LMFDB - Embedded newform 471.4.b.a.313.1

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			313.1
Character	$\chi$	$=$	471.313
Dual form			471.4.b.a.313.40

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.39623i q^{2} -3.00000 q^{3} -21.1193 q^{4} -16.5163i q^{5} +16.1887i q^{6} -25.8596i q^{7} +70.7949i q^{8} +9.00000 q^{9} +O(q^{10})$	\(q-5.39623i q^{2} -3.00000 q^{3} -21.1193 q^{4} -16.5163i q^{5} +16.1887i q^{6} -25.8596i q^{7} +70.7949i q^{8} +9.00000 q^{9} -89.1257 q^{10} +42.1429 q^{11} +63.3579 q^{12} -86.3665 q^{13} -139.544 q^{14} +49.5488i q^{15} +213.071 q^{16} -138.776 q^{17} -48.5661i q^{18} +58.3740 q^{19} +348.812i q^{20} +77.5789i q^{21} -227.413i q^{22} +80.8107i q^{23} -212.385i q^{24} -147.787 q^{25} +466.053i q^{26} -27.0000 q^{27} +546.137i q^{28} -128.639i q^{29} +267.377 q^{30} +199.648 q^{31} -583.421i q^{32} -126.429 q^{33} +748.865i q^{34} -427.105 q^{35} -190.074 q^{36} -193.390 q^{37} -315.000i q^{38} +259.099 q^{39} +1169.27 q^{40} +79.7126i q^{41} +418.633 q^{42} -380.057i q^{43} -890.029 q^{44} -148.647i q^{45} +436.074 q^{46} +219.285 q^{47} -639.213 q^{48} -325.720 q^{49} +797.495i q^{50} +416.327 q^{51} +1824.00 q^{52} +69.8915i q^{53} +145.698i q^{54} -696.044i q^{55} +1830.73 q^{56} -175.122 q^{57} -694.168 q^{58} -755.221i q^{59} -1046.44i q^{60} -529.188i q^{61} -1077.35i q^{62} -232.737i q^{63} -1443.71 q^{64} +1426.45i q^{65} +682.238i q^{66} +769.183 q^{67} +2930.85 q^{68} -242.432i q^{69} +2304.76i q^{70} -312.298 q^{71} +637.154i q^{72} +357.763i q^{73} +1043.58i q^{74} +443.362 q^{75} -1232.82 q^{76} -1089.80i q^{77} -1398.16i q^{78} +395.416i q^{79} -3519.14i q^{80} +81.0000 q^{81} +430.148 q^{82} +390.338i q^{83} -1638.41i q^{84} +2292.06i q^{85} -2050.88 q^{86} +385.918i q^{87} +2983.50i q^{88} +866.368 q^{89} -802.131 q^{90} +2233.40i q^{91} -1706.67i q^{92} -598.943 q^{93} -1183.31i q^{94} -964.122i q^{95} +1750.26i q^{96} -718.346i q^{97} +1757.66i q^{98} +379.286 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$		−	5.39623i		−	1.90786i	−0.300036	−	0.953928i	$-0.596999\pi$
$2$		−	5.39623i		−	1.90786i	0.300036	−	0.953928i	$-0.403001\pi$
$3$	−3.00000			−0.577350
$3$	−3.00000			−0.577350
$4$	−21.1193			−2.63991
$5$		−	16.5163i		−	1.47726i	−0.674111	−	0.738630i	$-0.735473\pi$
$5$		−	16.5163i		−	1.47726i	0.674111	−	0.738630i	$-0.264527\pi$
$6$			16.1887i			1.10150i
$7$		−	25.8596i		−	1.39629i	−0.715957	−	0.698144i	$-0.754009\pi$
$7$		−	25.8596i		−	1.39629i	0.715957	−	0.698144i	$-0.245991\pi$
$8$			70.7949i			3.12872i
$9$	9.00000			0.333333
$10$	−89.1257			−2.81840
$11$	42.1429			1.15514			0.577571	−	0.816341i	$-0.304001\pi$
$11$	42.1429			1.15514			0.577571	+	0.816341i	$0.304001\pi$
$12$	63.3579			1.52416
$13$	−86.3665			−1.84260			−0.921298	−	0.388858i	$-0.872870\pi$
$13$	−86.3665			−1.84260			−0.921298	+	0.388858i	$0.872870\pi$
$14$	−139.544			−2.66392
$15$			49.5488i			0.852897i
$16$	213.071			3.32923
$17$	−138.776			−1.97988			−0.989942	−	0.141474i	$-0.954816\pi$
$17$	−138.776			−1.97988			−0.989942	+	0.141474i	$0.954816\pi$
$18$		−	48.5661i		−	0.635952i
$19$	58.3740			0.704838			0.352419	−	0.935842i	$-0.385359\pi$
$19$	58.3740			0.704838			0.352419	+	0.935842i	$0.385359\pi$
$20$			348.812i			3.89984i
$21$			77.5789i			0.806147i
$22$		−	227.413i		−	2.20384i
$23$			80.8107i			0.732618i	0.930493	+	0.366309i	$0.119379\pi$
$23$			80.8107i			0.732618i	−0.930493	+	0.366309i	$0.880621\pi$
$24$		−	212.385i		−	1.80637i
$25$	−147.787			−1.18230
$26$			466.053i			3.51541i
$27$	−27.0000			−0.192450
$28$			546.137i			3.68608i
$29$		−	128.639i		−	0.823715i	−0.911248	−	0.411858i	$-0.864880\pi$
$29$		−	128.639i		−	0.823715i	0.911248	−	0.411858i	$-0.135120\pi$
$30$	267.377			1.62720
$31$	199.648			1.15670			0.578352	−	0.815788i	$-0.303696\pi$
$31$	199.648			1.15670			0.578352	+	0.815788i	$0.303696\pi$
$32$		−	583.421i		−	3.22298i
$33$	−126.429			−0.666921
$34$			748.865i			3.77733i
$35$	−427.105			−2.06268
$36$	−190.074			−0.879971
$37$	−193.390			−0.859275			−0.429638	−	0.903001i	$-0.641359\pi$
$37$	−193.390			−0.859275			−0.429638	+	0.903001i	$0.641359\pi$
$38$		−	315.000i		−	1.34473i
$39$	259.099			1.06382
$40$	1169.27			4.62194
$41$			79.7126i			0.303635i	0.988409	+	0.151817i	$0.0485125\pi$
$41$			79.7126i			0.303635i	−0.988409	+	0.151817i	$0.951487\pi$
$42$	418.633			1.53801
$43$		−	380.057i		−	1.34786i	−0.738793	−	0.673932i	$-0.764604\pi$
$43$		−	380.057i		−	1.34786i	0.738793	−	0.673932i	$-0.235396\pi$
$44$	−890.029			−3.04948
$45$		−	148.647i		−	0.492420i
$46$	436.074			1.39773
$47$	219.285			0.680553			0.340277	−	0.940325i	$-0.389479\pi$
$47$	219.285			0.680553			0.340277	+	0.940325i	$0.389479\pi$
$48$	−639.213			−1.92213
$49$	−325.720			−0.949621
$50$			797.495i			2.25566i
$51$	416.327			1.14309
$52$	1824.00			4.86429
$53$			69.8915i			0.181138i	0.995890	+	0.0905692i	$0.0288686\pi$
$53$			69.8915i			0.181138i	−0.995890	+	0.0905692i	$0.971131\pi$
$54$			145.698i			0.367167i
$55$		−	696.044i		−	1.70645i
$56$	1830.73			4.36860
$57$	−175.122			−0.406938
$58$	−694.168			−1.57153
$59$		−	755.221i		−	1.66646i	−0.552923	−	0.833232i	$-0.686488\pi$
$59$		−	755.221i		−	1.66646i	0.552923	−	0.833232i	$-0.313512\pi$
$60$		−	1046.44i		−	2.25157i
$61$		−	529.188i		−	1.11075i	−0.831601	−	0.555373i	$-0.812575\pi$
$61$		−	529.188i		−	1.11075i	0.831601	−	0.555373i	$-0.187425\pi$
$62$		−	1077.35i		−	2.20682i
$63$		−	232.737i		−	0.465429i
$64$	−1443.71			−2.81974
$65$			1426.45i			2.72199i
$66$			682.238i			1.27239i
$67$	769.183			1.40255			0.701273	−	0.712893i	$-0.252615\pi$
$67$	769.183			1.40255			0.701273	+	0.712893i	$0.252615\pi$
$68$	2930.85			5.22672
$69$		−	242.432i		−	0.422977i
$70$			2304.76i			3.93530i
$71$	−312.298			−0.522013			−0.261006	−	0.965337i	$-0.584054\pi$
$71$	−312.298			−0.522013			−0.261006	+	0.965337i	$0.584054\pi$
$72$			637.154i			1.04291i
$73$			357.763i			0.573602i	0.957990	+	0.286801i	$0.0925919\pi$
$73$			357.763i			0.573602i	−0.957990	+	0.286801i	$0.907408\pi$
$74$			1043.58i			1.63937i
$75$	443.362			0.682601
$76$	−1232.82			−1.86071
$77$		−	1089.80i		−	1.61291i
$78$		−	1398.16i		−	2.02962i
$79$			395.416i			0.563136i	0.959541	+	0.281568i	$0.0908545\pi$
$79$			395.416i			0.563136i	−0.959541	+	0.281568i	$0.909145\pi$
$80$		−	3519.14i		−	4.91815i
$81$	81.0000			0.111111
$82$	430.148			0.579291
$83$			390.338i			0.516206i	0.966117	+	0.258103i	$0.0830974\pi$
$83$			390.338i			0.516206i	−0.966117	+	0.258103i	$0.916903\pi$
$84$		−	1638.41i		−	2.12816i
$85$			2292.06i			2.92481i
$86$	−2050.88			−2.57153
$87$			385.918i			0.475572i
$88$			2983.50i			3.61412i
$89$	866.368			1.03185			0.515926	−	0.856633i	$-0.327448\pi$
$89$	866.368			1.03185			0.515926	+	0.856633i	$0.327448\pi$
$90$	−802.131			−0.939467
$91$			2233.40i			2.57279i
$92$		−	1706.67i		−	1.93405i
$93$	−598.943			−0.667823
$94$		−	1183.31i		−	1.29840i
$95$		−	964.122i		−	1.04123i
$96$			1750.26i			1.86079i
$97$		−	718.346i		−	0.751928i	−0.926634	−	0.375964i	$-0.877312\pi$
$97$		−	718.346i		−	0.751928i	0.926634	−	0.375964i	$-0.122688\pi$
$98$			1757.66i			1.81174i
$99$	379.286			0.385047
$100$	3121.17			3.12117
$101$	−874.388			−0.861434			−0.430717	−	0.902487i	$-0.641739\pi$
$101$	−874.388			−0.861434			−0.430717	+	0.902487i	$0.641739\pi$
$102$		−	2246.60i		−	2.18084i
$103$			1044.94i			0.999623i	0.866134	+	0.499812i	$0.166597\pi$
$103$			1044.94i			0.999623i	−0.866134	+	0.499812i	$0.833403\pi$
$104$		−	6114.30i		−	5.76497i
$105$	1281.31			1.19089
$106$	377.151			0.345586
$107$		−	1390.10i		−	1.25594i	−0.778237	−	0.627971i	$-0.783886\pi$
$107$		−	1390.10i		−	1.25594i	0.778237	−	0.627971i	$-0.216114\pi$
$108$	570.221			0.508052
$109$	−1762.64			−1.54890			−0.774451	−	0.632634i	$-0.781974\pi$
$109$	−1762.64			−1.54890			−0.774451	+	0.632634i	$0.781974\pi$
$110$	−3756.01			−3.25565
$111$	580.171			0.496103
$112$		−	5509.93i		−	4.64857i
$113$	131.000			0.109057			0.0545285	−	0.998512i	$-0.482634\pi$
$113$	131.000			0.109057			0.0545285	+	0.998512i	$0.482634\pi$
$114$			944.999i			0.776380i
$115$	1334.69			1.08227
$116$			2716.78i			2.17454i
$117$	−777.298			−0.614199
$118$	−4075.35			−3.17937
$119$			3588.68i			2.76449i
$120$	−3507.80			−2.66848
$121$	445.023			0.334353
$122$	−2855.62			−2.11914
$123$		−	239.138i		−	0.175303i
$124$	−4216.42			−3.05360
$125$			376.364i			0.269304i
$126$	−1255.90			−0.887972
$127$	−224.969			−0.157187			−0.0785935	−	0.996907i	$-0.525043\pi$
$127$	−224.969			−0.157187			−0.0785935	+	0.996907i	$0.525043\pi$
$128$			3123.22i			2.15669i
$129$			1140.17i			0.778190i
$130$	7697.47			5.19317
$131$			207.278i			0.138244i	0.997608	+	0.0691221i	$0.0220198\pi$
$131$			207.278i			0.138244i	−0.997608	+	0.0691221i	$0.977980\pi$
$132$	2670.09			1.76062
$133$		−	1509.53i		−	0.984157i
$134$		−	4150.69i		−	2.67586i
$135$			445.940i			0.284299i
$136$		−	9824.60i		−	6.19450i
$137$			285.336i			0.177941i	0.996034	+	0.0889704i	$0.0283577\pi$
$137$			285.336i			0.177941i	−0.996034	+	0.0889704i	$0.971642\pi$
$138$	−1308.22			−0.806979
$139$			430.343i			0.262599i	0.991343	+	0.131299i	$0.0419149\pi$
$139$			430.343i			0.262599i	−0.991343	+	0.131299i	$0.958085\pi$
$140$	9020.16			5.44530
$141$	−657.855			−0.392918
$142$			1685.23i			0.995925i
$143$	−3639.73			−2.12846
$144$	1917.64			1.10974
$145$	−2124.64			−1.21684
$146$	1930.57			1.09435
$147$	977.160			0.548264
$148$	4084.27			2.26841
$149$			799.365i			0.439507i	0.975555	+	0.219753i	$0.0705253\pi$
$149$			799.365i			0.439507i	−0.975555	+	0.219753i	$0.929475\pi$
$150$		−	2392.49i		−	1.30230i
$151$		−	873.212i		−	0.470603i	−0.971922	−	0.235301i	$-0.924392\pi$
$151$		−	873.212i		−	0.470603i	0.971922	−	0.235301i	$-0.0756077\pi$
$152$			4132.58i			2.20524i
$153$	−1248.98			−0.659961
$154$	−5880.81			−3.07720
$155$		−	3297.44i		−	1.70875i
$156$	−5472.00			−2.80840
$157$	1933.25	+	363.928i	0.982739	+	0.184998i
$157$	1933.25	+	363.928i	0.982739	+	0.184998i
$158$	2133.76			1.07438
$159$		−	209.674i		−	0.104580i
$160$	−9635.95			−4.76118
$161$	2089.74			1.02295
$162$		−	437.095i		−	0.211984i
$163$			3846.07i			1.84814i	0.382218	+	0.924072i	$0.375160\pi$
$163$			3846.07i			1.84814i	−0.382218	+	0.924072i	$0.624840\pi$
$164$		−	1683.48i		−	0.801569i
$165$			2088.13i			0.985217i
$166$	2106.35			0.984847
$167$	−1621.89			−0.751530			−0.375765	−	0.926715i	$-0.622620\pi$
$167$	−1621.89			−0.751530			−0.375765	+	0.926715i	$0.622620\pi$
$168$	−5492.18			−2.52221
$169$	5262.16			2.39516
$170$	12368.5			5.58011
$171$	525.366			0.234946
$172$			8026.54i			3.55825i
$173$	−1609.12			−0.707162			−0.353581	−	0.935404i	$-0.615036\pi$
$173$	−1609.12			−0.707162			−0.353581	+	0.935404i	$0.615036\pi$
$174$	2082.50			0.907323
$175$			3821.73i			1.65083i
$176$	8979.42			3.84574
$177$			2265.66i			0.962134i
$178$		−	4675.12i		−	1.96862i
$179$			3325.69i			1.38868i	0.719647	+	0.694340i	$0.244304\pi$
$179$			3325.69i			1.38868i	−0.719647	+	0.694340i	$0.755696\pi$
$180$			3139.31i			1.29995i
$181$			2451.19i			1.00660i	0.864111	+	0.503302i	$0.167882\pi$
$181$			2451.19i			1.00660i	−0.864111	+	0.503302i	$0.832118\pi$
$182$	12052.0			4.90852
$183$			1587.56i			0.641290i
$184$	−5720.99			−2.29216
$185$			3194.09i			1.26937i
$186$			3232.04i			1.27411i
$187$	−5848.40			−2.28705
$188$	−4631.15			−1.79660
$189$			698.210i			0.268716i
$190$	−5202.62			−1.98652
$191$			1405.80i			0.532565i	0.963895	+	0.266283i	$0.0857955\pi$
$191$			1405.80i			0.532565i	−0.963895	+	0.266283i	$0.914205\pi$
$192$	4331.13			1.62798
$193$	1218.52			0.454460			0.227230	−	0.973841i	$-0.427033\pi$
$193$	1218.52			0.454460			0.227230	+	0.973841i	$0.427033\pi$
$194$	−3876.36			−1.43457
$195$		−	4279.36i		−	1.57154i
$196$	6878.98			2.50692
$197$	−1670.06			−0.603994			−0.301997	−	0.953309i	$-0.597653\pi$
$197$	−1670.06			−0.603994			−0.301997	+	0.953309i	$0.597653\pi$
$198$		−	2046.72i		−	0.734615i
$199$	−77.4212			−0.0275791			−0.0137896	−	0.999905i	$-0.504389\pi$
$199$	−77.4212			−0.0275791			−0.0137896	+	0.999905i	$0.504389\pi$
$200$		−	10462.6i		−	3.69908i
$201$	−2307.55			−0.809760
$202$			4718.40i			1.64349i
$203$	−3326.57			−1.15014
$204$	−8792.54			−3.01765
$205$	1316.56			0.448547
$206$	5638.75			1.90714
$207$			727.297i			0.244206i
$208$	−18402.2			−6.13443
$209$	2460.05			0.814188
$210$		−	6914.27i		−	2.27205i
$211$			903.116i			0.294659i	0.989087	+	0.147330i	$0.0470678\pi$
$211$			903.116i			0.294659i	−0.989087	+	0.147330i	$0.952932\pi$
$212$		−	1476.06i		−	0.478190i
$213$	936.893			0.301384
$214$	−7501.28			−2.39615
$215$	−6277.13			−1.99115
$216$		−	1911.46i		−	0.602123i
$217$		−	5162.82i		−	1.61509i
$218$			9511.61i			2.95508i
$219$		−	1073.29i		−	0.331169i
$220$			14700.0i			4.50487i
$221$	11985.6			3.64813
$222$		−	3130.74i		−	0.946493i
$223$		−	4825.06i		−	1.44892i	−0.689315	−	0.724461i	$-0.742089\pi$
$223$		−	4825.06i		−	1.44892i	0.689315	−	0.724461i	$-0.257911\pi$
$224$	−15087.1			−4.50021
$225$	−1330.09			−0.394100
$226$		−	706.906i		−	0.208065i
$227$		−	739.344i		−	0.216176i	−0.994141	−	0.108088i	$-0.965527\pi$
$227$		−	739.344i		−	0.216176i	0.994141	−	0.108088i	$-0.0344728\pi$
$228$	3698.46			1.07428
$229$		−	2101.60i		−	0.606453i	−0.952918	−	0.303227i	$-0.901936\pi$
$229$		−	2101.60i		−	0.606453i	0.952918	−	0.303227i	$-0.0980639\pi$
$230$		−	7202.31i		−	2.06481i
$231$			3269.40i			0.931215i
$232$	9107.01			2.57717
$233$	−3406.34			−0.957755			−0.478877	−	0.877882i	$-0.658956\pi$
$233$	−3406.34			−0.957755			−0.478877	+	0.877882i	$0.658956\pi$
$234$			4194.48i			1.17180i
$235$		−	3621.77i		−	1.00535i
$236$			15949.8i			4.39932i
$237$		−	1186.25i		−	0.325127i
$238$	19365.4			5.27425
$239$	3211.81			0.869268			0.434634	−	0.900607i	$-0.356878\pi$
$239$	3211.81			0.869268			0.434634	+	0.900607i	$0.356878\pi$
$240$			10557.4i			2.83949i
$241$		−	3347.17i		−	0.894648i	−0.894372	−	0.447324i	$-0.852377\pi$
$241$		−	3347.17i		−	0.894648i	0.894372	−	0.447324i	$-0.147623\pi$
$242$		−	2401.45i		−	0.637896i
$243$	−243.000			−0.0641500
$244$			11176.1i			2.93228i
$245$			5379.68i			1.40284i
$246$	−1290.44			−0.334454
$247$	−5041.56			−1.29873
$248$			14134.0i			3.61900i
$249$		−	1171.01i		−	0.298032i
$250$	2030.95			0.513793
$251$		−	466.110i		−	0.117214i	−0.998281	−	0.0586068i	$-0.981334\pi$
$251$		−	466.110i		−	0.117214i	0.998281	−	0.0586068i	$-0.0186658\pi$
$252$			4915.24i			1.22869i
$253$			3405.60i			0.846277i
$254$			1213.98i			0.299890i
$255$		−	6876.17i		−	1.68864i
$256$	5303.93			1.29490
$257$	−3450.25			−0.837434			−0.418717	−	0.908117i	$-0.637520\pi$
$257$	−3450.25			−0.837434			−0.418717	+	0.908117i	$0.637520\pi$
$258$	6152.63			1.48467
$259$			5001.00i			1.19980i
$260$		−	30125.7i		−	7.18583i
$261$		−	1157.75i		−	0.274572i
$262$	1118.52			0.263750
$263$	4443.08			1.04172			0.520860	−	0.853642i	$-0.325611\pi$
$263$	4443.08			1.04172			0.520860	+	0.853642i	$0.325611\pi$
$264$		−	8950.50i		−	2.08661i
$265$	1154.35			0.267589
$266$	−8145.78			−1.87763
$267$	−2599.10			−0.595740
$268$	−16244.6			−3.70260
$269$		−	2548.78i		−	0.577702i	−0.957374	−	0.288851i	$-0.906727\pi$
$269$		−	2548.78i		−	0.577702i	0.957374	−	0.288851i	$-0.0932731\pi$
$270$	2406.39			0.542401
$271$			2977.11i			0.667330i	0.942692	+	0.333665i	$0.108285\pi$
$271$			2977.11i			0.667330i	−0.942692	+	0.333665i	$0.891715\pi$
$272$	−29569.0			−6.59150
$273$		−	6700.21i		−	1.48540i
$274$	1539.74			0.339486
$275$	−6228.19			−1.36572
$276$			5120.00i			1.11662i
$277$	2888.16			0.626471			0.313236	−	0.949675i	$-0.398587\pi$
$277$	2888.16			0.626471			0.313236	+	0.949675i	$0.398587\pi$
$278$	2322.23			0.501001
$279$	1796.83			0.385568
$280$		−	30236.8i		−	6.45355i
$281$	−2795.10			−0.593386			−0.296693	−	0.954973i	$-0.595884\pi$
$281$	−2795.10			−0.593386			−0.296693	+	0.954973i	$0.595884\pi$
$282$			3549.94i			0.749630i
$283$	68.0544			0.0142948			0.00714738	−	0.999974i	$-0.497725\pi$
$283$	68.0544			0.0142948			0.00714738	+	0.999974i	$0.497725\pi$
$284$	6595.51			1.37807
$285$			2892.37i			0.601154i
$286$			19640.8i			4.06079i
$287$	2061.34			0.423961
$288$		−	5250.79i		−	1.07433i
$289$	14345.7			2.91994
$290$			11465.1i			2.32156i
$291$			2155.04i			0.434126i
$292$		−	7555.70i		−	1.51426i
$293$		−	3141.89i		−	0.626454i	−0.949678	−	0.313227i	$-0.898590\pi$
$293$		−	3141.89i		−	0.626454i	0.949678	−	0.313227i	$-0.101410\pi$
$294$		−	5272.98i		−	1.04601i
$295$	−12473.4			−2.46180
$296$		−	13691.0i		−	2.68843i
$297$	−1137.86			−0.222307
$298$	4313.56			0.838516
$299$		−	6979.34i		−	1.34992i
$300$	−9363.51			−1.80201
$301$	−9828.13			−1.88201
$302$	−4712.05			−0.897842
$303$	2623.16			0.497349
$304$	12437.8			2.34657
$305$	−8740.21			−1.64086
$306$			6739.79i			1.25911i
$307$			4101.66i			0.762522i	0.924467	+	0.381261i	$0.124510\pi$
$307$			4101.66i			0.762522i	−0.924467	+	0.381261i	$0.875490\pi$
$308$			23015.8i			4.25795i
$309$		−	3134.83i		−	0.577133i
$310$	−17793.7			−3.26005
$311$	4178.99			0.761957			0.380979	−	0.924584i	$-0.375587\pi$
$311$	4178.99			0.761957			0.380979	+	0.924584i	$0.375587\pi$
$312$			18342.9i			3.32841i
$313$	3899.58			0.704208			0.352104	−	0.935961i	$-0.385466\pi$
$313$	3899.58			0.704208			0.352104	+	0.935961i	$0.385466\pi$
$314$	1963.84	−	10432.3i	0.352949	−	1.87492i
$315$	−3843.94			−0.687561
$316$		−	8350.91i		−	1.48663i
$317$	−4925.68			−0.872724			−0.436362	−	0.899771i	$-0.643733\pi$
$317$	−4925.68			−0.872724			−0.436362	+	0.899771i	$0.643733\pi$
$318$	−1131.45			−0.199524
$319$		−	5421.24i		−	0.951508i
$320$			23844.7i			4.16550i
$321$			4170.29i			0.725118i
$322$		−	11276.7i		−	1.95163i
$323$	−8100.89			−1.39550
$324$	−1710.66			−0.293324
$325$	12763.9			2.17850
$326$	20754.3			3.52599
$327$	5287.92			0.894258
$328$	−5643.24			−0.949988
$329$		−	5670.63i		−	0.950248i
$330$	11268.0			1.87965
$331$	2723.61			0.452276			0.226138	−	0.974095i	$-0.427390\pi$
$331$	2723.61			0.452276			0.226138	+	0.974095i	$0.427390\pi$
$332$		−	8243.66i		−	1.36274i
$333$	−1740.51			−0.286425
$334$			8752.08i			1.43381i
$335$		−	12704.0i		−	2.07193i
$336$			16529.8i			2.68385i
$337$			9356.88i			1.51247i	0.654301	+	0.756234i	$0.272963\pi$
$337$			9356.88i			1.51247i	−0.654301	+	0.756234i	$0.727037\pi$
$338$		−	28395.9i		−	4.56962i
$339$	−393.000			−0.0629641
$340$		−	48406.7i		−	7.72123i
$341$	8413.73			1.33616
$342$		−	2835.00i		−	0.448243i
$343$		−	446.856i		−	0.0703439i
$344$	26906.1			4.21709
$345$	−4004.08			−0.624847
$346$			8683.18i			1.34916i
$347$	−12909.0			−1.99709			−0.998544	−	0.0539398i	$-0.982822\pi$
$347$	−12909.0			−1.99709			−0.998544	+	0.0539398i	$0.982822\pi$
$348$		−	8150.33i		−	1.25547i
$349$	−4094.37			−0.627984			−0.313992	−	0.949426i	$-0.601667\pi$
$349$	−4094.37			−0.627984			−0.313992	+	0.949426i	$0.601667\pi$
$350$	20622.9			3.14955
$351$	2331.89			0.354608
$352$		−	24587.1i		−	3.72300i
$353$	433.401			0.0653474			0.0326737	−	0.999466i	$-0.489598\pi$
$353$	433.401			0.0653474			0.0326737	+	0.999466i	$0.489598\pi$
$354$	12226.0			1.83561
$355$			5157.99i			0.771149i
$356$	−18297.1			−2.72400
$357$		−	10766.1i		−	1.59608i
$358$	17946.2			2.64940
$359$		−	891.622i		−	0.131081i	−0.997850	−	0.0655404i	$-0.979123\pi$
$359$		−	891.622i		−	0.131081i	0.997850	−	0.0655404i	$-0.0208771\pi$
$360$	10523.4			1.54065
$361$	−3451.47			−0.503203
$362$	13227.2			1.92046
$363$	−1335.07			−0.193039
$364$		−	47168.0i		−	6.79196i
$365$	5908.91			0.847360
$366$	8566.86			1.22349
$367$			1223.48i			0.174019i	0.996207	+	0.0870097i	$0.0277311\pi$
$367$			1223.48i			0.174019i	−0.996207	+	0.0870097i	$0.972269\pi$
$368$			17218.4i			2.43906i
$369$			717.413i			0.101212i
$370$	17236.0			2.42178
$371$	1807.37			0.252921
$372$	12649.3			1.76300
$373$			6750.06i			0.937010i	0.883461	+	0.468505i	$0.155207\pi$
$373$			6750.06i			0.937010i	−0.883461	+	0.468505i	$0.844793\pi$
$374$			31559.3i			4.36336i
$375$		−	1129.09i		−	0.155483i
$376$			15524.2i			2.12926i
$377$			11110.1i			1.51777i
$378$	3767.70			0.512671
$379$			3723.83i			0.504697i	0.967636	+	0.252349i	$0.0812030\pi$
$379$			3723.83i			0.504697i	−0.967636	+	0.252349i	$0.918797\pi$
$380$			20361.6i			2.74876i
$381$	674.907			0.0907520
$382$	7586.01			1.01606
$383$		−	12247.8i		−	1.63403i	−0.576619	−	0.817013i	$-0.695628\pi$
$383$		−	12247.8i		−	1.63403i	0.576619	−	0.817013i	$-0.304372\pi$
$384$		−	9369.65i		−	1.24516i
$385$	−17999.4			−2.38269
$386$		−	6575.40i		−	0.867044i
$387$		−	3420.51i		−	0.449288i
$388$			15171.0i			1.98503i
$389$	−6057.84			−0.789576			−0.394788	−	0.918772i	$-0.629182\pi$
$389$	−6057.84			−0.789576			−0.394788	+	0.918772i	$0.629182\pi$
$390$	−23092.4			−2.99828
$391$		−	11214.6i		−	1.45050i
$392$		−	23059.3i		−	2.97110i
$393$		−	621.835i		−	0.0798153i
$394$			9012.03i			1.15233i
$395$	6530.80			0.831899
$396$	−8010.26			−1.01649
$397$		−	10540.7i		−	1.33256i	−0.745704	−	0.666278i	$-0.767886\pi$
$397$		−	10540.7i		−	1.33256i	0.745704	−	0.666278i	$-0.232114\pi$
$398$			417.783i			0.0526170i
$399$			4528.59i			0.568203i
$400$	−31489.2			−3.93615
$401$			1086.37i			0.135288i	0.997710	+	0.0676440i	$0.0215482\pi$
$401$			1086.37i			0.135288i	−0.997710	+	0.0676440i	$0.978452\pi$
$402$			12452.1i			1.54491i
$403$	−17242.9			−2.13134
$404$	18466.5			2.27411
$405$		−	1337.82i		−	0.164140i
$406$			17950.9i			2.19431i
$407$	−8150.03			−0.992585
$408$			29473.8i			3.57640i
$409$			2329.93i			0.281681i	0.990032	+	0.140840i	$0.0449805\pi$
$409$			2329.93i			0.281681i	−0.990032	+	0.140840i	$0.955020\pi$
$410$		−	7104.44i		−	0.855764i
$411$		−	856.008i		−	0.102734i
$412$		−	22068.5i		−	2.63892i
$413$	−19529.7			−2.32687
$414$	3924.66			0.465910
$415$	6446.93			0.762571
$416$			50388.0i			5.93864i
$417$		−	1291.03i		−	0.151612i
$418$		−	13275.0i		−	1.55335i
$419$	−13456.0			−1.56890			−0.784449	−	0.620193i	$-0.787054\pi$
$419$	−13456.0			−1.56890			−0.784449	+	0.620193i	$0.787054\pi$
$420$	−27060.5			−3.14385
$421$			498.561i			0.0577159i	0.999584	+	0.0288579i	$0.00918704\pi$
$421$			498.561i			0.0577159i	−0.999584	+	0.0288579i	$0.990813\pi$
$422$	4873.42			0.562167
$423$	1973.56			0.226851
$424$	−4947.96			−0.566731
$425$	20509.3			2.34082
$426$		−	5055.69i		−	0.574997i
$427$	−13684.6			−1.55092
$428$			29357.9i			3.31558i
$429$	10919.2			1.22887
$430$			33872.8i			3.79882i
$431$	−7213.00			−0.806121			−0.403061	−	0.915173i	$-0.632054\pi$
$431$	−7213.00			−0.806121			−0.403061	+	0.915173i	$0.632054\pi$
$432$	−5752.92			−0.640711
$433$			4037.09i			0.448061i	0.974582	+	0.224030i	$0.0719215\pi$
$433$			4037.09i			0.448061i	−0.974582	+	0.224030i	$0.928079\pi$
$434$	−27859.8			−3.08136
$435$	6373.93			0.702544
$436$	37225.7			4.08897
$437$			4717.25i			0.516377i
$438$	−5791.71			−0.631824
$439$		−	902.097i		−	0.0980746i	−0.998797	−	0.0490373i	$-0.984385\pi$
$439$		−	902.097i		−	0.0980746i	0.998797	−	0.0490373i	$-0.0156153\pi$
$440$	49276.3			5.33899
$441$	−2931.48			−0.316540
$442$		−	64676.8i		−	6.96010i
$443$			8108.69i			0.869651i	0.900515	+	0.434826i	$0.143190\pi$
$443$			8108.69i			0.869651i	−0.900515	+	0.434826i	$0.856810\pi$
$444$	−12252.8			−1.30967
$445$		−	14309.2i		−	1.52431i
$446$	−26037.1			−2.76434
$447$		−	2398.09i		−	0.253749i
$448$			37333.8i			3.93717i
$449$			15568.0i			1.63630i	0.575003	+	0.818151i	$0.305001\pi$
$449$			15568.0i			1.63630i	−0.575003	+	0.818151i	$0.694999\pi$
$450$			7177.46i			0.751886i
$451$			3359.32i			0.350741i
$452$	−2766.63			−0.287901
$453$			2619.64i			0.271702i
$454$	−3989.67			−0.412433
$455$	36887.5			3.80069
$456$		−	12397.7i		−	1.27320i
$457$	−11589.6			−1.18630			−0.593148	−	0.805093i	$-0.702115\pi$
$457$	−11589.6			−1.18630			−0.593148	+	0.805093i	$0.702115\pi$
$458$	−11340.7			−1.15703
$459$	3746.94			0.381029
$460$	−28187.8			−2.85709
$461$	−2404.72			−0.242948			−0.121474	−	0.992595i	$-0.538762\pi$
$461$	−2404.72			−0.242948			−0.121474	+	0.992595i	$0.538762\pi$
$462$	17642.4			1.77662
$463$			19450.8i			1.95239i	0.216904	+	0.976193i	$0.430404\pi$
$463$			19450.8i			1.95239i	−0.216904	+	0.976193i	$0.569596\pi$
$464$		−	27409.3i		−	2.74234i
$465$			9892.32i			0.986549i
$466$			18381.4i			1.82726i
$467$	−2691.32			−0.266680			−0.133340	−	0.991070i	$-0.542570\pi$
$467$	−2691.32			−0.266680			−0.133340	+	0.991070i	$0.542570\pi$
$468$	16416.0			1.62143
$469$		−	19890.8i		−	1.95836i
$470$	−19543.9			−1.91807
$471$	−5799.75	−	1091.78i	−0.567385	−	0.106808i
$472$	53465.8			5.21390
$473$		−	16016.7i		−	1.55697i
$474$	−6401.27			−0.620295
$475$	−8626.95			−0.833330
$476$		−	75790.6i		−	7.29801i
$477$			629.023i			0.0603795i
$478$		−	17331.7i		−	1.65844i
$479$		−	13478.9i		−	1.28573i	−0.765980	−	0.642865i	$-0.777746\pi$
$479$		−	13478.9i		−	1.28573i	0.765980	−	0.642865i	$-0.222254\pi$
$480$	28907.8			2.74887
$481$	16702.4			1.58330
$482$	−18062.1			−1.70686
$483$	−6269.21			−0.590598
$484$	−9398.58			−0.882662
$485$	−11864.4			−1.11079
$486$			1311.28i			0.122389i
$487$	17171.7			1.59779			0.798896	−	0.601469i	$-0.205418\pi$
$487$	17171.7			1.59779			0.798896	+	0.601469i	$0.205418\pi$
$488$	37463.8			3.47521
$489$		−	11538.2i		−	1.06703i
$490$	29030.0			2.67641
$491$		−	6766.94i		−	0.621971i	−0.950415	−	0.310986i	$-0.899341\pi$
$491$		−	6766.94i		−	0.621971i	0.950415	−	0.310986i	$-0.100659\pi$
$492$			5050.43i			0.462786i
$493$			17852.0i			1.63086i
$494$			27205.4i			2.47779i
$495$		−	6264.39i		−	0.568815i
$496$	42539.1			3.85094
$497$			8075.89i			0.728880i
$498$	−6319.06			−0.568602
$499$			11543.6i			1.03560i	0.855502	+	0.517800i	$0.173249\pi$
$499$			11543.6i			1.03560i	−0.855502	+	0.517800i	$0.826751\pi$
$500$		−	7948.54i		−	0.710939i
$501$	4865.66			0.433896
$502$	−2515.24			−0.223627
$503$		−	6035.16i		−	0.534979i	−0.963561	−	0.267489i	$-0.913806\pi$
$503$		−	6035.16i		−	0.534979i	0.963561	−	0.267489i	$-0.0861940\pi$
$504$	16476.6			1.45620
$505$			14441.6i			1.27256i
$506$	18377.4			1.61458
$507$	−15786.5			−1.38285
$508$	4751.19			0.414960
$509$		−	3931.99i		−	0.342402i	−0.985236	−	0.171201i	$-0.945235\pi$
$509$		−	3931.99i		−	0.342402i	0.985236	−	0.171201i	$-0.0547647\pi$
$510$	−37105.4			−3.22168
$511$	9251.61			0.800914
$512$		−	3635.49i		−	0.313803i
$513$	−1576.10			−0.135646
$514$			18618.3i			1.59770i
$515$	17258.6			1.47670
$516$		−	24079.6i		−	2.05435i
$517$	9241.30			0.786135
$518$	26986.6			2.28904
$519$	4827.36			0.408280
$520$	−100985.			−8.51636
$521$			6810.70i			0.572711i	0.958123	+	0.286355i	$0.0924438\pi$
$521$			6810.70i			0.572711i	−0.958123	+	0.286355i	$0.907556\pi$
$522$	−6247.51			−0.523843
$523$	−2557.35			−0.213814			−0.106907	−	0.994269i	$-0.534095\pi$
$523$	−2557.35			−0.213814			−0.106907	+	0.994269i	$0.534095\pi$
$524$		−	4377.57i		−	0.364953i
$525$		−	11465.2i		−	0.953108i
$526$		−	23975.9i		−	1.98745i
$527$	−27706.2			−2.29014
$528$	−26938.3			−2.22034
$529$	5636.62			0.463271
$530$		−	6229.13i		−	0.510521i
$531$		−	6796.99i		−	0.555488i
$532$			31880.2i			2.59809i
$533$		−	6884.49i		−	0.559476i
$534$			14025.4i			1.13659i
$535$	−22959.2			−1.85535
$536$			54454.2i			4.38818i
$537$		−	9977.07i		−	0.801755i
$538$	−13753.8			−1.10217
$539$	−13726.8			−1.09695
$540$		−	9417.94i		−	0.750525i
$541$			21014.8i			1.67005i	0.550215	+	0.835023i	$0.314546\pi$
$541$			21014.8i			1.67005i	−0.550215	+	0.835023i	$0.685454\pi$
$542$	16065.2			1.27317
$543$		−	7353.56i		−	0.581163i
$544$			80964.6i			6.38112i
$545$			29112.2i			2.28813i
$546$	−36155.9			−2.83394
$547$	19759.7			1.54454			0.772271	−	0.635293i	$-0.219121\pi$
$547$	19759.7			1.54454			0.772271	+	0.635293i	$0.219121\pi$
$548$		−	6026.10i		−	0.469749i
$549$		−	4762.69i		−	0.370249i
$550$			33608.8i			2.60560i
$551$		−	7509.20i		−	0.580586i
$552$	17163.0			1.32338
$553$	10225.3			0.786300
$554$		−	15585.2i		−	1.19522i
$555$		−	9582.27i		−	0.732873i
$556$		−	9088.56i		−	0.693239i
$557$	15982.4			1.21579			0.607897	−	0.794016i	$-0.292013\pi$
$557$	15982.4			1.21579			0.607897	+	0.794016i	$0.292013\pi$
$558$		−	9696.11i		−	0.735608i
$559$			32824.2i			2.48357i
$560$	−91003.6			−6.86715
$561$	17545.2			1.32043
$562$			15083.0i			1.13210i
$563$		−	3861.48i		−	0.289062i	−0.989500	−	0.144531i	$-0.953833\pi$
$563$		−	3861.48i		−	0.289062i	0.989500	−	0.144531i	$-0.0461674\pi$
$564$	13893.4			1.03727
$565$		−	2163.63i		−	0.161106i
$566$		−	367.237i		−	0.0272723i
$567$		−	2094.63i		−	0.155143i
$568$		−	22109.1i		−	1.63323i
$569$		−	14475.6i		−	1.06652i	−0.845952	−	0.533259i	$-0.820967\pi$
$569$		−	14475.6i		−	1.06652i	0.845952	−	0.533259i	$-0.179033\pi$
$570$	15607.9			1.14692
$571$	16276.0			1.19287			0.596436	−	0.802661i	$-0.296583\pi$
$571$	16276.0			1.19287			0.596436	+	0.802661i	$0.296583\pi$
$572$	76868.6			5.61895
$573$		−	4217.39i		−	0.307477i
$574$		−	11123.5i		−	0.808857i
$575$		−	11942.8i		−	0.866173i
$576$	−12993.4			−0.939914
$577$	−19274.2			−1.39063			−0.695317	−	0.718703i	$-0.744736\pi$
$577$	−19274.2			−1.39063			−0.695317	+	0.718703i	$0.744736\pi$
$578$		−	77412.5i		−	5.57083i
$579$	−3655.55			−0.262383
$580$	44871.0			3.21236
$581$	10094.0			0.720773
$582$	11629.1			0.828250
$583$			2945.43i			0.209241i
$584$	−25327.8			−1.79464
$585$			12838.1i			0.907332i
$586$	−16954.3			−1.19518
$587$		−	7296.12i		−	0.513021i	−0.966541	−	0.256510i	$-0.917427\pi$
$587$		−	7296.12i		−	0.513021i	0.966541	−	0.256510i	$-0.0825728\pi$
$588$	−20636.9			−1.44737
$589$	11654.2			0.815288
$590$			67309.6i			4.69677i
$591$	5010.18			0.348716
$592$	−41205.9			−2.86073
$593$	−7530.87			−0.521511			−0.260755	−	0.965405i	$-0.583972\pi$
$593$	−7530.87			−0.521511			−0.260755	+	0.965405i	$0.583972\pi$
$594$			6140.15i			0.424130i
$595$	59271.7			4.08387
$596$		−	16882.0i		−	1.16026i
$597$	232.264			0.0159228
$598$	−37662.1			−2.57545
$599$			16792.7i			1.14546i	0.819744	+	0.572731i	$0.194116\pi$
$599$			16792.7i			1.14546i	−0.819744	+	0.572731i	$0.805884\pi$
$600$			31387.8i			2.13567i
$601$	−18842.4			−1.27887			−0.639433	−	0.768847i	$-0.720831\pi$
$601$	−18842.4			−1.27887			−0.639433	+	0.768847i	$0.720831\pi$
$602$			53034.9i			3.59060i
$603$	6922.64			0.467515
$604$			18441.6i			1.24235i
$605$		−	7350.13i		−	0.493926i
$606$		−	14155.2i		−	0.948871i
$607$		−	131.812i		−	0.00881395i	−0.999990	−	0.00440698i	$-0.998597\pi$
$607$		−	131.812i		−	0.00881395i	0.999990	−	0.00440698i	$-0.00140279\pi$
$608$		−	34056.7i		−	2.27168i
$609$	9979.70			0.664036
$610$			47164.2i			3.13053i
$611$	−18938.9			−1.25398
$612$	26377.6			1.74224
$613$		−	26388.6i		−	1.73871i	−0.494192	−	0.869353i	$-0.664536\pi$
$613$		−	26388.6i		−	1.73871i	0.494192	−	0.869353i	$-0.335464\pi$
$614$	22133.5			1.45478
$615$	−3949.67			−0.258969
$616$	77152.2			5.04635
$617$	−25552.9			−1.66729			−0.833646	−	0.552299i	$-0.813751\pi$
$617$	−25552.9			−1.66729			−0.833646	+	0.552299i	$0.813751\pi$
$618$	−16916.2			−1.10109
$619$	21068.1			1.36801			0.684006	−	0.729476i	$-0.260236\pi$
$619$	21068.1			1.36801			0.684006	+	0.729476i	$0.260236\pi$
$620$			69639.6i			4.51096i
$621$		−	2181.89i		−	0.140992i
$622$		−	22550.8i		−	1.45370i
$623$		−	22403.9i		−	1.44076i
$624$	55206.5			3.54172
$625$	−12257.3			−0.784467
$626$		−	21043.0i		−	1.34353i
$627$	−7380.15			−0.470072
$628$	−40828.9	−	7685.92i	−2.59435	−	0.488378i
$629$	26837.9			1.70127
$630$			20742.8i			1.31177i
$631$	16027.1			1.01114			0.505569	−	0.862786i	$-0.331282\pi$
$631$	16027.1			1.01114			0.505569	+	0.862786i	$0.331282\pi$
$632$	−27993.4			−1.76190
$633$		−	2709.35i		−	0.170122i
$634$			26580.1i			1.66503i
$635$			3715.65i			0.232206i
$636$			4428.18i			0.276083i
$637$	28131.3			1.74977
$638$	−29254.2			−1.81534
$639$	−2810.68			−0.174004
$640$	51583.9			3.18599
$641$	−14186.9			−0.874180			−0.437090	−	0.899418i	$-0.643991\pi$
$641$	−14186.9			−0.874180			−0.437090	+	0.899418i	$0.643991\pi$
$642$	22503.8			1.38342
$643$		−	4399.68i		−	0.269839i	−0.990857	−	0.134919i	$-0.956922\pi$
$643$		−	4399.68i		−	0.269839i	0.990857	−	0.134919i	$-0.0430776\pi$
$644$	−44133.8			−2.70049
$645$	18831.4			1.14959
$646$			43714.3i			2.66241i
$647$	−20343.7			−1.23616			−0.618079	−	0.786116i	$-0.712089\pi$
$647$	−20343.7			−1.23616			−0.618079	+	0.786116i	$0.712089\pi$
$648$			5734.38i			0.347636i
$649$		−	31827.2i		−	1.92500i
$650$		−	68876.8i		−	4.15626i
$651$			15488.4i			0.932473i
$652$		−	81226.4i		−	4.87894i
$653$	17220.1			1.03197			0.515985	−	0.856598i	$-0.327426\pi$
$653$	17220.1			1.03197			0.515985	+	0.856598i	$0.327426\pi$
$654$		−	28534.8i		−	1.70612i
$655$	3423.46			0.204223
$656$			16984.4i			1.01087i
$657$			3219.86i			0.191201i
$658$	−30600.0			−1.81294
$659$	−13853.6			−0.818907			−0.409454	−	0.912331i	$-0.634281\pi$
$659$	−13853.6			−0.818907			−0.409454	+	0.912331i	$0.634281\pi$
$660$		−	44099.9i		−	2.60089i
$661$	11153.4			0.656306			0.328153	−	0.944625i	$-0.393574\pi$
$661$	11153.4			0.656306			0.328153	+	0.944625i	$0.393574\pi$
$662$		−	14697.2i		−	0.862877i
$663$	−35956.7			−2.10625
$664$	−27633.9			−1.61507
$665$	−24931.8			−1.45386
$666$			9392.22i			0.546458i
$667$	10395.4			0.603468
$668$	34253.2			1.98397
$669$			14475.2i			0.836536i
$670$	−68553.9			−3.95294
$671$		−	22301.5i		−	1.28307i
$672$	45261.2			2.59819
$673$		−	20235.3i		−	1.15901i	−0.814969	−	0.579505i	$-0.803246\pi$
$673$		−	20235.3i		−	1.15901i	0.814969	−	0.579505i	$-0.196754\pi$
$674$	50491.9			2.88557
$675$	3990.26			0.227534
$676$	−111133.			−6.32302
$677$	16644.0			0.944875			0.472437	−	0.881364i	$-0.343374\pi$
$677$	16644.0			0.944875			0.472437	+	0.881364i	$0.343374\pi$
$678$			2120.72i			0.120126i
$679$	−18576.2			−1.04991
$680$	−162266.			−9.15090
$681$			2218.03i			0.124809i
$682$		−	45402.5i		−	2.54919i
$683$		−	23187.6i		−	1.29905i	−0.760340	−	0.649525i	$-0.774968\pi$
$683$		−	23187.6i		−	1.29905i	0.760340	−	0.649525i	$-0.225032\pi$
$684$	−11095.4			−0.620237
$685$	4712.69			0.262865
$686$	−2411.34			−0.134206
$687$			6304.81i			0.350136i
$688$		−	80979.1i		−	4.48735i
$689$		−	6036.28i		−	0.333765i
$690$			21606.9i			1.19212i
$691$			367.018i			0.0202055i	0.999949	+	0.0101028i	$0.00321586\pi$
$691$			367.018i			0.0202055i	−0.999949	+	0.0101028i	$0.996784\pi$
$692$	33983.5			1.86685
$693$		−	9808.19i		−	0.537637i
$694$			69659.8i			3.81016i
$695$	7107.67			0.387927
$696$	−27321.0			−1.48793
$697$		−	11062.2i		−	0.601161i
$698$			22094.2i			1.19810i
$699$	10219.0			0.552960
$700$		−	80712.3i		−	4.35805i
$701$		−	35552.6i		−	1.91555i	−0.287516	−	0.957776i	$-0.592829\pi$
$701$		−	35552.6i		−	1.91555i	0.287516	−	0.957776i	$-0.407171\pi$
$702$		−	12583.4i		−	0.676540i
$703$	−11289.0			−0.605650
$704$	−60842.0			−3.25720
$705$			10865.3i			0.580442i
$706$		−	2338.73i		−	0.124673i
$707$			22611.3i			1.20281i
$708$		−	47849.3i		−	2.53995i
$709$	−13540.2			−0.717225			−0.358612	−	0.933487i	$-0.616750\pi$
$709$	−13540.2			−0.717225			−0.358612	+	0.933487i	$0.616750\pi$
$710$	27833.7			1.47124
$711$			3558.74i			0.187712i
$712$			61334.4i			3.22837i
$713$			16133.7i			0.847421i
$714$	−58096.1			−3.04509
$715$			60114.8i			3.14429i
$716$		−	70236.3i		−	3.66600i
$717$	−9635.44			−0.501872
$718$	−4811.40			−0.250083
$719$		−	20980.5i		−	1.08823i	−0.839010	−	0.544116i	$-0.816865\pi$
$719$		−	20980.5i		−	1.08823i	0.839010	−	0.544116i	$-0.183135\pi$
$720$		−	31672.2i		−	1.63938i
$721$	27021.8			1.39576
$722$			18624.9i			0.960040i
$723$			10041.5i			0.516525i
$724$		−	51767.4i		−	2.65735i
$725$			19011.3i			0.973878i
$726$			7204.34i			0.368290i
$727$	−28427.1			−1.45021			−0.725104	−	0.688639i	$-0.758208\pi$
$727$	−28427.1			−1.45021			−0.725104	+	0.688639i	$0.758208\pi$
$728$	−158114.			−8.04956
$729$	729.000			0.0370370
$730$		−	31885.8i		−	1.61664i
$731$			52742.6i			2.66861i
$732$		−	33528.2i		−	1.69295i
$733$	9031.16			0.455080			0.227540	−	0.973769i	$-0.426932\pi$
$733$	9031.16			0.455080			0.227540	+	0.973769i	$0.426932\pi$
$734$	6602.18			0.332004
$735$		−	16139.0i		−	0.809929i
$736$	47146.7			2.36121
$737$	32415.6			1.62014
$738$	3871.33			0.193097
$739$	13765.7			0.685224			0.342612	−	0.939477i	$-0.388688\pi$
$739$	13765.7			0.685224			0.342612	+	0.939477i	$0.388688\pi$
$740$		−	67457.0i		−	3.35104i
$741$	15124.7			0.749823
$742$		−	9752.97i		−	0.482538i
$743$	−36737.6			−1.81396			−0.906980	−	0.421173i	$-0.861619\pi$
$743$	−36737.6			−1.81396			−0.906980	+	0.421173i	$0.861619\pi$
$744$		−	42402.1i		−	2.08943i
$745$	13202.5			0.649266
$746$	36424.9			1.78768
$747$			3513.04i			0.172069i
$748$	123514.			6.03761
$749$	−35947.4			−1.75366
$750$	−6092.84			−0.296639
$751$			31403.0i			1.52585i	0.646488	+	0.762924i	$0.276237\pi$
$751$			31403.0i			1.52585i	−0.646488	+	0.762924i	$0.723763\pi$
$752$	46723.3			2.26572
$753$			1398.33i			0.0676733i
$754$	59952.8			2.89569
$755$	−14422.2			−0.695203
$756$		−	14745.7i		−	0.709387i
$757$			4918.53i			0.236152i	0.993005	+	0.118076i	$0.0376727\pi$
$757$			4918.53i			0.236152i	−0.993005	+	0.118076i	$0.962327\pi$
$758$	20094.6			0.962889
$759$		−	10216.8i		−	0.488598i
$760$	68254.9			3.25772
$761$		−	13455.7i		−	0.640958i	−0.947256	−	0.320479i	$-0.896156\pi$
$761$		−	13455.7i		−	0.640958i	0.947256	−	0.320479i	$-0.103844\pi$
$762$		−	3641.95i		−	0.173142i
$763$			45581.2i			2.16271i
$764$		−	29689.5i		−	1.40593i
$765$			20628.5i			0.974935i
$766$	−66091.9			−3.11749
$767$			65225.8i			3.07062i
$768$	−15911.8			−0.747613
$769$	−18730.8			−0.878349			−0.439174	−	0.898402i	$-0.644729\pi$
$769$	−18730.8			−0.878349			−0.439174	+	0.898402i	$0.644729\pi$
$770$			97129.1i			4.54583i
$771$	10350.7			0.483493
$772$	−25734.2			−1.19974
$773$	−8299.90			−0.386192			−0.193096	−	0.981180i	$-0.561853\pi$
$773$	−8299.90			−0.386192			−0.193096	+	0.981180i	$0.561853\pi$
$774$	−18457.9			−0.857177
$775$	−29505.4			−1.36757
$776$	50855.2			2.35257
$777$		−	15003.0i		−	0.692703i
$778$			32689.5i			1.50640i
$779$			4653.15i			0.214013i
$780$			90377.1i			4.14874i
$781$	−13161.1			−0.602999
$782$	−60516.4			−2.76734
$783$			3473.26i			0.158524i
$784$	−69401.5			−3.16151
$785$	6010.74	−	31930.1i	0.273290	−	1.45176i
$786$	−3355.56			−0.152276
$787$		−	6033.50i		−	0.273279i	−0.990621	−	0.136640i	$-0.956370\pi$
$787$		−	6033.50i		−	0.273279i	0.990621	−	0.136640i	$-0.0436303\pi$
$788$	35270.5			1.59449
$789$	−13329.2			−0.601437
$790$		−	35241.7i		−	1.58714i
$791$		−	3387.61i		−	0.152275i
$792$			26851.5i			1.20471i
$793$			45704.1i			2.04666i
$794$	−56880.3			−2.54232
$795$	−3463.04			−0.154492
$796$	1635.08			0.0728065
$797$	−1353.35			−0.0601483			−0.0300742	−	0.999548i	$-0.509574\pi$
$797$	−1353.35			−0.0601483			−0.0300742	+	0.999548i	$0.509574\pi$
$798$	24437.3			1.08405
$799$	−30431.4			−1.34742
$800$			86222.3i			3.81052i
$801$	7797.31			0.343950
$802$	5862.28			0.258110
$803$			15077.2i			0.662592i
$804$	48733.8			2.13770
$805$		−	34514.6i		−	1.51116i
$806$			93046.5i			4.06628i
$807$			7646.33i			0.333536i
$808$		−	61902.2i		−	2.69519i
$809$			19280.9i			0.837924i	0.908004	+	0.418962i	$0.137606\pi$
$809$			19280.9i			0.837924i	−0.908004	+	0.418962i	$0.862394\pi$
$810$	−7219.18			−0.313156
$811$		−	21618.2i		−	0.936026i	−0.883722	−	0.468013i	$-0.844970\pi$
$811$		−	21618.2i		−	0.936026i	0.883722	−	0.468013i	$-0.155030\pi$
$812$	70254.8			3.03628
$813$		−	8931.32i		−	0.385283i
$814$			43979.5i			1.89371i
$815$	63522.8			2.73019
$816$	88707.1			3.80560
$817$		−	22185.5i		−	0.950026i
$818$	12572.8			0.537407
$819$			20100.6i			0.857598i
$820$	−27804.7			−1.18413
$821$	3290.36			0.139871			0.0699356	−	0.997552i	$-0.477721\pi$
$821$	3290.36			0.139871			0.0699356	+	0.997552i	$0.477721\pi$
$822$	−4619.22			−0.196002
$823$		−	15314.1i		−	0.648622i	−0.945950	−	0.324311i	$-0.894867\pi$
$823$		−	15314.1i		−	0.648622i	0.945950	−	0.324311i	$-0.105133\pi$
$824$	−73976.5			−3.12754
$825$	18684.6			0.788501
$826$			105387.i			4.43932i
$827$	22934.1			0.964323			0.482162	−	0.876082i	$-0.339852\pi$
$827$	22934.1			0.964323			0.482162	+	0.876082i	$0.339852\pi$
$828$		−	15360.0i		−	0.644683i
$829$	26543.0			1.11204			0.556018	−	0.831171i	$-0.312329\pi$
$829$	26543.0			1.11204			0.556018	+	0.831171i	$0.312329\pi$
$830$		−	34789.1i		−	1.45488i
$831$	−8664.47			−0.361693
$832$	124688.			5.19565
$833$	45202.0			1.88014
$834$	−6966.70			−0.289253
$835$			26787.5i			1.11021i
$836$	−51954.6			−2.14939
$837$	−5390.49			−0.222608
$838$			72611.7i			2.99323i
$839$			9186.86i			0.378028i	0.981974	+	0.189014i	$0.0605291\pi$
$839$			9186.86i			0.378028i	−0.981974	+	0.189014i	$0.939471\pi$
$840$			90710.4i			3.72596i
$841$	7840.90			0.321493
$842$	2690.35			0.110114
$843$	8385.29			0.342592
$844$		−	19073.2i		−	0.777875i
$845$		−	86911.4i		−	3.53827i
$846$		−	10649.8i		−	0.432799i
$847$		−	11508.1i		−	0.466852i
$848$			14891.8i			0.603052i
$849$	−204.163			−0.00825308
$850$		−	110673.i		−	4.46594i
$851$		−	15628.0i		−	0.629520i
$852$	−19786.5			−0.795628
$853$	−3728.25			−0.149652			−0.0748259	−	0.997197i	$-0.523840\pi$
$853$	−3728.25			−0.149652			−0.0748259	+	0.997197i	$0.523840\pi$
$854$			73845.2i			2.95894i
$855$		−	8677.10i		−	0.347077i
$856$	98411.7			3.92949
$857$			36740.6i			1.46445i	0.681062	+	0.732226i	$0.261519\pi$
$857$			36740.6i			1.46445i	−0.681062	+	0.732226i	$0.738481\pi$
$858$		−	58922.5i		−	2.34450i
$859$		−	12836.7i		−	0.509876i	−0.966957	−	0.254938i	$-0.917945\pi$
$859$		−	12836.7i		−	0.509876i	0.966957	−	0.254938i	$-0.0820551\pi$
$860$	132569.			5.25646
$861$	−6184.01			−0.244774
$862$			38923.0i			1.53796i
$863$		−	17268.5i		−	0.681143i	−0.940219	−	0.340571i	$-0.889380\pi$
$863$		−	17268.5i		−	0.681143i	0.940219	−	0.340571i	$-0.110620\pi$
$864$			15752.4i			0.620262i
$865$			26576.7i			1.04466i
$866$	21785.1			0.854835
$867$	−43037.0			−1.68583
$868$			109035.i			4.26370i
$869$			16664.0i			0.650502i
$870$		−	34395.2i		−	1.34035i
$871$	−66431.6			−2.58433
$872$		−	124786.i		−	4.84608i
$873$		−	6465.12i		−	0.250643i
$874$	25455.4			0.985173
$875$	9732.62			0.376026
$876$			22667.1i			0.874259i
$877$		−	42304.7i		−	1.62888i	−0.580247	−	0.814441i	$-0.697044\pi$
$877$		−	42304.7i		−	1.62888i	0.580247	−	0.814441i	$-0.302956\pi$
$878$	−4867.92			−0.187112
$879$			9425.66i			0.361683i
$880$		−	148307.i		−	5.68116i
$881$			36672.6i			1.40242i	0.712954	+	0.701210i	$0.247357\pi$
$881$			36672.6i			1.40242i	−0.712954	+	0.701210i	$0.752643\pi$
$882$			15818.9i			0.603913i
$883$		−	12260.8i		−	0.467279i	−0.972323	−	0.233640i	$-0.924936\pi$
$883$		−	12260.8i		−	0.467279i	0.972323	−	0.233640i	$-0.0750636\pi$
$884$	−253127.			−9.63074
$885$	37420.3			1.42132
$886$	43756.4			1.65917
$887$			7945.77i			0.300781i	0.988627	+	0.150391i	$0.0480531\pi$
$887$			7945.77i			0.300781i	−0.988627	+	0.150391i	$0.951947\pi$
$888$			41073.1i			1.55217i
$889$			5817.61i			0.219478i
$890$	−77215.6			−2.90817
$891$	3413.57			0.128349
$892$			101902.i			3.82503i
$893$	12800.5			0.479680
$894$	−12940.7			−0.484117
$895$	54928.0			2.05144
$896$	80765.2			3.01136
$897$			20938.0i			0.779376i
$898$	84008.6			3.12183
$899$		−	25682.6i		−	0.952794i
$900$	28090.5			1.04039
$901$		−	9699.23i		−	0.358633i
$902$	18127.7			0.669163
$903$	29484.4			1.08658
$904$			9274.13i			0.341209i
$905$	40484.5			1.48702
$906$	14136.2			0.518369
$907$	17146.1			0.627705			0.313852	−	0.949472i	$-0.398380\pi$
$907$	17146.1			0.627705			0.313852	+	0.949472i	$0.398380\pi$
$908$			15614.4i			0.570686i
$909$	−7869.49			−0.287145
$910$		−	199054.i		−	7.25117i
$911$	−1380.32			−0.0501999			−0.0250999	−	0.999685i	$-0.507990\pi$
$911$	−1380.32			−0.0501999			−0.0250999	+	0.999685i	$0.507990\pi$
$912$	−37313.4			−1.35479
$913$			16450.0i			0.596292i
$914$			62540.1i			2.26328i
$915$	26220.6			0.947352
$916$			44384.4i			1.60098i
$917$	5360.14			0.193029
$918$		−	20219.4i		−	0.726948i
$919$		−	46778.7i		−	1.67909i	−0.543287	−	0.839547i	$-0.682821\pi$
$919$		−	46778.7i		−	1.67909i	0.543287	−	0.839547i	$-0.317179\pi$
$920$			94489.4i			3.38611i
$921$		−	12305.0i		−	0.440242i
$922$			12976.4i			0.463510i
$923$	26972.0			0.961858
$924$		−	69047.4i		−	2.45833i
$925$	28580.7			1.01592
$926$	104961.			3.72487
$927$			9404.48i			0.333208i
$928$	−75051.0			−2.65482
$929$	43258.1			1.52772			0.763860	−	0.645382i	$-0.223302\pi$
$929$	43258.1			1.52772			0.763860	+	0.645382i	$0.223302\pi$
$930$	53381.2			1.88219
$931$	−19013.6			−0.669329
$932$	71939.6			2.52839
$933$	−12537.0			−0.439916
$934$			14523.0i			0.508787i
$935$			96593.9i			3.37856i
$936$		−	55028.7i		−	1.92166i
$937$		−	6525.59i		−	0.227515i	−0.993509	−	0.113758i	$-0.963711\pi$
$937$		−	6525.59i		−	0.227515i	0.993509	−	0.113758i	$-0.0362887\pi$
$938$	−107335.			−3.73627
$939$	−11698.7			−0.406575
$940$			76489.3i			2.65405i
$941$	−52098.9			−1.80486			−0.902431	−	0.430833i	$-0.858220\pi$
$941$	−52098.9			−1.80486			−0.902431	+	0.430833i	$0.858220\pi$
$942$	−5891.52	+	31296.8i	−0.203775	+	1.08249i
$943$	−6441.63			−0.222448
$944$		−	160916.i		−	5.54805i
$945$	11531.8			0.396963
$946$	−86429.8			−2.97048
$947$		−	15186.4i		−	0.521111i	−0.965459	−	0.260556i	$-0.916094\pi$
$947$		−	15186.4i		−	0.521111i	0.965459	−	0.260556i	$-0.0839057\pi$
$948$			25052.7i			0.858307i
$949$		−	30898.7i		−	1.05692i
$950$			46553.0i			1.58987i
$951$	14777.0			0.503868
$952$	−254060.			−8.64931
$953$	−40030.6			−1.36067			−0.680336	−	0.732901i	$-0.738166\pi$
$953$	−40030.6			−1.36067			−0.680336	+	0.732901i	$0.738166\pi$
$954$	3394.36			0.115195
$955$	23218.6			0.786738
$956$	−67831.3			−2.29479
$957$			16263.7i			0.549353i
$958$	−72735.0			−2.45299
$959$	7378.68			0.248457
$960$		−	71534.1i		−	2.40495i
$961$	10068.2			0.337962
$962$		−	90130.3i		−	3.02070i
$963$		−	12510.9i		−	0.418647i
$964$			70689.9i			2.36179i
$965$		−	20125.4i		−	0.671356i
$966$			33830.1i			1.12678i
$967$	−44253.5			−1.47166			−0.735832	−	0.677165i	$-0.763208\pi$
$967$	−44253.5			−1.47166			−0.735832	+	0.677165i	$0.763208\pi$
$968$			31505.4i			1.04610i
$969$	24302.7			0.805691
$970$			64023.1i			2.11923i
$971$		−	17941.4i		−	0.592962i	−0.955039	−	0.296481i	$-0.904187\pi$
$971$		−	17941.4i		−	0.592962i	0.955039	−	0.296481i	$-0.0958131\pi$
$972$	5131.99			0.169351
$973$	11128.5			0.366664
$974$		−	92662.6i		−	3.04836i
$975$	−38291.6			−1.25776
$976$		−	112754.i		−	3.69793i
$977$	−24580.0			−0.804896			−0.402448	−	0.915443i	$-0.631841\pi$
$977$	−24580.0			−0.804896			−0.402448	+	0.915443i	$0.631841\pi$
$978$	−62262.9			−2.03573
$979$	36511.2			1.19193
$980$		−	113615.i		−	3.70337i
$981$	−15863.8			−0.516300
$982$	−36516.0			−1.18663
$983$		−	6697.36i		−	0.217307i	−0.994080	−	0.108653i	$-0.965346\pi$
$983$		−	6697.36i		−	0.217307i	0.994080	−	0.108653i	$-0.0346539\pi$
$984$	16929.7			0.548476
$985$			27583.2i			0.892257i
$986$	96333.6			3.11145
$987$			17011.9i			0.548626i
$988$	106474.			3.42854
$989$	30712.7			0.987469
$990$	−33804.1			−1.08522
$991$	8472.94			0.271596			0.135798	−	0.990737i	$-0.456640\pi$
$991$	8472.94			0.271596			0.135798	+	0.990737i	$0.456640\pi$
$992$		−	116479.i		−	3.72803i
$993$	−8170.83			−0.261121
$994$	43579.4			1.39060
$995$			1278.71i			0.0407415i
$996$			24731.0i			0.786779i
$997$			1344.66i			0.0427139i	0.999772	+	0.0213570i	$0.00679865\pi$
$997$			1344.66i			0.0427139i	−0.999772	+	0.0213570i	$0.993201\pi$
$998$	62292.1			1.97577
$999$	5221.54			0.165368

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	471.4.b.a.313.1	✓	40
157.156	even	2	inner	471.4.b.a.313.40	yes	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
471.4.b.a.313.1	✓	40	1.1	even	1	trivial
471.4.b.a.313.40	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)

\(f(q)\)	\(=\)	\(q-5.39623i q^{2} -3.00000 q^{3} -21.1193 q^{4} -16.5163i q^{5} +16.1887i q^{6} -25.8596i q^{7} +70.7949i q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q-5.39623i q^{2} -3.00000 q^{3} -21.1193 q^{4} -16.5163i q^{5} +16.1887i q^{6} -25.8596i q^{7} +70.7949i q^{8} +9.00000 q^{9} -89.1257 q^{10} +42.1429 q^{11} +63.3579 q^{12} -86.3665 q^{13} -139.544 q^{14} +49.5488i q^{15} +213.071 q^{16} -138.776 q^{17} -48.5661i q^{18} +58.3740 q^{19} +348.812i q^{20} +77.5789i q^{21} -227.413i q^{22} +80.8107i q^{23} -212.385i q^{24} -147.787 q^{25} +466.053i q^{26} -27.0000 q^{27} +546.137i q^{28} -128.639i q^{29} +267.377 q^{30} +199.648 q^{31} -583.421i q^{32} -126.429 q^{33} +748.865i q^{34} -427.105 q^{35} -190.074 q^{36} -193.390 q^{37} -315.000i q^{38} +259.099 q^{39} +1169.27 q^{40} +79.7126i q^{41} +418.633 q^{42} -380.057i q^{43} -890.029 q^{44} -148.647i q^{45} +436.074 q^{46} +219.285 q^{47} -639.213 q^{48} -325.720 q^{49} +797.495i q^{50} +416.327 q^{51} +1824.00 q^{52} +69.8915i q^{53} +145.698i q^{54} -696.044i q^{55} +1830.73 q^{56} -175.122 q^{57} -694.168 q^{58} -755.221i q^{59} -1046.44i q^{60} -529.188i q^{61} -1077.35i q^{62} -232.737i q^{63} -1443.71 q^{64} +1426.45i q^{65} +682.238i q^{66} +769.183 q^{67} +2930.85 q^{68} -242.432i q^{69} +2304.76i q^{70} -312.298 q^{71} +637.154i q^{72} +357.763i q^{73} +1043.58i q^{74} +443.362 q^{75} -1232.82 q^{76} -1089.80i q^{77} -1398.16i q^{78} +395.416i q^{79} -3519.14i q^{80} +81.0000 q^{81} +430.148 q^{82} +390.338i q^{83} -1638.41i q^{84} +2292.06i q^{85} -2050.88 q^{86} +385.918i q^{87} +2983.50i q^{88} +866.368 q^{89} -802.131 q^{90} +2233.40i q^{91} -1706.67i q^{92} -598.943 q^{93} -1183.31i q^{94} -964.122i q^{95} +1750.26i q^{96} -718.346i q^{97} +1757.66i q^{98} +379.286 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 120 q^{3} - 164 q^{4} + 360 q^{9}+O(q^{10}) \) 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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