.49.3		4	20.19	odd	2
605.8.a.d.1.1		2	55.43	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:	\( N \)	\(=\)	\( 25 = 5^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	25.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+18.7178i q^{2} +59.7424i q^{3} -222.356 q^{4} -1118.25 q^{6} +438.197i q^{7} -1766.14i q^{8} -1382.15 q^{9} +O(q^{10})\)	\(q+18.7178i q^{2} +59.7424i q^{3} -222.356 q^{4} -1118.25 q^{6} +438.197i q^{7} -1766.14i q^{8} -1382.15 q^{9} +5759.12 q^{11} -13284.1i q^{12} +3530.42i q^{13} -8202.08 q^{14} +4596.61 q^{16} -23991.9i q^{17} -25870.8i q^{18} -16590.3 q^{19} -26178.9 q^{21} +107798. i q^{22} +65626.9i q^{23} +105513. q^{24} -66081.7 q^{26} +48083.5i q^{27} -97435.6i q^{28} -134041. q^{29} +129002. q^{31} -140027. i q^{32} +344064. i q^{33} +449075. q^{34} +307330. q^{36} +161108. i q^{37} -310534. i q^{38} -210916. q^{39} -362989. q^{41} -490012. i q^{42} -588189. i q^{43} -1.28057e6 q^{44} -1.22839e6 q^{46} +343895. i q^{47} +274612. i q^{48} +631527. q^{49} +1.43333e6 q^{51} -785010. i q^{52} +1.66139e6i q^{53} -900018. q^{54} +773915. q^{56} -991144. i q^{57} -2.50896e6i q^{58} +2.54214e6 q^{59} +2.52337e6 q^{61} +2.41464e6i q^{62} -605655. i q^{63} +3.20936e6 q^{64} -6.44011e6 q^{66} +1.56618e6i q^{67} +5.33474e6i q^{68} -3.92071e6 q^{69} -299354. q^{71} +2.44107e6i q^{72} -312494. i q^{73} -3.01558e6 q^{74} +3.68895e6 q^{76} +2.52363e6i q^{77} -3.94788e6i q^{78} +1.95247e6 q^{79} -5.89539e6 q^{81} -6.79435e6i q^{82} +621372. i q^{83} +5.82104e6 q^{84} +1.10096e7 q^{86} -8.00795e6i q^{87} -1.01714e7i q^{88} -5.78298e6 q^{89} -1.54702e6 q^{91} -1.45925e7i q^{92} +7.70690e6i q^{93} -6.43696e6 q^{94} +8.36554e6 q^{96} +7.20152e6i q^{97} +1.18208e7i q^{98} -7.95998e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 192 q^{4} - 2032 q^{6} - 11108 q^{9}+O(q^{10}) \) 4 * q - 192 * q^4 - 2032 * q^6 - 11108 * q^9	\( 4 q - 192 q^{4} - 2032 q^{6} - 11108 q^{9} + 9088 q^{11} - 15024 q^{14} + 40704 q^{16} - 77520 q^{19} - 138192 q^{21} + 263040 q^{24} - 114032 q^{26} + 144520 q^{29} + 613648 q^{31} + 906736 q^{34} - 439616 q^{36} - 1549456 q^{39} + 528728 q^{41} - 2868224 q^{44} - 2606832 q^{46} + 2330828 q^{49} + 2332688 q^{51} - 2205920 q^{54} + 1898880 q^{56} + 2240240 q^{59} + 4514088 q^{61} + 10293248 q^{64} - 13128704 q^{66} + 1490544 q^{69} + 1243568 q^{71} - 5302544 q^{74} + 1775360 q^{76} - 8666080 q^{79} - 4795756 q^{81} + 796416 q^{84} + 21596368 q^{86} - 12051240 q^{89} - 10704592 q^{91} - 11721584 q^{94} + 26610688 q^{96} - 5781376 q^{99}+O(q^{100}) \) 4 * q - 192 * q^4 - 2032 * q^6 - 11108 * q^9 + 9088 * q^11 - 15024 * q^14 + 40704 * q^16 - 77520 * q^19 - 138192 * q^21 + 263040 * q^24 - 114032 * q^26 + 144520 * q^29 + 613648 * q^31 + 906736 * q^34 - 439616 * q^36 - 1549456 * q^39 + 528728 * q^41 - 2868224 * q^44 - 2606832 * q^46 + 2330828 * q^49 + 2332688 * q^51 - 2205920 * q^54 + 1898880 * q^56 + 2240240 * q^59 + 4514088 * q^61 + 10293248 * q^64 - 13128704 * q^66 + 1490544 * q^69 + 1243568 * q^71 - 5302544 * q^74 + 1775360 * q^76 - 8666080 * q^79 - 4795756 * q^81 + 796416 * q^84 + 21596368 * q^86 - 12051240 * q^89 - 10704592 * q^91 - 11721584 * q^94 + 26610688 * q^96 - 5781376 * q^99

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