LMFDB - Embedded newform 471.4.b.a.313.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [471,4,Mod(313,471)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(471, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("471.313");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 471 = 3 \cdot 157 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	471.b (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$27.7898996127$
Analytic rank:	$0$
Dimension:	$40$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			313.6
Character	$\chi$	$=$	471.313
Dual form			471.4.b.a.313.35

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.30375i q^{2} -3.00000 q^{3} -10.5223 q^{4} -4.32730i q^{5} +12.9113i q^{6} +21.9616i q^{7} +10.8553i q^{8} +9.00000 q^{9} +O(q^{10})$	\(q-4.30375i q^{2} -3.00000 q^{3} -10.5223 q^{4} -4.32730i q^{5} +12.9113i q^{6} +21.9616i q^{7} +10.8553i q^{8} +9.00000 q^{9} -18.6236 q^{10} +64.5166 q^{11} +31.5669 q^{12} -82.9568 q^{13} +94.5172 q^{14} +12.9819i q^{15} -37.4597 q^{16} +98.4405 q^{17} -38.7338i q^{18} +51.7876 q^{19} +45.5331i q^{20} -65.8847i q^{21} -277.664i q^{22} -83.6546i q^{23} -32.5659i q^{24} +106.274 q^{25} +357.026i q^{26} -27.0000 q^{27} -231.086i q^{28} -285.149i q^{29} +55.8709 q^{30} -237.068 q^{31} +248.060i q^{32} -193.550 q^{33} -423.664i q^{34} +95.0344 q^{35} -94.7006 q^{36} +298.808 q^{37} -222.881i q^{38} +248.870 q^{39} +46.9742 q^{40} -362.253i q^{41} -283.552 q^{42} +280.142i q^{43} -678.862 q^{44} -38.9457i q^{45} -360.029 q^{46} -348.798 q^{47} +112.379 q^{48} -139.311 q^{49} -457.379i q^{50} -295.322 q^{51} +872.896 q^{52} -271.688i q^{53} +116.201i q^{54} -279.183i q^{55} -238.400 q^{56} -155.363 q^{57} -1227.21 q^{58} +546.326i q^{59} -136.599i q^{60} -128.182i q^{61} +1020.28i q^{62} +197.654i q^{63} +767.911 q^{64} +358.979i q^{65} +832.991i q^{66} +64.8197 q^{67} -1035.82 q^{68} +250.964i q^{69} -409.004i q^{70} -561.033 q^{71} +97.6978i q^{72} -152.189i q^{73} -1285.99i q^{74} -318.823 q^{75} -544.925 q^{76} +1416.89i q^{77} -1071.08i q^{78} -994.744i q^{79} +162.100i q^{80} +81.0000 q^{81} -1559.05 q^{82} -848.450i q^{83} +693.258i q^{84} -425.982i q^{85} +1205.66 q^{86} +855.447i q^{87} +700.348i q^{88} +749.365 q^{89} -167.613 q^{90} -1821.86i q^{91} +880.238i q^{92} +711.205 q^{93} +1501.14i q^{94} -224.101i q^{95} -744.180i q^{96} -901.408i q^{97} +599.560i q^{98} +580.649 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q - 120 q^{3} - 164 q^{4} + 360 q^{9}+O(q^{10}) $ 40 * q - 120 * q^3 - 164 * q^4 + 360 * q^9	$ 40 q - 120 q^{3} - 164 q^{4} + 360 q^{9} - 174 q^{10} + 110 q^{11} + 492 q^{12} - 194 q^{13} - 78 q^{14} + 796 q^{16} - 150 q^{17} + 172 q^{19} - 668 q^{25} - 1080 q^{27} + 522 q^{30} + 66 q^{31} - 330 q^{33} - 400 q^{35} - 1476 q^{36} - 142 q^{37} + 582 q^{39} + 1160 q^{40} + 234 q^{42} - 1182 q^{44} + 132 q^{46} - 244 q^{47} - 2388 q^{48} - 3786 q^{49} + 450 q^{51} + 1596 q^{52} - 256 q^{56} - 516 q^{57} - 1780 q^{58} - 1790 q^{64} - 320 q^{67} + 1646 q^{68} + 712 q^{71} + 2004 q^{75} - 3004 q^{76} + 3240 q^{81} + 4112 q^{82} - 4198 q^{86} + 366 q^{89} - 1566 q^{90} - 198 q^{93} + 990 q^{99}+O(q^{100}) $ 40 * q - 120 * q^3 - 164 * q^4 + 360 * q^9 - 174 * q^10 + 110 * q^11 + 492 * q^12 - 194 * q^13 - 78 * q^14 + 796 * q^16 - 150 * q^17 + 172 * q^19 - 668 * q^25 - 1080 * q^27 + 522 * q^30 + 66 * q^31 - 330 * q^33 - 400 * q^35 - 1476 * q^36 - 142 * q^37 + 582 * q^39 + 1160 * q^40 + 234 * q^42 - 1182 * q^44 + 132 * q^46 - 244 * q^47 - 2388 * q^48 - 3786 * q^49 + 450 * q^51 + 1596 * q^52 - 256 * q^56 - 516 * q^57 - 1780 * q^58 - 1790 * q^64 - 320 * q^67 + 1646 * q^68 + 712 * q^71 + 2004 * q^75 - 3004 * q^76 + 3240 * q^81 + 4112 * q^82 - 4198 * q^86 + 366 * q^89 - 1566 * q^90 - 198 * q^93 + 990 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/471\mathbb{Z}\right)^\times$.

$n$	$158$	$319$
$\chi(n)$	$1$	$-1$

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 $

$2$		−	4.30375i		−	1.52161i	−0.648983	−	0.760803i	$-0.724805\pi$
$2$		−	4.30375i		−	1.52161i	0.648983	−	0.760803i	$-0.275195\pi$
$3$	−3.00000			−0.577350
$3$	−3.00000			−0.577350
$4$	−10.5223			−1.31529
$5$		−	4.32730i		−	0.387046i	−0.981096	−	0.193523i	$-0.938009\pi$
$5$		−	4.32730i		−	0.387046i	0.981096	−	0.193523i	$-0.0619913\pi$
$6$			12.9113i			0.878500i
$7$			21.9616i			1.18581i	0.805271	+	0.592907i	$0.202020\pi$
$7$			21.9616i			1.18581i	−0.805271	+	0.592907i	$0.797980\pi$
$8$			10.8553i			0.479742i
$9$	9.00000			0.333333
$10$	−18.6236			−0.588931
$11$	64.5166			1.76841			0.884204	−	0.467101i	$-0.154702\pi$
$11$	64.5166			1.76841			0.884204	+	0.467101i	$0.154702\pi$
$12$	31.5669			0.759381
$13$	−82.9568			−1.76985			−0.884926	−	0.465731i	$-0.845791\pi$
$13$	−82.9568			−1.76985			−0.884926	+	0.465731i	$0.845791\pi$
$14$	94.5172			1.80434
$15$			12.9819i			0.223461i
$16$	−37.4597			−0.585308
$17$	98.4405			1.40443			0.702216	−	0.711964i	$-0.252194\pi$
$17$	98.4405			1.40443			0.702216	+	0.711964i	$0.252194\pi$
$18$		−	38.7338i		−	0.507202i
$19$	51.7876			0.625310			0.312655	−	0.949867i	$-0.398782\pi$
$19$	51.7876			0.625310			0.312655	+	0.949867i	$0.398782\pi$
$20$			45.5331i			0.509076i
$21$		−	65.8847i		−	0.684630i
$22$		−	277.664i		−	2.69082i
$23$		−	83.6546i		−	0.758400i	−0.925315	−	0.379200i	$-0.876199\pi$
$23$		−	83.6546i		−	0.758400i	0.925315	−	0.379200i	$-0.123801\pi$
$24$		−	32.5659i		−	0.276979i
$25$	106.274			0.850196
$26$			357.026i			2.69302i
$27$	−27.0000			−0.192450
$28$		−	231.086i		−	1.55968i
$29$		−	285.149i		−	1.82589i	−0.408080	−	0.912946i	$-0.633802\pi$
$29$		−	285.149i		−	1.82589i	0.408080	−	0.912946i	$-0.366198\pi$
$30$	55.8709			0.340020
$31$	−237.068			−1.37351			−0.686754	−	0.726890i	$-0.740965\pi$
$31$	−237.068			−1.37351			−0.686754	+	0.726890i	$0.740965\pi$
$32$			248.060i			1.37035i
$33$	−193.550			−1.02099
$34$		−	423.664i		−	2.13699i
$35$	95.0344			0.458964
$36$	−94.7006			−0.438429
$37$	298.808			1.32767			0.663833	−	0.747881i	$-0.268928\pi$
$37$	298.808			1.32767			0.663833	+	0.747881i	$0.268928\pi$
$38$		−	222.881i		−	0.951476i
$39$	248.870			1.02182
$40$	46.9742			0.185682
$41$		−	362.253i		−	1.37986i	−0.723874	−	0.689932i	$-0.757640\pi$
$41$		−	362.253i		−	1.37986i	0.723874	−	0.689932i	$-0.242360\pi$
$42$	−283.552			−1.04174
$43$			280.142i			0.993517i	0.867889	+	0.496758i	$0.165476\pi$
$43$			280.142i			0.993517i	−0.867889	+	0.496758i	$0.834524\pi$
$44$	−678.862			−2.32596
$45$		−	38.9457i		−	0.129015i
$46$	−360.029			−1.15399
$47$	−348.798			−1.08250			−0.541249	−	0.840862i	$-0.682049\pi$
$47$	−348.798			−1.08250			−0.541249	+	0.840862i	$0.682049\pi$
$48$	112.379			0.337928
$49$	−139.311			−0.406154
$50$		−	457.379i		−	1.29366i
$51$	−295.322			−0.810849
$52$	872.896			2.32786
$53$		−	271.688i		−	0.704136i	−0.935974	−	0.352068i	$-0.885479\pi$
$53$		−	271.688i		−	0.704136i	0.935974	−	0.352068i	$-0.114521\pi$
$54$			116.201i			0.292833i
$55$		−	279.183i		−	0.684455i
$56$	−238.400			−0.568884
$57$	−155.363			−0.361023
$58$	−1227.21			−2.77829
$59$			546.326i			1.20552i	0.797924	+	0.602759i	$0.205932\pi$
$59$			546.326i			1.20552i	−0.797924	+	0.602759i	$0.794068\pi$
$60$		−	136.599i		−	0.293915i
$61$		−	128.182i		−	0.269050i	−0.990910	−	0.134525i	$-0.957049\pi$
$61$		−	128.182i		−	0.269050i	0.990910	−	0.134525i	$-0.0429509\pi$
$62$			1020.28i			2.08994i
$63$			197.654i			0.395271i
$64$	767.911			1.49983
$65$			358.979i			0.685014i
$66$			832.991i			1.55355i
$67$	64.8197			0.118194			0.0590969	−	0.998252i	$-0.481178\pi$
$67$	64.8197			0.118194			0.0590969	+	0.998252i	$0.481178\pi$
$68$	−1035.82			−1.84723
$69$			250.964i			0.437862i
$70$		−	409.004i		−	0.698363i
$71$	−561.033			−0.937779			−0.468890	−	0.883257i	$-0.655346\pi$
$71$	−561.033			−0.937779			−0.468890	+	0.883257i	$0.655346\pi$
$72$			97.6978i			0.159914i
$73$		−	152.189i		−	0.244006i	−0.992530	−	0.122003i	$-0.961068\pi$
$73$		−	152.189i		−	0.244006i	0.992530	−	0.122003i	$-0.0389317\pi$
$74$		−	1285.99i		−	2.02019i
$75$	−318.823			−0.490861
$76$	−544.925			−0.822462
$77$			1416.89i			2.09700i
$78$		−	1071.08i		−	1.55482i
$79$		−	994.744i		−	1.41668i	−0.705873	−	0.708338i	$-0.749445\pi$
$79$		−	994.744i		−	1.41668i	0.705873	−	0.708338i	$-0.250555\pi$
$80$			162.100i			0.226541i
$81$	81.0000			0.111111
$82$	−1559.05			−2.09961
$83$		−	848.450i		−	1.12204i	−0.827801	−	0.561021i	$-0.810409\pi$
$83$		−	848.450i		−	1.12204i	0.827801	−	0.561021i	$-0.189591\pi$
$84$			693.258i			0.900484i
$85$		−	425.982i		−	0.543579i
$86$	1205.66			1.51174
$87$			855.447i			1.05418i
$88$			700.348i			0.848379i
$89$	749.365			0.892500			0.446250	−	0.894908i	$-0.352759\pi$
$89$	749.365			0.892500			0.446250	+	0.894908i	$0.352759\pi$
$90$	−167.613			−0.196310
$91$		−	1821.86i		−	2.09872i
$92$			880.238i			0.997513i
$93$	711.205			0.792995
$94$			1501.14i			1.64714i
$95$		−	224.101i		−	0.242024i
$96$		−	744.180i		−	0.791172i
$97$		−	901.408i		−	0.943547i	−0.881720	−	0.471774i	$-0.843614\pi$
$97$		−	901.408i		−	0.943547i	0.881720	−	0.471774i	$-0.156386\pi$
$98$			599.560i			0.618007i
$99$	580.649			0.589469
$100$	−1118.25			−1.11825
$101$	931.144			0.917350			0.458675	−	0.888604i	$-0.348324\pi$
$101$	931.144			0.917350			0.458675	+	0.888604i	$0.348324\pi$
$102$			1270.99i			1.23379i
$103$			1305.98i			1.24934i	0.780889	+	0.624670i	$0.214766\pi$
$103$			1305.98i			1.24934i	−0.780889	+	0.624670i	$0.785234\pi$
$104$		−	900.522i		−	0.849072i
$105$	−285.103			−0.264983
$106$	−1169.28			−1.07142
$107$		−	1032.22i		−	0.932603i	−0.884626	−	0.466302i	$-0.845586\pi$
$107$		−	1032.22i		−	0.932603i	0.884626	−	0.466302i	$-0.154414\pi$
$108$	284.102			0.253127
$109$	−433.778			−0.381178			−0.190589	−	0.981670i	$-0.561040\pi$
$109$	−433.778			−0.381178			−0.190589	+	0.981670i	$0.561040\pi$
$110$	−1201.53			−1.04147
$111$	−896.423			−0.766529
$112$		−	822.675i		−	0.694067i
$113$	−1233.25			−1.02667			−0.513337	−	0.858187i	$-0.671591\pi$
$113$	−1233.25			−1.02667			−0.513337	+	0.858187i	$0.671591\pi$
$114$			668.644i			0.549335i
$115$	−361.999			−0.293535
$116$			3000.42i			2.40157i
$117$	−746.611			−0.589951
$118$	2351.25			1.83432
$119$			2161.91i			1.66539i
$120$	−140.923			−0.107203
$121$	2831.39			2.12727
$122$	−551.665			−0.409389
$123$			1086.76i			0.796665i
$124$	2494.50			1.80656
$125$		−	1000.79i		−	0.716110i
$126$	850.655			0.601447
$127$	170.563			0.119174			0.0595868	−	0.998223i	$-0.481022\pi$
$127$	170.563			0.119174			0.0595868	+	0.998223i	$0.481022\pi$
$128$		−	1320.42i		−	0.911794i
$129$		−	840.425i		−	0.573607i
$130$	1544.96			1.04232
$131$			1647.93i			1.09908i	0.835466	+	0.549542i	$0.185198\pi$
$131$			1647.93i			1.09908i	−0.835466	+	0.549542i	$0.814802\pi$
$132$	2036.59			1.34290
$133$			1137.34i			0.741502i
$134$		−	278.968i		−	0.179844i
$135$			116.837i			0.0744870i
$136$			1068.60i			0.673764i
$137$		−	2162.60i		−	1.34864i	−0.738440	−	0.674319i	$-0.764437\pi$
$137$		−	2162.60i		−	1.34864i	0.738440	−	0.674319i	$-0.235563\pi$
$138$	1080.09			0.666254
$139$		−	750.212i		−	0.457785i	−0.973452	−	0.228893i	$-0.926489\pi$
$139$		−	750.212i		−	0.457785i	0.973452	−	0.228893i	$-0.0735105\pi$
$140$	−999.979			−0.603669
$141$	1046.39			0.624981
$142$			2414.55i			1.42693i
$143$	−5352.09			−3.12982
$144$	−337.138			−0.195103
$145$	−1233.93			−0.706703
$146$	−654.986			−0.371281
$147$	417.933			0.234493
$148$	−3144.14			−1.74626
$149$		−	1013.38i		−	0.557177i	−0.960411	−	0.278589i	$-0.910133\pi$
$149$		−	1013.38i		−	0.557177i	0.960411	−	0.278589i	$-0.0898666\pi$
$150$			1372.14i			0.746897i
$151$		−	1933.66i		−	1.04211i	−0.853523	−	0.521055i	$-0.825539\pi$
$151$		−	1933.66i		−	1.04211i	0.853523	−	0.521055i	$-0.174461\pi$
$152$			562.171i			0.299987i
$153$	885.965			0.468144
$154$	6097.93			3.19081
$155$			1025.87i			0.531610i
$156$	−2618.69			−1.34399
$157$	−1930.71	+	377.186i	−0.981446	+	0.191737i
$157$	−1930.71	+	377.186i	−0.981446	+	0.191737i
$158$	−4281.13			−2.15562
$159$			815.063i			0.406533i
$160$	1073.43			0.530388
$161$	1837.19			0.899321
$162$		−	348.604i		−	0.169067i
$163$			675.639i			0.324663i	0.986736	+	0.162332i	$0.0519015\pi$
$163$			675.639i			0.324663i	−0.986736	+	0.162332i	$0.948099\pi$
$164$			3811.73i			1.81492i
$165$			837.548i			0.395170i
$166$	−3651.52			−1.70731
$167$	4022.77			1.86402			0.932011	−	0.362431i	$-0.118053\pi$
$167$	4022.77			1.86402			0.932011	+	0.362431i	$0.118053\pi$
$168$	715.199			0.328445
$169$	4684.83			2.13238
$170$	−1833.32			−0.827113
$171$	466.089			0.208437
$172$		−	2947.73i		−	1.30676i
$173$	−1148.22			−0.504610			−0.252305	−	0.967648i	$-0.581189\pi$
$173$	−1148.22			−0.504610			−0.252305	+	0.967648i	$0.581189\pi$
$174$	3681.63			1.60405
$175$			2333.95i			1.00817i
$176$	−2416.78			−1.03506
$177$		−	1638.98i		−	0.696006i
$178$		−	3225.08i		−	1.35803i
$179$		−	2952.37i		−	1.23280i	−0.787434	−	0.616399i	$-0.788591\pi$
$179$		−	2952.37i		−	1.23280i	0.787434	−	0.616399i	$-0.211409\pi$
$180$			409.798i			0.169692i
$181$		−	3306.30i		−	1.35777i	−0.734247	−	0.678883i	$-0.762465\pi$
$181$		−	3306.30i		−	1.35777i	0.734247	−	0.678883i	$-0.237535\pi$
$182$	−7840.85			−3.19342
$183$			384.547i			0.155336i
$184$	908.097			0.363836
$185$		−	1293.03i		−	0.513867i
$186$		−	3060.85i		−	1.20663i
$187$	6351.05			2.48361
$188$	3670.16			1.42380
$189$		−	592.963i		−	0.228210i
$190$	−964.474			−0.368265
$191$		−	5047.97i		−	1.91235i	−0.292803	−	0.956173i	$-0.594588\pi$
$191$		−	5047.97i		−	1.91235i	0.292803	−	0.956173i	$-0.405412\pi$
$192$	−2303.73			−0.865925
$193$	228.756			0.0853172			0.0426586	−	0.999090i	$-0.486417\pi$
$193$	228.756			0.0853172			0.0426586	+	0.999090i	$0.486417\pi$
$194$	−3879.44			−1.43571
$195$		−	1076.94i		−	0.395493i
$196$	1465.87			0.534209
$197$	1278.99			0.462560			0.231280	−	0.972887i	$-0.425709\pi$
$197$	1278.99			0.462560			0.231280	+	0.972887i	$0.425709\pi$
$198$		−	2498.97i		−	0.896940i
$199$	423.523			0.150868			0.0754340	−	0.997151i	$-0.475966\pi$
$199$	423.523			0.150868			0.0754340	+	0.997151i	$0.475966\pi$
$200$			1153.64i			0.407874i
$201$	−194.459			−0.0682392
$202$		−	4007.42i		−	1.39585i
$203$	6262.33			2.16517
$204$	3107.46			1.06650
$205$	−1567.58			−0.534070
$206$	5620.61			1.90100
$207$		−	752.892i		−	0.252800i
$208$	3107.54			1.03591
$209$	3341.16			1.10580
$210$			1227.01i			0.403200i
$211$			715.224i			0.233356i	0.993170	+	0.116678i	$0.0372245\pi$
$211$			715.224i			0.233356i	−0.993170	+	0.116678i	$0.962775\pi$
$212$			2858.78i			0.926140i
$213$	1683.10			0.541427
$214$	−4442.42			−1.41906
$215$	1212.26			0.384536
$216$		−	293.093i		−	0.0923263i
$217$		−	5206.39i		−	1.62872i
$218$			1866.87i			0.580002i
$219$			456.568i			0.140877i
$220$			2937.64i			0.900254i
$221$	−8166.31			−2.48564
$222$			3857.98i			1.16635i
$223$			1037.42i			0.311528i	0.987794	+	0.155764i	$0.0497840\pi$
$223$			1037.42i			0.311528i	−0.987794	+	0.155764i	$0.950216\pi$
$224$	−5447.79			−1.62498
$225$	956.470			0.283399
$226$			5307.60i			1.56219i
$227$		−	1298.59i		−	0.379694i	−0.981814	−	0.189847i	$-0.939201\pi$
$227$		−	1298.59i		−	0.379694i	0.981814	−	0.189847i	$-0.0607992\pi$
$228$	1634.77			0.474849
$229$		−	562.065i		−	0.162193i	−0.996706	−	0.0810967i	$-0.974158\pi$
$229$		−	562.065i		−	0.162193i	0.996706	−	0.0810967i	$-0.0258423\pi$
$230$			1557.95i			0.446645i
$231$		−	4250.66i		−	1.21070i
$232$	3095.38			0.875956
$233$	2215.21			0.622847			0.311423	−	0.950271i	$-0.399194\pi$
$233$	2215.21			0.622847			0.311423	+	0.950271i	$0.399194\pi$
$234$			3213.23i			0.897673i
$235$			1509.35i			0.418976i
$236$		−	5748.60i		−	1.58560i
$237$			2984.23i			0.817919i
$238$	9304.32			2.53407
$239$	4375.88			1.18432			0.592159	−	0.805821i	$-0.298276\pi$
$239$	4375.88			1.18432			0.592159	+	0.805821i	$0.298276\pi$
$240$		−	486.299i		−	0.130794i
$241$		−	2696.45i		−	0.720721i	−0.932813	−	0.360360i	$-0.882654\pi$
$241$		−	2696.45i		−	0.720721i	0.932813	−	0.360360i	$-0.117346\pi$
$242$		−	12185.6i		−	3.23686i
$243$	−243.000			−0.0641500
$244$			1348.77i			0.353878i
$245$			602.840i			0.157200i
$246$	4677.14			1.21221
$247$	−4296.14			−1.10671
$248$		−	2573.45i		−	0.658929i
$249$			2545.35i			0.647811i
$250$	−4307.17			−1.08964
$251$			6874.87i			1.72884i	0.502774	+	0.864418i	$0.332313\pi$
$251$			6874.87i			1.72884i	−0.502774	+	0.864418i	$0.667687\pi$
$252$		−	2079.77i		−	0.519895i
$253$		−	5397.11i		−	1.34116i
$254$		−	734.062i		−	0.181335i
$255$			1277.95i			0.313835i
$256$	460.530			0.112434
$257$	−3452.31			−0.837936			−0.418968	−	0.908001i	$-0.637608\pi$
$257$	−3452.31			−0.837936			−0.418968	+	0.908001i	$0.637608\pi$
$258$	−3616.98			−0.872804
$259$			6562.29i			1.57437i
$260$		−	3777.28i		−	0.900989i
$261$		−	2566.34i		−	0.608631i
$262$	7092.27			1.67237
$263$	4822.18			1.13060			0.565301	−	0.824884i	$-0.308760\pi$
$263$	4822.18			1.13060			0.565301	+	0.824884i	$0.308760\pi$
$264$		−	2101.04i		−	0.489812i
$265$	−1175.67			−0.272533
$266$	4894.82			1.12827
$267$	−2248.09			−0.515285
$268$	−682.051			−0.155459
$269$			1177.30i			0.266845i	0.991059	+	0.133422i	$0.0425967\pi$
$269$			1177.30i			0.266845i	−0.991059	+	0.133422i	$0.957403\pi$
$270$	502.838			0.113340
$271$			1331.93i			0.298558i	0.988795	+	0.149279i	$0.0476952\pi$
$271$			1331.93i			0.298558i	−0.988795	+	0.149279i	$0.952305\pi$
$272$	−3687.56			−0.822025
$273$			5465.59i			1.21169i
$274$	−9307.30			−2.05210
$275$	6856.47			1.50349
$276$		−	2640.72i		−	0.575914i
$277$	−2113.48			−0.458436			−0.229218	−	0.973375i	$-0.573617\pi$
$277$	−2113.48			−0.458436			−0.229218	+	0.973375i	$0.573617\pi$
$278$	−3228.73			−0.696569
$279$	−2133.61			−0.457836
$280$			1031.63i			0.220184i
$281$	−2797.18			−0.593829			−0.296915	−	0.954904i	$-0.595958\pi$
$281$	−2797.18			−0.593829			−0.296915	+	0.954904i	$0.595958\pi$
$282$		−	4503.42i		−	0.950975i
$283$	2916.58			0.612624			0.306312	−	0.951931i	$-0.400905\pi$
$283$	2916.58			0.612624			0.306312	+	0.951931i	$0.400905\pi$
$284$	5903.35			1.23345
$285$			672.302i			0.139732i
$286$			23034.1i			4.76236i
$287$	7955.65			1.63626
$288$			2232.54i			0.456784i
$289$	4777.53			0.972427
$290$			5310.51i			1.07532i
$291$			2704.22i			0.544757i
$292$			1601.38i			0.320938i
$293$			7178.44i			1.43129i	0.698462	+	0.715647i	$0.253868\pi$
$293$			7178.44i			1.43129i	−0.698462	+	0.715647i	$0.746132\pi$
$294$		−	1798.68i		−	0.356806i
$295$	2364.12			0.466590
$296$			3243.65i			0.636937i
$297$	−1741.95			−0.340330
$298$	−4361.34			−0.847804
$299$			6939.72i			1.34226i
$300$	3354.75			0.645622
$301$	−6152.36			−1.17813
$302$	−8321.98			−1.58568
$303$	−2793.43			−0.529632
$304$	−1939.95			−0.365999
$305$	−554.684			−0.104135
$306$		−	3812.97i		−	0.712331i
$307$			5520.41i			1.02627i	0.858307	+	0.513137i	$0.171517\pi$
$307$			5520.41i			1.02627i	−0.858307	+	0.513137i	$0.828483\pi$
$308$		−	14908.9i		−	2.75816i
$309$		−	3917.94i		−	0.721306i
$310$	4415.07			0.808901
$311$	−5187.92			−0.945916			−0.472958	−	0.881085i	$-0.656814\pi$
$311$	−5187.92			−0.945916			−0.472958	+	0.881085i	$0.656814\pi$
$312$			2701.57i			0.490212i
$313$	5897.70			1.06504			0.532520	−	0.846417i	$-0.321245\pi$
$313$	5897.70			1.06504			0.532520	+	0.846417i	$0.321245\pi$
$314$	1623.32	+	8309.28i	0.291748	+	1.49338i
$315$	855.309			0.152988
$316$			10467.0i			1.86334i
$317$	−11044.6			−1.95687			−0.978435	−	0.206555i	$-0.933775\pi$
$317$	−11044.6			−1.95687			−0.978435	+	0.206555i	$0.933775\pi$
$318$	3507.83			0.618583
$319$		−	18396.9i		−	3.22892i
$320$		−	3322.98i		−	0.580501i
$321$			3096.66i			0.538439i
$322$		−	7906.80i		−	1.36841i
$323$	5098.00			0.878205
$324$	−852.305			−0.146143
$325$	−8816.19			−1.50472
$326$	2907.78			0.494010
$327$	1301.33			0.220073
$328$	3932.37			0.661978
$329$		−	7660.16i		−	1.28364i
$330$	3604.60			0.601293
$331$	3836.63			0.637100			0.318550	−	0.947906i	$-0.396804\pi$
$331$	3836.63			0.637100			0.318550	+	0.947906i	$0.396804\pi$
$332$			8927.64i			1.47581i
$333$	2689.27			0.442556
$334$		−	17313.0i		−	2.83631i
$335$		−	280.494i		−	0.0457464i
$336$			2468.02i			0.400720i
$337$			9027.31i			1.45920i	0.683876	+	0.729598i	$0.260293\pi$
$337$			9027.31i			1.45920i	−0.683876	+	0.729598i	$0.739707\pi$
$338$		−	20162.4i		−	3.24464i
$339$	3699.74			0.592751
$340$			4482.30i			0.714962i
$341$	−15294.8			−2.42892
$342$		−	2005.93i		−	0.317159i
$343$			4473.33i			0.704190i
$344$	−3041.03			−0.476631
$345$	1086.00			0.169473
$346$			4941.65i			0.767817i
$347$	116.613			0.0180407			0.00902036	−	0.999959i	$-0.497129\pi$
$347$	116.613			0.0180407			0.00902036	+	0.999959i	$0.497129\pi$
$348$		−	9001.27i		−	1.38655i
$349$	−4583.01			−0.702931			−0.351465	−	0.936201i	$-0.614316\pi$
$349$	−4583.01			−0.702931			−0.351465	+	0.936201i	$0.614316\pi$
$350$	10044.8			1.53404
$351$	2239.83			0.340608
$352$			16004.0i			2.42334i
$353$	13035.1			1.96541			0.982705	−	0.185179i	$-0.0592867\pi$
$353$	13035.1			1.96541			0.982705	+	0.185179i	$0.0592867\pi$
$354$	−7053.75			−1.05905
$355$			2427.76i			0.362963i
$356$	−7885.03			−1.17389
$357$		−	6485.73i		−	0.961516i
$358$	−12706.3			−1.87583
$359$			9089.43i			1.33627i	0.744039	+	0.668136i	$0.232908\pi$
$359$			9089.43i			1.33627i	−0.744039	+	0.668136i	$0.767092\pi$
$360$	422.768			0.0618939
$361$	−4177.04			−0.608987
$362$	−14229.5			−2.06598
$363$	−8494.18			−1.22818
$364$			19170.2i			2.76041i
$365$	−658.570			−0.0944414
$366$	1655.00			0.236361
$367$			9184.41i			1.30633i	0.757217	+	0.653164i	$0.226559\pi$
$367$			9184.41i			1.30633i	−0.757217	+	0.653164i	$0.773441\pi$
$368$			3133.68i			0.443898i
$369$		−	3260.28i		−	0.459955i
$370$	−5564.88			−0.781904
$371$	5966.69			0.834974
$372$	−7483.50			−1.04302
$373$			11318.1i			1.57112i	0.618787	+	0.785559i	$0.287624\pi$
$373$			11318.1i			1.57112i	−0.618787	+	0.785559i	$0.712376\pi$
$374$		−	27333.3i		−	3.77907i
$375$			3002.38i			0.413446i
$376$		−	3786.31i		−	0.519320i
$377$			23655.1i			3.23156i
$378$	−2551.96			−0.347246
$379$		−	3742.05i		−	0.507166i	−0.967314	−	0.253583i	$-0.918391\pi$
$379$		−	3742.05i		−	0.507166i	0.967314	−	0.253583i	$-0.0816091\pi$
$380$			2358.05i			0.318330i
$381$	−511.690			−0.0688049
$382$	−21725.2			−2.90984
$383$		−	235.431i		−	0.0314098i	−0.999877	−	0.0157049i	$-0.995001\pi$
$383$		−	235.431i		−	0.0314098i	0.999877	−	0.0157049i	$-0.00499924\pi$
$384$			3961.26i			0.526425i
$385$	6131.29			0.811636
$386$		−	984.509i		−	0.129819i
$387$			2521.28i			0.331172i
$388$			9484.87i			1.24103i
$389$	6829.09			0.890099			0.445050	−	0.895506i	$-0.353186\pi$
$389$	6829.09			0.890099			0.445050	+	0.895506i	$0.353186\pi$
$390$	−4634.87			−0.601784
$391$		−	8235.01i		−	1.06512i
$392$		−	1512.26i		−	0.194849i
$393$		−	4943.78i		−	0.634557i
$394$		−	5504.46i		−	0.703834i
$395$	−4304.56			−0.548319
$396$	−6109.76			−0.775321
$397$			454.543i			0.0574632i	0.999587	+	0.0287316i	$0.00914680\pi$
$397$			454.543i			0.0574632i	−0.999587	+	0.0287316i	$0.990853\pi$
$398$		−	1822.74i		−	0.229562i
$399$		−	3412.01i		−	0.428106i
$400$	−3981.01			−0.497627
$401$			3116.01i			0.388045i	0.980997	+	0.194022i	$0.0621535\pi$
$401$			3116.01i			0.388045i	−0.980997	+	0.194022i	$0.937847\pi$
$402$			836.903i			0.103833i
$403$	19666.4			2.43091
$404$	−9797.77			−1.20658
$405$		−	350.511i		−	0.0430051i
$406$		−	26951.5i		−	3.29453i
$407$	19278.1			2.34786
$408$		−	3205.81i		−	0.388998i
$409$		−	6461.39i		−	0.781161i	−0.920569	−	0.390581i	$-0.872274\pi$
$409$		−	6461.39i		−	0.781161i	0.920569	−	0.390581i	$-0.127726\pi$
$410$			6746.47i			0.812645i
$411$			6487.81i			0.778637i
$412$		−	13741.9i		−	1.64324i
$413$	−11998.2			−1.42952
$414$	−3240.26			−0.384662
$415$	−3671.50			−0.434282
$416$		−	20578.3i		−	2.42532i
$417$			2250.64i			0.264303i
$418$		−	14379.5i		−	1.68260i
$419$	−3580.54			−0.417473			−0.208736	−	0.977972i	$-0.566935\pi$
$419$	−3580.54			−0.417473			−0.208736	+	0.977972i	$0.566935\pi$
$420$	2999.94			0.348528
$421$			12837.5i			1.48614i	0.669215	+	0.743068i	$0.266630\pi$
$421$			12837.5i			1.48614i	−0.669215	+	0.743068i	$0.733370\pi$
$422$	3078.15			0.355076
$423$	−3139.18			−0.360833
$424$	2949.26			0.337803
$425$	10461.7			1.19404
$426$		−	7243.64i		−	0.823839i
$427$	2815.09			0.319044
$428$			10861.3i			1.22664i
$429$	16056.3			1.80700
$430$		−	5217.26i		−	0.585113i
$431$	−7138.93			−0.797842			−0.398921	−	0.916985i	$-0.630615\pi$
$431$	−7138.93			−0.797842			−0.398921	+	0.916985i	$0.630615\pi$
$432$	1011.41			0.112643
$433$		−	4571.48i		−	0.507370i	−0.967287	−	0.253685i	$-0.918357\pi$
$433$		−	4571.48i		−	0.507370i	0.967287	−	0.253685i	$-0.0816427\pi$
$434$	−22407.0			−2.47828
$435$	3701.78			0.408015
$436$	4564.33			0.501358
$437$		−	4332.28i		−	0.474235i
$438$	1964.96			0.214359
$439$			8439.12i			0.917488i	0.888568	+	0.458744i	$0.151700\pi$
$439$			8439.12i			0.917488i	−0.888568	+	0.458744i	$0.848300\pi$
$440$	3030.62			0.328361
$441$	−1253.80			−0.135385
$442$			35145.8i			3.78216i
$443$			8263.05i			0.886206i	0.896471	+	0.443103i	$0.146122\pi$
$443$			8263.05i			0.886206i	−0.896471	+	0.443103i	$0.853878\pi$
$444$	9432.42			1.00820
$445$		−	3242.73i		−	0.345438i
$446$	4464.80			0.474023
$447$			3040.14i			0.321686i
$448$			16864.5i			1.77851i
$449$			7863.65i			0.826522i	0.910613	+	0.413261i	$0.135610\pi$
$449$			7863.65i			0.826522i	−0.910613	+	0.413261i	$0.864390\pi$
$450$		−	4116.41i		−	0.431221i
$451$		−	23371.3i		−	2.44016i
$452$	12976.6			1.35037
$453$			5800.97i			0.601663i
$454$	−5588.82			−0.577745
$455$	−7883.75			−0.812298
$456$		−	1686.51i		−	0.173198i
$457$	13472.7			1.37905			0.689525	−	0.724262i	$-0.257819\pi$
$457$	13472.7			1.37905			0.689525	+	0.724262i	$0.257819\pi$
$458$	−2418.99			−0.246795
$459$	−2657.89			−0.270283
$460$	3809.06			0.386083
$461$	7642.34			0.772102			0.386051	−	0.922477i	$-0.373839\pi$
$461$	7642.34			0.772102			0.386051	+	0.922477i	$0.373839\pi$
$462$	−18293.8			−1.84222
$463$			16014.0i			1.60741i	0.595026	+	0.803706i	$0.297142\pi$
$463$			16014.0i			1.60741i	−0.595026	+	0.803706i	$0.702858\pi$
$464$			10681.6i			1.06871i
$465$		−	3077.60i		−	0.306925i
$466$		−	9533.72i		−	0.947728i
$467$	−3657.99			−0.362466			−0.181233	−	0.983440i	$-0.558009\pi$
$467$	−3657.99			−0.362466			−0.181233	+	0.983440i	$0.558009\pi$
$468$	7856.06			0.775954
$469$			1423.54i			0.140156i
$470$	6495.89			0.637517
$471$	5792.12	−	1131.56i	0.566638	−	0.110699i
$472$	−5930.53			−0.578337
$473$			18073.8i			1.75694i
$474$	12843.4			1.24455
$475$	5503.70			0.531636
$476$		−	22748.2i		−	2.19047i
$477$		−	2445.19i		−	0.234712i
$478$		−	18832.7i		−	1.80207i
$479$		−	16885.2i		−	1.61066i	−0.592828	−	0.805329i	$-0.701989\pi$
$479$		−	16885.2i		−	1.61066i	0.592828	−	0.805329i	$-0.298011\pi$
$480$	−3220.29			−0.306220
$481$	−24788.1			−2.34977
$482$	−11604.9			−1.09665
$483$	−5511.56			−0.519223
$484$	−29792.7			−2.79796
$485$	−3900.66			−0.365196
$486$			1045.81i			0.0976111i
$487$	270.659			0.0251842			0.0125921	−	0.999921i	$-0.495992\pi$
$487$	270.659			0.0251842			0.0125921	+	0.999921i	$0.495992\pi$
$488$	1391.46			0.129075
$489$		−	2026.92i		−	0.187444i
$490$	2594.48			0.239197
$491$		−	13645.7i		−	1.25422i	−0.778931	−	0.627110i	$-0.784238\pi$
$491$		−	13645.7i		−	1.25422i	0.778931	−	0.627110i	$-0.215762\pi$
$492$		−	11435.2i		−	1.04784i
$493$		−	28070.2i		−	2.56434i
$494$			18489.5i			1.68397i
$495$		−	2512.65i		−	0.228152i
$496$	8880.52			0.803925
$497$		−	12321.2i		−	1.11203i
$498$	10954.6			0.985714
$499$		−	3881.24i		−	0.348193i	−0.984729	−	0.174096i	$-0.944300\pi$
$499$		−	3881.24i		−	0.348193i	0.984729	−	0.174096i	$-0.0557004\pi$
$500$			10530.6i			0.941890i
$501$	−12068.3			−1.07619
$502$	29587.7			2.63061
$503$		−	4969.52i		−	0.440517i	−0.975442	−	0.220258i	$-0.929310\pi$
$503$		−	4969.52i		−	0.440517i	0.975442	−	0.220258i	$-0.0706901\pi$
$504$	−2145.60			−0.189628
$505$		−	4029.34i		−	0.355056i
$506$	−23227.8			−2.04072
$507$	−14054.5			−1.23113
$508$	−1794.72			−0.156747
$509$		−	7013.60i		−	0.610751i	−0.952232	−	0.305375i	$-0.901218\pi$
$509$		−	7013.60i		−	0.610751i	0.952232	−	0.305375i	$-0.0987820\pi$
$510$	5499.96			0.477534
$511$	3342.32			0.289345
$512$		−	12545.4i		−	1.08287i
$513$	−1398.27			−0.120341
$514$			14857.9i			1.27501i
$515$	5651.37			0.483551
$516$			8843.20i			0.754458i
$517$	−22503.3			−1.91430
$518$	28242.5			2.39556
$519$	3444.66			0.291336
$520$	−3896.83			−0.328629
$521$		−	15540.8i		−	1.30683i	−0.757002	−	0.653413i	$-0.773337\pi$
$521$		−	15540.8i		−	1.30683i	0.757002	−	0.653413i	$-0.226663\pi$
$522$	−11044.9			−0.926096
$523$	−4516.14			−0.377585			−0.188792	−	0.982017i	$-0.560457\pi$
$523$	−4516.14			−0.377585			−0.188792	+	0.982017i	$0.560457\pi$
$524$		−	17340.0i		−	1.44561i
$525$		−	7001.86i		−	0.582069i
$526$		−	20753.5i		−	1.72033i
$527$	−23337.1			−1.92900
$528$	7250.33			0.597595
$529$	5168.90			0.424829
$530$			5059.81i			0.414687i
$531$			4916.93i			0.401839i
$532$		−	11967.4i		−	0.975287i
$533$			30051.4i			2.44216i
$534$			9675.24i			0.784061i
$535$	−4466.73			−0.360960
$536$			703.638i			0.0567025i
$537$			8857.12i			0.711756i
$538$	5066.81			0.406033
$539$	−8987.87			−0.718246
$540$		−	1229.39i		−	0.0979717i
$541$		−	12918.3i		−	1.02662i	−0.858204	−	0.513309i	$-0.828419\pi$
$541$		−	12918.3i		−	1.02662i	0.858204	−	0.513309i	$-0.171581\pi$
$542$	5732.31			0.454287
$543$			9918.91i			0.783906i
$544$			24419.1i			1.92456i
$545$			1877.09i			0.147533i
$546$	23522.5			1.84372
$547$	−11760.5			−0.919276			−0.459638	−	0.888106i	$-0.652021\pi$
$547$	−11760.5			−0.919276			−0.459638	+	0.888106i	$0.652021\pi$
$548$			22755.5i			1.77385i
$549$		−	1153.64i		−	0.0896835i
$550$		−	29508.5i		−	2.28772i
$551$		−	14767.2i		−	1.14175i
$552$	−2724.29			−0.210061
$553$	21846.2			1.67991
$554$			9095.89i			0.697559i
$555$			3879.09i			0.296682i
$556$			7893.95i			0.602119i
$557$	−6544.64			−0.497855			−0.248928	−	0.968522i	$-0.580078\pi$
$557$	−6544.64			−0.497855			−0.248928	+	0.968522i	$0.580078\pi$
$558$			9182.55i			0.696646i
$559$		−	23239.7i		−	1.75838i
$560$	−3559.96			−0.268635
$561$	−19053.1			−1.43391
$562$			12038.4i			0.903574i
$563$			5288.01i			0.395849i	0.980217	+	0.197924i	$0.0634201\pi$
$563$			5288.01i			0.395849i	−0.980217	+	0.197924i	$0.936580\pi$
$564$	−11010.5			−0.822029
$565$			5336.64i			0.397370i
$566$		−	12552.2i		−	0.932173i
$567$			1778.89i			0.131757i
$568$		−	6090.18i		−	0.449892i
$569$			11773.5i			0.867439i	0.901048	+	0.433719i	$0.142799\pi$
$569$			11773.5i			0.867439i	−0.901048	+	0.433719i	$0.857201\pi$
$570$	2893.42			0.212618
$571$	2014.19			0.147620			0.0738102	−	0.997272i	$-0.476484\pi$
$571$	2014.19			0.147620			0.0738102	+	0.997272i	$0.476484\pi$
$572$	56316.3			4.11661
$573$			15143.9i			1.10409i
$574$		−	34239.1i		−	2.48975i
$575$		−	8890.35i		−	0.644788i
$576$	6911.20			0.499942
$577$	−15200.8			−1.09674			−0.548371	−	0.836235i	$-0.684752\pi$
$577$	−15200.8			−1.09674			−0.548371	+	0.836235i	$0.684752\pi$
$578$		−	20561.3i		−	1.47965i
$579$	−686.268			−0.0492579
$580$	12983.7			0.929517
$581$	18633.3			1.33053
$582$	11638.3			0.828906
$583$		−	17528.4i		−	1.24520i
$584$	1652.06			0.117060
$585$			3230.81i			0.228338i
$586$	30894.2			2.17787
$587$			16405.6i			1.15355i	0.816904	+	0.576774i	$0.195689\pi$
$587$			16405.6i			1.15355i	−0.816904	+	0.576774i	$0.804311\pi$
$588$	−4397.61			−0.308426
$589$	−12277.2			−0.858869
$590$		−	10174.6i		−	0.709967i
$591$	−3836.97			−0.267059
$592$	−11193.3			−0.777094
$593$	−77.2340			−0.00534843			−0.00267421	−	0.999996i	$-0.500851\pi$
$593$	−77.2340			−0.00534843			−0.00267421	+	0.999996i	$0.500851\pi$
$594$			7496.92i			0.517849i
$595$	9355.23			0.644583
$596$			10663.1i			0.732848i
$597$	−1270.57			−0.0871036
$598$	29866.9			2.04239
$599$			115.470i			0.00787641i	0.999992	+	0.00393821i	$0.00125357\pi$
$599$			115.470i			0.00787641i	−0.999992	+	0.00393821i	$0.998746\pi$
$600$		−	3460.93i		−	0.235486i
$601$	−23864.8			−1.61974			−0.809871	−	0.586608i	$-0.800463\pi$
$601$	−23864.8			−1.61974			−0.809871	+	0.586608i	$0.800463\pi$
$602$			26478.2i			1.79264i
$603$	583.377			0.0393979
$604$			20346.5i			1.37067i
$605$		−	12252.3i		−	0.823349i
$606$			12022.2i			0.805892i
$607$		−	10480.7i		−	0.700819i	−0.936597	−	0.350410i	$-0.886042\pi$
$607$		−	10480.7i		−	0.700819i	0.936597	−	0.350410i	$-0.113958\pi$
$608$			12846.4i			0.856895i
$609$	−18787.0			−1.25006
$610$			2387.22i			0.158452i
$611$	28935.2			1.91586
$612$	−9322.38			−0.615743
$613$			8340.92i			0.549570i	0.961506	+	0.274785i	$0.0886068\pi$
$613$			8340.92i			0.549570i	−0.961506	+	0.274785i	$0.911393\pi$
$614$	23758.5			1.56159
$615$	4702.73			0.308346
$616$	−15380.7			−1.00602
$617$	23461.6			1.53084			0.765420	−	0.643531i	$-0.222531\pi$
$617$	23461.6			1.53084			0.765420	+	0.643531i	$0.222531\pi$
$618$	−16861.8			−1.09754
$619$	5561.50			0.361124			0.180562	−	0.983564i	$-0.442208\pi$
$619$	5561.50			0.361124			0.180562	+	0.983564i	$0.442208\pi$
$620$		−	10794.5i		−	0.699219i
$621$			2258.68i			0.145954i
$622$			22327.5i			1.43931i
$623$			16457.2i			1.05834i
$624$	−9322.62			−0.598083
$625$	8953.57			0.573028
$626$		−	25382.2i		−	1.62057i
$627$	−10023.5			−0.638436
$628$	20315.4	−	3968.86i	1.29088	−	0.252189i
$629$	29414.8			1.86462
$630$		−	3681.04i		−	0.232788i
$631$	18184.1			1.14722			0.573612	−	0.819127i	$-0.305542\pi$
$631$	18184.1			1.14722			0.573612	+	0.819127i	$0.305542\pi$
$632$	10798.3			0.679639
$633$		−	2145.67i		−	0.134728i
$634$			47533.3i			2.97759i
$635$		−	738.078i		−	0.0461256i
$636$		−	8576.33i		−	0.534707i
$637$	11556.8			0.718833
$638$	−79175.5			−4.91315
$639$	−5049.29			−0.312593
$640$	−5713.85			−0.352906
$641$	−20965.6			−1.29188			−0.645938	−	0.763390i	$-0.723534\pi$
$641$	−20965.6			−1.29188			−0.645938	+	0.763390i	$0.723534\pi$
$642$	13327.3			0.819292
$643$		−	15162.0i		−	0.929909i	−0.885334	−	0.464955i	$-0.846071\pi$
$643$		−	15162.0i		−	0.929909i	0.885334	−	0.464955i	$-0.153929\pi$
$644$	−19331.4			−1.18286
$645$	−3636.77			−0.222012
$646$		−	21940.5i		−	1.33628i
$647$	−15800.3			−0.960084			−0.480042	−	0.877245i	$-0.659379\pi$
$647$	−15800.3			−0.960084			−0.480042	+	0.877245i	$0.659379\pi$
$648$			879.280i			0.0533046i
$649$			35247.1i			2.13185i
$650$			37942.7i			2.28959i
$651$			15619.2i			0.940344i
$652$		−	7109.27i		−	0.427025i
$653$	1564.73			0.0937713			0.0468856	−	0.998900i	$-0.485070\pi$
$653$	1564.73			0.0937713			0.0468856	+	0.998900i	$0.485070\pi$
$654$		−	5600.62i		−	0.334865i
$655$	7131.08			0.425396
$656$			13569.9i			0.807646i
$657$		−	1369.70i		−	0.0813353i
$658$	−32967.4			−1.95320
$659$	4503.99			0.266237			0.133119	−	0.991100i	$-0.457501\pi$
$659$	4503.99			0.266237			0.133119	+	0.991100i	$0.457501\pi$
$660$		−	8812.93i		−	0.519762i
$661$	−7148.91			−0.420666			−0.210333	−	0.977630i	$-0.567455\pi$
$661$	−7148.91			−0.420666			−0.210333	+	0.977630i	$0.567455\pi$
$662$		−	16511.9i		−	0.969415i
$663$	24498.9			1.43508
$664$	9210.19			0.538290
$665$	4921.61			0.286995
$666$		−	11573.9i		−	0.673395i
$667$	−23854.0			−1.38476
$668$	−42328.8			−2.45172
$669$		−	3112.26i		−	0.179861i
$670$	−1207.18			−0.0696080
$671$		−	8269.89i		−	0.475791i
$672$	16343.4			0.938183
$673$		−	28417.6i		−	1.62767i	−0.581099	−	0.813833i	$-0.697377\pi$
$673$		−	28417.6i		−	1.62767i	0.581099	−	0.813833i	$-0.302623\pi$
$674$	38851.3			2.22032
$675$	−2869.41			−0.163620
$676$	−49295.2			−2.80469
$677$	29777.4			1.69046			0.845229	−	0.534405i	$-0.179464\pi$
$677$	29777.4			1.69046			0.845229	+	0.534405i	$0.179464\pi$
$678$		−	15922.8i		−	0.901934i
$679$	19796.3			1.11887
$680$	4624.16			0.260777
$681$			3895.78i			0.219217i
$682$			65825.2i			3.69586i
$683$			1246.55i			0.0698360i	0.999390	+	0.0349180i	$0.0111170\pi$
$683$			1246.55i			0.0698360i	−0.999390	+	0.0349180i	$0.988883\pi$
$684$	−4904.32			−0.274154
$685$	−9358.23			−0.521985
$686$	19252.1			1.07150
$687$			1686.19i			0.0936424i
$688$		−	10494.0i		−	0.581514i
$689$			22538.4i			1.24622i
$690$		−	4673.86i		−	0.257871i
$691$			25092.8i			1.38144i	0.723122	+	0.690720i	$0.242706\pi$
$691$			25092.8i			1.38144i	−0.723122	+	0.690720i	$0.757294\pi$
$692$	12081.9			0.663706
$693$			12752.0i			0.699001i
$694$		−	501.875i		−	0.0274509i
$695$	−3246.39			−0.177184
$696$	−9286.15			−0.505734
$697$		−	35660.4i		−	1.93792i
$698$			19724.1i			1.06958i
$699$	−6645.64			−0.359601
$700$		−	24558.6i		−	1.32604i
$701$			24888.5i			1.34098i	0.741920	+	0.670489i	$0.233916\pi$
$701$			24888.5i			1.34098i	−0.741920	+	0.670489i	$0.766084\pi$
$702$		−	9639.69i		−	0.518272i
$703$	15474.5			0.830204
$704$	49543.0			2.65230
$705$		−	4528.06i		−	0.241896i
$706$		−	56099.9i		−	2.99058i
$707$			20449.4i			1.08781i
$708$			17245.8i			0.915447i
$709$	25965.7			1.37541			0.687704	−	0.725992i	$-0.258619\pi$
$709$	25965.7			1.37541			0.687704	+	0.725992i	$0.258619\pi$
$710$	10448.5			0.552287
$711$		−	8952.70i		−	0.472226i
$712$			8134.59i			0.428169i
$713$			19831.9i			1.04167i
$714$	−27913.0			−1.46305
$715$			23160.1i			1.21138i
$716$			31065.7i			1.62148i
$717$	−13127.6			−0.683766
$718$	39118.7			2.03328
$719$			10146.0i			0.526261i	0.964760	+	0.263130i	$0.0847549\pi$
$719$			10146.0i			0.526261i	−0.964760	+	0.263130i	$0.915245\pi$
$720$			1458.90i			0.0755137i
$721$	−28681.4			−1.48148
$722$			17977.0i			0.926638i
$723$			8089.36i			0.416108i
$724$			34789.9i			1.78585i
$725$		−	30304.1i		−	1.55237i
$726$			36556.8i			1.86880i
$727$	27329.4			1.39421			0.697104	−	0.716970i	$-0.254471\pi$
$727$	27329.4			1.39421			0.697104	+	0.716970i	$0.254471\pi$
$728$	19776.9			1.00684
$729$	729.000			0.0370370
$730$			2834.32i			0.143703i
$731$			27577.3i			1.39533i
$732$		−	4046.32i		−	0.204312i
$733$	−4203.42			−0.211810			−0.105905	−	0.994376i	$-0.533774\pi$
$733$	−4203.42			−0.211810			−0.105905	+	0.994376i	$0.533774\pi$
$734$	39527.4			1.98772
$735$		−	1808.52i		−	0.0907596i
$736$	20751.4			1.03927
$737$	4181.94			0.209015
$738$	−14031.4			−0.699870
$739$	−4930.88			−0.245447			−0.122724	−	0.992441i	$-0.539163\pi$
$739$	−4930.88			−0.245447			−0.122724	+	0.992441i	$0.539163\pi$
$740$			13605.6i			0.675883i
$741$	12888.4			0.638958
$742$		−	25679.2i		−	1.27050i
$743$	11973.0			0.591182			0.295591	−	0.955315i	$-0.404483\pi$
$743$	11973.0			0.591182			0.295591	+	0.955315i	$0.404483\pi$
$744$			7720.35i			0.380433i
$745$	−4385.20			−0.215653
$746$	48710.1			2.39062
$747$		−	7636.05i		−	0.374014i
$748$	−66827.6			−3.26665
$749$	22669.2			1.10589
$750$	12921.5			0.629103
$751$		−	8652.06i		−	0.420397i	−0.977659	−	0.210199i	$-0.932589\pi$
$751$		−	8652.06i		−	0.420397i	0.977659	−	0.210199i	$-0.0674110\pi$
$752$	13065.9			0.633596
$753$		−	20624.6i		−	0.998144i
$754$	101806.			4.91716
$755$	−8367.51			−0.403344
$756$			6239.32i			0.300161i
$757$			7085.77i			0.340207i	0.985426	+	0.170103i	$0.0544102\pi$
$757$			7085.77i			0.340207i	−0.985426	+	0.170103i	$0.945590\pi$
$758$	−16104.8			−0.771707
$759$			16191.3i			0.774319i
$760$	2432.68			0.116109
$761$		−	12931.6i		−	0.615993i	−0.951387	−	0.307997i	$-0.900341\pi$
$761$		−	12931.6i		−	0.615993i	0.951387	−	0.307997i	$-0.0996586\pi$
$762$			2202.19i			0.104694i
$763$		−	9526.44i		−	0.452006i
$764$			53116.2i			2.51528i
$765$		−	3833.84i		−	0.181193i
$766$	−1013.24			−0.0477934
$767$		−	45321.4i		−	2.13359i
$768$	−1381.59			−0.0649138
$769$	−20606.1			−0.966285			−0.483143	−	0.875542i	$-0.660505\pi$
$769$	−20606.1			−0.966285			−0.483143	+	0.875542i	$0.660505\pi$
$770$		−	26387.6i		−	1.23499i
$771$	10356.9			0.483782
$772$	−2407.04			−0.112216
$773$	42.0384			0.00195603			0.000978017	−	1.00000i	$-0.499689\pi$
$773$	42.0384			0.00195603			0.000978017		1.00000i	$0.499689\pi$
$774$	10850.9			0.503914
$775$	−25194.3			−1.16775
$776$	9785.06			0.452659
$777$		−	19686.9i		−	0.908960i
$778$		−	29390.7i		−	1.35438i
$779$		−	18760.2i		−	0.862843i
$780$			11331.8i			0.520186i
$781$	−36195.9			−1.65838
$782$	−35441.4			−1.62069
$783$			7699.03i			0.351393i
$784$	5218.55			0.237726
$785$	1632.20	+	8354.74i	0.0742110	+	0.379864i
$786$	−21276.8			−0.965546
$787$		−	10165.4i		−	0.460430i	−0.973140	−	0.230215i	$-0.926057\pi$
$787$		−	10165.4i		−	0.460430i	0.973140	−	0.230215i	$-0.0739430\pi$
$788$	−13457.9			−0.608399
$789$	−14466.6			−0.652754
$790$			18525.8i			0.834325i
$791$		−	27084.1i		−	1.21745i
$792$			6303.13i			0.282793i
$793$			10633.6i			0.476179i
$794$	1956.24			0.0874363
$795$	3527.02			0.157347
$796$	−4456.43			−0.198434
$797$	−14355.9			−0.638032			−0.319016	−	0.947749i	$-0.603352\pi$
$797$	−14355.9			−0.638032			−0.319016	+	0.947749i	$0.603352\pi$
$798$	−14684.5			−0.651409
$799$	−34335.9			−1.52030
$800$			26362.4i			1.16507i
$801$	6744.28			0.297500
$802$	13410.5			0.590452
$803$		−	9818.75i		−	0.431502i
$804$	2046.15			0.0897541
$805$		−	7950.07i		−	0.348078i
$806$		−	84639.5i		−	3.69888i
$807$		−	3531.90i		−	0.154063i
$808$			10107.9i			0.440091i
$809$			23132.7i			1.00532i	0.864485	+	0.502658i	$0.167645\pi$
$809$			23132.7i			1.00532i	−0.864485	+	0.502658i	$0.832355\pi$
$810$	−1508.51			−0.0654368
$811$		−	9932.49i		−	0.430058i	−0.976608	−	0.215029i	$-0.931015\pi$
$811$		−	9932.49i		−	0.430058i	0.976608	−	0.215029i	$-0.0689845\pi$
$812$	−65894.0			−2.84782
$813$		−	3995.80i		−	0.172372i
$814$		−	82968.0i		−	3.57251i
$815$	2923.69			0.125660
$816$	11062.7			0.474597
$817$			14507.9i			0.621256i
$818$	−27808.2			−1.18862
$819$		−	16396.8i		−	0.699572i
$820$	16494.5			0.702455
$821$	13529.7			0.575139			0.287570	−	0.957760i	$-0.407153\pi$
$821$	13529.7			0.575139			0.287570	+	0.957760i	$0.407153\pi$
$822$	27921.9			1.18478
$823$		−	2525.58i		−	0.106970i	−0.998569	−	0.0534849i	$-0.982967\pi$
$823$		−	2525.58i		−	0.106970i	0.998569	−	0.0534849i	$-0.0170329\pi$
$824$	−14176.8			−0.599360
$825$	−20569.4			−0.868042
$826$			51637.2i			2.17517i
$827$	419.465			0.0176375			0.00881875	−	0.999961i	$-0.497193\pi$
$827$	419.465			0.0176375			0.00881875	+	0.999961i	$0.497193\pi$
$828$			7922.15i			0.332504i
$829$	8813.87			0.369262			0.184631	−	0.982808i	$-0.440891\pi$
$829$	8813.87			0.369262			0.184631	+	0.982808i	$0.440891\pi$
$830$			15801.2i			0.660806i
$831$	6340.44			0.264678
$832$	−63703.4			−2.65447
$833$	−13713.8			−0.570416
$834$	9686.19			0.402164
$835$		−	17407.8i		−	0.721461i
$836$	−35156.7			−1.45445
$837$	6400.84			0.264332
$838$			15409.8i			0.635229i
$839$		−	8648.46i		−	0.355874i	−0.984042	−	0.177937i	$-0.943058\pi$
$839$		−	8648.46i		−	0.355874i	0.984042	−	0.177937i	$-0.0569423\pi$
$840$		−	3094.88i		−	0.127123i
$841$	−56921.0			−2.33388
$842$	55249.6			2.26132
$843$	8391.55			0.342847
$844$		−	7525.80i		−	0.306930i
$845$		−	20272.7i		−	0.825327i
$846$			13510.3i			0.549046i
$847$			62181.8i			2.52254i
$848$			10177.4i			0.412136i
$849$	−8749.74			−0.353699
$850$		−	45024.6i		−	1.81686i
$851$		−	24996.6i		−	1.00690i
$852$	−17710.0			−0.712132
$853$	−37998.9			−1.52527			−0.762637	−	0.646827i	$-0.776096\pi$
$853$	−37998.9			−1.52527			−0.762637	+	0.646827i	$0.776096\pi$
$854$		−	12115.4i		−	0.485459i
$855$		−	2016.91i		−	0.0806745i
$856$	11205.1			0.447409
$857$			10049.0i			0.400547i	0.979740	+	0.200274i	$0.0641831\pi$
$857$			10049.0i			0.400547i	−0.979740	+	0.200274i	$0.935817\pi$
$858$		−	69102.2i		−	2.74955i
$859$			5921.99i			0.235222i	0.993060	+	0.117611i	$0.0375236\pi$
$859$			5921.99i			0.235222i	−0.993060	+	0.117611i	$0.962476\pi$
$860$	−12755.7			−0.505775
$861$	−23866.9			−0.944696
$862$			30724.2i			1.21400i
$863$			28489.2i			1.12374i	0.827227	+	0.561868i	$0.189917\pi$
$863$			28489.2i			1.12374i	−0.827227	+	0.561868i	$0.810083\pi$
$864$		−	6697.62i		−	0.263724i
$865$			4968.69i			0.195307i
$866$	−19674.5			−0.772018
$867$	−14332.6			−0.561431
$868$			54783.2i			2.14224i
$869$		−	64177.5i		−	2.50526i
$870$		−	15931.5i		−	0.620839i
$871$	−5377.23			−0.209186
$872$		−	4708.79i		−	0.182867i
$873$		−	8112.67i		−	0.314516i
$874$	−18645.0			−0.721600
$875$	21979.0			0.849173
$876$		−	4804.14i		−	0.185293i
$877$			25639.7i			0.987221i	0.869683	+	0.493610i	$0.164323\pi$
$877$			25639.7i			0.987221i	−0.869683	+	0.493610i	$0.835677\pi$
$878$	36319.9			1.39606
$879$		−	21535.3i		−	0.826358i
$880$			10458.1i			0.400617i
$881$		−	28719.9i		−	1.09830i	−0.835725	−	0.549148i	$-0.814952\pi$
$881$		−	28719.9i		−	1.09830i	0.835725	−	0.549148i	$-0.185048\pi$
$882$			5396.04i			0.206002i
$883$		−	42068.5i		−	1.60330i	−0.597790	−	0.801652i	$-0.703955\pi$
$883$		−	42068.5i		−	1.60330i	0.597790	−	0.801652i	$-0.296045\pi$
$884$	85928.3			3.26932
$885$	−7092.35			−0.269386
$886$	35562.1			1.34846
$887$			5436.89i			0.205809i	0.994691	+	0.102905i	$0.0328136\pi$
$887$			5436.89i			0.205809i	−0.994691	+	0.102905i	$0.967186\pi$
$888$		−	9730.95i		−	0.367736i
$889$			3745.84i			0.141318i
$890$	−13955.9			−0.525621
$891$	5225.84			0.196490
$892$		−	10916.0i		−	0.409749i
$893$	−18063.4			−0.676898
$894$	13084.0			0.489480
$895$	−12775.8			−0.477149
$896$	28998.5			1.08122
$897$		−	20819.2i		−	0.774952i
$898$	33843.2			1.25764
$899$			67599.8i			2.50788i
$900$	−10064.3			−0.372750
$901$		−	26745.1i		−	0.988910i
$902$	−100584.			−3.71297
$903$	18457.1			0.680191
$904$		−	13387.3i		−	0.492538i
$905$	−14307.4			−0.525517
$906$	24965.9			0.915494
$907$	5167.65			0.189183			0.0945915	−	0.995516i	$-0.469846\pi$
$907$	5167.65			0.189183			0.0945915	+	0.995516i	$0.469846\pi$
$908$			13664.2i			0.499407i
$909$	8380.30			0.305783
$910$			33929.7i			1.23600i
$911$	17686.5			0.643226			0.321613	−	0.946871i	$-0.395775\pi$
$911$	17686.5			0.643226			0.321613	+	0.946871i	$0.395775\pi$
$912$	5819.85			0.211310
$913$		−	54739.1i		−	1.98423i
$914$		−	57983.1i		−	2.09837i
$915$	1664.05			0.0601222
$916$			5914.21i			0.213331i
$917$	−36191.1			−1.30331
$918$			11438.9i			0.411264i
$919$		−	18229.7i		−	0.654346i	−0.944964	−	0.327173i	$-0.893904\pi$
$919$		−	18229.7i		−	0.654346i	0.944964	−	0.327173i	$-0.106096\pi$
$920$		−	3929.61i		−	0.140821i
$921$		−	16561.2i		−	0.592520i
$922$		−	32890.7i		−	1.17484i
$923$	46541.5			1.65973
$924$			44726.7i			1.59242i
$925$	31755.6			1.12878
$926$	68920.1			2.44585
$927$			11753.8i			0.416446i
$928$	70734.1			2.50211
$929$	−39567.2			−1.39737			−0.698685	−	0.715430i	$-0.746231\pi$
$929$	−39567.2			−1.39737			−0.698685	+	0.715430i	$0.746231\pi$
$930$	−13245.2			−0.467019
$931$	−7214.58			−0.253973
$932$	−23309.1			−0.819222
$933$	15563.8			0.546125
$934$			15743.1i			0.551531i
$935$		−	27482.9i		−	0.961269i
$936$		−	8104.70i		−	0.283024i
$937$		−	5335.61i		−	0.186027i	−0.995665	−	0.0930133i	$-0.970350\pi$
$937$		−	5335.61i		−	0.186027i	0.995665	−	0.0930133i	$-0.0296499\pi$
$938$	6126.57			0.213262
$939$	−17693.1			−0.614902
$940$		−	15881.9i		−	0.551074i
$941$	−28329.4			−0.981417			−0.490708	−	0.871324i	$-0.663262\pi$
$941$	−28329.4			−0.981417			−0.490708	+	0.871324i	$0.663262\pi$
$942$	−4869.95	−	24927.8i	−0.168441	−	0.862201i
$943$	−30304.2			−1.04649
$944$		−	20465.2i		−	0.705600i
$945$	−2565.93			−0.0883277
$946$	77785.1			2.67338
$947$		−	32569.1i		−	1.11759i	−0.829307	−	0.558793i	$-0.811265\pi$
$947$		−	32569.1i		−	1.11759i	0.829307	−	0.558793i	$-0.188735\pi$
$948$		−	31401.0i		−	1.07580i
$949$			12625.2i			0.431854i
$950$		−	23686.6i		−	0.808941i
$951$	33133.9			1.12980
$952$	−23468.2			−0.798959
$953$	−10798.0			−0.367032			−0.183516	−	0.983017i	$-0.558748\pi$
$953$	−10798.0			−0.367032			−0.183516	+	0.983017i	$0.558748\pi$
$954$	−10523.5			−0.357139
$955$	−21844.1			−0.740165
$956$	−46044.3			−1.55772
$957$			55190.6i			1.86422i
$958$	−72669.8			−2.45079
$959$	47494.1			1.59923
$960$			9968.94i			0.335152i
$961$	26410.4			0.886522
$962$			106682.i			3.57543i
$963$		−	9289.99i		−	0.310868i
$964$			28372.8i			0.947954i
$965$		−	989.896i		−	0.0330216i
$966$			23720.4i			0.790054i
$967$	38257.4			1.27226			0.636130	−	0.771582i	$-0.280534\pi$
$967$	38257.4			1.27226			0.636130	+	0.771582i	$0.280534\pi$
$968$			30735.6i			1.02054i
$969$	−15294.0			−0.507032
$970$			16787.5i			0.555684i
$971$			10914.1i			0.360710i	0.983602	+	0.180355i	$0.0577246\pi$
$971$			10914.1i			0.360710i	−0.983602	+	0.180355i	$0.942275\pi$
$972$	2556.92			0.0843756
$973$	16475.8			0.542848
$974$		−	1164.85i		−	0.0383205i
$975$	26448.6			0.868751
$976$			4801.68i			0.157477i
$977$	−29400.7			−0.962754			−0.481377	−	0.876514i	$-0.659863\pi$
$977$	−29400.7			−0.962754			−0.481377	+	0.876514i	$0.659863\pi$
$978$	−8723.35			−0.285217
$979$	48346.5			1.57830
$980$		−	6343.26i		−	0.206763i
$981$	−3904.00			−0.127059
$982$	−58727.7			−1.90843
$983$			36060.5i			1.17004i	0.811018	+	0.585021i	$0.198914\pi$
$983$			36060.5i			1.17004i	−0.811018	+	0.585021i	$0.801086\pi$
$984$	−11797.1			−0.382193
$985$		−	5534.58i		−	0.179032i
$986$	−120807.			−3.90192
$987$			22980.5i			0.741111i
$988$	45205.2			1.45564
$989$	23435.2			0.753483
$990$	−10813.8			−0.347157
$991$	−18440.4			−0.591098			−0.295549	−	0.955328i	$-0.595502\pi$
$991$	−18440.4			−0.591098			−0.295549	+	0.955328i	$0.595502\pi$
$992$		−	58807.2i		−	1.88219i
$993$	−11509.9			−0.367830
$994$	−53027.2			−1.69207
$995$		−	1832.71i		−	0.0583928i
$996$		−	26782.9i		−	0.852057i
$997$		−	3986.51i		−	0.126634i	−0.997993	−	0.0633169i	$-0.979832\pi$
$997$		−	3986.51i		−	0.126634i	0.997993	−	0.0633169i	$-0.0201679\pi$
$998$	−16703.9			−0.529812
$999$	−8067.80			−0.255510

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	471.4.b.a.313.6	✓	40
157.156	even	2	inner	471.4.b.a.313.35	yes	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
471.4.b.a.313.6	✓	40	1.1	even	1	trivial
471.4.b.a.313.35	yes	40	157.156	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 471 = 3 \cdot 157 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	471.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-4.30375i q^{2} -3.00000 q^{3} -10.5223 q^{4} -4.32730i q^{5} +12.9113i q^{6} +21.9616i q^{7} +10.8553i q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q-4.30375i q^{2} -3.00000 q^{3} -10.5223 q^{4} -4.32730i q^{5} +12.9113i q^{6} +21.9616i q^{7} +10.8553i q^{8} +9.00000 q^{9} -18.6236 q^{10} +64.5166 q^{11} +31.5669 q^{12} -82.9568 q^{13} +94.5172 q^{14} +12.9819i q^{15} -37.4597 q^{16} +98.4405 q^{17} -38.7338i q^{18} +51.7876 q^{19} +45.5331i q^{20} -65.8847i q^{21} -277.664i q^{22} -83.6546i q^{23} -32.5659i q^{24} +106.274 q^{25} +357.026i q^{26} -27.0000 q^{27} -231.086i q^{28} -285.149i q^{29} +55.8709 q^{30} -237.068 q^{31} +248.060i q^{32} -193.550 q^{33} -423.664i q^{34} +95.0344 q^{35} -94.7006 q^{36} +298.808 q^{37} -222.881i q^{38} +248.870 q^{39} +46.9742 q^{40} -362.253i q^{41} -283.552 q^{42} +280.142i q^{43} -678.862 q^{44} -38.9457i q^{45} -360.029 q^{46} -348.798 q^{47} +112.379 q^{48} -139.311 q^{49} -457.379i q^{50} -295.322 q^{51} +872.896 q^{52} -271.688i q^{53} +116.201i q^{54} -279.183i q^{55} -238.400 q^{56} -155.363 q^{57} -1227.21 q^{58} +546.326i q^{59} -136.599i q^{60} -128.182i q^{61} +1020.28i q^{62} +197.654i q^{63} +767.911 q^{64} +358.979i q^{65} +832.991i q^{66} +64.8197 q^{67} -1035.82 q^{68} +250.964i q^{69} -409.004i q^{70} -561.033 q^{71} +97.6978i q^{72} -152.189i q^{73} -1285.99i q^{74} -318.823 q^{75} -544.925 q^{76} +1416.89i q^{77} -1071.08i q^{78} -994.744i q^{79} +162.100i q^{80} +81.0000 q^{81} -1559.05 q^{82} -848.450i q^{83} +693.258i q^{84} -425.982i q^{85} +1205.66 q^{86} +855.447i q^{87} +700.348i q^{88} +749.365 q^{89} -167.613 q^{90} -1821.86i q^{91} +880.238i q^{92} +711.205 q^{93} +1501.14i q^{94} -224.101i q^{95} -744.180i q^{96} -901.408i q^{97} +599.560i q^{98} +580.649 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 120 q^{3} - 164 q^{4} + 360 q^{9}+O(q^{10}) \) 40 * q - 120 * q^3 - 164 * q^4 + 360 * q^9	\( 40 q - 120 q^{3} - 164 q^{4} + 360 q^{9} - 174 q^{10} + 110 q^{11} + 492 q^{12} - 194 q^{13} - 78 q^{14} + 796 q^{16} - 150 q^{17} + 172 q^{19} - 668 q^{25} - 1080 q^{27} + 522 q^{30} + 66 q^{31} - 330 q^{33} - 400 q^{35} - 1476 q^{36} - 142 q^{37} + 582 q^{39} + 1160 q^{40} + 234 q^{42} - 1182 q^{44} + 132 q^{46} - 244 q^{47} - 2388 q^{48} - 3786 q^{49} + 450 q^{51} + 1596 q^{52} - 256 q^{56} - 516 q^{57} - 1780 q^{58} - 1790 q^{64} - 320 q^{67} + 1646 q^{68} + 712 q^{71} + 2004 q^{75} - 3004 q^{76} + 3240 q^{81} + 4112 q^{82} - 4198 q^{86} + 366 q^{89} - 1566 q^{90} - 198 q^{93} + 990 q^{99}+O(q^{100}) \) 40 * q - 120 * q^3 - 164 * q^4 + 360 * q^9 - 174 * q^10 + 110 * q^11 + 492 * q^12 - 194 * q^13 - 78 * q^14 + 796 * q^16 - 150 * q^17 + 172 * q^19 - 668 * q^25 - 1080 * q^27 + 522 * q^30 + 66 * q^31 - 330 * q^33 - 400 * q^35 - 1476 * q^36 - 142 * q^37 + 582 * q^39 + 1160 * q^40 + 234 * q^42 - 1182 * q^44 + 132 * q^46 - 244 * q^47 - 2388 * q^48 - 3786 * q^49 + 450 * q^51 + 1596 * q^52 - 256 * q^56 - 516 * q^57 - 1780 * q^58 - 1790 * q^64 - 320 * q^67 + 1646 * q^68 + 712 * q^71 + 2004 * q^75 - 3004 * q^76 + 3240 * q^81 + 4112 * q^82 - 4198 * q^86 + 366 * q^89 - 1566 * q^90 - 198 * q^93 + 990 * q^99

Introduction

L-functions

Modular forms

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