LMFDB - Embedded newform 471.4.b.a.313.15

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			313.15
Character	$\chi$	$=$	471.313
Dual form			471.4.b.a.313.26

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.46936i q^{2} -3.00000 q^{3} +5.84098 q^{4} -4.94652i q^{5} +4.40809i q^{6} +12.8754i q^{7} -20.3374i q^{8} +9.00000 q^{9} +O(q^{10})$	\(q-1.46936i q^{2} -3.00000 q^{3} +5.84098 q^{4} -4.94652i q^{5} +4.40809i q^{6} +12.8754i q^{7} -20.3374i q^{8} +9.00000 q^{9} -7.26823 q^{10} -22.3563 q^{11} -17.5229 q^{12} +72.1239 q^{13} +18.9187 q^{14} +14.8396i q^{15} +16.8448 q^{16} +19.7682 q^{17} -13.2243i q^{18} -82.0165 q^{19} -28.8925i q^{20} -38.6263i q^{21} +32.8496i q^{22} +50.1678i q^{23} +61.0122i q^{24} +100.532 q^{25} -105.976i q^{26} -27.0000 q^{27} +75.2052i q^{28} -147.572i q^{29} +21.8047 q^{30} +144.553 q^{31} -187.450i q^{32} +67.0690 q^{33} -29.0466i q^{34} +63.6887 q^{35} +52.5688 q^{36} +221.972 q^{37} +120.512i q^{38} -216.372 q^{39} -100.599 q^{40} +22.4304i q^{41} -56.7561 q^{42} +204.958i q^{43} -130.583 q^{44} -44.5187i q^{45} +73.7146 q^{46} -84.4528 q^{47} -50.5344 q^{48} +177.223 q^{49} -147.718i q^{50} -59.3045 q^{51} +421.274 q^{52} -213.343i q^{53} +39.6728i q^{54} +110.586i q^{55} +261.853 q^{56} +246.050 q^{57} -216.836 q^{58} -14.1094i q^{59} +86.6776i q^{60} -811.300i q^{61} -212.401i q^{62} +115.879i q^{63} -140.674 q^{64} -356.763i q^{65} -98.5487i q^{66} +19.0720 q^{67} +115.465 q^{68} -150.503i q^{69} -93.5817i q^{70} +217.768 q^{71} -183.037i q^{72} -235.896i q^{73} -326.157i q^{74} -301.596 q^{75} -479.056 q^{76} -287.848i q^{77} +317.928i q^{78} -1094.10i q^{79} -83.3233i q^{80} +81.0000 q^{81} +32.9584 q^{82} -1229.87i q^{83} -225.615i q^{84} -97.7838i q^{85} +301.158 q^{86} +442.715i q^{87} +454.670i q^{88} +1399.40 q^{89} -65.4141 q^{90} +928.627i q^{91} +293.029i q^{92} -433.659 q^{93} +124.092i q^{94} +405.697i q^{95} +562.351i q^{96} -698.636i q^{97} -260.405i q^{98} -201.207 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$		−	1.46936i		−	0.519498i	−0.965676	−	0.259749i	$-0.916360\pi$
$2$		−	1.46936i		−	0.519498i	0.965676	−	0.259749i	$-0.0836398\pi$
$3$	−3.00000			−0.577350
$3$	−3.00000			−0.577350
$4$	5.84098			0.730122
$5$		−	4.94652i		−	0.442431i	−0.975225	−	0.221215i	$-0.928998\pi$
$5$		−	4.94652i		−	0.442431i	0.975225	−	0.221215i	$-0.0710024\pi$
$6$			4.40809i			0.299932i
$7$			12.8754i			0.695209i	0.937641	+	0.347604i	$0.113005\pi$
$7$			12.8754i			0.695209i	−0.937641	+	0.347604i	$0.886995\pi$
$8$		−	20.3374i		−	0.898795i
$9$	9.00000			0.333333
$10$	−7.26823			−0.229842
$11$	−22.3563			−0.612790			−0.306395	−	0.951904i	$-0.599123\pi$
$11$	−22.3563			−0.612790			−0.306395	+	0.951904i	$0.599123\pi$
$12$	−17.5229			−0.421536
$13$	72.1239			1.53874			0.769368	−	0.638806i	$-0.220571\pi$
$13$	72.1239			1.53874			0.769368	+	0.638806i	$0.220571\pi$
$14$	18.9187			0.361159
$15$			14.8396i			0.255437i
$16$	16.8448			0.263200
$17$	19.7682			0.282029			0.141014	−	0.990008i	$-0.454964\pi$
$17$	19.7682			0.282029			0.141014	+	0.990008i	$0.454964\pi$
$18$		−	13.2243i		−	0.173166i
$19$	−82.0165			−0.990309			−0.495155	−	0.868805i	$-0.664889\pi$
$19$	−82.0165			−0.990309			−0.495155	+	0.868805i	$0.664889\pi$
$20$		−	28.8925i		−	0.323028i
$21$		−	38.6263i		−	0.401379i
$22$			32.8496i			0.318343i
$23$			50.1678i			0.454813i	0.973800	+	0.227407i	$0.0730247\pi$
$23$			50.1678i			0.454813i	−0.973800	+	0.227407i	$0.926975\pi$
$24$			61.0122i			0.518919i
$25$	100.532			0.804255
$26$		−	105.976i		−	0.799370i
$27$	−27.0000			−0.192450
$28$			75.2052i			0.507587i
$29$		−	147.572i		−	0.944943i	−0.881346	−	0.472471i	$-0.843362\pi$
$29$		−	147.572i		−	0.944943i	0.881346	−	0.472471i	$-0.156638\pi$
$30$	21.8047			0.132699
$31$	144.553			0.837500			0.418750	−	0.908102i	$-0.362468\pi$
$31$	144.553			0.837500			0.418750	+	0.908102i	$0.362468\pi$
$32$		−	187.450i		−	1.03553i
$33$	67.0690			0.353795
$34$		−	29.0466i		−	0.146513i
$35$	63.6887			0.307582
$36$	52.5688			0.243374
$37$	221.972			0.986269			0.493135	−	0.869953i	$-0.335851\pi$
$37$	221.972			0.986269			0.493135	+	0.869953i	$0.335851\pi$
$38$			120.512i			0.514464i
$39$	−216.372			−0.888390
$40$	−100.599			−0.397654
$41$			22.4304i			0.0854402i	0.999087	+	0.0427201i	$0.0136024\pi$
$41$			22.4304i			0.0854402i	−0.999087	+	0.0427201i	$0.986398\pi$
$42$	−56.7561			−0.208515
$43$			204.958i			0.726879i	0.931618	+	0.363440i	$0.118398\pi$
$43$			204.958i			0.726879i	−0.931618	+	0.363440i	$0.881602\pi$
$44$	−130.583			−0.447412
$45$		−	44.5187i		−	0.147477i
$46$	73.7146			0.236275
$47$	−84.4528			−0.262100			−0.131050	−	0.991376i	$-0.541835\pi$
$47$	−84.4528			−0.262100			−0.131050	+	0.991376i	$0.541835\pi$
$48$	−50.5344			−0.151959
$49$	177.223			0.516685
$50$		−	147.718i		−	0.417809i
$51$	−59.3045			−0.162829
$52$	421.274			1.12346
$53$		−	213.343i		−	0.552924i	−0.961025	−	0.276462i	$-0.910838\pi$
$53$		−	213.343i		−	0.552924i	0.961025	−	0.276462i	$-0.0891620\pi$
$54$			39.6728i			0.0999774i
$55$			110.586i			0.271117i
$56$	261.853			0.624850
$57$	246.050			0.571755
$58$	−216.836			−0.490896
$59$		−	14.1094i		−	0.0311336i	−0.999879	−	0.0155668i	$-0.995045\pi$
$59$		−	14.1094i		−	0.0311336i	0.999879	−	0.0155668i	$-0.00495527\pi$
$60$			86.6776i			0.186500i
$61$		−	811.300i		−	1.70289i	−0.524444	−	0.851445i	$-0.675727\pi$
$61$		−	811.300i		−	1.70289i	0.524444	−	0.851445i	$-0.324273\pi$
$62$		−	212.401i		−	0.435079i
$63$			115.879i			0.231736i
$64$	−140.674			−0.274754
$65$		−	356.763i		−	0.680784i
$66$		−	98.5487i		−	0.183795i
$67$	19.0720			0.0347764			0.0173882	−	0.999849i	$-0.494465\pi$
$67$	19.0720			0.0347764			0.0173882	+	0.999849i	$0.494465\pi$
$68$	115.465			0.205915
$69$		−	150.503i		−	0.262587i
$70$		−	93.5817i		−	0.159788i
$71$	217.768			0.364005			0.182003	−	0.983298i	$-0.441742\pi$
$71$	217.768			0.364005			0.182003	+	0.983298i	$0.441742\pi$
$72$		−	183.037i		−	0.299598i
$73$		−	235.896i		−	0.378213i	−0.981957	−	0.189106i	$-0.939441\pi$
$73$		−	235.896i		−	0.378213i	0.981957	−	0.189106i	$-0.0605591\pi$
$74$		−	326.157i		−	0.512365i
$75$	−301.596			−0.464337
$76$	−479.056			−0.723047
$77$		−	287.848i		−	0.426017i
$78$			317.928i			0.461516i
$79$		−	1094.10i		−	1.55818i	−0.626911	−	0.779091i	$-0.715681\pi$
$79$		−	1094.10i		−	1.55818i	0.626911	−	0.779091i	$-0.284319\pi$
$80$		−	83.3233i		−	0.116448i
$81$	81.0000			0.111111
$82$	32.9584			0.0443860
$83$		−	1229.87i		−	1.62645i	−0.581947	−	0.813226i	$-0.697709\pi$
$83$		−	1229.87i		−	1.62645i	0.581947	−	0.813226i	$-0.302291\pi$
$84$		−	225.615i		−	0.293056i
$85$		−	97.7838i		−	0.124778i
$86$	301.158			0.377612
$87$			442.715i			0.545563i
$88$			454.670i			0.550772i
$89$	1399.40			1.66670			0.833349	−	0.552747i	$-0.186421\pi$
$89$	1399.40			1.66670			0.833349	+	0.552747i	$0.186421\pi$
$90$	−65.4141			−0.0766139
$91$			928.627i			1.06974i
$92$			293.029i			0.332069i
$93$	−433.659			−0.483531
$94$			124.092i			0.136160i
$95$			405.697i			0.438143i
$96$			562.351i			0.597862i
$97$		−	698.636i		−	0.731296i	−0.930753	−	0.365648i	$-0.880847\pi$
$97$		−	698.636i		−	0.731296i	0.930753	−	0.365648i	$-0.119153\pi$
$98$		−	260.405i		−	0.268417i
$99$	−201.207			−0.204263
$100$	587.204			0.587204
$101$	−768.315			−0.756933			−0.378467	−	0.925615i	$-0.623548\pi$
$101$	−768.315			−0.756933			−0.378467	+	0.925615i	$0.623548\pi$
$102$			87.1398i			0.0845895i
$103$			1193.25i			1.14150i	0.821123	+	0.570751i	$0.193348\pi$
$103$			1193.25i			1.14150i	−0.821123	+	0.570751i	$0.806652\pi$
$104$		−	1466.81i		−	1.38301i
$105$	−191.066			−0.177582
$106$	−313.479			−0.287243
$107$		−	1391.57i		−	1.25727i	−0.777701	−	0.628635i	$-0.783614\pi$
$107$		−	1391.57i		−	1.25727i	0.777701	−	0.628635i	$-0.216386\pi$
$108$	−157.706			−0.140512
$109$	580.300			0.509932			0.254966	−	0.966950i	$-0.417936\pi$
$109$	580.300			0.509932			0.254966	+	0.966950i	$0.417936\pi$
$110$	162.491			0.140845
$111$	−665.916			−0.569423
$112$			216.884i			0.182979i
$113$	1325.78			1.10371			0.551855	−	0.833940i	$-0.313920\pi$
$113$	1325.78			1.10371			0.551855	+	0.833940i	$0.313920\pi$
$114$		−	361.536i		−	0.297026i
$115$	248.156			0.201223
$116$		−	861.962i		−	0.689924i
$117$	649.115			0.512912
$118$	−20.7318			−0.0161739
$119$			254.524i			0.196069i
$120$	301.798			0.229586
$121$	−831.194			−0.624488
$122$	−1192.09			−0.884647
$123$		−	67.2913i		−	0.0493289i
$124$	844.331			0.611477
$125$		−	1115.60i		−	0.798258i
$126$	170.268			0.120386
$127$	−1303.18			−0.910540			−0.455270	−	0.890353i	$-0.650457\pi$
$127$	−1303.18			−0.910540			−0.455270	+	0.890353i	$0.650457\pi$
$128$		−	1292.90i		−	0.892793i
$129$		−	614.874i		−	0.419664i
$130$	−524.213			−0.353666
$131$			1368.57i			0.912768i	0.889783	+	0.456384i	$0.150856\pi$
$131$			1368.57i			0.912768i	−0.889783	+	0.456384i	$0.849144\pi$
$132$	391.749			0.258313
$133$		−	1056.00i		−	0.688472i
$134$		−	28.0237i		−	0.0180663i
$135$			133.556i			0.0851458i
$136$		−	402.033i		−	0.253486i
$137$			2346.90i			1.46357i	0.681534	+	0.731786i	$0.261313\pi$
$137$			2346.90i			1.46357i	−0.681534	+	0.731786i	$0.738687\pi$
$138$	−221.144			−0.136413
$139$			1267.34i			0.773344i	0.922217	+	0.386672i	$0.126375\pi$
$139$			1267.34i			0.773344i	−0.922217	+	0.386672i	$0.873625\pi$
$140$	372.004			0.224572
$141$	253.358			0.151324
$142$		−	319.981i		−	0.189100i
$143$	−1612.43			−0.942922
$144$	151.603			0.0877334
$145$	−729.966			−0.418072
$146$	−346.616			−0.196481
$147$	−531.669			−0.298308
$148$	1296.53			0.720097
$149$			2228.71i			1.22539i	0.790320	+	0.612694i	$0.209914\pi$
$149$			2228.71i			1.22539i	−0.790320	+	0.612694i	$0.790086\pi$
$150$			443.153i			0.241222i
$151$			595.191i			0.320768i	0.987055	+	0.160384i	$0.0512732\pi$
$151$			595.191i			0.320768i	−0.987055	+	0.160384i	$0.948727\pi$
$152$			1668.00i			0.890085i
$153$	177.914			0.0940096
$154$	−422.953			−0.221315
$155$		−	715.035i		−	0.370535i
$156$	−1263.82			−0.648633
$157$	729.836	+	1826.81i	0.371002	+	0.928632i
$157$	729.836	+	1826.81i	0.371002	+	0.928632i
$158$	−1607.64			−0.809472
$159$			640.030i			0.319231i
$160$	−927.228			−0.458149
$161$	−645.933			−0.316190
$162$		−	119.018i		−	0.0577220i
$163$			865.206i			0.415756i	0.978155	+	0.207878i	$0.0666556\pi$
$163$			865.206i			0.415756i	−0.978155	+	0.207878i	$0.933344\pi$
$164$			131.016i			0.0623817i
$165$		−	331.759i		−	0.156530i
$166$	−1807.12			−0.844939
$167$	1171.86			0.543001			0.271501	−	0.962438i	$-0.412480\pi$
$167$	1171.86			0.543001			0.271501	+	0.962438i	$0.412480\pi$
$168$	−785.559			−0.360757
$169$	3004.85			1.36771
$170$	−143.680			−0.0648220
$171$	−738.149			−0.330103
$172$			1197.16i			0.530711i
$173$	−3542.12			−1.55666			−0.778330	−	0.627855i	$-0.783933\pi$
$173$	−3542.12			−1.55666			−0.778330	+	0.627855i	$0.783933\pi$
$174$	650.508			0.283419
$175$			1294.39i			0.559125i
$176$	−376.588			−0.161286
$177$			42.3281i			0.0179750i
$178$		−	2056.22i		−	0.865846i
$179$			571.501i			0.238637i	0.992856	+	0.119318i	$0.0380709\pi$
$179$			571.501i			0.238637i	−0.992856	+	0.119318i	$0.961929\pi$
$180$		−	260.033i		−	0.107676i
$181$			3873.10i			1.59053i	0.606264	+	0.795263i	$0.292667\pi$
$181$			3873.10i			1.59053i	−0.606264	+	0.795263i	$0.707333\pi$
$182$	1364.49			0.555729
$183$			2433.90i			0.983164i
$184$	1020.28			0.408784
$185$		−	1097.99i		−	0.436356i
$186$			637.202i			0.251193i
$187$	−441.944			−0.172824
$188$	−493.287			−0.191365
$189$		−	347.637i		−	0.133793i
$190$	596.115			0.227614
$191$			283.289i			0.107320i	0.998559	+	0.0536599i	$0.0170887\pi$
$191$			283.289i			0.107320i	−0.998559	+	0.0536599i	$0.982911\pi$
$192$	422.022			0.158629
$193$	−1789.06			−0.667250			−0.333625	−	0.942706i	$-0.608272\pi$
$193$	−1789.06			−0.667250			−0.333625	+	0.942706i	$0.608272\pi$
$194$	−1026.55			−0.379907
$195$			1070.29i			0.393051i
$196$	1035.15			0.377243
$197$	967.408			0.349873			0.174936	−	0.984580i	$-0.444028\pi$
$197$	967.408			0.349873			0.174936	+	0.984580i	$0.444028\pi$
$198$			295.646i			0.106114i
$199$	3613.64			1.28726			0.643628	−	0.765339i	$-0.277429\pi$
$199$	3613.64			1.28726			0.643628	+	0.765339i	$0.277429\pi$
$200$		−	2044.56i		−	0.722860i
$201$	−57.2161			−0.0200782
$202$			1128.93i			0.393225i
$203$	1900.05			0.656932
$204$	−346.396			−0.118885
$205$	110.953			0.0378013
$206$	1753.32			0.593008
$207$			451.510i			0.151604i
$208$	1214.91			0.404996
$209$	1833.59			0.606852
$210$			280.745i			0.0922536i
$211$			306.600i			0.100034i	0.998748	+	0.0500171i	$0.0159276\pi$
$211$			306.600i			0.100034i	−0.998748	+	0.0500171i	$0.984072\pi$
$212$		−	1246.13i		−	0.403702i
$213$	−653.305			−0.210158
$214$	−2044.71			−0.653149
$215$	1013.83			0.321594
$216$			549.110i			0.172973i
$217$			1861.18i			0.582237i
$218$		−	852.670i		−	0.264909i
$219$			707.688i			0.218361i
$220$			645.931i			0.197949i
$221$	1425.76			0.433968
$222$			978.471i			0.295814i
$223$		−	131.482i		−	0.0394830i	−0.999805	−	0.0197415i	$-0.993716\pi$
$223$		−	131.482i		−	0.0394830i	0.999805	−	0.0197415i	$-0.00628433\pi$
$224$	2413.51			0.719907
$225$	904.787			0.268085
$226$		−	1948.05i		−	0.573374i
$227$		−	4436.26i		−	1.29711i	−0.761166	−	0.648557i	$-0.775373\pi$
$227$		−	4436.26i		−	1.29711i	0.761166	−	0.648557i	$-0.224627\pi$
$228$	1437.17			0.417451
$229$			2543.10i			0.733854i	0.930250	+	0.366927i	$0.119590\pi$
$229$			2543.10i			0.733854i	−0.930250	+	0.366927i	$0.880410\pi$
$230$		−	364.631i		−	0.104535i
$231$			863.544i			0.245961i
$232$	−3001.22			−0.849310
$233$	6042.00			1.69882			0.849409	−	0.527736i	$-0.176959\pi$
$233$	6042.00			1.69882			0.849409	+	0.527736i	$0.176959\pi$
$234$		−	953.785i		−	0.266457i
$235$			417.748i			0.115961i
$236$		−	82.4125i		−	0.0227314i
$237$			3282.31i			0.899617i
$238$	373.988			0.101857
$239$	−1992.67			−0.539311			−0.269655	−	0.962957i	$-0.586910\pi$
$239$	−1992.67			−0.539311			−0.269655	+	0.962957i	$0.586910\pi$
$240$			249.970i			0.0672312i
$241$			4488.35i			1.19967i	0.800125	+	0.599834i	$0.204767\pi$
$241$			4488.35i			1.19967i	−0.800125	+	0.599834i	$0.795233\pi$
$242$			1221.32i			0.324420i
$243$	−243.000			−0.0641500
$244$		−	4738.78i		−	1.24332i
$245$		−	876.638i		−	0.228597i
$246$	−98.8753			−0.0256263
$247$	−5915.35			−1.52382
$248$		−	2939.83i		−	0.752740i
$249$			3689.60i			0.939033i
$250$	−1639.22			−0.414693
$251$		−	4885.89i		−	1.22866i	−0.789048	−	0.614332i	$-0.789426\pi$
$251$		−	4885.89i		−	1.22866i	0.789048	−	0.614332i	$-0.210574\pi$
$252$			676.846i			0.169196i
$253$		−	1121.57i		−	0.278705i
$254$			1914.84i			0.473023i
$255$			293.351i			0.0720407i
$256$	−3025.13			−0.738557
$257$	−3532.16			−0.857314			−0.428657	−	0.903467i	$-0.641013\pi$
$257$	−3532.16			−0.857314			−0.428657	+	0.903467i	$0.641013\pi$
$258$	−903.473			−0.218015
$259$			2857.99i			0.685663i
$260$		−	2083.84i		−	0.497055i
$261$		−	1328.14i		−	0.314981i
$262$	2010.93			0.474181
$263$	1901.34			0.445786			0.222893	−	0.974843i	$-0.428450\pi$
$263$	1901.34			0.445786			0.222893	+	0.974843i	$0.428450\pi$
$264$		−	1364.01i		−	0.317989i
$265$	−1055.31			−0.244630
$266$	−1551.64			−0.357659
$267$	−4198.20			−0.962268
$268$	111.399			0.0253910
$269$			6193.20i			1.40374i	0.712305	+	0.701870i	$0.247651\pi$
$269$			6193.20i			1.40374i	−0.712305	+	0.701870i	$0.752349\pi$
$270$	196.242			0.0442331
$271$			7632.45i			1.71084i	0.517933	+	0.855421i	$0.326702\pi$
$271$			7632.45i			1.71084i	−0.517933	+	0.855421i	$0.673298\pi$
$272$	332.991			0.0742300
$273$		−	2785.88i		−	0.617616i
$274$	3448.45			0.760323
$275$	−2247.53			−0.492840
$276$		−	879.087i		−	0.191720i
$277$	−325.575			−0.0706205			−0.0353103	−	0.999376i	$-0.511242\pi$
$277$	−325.575			−0.0706205			−0.0353103	+	0.999376i	$0.511242\pi$
$278$	1862.19			0.401750
$279$	1300.98			0.279167
$280$		−	1295.26i		−	0.276453i
$281$	521.494			0.110711			0.0553554	−	0.998467i	$-0.482371\pi$
$281$	521.494			0.110711			0.0553554	+	0.998467i	$0.482371\pi$
$282$		−	372.275i		−	0.0786123i
$283$	−6983.45			−1.46687			−0.733433	−	0.679762i	$-0.762083\pi$
$283$	−6983.45			−1.46687			−0.733433	+	0.679762i	$0.762083\pi$
$284$	1271.98			0.265768
$285$		−	1217.09i		−	0.252962i
$286$			2369.24i			0.489846i
$287$	−288.802			−0.0593987
$288$		−	1687.05i		−	0.345176i
$289$	−4522.22			−0.920460
$290$			1072.58i			0.217187i
$291$			2095.91i			0.422214i
$292$		−	1377.86i		−	0.276141i
$293$			637.031i			0.127016i	0.997981	+	0.0635081i	$0.0202289\pi$
$293$			637.031i			0.127016i	−0.997981	+	0.0635081i	$0.979771\pi$
$294$			781.214i			0.154970i
$295$	−69.7924			−0.0137745
$296$		−	4514.33i		−	0.886454i
$297$	603.621			0.117932
$298$	3274.78			0.636587
$299$			3618.30i			0.699838i
$300$	−1761.61			−0.339023
$301$	−2638.93			−0.505333
$302$	874.550			0.166638
$303$	2304.95			0.437015
$304$	−1381.55			−0.260650
$305$	−4013.11			−0.753411
$306$		−	261.419i		−	0.0488378i
$307$		−	9552.77i		−	1.77591i	−0.459927	−	0.887957i	$-0.652125\pi$
$307$		−	9552.77i		−	1.77591i	0.459927	−	0.887957i	$-0.347875\pi$
$308$		−	1681.31i		−	0.311044i
$309$		−	3579.76i		−	0.659047i
$310$	−1050.65			−0.192492
$311$	−2679.95			−0.488636			−0.244318	−	0.969695i	$-0.578564\pi$
$311$	−2679.95			−0.488636			−0.244318	+	0.969695i	$0.578564\pi$
$312$			4400.44i			0.798480i
$313$	1862.35			0.336314			0.168157	−	0.985760i	$-0.446218\pi$
$313$	1862.35			0.336314			0.168157	+	0.985760i	$0.446218\pi$
$314$	2684.24	−	1072.39i	0.482422	−	0.192735i
$315$	573.198			0.102527
$316$		−	6390.64i		−	1.13766i
$317$	616.031			0.109147			0.0545737	−	0.998510i	$-0.482620\pi$
$317$	616.031			0.109147			0.0545737	+	0.998510i	$0.482620\pi$
$318$	940.436			0.165840
$319$			3299.16i			0.579052i
$320$			695.847i			0.121559i
$321$			4174.70i			0.725885i
$322$			949.109i			0.164260i
$323$	−1621.32			−0.279296
$324$	473.119			0.0811247
$325$	7250.75			1.23754
$326$	1271.30			0.215984
$327$	−1740.90			−0.294410
$328$	456.177			0.0767932
$329$		−	1087.37i		−	0.182214i
$330$	−487.473			−0.0813168
$331$	−4652.55			−0.772590			−0.386295	−	0.922375i	$-0.626245\pi$
$331$	−4652.55			−0.772590			−0.386295	+	0.922375i	$0.626245\pi$
$332$		−	7183.63i		−	1.18751i
$333$	1997.75			0.328756
$334$		−	1721.89i		−	0.282088i
$335$		−	94.3403i		−	0.0153862i
$336$		−	650.653i		−	0.105643i
$337$			477.677i			0.0772128i	0.999254	+	0.0386064i	$0.0122919\pi$
$337$			477.677i			0.0772128i	−0.999254	+	0.0386064i	$0.987708\pi$
$338$		−	4415.22i		−	0.710521i
$339$	−3977.35			−0.637227
$340$		−	571.153i		−	0.0911032i
$341$	−3231.68			−0.513211
$342$			1084.61i			0.171488i
$343$			6698.10i			1.05441i
$344$	4168.31			0.653315
$345$	−744.469			−0.116176
$346$			5204.65i			0.808682i
$347$	7159.31			1.10758			0.553792	−	0.832655i	$-0.313180\pi$
$347$	7159.31			1.10758			0.553792	+	0.832655i	$0.313180\pi$
$348$			2585.89i			0.398328i
$349$	−4040.57			−0.619733			−0.309867	−	0.950780i	$-0.600284\pi$
$349$	−4040.57			−0.619733			−0.309867	+	0.950780i	$0.600284\pi$
$350$	1901.93			0.290464
$351$	−1947.34			−0.296130
$352$			4190.70i			0.634560i
$353$	−3064.42			−0.462047			−0.231023	−	0.972948i	$-0.574207\pi$
$353$	−3064.42			−0.462047			−0.231023	+	0.972948i	$0.574207\pi$
$354$	62.1953			0.00933798
$355$		−	1077.20i		−	0.161047i
$356$	8173.86			1.21689
$357$		−	763.572i		−	0.113200i
$358$	839.741			0.123971
$359$			11621.5i			1.70852i	0.519843	+	0.854262i	$0.325990\pi$
$359$			11621.5i			1.70852i	−0.519843	+	0.854262i	$0.674010\pi$
$360$	−905.395			−0.132551
$361$	−132.293			−0.0192875
$362$	5690.99			0.826275
$363$	2493.58			0.360548
$364$			5424.09i			0.781043i
$365$	−1166.87			−0.167333
$366$	3576.28			0.510751
$367$		−	1691.85i		−	0.240637i	−0.992735	−	0.120319i	$-0.961608\pi$
$367$		−	1691.85i		−	0.240637i	0.992735	−	0.120319i	$-0.0383916\pi$
$368$			845.067i			0.119707i
$369$			201.874i			0.0284801i
$370$	−1613.34			−0.226686
$371$	2746.89			0.384397
$372$	−2532.99			−0.353036
$373$		−	12113.6i		−	1.68155i	−0.541387	−	0.840773i	$-0.682101\pi$
$373$		−	12113.6i		−	1.68155i	0.541387	−	0.840773i	$-0.317899\pi$
$374$			649.376i			0.0897819i
$375$			3346.80i			0.460874i
$376$			1717.55i			0.235574i
$377$		−	10643.4i		−	1.45402i
$378$	−510.804			−0.0695052
$379$		−	4764.05i		−	0.645681i	−0.946453	−	0.322840i	$-0.895362\pi$
$379$		−	4764.05i		−	0.645681i	0.946453	−	0.322840i	$-0.104638\pi$
$380$			2369.66i			0.319898i
$381$	3909.54			0.525700
$382$	416.255			0.0557524
$383$			307.764i			0.0410601i	0.999789	+	0.0205301i	$0.00653538\pi$
$383$			307.764i			0.0410601i	−0.999789	+	0.0205301i	$0.993465\pi$
$384$			3878.71i			0.515454i
$385$	−1423.85			−0.188483
$386$			2628.77i			0.346635i
$387$			1844.62i			0.242293i
$388$		−	4080.72i		−	0.533936i
$389$	−10499.8			−1.36854			−0.684268	−	0.729230i	$-0.739878\pi$
$389$	−10499.8			−1.36854			−0.684268	+	0.729230i	$0.739878\pi$
$390$	1572.64			0.204189
$391$			991.726i			0.128270i
$392$		−	3604.25i		−	0.464394i
$393$		−	4105.71i		−	0.526987i
$394$		−	1421.47i		−	0.181758i
$395$	−5412.01			−0.689387
$396$	−1175.25			−0.149137
$397$			4285.68i			0.541794i	0.962608	+	0.270897i	$0.0873203\pi$
$397$			4285.68i			0.541794i	−0.962608	+	0.270897i	$0.912680\pi$
$398$		−	5309.74i		−	0.668726i
$399$			3168.00i			0.397489i
$400$	1693.44			0.211680
$401$		−	1402.14i		−	0.174612i	−0.996182	−	0.0873062i	$-0.972174\pi$
$401$		−	1402.14i		−	0.174612i	0.996182	−	0.0873062i	$-0.0278259\pi$
$402$			84.0712i			0.0104306i
$403$	10425.7			1.28869
$404$	−4487.71			−0.552653
$405$		−	400.668i		−	0.0491590i
$406$		−	2791.86i		−	0.341275i
$407$	−4962.48			−0.604376
$408$			1206.10i			0.146350i
$409$		−	8387.78i		−	1.01406i	−0.861929	−	0.507028i	$-0.830744\pi$
$409$		−	8387.78i		−	1.01406i	0.861929	−	0.507028i	$-0.169256\pi$
$410$		−	163.030i		−	0.0196377i
$411$		−	7040.71i		−	0.844994i
$412$			6969.77i			0.833436i
$413$	181.665			0.0216444
$414$	663.432			0.0787582
$415$	−6083.57			−0.719592
$416$		−	13519.6i		−	1.59340i
$417$		−	3802.03i		−	0.446490i
$418$		−	2694.21i		−	0.315258i
$419$	−6080.91			−0.709002			−0.354501	−	0.935056i	$-0.615349\pi$
$419$	−6080.91			−0.709002			−0.354501	+	0.935056i	$0.615349\pi$
$420$	−1116.01			−0.129657
$421$			1098.58i			0.127177i	0.997976	+	0.0635883i	$0.0202545\pi$
$421$			1098.58i			0.127177i	−0.997976	+	0.0635883i	$0.979746\pi$
$422$	450.506			0.0519675
$423$	−760.075			−0.0873667
$424$	−4338.85			−0.496965
$425$	1987.33			0.226823
$426$			959.942i			0.109177i
$427$	10445.8			1.18386
$428$		−	8128.11i		−	0.917960i
$429$	4837.28			0.544396
$430$		−	1489.68i		−	0.167067i
$431$	7160.80			0.800287			0.400143	−	0.916453i	$-0.368960\pi$
$431$	7160.80			0.800287			0.400143	+	0.916453i	$0.368960\pi$
$432$	−454.810			−0.0506529
$433$		−	9273.03i		−	1.02918i	−0.857437	−	0.514588i	$-0.827945\pi$
$433$		−	9273.03i		−	1.02918i	0.857437	−	0.514588i	$-0.172055\pi$
$434$	2734.75			0.302471
$435$	2189.90			0.241374
$436$	3389.52			0.372313
$437$		−	4114.59i		−	0.450406i
$438$	1039.85			0.113438
$439$		−	8210.41i		−	0.892623i	−0.894878	−	0.446312i	$-0.852737\pi$
$439$		−	8210.41i		−	0.892623i	0.894878	−	0.446312i	$-0.147263\pi$
$440$	2249.04			0.243679
$441$	1595.01			0.172228
$442$		−	2094.95i		−	0.225445i
$443$			116.027i			0.0124438i	0.999981	+	0.00622188i	$0.00198050\pi$
$443$			116.027i			0.0124438i	−0.999981	+	0.00622188i	$0.998020\pi$
$444$	−3889.60			−0.415748
$445$		−	6922.17i		−	0.737398i
$446$	−193.195			−0.0205114
$447$		−	6686.12i		−	0.707478i
$448$		−	1811.24i		−	0.191011i
$449$		−	432.121i		−	0.0454188i	−0.999742	−	0.0227094i	$-0.992771\pi$
$449$		−	432.121i		−	0.0454188i	0.999742	−	0.0227094i	$-0.00722925\pi$
$450$		−	1329.46i		−	0.139270i
$451$		−	501.463i		−	0.0523569i
$452$	7743.86			0.805842
$453$		−	1785.57i		−	0.185195i
$454$	−6518.47			−0.673848
$455$	4593.48			0.473287
$456$		−	5004.01i		−	0.513891i
$457$	9018.04			0.923076			0.461538	−	0.887120i	$-0.347298\pi$
$457$	9018.04			0.923076			0.461538	+	0.887120i	$0.347298\pi$
$458$	3736.73			0.381236
$459$	−533.741			−0.0542764
$460$	1449.47			0.146918
$461$	625.625			0.0632066			0.0316033	−	0.999500i	$-0.489939\pi$
$461$	625.625			0.0632066			0.0316033	+	0.999500i	$0.489939\pi$
$462$	1268.86			0.127776
$463$		−	7755.27i		−	0.778441i	−0.921145	−	0.389220i	$-0.872744\pi$
$463$		−	7755.27i		−	0.778441i	0.921145	−	0.389220i	$-0.127256\pi$
$464$		−	2485.81i		−	0.248709i
$465$			2145.10i			0.213929i
$466$		−	8877.88i		−	0.882532i
$467$	−8505.55			−0.842805			−0.421402	−	0.906874i	$-0.638462\pi$
$467$	−8505.55			−0.842805			−0.421402	+	0.906874i	$0.638462\pi$
$468$	3791.46			0.374488
$469$			245.561i			0.0241769i
$470$	613.823			0.0602415
$471$	−2189.51	−	5480.43i	−0.214198	−	0.536146i
$472$	−286.948			−0.0279827
$473$		−	4582.11i		−	0.445424i
$474$	4822.91			0.467349
$475$	−8245.27			−0.796461
$476$			1486.67i			0.143154i
$477$		−	1920.09i		−	0.184308i
$478$			2927.96i			0.280171i
$479$			10755.4i			1.02594i	0.858407	+	0.512970i	$0.171455\pi$
$479$			10755.4i			1.02594i	−0.858407	+	0.512970i	$0.828545\pi$
$480$	2781.68			0.264512
$481$	16009.5			1.51761
$482$	6595.01			0.623225
$483$	1937.80			0.182552
$484$	−4854.98			−0.455953
$485$	−3455.82			−0.323548
$486$			357.055i			0.0333258i
$487$	7624.10			0.709406			0.354703	−	0.934979i	$-0.384582\pi$
$487$	7624.10			0.709406			0.354703	+	0.934979i	$0.384582\pi$
$488$	−16499.7			−1.53055
$489$		−	2595.62i		−	0.240037i
$490$	−1288.10			−0.118756
$491$			18175.8i			1.67060i	0.549797	+	0.835298i	$0.314705\pi$
$491$			18175.8i			1.67060i	−0.549797	+	0.835298i	$0.685295\pi$
$492$		−	393.047i		−	0.0360161i
$493$		−	2917.22i		−	0.266501i
$494$			8691.79i			0.791623i
$495$			995.276i			0.0903724i
$496$	2434.97			0.220430
$497$			2803.87i			0.253060i
$498$	5421.36			0.487826
$499$			14506.9i			1.30144i	0.759320	+	0.650718i	$0.225532\pi$
$499$			14506.9i			1.30144i	−0.759320	+	0.650718i	$0.774468\pi$
$500$		−	6516.19i		−	0.582826i
$501$	−3515.58			−0.313502
$502$	−7179.14			−0.638288
$503$			5062.63i			0.448771i	0.974500	+	0.224385i	$0.0720374\pi$
$503$			5062.63i			0.448771i	−0.974500	+	0.224385i	$0.927963\pi$
$504$	2356.68			0.208283
$505$			3800.49i			0.334890i
$506$	−1647.99			−0.144787
$507$	−9014.56			−0.789647
$508$	−7611.85			−0.664805
$509$			21986.8i			1.91463i	0.289050	+	0.957314i	$0.406661\pi$
$509$			21986.8i			1.91463i	−0.289050	+	0.957314i	$0.593339\pi$
$510$	431.039			0.0374250
$511$	3037.27			0.262937
$512$		−	5898.20i		−	0.509114i
$513$	2214.45			0.190585
$514$			5190.01i			0.445373i
$515$	5902.46			0.505036
$516$		−	3591.47i		−	0.306406i
$517$	1888.06			0.160612
$518$	4199.42			0.356200
$519$	10626.4			0.898738
$520$	−7255.62			−0.611885
$521$		−	1498.57i		−	0.126014i	−0.998013	−	0.0630071i	$-0.979931\pi$
$521$		−	1498.57i		−	0.126014i	0.998013	−	0.0630071i	$-0.0200691\pi$
$522$	−1951.52			−0.163632
$523$	−1153.31			−0.0964255			−0.0482128	−	0.998837i	$-0.515353\pi$
$523$	−1153.31			−0.0964255			−0.0482128	+	0.998837i	$0.515353\pi$
$524$			7993.79i			0.666432i
$525$		−	3883.18i		−	0.322811i
$526$		−	2793.76i		−	0.231585i
$527$	2857.55			0.236199
$528$	1129.77			0.0931188
$529$	9650.19			0.793145
$530$			1550.63i			0.127085i
$531$		−	126.984i		−	0.0103779i
$532$		−	6168.06i		−	0.502668i
$533$			1617.77i			0.131470i
$534$			6168.67i			0.499896i
$535$	−6883.42			−0.556254
$536$		−	387.876i		−	0.0312569i
$537$		−	1714.50i		−	0.137777i
$538$	9100.05			0.729240
$539$	−3962.06			−0.316619
$540$			780.098i			0.0621668i
$541$		−	12206.4i		−	0.970045i	−0.874502	−	0.485023i	$-0.838811\pi$
$541$		−	12206.4i		−	0.970045i	0.874502	−	0.485023i	$-0.161189\pi$
$542$	11214.8			0.888779
$543$		−	11619.3i		−	0.918291i
$544$		−	3705.55i		−	0.292048i
$545$		−	2870.47i		−	0.225610i
$546$	−4093.47			−0.320850
$547$	15505.1			1.21197			0.605987	−	0.795475i	$-0.292778\pi$
$547$	15505.1			1.21197			0.605987	+	0.795475i	$0.292778\pi$
$548$			13708.2i			1.06859i
$549$		−	7301.70i		−	0.567630i
$550$			3302.43i			0.256029i
$551$			12103.3i			0.935786i
$552$	−3060.85			−0.236011
$553$	14087.1			1.08326
$554$			478.387i			0.0366872i
$555$			3293.97i			0.251930i
$556$			7402.53i			0.564635i
$557$	−12453.1			−0.947317			−0.473658	−	0.880709i	$-0.657067\pi$
$557$	−12453.1			−0.947317			−0.473658	+	0.880709i	$0.657067\pi$
$558$		−	1911.61i		−	0.145026i
$559$			14782.4i			1.11848i
$560$	1072.82			0.0809555
$561$	1325.83			0.0997802
$562$		−	766.264i		−	0.0575140i
$563$			1763.65i			0.132023i	0.997819	+	0.0660116i	$0.0210274\pi$
$563$			1763.65i			0.132023i	−0.997819	+	0.0660116i	$0.978973\pi$
$564$	1479.86			0.110485
$565$		−	6558.02i		−	0.488315i
$566$			10261.2i			0.762034i
$567$			1042.91i			0.0772454i
$568$		−	4428.84i		−	0.327166i
$569$		−	14445.7i		−	1.06431i	−0.846646	−	0.532156i	$-0.821382\pi$
$569$		−	14445.7i		−	1.06431i	0.846646	−	0.532156i	$-0.178618\pi$
$570$	−1788.35			−0.131413
$571$	837.220			0.0613600			0.0306800	−	0.999529i	$-0.490233\pi$
$571$	837.220			0.0613600			0.0306800	+	0.999529i	$0.490233\pi$
$572$	−9418.14			−0.688448
$573$		−	849.868i		−	0.0619612i
$574$			424.354i			0.0308575i
$575$			5043.46i			0.365786i
$576$	−1266.06			−0.0915845
$577$	−3624.16			−0.261483			−0.130741	−	0.991416i	$-0.541736\pi$
$577$	−3624.16			−0.261483			−0.130741	+	0.991416i	$0.541736\pi$
$578$			6644.78i			0.478177i
$579$	5367.17			0.385237
$580$	−4263.71			−0.305243
$581$	15835.1			1.13072
$582$	3079.65			0.219339
$583$			4769.58i			0.338826i
$584$	−4797.51			−0.339936
$585$		−	3210.86i		−	0.226928i
$586$	936.029			0.0659846
$587$			7431.17i			0.522516i	0.965269	+	0.261258i	$0.0841374\pi$
$587$			7431.17i			0.522516i	−0.965269	+	0.261258i	$0.915863\pi$
$588$	−3105.46			−0.217801
$589$	−11855.7			−0.829384
$590$			102.550i			0.00715581i
$591$	−2902.22			−0.201999
$592$	3739.08			0.259586
$593$	−20235.7			−1.40132			−0.700659	−	0.713496i	$-0.747111\pi$
$593$	−20235.7			−1.40132			−0.700659	+	0.713496i	$0.747111\pi$
$594$		−	886.938i		−	0.0612652i
$595$	1259.01			0.0867468
$596$			13017.8i			0.894683i
$597$	−10840.9			−0.743197
$598$	5316.59			0.363564
$599$			2676.96i			0.182601i	0.995823	+	0.0913003i	$0.0291023\pi$
$599$			2676.96i			0.182601i	−0.995823	+	0.0913003i	$0.970898\pi$
$600$			6133.67i			0.417344i
$601$	−26488.6			−1.79783			−0.898914	−	0.438125i	$-0.855643\pi$
$601$	−26488.6			−1.79783			−0.898914	+	0.438125i	$0.855643\pi$
$602$			3877.54i			0.262519i
$603$	171.648			0.0115921
$604$			3476.49i			0.234200i
$605$			4111.52i			0.276293i
$606$		−	3386.80i		−	0.227029i
$607$		−	12988.5i		−	0.868514i	−0.900789	−	0.434257i	$-0.857011\pi$
$607$		−	12988.5i		−	0.868514i	0.900789	−	0.434257i	$-0.142989\pi$
$608$			15374.0i			1.02549i
$609$	−5700.15			−0.379280
$610$			5896.72i			0.391395i
$611$	−6091.06			−0.403303
$612$	1039.19			0.0686384
$613$			3157.14i			0.208019i	0.994576	+	0.104010i	$0.0331673\pi$
$613$			3157.14i			0.208019i	−0.994576	+	0.104010i	$0.966833\pi$
$614$	−14036.5			−0.922583
$615$	−332.858			−0.0218246
$616$	−5854.08			−0.382902
$617$	−17269.9			−1.12684			−0.563421	−	0.826170i	$-0.690515\pi$
$617$	−17269.9			−1.12684			−0.563421	+	0.826170i	$0.690515\pi$
$618$	−5259.96			−0.342373
$619$	−4562.08			−0.296228			−0.148114	−	0.988970i	$-0.547320\pi$
$619$	−4562.08			−0.296228			−0.148114	+	0.988970i	$0.547320\pi$
$620$		−	4176.50i		−	0.270536i
$621$		−	1354.53i		−	0.0875289i
$622$			3937.81i			0.253846i
$623$			18017.9i			1.15870i
$624$	−3644.74			−0.233824
$625$	7048.15			0.451082
$626$		−	2736.47i		−	0.174715i
$627$	−5500.77			−0.350366
$628$	4262.96	+	10670.3i	0.270876	+	0.678015i
$629$	4387.98			0.278156
$630$		−	842.236i		−	0.0532627i
$631$	21537.0			1.35876			0.679378	−	0.733788i	$-0.262250\pi$
$631$	21537.0			1.35876			0.679378	+	0.733788i	$0.262250\pi$
$632$	−22251.2			−1.40049
$633$		−	919.799i		−	0.0577547i
$634$		−	905.173i		−	0.0567019i
$635$			6446.21i			0.402851i
$636$			3738.40i			0.233077i
$637$	12782.0			0.795042
$638$	4847.66			0.300816
$639$	1959.92			0.121335
$640$	−6395.37			−0.394999
$641$	23835.8			1.46873			0.734365	−	0.678755i	$-0.237480\pi$
$641$	23835.8			1.46873			0.734365	+	0.678755i	$0.237480\pi$
$642$	6134.14			0.377096
$643$			16629.0i			1.01988i	0.860210	+	0.509939i	$0.170332\pi$
$643$			16629.0i			1.01988i	−0.860210	+	0.509939i	$0.829668\pi$
$644$	−3772.88			−0.230857
$645$	−3041.49			−0.185672
$646$			2382.30i			0.145093i
$647$	−20555.7			−1.24904			−0.624519	−	0.781010i	$-0.714705\pi$
$647$	−20555.7			−1.24904			−0.624519	+	0.781010i	$0.714705\pi$
$648$		−	1647.33i		−	0.0998661i
$649$			315.434i			0.0190784i
$650$		−	10654.0i		−	0.642897i
$651$		−	5583.55i		−	0.336155i
$652$			5053.65i			0.303552i
$653$	−7123.93			−0.426923			−0.213462	−	0.976951i	$-0.568474\pi$
$653$	−7123.93			−0.426923			−0.213462	+	0.976951i	$0.568474\pi$
$654$			2558.01i			0.152945i
$655$	6769.67			0.403836
$656$			377.837i			0.0224879i
$657$		−	2123.06i		−	0.126071i
$658$	−1597.74			−0.0946599
$659$	−27980.6			−1.65398			−0.826988	−	0.562220i	$-0.809948\pi$
$659$	−27980.6			−1.65398			−0.826988	+	0.562220i	$0.809948\pi$
$660$		−	1937.79i		−	0.114286i
$661$	5269.16			0.310055			0.155028	−	0.987910i	$-0.450453\pi$
$661$	5269.16			0.310055			0.155028	+	0.987910i	$0.450453\pi$
$662$			6836.28i			0.401359i
$663$	−4277.27			−0.250551
$664$	−25012.3			−1.46185
$665$	−5223.52			−0.304601
$666$		−	2935.41i		−	0.170788i
$667$	7403.34			0.429773
$668$	6844.80			0.396457
$669$			394.447i			0.0227955i
$670$	−138.620			−0.00799307
$671$			18137.7i			1.04351i
$672$	−7240.52			−0.415638
$673$		−	3444.16i		−	0.197270i	−0.995124	−	0.0986350i	$-0.968552\pi$
$673$		−	3444.16i		−	0.197270i	0.995124	−	0.0986350i	$-0.0314476\pi$
$674$	701.880			0.0401119
$675$	−2714.36			−0.154779
$676$	17551.3			0.998594
$677$	31855.2			1.80841			0.904206	−	0.427097i	$-0.140464\pi$
$677$	31855.2			1.80841			0.904206	+	0.427097i	$0.140464\pi$
$678$			5844.16i			0.331038i
$679$	8995.25			0.508404
$680$	−1988.67			−0.112150
$681$			13308.8i			0.748889i
$682$			4748.50i			0.266612i
$683$		−	5410.31i		−	0.303103i	−0.988449	−	0.151552i	$-0.951573\pi$
$683$		−	5410.31i		−	0.303103i	0.988449	−	0.151552i	$-0.0484270\pi$
$684$	−4311.51			−0.241016
$685$	11609.0			0.647529
$686$	9841.93			0.547765
$687$		−	7629.29i		−	0.423691i
$688$			3452.48i			0.191315i
$689$		−	15387.2i		−	0.850804i
$690$			1093.89i			0.0603534i
$691$		−	3626.50i		−	0.199651i	−0.995005	−	0.0998253i	$-0.968172\pi$
$691$		−	3626.50i		−	0.199651i	0.995005	−	0.0998253i	$-0.0318284\pi$
$692$	−20689.4			−1.13655
$693$		−	2590.63i		−	0.142006i
$694$		−	10519.6i		−	0.575388i
$695$	6268.95			0.342151
$696$	9003.66			0.490349
$697$			443.409i			0.0240966i
$698$			5937.06i			0.321950i
$699$	−18126.0			−0.980812
$700$			7560.52i			0.408230i
$701$		−	9984.62i		−	0.537966i	−0.963145	−	0.268983i	$-0.913312\pi$
$701$		−	9984.62i		−	0.537966i	0.963145	−	0.268983i	$-0.0866875\pi$
$702$			2861.35i			0.153839i
$703$	−18205.4			−0.976712
$704$	3144.95			0.168366
$705$		−	1253.24i		−	0.0669502i
$706$			4502.74i			0.240032i
$707$		−	9892.40i		−	0.526226i
$708$			247.238i			0.0131240i
$709$	−2339.91			−0.123945			−0.0619725	−	0.998078i	$-0.519739\pi$
$709$	−2339.91			−0.123945			−0.0619725	+	0.998078i	$0.519739\pi$
$710$	−1582.79			−0.0836636
$711$		−	9846.94i		−	0.519394i
$712$		−	28460.2i		−	1.49802i
$713$			7251.90i			0.380906i
$714$	−1121.96			−0.0588073
$715$			7975.91i			0.417178i
$716$			3338.12i			0.174234i
$717$	5978.02			0.311371
$718$	17076.2			0.887574
$719$		−	5156.13i		−	0.267442i	−0.991019	−	0.133721i	$-0.957307\pi$
$719$		−	5156.13i		−	0.267442i	0.991019	−	0.133721i	$-0.0426927\pi$
$720$		−	749.909i		−	0.0388159i
$721$	−15363.7			−0.793583
$722$			194.386i			0.0100198i
$723$		−	13465.0i		−	0.692628i
$724$			22622.7i			1.16128i
$725$		−	14835.6i		−	0.759975i
$726$		−	3663.97i		−	0.187304i
$727$	−16606.9			−0.847202			−0.423601	−	0.905849i	$-0.639234\pi$
$727$	−16606.9			−0.847202			−0.423601	+	0.905849i	$0.639234\pi$
$728$	18885.9			0.961479
$729$	729.000			0.0370370
$730$			1714.55i			0.0869291i
$731$			4051.65i			0.205001i
$732$			14216.3i			0.717830i
$733$	−8724.09			−0.439607			−0.219803	−	0.975544i	$-0.570542\pi$
$733$	−8724.09			−0.439607			−0.219803	+	0.975544i	$0.570542\pi$
$734$	−2485.94			−0.125011
$735$			2629.91i			0.131981i
$736$	9403.97			0.470971
$737$	−426.381			−0.0213106
$738$	296.626			0.0147953
$739$	515.320			0.0256514			0.0128257	−	0.999918i	$-0.495917\pi$
$739$	515.320			0.0256514			0.0128257	+	0.999918i	$0.495917\pi$
$740$		−	6413.33i		−	0.318593i
$741$	17746.0			0.879780
$742$		−	4036.18i		−	0.199694i
$743$	−13435.0			−0.663369			−0.331684	−	0.943390i	$-0.607617\pi$
$743$	−13435.0			−0.663369			−0.331684	+	0.943390i	$0.607617\pi$
$744$			8819.50i			0.434595i
$745$	11024.4			0.542149
$746$	−17799.2			−0.873560
$747$		−	11068.8i		−	0.542151i
$748$	−2581.39			−0.126183
$749$	17917.0			0.874064
$750$	4917.66			0.239423
$751$			9825.29i			0.477403i	0.971093	+	0.238702i	$0.0767218\pi$
$751$			9825.29i			0.477403i	−0.971093	+	0.238702i	$0.923278\pi$
$752$	−1422.59			−0.0689848
$753$			14657.7i			0.709369i
$754$	−15639.0			−0.755359
$755$	2944.13			0.141917
$756$		−	2030.54i		−	0.0976852i
$757$		−	34881.0i		−	1.67473i	−0.546643	−	0.837366i	$-0.684095\pi$
$757$		−	34881.0i		−	1.67473i	0.546643	−	0.837366i	$-0.315905\pi$
$758$	−7000.12			−0.335430
$759$			3364.71i			0.160910i
$760$	8250.82			0.393801
$761$			20122.5i			0.958529i	0.877670	+	0.479265i	$0.159097\pi$
$761$			20122.5i			0.958529i	−0.877670	+	0.479265i	$0.840903\pi$
$762$		−	5744.53i		−	0.273100i
$763$			7471.62i			0.354509i
$764$			1654.69i			0.0783566i
$765$		−	880.054i		−	0.0415927i
$766$	452.217			0.0213306
$767$		−	1017.62i		−	0.0479064i
$768$	9075.39			0.426406
$769$	−17295.1			−0.811024			−0.405512	−	0.914090i	$-0.632907\pi$
$769$	−17295.1			−0.811024			−0.405512	+	0.914090i	$0.632907\pi$
$770$			2092.15i			0.0979165i
$771$	10596.5			0.494971
$772$	−10449.8			−0.487174
$773$	−21185.8			−0.985772			−0.492886	−	0.870094i	$-0.664058\pi$
$773$	−21185.8			−0.985772			−0.492886	+	0.870094i	$0.664058\pi$
$774$	2710.42			0.125871
$775$	14532.2			0.673563
$776$	−14208.4			−0.657285
$777$		−	8573.96i		−	0.395868i
$778$			15428.0i			0.710952i
$779$		−	1839.67i		−	0.0846122i
$780$			6251.52i			0.286975i
$781$	−4868.51			−0.223059
$782$	1457.20			0.0666362
$783$			3984.43i			0.181854i
$784$	2985.29			0.135992
$785$	9036.36	−	3610.15i	0.410855	−	0.164142i
$786$	−6032.78			−0.273768
$787$			27453.2i			1.24346i	0.783232	+	0.621729i	$0.213570\pi$
$787$			27453.2i			1.24346i	−0.783232	+	0.621729i	$0.786430\pi$
$788$	5650.61			0.255450
$789$	−5704.03			−0.257375
$790$			7952.21i			0.358135i
$791$			17070.0i			0.767308i
$792$			4092.03i			0.183591i
$793$		−	58514.1i		−	2.62030i
$794$	6297.22			0.281461
$795$	3165.93			0.141237
$796$	21107.2			0.939854
$797$	−33862.1			−1.50497			−0.752483	−	0.658612i	$-0.771144\pi$
$797$	−33862.1			−1.50497			−0.752483	+	0.658612i	$0.771144\pi$
$798$	4654.93			0.206495
$799$	−1669.48			−0.0739197
$800$		−	18844.7i		−	0.832828i
$801$	12594.6			0.555566
$802$	−2060.25			−0.0907108
$803$			5273.77i			0.231765i
$804$	−334.198			−0.0146595
$805$			3195.12i			0.139892i
$806$		−	15319.2i		−	0.669472i
$807$		−	18579.6i		−	0.810449i
$808$			15625.5i			0.680327i
$809$			13456.2i			0.584790i	0.956298	+	0.292395i	$0.0944522\pi$
$809$			13456.2i			0.584790i	−0.956298	+	0.292395i	$0.905548\pi$
$810$	−588.727			−0.0255380
$811$			26653.7i			1.15405i	0.816725	+	0.577027i	$0.195787\pi$
$811$			26653.7i			1.15405i	−0.816725	+	0.577027i	$0.804213\pi$
$812$	11098.1			0.479641
$813$		−	22897.3i		−	0.987755i
$814$			7291.68i			0.313972i
$815$	4279.76			0.183943
$816$	−998.974			−0.0428567
$817$		−	16809.9i		−	0.719835i
$818$	−12324.7			−0.526800
$819$			8357.64i			0.356581i
$820$	648.072			0.0275996
$821$	−19565.6			−0.831721			−0.415860	−	0.909428i	$-0.636520\pi$
$821$	−19565.6			−0.831721			−0.415860	+	0.909428i	$0.636520\pi$
$822$	−10345.3			−0.438973
$823$		−	34235.1i		−	1.45001i	−0.688742	−	0.725007i	$-0.741837\pi$
$823$		−	34235.1i		−	1.45001i	0.688742	−	0.725007i	$-0.258163\pi$
$824$	24267.7			1.02598
$825$	6742.58			0.284541
$826$		−	266.931i		−	0.0112442i
$827$	17627.0			0.741176			0.370588	−	0.928797i	$-0.379156\pi$
$827$	17627.0			0.741176			0.370588	+	0.928797i	$0.379156\pi$
$828$			2637.26i			0.110690i
$829$	−25243.5			−1.05759			−0.528796	−	0.848749i	$-0.677356\pi$
$829$	−25243.5			−1.05759			−0.528796	+	0.848749i	$0.677356\pi$
$830$			8938.97i			0.373827i
$831$	976.724			0.0407728
$832$	−10145.9			−0.422773
$833$	3503.37			0.145720
$834$	−5586.56			−0.231951
$835$		−	5796.63i		−	0.240240i
$836$	10710.0			0.443076
$837$	−3902.93			−0.161177
$838$			8935.06i			0.368325i
$839$			22414.6i			0.922335i	0.887313	+	0.461167i	$0.152569\pi$
$839$			22414.6i			0.922335i	−0.887313	+	0.461167i	$0.847431\pi$
$840$			3885.79i			0.159610i
$841$	2611.65			0.107083
$842$	1614.21			0.0660680
$843$	−1564.48			−0.0639189
$844$			1790.84i			0.0730371i
$845$		−	14863.6i		−	0.605116i
$846$			1116.83i			0.0453868i
$847$		−	10702.0i		−	0.434150i
$848$		−	3593.73i		−	0.145530i
$849$	20950.3			0.846895
$850$		−	2920.11i		−	0.117834i
$851$			11135.8i			0.448568i
$852$	−3815.94			−0.153441
$853$	39464.0			1.58408			0.792040	−	0.610469i	$-0.209019\pi$
$853$	39464.0			1.58408			0.792040	+	0.610469i	$0.209019\pi$
$854$		−	15348.7i		−	0.615015i
$855$			3651.27i			0.146048i
$856$	−28300.8			−1.13003
$857$			22210.2i			0.885282i	0.896699	+	0.442641i	$0.145958\pi$
$857$			22210.2i			0.885282i	−0.896699	+	0.442641i	$0.854042\pi$
$858$		−	7107.71i		−	0.282813i
$859$		−	15948.7i		−	0.633485i	−0.948512	−	0.316743i	$-0.897411\pi$
$859$		−	15948.7i		−	0.633485i	0.948512	−	0.316743i	$-0.102589\pi$
$860$	5921.76			0.234803
$861$	866.406			0.0342939
$862$		−	10521.8i		−	0.415747i
$863$			15243.5i			0.601268i	0.953740	+	0.300634i	$0.0971982\pi$
$863$			15243.5i			0.601268i	−0.953740	+	0.300634i	$0.902802\pi$
$864$			5061.16i			0.199287i
$865$			17521.2i			0.688714i
$866$	−13625.4			−0.534655
$867$	13566.7			0.531428
$868$			10871.1i			0.425104i
$869$			24460.2i			0.954839i
$870$		−	3217.75i		−	0.125393i
$871$	1375.55			0.0535117
$872$		−	11801.8i		−	0.458325i
$873$		−	6287.73i		−	0.243765i
$874$	−6045.82			−0.233985
$875$	14363.8			0.554956
$876$			4133.59i			0.159430i
$877$			21667.7i			0.834282i	0.908842	+	0.417141i	$0.136968\pi$
$877$			21667.7i			0.834282i	−0.908842	+	0.417141i	$0.863032\pi$
$878$	−12064.1			−0.463716
$879$		−	1911.09i		−	0.0733328i
$880$			1862.80i			0.0713581i
$881$			21646.9i			0.827814i	0.910319	+	0.413907i	$0.135836\pi$
$881$			21646.9i			0.827814i	−0.910319	+	0.413907i	$0.864164\pi$
$882$		−	2343.64i		−	0.0894722i
$883$		−	25828.8i		−	0.984379i	−0.870488	−	0.492190i	$-0.836197\pi$
$883$		−	25828.8i		−	0.984379i	0.870488	−	0.492190i	$-0.163803\pi$
$884$	8327.82			0.316849
$885$	209.377			0.00795270
$886$	170.485			0.00646451
$887$			11755.9i			0.445012i	0.974931	+	0.222506i	$0.0714236\pi$
$887$			11755.9i			0.445012i	−0.974931	+	0.222506i	$0.928576\pi$
$888$			13543.0i			0.511794i
$889$		−	16779.0i		−	0.633015i
$890$	−10171.2			−0.383077
$891$	−1810.86			−0.0680878
$892$		−	767.986i		−	0.0288274i
$893$	6926.52			0.259560
$894$	−9824.33			−0.367533
$895$	2826.94			0.105580
$896$	16646.7			0.620677
$897$		−	10854.9i		−	0.404051i
$898$	−634.942			−0.0235950
$899$		−	21331.9i		−	0.791389i
$900$	5284.84			0.195735
$901$		−	4217.41i		−	0.155940i
$902$	−736.830			−0.0271993
$903$	7916.78			0.291754
$904$		−	26963.0i		−	0.992008i
$905$	19158.4			0.703698
$906$	−2623.65			−0.0962086
$907$	24991.9			0.914932			0.457466	−	0.889227i	$-0.348757\pi$
$907$	24991.9			0.914932			0.457466	+	0.889227i	$0.348757\pi$
$908$		−	25912.1i		−	0.947051i
$909$	−6914.84			−0.252311
$910$		−	6749.48i		−	0.245871i
$911$	21982.0			0.799448			0.399724	−	0.916636i	$-0.369106\pi$
$911$	21982.0			0.799448			0.399724	+	0.916636i	$0.369106\pi$
$912$	4144.66			0.150486
$913$			27495.4i			0.996674i
$914$		−	13250.8i		−	0.479536i
$915$	12039.3			0.434982
$916$			14854.2i			0.535803i
$917$	−17621.0			−0.634564
$918$			784.258i			0.0281965i
$919$		−	30542.3i		−	1.09630i	−0.836381	−	0.548149i	$-0.815333\pi$
$919$		−	30542.3i		−	1.09630i	0.836381	−	0.548149i	$-0.184667\pi$
$920$		−	5046.85i		−	0.180858i
$921$			28658.3i			1.02532i
$922$		−	919.269i		−	0.0328357i
$923$	15706.3			0.560108
$924$			5043.94i			0.179582i
$925$	22315.3			0.793212
$926$	−11395.3			−0.404398
$927$			10739.3i			0.380501i
$928$	−27662.3			−0.978513
$929$	23792.9			0.840280			0.420140	−	0.907459i	$-0.361981\pi$
$929$	23792.9			0.840280			0.420140	+	0.907459i	$0.361981\pi$
$930$	3151.94			0.111136
$931$	−14535.2			−0.511678
$932$	35291.2			1.24034
$933$	8039.85			0.282114
$934$			12497.7i			0.437835i
$935$			2186.09i			0.0764628i
$936$		−	13201.3i		−	0.461003i
$937$		−	1866.82i		−	0.0650869i	−0.999470	−	0.0325435i	$-0.989639\pi$
$937$		−	1866.82i		−	0.0650869i	0.999470	−	0.0325435i	$-0.0103607\pi$
$938$	360.818			0.0125598
$939$	−5587.06			−0.194171
$940$			2440.05i			0.0846657i
$941$	48540.7			1.68159			0.840797	−	0.541350i	$-0.182087\pi$
$941$	48540.7			1.68159			0.840797	+	0.541350i	$0.182087\pi$
$942$	−8052.73	+	3217.18i	−0.278527	+	0.111275i
$943$	−1125.29			−0.0388593
$944$		−	237.670i		−	0.00819438i
$945$	−1719.59			−0.0591941
$946$	−6732.78			−0.231397
$947$		−	20709.2i		−	0.710620i	−0.934748	−	0.355310i	$-0.884375\pi$
$947$		−	20709.2i		−	0.710620i	0.934748	−	0.355310i	$-0.115625\pi$
$948$			19171.9i			0.656830i
$949$		−	17013.7i		−	0.581970i
$950$			12115.3i			0.413760i
$951$	−1848.09			−0.0630163
$952$	5176.36			0.176226
$953$	13499.4			0.458856			0.229428	−	0.973326i	$-0.426315\pi$
$953$	13499.4			0.458856			0.229428	+	0.973326i	$0.426315\pi$
$954$	−2821.31			−0.0957476
$955$	1401.30			0.0474816
$956$	−11639.2			−0.393763
$957$		−	9897.48i		−	0.334316i
$958$	15803.5			0.532973
$959$	−30217.4			−1.01749
$960$		−	2087.54i		−	0.0701824i
$961$	−8895.43			−0.298595
$962$		−	23523.7i		−	0.788394i
$963$		−	12524.1i		−	0.419090i
$964$			26216.3i			0.875904i
$965$			8849.62i			0.295212i
$966$		−	2847.33i		−	0.0948356i
$967$	−31713.1			−1.05463			−0.527313	−	0.849671i	$-0.676801\pi$
$967$	−31713.1			−1.05463			−0.527313	+	0.849671i	$0.676801\pi$
$968$			16904.3i			0.561287i
$969$	4863.95			0.161251
$970$			5077.85i			0.168082i
$971$		−	816.592i		−	0.0269883i	−0.999909	−	0.0134942i	$-0.995705\pi$
$971$		−	816.592i		−	0.0269883i	0.999909	−	0.0134942i	$-0.00429546\pi$
$972$	−1419.36			−0.0468373
$973$	−16317.6			−0.537635
$974$		−	11202.6i		−	0.368535i
$975$	−21752.3			−0.714492
$976$		−	13666.2i		−	0.448201i
$977$	−12860.8			−0.421138			−0.210569	−	0.977579i	$-0.567532\pi$
$977$	−12860.8			−0.421138			−0.210569	+	0.977579i	$0.567532\pi$
$978$	−3813.90			−0.124698
$979$	−31285.5			−1.02134
$980$		−	5120.42i		−	0.166904i
$981$	5222.70			0.169977
$982$	26706.8			0.867871
$983$			6932.91i			0.224949i	0.993655	+	0.112475i	$0.0358777\pi$
$983$			6932.91i			0.224949i	−0.993655	+	0.112475i	$0.964122\pi$
$984$	−1368.53			−0.0443365
$985$		−	4785.31i		−	0.154795i
$986$	−4286.45			−0.138447
$987$			3262.10i			0.105201i
$988$	−34551.4			−1.11258
$989$	−10282.3			−0.330594
$990$	1462.42			0.0469482
$991$	28801.9			0.923233			0.461617	−	0.887080i	$-0.347270\pi$
$991$	28801.9			0.923233			0.461617	+	0.887080i	$0.347270\pi$
$992$		−	27096.5i		−	0.867253i
$993$	13957.7			0.446055
$994$	4119.89			0.131464
$995$		−	17874.9i		−	0.569521i
$996$			21550.9i			0.685609i
$997$			5050.70i			0.160438i	0.996777	+	0.0802192i	$0.0255620\pi$
$997$			5050.70i			0.160438i	−0.996777	+	0.0802192i	$0.974438\pi$
$998$	21315.8			0.676093
$999$	−5993.24			−0.189808

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	471.4.b.a.313.15	✓	40
157.156	even	2	inner	471.4.b.a.313.26	yes	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
471.4.b.a.313.15	✓	40	1.1	even	1	trivial
471.4.b.a.313.26	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)

\(f(q)\)	\(=\)	\(q-1.46936i q^{2} -3.00000 q^{3} +5.84098 q^{4} -4.94652i q^{5} +4.40809i q^{6} +12.8754i q^{7} -20.3374i q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q-1.46936i q^{2} -3.00000 q^{3} +5.84098 q^{4} -4.94652i q^{5} +4.40809i q^{6} +12.8754i q^{7} -20.3374i q^{8} +9.00000 q^{9} -7.26823 q^{10} -22.3563 q^{11} -17.5229 q^{12} +72.1239 q^{13} +18.9187 q^{14} +14.8396i q^{15} +16.8448 q^{16} +19.7682 q^{17} -13.2243i q^{18} -82.0165 q^{19} -28.8925i q^{20} -38.6263i q^{21} +32.8496i q^{22} +50.1678i q^{23} +61.0122i q^{24} +100.532 q^{25} -105.976i q^{26} -27.0000 q^{27} +75.2052i q^{28} -147.572i q^{29} +21.8047 q^{30} +144.553 q^{31} -187.450i q^{32} +67.0690 q^{33} -29.0466i q^{34} +63.6887 q^{35} +52.5688 q^{36} +221.972 q^{37} +120.512i q^{38} -216.372 q^{39} -100.599 q^{40} +22.4304i q^{41} -56.7561 q^{42} +204.958i q^{43} -130.583 q^{44} -44.5187i q^{45} +73.7146 q^{46} -84.4528 q^{47} -50.5344 q^{48} +177.223 q^{49} -147.718i q^{50} -59.3045 q^{51} +421.274 q^{52} -213.343i q^{53} +39.6728i q^{54} +110.586i q^{55} +261.853 q^{56} +246.050 q^{57} -216.836 q^{58} -14.1094i q^{59} +86.6776i q^{60} -811.300i q^{61} -212.401i q^{62} +115.879i q^{63} -140.674 q^{64} -356.763i q^{65} -98.5487i q^{66} +19.0720 q^{67} +115.465 q^{68} -150.503i q^{69} -93.5817i q^{70} +217.768 q^{71} -183.037i q^{72} -235.896i q^{73} -326.157i q^{74} -301.596 q^{75} -479.056 q^{76} -287.848i q^{77} +317.928i q^{78} -1094.10i q^{79} -83.3233i q^{80} +81.0000 q^{81} +32.9584 q^{82} -1229.87i q^{83} -225.615i q^{84} -97.7838i q^{85} +301.158 q^{86} +442.715i q^{87} +454.670i q^{88} +1399.40 q^{89} -65.4141 q^{90} +928.627i q^{91} +293.029i q^{92} -433.659 q^{93} +124.092i q^{94} +405.697i q^{95} +562.351i q^{96} -698.636i q^{97} -260.405i q^{98} -201.207 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 120 q^{3} - 164 q^{4} + 360 q^{9}+O(q^{10}) \) 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

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