LMFDB - Embedded newform 471.4.b.a.313.3

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			313.3
Character	$\chi$	$=$	471.313
Dual form			471.4.b.a.313.38

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.19343i q^{2} -3.00000 q^{3} -18.9718 q^{4} -15.3454i q^{5} +15.5803i q^{6} +23.0710i q^{7} +56.9811i q^{8} +9.00000 q^{9} +O(q^{10})$	\(q-5.19343i q^{2} -3.00000 q^{3} -18.9718 q^{4} -15.3454i q^{5} +15.5803i q^{6} +23.0710i q^{7} +56.9811i q^{8} +9.00000 q^{9} -79.6953 q^{10} +10.8653 q^{11} +56.9153 q^{12} +71.2171 q^{13} +119.818 q^{14} +46.0362i q^{15} +144.153 q^{16} +78.7769 q^{17} -46.7409i q^{18} -14.6390 q^{19} +291.129i q^{20} -69.2130i q^{21} -56.4280i q^{22} +30.0448i q^{23} -170.943i q^{24} -110.481 q^{25} -369.861i q^{26} -27.0000 q^{27} -437.697i q^{28} +158.532i q^{29} +239.086 q^{30} +76.7777 q^{31} -292.803i q^{32} -32.5958 q^{33} -409.123i q^{34} +354.033 q^{35} -170.746 q^{36} +97.2078 q^{37} +76.0265i q^{38} -213.651 q^{39} +874.397 q^{40} +411.618i q^{41} -359.453 q^{42} -144.070i q^{43} -206.133 q^{44} -138.108i q^{45} +156.036 q^{46} +378.542 q^{47} -432.460 q^{48} -189.271 q^{49} +573.775i q^{50} -236.331 q^{51} -1351.11 q^{52} +141.751i q^{53} +140.223i q^{54} -166.732i q^{55} -1314.61 q^{56} +43.9169 q^{57} +823.324 q^{58} +78.0430i q^{59} -873.387i q^{60} -434.066i q^{61} -398.740i q^{62} +207.639i q^{63} -367.424 q^{64} -1092.85i q^{65} +169.284i q^{66} -889.877 q^{67} -1494.54 q^{68} -90.1344i q^{69} -1838.65i q^{70} -408.257 q^{71} +512.830i q^{72} -1051.32i q^{73} -504.842i q^{74} +331.443 q^{75} +277.727 q^{76} +250.672i q^{77} +1109.58i q^{78} +1165.41i q^{79} -2212.09i q^{80} +81.0000 q^{81} +2137.71 q^{82} +1400.98i q^{83} +1313.09i q^{84} -1208.86i q^{85} -748.217 q^{86} -475.595i q^{87} +619.114i q^{88} +680.367 q^{89} -717.257 q^{90} +1643.05i q^{91} -570.003i q^{92} -230.333 q^{93} -1965.93i q^{94} +224.641i q^{95} +878.408i q^{96} -1084.25i q^{97} +982.966i q^{98} +97.7873 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$		−	5.19343i		−	1.83616i	−0.396400	−	0.918078i	$-0.629741\pi$
$2$		−	5.19343i		−	1.83616i	0.396400	−	0.918078i	$-0.370259\pi$
$3$	−3.00000			−0.577350
$3$	−3.00000			−0.577350
$4$	−18.9718			−2.37147
$5$		−	15.3454i		−	1.37253i	−0.727350	−	0.686267i	$-0.759248\pi$
$5$		−	15.3454i		−	1.37253i	0.727350	−	0.686267i	$-0.240752\pi$
$6$			15.5803i			1.06011i
$7$			23.0710i			1.24572i	0.782335	+	0.622858i	$0.214029\pi$
$7$			23.0710i			1.24572i	−0.782335	+	0.622858i	$0.785971\pi$
$8$			56.9811i			2.51823i
$9$	9.00000			0.333333
$10$	−79.6953			−2.52019
$11$	10.8653			0.297818			0.148909	−	0.988851i	$-0.452424\pi$
$11$	10.8653			0.297818			0.148909	+	0.988851i	$0.452424\pi$
$12$	56.9153			1.36917
$13$	71.2171			1.51939			0.759695	−	0.650280i	$-0.225348\pi$
$13$	71.2171			1.51939			0.759695	+	0.650280i	$0.225348\pi$
$14$	119.818			2.28733
$15$			46.0362i			0.792432i
$16$	144.153			2.25240
$17$	78.7769			1.12389			0.561947	−	0.827173i	$-0.310052\pi$
$17$	78.7769			1.12389			0.561947	+	0.827173i	$0.310052\pi$
$18$		−	46.7409i		−	0.612052i
$19$	−14.6390			−0.176758			−0.0883792	−	0.996087i	$-0.528169\pi$
$19$	−14.6390			−0.176758			−0.0883792	+	0.996087i	$0.528169\pi$
$20$			291.129i			3.25492i
$21$		−	69.2130i		−	0.719215i
$22$		−	56.4280i		−	0.546840i
$23$			30.0448i			0.272381i	0.990683	+	0.136191i	$0.0434860\pi$
$23$			30.0448i			0.272381i	−0.990683	+	0.136191i	$0.956514\pi$
$24$		−	170.943i		−	1.45390i
$25$	−110.481			−0.883847
$26$		−	369.861i		−	2.78984i
$27$	−27.0000			−0.192450
$28$		−	437.697i		−	2.95418i
$29$			158.532i			1.01512i	0.861615	+	0.507562i	$0.169453\pi$
$29$			158.532i			1.01512i	−0.861615	+	0.507562i	$0.830547\pi$
$30$	239.086			1.45503
$31$	76.7777			0.444828			0.222414	−	0.974952i	$-0.428606\pi$
$31$	76.7777			0.444828			0.222414	+	0.974952i	$0.428606\pi$
$32$		−	292.803i		−	1.61752i
$33$	−32.5958			−0.171945
$34$		−	409.123i		−	2.06365i
$35$	354.033			1.70979
$36$	−170.746			−0.790490
$37$	97.2078			0.431915			0.215958	−	0.976403i	$-0.430713\pi$
$37$	97.2078			0.431915			0.215958	+	0.976403i	$0.430713\pi$
$38$			76.0265i			0.324556i
$39$	−213.651			−0.877220
$40$	874.397			3.45636
$41$			411.618i			1.56790i	0.620822	+	0.783951i	$0.286799\pi$
$41$			411.618i			1.56790i	−0.620822	+	0.783951i	$0.713201\pi$
$42$	−359.453			−1.32059
$43$		−	144.070i		−	0.510940i	−0.966817	−	0.255470i	$-0.917770\pi$
$43$		−	144.070i		−	0.510940i	0.966817	−	0.255470i	$-0.0822302\pi$
$44$	−206.133			−0.706266
$45$		−	138.108i		−	0.457511i
$46$	156.036			0.500135
$47$	378.542			1.17481			0.587404	−	0.809294i	$-0.300150\pi$
$47$	378.542			1.17481			0.587404	+	0.809294i	$0.300150\pi$
$48$	−432.460			−1.30042
$49$	−189.271			−0.551810
$50$			573.775i			1.62288i
$51$	−236.331			−0.648881
$52$	−1351.11			−3.60319
$53$			141.751i			0.367377i	0.982985	+	0.183688i	$0.0588037\pi$
$53$			141.751i			0.367377i	−0.982985	+	0.183688i	$0.941196\pi$
$54$			140.223i			0.353368i
$55$		−	166.732i		−	0.408765i
$56$	−1314.61			−3.13700
$57$	43.9169			0.102051
$58$	823.324			1.86393
$59$			78.0430i			0.172209i	0.996286	+	0.0861046i	$0.0274419\pi$
$59$			78.0430i			0.172209i	−0.996286	+	0.0861046i	$0.972558\pi$
$60$		−	873.387i		−	1.87923i
$61$		−	434.066i		−	0.911089i	−0.890213	−	0.455544i	$-0.849445\pi$
$61$		−	434.066i		−	0.911089i	0.890213	−	0.455544i	$-0.150555\pi$
$62$		−	398.740i		−	0.816774i
$63$			207.639i			0.415239i
$64$	−367.424			−0.717626
$65$		−	1092.85i		−	2.08541i
$66$			169.284i			0.315718i
$67$	−889.877			−1.62262			−0.811312	−	0.584613i	$-0.801246\pi$
$67$	−889.877			−1.62262			−0.811312	+	0.584613i	$0.801246\pi$
$68$	−1494.54			−2.66528
$69$		−	90.1344i		−	0.157260i
$70$		−	1838.65i		−	3.13944i
$71$	−408.257			−0.682412			−0.341206	−	0.939989i	$-0.610835\pi$
$71$	−408.257			−0.682412			−0.341206	+	0.939989i	$0.610835\pi$
$72$			512.830i			0.839411i
$73$		−	1051.32i		−	1.68558i	−0.538239	−	0.842792i	$-0.680910\pi$
$73$		−	1051.32i		−	1.68558i	0.538239	−	0.842792i	$-0.319090\pi$
$74$		−	504.842i		−	0.793064i
$75$	331.443			0.510289
$76$	277.727			0.419177
$77$			250.672i			0.370997i
$78$			1109.58i			1.61071i
$79$			1165.41i			1.65974i	0.557960	+	0.829868i	$0.311584\pi$
$79$			1165.41i			1.65974i	−0.557960	+	0.829868i	$0.688416\pi$
$80$		−	2212.09i		−	3.09149i
$81$	81.0000			0.111111
$82$	2137.71			2.87891
$83$			1400.98i			1.85274i	0.376615	+	0.926370i	$0.377088\pi$
$83$			1400.98i			1.85274i	−0.376615	+	0.926370i	$0.622912\pi$
$84$			1313.09i			1.70560i
$85$		−	1208.86i		−	1.54258i
$86$	−748.217			−0.938166
$87$		−	475.595i		−	0.586082i
$88$			619.114i			0.749975i
$89$	680.367			0.810323			0.405162	−	0.914245i	$-0.367215\pi$
$89$	680.367			0.810323			0.405162	+	0.914245i	$0.367215\pi$
$90$	−717.257			−0.840062
$91$			1643.05i			1.89273i
$92$		−	570.003i		−	0.645944i
$93$	−230.333			−0.256822
$94$		−	1965.93i		−	2.15713i
$95$			224.641i			0.242607i
$96$			878.408i			0.933877i
$97$		−	1084.25i		−	1.13494i	−0.823395	−	0.567469i	$-0.807923\pi$
$97$		−	1084.25i		−	1.13494i	0.823395	−	0.567469i	$-0.192077\pi$
$98$			982.966i			1.01321i
$99$	97.7873			0.0992727
$100$	2096.02			2.09602
$101$	1235.62			1.21731			0.608657	−	0.793433i	$-0.291709\pi$
$101$	1235.62			1.21731			0.608657	+	0.793433i	$0.291709\pi$
$102$			1227.37i			1.19145i
$103$		−	618.765i		−	0.591929i	−0.955199	−	0.295964i	$-0.904359\pi$
$103$		−	618.765i		−	0.591929i	0.955199	−	0.295964i	$-0.0956410\pi$
$104$			4058.03i			3.82618i
$105$	−1062.10			−0.987146
$106$	736.173			0.674561
$107$		−	2.17878i		−	0.00196851i	−1.00000		0.000984254i	$-0.999687\pi$
$107$		−	2.17878i		−	0.00196851i	1.00000		0.000984254i	$-0.000313298\pi$
$108$	512.237			0.456390
$109$	310.887			0.273189			0.136594	−	0.990627i	$-0.456384\pi$
$109$	310.887			0.273189			0.136594	+	0.990627i	$0.456384\pi$
$110$	−865.909			−0.750556
$111$	−291.623			−0.249366
$112$			3325.76i			2.80585i
$113$	1267.17			1.05491			0.527456	−	0.849582i	$-0.323146\pi$
$113$	1267.17			1.05491			0.527456	+	0.849582i	$0.323146\pi$
$114$		−	228.080i		−	0.187382i
$115$	461.049			0.373853
$116$		−	3007.62i		−	2.40734i
$117$	640.954			0.506463
$118$	405.311			0.316203
$119$			1817.46i			1.40005i
$120$	−2623.19			−1.99553
$121$	−1212.95			−0.911304
$122$	−2254.29			−1.67290
$123$		−	1234.86i		−	0.905229i
$124$	−1456.61			−1.05490
$125$		−	222.801i		−	0.159424i
$126$	1078.36			0.762443
$127$	1203.75			0.841067			0.420534	−	0.907277i	$-0.361843\pi$
$127$	1203.75			0.841067			0.420534	+	0.907277i	$0.361843\pi$
$128$		−	434.229i		−	0.299850i
$129$			432.209i			0.294991i
$130$	−5675.66			−3.82914
$131$		−	1363.61i		−	0.909460i	−0.890629	−	0.454730i	$-0.849736\pi$
$131$		−	1363.61i		−	0.909460i	0.890629	−	0.454730i	$-0.150264\pi$
$132$	618.399			0.407763
$133$		−	337.736i		−	0.220191i
$134$			4621.52i			2.97939i
$135$			414.325i			0.264144i
$136$			4488.79i			2.83023i
$137$		−	2838.13i		−	1.76991i	−0.465678	−	0.884954i	$-0.654189\pi$
$137$		−	2838.13i		−	1.76991i	0.465678	−	0.884954i	$-0.345811\pi$
$138$	−468.107			−0.288753
$139$			1649.77i			1.00670i	0.864082	+	0.503351i	$0.167900\pi$
$139$			1649.77i			1.00670i	−0.864082	+	0.503351i	$0.832100\pi$
$140$	−6716.63			−4.05471
$141$	−1135.63			−0.678276
$142$			2120.26i			1.25301i
$143$	773.792			0.452502
$144$	1297.38			0.750799
$145$	2432.73			1.39329
$146$	−5459.96			−3.09500
$147$	567.813			0.318588
$148$	−1844.20			−1.02427
$149$		−	330.874i		−	0.181921i	−0.995854	−	0.0909606i	$-0.971006\pi$
$149$		−	330.874i		−	0.181921i	0.995854	−	0.0909606i	$-0.0289937\pi$
$150$		−	1721.33i		−	0.936971i
$151$		−	1576.13i		−	0.849426i	−0.905328	−	0.424713i	$-0.860375\pi$
$151$		−	1576.13i		−	0.849426i	0.905328	−	0.424713i	$-0.139625\pi$
$152$		−	834.144i		−	0.445119i
$153$	708.992			0.374632
$154$	1301.85			0.681208
$155$		−	1178.18i		−	0.610541i
$156$	4053.34			2.08030
$157$	−1328.62	+	1450.75i	−0.675384	+	0.737466i
$157$	−1328.62	+	1450.75i	−0.675384	+	0.737466i
$158$	6052.49			3.04753
$159$		−	425.252i		−	0.212105i
$160$	−4493.17			−2.22010
$161$	−693.163			−0.339310
$162$		−	420.668i		−	0.204017i
$163$			348.736i			0.167577i	0.996484	+	0.0837887i	$0.0267021\pi$
$163$			348.736i			0.167577i	−0.996484	+	0.0837887i	$0.973298\pi$
$164$		−	7809.12i		−	3.71823i
$165$			500.195i			0.236001i
$166$	7275.89			3.40192
$167$	3383.42			1.56776			0.783882	−	0.620910i	$-0.213237\pi$
$167$	3383.42			1.56776			0.783882	+	0.620910i	$0.213237\pi$
$168$	3943.83			1.81115
$169$	2874.87			1.30855
$170$	−6278.15			−2.83242
$171$	−131.751			−0.0589195
$172$			2733.26i			1.21168i
$173$	4051.79			1.78065			0.890323	−	0.455329i	$-0.150478\pi$
$173$	4051.79			1.78065			0.890323	+	0.455329i	$0.150478\pi$
$174$	−2469.97			−1.07614
$175$		−	2548.90i		−	1.10102i
$176$	1566.26			0.670805
$177$		−	234.129i		−	0.0994250i
$178$		−	3533.44i		−	1.48788i
$179$			852.549i			0.355991i	0.984031	+	0.177996i	$0.0569613\pi$
$179$			852.549i			0.355991i	−0.984031	+	0.177996i	$0.943039\pi$
$180$			2620.16i			1.08497i
$181$			644.164i			0.264532i	0.991214	+	0.132266i	$0.0422253\pi$
$181$			644.164i			0.264532i	−0.991214	+	0.132266i	$0.957775\pi$
$182$	8533.07			3.47535
$183$			1302.20i			0.526017i
$184$	−1711.99			−0.685920
$185$		−	1491.69i		−	0.592818i
$186$			1196.22i			0.471565i
$187$	855.931			0.334716
$188$	−7181.60			−2.78602
$189$		−	622.917i		−	0.239738i
$190$	1166.66			0.445464
$191$		−	3455.87i		−	1.30920i	−0.755974	−	0.654601i	$-0.772837\pi$
$191$		−	3455.87i		−	1.30920i	0.755974	−	0.654601i	$-0.227163\pi$
$192$	1102.27			0.414321
$193$	3160.35			1.17869			0.589345	−	0.807881i	$-0.299386\pi$
$193$	3160.35			1.17869			0.589345	+	0.807881i	$0.299386\pi$
$194$	−5630.98			−2.08392
$195$			3278.56i			1.20401i
$196$	3590.80			1.30860
$197$	−5436.08			−1.96602			−0.983008	−	0.183565i	$-0.941236\pi$
$197$	−5436.08			−1.96602			−0.983008	+	0.183565i	$0.941236\pi$
$198$		−	507.852i		−	0.182280i
$199$	397.144			0.141471			0.0707357	−	0.997495i	$-0.477465\pi$
$199$	397.144			0.141471			0.0707357	+	0.997495i	$0.477465\pi$
$200$		−	6295.32i		−	2.22573i
$201$	2669.63			0.936823
$202$		−	6417.11i		−	2.23518i
$203$	−3657.48			−1.26456
$204$	4483.61			1.53880
$205$	6316.44			2.15200
$206$	−3213.51			−1.08687
$207$			270.403i			0.0907938i
$208$	10266.2			3.42227
$209$	−159.056			−0.0526418
$210$			5515.95i			1.81255i
$211$			4578.89i			1.49395i	0.664851	+	0.746976i	$0.268495\pi$
$211$			4578.89i			1.49395i	−0.664851	+	0.746976i	$0.731505\pi$
$212$		−	2689.26i		−	0.871222i
$213$	1224.77			0.393991
$214$	−11.3153			−0.00361449
$215$	−2210.81			−0.701282
$216$		−	1538.49i		−	0.484634i
$217$			1771.34i			0.554130i
$218$		−	1614.57i		−	0.501617i
$219$			3153.96i			0.973172i
$220$			3163.19i			0.969374i
$221$	5610.26			1.70763
$222$			1514.53i			0.457876i
$223$			892.945i			0.268144i	0.990972	+	0.134072i	$0.0428053\pi$
$223$			892.945i			0.268144i	−0.990972	+	0.134072i	$0.957195\pi$
$224$	6755.25			2.01497
$225$	−994.328			−0.294616
$226$		−	6580.95i		−	1.93698i
$227$			1654.76i			0.483834i	0.970297	+	0.241917i	$0.0777761\pi$
$227$			1654.76i			0.483834i	−0.970297	+	0.241917i	$0.922224\pi$
$228$	−833.181			−0.242012
$229$		−	1944.60i		−	0.561146i	−0.959833	−	0.280573i	$-0.909475\pi$
$229$		−	1944.60i		−	0.561146i	0.959833	−	0.280573i	$-0.0905245\pi$
$230$		−	2394.43i		−	0.686452i
$231$		−	752.017i		−	0.214195i
$232$	−9033.31			−2.55632
$233$	−4254.26			−1.19616			−0.598081	−	0.801436i	$-0.704070\pi$
$233$	−4254.26			−1.19616			−0.598081	+	0.801436i	$0.704070\pi$
$234$		−	3328.75i		−	0.929946i
$235$		−	5808.87i		−	1.61246i
$236$		−	1480.61i		−	0.408389i
$237$		−	3496.24i		−	0.958249i
$238$	9438.87			2.57072
$239$	2711.75			0.733926			0.366963	−	0.930235i	$-0.380398\pi$
$239$	2711.75			0.733926			0.366963	+	0.930235i	$0.380398\pi$
$240$			6636.27i			1.78487i
$241$		−	776.084i		−	0.207435i	−0.994607	−	0.103718i	$-0.966926\pi$
$241$		−	776.084i		−	0.207435i	0.994607	−	0.103718i	$-0.0330739\pi$
$242$			6299.36i			1.67330i
$243$	−243.000			−0.0641500
$244$			8234.99i			2.16062i
$245$			2904.43i			0.757378i
$246$	−6413.14			−1.66214
$247$	−1042.54			−0.268565
$248$			4374.87i			1.12018i
$249$		−	4202.94i		−	1.06968i
$250$	−1157.10			−0.292727
$251$		−	2669.37i		−	0.671272i	−0.941992	−	0.335636i	$-0.891049\pi$
$251$		−	2669.37i		−	0.671272i	0.941992	−	0.335636i	$-0.108951\pi$
$252$		−	3939.28i		−	0.984726i
$253$			326.444i			0.0811201i
$254$		−	6251.59i		−	1.54433i
$255$			3626.59i			0.890611i
$256$	−5194.53			−1.26820
$257$	5486.57			1.33168			0.665842	−	0.746093i	$-0.268073\pi$
$257$	5486.57			1.33168			0.665842	+	0.746093i	$0.268073\pi$
$258$	2244.65			0.541650
$259$			2242.68i			0.538044i
$260$			20733.4i			4.94549i
$261$			1426.79i			0.338375i
$262$	−7081.83			−1.66991
$263$	3206.38			0.751764			0.375882	−	0.926667i	$-0.377340\pi$
$263$	3206.38			0.751764			0.375882	+	0.926667i	$0.377340\pi$
$264$		−	1857.34i		−	0.432998i
$265$	2175.22			0.504236
$266$	−1754.01			−0.404305
$267$	−2041.10			−0.467840
$268$	16882.5			3.84800
$269$			7176.44i			1.62660i	0.581844	+	0.813300i	$0.302331\pi$
$269$			7176.44i			1.62660i	−0.581844	+	0.813300i	$0.697669\pi$
$270$	2151.77			0.485010
$271$		−	4626.73i		−	1.03710i	−0.855047	−	0.518550i	$-0.826472\pi$
$271$		−	4626.73i		−	1.03710i	0.855047	−	0.518550i	$-0.173528\pi$
$272$	11356.0			2.53146
$273$		−	4929.15i		−	1.09277i
$274$	−14739.6			−3.24983
$275$	−1200.40			−0.263226
$276$			1710.01i			0.372936i
$277$	−7455.54			−1.61718			−0.808592	−	0.588370i	$-0.799770\pi$
$277$	−7455.54			−1.61718			−0.808592	+	0.588370i	$0.799770\pi$
$278$	8567.97			1.84846
$279$	690.999			0.148276
$280$			20173.2i			4.30564i
$281$	−1812.54			−0.384794			−0.192397	−	0.981317i	$-0.561626\pi$
$281$	−1812.54			−0.384794			−0.192397	+	0.981317i	$0.561626\pi$
$282$			5897.80i			1.24542i
$283$	4676.18			0.982226			0.491113	−	0.871096i	$-0.336590\pi$
$283$	4676.18			0.982226			0.491113	+	0.871096i	$0.336590\pi$
$284$	7745.36			1.61832
$285$		−	673.922i		−	0.140069i
$286$		−	4018.64i		−	0.830864i
$287$	−9496.45			−1.95316
$288$		−	2635.23i		−	0.539174i
$289$	1292.80			0.263139
$290$		−	12634.2i		−	2.55830i
$291$			3252.75i			0.655256i
$292$			19945.4i			3.99731i
$293$		−	3010.18i		−	0.600193i	−0.953909	−	0.300096i	$-0.902981\pi$
$293$		−	3010.18i		−	0.600193i	0.953909	−	0.300096i	$-0.0970188\pi$
$294$		−	2948.90i		−	0.584977i
$295$	1197.60			0.236363
$296$			5539.01i			1.08766i
$297$	−293.362			−0.0573151
$298$	−1718.37			−0.334036
$299$			2139.70i			0.413854i
$300$	−6288.05			−1.21014
$301$	3323.83			0.636487
$302$	−8185.51			−1.55968
$303$	−3706.86			−0.702817
$304$	−2110.26			−0.398130
$305$	−6660.91			−1.25050
$306$		−	3682.10i		−	0.687882i
$307$			5573.04i			1.03606i	0.855363	+	0.518030i	$0.173334\pi$
$307$			5573.04i			1.03606i	−0.855363	+	0.518030i	$0.826666\pi$
$308$		−	4755.69i		−	0.879808i
$309$			1856.29i			0.341750i
$310$	−6118.81			−1.12105
$311$	5022.28			0.915715			0.457857	−	0.889026i	$-0.348617\pi$
$311$	5022.28			0.915715			0.457857	+	0.889026i	$0.348617\pi$
$312$		−	12174.1i		−	2.20904i
$313$	−763.035			−0.137793			−0.0688967	−	0.997624i	$-0.521948\pi$
$313$	−763.035			−0.137793			−0.0688967	+	0.997624i	$0.521948\pi$
$314$	7534.35	+	6900.10i	1.35410	+	1.24011i
$315$	3186.30			0.569929
$316$		−	22109.9i		−	3.93601i
$317$	10398.1			1.84232			0.921158	−	0.389189i	$-0.127245\pi$
$317$	10398.1			1.84232			0.921158	+	0.389189i	$0.127245\pi$
$318$	−2208.52			−0.389458
$319$			1722.49i			0.302322i
$320$			5638.27i			0.984965i
$321$			6.53633i			0.00113652i
$322$			3599.90i			0.623026i
$323$	−1153.21			−0.198658
$324$	−1536.71			−0.263497
$325$	−7868.13			−1.34291
$326$	1811.14			0.307698
$327$	−932.661			−0.157726
$328$	−23454.5			−3.94834
$329$			8733.34i			1.46348i
$330$	2597.73			0.433334
$331$	−6685.36			−1.11015			−0.555077	−	0.831799i	$-0.687311\pi$
$331$	−6685.36			−1.11015			−0.555077	+	0.831799i	$0.687311\pi$
$332$		−	26579.0i		−	4.39372i
$333$	874.870			0.143972
$334$		−	17571.5i		−	2.87866i
$335$			13655.5i			2.22711i
$336$		−	9977.29i		−	1.61996i
$337$		−	676.096i		−	0.109286i	−0.998506	−	0.0546428i	$-0.982598\pi$
$337$		−	676.096i		−	0.109286i	0.998506	−	0.0546428i	$-0.0174020\pi$
$338$		−	14930.5i		−	2.40269i
$339$	−3801.50			−0.609054
$340$			22934.2i			3.65819i
$341$	834.209			0.132478
$342$			684.239i			0.108185i
$343$			3546.68i			0.558318i
$344$	8209.25			1.28667
$345$	−1383.15			−0.215844
$346$		−	21042.7i		−	3.26954i
$347$	−8347.73			−1.29144			−0.645720	−	0.763574i	$-0.723443\pi$
$347$	−8347.73			−1.29144			−0.645720	+	0.763574i	$0.723443\pi$
$348$			9022.87i			1.38988i
$349$	−9539.24			−1.46311			−0.731553	−	0.681784i	$-0.761204\pi$
$349$	−9539.24			−1.46311			−0.731553	+	0.681784i	$0.761204\pi$
$350$	−13237.6			−2.02165
$351$	−1922.86			−0.292407
$352$		−	3181.38i		−	0.481727i
$353$	6186.49			0.932786			0.466393	−	0.884578i	$-0.345553\pi$
$353$	6186.49			0.932786			0.466393	+	0.884578i	$0.345553\pi$
$354$	−1215.93			−0.182560
$355$			6264.87i			0.936633i
$356$	−12907.8			−1.92166
$357$		−	5452.39i		−	0.808322i
$358$	4427.65			0.653656
$359$			6264.83i			0.921017i	0.887655	+	0.460509i	$0.152333\pi$
$359$			6264.83i			0.921017i	−0.887655	+	0.460509i	$0.847667\pi$
$360$	7869.57			1.15212
$361$	−6644.70			−0.968756
$362$	3345.42			0.485722
$363$	3638.84			0.526142
$364$		−	31171.5i		−	4.48855i
$365$	−16132.9			−2.31352
$366$	6762.87			0.965850
$367$		−	4660.22i		−	0.662838i	−0.943484	−	0.331419i	$-0.892473\pi$
$367$		−	4660.22i		−	0.662838i	0.943484	−	0.331419i	$-0.107527\pi$
$368$			4331.06i			0.613511i
$369$			3704.57i			0.522634i
$370$	−7747.00			−1.08851
$371$	−3270.33			−0.457647
$372$	4369.82			0.609045
$373$			5526.35i			0.767141i	0.923512	+	0.383571i	$0.125306\pi$
$373$			5526.35i			0.767141i	−0.923512	+	0.383571i	$0.874694\pi$
$374$		−	4445.22i		−	0.614591i
$375$			668.404i			0.0920432i
$376$			21569.7i			2.95844i
$377$			11290.2i			1.54237i
$378$	−3235.08			−0.440197
$379$		−	777.281i		−	0.105346i	−0.998612	−	0.0526731i	$-0.983226\pi$
$379$		−	777.281i		−	0.105346i	0.998612	−	0.0526731i	$-0.0167741\pi$
$380$		−	4261.83i		−	0.575334i
$381$	−3611.25			−0.485590
$382$	−17947.8			−2.40390
$383$			14067.9i			1.87686i	0.345470	+	0.938430i	$0.387719\pi$
$383$			14067.9i			1.87686i	−0.345470	+	0.938430i	$0.612281\pi$
$384$			1302.69i			0.173118i
$385$	3846.66			0.509205
$386$		−	16413.1i		−	2.16426i
$387$		−	1296.63i		−	0.170313i
$388$			20570.1i			2.69147i
$389$	13173.5			1.71702			0.858511	−	0.512796i	$-0.171390\pi$
$389$	13173.5			1.71702			0.858511	+	0.512796i	$0.171390\pi$
$390$	17027.0			2.21076
$391$			2366.84i			0.306128i
$392$		−	10784.9i		−	1.38959i
$393$			4090.83i			0.525077i
$394$			28231.9i			3.60991i
$395$	17883.7			2.27804
$396$	−1855.20			−0.235422
$397$			1600.28i			0.202307i	0.994871	+	0.101154i	$0.0322533\pi$
$397$			1600.28i			0.202307i	−0.994871	+	0.101154i	$0.967747\pi$
$398$		−	2062.54i		−	0.259764i
$399$			1013.21i			0.127127i
$400$	−15926.2			−1.99078
$401$			640.040i			0.0797059i	0.999206	+	0.0398530i	$0.0126889\pi$
$401$			640.040i			0.0797059i	−0.999206	+	0.0398530i	$0.987311\pi$
$402$		−	13864.6i		−	1.72015i
$403$	5467.88			0.675867
$404$	−23441.9			−2.88682
$405$		−	1242.98i		−	0.152504i
$406$			18994.9i			2.32192i
$407$	1056.19			0.128632
$408$		−	13466.4i		−	1.63403i
$409$		−	10198.3i		−	1.23294i	−0.787376	−	0.616472i	$-0.788561\pi$
$409$		−	10198.3i		−	1.23294i	0.787376	−	0.616472i	$-0.211439\pi$
$410$		−	32804.0i		−	3.95140i
$411$			8514.38i			1.02186i
$412$			11739.0i			1.40374i
$413$	−1800.53			−0.214524
$414$	1404.32			0.166712
$415$	21498.6			2.54295
$416$		−	20852.6i		−	2.45765i
$417$		−	4949.31i		−	0.581220i
$418$			826.047i			0.0966586i
$419$	−10154.8			−1.18399			−0.591997	−	0.805940i	$-0.701660\pi$
$419$	−10154.8			−1.18399			−0.591997	+	0.805940i	$0.701660\pi$
$420$	20149.9			2.34099
$421$		−	4519.57i		−	0.523207i	−0.965175	−	0.261603i	$-0.915749\pi$
$421$		−	4519.57i		−	0.523207i	0.965175	−	0.261603i	$-0.0842513\pi$
$422$	23780.2			2.74313
$423$	3406.88			0.391603
$424$	−8077.11			−0.925139
$425$	−8703.34			−0.993351
$426$		−	6360.77i		−	0.723428i
$427$	10014.3			1.13496
$428$			41.3352i			0.00466826i
$429$	−2321.38			−0.261252
$430$			11481.7i			1.28766i
$431$	6365.55			0.711410			0.355705	−	0.934598i	$-0.384241\pi$
$431$	6365.55			0.711410			0.355705	+	0.934598i	$0.384241\pi$
$432$	−3892.14			−0.433474
$433$			1747.09i			0.193902i	0.995289	+	0.0969510i	$0.0309090\pi$
$433$			1747.09i			0.193902i	−0.995289	+	0.0969510i	$0.969091\pi$
$434$	9199.32			1.01747
$435$	−7298.19			−0.804417
$436$	−5898.07			−0.647859
$437$		−	439.825i		−	0.0481457i
$438$	16379.9			1.78690
$439$		−	11199.1i		−	1.21755i	−0.793342	−	0.608777i	$-0.791661\pi$
$439$		−	11199.1i		−	1.21755i	0.793342	−	0.608777i	$-0.208339\pi$
$440$	9500.54			1.02937
$441$	−1703.44			−0.183937
$442$		−	29136.5i		−	3.13548i
$443$			13058.0i			1.40046i	0.713917	+	0.700230i	$0.246919\pi$
$443$			13058.0i			1.40046i	−0.713917	+	0.700230i	$0.753081\pi$
$444$	5532.61			0.591365
$445$		−	10440.5i		−	1.11220i
$446$	4637.45			0.492353
$447$			992.623i			0.105032i
$448$		−	8476.85i		−	0.893958i
$449$			11012.7i			1.15751i	0.815501	+	0.578756i	$0.196462\pi$
$449$			11012.7i			1.15751i	−0.815501	+	0.578756i	$0.803538\pi$
$450$			5163.98i			0.540960i
$451$			4472.34i			0.466950i
$452$	−24040.4			−2.50169
$453$			4728.38i			0.490417i
$454$	8593.88			0.888394
$455$	25213.2			2.59783
$456$			2502.43i			0.256989i
$457$	−4490.17			−0.459609			−0.229805	−	0.973237i	$-0.573809\pi$
$457$	−4490.17			−0.459609			−0.229805	+	0.973237i	$0.573809\pi$
$458$	−10099.1			−1.03035
$459$	−2126.98			−0.216294
$460$	−8746.91			−0.886580
$461$	−1277.94			−0.129109			−0.0645547	−	0.997914i	$-0.520563\pi$
$461$	−1277.94			−0.129109			−0.0645547	+	0.997914i	$0.520563\pi$
$462$	−3905.55			−0.393296
$463$			7909.73i			0.793944i	0.917831	+	0.396972i	$0.129939\pi$
$463$			7909.73i			0.793944i	−0.917831	+	0.396972i	$0.870061\pi$
$464$			22852.9i			2.28646i
$465$			3534.55i			0.352496i
$466$			22094.2i			2.19634i
$467$	−16754.1			−1.66015			−0.830074	−	0.557653i	$-0.811702\pi$
$467$	−16754.1			−1.66015			−0.830074	+	0.557653i	$0.811702\pi$
$468$	−12160.0			−1.20106
$469$		−	20530.4i		−	2.02133i
$470$	−30168.0			−2.96074
$471$	3985.86	−	4352.24i	0.389933	−	0.425776i
$472$	−4446.98			−0.433663
$473$		−	1565.35i		−	0.152167i
$474$	−18157.5			−1.75949
$475$	1617.33			0.156227
$476$		−	34480.4i		−	3.32019i
$477$			1275.76i			0.122459i
$478$		−	14083.3i		−	1.34760i
$479$		−	409.223i		−	0.0390352i	−0.999810	−	0.0195176i	$-0.993787\pi$
$479$		−	409.223i		−	0.0390352i	0.999810	−	0.0195176i	$-0.00621305\pi$
$480$	13479.5			1.28178
$481$	6922.86			0.656248
$482$	−4030.54			−0.380884
$483$	2079.49			0.195901
$484$	23011.7			2.16113
$485$	−16638.2			−1.55774
$486$			1262.00i			0.117789i
$487$	−7774.39			−0.723390			−0.361695	−	0.932296i	$-0.617802\pi$
$487$	−7774.39			−0.723390			−0.361695	+	0.932296i	$0.617802\pi$
$488$	24733.5			2.29433
$489$		−	1046.21i		−	0.0967509i
$490$	15084.0			1.39066
$491$			12556.9i			1.15414i	0.816694	+	0.577071i	$0.195804\pi$
$491$			12556.9i			1.15414i	−0.816694	+	0.577071i	$0.804196\pi$
$492$			23427.4i			2.14672i
$493$			12488.6i			1.14089i
$494$			5414.39i			0.493127i
$495$		−	1500.58i		−	0.136255i
$496$	11067.8			1.00193
$497$		−	9418.90i		−	0.850092i
$498$	−21827.7			−1.96410
$499$		−	16630.5i		−	1.49195i	−0.665976	−	0.745973i	$-0.731985\pi$
$499$		−	16630.5i		−	1.49195i	0.665976	−	0.745973i	$-0.268015\pi$
$500$			4226.93i			0.378068i
$501$	−10150.2			−0.905149
$502$	−13863.2			−1.23256
$503$		−	21272.4i		−	1.88566i	−0.333273	−	0.942830i	$-0.608153\pi$
$503$		−	21272.4i		−	1.88566i	0.333273	−	0.942830i	$-0.391847\pi$
$504$	−11831.5			−1.04567
$505$		−	18961.1i		−	1.67080i
$506$	1695.37			0.148949
$507$	−8624.62			−0.755489
$508$	−22837.2			−1.99456
$509$			8889.35i			0.774093i	0.922060	+	0.387047i	$0.126505\pi$
$509$			8889.35i			0.774093i	−0.922060	+	0.387047i	$0.873495\pi$
$510$	18834.4			1.63530
$511$	24255.0			2.09976
$512$			23503.6i			2.02876i
$513$	395.252			0.0340172
$514$		−	28494.1i		−	2.44518i
$515$	−9495.18			−0.812442
$516$		−	8199.77i		−	0.699563i
$517$	4112.95			0.349879
$518$	11647.2			0.987933
$519$	−12155.4			−1.02806
$520$	62272.0			5.25155
$521$		−	9502.63i		−	0.799075i	−0.916717	−	0.399537i	$-0.869171\pi$
$521$		−	9502.63i		−	0.799075i	0.916717	−	0.399537i	$-0.130829\pi$
$522$	7409.91			0.621309
$523$	12670.6			1.05937			0.529683	−	0.848196i	$-0.322311\pi$
$523$	12670.6			1.05937			0.529683	+	0.848196i	$0.322311\pi$
$524$			25870.1i			2.15676i
$525$			7646.71i			0.635676i
$526$		−	16652.1i		−	1.38036i
$527$	6048.31			0.499940
$528$	−4698.79			−0.387289
$529$	11264.3			0.925808
$530$		−	11296.9i		−	0.925857i
$531$			702.387i			0.0574030i
$532$			6407.44i			0.522176i
$533$			29314.3i			2.38225i
$534$			10600.3i			0.859028i
$535$	−33.4342			−0.00270184
$536$		−	50706.2i		−	4.08614i
$537$		−	2557.65i		−	0.205532i
$538$	37270.4			2.98669
$539$	−2056.48			−0.164339
$540$		−	7860.48i		−	0.626410i
$541$		−	3781.46i		−	0.300513i	−0.988647	−	0.150257i	$-0.951990\pi$
$541$		−	3781.46i		−	0.300513i	0.988647	−	0.150257i	$-0.0480100\pi$
$542$	−24028.6			−1.90428
$543$		−	1932.49i		−	0.152728i
$544$		−	23066.1i		−	1.81792i
$545$		−	4770.68i		−	0.374960i
$546$	−25599.2			−2.00649
$547$	−4336.26			−0.338949			−0.169475	−	0.985535i	$-0.554207\pi$
$547$	−4336.26			−0.338949			−0.169475	+	0.985535i	$0.554207\pi$
$548$			53844.2i			4.19728i
$549$		−	3906.59i		−	0.303696i
$550$			6234.21i			0.483323i
$551$		−	2320.74i		−	0.179432i
$552$	5135.96			0.396016
$553$	−26887.2			−2.06756
$554$			38719.8i			2.96940i
$555$			4475.07i			0.342264i
$556$		−	31299.0i		−	2.38736i
$557$	−15913.6			−1.21056			−0.605278	−	0.796014i	$-0.706938\pi$
$557$	−15913.6			−1.21056			−0.605278	+	0.796014i	$0.706938\pi$
$558$		−	3588.66i		−	0.272258i
$559$		−	10260.2i		−	0.776317i
$560$	51035.1			3.85112
$561$	−2567.79			−0.193248
$562$			9413.32i			0.706543i
$563$		−	14301.9i		−	1.07061i	−0.844659	−	0.535305i	$-0.820197\pi$
$563$		−	14301.9i		−	1.07061i	0.844659	−	0.535305i	$-0.179803\pi$
$564$	21544.8			1.60851
$565$		−	19445.2i		−	1.44790i
$566$		−	24285.4i		−	1.80352i
$567$			1868.75i			0.138413i
$568$		−	23262.9i		−	1.71847i
$569$		−	13937.0i		−	1.02684i	−0.858138	−	0.513419i	$-0.828379\pi$
$569$		−	13937.0i		−	1.02684i	0.858138	−	0.513419i	$-0.171621\pi$
$570$	−3499.97			−0.257189
$571$	−12863.2			−0.942750			−0.471375	−	0.881933i	$-0.656242\pi$
$571$	−12863.2			−0.942750			−0.471375	+	0.881933i	$0.656242\pi$
$572$	−14680.2			−1.07309
$573$			10367.6i			0.755868i
$574$			49319.2i			3.58631i
$575$		−	3319.38i		−	0.240744i
$576$	−3306.82			−0.239209
$577$	−13300.7			−0.959649			−0.479824	−	0.877365i	$-0.659300\pi$
$577$	−13300.7			−0.959649			−0.479824	+	0.877365i	$0.659300\pi$
$578$		−	6714.09i		−	0.483165i
$579$	−9481.06			−0.680517
$580$	−46153.2			−3.30415
$581$	−32322.0			−2.30799
$582$	16892.9			1.20315
$583$			1540.16i			0.109411i
$584$	59905.3			4.24469
$585$		−	9835.68i		−	0.695138i
$586$	−15633.2			−1.10205
$587$		−	22760.8i		−	1.60040i	−0.599731	−	0.800202i	$-0.704726\pi$
$587$		−	22760.8i		−	1.60040i	0.599731	−	0.800202i	$-0.295274\pi$
$588$	−10772.4			−0.755521
$589$	−1123.95			−0.0786271
$590$		−	6219.66i		−	0.433999i
$591$	16308.3			1.13508
$592$	14012.8			0.972845
$593$	−9366.34			−0.648616			−0.324308	−	0.945951i	$-0.605132\pi$
$593$	−9366.34			−0.648616			−0.324308	+	0.945951i	$0.605132\pi$
$594$			1523.56i			0.105239i
$595$	27889.7			1.92162
$596$			6277.26i			0.431421i
$597$	−1191.43			−0.0816786
$598$	11112.4			0.759900
$599$		−	6352.01i		−	0.433282i	−0.976251	−	0.216641i	$-0.930490\pi$
$599$		−	6352.01i		−	0.433282i	0.976251	−	0.216641i	$-0.0695102\pi$
$600$			18886.0i			1.28503i
$601$	23689.9			1.60787			0.803937	−	0.594714i	$-0.202735\pi$
$601$	23689.9			1.60787			0.803937	+	0.594714i	$0.202735\pi$
$602$		−	17262.1i		−	1.16869i
$603$	−8008.90			−0.540875
$604$			29901.9i			2.01439i
$605$			18613.1i			1.25080i
$606$			19251.3i			1.29048i
$607$		−	13728.2i		−	0.917974i	−0.888443	−	0.458987i	$-0.848213\pi$
$607$		−	13728.2i		−	0.917974i	0.888443	−	0.458987i	$-0.151787\pi$
$608$			4286.33i			0.285911i
$609$	10972.5			0.730092
$610$			34593.0i			2.29611i
$611$	26958.7			1.78499
$612$	−13450.8			−0.888427
$613$			20562.4i			1.35483i	0.735602	+	0.677414i	$0.236899\pi$
$613$			20562.4i			1.35483i	−0.735602	+	0.677414i	$0.763101\pi$
$614$	28943.2			1.90237
$615$	−18949.3			−1.24246
$616$	−14283.6			−0.934256
$617$	20927.9			1.36552			0.682759	−	0.730644i	$-0.260780\pi$
$617$	20927.9			1.36552			0.682759	+	0.730644i	$0.260780\pi$
$618$	9640.54			0.627507
$619$	24651.7			1.60070			0.800351	−	0.599532i	$-0.204647\pi$
$619$	24651.7			1.60070			0.800351	+	0.599532i	$0.204647\pi$
$620$			22352.2i			1.44788i
$621$		−	811.210i		−	0.0524198i
$622$		−	26082.9i		−	1.68139i
$623$			15696.7i			1.00943i
$624$	−30798.6			−1.97585
$625$	−17229.1			−1.10266
$626$			3962.77i			0.253010i
$627$	477.168			0.0303928
$628$	25206.2	−	27523.2i	1.60165	−	1.74888i
$629$	7657.73			0.485427
$630$		−	16547.8i		−	1.04648i
$631$	−8756.15			−0.552419			−0.276210	−	0.961097i	$-0.589078\pi$
$631$	−8756.15			−0.552419			−0.276210	+	0.961097i	$0.589078\pi$
$632$	−66406.4			−4.17960
$633$		−	13736.7i		−	0.862533i
$634$		−	54001.7i		−	3.38278i
$635$		−	18472.0i		−	1.15439i
$636$			8067.78i			0.503000i
$637$	−13479.3			−0.838415
$638$	8945.62			0.555111
$639$	−3674.32			−0.227471
$640$	−6663.41			−0.411554
$641$	−1730.99			−0.106662			−0.0533308	−	0.998577i	$-0.516984\pi$
$641$	−1730.99			−0.106662			−0.0533308	+	0.998577i	$0.516984\pi$
$642$	33.9460			0.00208683
$643$			20087.1i			1.23197i	0.787758	+	0.615985i	$0.211242\pi$
$643$			20087.1i			1.23197i	−0.787758	+	0.615985i	$0.788758\pi$
$644$	13150.5			0.804664
$645$	6632.42			0.404885
$646$			5989.13i			0.364767i
$647$	17287.1			1.05042			0.525212	−	0.850971i	$-0.323986\pi$
$647$	17287.1			1.05042			0.525212	+	0.850971i	$0.323986\pi$
$648$			4615.47i			0.279804i
$649$			847.958i			0.0512870i
$650$			40862.6i			2.46579i
$651$		−	5314.01i		−	0.319927i
$652$		−	6616.14i		−	0.397405i
$653$	10503.8			0.629471			0.314735	−	0.949179i	$-0.398084\pi$
$653$	10503.8			0.629471			0.314735	+	0.949179i	$0.398084\pi$
$654$			4843.71i			0.289609i
$655$	−20925.1			−1.24826
$656$			59336.2i			3.53154i
$657$		−	9461.87i		−	0.561861i
$658$	45356.0			2.68718
$659$	5225.80			0.308905			0.154452	−	0.988000i	$-0.450639\pi$
$659$	5225.80			0.308905			0.154452	+	0.988000i	$0.450639\pi$
$660$		−	9489.57i		−	0.559668i
$661$	7445.66			0.438128			0.219064	−	0.975710i	$-0.429700\pi$
$661$	7445.66			0.438128			0.219064	+	0.975710i	$0.429700\pi$
$662$			34720.0i			2.03841i
$663$	−16830.8			−0.985903
$664$	−79829.3			−4.66563
$665$	−5182.68			−0.302219
$666$		−	4543.58i		−	0.264355i
$667$	−4763.05			−0.276501
$668$	−64189.3			−3.71790
$669$		−	2678.83i		−	0.154813i
$670$	70919.0			4.08931
$671$		−	4716.23i		−	0.271339i
$672$	−20265.8			−1.16335
$673$		−	3334.65i		−	0.190998i	−0.995430	−	0.0954989i	$-0.969555\pi$
$673$		−	3334.65i		−	0.190998i	0.995430	−	0.0954989i	$-0.0304446\pi$
$674$	−3511.26			−0.200666
$675$	2982.98			0.170096
$676$	−54541.4			−3.10317
$677$	−6634.10			−0.376616			−0.188308	−	0.982110i	$-0.560300\pi$
$677$	−6634.10			−0.376616			−0.188308	+	0.982110i	$0.560300\pi$
$678$			19742.8i			1.11832i
$679$	25014.7			1.41381
$680$	68882.3			3.88458
$681$		−	4964.28i		−	0.279341i
$682$		−	4332.41i		−	0.243250i
$683$			21586.5i			1.20935i	0.796473	+	0.604674i	$0.206697\pi$
$683$			21586.5i			1.20935i	−0.796473	+	0.604674i	$0.793303\pi$
$684$	2499.54			0.139726
$685$	−43552.1			−2.42926
$686$	18419.5			1.02516
$687$			5833.79i			0.323978i
$688$		−	20768.1i		−	1.15084i
$689$			10095.1i			0.558188i
$690$			7183.28i			0.396323i
$691$			7215.58i			0.397241i	0.980076	+	0.198621i	$0.0636461\pi$
$691$			7215.58i			0.397241i	−0.980076	+	0.198621i	$0.936354\pi$
$692$	−76869.5			−4.22275
$693$			2256.05i			0.123666i
$694$			43353.4i			2.37128i
$695$	25316.4			1.38173
$696$	27099.9			1.47589
$697$			32426.0i			1.76216i
$698$			49541.4i			2.68649i
$699$	12762.8			0.690605
$700$			48357.2i			2.61104i
$701$		−	18724.2i		−	1.00885i	−0.863455	−	0.504425i	$-0.831704\pi$
$701$		−	18724.2i		−	1.00885i	0.863455	−	0.504425i	$-0.168296\pi$
$702$			9986.25i			0.536904i
$703$	−1423.02			−0.0763446
$704$	−3992.16			−0.213722
$705$			17426.6i			0.930957i
$706$		−	32129.1i		−	1.71274i
$707$			28507.0i			1.51643i
$708$			4441.84i			0.235783i
$709$	24136.3			1.27850			0.639251	−	0.768998i	$-0.279244\pi$
$709$	24136.3			1.27850			0.639251	+	0.768998i	$0.279244\pi$
$710$	32536.2			1.71980
$711$			10488.7i			0.553245i
$712$			38768.1i			2.04058i
$713$			2306.77i			0.121163i
$714$	−28316.6			−1.48420
$715$		−	11874.1i		−	0.621073i
$716$		−	16174.3i		−	0.844223i
$717$	−8135.24			−0.423733
$718$	32536.0			1.69113
$719$			12037.8i			0.624389i	0.950018	+	0.312195i	$0.101064\pi$
$719$			12037.8i			0.624389i	−0.950018	+	0.312195i	$0.898936\pi$
$720$		−	19908.8i		−	1.03050i
$721$	14275.5			0.737376
$722$			34508.8i			1.77879i
$723$			2328.25i			0.119763i
$724$		−	12220.9i		−	0.627330i
$725$		−	17514.7i		−	0.897214i
$726$		−	18898.1i		−	0.966079i
$727$	−17802.8			−0.908209			−0.454104	−	0.890948i	$-0.650041\pi$
$727$	−17802.8			−0.908209			−0.454104	+	0.890948i	$0.650041\pi$
$728$	−93622.7			−4.76633
$729$	729.000			0.0370370
$730$			83785.2i			4.24798i
$731$		−	11349.4i		−	0.574243i
$732$		−	24705.0i		−	1.24743i
$733$	−34765.6			−1.75184			−0.875919	−	0.482459i	$-0.839744\pi$
$733$	−34765.6			−1.75184			−0.875919	+	0.482459i	$0.839744\pi$
$734$	−24202.5			−1.21707
$735$		−	8713.30i		−	0.437272i
$736$	8797.20			0.440583
$737$	−9668.75			−0.483247
$738$	19239.4			0.959638
$739$	18476.1			0.919695			0.459848	−	0.887998i	$-0.347904\pi$
$739$	18476.1			0.919695			0.459848	+	0.887998i	$0.347904\pi$
$740$			28300.0i			1.40585i
$741$	3127.63			0.155056
$742$			16984.2i			0.840311i
$743$	−14397.0			−0.710866			−0.355433	−	0.934702i	$-0.615667\pi$
$743$	−14397.0			−0.710866			−0.355433	+	0.934702i	$0.615667\pi$
$744$		−	13124.6i		−	0.646737i
$745$	−5077.39			−0.249693
$746$	28700.7			1.40859
$747$			12608.8i			0.617580i
$748$	−16238.5			−0.793769
$749$	50.2666			0.00245220
$750$	3471.31			0.169006
$751$		−	22680.7i		−	1.10204i	−0.834492	−	0.551020i	$-0.814239\pi$
$751$		−	22680.7i		−	1.10204i	0.834492	−	0.551020i	$-0.185761\pi$
$752$	54568.1			2.64614
$753$			8008.11i			0.387559i
$754$	58634.7			2.83203
$755$	−24186.3			−1.16587
$756$			11817.8i			0.568532i
$757$			10356.8i			0.497259i	0.968599	+	0.248629i	$0.0799801\pi$
$757$			10356.8i			0.497259i	−0.968599	+	0.248629i	$0.920020\pi$
$758$	−4036.76			−0.193432
$759$		−	979.333i		−	0.0468347i
$760$	−12800.3			−0.610940
$761$		−	22656.9i		−	1.07925i	−0.841904	−	0.539627i	$-0.818565\pi$
$761$		−	22656.9i		−	1.07925i	0.841904	−	0.539627i	$-0.181435\pi$
$762$			18754.8i			0.891620i
$763$			7172.47i			0.340316i
$764$			65563.8i			3.10473i
$765$		−	10879.8i		−	0.514194i
$766$	73060.8			3.44621
$767$			5558.00i			0.261653i
$768$	15583.6			0.732193
$769$	13322.8			0.624751			0.312375	−	0.949959i	$-0.398875\pi$
$769$	13322.8			0.624751			0.312375	+	0.949959i	$0.398875\pi$
$770$		−	19977.4i		−	0.934981i
$771$	−16459.7			−0.768848
$772$	−59957.5			−2.79523
$773$	−5272.57			−0.245331			−0.122666	−	0.992448i	$-0.539144\pi$
$773$	−5272.57			−0.245331			−0.122666	+	0.992448i	$0.539144\pi$
$774$	−6733.95			−0.312722
$775$	−8482.46			−0.393160
$776$	61781.7			2.85804
$777$		−	6728.04i		−	0.310640i
$778$		−	68415.5i		−	3.15272i
$779$		−	6025.67i		−	0.277140i
$780$		−	62200.1i		−	2.85528i
$781$	−4435.82			−0.203234
$782$	12292.0			0.562099
$783$		−	4280.36i		−	0.195361i
$784$	−27284.1			−1.24290
$785$	22262.3	+	20388.2i	1.01220	+	0.926987i
$786$	21245.5			0.964124
$787$			21639.7i			0.980141i	0.871683	+	0.490070i	$0.163029\pi$
$787$			21639.7i			0.980141i	−0.871683	+	0.490070i	$0.836971\pi$
$788$	103132.			4.66234
$789$	−9619.15			−0.434031
$790$		−	92877.8i		−	4.18284i
$791$			29234.8i			1.31412i
$792$			5572.03i			0.249992i
$793$		−	30912.9i		−	1.38430i
$794$	8310.97			0.371468
$795$	−6525.66			−0.291121
$796$	−7534.53			−0.335495
$797$	−4494.86			−0.199769			−0.0998847	−	0.994999i	$-0.531847\pi$
$797$	−4494.86			−0.199769			−0.0998847	+	0.994999i	$0.531847\pi$
$798$	5262.02			0.233425
$799$	29820.4			1.32036
$800$			32349.1i			1.42964i
$801$	6123.30			0.270108
$802$	3324.00			0.146352
$803$		−	11422.9i		−	0.501997i
$804$	−50647.6			−2.22165
$805$			10636.9i			0.465714i
$806$		−	28397.1i		−	1.24100i
$807$		−	21529.3i		−	0.939118i
$808$			70406.9i			3.06548i
$809$		−	35126.8i		−	1.52657i	−0.646063	−	0.763284i	$-0.723586\pi$
$809$		−	35126.8i		−	1.52657i	0.646063	−	0.763284i	$-0.276414\pi$
$810$	−6455.32			−0.280021
$811$		−	22730.0i		−	0.984164i	−0.870549	−	0.492082i	$-0.836236\pi$
$811$		−	22730.0i		−	0.984164i	0.870549	−	0.492082i	$-0.163764\pi$
$812$	69388.9			2.99886
$813$			13880.2i			0.598770i
$814$		−	5485.24i		−	0.236189i
$815$	5351.49			0.230006
$816$	−34067.9			−1.46154
$817$			2109.03i			0.0903129i
$818$	−52964.3			−2.26388
$819$			14787.4i			0.630910i
$820$	−119834.			−5.10340
$821$	723.202			0.0307429			0.0153714	−	0.999882i	$-0.495107\pi$
$821$	723.202			0.0307429			0.0153714	+	0.999882i	$0.495107\pi$
$822$	44218.9			1.87629
$823$		−	31537.4i		−	1.33575i	−0.744272	−	0.667876i	$-0.767203\pi$
$823$		−	31537.4i		−	1.33575i	0.744272	−	0.667876i	$-0.232797\pi$
$824$	35257.9			1.49061
$825$	3601.21			0.151973
$826$			9350.94i			0.393899i
$827$	−841.706			−0.0353918			−0.0176959	−	0.999843i	$-0.505633\pi$
$827$	−841.706			−0.0353918			−0.0176959	+	0.999843i	$0.505633\pi$
$828$		−	5130.02i		−	0.215315i
$829$	3609.59			0.151226			0.0756129	−	0.997137i	$-0.475909\pi$
$829$	3609.59			0.151226			0.0756129	+	0.997137i	$0.475909\pi$
$830$		−	111651.i		−	4.66925i
$831$	22366.6			0.933681
$832$	−26166.9			−1.09035
$833$	−14910.2			−0.620176
$834$	−25703.9			−1.06721
$835$		−	51919.8i		−	2.15181i
$836$	3017.57			0.124838
$837$	−2073.00			−0.0856072
$838$			52738.2i			2.17400i
$839$			12468.4i			0.513059i	0.966536	+	0.256530i	$0.0825792\pi$
$839$			12468.4i			0.513059i	−0.966536	+	0.256530i	$0.917421\pi$
$840$		−	60519.6i		−	2.48586i
$841$	−743.296			−0.0304767
$842$	−23472.1			−0.960690
$843$	5437.63			0.222161
$844$		−	86869.6i		−	3.54286i
$845$		−	44116.0i		−	1.79602i
$846$		−	17693.4i		−	0.719044i
$847$		−	27983.9i		−	1.13523i
$848$			20433.9i			0.827478i
$849$	−14028.5			−0.567089
$850$			45200.2i			1.82395i
$851$			2920.59i			0.117646i
$852$	−23236.1			−0.934337
$853$	−9877.50			−0.396482			−0.198241	−	0.980153i	$-0.563523\pi$
$853$	−9877.50			−0.396482			−0.198241	+	0.980153i	$0.563523\pi$
$854$		−	52008.8i		−	2.08396i
$855$			2021.77i			0.0808689i
$856$	124.149			0.00495716
$857$		−	14801.5i		−	0.589976i	−0.955501	−	0.294988i	$-0.904684\pi$
$857$		−	14801.5i		−	0.589976i	0.955501	−	0.294988i	$-0.0953157\pi$
$858$			12055.9i			0.479699i
$859$		−	4743.13i		−	0.188397i	−0.995553	−	0.0941987i	$-0.969971\pi$
$859$		−	4743.13i		−	0.188397i	0.995553	−	0.0941987i	$-0.0300289\pi$
$860$	41942.9			1.66307
$861$	28489.3			1.12766
$862$		−	33059.1i		−	1.30626i
$863$			16418.7i			0.647624i	0.946121	+	0.323812i	$0.104965\pi$
$863$			16418.7i			0.647624i	−0.946121	+	0.323812i	$0.895035\pi$
$864$			7905.68i			0.311292i
$865$		−	62176.3i		−	2.44400i
$866$	9073.37			0.356034
$867$	−3878.41			−0.151924
$868$		−	33605.4i		−	1.31410i
$869$			12662.5i			0.494299i
$870$			37902.7i			1.47704i
$871$	−63374.5			−2.46540
$872$			17714.7i			0.687952i
$873$		−	9758.25i		−	0.378312i
$874$	−2284.20			−0.0884030
$875$	5140.25			0.198597
$876$		−	59836.1i		−	2.30785i
$877$		−	19238.8i		−	0.740762i	−0.928880	−	0.370381i	$-0.879227\pi$
$877$		−	19238.8i		−	0.740762i	0.928880	−	0.370381i	$-0.120773\pi$
$878$	−58162.0			−2.23562
$879$			9030.53i			0.346521i
$880$		−	24034.9i		−	0.920702i
$881$			31075.0i			1.18836i	0.804333	+	0.594179i	$0.202523\pi$
$881$			31075.0i			1.18836i	−0.804333	+	0.594179i	$0.797477\pi$
$882$			8846.69i			0.337737i
$883$			32968.0i			1.25647i	0.778024	+	0.628235i	$0.216222\pi$
$883$			32968.0i			1.25647i	−0.778024	+	0.628235i	$0.783778\pi$
$884$	−106437.			−4.04960
$885$	−3592.80			−0.136464
$886$	67815.8			2.57146
$887$			5408.25i			0.204725i	0.994747	+	0.102363i	$0.0326402\pi$
$887$			5408.25i			0.204725i	−0.994747	+	0.102363i	$0.967360\pi$
$888$		−	16617.0i		−	0.627962i
$889$			27771.7i			1.04773i
$890$	−54222.0			−2.04216
$891$	880.086			0.0330909
$892$		−	16940.7i		−	0.635894i
$893$	−5541.46			−0.207657
$894$	5155.12			0.192856
$895$	13082.7			0.488610
$896$	10018.1			0.373528
$897$		−	6419.11i		−	0.238938i
$898$	57193.9			2.12537
$899$			12171.7i			0.451556i
$900$	18864.1			0.698672
$901$			11166.7i			0.412893i
$902$	23226.8			0.857392
$903$	−9971.50			−0.367476
$904$			72204.5i			2.65651i
$905$	9884.94			0.363079
$906$	24556.5			0.900481
$907$	42827.2			1.56787			0.783933	−	0.620846i	$-0.213211\pi$
$907$	42827.2			1.56787			0.783933	+	0.620846i	$0.213211\pi$
$908$		−	31393.7i		−	1.14740i
$909$	11120.6			0.405771
$910$		−	130943.i		−	4.77003i
$911$	47620.4			1.73187			0.865936	−	0.500156i	$-0.166724\pi$
$911$	47620.4			1.73187			0.865936	+	0.500156i	$0.166724\pi$
$912$	6330.77			0.229861
$913$			15222.0i			0.551779i
$914$			23319.4i			0.843914i
$915$	19982.7			0.721976
$916$			36892.4i			1.33074i
$917$	31459.9			1.13293
$918$			11046.3i			0.397149i
$919$		−	35372.8i		−	1.26969i	−0.772642	−	0.634843i	$-0.781065\pi$
$919$		−	35372.8i		−	1.26969i	0.772642	−	0.634843i	$-0.218935\pi$
$920$			26271.1i			0.941448i
$921$		−	16719.1i		−	0.598169i
$922$			6636.87i			0.237065i
$923$	−29074.9			−1.03685
$924$			14267.1i			0.507957i
$925$	−10739.6			−0.381747
$926$	41078.7			1.45781
$927$		−	5568.88i		−	0.197310i
$928$	46418.5			1.64199
$929$	−29994.8			−1.05931			−0.529653	−	0.848214i	$-0.677678\pi$
$929$	−29994.8			−1.05931			−0.529653	+	0.848214i	$0.677678\pi$
$930$	18356.4			0.647238
$931$	2770.73			0.0975371
$932$	80710.8			2.83666
$933$	−15066.8			−0.528688
$934$			87011.5i			3.04829i
$935$		−	13134.6i		−	0.459409i
$936$			36522.2i			1.27539i
$937$			11068.1i			0.385892i	0.981209	+	0.192946i	$0.0618042\pi$
$937$			11068.1i			0.385892i	−0.981209	+	0.192946i	$0.938196\pi$
$938$	−106623.			−3.71148
$939$	2289.11			0.0795550
$940$			110205.i			3.82391i
$941$	42018.6			1.45565			0.727825	−	0.685763i	$-0.240531\pi$
$941$	42018.6			1.45565			0.727825	+	0.685763i	$0.240531\pi$
$942$	−22603.1	−	20700.3i	−0.781791	−	0.715979i
$943$	−12367.0			−0.427068
$944$			11250.2i			0.387884i
$945$	−9558.90			−0.329049
$946$	−8129.56			−0.279403
$947$			1876.79i			0.0644006i	0.999481	+	0.0322003i	$0.0102514\pi$
$947$			1876.79i			0.0644006i	−0.999481	+	0.0322003i	$0.989749\pi$
$948$			66329.7i			2.27246i
$949$		−	74871.9i		−	2.56106i
$950$		−	8399.48i		−	0.286858i
$951$	−31194.2			−1.06366
$952$	−103561.			−3.52566
$953$	41082.3			1.39642			0.698208	−	0.715895i	$-0.253981\pi$
$953$	41082.3			1.39642			0.698208	+	0.715895i	$0.253981\pi$
$954$	6625.56			0.224854
$955$	−53031.6			−1.79692
$956$	−51446.6			−1.74048
$957$		−	5167.46i		−	0.174546i
$958$	−2125.27			−0.0716748
$959$	65478.4			2.20480
$960$		−	16914.8i		−	0.568670i
$961$	−23896.2			−0.802128
$962$		−	35953.4i		−	1.20497i
$963$		−	19.6090i		−	0.000656170i
$964$			14723.7i			0.491927i
$965$		−	48496.8i		−	1.61779i
$966$		−	10799.7i		−	0.359704i
$967$	−3834.97			−0.127533			−0.0637664	−	0.997965i	$-0.520311\pi$
$967$	−3834.97			−0.127533			−0.0637664	+	0.997965i	$0.520311\pi$
$968$		−	69115.0i		−	2.29488i
$969$	3459.64			0.114695
$970$			86409.6i			2.86025i
$971$		−	6631.61i		−	0.219175i	−0.993977	−	0.109587i	$-0.965047\pi$
$971$		−	6631.61i		−	0.219175i	0.993977	−	0.109587i	$-0.0349529\pi$
$972$	4610.14			0.152130
$973$	−38061.8			−1.25407
$974$			40375.8i			1.32826i
$975$	23604.4			0.775328
$976$		−	62572.1i		−	2.05213i
$977$	−11815.8			−0.386920			−0.193460	−	0.981108i	$-0.561971\pi$
$977$	−11815.8			−0.386920			−0.193460	+	0.981108i	$0.561971\pi$
$978$	−5433.41			−0.177650
$979$	7392.36			0.241329
$980$		−	55102.2i		−	1.79610i
$981$	2797.98			0.0910629
$982$	65213.2			2.11918
$983$		−	35962.1i		−	1.16685i	−0.812167	−	0.583425i	$-0.801712\pi$
$983$		−	35962.1i		−	1.16685i	0.812167	−	0.583425i	$-0.198288\pi$
$984$	70363.4			2.27958
$985$			83418.8i			2.69842i
$986$	64858.9			2.09486
$987$		−	26200.0i		−	0.844940i
$988$	19778.9			0.636893
$989$	4328.55			0.139171
$990$	−7793.18			−0.250185
$991$	−6169.69			−0.197767			−0.0988833	−	0.995099i	$-0.531527\pi$
$991$	−6169.69			−0.197767			−0.0988833	+	0.995099i	$0.531527\pi$
$992$		−	22480.7i		−	0.719520i
$993$	20056.1			0.640947
$994$	−48916.4			−1.56090
$995$		−	6094.34i		−	0.194174i
$996$			79737.1i			2.53671i
$997$			60155.6i			1.91088i	0.295184	+	0.955441i	$0.404619\pi$
$997$			60155.6i			1.91088i	−0.295184	+	0.955441i	$0.595381\pi$
$998$	−86369.2			−2.73945
$999$	−2624.61			−0.0831221

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	471.4.b.a.313.3	✓	40
157.156	even	2	inner	471.4.b.a.313.38	yes	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
471.4.b.a.313.3	✓	40	1.1	even	1	trivial
471.4.b.a.313.38	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)

\(f(q)\)	\(=\)	\(q-5.19343i q^{2} -3.00000 q^{3} -18.9718 q^{4} -15.3454i q^{5} +15.5803i q^{6} +23.0710i q^{7} +56.9811i q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q-5.19343i q^{2} -3.00000 q^{3} -18.9718 q^{4} -15.3454i q^{5} +15.5803i q^{6} +23.0710i q^{7} +56.9811i q^{8} +9.00000 q^{9} -79.6953 q^{10} +10.8653 q^{11} +56.9153 q^{12} +71.2171 q^{13} +119.818 q^{14} +46.0362i q^{15} +144.153 q^{16} +78.7769 q^{17} -46.7409i q^{18} -14.6390 q^{19} +291.129i q^{20} -69.2130i q^{21} -56.4280i q^{22} +30.0448i q^{23} -170.943i q^{24} -110.481 q^{25} -369.861i q^{26} -27.0000 q^{27} -437.697i q^{28} +158.532i q^{29} +239.086 q^{30} +76.7777 q^{31} -292.803i q^{32} -32.5958 q^{33} -409.123i q^{34} +354.033 q^{35} -170.746 q^{36} +97.2078 q^{37} +76.0265i q^{38} -213.651 q^{39} +874.397 q^{40} +411.618i q^{41} -359.453 q^{42} -144.070i q^{43} -206.133 q^{44} -138.108i q^{45} +156.036 q^{46} +378.542 q^{47} -432.460 q^{48} -189.271 q^{49} +573.775i q^{50} -236.331 q^{51} -1351.11 q^{52} +141.751i q^{53} +140.223i q^{54} -166.732i q^{55} -1314.61 q^{56} +43.9169 q^{57} +823.324 q^{58} +78.0430i q^{59} -873.387i q^{60} -434.066i q^{61} -398.740i q^{62} +207.639i q^{63} -367.424 q^{64} -1092.85i q^{65} +169.284i q^{66} -889.877 q^{67} -1494.54 q^{68} -90.1344i q^{69} -1838.65i q^{70} -408.257 q^{71} +512.830i q^{72} -1051.32i q^{73} -504.842i q^{74} +331.443 q^{75} +277.727 q^{76} +250.672i q^{77} +1109.58i q^{78} +1165.41i q^{79} -2212.09i q^{80} +81.0000 q^{81} +2137.71 q^{82} +1400.98i q^{83} +1313.09i q^{84} -1208.86i q^{85} -748.217 q^{86} -475.595i q^{87} +619.114i q^{88} +680.367 q^{89} -717.257 q^{90} +1643.05i q^{91} -570.003i q^{92} -230.333 q^{93} -1965.93i q^{94} +224.641i q^{95} +878.408i q^{96} -1084.25i q^{97} +982.966i q^{98} +97.7873 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 120 q^{3} - 164 q^{4} + 360 q^{9}+O(q^{10}) \) 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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