LMFDB - Embedded newform 269.4.b.a.268.9

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [269,4,Mod(268,269)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("269.268"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(269, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 269 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	269.b (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)

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

Self dual:	no
Analytic conductor:	$15.8715137915$
Analytic rank:	$0$
Dimension:	$66$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			268.9
Character	$\chi$	$=$	269.268
Dual form			269.4.b.a.268.58

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-4.45843i q^{2} +0.747483i q^{3} -11.8776 q^{4} -16.4504 q^{5} +3.33261 q^{6} +20.0331i q^{7} +17.2882i q^{8} +26.4413 q^{9} +73.3432i q^{10} +19.7959 q^{11} -8.87834i q^{12} +53.8132 q^{13} +89.3161 q^{14} -12.2964i q^{15} -17.9428 q^{16} +79.4618i q^{17} -117.887i q^{18} -108.218i q^{19} +195.392 q^{20} -14.9744 q^{21} -88.2586i q^{22} -140.315 q^{23} -12.9226 q^{24} +145.617 q^{25} -239.922i q^{26} +39.9465i q^{27} -237.946i q^{28} +32.4595i q^{29} -54.8228 q^{30} -174.877i q^{31} +218.302i q^{32} +14.7971i q^{33} +354.275 q^{34} -329.553i q^{35} -314.060 q^{36} +391.562 q^{37} -482.483 q^{38} +40.2244i q^{39} -284.398i q^{40} +234.544 q^{41} +66.7623i q^{42} +458.267 q^{43} -235.128 q^{44} -434.971 q^{45} +625.585i q^{46} +522.533 q^{47} -13.4120i q^{48} -58.3240 q^{49} -649.224i q^{50} -59.3964 q^{51} -639.173 q^{52} -235.184 q^{53} +178.099 q^{54} -325.651 q^{55} -346.336 q^{56} +80.8912 q^{57} +144.718 q^{58} +714.231i q^{59} +146.053i q^{60} +541.701 q^{61} -779.675 q^{62} +529.700i q^{63} +829.745 q^{64} -885.250 q^{65} +65.9718 q^{66} -410.309 q^{67} -943.819i q^{68} -104.883i q^{69} -1469.29 q^{70} -1040.22i q^{71} +457.122i q^{72} -284.957 q^{73} -1745.75i q^{74} +108.846i q^{75} +1285.37i q^{76} +396.572i q^{77} +179.338 q^{78} +1066.64 q^{79} +295.168 q^{80} +684.055 q^{81} -1045.70i q^{82} +1201.98i q^{83} +177.860 q^{84} -1307.18i q^{85} -2043.15i q^{86} -24.2629 q^{87} +342.235i q^{88} -884.858 q^{89} +1939.29i q^{90} +1078.04i q^{91} +1666.61 q^{92} +130.717 q^{93} -2329.68i q^{94} +1780.23i q^{95} -163.177 q^{96} -738.008 q^{97} +260.034i q^{98} +523.428 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 66 q - 258 q^{4} - 34 q^{5} + 48 q^{6} - 564 q^{9} + 22 q^{11} + 118 q^{13} + 60 q^{14} + 1030 q^{16} + 144 q^{20} - 64 q^{21} - 80 q^{23} - 778 q^{24} + 1676 q^{25} - 1146 q^{30} + 308 q^{34} + 2030 q^{36}+ \cdots - 3982 q^{99}+O(q^{100}) $ 66 * q - 258 * q^4 - 34 * q^5 + 48 * q^6 - 564 * q^9 + 22 * q^11 + 118 * q^13 + 60 * q^14 + 1030 * q^16 + 144 * q^20 - 64 * q^21 - 80 * q^23 - 778 * q^24 + 1676 * q^25 - 1146 * q^30 + 308 * q^34 + 2030 * q^36 + 470 * q^37 + 696 * q^38 + 76 * q^41 + 426 * q^43 + 698 * q^44 + 1254 * q^45 - 1280 * q^47 - 3502 * q^49 - 640 * q^51 - 1924 * q^52 - 618 * q^53 - 3766 * q^54 + 176 * q^55 + 3164 * q^56 + 1668 * q^57 - 2170 * q^58 - 1026 * q^61 + 516 * q^62 - 2936 * q^64 - 900 * q^65 - 5296 * q^66 + 862 * q^67 - 1776 * q^70 + 3560 * q^73 + 8818 * q^78 + 2064 * q^79 - 3966 * q^80 + 7446 * q^81 - 3304 * q^84 - 776 * q^87 - 892 * q^89 - 6402 * q^92 + 7388 * q^93 + 8632 * q^96 + 4552 * q^97 - 3982 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/269\mathbb{Z}\right)^\times$.

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

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		−	4.45843i		−	1.57629i	−0.615487	−	0.788147i	$-0.711041\pi$
$2$		−	4.45843i		−	1.57629i	0.615487	−	0.788147i	$-0.288959\pi$
$3$			0.747483i			0.143853i	0.997410	+	0.0719266i	$0.0229147\pi$
$3$			0.747483i			0.143853i	−0.997410	+	0.0719266i	$0.977085\pi$
$4$	−11.8776			−1.48470
$5$	−16.4504			−1.47137			−0.735686	−	0.677323i	$-0.763140\pi$
$5$	−16.4504			−1.47137			−0.735686	+	0.677323i	$0.763140\pi$
$6$	3.33261			0.226755
$7$			20.0331i			1.08168i	0.841124	+	0.540842i	$0.181894\pi$
$7$			20.0331i			1.08168i	−0.841124	+	0.540842i	$0.818106\pi$
$8$			17.2882i			0.764037i
$9$	26.4413			0.979306
$10$			73.3432i			2.31932i
$11$	19.7959			0.542607			0.271304	−	0.962494i	$-0.412545\pi$
$11$	19.7959			0.542607			0.271304	+	0.962494i	$0.412545\pi$
$12$		−	8.87834i		−	0.213580i
$13$	53.8132			1.14808			0.574042	−	0.818826i	$-0.305375\pi$
$13$	53.8132			1.14808			0.574042	+	0.818826i	$0.305375\pi$
$14$	89.3161			1.70505
$15$		−	12.2964i		−	0.211662i
$16$	−17.9428			−0.280357
$17$			79.4618i			1.13367i	0.823833	+	0.566833i	$0.191832\pi$
$17$			79.4618i			1.13367i	−0.823833	+	0.566833i	$0.808168\pi$
$18$		−	117.887i		−	1.54368i
$19$		−	108.218i		−	1.30668i	−0.757065	−	0.653340i	$-0.773367\pi$
$19$		−	108.218i		−	1.30668i	0.757065	−	0.653340i	$-0.226633\pi$
$20$	195.392			2.18455
$21$	−14.9744			−0.155604
$22$		−	88.2586i		−	0.855309i
$23$	−140.315			−1.27207			−0.636037	−	0.771659i	$-0.719427\pi$
$23$	−140.315			−1.27207			−0.636037	+	0.771659i	$0.719427\pi$
$24$	−12.9226			−0.109909
$25$	145.617			1.16494
$26$		−	239.922i		−	1.80972i
$27$			39.9465i			0.284730i
$28$		−	237.946i		−	1.60598i
$29$			32.4595i			0.207847i	0.994585	+	0.103924i	$0.0331398\pi$
$29$			32.4595i			0.207847i	−0.994585	+	0.103924i	$0.966860\pi$
$30$	−54.8228			−0.333641
$31$		−	174.877i		−	1.01319i	−0.862185	−	0.506593i	$-0.830905\pi$
$31$		−	174.877i		−	1.01319i	0.862185	−	0.506593i	$-0.169095\pi$
$32$			218.302i			1.20596i
$33$			14.7971i			0.0780558i
$34$	354.275			1.78699
$35$		−	329.553i		−	1.59156i
$36$	−314.060			−1.45398
$37$	391.562			1.73980			0.869898	−	0.493232i	$-0.164185\pi$
$37$	391.562			1.73980			0.869898	+	0.493232i	$0.164185\pi$
$38$	−482.483			−2.05971
$39$			40.2244i			0.165156i
$40$		−	284.398i		−	1.12418i
$41$	234.544			0.893404			0.446702	−	0.894683i	$-0.352598\pi$
$41$	234.544			0.893404			0.446702	+	0.894683i	$0.352598\pi$
$42$			66.7623i			0.245277i
$43$	458.267			1.62523			0.812616	−	0.582799i	$-0.198043\pi$
$43$	458.267			1.62523			0.812616	+	0.582799i	$0.198043\pi$
$44$	−235.128			−0.805611
$45$	−434.971			−1.44092
$46$			625.585i			2.00516i
$47$	522.533			1.62169			0.810844	−	0.585263i	$-0.199009\pi$
$47$	522.533			1.62169			0.810844	+	0.585263i	$0.199009\pi$
$48$		−	13.4120i		−	0.0403303i
$49$	−58.3240			−0.170041
$50$		−	649.224i		−	1.83628i
$51$	−59.3964			−0.163082
$52$	−639.173			−1.70457
$53$	−235.184			−0.609528			−0.304764	−	0.952428i	$-0.598578\pi$
$53$	−235.184			−0.609528			−0.304764	+	0.952428i	$0.598578\pi$
$54$	178.099			0.448818
$55$	−325.651			−0.798377
$56$	−346.336			−0.826447
$57$	80.8912			0.187970
$58$	144.718			0.327629
$59$			714.231i			1.57602i	0.615665	+	0.788008i	$0.288888\pi$
$59$			714.231i			1.57602i	−0.615665	+	0.788008i	$0.711112\pi$
$60$			146.053i			0.314255i
$61$	541.701			1.13701			0.568506	−	0.822679i	$-0.307522\pi$
$61$	541.701			1.13701			0.568506	+	0.822679i	$0.307522\pi$
$62$	−779.675			−1.59708
$63$			529.700i			1.05930i
$64$	829.745			1.62059
$65$	−885.250			−1.68926
$66$	65.9718			0.123039
$67$	−410.309			−0.748167			−0.374083	−	0.927395i	$-0.622043\pi$
$67$	−410.309			−0.748167			−0.374083	+	0.927395i	$0.622043\pi$
$68$		−	943.819i		−	1.68316i
$69$		−	104.883i		−	0.182992i
$70$	−1469.29			−2.50877
$71$		−	1040.22i		−	1.73875i	−0.494151	−	0.869376i	$-0.664521\pi$
$71$		−	1040.22i		−	1.73875i	0.494151	−	0.869376i	$-0.335479\pi$
$72$			457.122i			0.748226i
$73$	−284.957			−0.456873			−0.228436	−	0.973559i	$-0.573361\pi$
$73$	−284.957			−0.456873			−0.228436	+	0.973559i	$0.573361\pi$
$74$		−	1745.75i		−	2.74243i
$75$			108.846i			0.167580i
$76$			1285.37i			1.94003i
$77$			396.572i			0.586930i
$78$	179.338			0.260334
$79$	1066.64			1.51906			0.759531	−	0.650472i	$-0.225429\pi$
$79$	1066.64			1.51906			0.759531	+	0.650472i	$0.225429\pi$
$80$	295.168			0.412509
$81$	684.055			0.938347
$82$		−	1045.70i		−	1.40827i
$83$			1201.98i			1.58957i	0.606890	+	0.794786i	$0.292417\pi$
$83$			1201.98i			1.58957i	−0.606890	+	0.794786i	$0.707583\pi$
$84$	177.860			0.231026
$85$		−	1307.18i		−	1.66805i
$86$		−	2043.15i		−	2.56185i
$87$	−24.2629			−0.0298995
$88$			342.235i			0.414572i
$89$	−884.858			−1.05387			−0.526937	−	0.849904i	$-0.676660\pi$
$89$	−884.858			−1.05387			−0.526937	+	0.849904i	$0.676660\pi$
$90$			1939.29i			2.27132i
$91$			1078.04i			1.24186i
$92$	1666.61			1.88865
$93$	130.717			0.145750
$94$		−	2329.68i		−	2.55626i
$95$			1780.23i			1.92261i
$96$	−163.177			−0.173482
$97$	−738.008			−0.772509			−0.386254	−	0.922392i	$-0.626231\pi$
$97$	−738.008			−0.772509			−0.386254	+	0.922392i	$0.626231\pi$
$98$			260.034i			0.268034i
$99$	523.428			0.531379
$100$	−1729.59			−1.72959
$101$			818.116i			0.805996i	0.915201	+	0.402998i	$0.132032\pi$
$101$			818.116i			0.805996i	−0.915201	+	0.402998i	$0.867968\pi$
$102$			264.815i			0.257065i
$103$	159.594			0.152672			0.0763361	−	0.997082i	$-0.475678\pi$
$103$	159.594			0.152672			0.0763361	+	0.997082i	$0.475678\pi$
$104$			930.332i			0.877179i
$105$	246.335			0.228951
$106$			1048.55i			0.960796i
$107$		−	561.611i		−	0.507411i	−0.967281	−	0.253706i	$-0.918351\pi$
$107$		−	561.611i		−	0.507411i	0.967281	−	0.253706i	$-0.0816494\pi$
$108$		−	474.470i		−	0.422739i
$109$		−	1021.01i		−	0.897206i	−0.893731	−	0.448603i	$-0.851922\pi$
$109$		−	1021.01i		−	0.897206i	0.893731	−	0.448603i	$-0.148078\pi$
$110$			1451.89i			1.25848i
$111$			292.686i			0.250275i
$112$		−	359.450i		−	0.303258i
$113$			2232.65i			1.85867i	0.369236	+	0.929336i	$0.379619\pi$
$113$			2232.65i			1.85867i	−0.369236	+	0.929336i	$0.620381\pi$
$114$		−	360.648i		−	0.296296i
$115$	2308.24			1.87169
$116$		−	385.542i		−	0.308592i
$117$	1422.89			1.12433
$118$	3184.35			2.48427
$119$	−1591.86			−1.22627
$120$	212.583			0.161717
$121$	−939.124			−0.705577
$122$		−	2415.14i		−	1.79227i
$123$			175.318i			0.128519i
$124$			2077.12i			1.50428i
$125$	−339.159			−0.242682
$126$	2361.63			1.66977
$127$	324.230			0.226541			0.113271	−	0.993564i	$-0.463867\pi$
$127$	324.230			0.226541			0.113271	+	0.993564i	$0.463867\pi$
$128$		−	1952.94i		−	1.34857i
$129$			342.547i			0.233795i
$130$			3946.83i			2.66277i
$131$	655.006			0.436856			0.218428	−	0.975853i	$-0.429907\pi$
$131$	655.006			0.436856			0.218428	+	0.975853i	$0.429907\pi$
$132$		−	175.754i		−	0.115890i
$133$	2167.94			1.41341
$134$			1829.33i			1.17933i
$135$		−	657.137i		−	0.418943i
$136$	−1373.75			−0.866163
$137$			1003.06i			0.625527i	0.949831	+	0.312763i	$0.101255\pi$
$137$			1003.06i			0.625527i	−0.949831	+	0.312763i	$0.898745\pi$
$138$	−467.614			−0.288449
$139$			1487.82i			0.907880i	0.891032	+	0.453940i	$0.149982\pi$
$139$			1487.82i			0.907880i	−0.891032	+	0.453940i	$0.850018\pi$
$140$			3914.31i			2.36300i
$141$			390.585i			0.233285i
$142$	−4637.75			−2.74079
$143$	1065.28			0.622958
$144$	−474.432			−0.274555
$145$		−	533.973i		−	0.305821i
$146$			1270.46i			0.720166i
$147$		−	43.5962i		−	0.0244609i
$148$	−4650.83			−2.58308
$149$	−1381.96			−0.759830			−0.379915	−	0.925021i	$-0.624047\pi$
$149$	−1381.96			−0.759830			−0.379915	+	0.925021i	$0.624047\pi$
$150$	485.284			0.264155
$151$	632.461			0.340854			0.170427	−	0.985370i	$-0.445485\pi$
$151$	632.461			0.340854			0.170427	+	0.985370i	$0.445485\pi$
$152$	1870.89			0.998352
$153$			2101.07i			1.11021i
$154$	1768.09			0.925174
$155$			2876.80i			1.49077i
$156$		−	477.771i		−	0.245207i
$157$			555.403i			0.282331i	0.989986	+	0.141166i	$0.0450850\pi$
$157$			555.403i			0.282331i	−0.989986	+	0.141166i	$0.954915\pi$
$158$		−	4755.52i		−	2.39449i
$159$		−	175.796i		−	0.0876826i
$160$		−	3591.17i		−	1.77442i
$161$		−	2810.94i		−	1.37598i
$162$		−	3049.81i		−	1.47911i
$163$			211.438i			0.101602i	0.998709	+	0.0508009i	$0.0161774\pi$
$163$			211.438i			0.101602i	−0.998709	+	0.0508009i	$0.983823\pi$
$164$	−2785.83			−1.32644
$165$		−	243.419i		−	0.114849i
$166$	5358.95			2.50563
$167$		−	3219.33i		−	1.49173i	−0.666096	−	0.745866i	$-0.732036\pi$
$167$		−	3219.33i		−	1.49173i	0.666096	−	0.745866i	$-0.267964\pi$
$168$		−	258.880i		−	0.118887i
$169$	698.857			0.318096
$170$	−5827.99			−2.62933
$171$		−	2861.42i		−	1.27964i
$172$	−5443.12			−2.41299
$173$	1481.90			0.651253			0.325626	−	0.945498i	$-0.394425\pi$
$173$	1481.90			0.651253			0.325626	+	0.945498i	$0.394425\pi$
$174$			108.175i			0.0471305i
$175$			2917.16i			1.26009i
$176$	−355.194			−0.152124
$177$	−533.876			−0.226715
$178$			3945.08i			1.66122i
$179$			684.319i			0.285745i	0.989741	+	0.142873i	$0.0456339\pi$
$179$			684.319i			0.285745i	−0.989741	+	0.142873i	$0.954366\pi$
$180$	5166.42			2.13935
$181$		−	2806.64i		−	1.15258i	−0.817247	−	0.576288i	$-0.804501\pi$
$181$		−	2806.64i		−	1.15258i	0.817247	−	0.576288i	$-0.195499\pi$
$182$	4806.38			1.95754
$183$			404.913i			0.163563i
$184$		−	2425.79i		−	0.971912i
$185$	−6441.37			−2.55989
$186$		−	582.794i		−	0.229745i
$187$			1573.02i			0.615136i
$188$	−6206.46			−2.40773
$189$	−800.250			−0.307988
$190$	7937.06			3.03060
$191$	−2094.78			−0.793575			−0.396787	−	0.917911i	$-0.629875\pi$
$191$	−2094.78			−0.793575			−0.396787	+	0.917911i	$0.629875\pi$
$192$			620.220i			0.233128i
$193$		−	3695.29i		−	1.37820i	−0.724667	−	0.689100i	$-0.758006\pi$
$193$		−	3695.29i		−	1.37820i	0.724667	−	0.689100i	$-0.241994\pi$
$194$			3290.36i			1.21770i
$195$		−	661.710i		−	0.243005i
$196$	692.751			0.252460
$197$		−	2454.57i		−	0.887720i	−0.896096	−	0.443860i	$-0.853609\pi$
$197$		−	2454.57i		−	0.887720i	0.896096	−	0.443860i	$-0.146391\pi$
$198$		−	2333.67i		−	0.837609i
$199$	1001.24			0.356663			0.178331	−	0.983970i	$-0.442930\pi$
$199$	1001.24			0.356663			0.178331	+	0.983970i	$0.442930\pi$
$200$			2517.45i			0.890054i
$201$		−	306.699i		−	0.107626i
$202$	3647.52			1.27049
$203$	−650.263			−0.224825
$204$	705.489			0.242128
$205$	−3858.35			−1.31453
$206$		−	711.539i		−	0.240656i
$207$	−3710.11			−1.24575
$208$	−965.561			−0.321873
$209$		−	2142.27i		−	0.709014i
$210$		−	1098.27i		−	0.360894i
$211$	−2893.77			−0.944149			−0.472074	−	0.881559i	$-0.656495\pi$
$211$	−2893.77			−0.944149			−0.472074	+	0.881559i	$0.656495\pi$
$212$	2793.43			0.904969
$213$	777.547			0.250125
$214$	−2503.91			−0.799829
$215$	−7538.69			−2.39132
$216$	−690.602			−0.217544
$217$	3503.31			1.09595
$218$	−4552.13			−1.41426
$219$		−	213.001i		−	0.0657226i
$220$	3867.96			1.18535
$221$			4276.09i			1.30154i
$222$	1304.92			0.394508
$223$			4328.16i			1.29971i	0.760058	+	0.649855i	$0.225170\pi$
$223$			4328.16i			1.29971i	−0.760058	+	0.649855i	$0.774830\pi$
$224$	−4373.27			−1.30447
$225$	3850.30			1.14083
$226$	9954.11			2.92981
$227$		−	4113.13i		−	1.20263i	−0.799011	−	0.601317i	$-0.794643\pi$
$227$		−	4113.13i		−	1.20263i	0.799011	−	0.601317i	$-0.205357\pi$
$228$	−960.796			−0.279080
$229$			2704.96i			0.780563i	0.920696	+	0.390282i	$0.127622\pi$
$229$			2704.96i			0.780563i	−0.920696	+	0.390282i	$0.872378\pi$
$230$		−	10291.2i		−	2.95034i
$231$	−296.431			−0.0844317
$232$	−561.166			−0.158803
$233$	1882.39			0.529269			0.264635	−	0.964349i	$-0.414749\pi$
$233$	1882.39			0.529269			0.264635	+	0.964349i	$0.414749\pi$
$234$		−	6343.85i		−	1.77227i
$235$	−8595.90			−2.38611
$236$		−	8483.38i		−	2.33992i
$237$			797.292i			0.218522i
$238$			7097.23i			1.93296i
$239$	2075.90			0.561835			0.280918	−	0.959732i	$-0.409361\pi$
$239$	2075.90			0.561835			0.280918	+	0.959732i	$0.409361\pi$
$240$			220.633i			0.0593408i
$241$			7100.72i			1.89792i	0.315404	+	0.948958i	$0.397860\pi$
$241$			7100.72i			1.89792i	−0.315404	+	0.948958i	$0.602140\pi$
$242$			4187.02i			1.11220i
$243$			1589.87i			0.419714i
$244$	−6434.13			−1.68813
$245$	959.455			0.250193
$246$	781.642			0.202584
$247$		−	5823.55i		−	1.50018i
$248$	3023.30			0.774111
$249$	−898.460			−0.228665
$250$			1512.12i			0.382539i
$251$			424.166i			0.106666i	0.998577	+	0.0533330i	$0.0169845\pi$
$251$			424.166i			0.106666i	−0.998577	+	0.0533330i	$0.983016\pi$
$252$		−	6291.58i		−	1.57275i
$253$	−2777.66			−0.690236
$254$		−	1445.56i		−	0.357096i
$255$	977.097			0.239954
$256$	−2069.11			−0.505153
$257$		−	1365.07i		−	0.331326i	−0.986182	−	0.165663i	$-0.947024\pi$
$257$		−	1365.07i		−	0.331326i	0.986182	−	0.165663i	$-0.0529764\pi$
$258$	1527.22			0.368530
$259$			7844.20i			1.88191i
$260$	10514.7			2.50805
$261$			858.270i			0.203546i
$262$		−	2920.30i		−	0.688614i
$263$	2909.24			0.682096			0.341048	−	0.940046i	$-0.389218\pi$
$263$	2909.24			0.682096			0.341048	+	0.940046i	$0.389218\pi$
$264$	−255.815			−0.0596375
$265$	3868.88			0.896843
$266$		−	9665.62i		−	2.22796i
$267$		−	661.417i		−	0.151603i
$268$	4873.50			1.11081
$269$	756.176	+	4346.64i	0.171393	+	0.985203i
$269$	756.176	+	4346.64i	0.171393	+	0.985203i
$270$	−2929.80			−0.660378
$271$		−	4670.81i		−	1.04698i	−0.852032	−	0.523490i	$-0.824630\pi$
$271$		−	4670.81i		−	1.04698i	0.852032	−	0.523490i	$-0.175370\pi$
$272$		−	1425.77i		−	0.317831i
$273$	−805.819			−0.178646
$274$	4472.08			0.986015
$275$	2882.62			0.632103
$276$			1245.76i			0.271689i
$277$		−	2417.22i		−	0.524320i	−0.965024	−	0.262160i	$-0.915565\pi$
$277$		−	2417.22i		−	0.524320i	0.965024	−	0.262160i	$-0.0844348\pi$
$278$	6633.35			1.43109
$279$		−	4623.96i		−	0.992219i
$280$	5697.37			1.21601
$281$			2877.15i			0.610805i	0.952223	+	0.305402i	$0.0987910\pi$
$281$			2877.15i			0.610805i	−0.952223	+	0.305402i	$0.901209\pi$
$282$	1741.40			0.367726
$283$	5652.21			1.18724			0.593620	−	0.804745i	$-0.297698\pi$
$283$	5652.21			1.18724			0.593620	+	0.804745i	$0.297698\pi$
$284$			12355.4i			2.58153i
$285$	−1330.70			−0.276574
$286$		−	4749.47i		−	0.981966i
$287$			4698.63i			0.966381i
$288$			5772.19i			1.18101i
$289$	−1401.19			−0.285200
$290$	−2380.68			−0.482064
$291$		−	551.649i		−	0.111128i
$292$	3384.62			0.678321
$293$	4767.45			0.950571			0.475285	−	0.879832i	$-0.342345\pi$
$293$	4767.45			0.950571			0.475285	+	0.879832i	$0.342345\pi$
$294$	−194.371			−0.0385576
$295$		−	11749.4i		−	2.31891i
$296$			6769.40i			1.32927i
$297$			790.775i			0.154496i
$298$			6161.38i			1.19772i
$299$	−7550.79			−1.46045
$300$		−	1292.84i		−	0.248807i
$301$			9180.49i			1.75799i
$302$		−	2819.78i		−	0.537286i
$303$	−611.528			−0.115945
$304$			1941.74i			0.366337i
$305$	−8911.22			−1.67297
$306$	9367.49			1.75001
$307$	7206.97			1.33982			0.669908	−	0.742444i	$-0.266334\pi$
$307$	7206.97			1.33982			0.669908	+	0.742444i	$0.266334\pi$
$308$		−	4710.34i		−	0.871417i
$309$			119.294i			0.0219624i
$310$	12826.0			2.34990
$311$		−	474.044i		−	0.0864326i	−0.999066	−	0.0432163i	$-0.986240\pi$
$311$		−	474.044i		−	0.0864326i	0.999066	−	0.0432163i	$-0.0137605\pi$
$312$	−695.408			−0.126185
$313$	−9439.98			−1.70473			−0.852363	−	0.522951i	$-0.824831\pi$
$313$	−9439.98			−1.70473			−0.852363	+	0.522951i	$0.824831\pi$
$314$	2476.23			0.445037
$315$		−	8713.80i		−	1.55862i
$316$	−12669.1			−2.25536
$317$		−	8030.93i		−	1.42291i	−0.702733	−	0.711454i	$-0.748037\pi$
$317$		−	8030.93i		−	1.42291i	0.702733	−	0.711454i	$-0.251963\pi$
$318$	−783.775			−0.138214
$319$			642.564i			0.112779i
$320$	−13649.7			−2.38450
$321$	419.795			0.0729927
$322$	−12532.4			−2.16895
$323$	8599.20			1.48134
$324$	−8124.96			−1.39317
$325$	7836.11			1.33744
$326$	942.683			0.160155
$327$	763.192			0.129066
$328$			4054.84i			0.682594i
$329$			10467.9i			1.75415i
$330$	−1085.27			−0.181036
$331$	−11177.9			−1.85618			−0.928089	−	0.372358i	$-0.878549\pi$
$331$	−11177.9			−1.85618			−0.928089	+	0.372358i	$0.878549\pi$
$332$		−	14276.7i		−	2.36004i
$333$	10353.4			1.70379
$334$	−14353.2			−2.35141
$335$	6749.76			1.10083
$336$	268.683			0.0436246
$337$			1348.61i			0.217993i	0.994042	+	0.108997i	$0.0347637\pi$
$337$			1348.61i			0.217993i	−0.994042	+	0.108997i	$0.965236\pi$
$338$		−	3115.81i		−	0.501413i
$339$	−1668.87			−0.267376
$340$			15526.2i			2.47655i
$341$		−	3461.83i		−	0.549762i
$342$	−12757.5			−2.01709
$343$			5702.94i			0.897754i
$344$			7922.60i			1.24174i
$345$			1725.37i			0.269249i
$346$		−	6606.95i		−	1.02657i
$347$	3923.90			0.607049			0.303525	−	0.952824i	$-0.401837\pi$
$347$	3923.90			0.607049			0.303525	+	0.952824i	$0.401837\pi$
$348$	288.186			0.0443920
$349$	−578.456			−0.0887222			−0.0443611	−	0.999016i	$-0.514125\pi$
$349$	−578.456			−0.0887222			−0.0443611	+	0.999016i	$0.514125\pi$
$350$	13005.9			1.98628
$351$			2149.65i			0.326893i
$352$			4321.49i			0.654364i
$353$	−3773.75			−0.568999			−0.284500	−	0.958676i	$-0.591827\pi$
$353$	−3773.75			−0.568999			−0.284500	+	0.958676i	$0.591827\pi$
$354$			2380.25i			0.357370i
$355$			17112.1i			2.55835i
$356$	10510.0			1.56469
$357$		−	1189.89i		−	0.176403i
$358$	3050.99			0.450418
$359$		−	3579.15i		−	0.526185i	−0.964771	−	0.263093i	$-0.915258\pi$
$359$		−	3579.15i		−	0.526185i	0.964771	−	0.263093i	$-0.0847425\pi$
$360$		−	7519.85i		−	1.10092i
$361$	−4852.14			−0.707412
$362$	−12513.2			−1.81680
$363$		−	701.979i		−	0.101500i
$364$		−	12804.6i		−	1.84380i
$365$	4687.67			0.672230
$366$	1805.28			0.257823
$367$		−	10255.3i		−	1.45864i	−0.684171	−	0.729321i	$-0.739836\pi$
$367$		−	10255.3i		−	1.45864i	0.684171	−	0.729321i	$-0.260164\pi$
$368$	2517.65			0.356635
$369$	6201.63			0.874916
$370$			28718.4i			4.03514i
$371$		−	4711.46i		−	0.659317i
$372$	−1552.61			−0.216396
$373$			7723.23i			1.07210i	0.844186	+	0.536051i	$0.180084\pi$
$373$			7723.23i			1.07210i	−0.844186	+	0.536051i	$0.819916\pi$
$374$	7013.19			0.969635
$375$		−	253.516i		−	0.0349106i
$376$			9033.65i			1.23903i
$377$			1746.75i			0.238626i
$378$			3567.86i			0.485479i
$379$		−	1335.38i		−	0.180987i	−0.995897	−	0.0904934i	$-0.971156\pi$
$379$		−	1335.38i		−	0.180987i	0.995897	−	0.0904934i	$-0.0288444\pi$
$380$		−	21145.0i		−	2.85451i
$381$			242.356i			0.0325887i
$382$			9339.43i			1.25091i
$383$		−	6313.05i		−	0.842250i	−0.907003	−	0.421125i	$-0.861635\pi$
$383$		−	6313.05i		−	0.842250i	0.907003	−	0.421125i	$-0.138365\pi$
$384$	1459.79			0.193997
$385$		−	6523.79i		−	0.863592i
$386$	−16475.2			−2.17245
$387$	12117.1			1.59160
$388$	8765.79			1.14695
$389$	−12377.7			−1.61330			−0.806652	−	0.591027i	$-0.798723\pi$
$389$	−12377.7			−1.61330			−0.806652	+	0.591027i	$0.798723\pi$
$390$	−2950.19			−0.383048
$391$		−	11149.7i		−	1.44211i
$392$		−	1008.32i		−	0.129917i
$393$			489.606i			0.0628432i
$394$	−10943.5			−1.39931
$395$	−17546.6			−2.23510
$396$	−6217.09			−0.788940
$397$			13116.5i			1.65818i	0.559113	+	0.829091i	$0.311142\pi$
$397$			13116.5i			1.65818i	−0.559113	+	0.829091i	$0.688858\pi$
$398$		−	4463.95i		−	0.562205i
$399$			1620.50i			0.203324i
$400$	−2612.78			−0.326598
$401$			1509.68i			0.188004i	0.995572	+	0.0940022i	$0.0299661\pi$
$401$			1509.68i			0.188004i	−0.995572	+	0.0940022i	$0.970034\pi$
$402$	−1367.40			−0.169651
$403$		−	9410.66i		−	1.16322i
$404$		−	9717.29i		−	1.19667i
$405$	−11253.0			−1.38066
$406$			2899.16i			0.354391i
$407$	7751.32			0.944026
$408$		−	1026.86i		−	0.124600i
$409$		−	9736.34i		−	1.17709i	−0.808463	−	0.588546i	$-0.799700\pi$
$409$		−	9736.34i		−	1.17709i	0.808463	−	0.588546i	$-0.200300\pi$
$410$			17202.2i			2.07209i
$411$	−749.771			−0.0899841
$412$	−1895.60			−0.226673
$413$	−14308.2			−1.70475
$414$			16541.3i			1.96367i
$415$		−	19773.1i		−	2.33885i
$416$			11747.5i			1.38455i
$417$	−1112.12			−0.130602
$418$	−9551.17			−1.11761
$419$	1550.89			0.180826			0.0904128	−	0.995904i	$-0.471181\pi$
$419$	1550.89			0.180826			0.0904128	+	0.995904i	$0.471181\pi$
$420$	−2925.88			−0.339925
$421$	32.8012			0.00379722			0.00189861	−	0.999998i	$-0.499396\pi$
$421$	32.8012			0.00379722			0.00189861	+	0.999998i	$0.499396\pi$
$422$			12901.7i			1.48826i
$423$	13816.4			1.58813
$424$		−	4065.90i		−	0.465702i
$425$			11571.0i			1.32065i
$426$		−	3466.64i		−	0.394271i
$427$			10851.9i			1.22989i
$428$			6670.61i			0.753356i
$429$			796.278i			0.0896146i
$430$			33610.7i			3.76943i
$431$		−	5213.64i		−	0.582673i	−0.956621	−	0.291336i	$-0.905900\pi$
$431$		−	5213.64i		−	0.582673i	0.956621	−	0.291336i	$-0.0940999\pi$
$432$		−	716.753i		−	0.0798259i
$433$	−1156.14			−0.128315			−0.0641575	−	0.997940i	$-0.520436\pi$
$433$	−1156.14			−0.128315			−0.0641575	+	0.997940i	$0.520436\pi$
$434$		−	15619.3i		−	1.72753i
$435$	399.136			0.0439933
$436$			12127.2i			1.33209i
$437$			15184.6i			1.66219i
$438$	−949.649			−0.103598
$439$	388.981			0.0422895			0.0211447	−	0.999776i	$-0.493269\pi$
$439$	388.981			0.0422895			0.0211447	+	0.999776i	$0.493269\pi$
$440$		−	5629.91i		−	0.609990i
$441$	−1542.16			−0.166522
$442$	19064.7			2.05162
$443$			2199.62i			0.235908i	0.993019	+	0.117954i	$0.0376336\pi$
$443$			2199.62i			0.235908i	−0.993019	+	0.117954i	$0.962366\pi$
$444$		−	3476.42i		−	0.371585i
$445$	14556.3			1.55064
$446$	19296.8			2.04872
$447$		−	1032.99i		−	0.109304i
$448$			16622.3i			1.75297i
$449$	−10483.3			−1.10187			−0.550935	−	0.834548i	$-0.685729\pi$
$449$	−10483.3			−1.10187			−0.550935	+	0.834548i	$0.685729\pi$
$450$		−	17166.3i		−	1.79828i
$451$	4643.00			0.484768
$452$		−	26518.6i		−	2.75958i
$453$			472.754i			0.0490329i
$454$	−18338.1			−1.89570
$455$		−	17734.3i		−	1.82724i
$456$			1398.46i			0.143616i
$457$	−4680.29			−0.479069			−0.239535	−	0.970888i	$-0.576995\pi$
$457$	−4680.29			−0.479069			−0.239535	+	0.970888i	$0.576995\pi$
$458$	12059.9			1.23040
$459$	−3174.22			−0.322788
$460$	−27416.5			−2.77891
$461$			8646.51i			0.873554i	0.899570	+	0.436777i	$0.143880\pi$
$461$			8646.51i			0.873554i	−0.899570	+	0.436777i	$0.856120\pi$
$462$			1321.62i			0.133089i
$463$			2096.75i			0.210462i	0.994448	+	0.105231i	$0.0335582\pi$
$463$			2096.75i			0.210462i	−0.994448	+	0.105231i	$0.966442\pi$
$464$		−	582.415i		−	0.0582715i
$465$	−2150.36			−0.214453
$466$		−	8392.53i		−	0.834284i
$467$		−	9740.06i		−	0.965131i	−0.875860	−	0.482565i	$-0.839705\pi$
$467$		−	9740.06i		−	0.965131i	0.875860	−	0.482565i	$-0.160295\pi$
$468$	−16900.6			−1.66929
$469$		−	8219.74i		−	0.809280i
$470$			38324.3i			3.76121i
$471$	−415.155			−0.0406143
$472$	−12347.8			−1.20414
$473$	9071.78			0.881863
$474$	3554.67			0.344455
$475$		−	15758.4i		−	1.52220i
$476$	18907.6			1.82065
$477$	−6218.56			−0.596915
$478$		−	9255.25i		−	0.885618i
$479$		−	5090.11i		−	0.485539i	−0.970084	−	0.242769i	$-0.921944\pi$
$479$		−	5090.11i		−	0.485539i	0.970084	−	0.242769i	$-0.0780558\pi$
$480$	2684.34			0.255256
$481$	21071.2			1.99743
$482$	31658.1			2.99167
$483$	2101.13			0.197939
$484$	11154.6			1.04757
$485$	12140.6			1.13665
$486$	7088.35			0.661593
$487$	5839.88			0.543389			0.271694	−	0.962384i	$-0.412416\pi$
$487$	5839.88			0.543389			0.271694	+	0.962384i	$0.412416\pi$
$488$			9365.03i			0.868719i
$489$	−158.046			−0.0146158
$490$		−	4277.67i		−	0.394378i
$491$	−2212.59			−0.203366			−0.101683	−	0.994817i	$-0.532423\pi$
$491$	−2212.59			−0.203366			−0.101683	+	0.994817i	$0.532423\pi$
$492$		−	2082.36i		−	0.190813i
$493$	−2579.29			−0.235630
$494$	−25963.9			−2.36472
$495$	−8610.62			−0.781856
$496$			3137.78i			0.284054i
$497$	20838.8			1.88078
$498$			4005.73i			0.360444i
$499$		−	13306.1i		−	1.19371i	−0.802348	−	0.596856i	$-0.796416\pi$
$499$		−	13306.1i		−	1.19371i	0.802348	−	0.596856i	$-0.203584\pi$
$500$	4028.40			0.360311
$501$	2406.40			0.214590
$502$	1891.12			0.168137
$503$		−	12126.7i		−	1.07496i	−0.843277	−	0.537479i	$-0.819377\pi$
$503$		−	12126.7i		−	1.07496i	0.843277	−	0.537479i	$-0.180623\pi$
$504$	−9157.55			−0.809345
$505$		−	13458.4i		−	1.18592i
$506$			12384.0i			1.08802i
$507$			522.384i			0.0457591i
$508$	−3851.08			−0.336347
$509$			1572.21i			0.136910i	0.997654	+	0.0684549i	$0.0218069\pi$
$509$			1572.21i			0.136910i	−0.997654	+	0.0684549i	$0.978193\pi$
$510$		−	4356.32i		−	0.378238i
$511$		−	5708.57i		−	0.494192i
$512$		−	6398.56i		−	0.552303i
$513$	4322.93			0.372050
$514$	−6086.08			−0.522268
$515$	−2625.39			−0.224638
$516$		−	4068.64i		−	0.347116i
$517$	10344.0			0.879939
$518$	34972.8			2.96644
$519$			1107.70i			0.0936849i
$520$		−	15304.4i		−	1.29066i
$521$			4338.75i			0.364845i	0.983220	+	0.182422i	$0.0583938\pi$
$521$			4338.75i			0.364845i	−0.983220	+	0.182422i	$0.941606\pi$
$522$	3826.54			0.320849
$523$			21042.1i			1.75929i	0.475634	+	0.879644i	$0.342219\pi$
$523$			21042.1i			1.75929i	−0.475634	+	0.879644i	$0.657781\pi$
$524$	−7779.93			−0.648603
$525$	−2180.53			−0.181268
$526$		−	12970.6i		−	1.07518i
$527$	13896.0			1.14861
$528$		−	265.502i		−	0.0218835i
$529$	7521.29			0.618171
$530$		−	17249.1i		−	1.41369i
$531$			18885.2i			1.54340i
$532$	−25750.0			−2.09850
$533$	12621.5			1.02570
$534$	−2948.88			−0.238971
$535$			9238.75i			0.746591i
$536$		−	7093.49i		−	0.571627i
$537$	−511.517			−0.0411054
$538$	19379.2	−	3371.36i	1.55297	−	0.270167i
$539$	−1154.57			−0.0922653
$540$			7805.23i			0.622007i
$541$		−	12490.1i		−	0.992586i	−0.868155	−	0.496293i	$-0.834694\pi$
$541$		−	12490.1i		−	0.992586i	0.868155	−	0.496293i	$-0.165306\pi$
$542$	−20824.5			−1.65035
$543$	2097.92			0.165802
$544$	−17346.7			−1.36716
$545$			16796.1i			1.32012i
$546$			3592.69i			0.281599i
$547$	−22040.9			−1.72286			−0.861428	−	0.507881i	$-0.830429\pi$
$547$	−22040.9			−1.72286			−0.861428	+	0.507881i	$0.830429\pi$
$548$		−	11914.0i		−	0.928723i
$549$	14323.3			1.11348
$550$		−	12851.9i		−	0.996380i
$551$	3512.70			0.271590
$552$	1813.24			0.139813
$553$			21368.0i			1.64314i
$554$	−10777.0			−0.826483
$555$		−	4814.82i		−	0.368248i
$556$		−	17671.8i		−	1.34793i
$557$			1291.94i			0.0982790i	0.998792	+	0.0491395i	$0.0156479\pi$
$557$			1291.94i			0.0982790i	−0.998792	+	0.0491395i	$0.984352\pi$
$558$	−20615.6			−1.56403
$559$	24660.8			1.86590
$560$			5913.12i			0.446205i
$561$	−1175.80			−0.0884892
$562$	12827.6			0.962808
$563$	5542.17			0.414875			0.207438	−	0.978248i	$-0.433488\pi$
$563$	5542.17			0.414875			0.207438	+	0.978248i	$0.433488\pi$
$564$		−	4639.23i		−	0.346359i
$565$		−	36728.0i		−	2.73480i
$566$		−	25200.0i		−	1.87144i
$567$			13703.7i			1.01500i
$568$	17983.5			1.32847
$569$		−	5251.84i		−	0.386940i	−0.981106	−	0.193470i	$-0.938026\pi$
$569$		−	5251.84i		−	0.386940i	0.981106	−	0.193470i	$-0.0619742\pi$
$570$			5932.82i			0.435962i
$571$			10156.2i			0.744349i	0.928163	+	0.372174i	$0.121388\pi$
$571$			10156.2i			0.744349i	−0.928163	+	0.372174i	$0.878612\pi$
$572$	−12653.0			−0.924909
$573$		−	1565.81i		−	0.114158i
$574$	20948.5			1.52330
$575$	−20432.2			−1.48188
$576$	21939.5			1.58706
$577$		−	21691.8i		−	1.56507i	−0.622609	−	0.782533i	$-0.713927\pi$
$577$		−	21691.8i		−	1.56507i	0.622609	−	0.782533i	$-0.286073\pi$
$578$			6247.09i			0.449558i
$579$	2762.16			0.198258
$580$			6342.33i			0.454054i
$581$	−24079.4			−1.71941
$582$	−2459.49			−0.175170
$583$	−4655.67			−0.330734
$584$		−	4926.39i		−	0.349068i
$585$	−23407.1			−1.65430
$586$		−	21255.3i		−	1.49838i
$587$	23401.1			1.64543			0.822715	−	0.568454i	$-0.192458\pi$
$587$	23401.1			1.64543			0.822715	+	0.568454i	$0.192458\pi$
$588$			517.820i			0.0363172i
$589$	−18924.8			−1.32391
$590$	−52384.0			−3.65528
$591$	1834.75			0.127701
$592$	−7025.74			−0.487764
$593$	18192.5			1.25983			0.629913	−	0.776666i	$-0.283090\pi$
$593$	18192.5			1.25983			0.629913	+	0.776666i	$0.283090\pi$
$594$	3525.62			0.243532
$595$	26186.9			1.80430
$596$	16414.4			1.12812
$597$			748.409i			0.0513071i
$598$			33664.7i			2.30209i
$599$	−4207.80			−0.287022			−0.143511	−	0.989649i	$-0.545839\pi$
$599$	−4207.80			−0.287022			−0.143511	+	0.989649i	$0.545839\pi$
$600$	−1881.76			−0.128037
$601$			2855.62i			0.193816i	0.995293	+	0.0969079i	$0.0308952\pi$
$601$			2855.62i			0.193816i	−0.995293	+	0.0969079i	$0.969105\pi$
$602$	40930.6			2.77111
$603$	−10849.1			−0.732684
$604$	−7512.14			−0.506067
$605$	15449.0			1.03817
$606$			2726.46i			0.182764i
$607$			5016.25i			0.335426i	0.985836	+	0.167713i	$0.0536381\pi$
$607$			5016.25i			0.335426i	−0.985836	+	0.167713i	$0.946362\pi$
$608$	23624.3			1.57581
$609$		−	486.061i		−	0.0323418i
$610$			39730.1i			2.63709i
$611$	28119.2			1.86183
$612$		−	24955.8i		−	1.64833i
$613$			26975.8i			1.77739i	0.458496	+	0.888697i	$0.348388\pi$
$613$			26975.8i			1.77739i	−0.458496	+	0.888697i	$0.651612\pi$
$614$		−	32131.8i		−	2.11194i
$615$		−	2884.05i		−	0.189099i
$616$	−6856.01			−0.448436
$617$	−18376.7			−1.19906			−0.599530	−	0.800353i	$-0.704646\pi$
$617$	−18376.7			−1.19906			−0.599530	+	0.800353i	$0.704646\pi$
$618$	531.863			0.0346192
$619$	−15077.1			−0.978995			−0.489498	−	0.872005i	$-0.662820\pi$
$619$	−15077.1			−0.978995			−0.489498	+	0.872005i	$0.662820\pi$
$620$		−	34169.5i		−	2.21336i
$621$		−	5605.09i		−	0.362197i
$622$	−2113.49			−0.136243
$623$		−	17726.4i		−	1.13996i
$624$		−	721.741i		−	0.0463025i
$625$	−12622.8			−0.807860
$626$			42087.5i			2.68715i
$627$	1601.31			0.101994
$628$		−	6596.88i		−	0.419178i
$629$			31114.3i			1.97235i
$630$	−38849.9			−2.45685
$631$	11180.8			0.705390			0.352695	−	0.935738i	$-0.385265\pi$
$631$	11180.8			0.705390			0.352695	+	0.935738i	$0.385265\pi$
$632$			18440.2i			1.16062i
$633$		−	2163.05i		−	0.135819i
$634$	−35805.4			−2.24292
$635$	−5333.72			−0.333327
$636$			2088.04i			0.130183i
$637$	−3138.60			−0.195221
$638$	2864.83			0.177774
$639$		−	27504.7i		−	1.70277i
$640$			32126.8i			1.98425i
$641$	−5636.00			−0.347283			−0.173642	−	0.984809i	$-0.555553\pi$
$641$	−5636.00			−0.347283			−0.173642	+	0.984809i	$0.555553\pi$
$642$		−	1871.63i		−	0.115058i
$643$	−14324.8			−0.878559			−0.439279	−	0.898351i	$-0.644766\pi$
$643$	−14324.8			−0.878559			−0.439279	+	0.898351i	$0.644766\pi$
$644$			33387.3i			2.04293i
$645$		−	5635.04i		−	0.343999i
$646$		−	38339.0i		−	2.33503i
$647$		−	10616.8i		−	0.645114i	−0.946550	−	0.322557i	$-0.895458\pi$
$647$		−	10616.8i		−	0.645114i	0.946550	−	0.322557i	$-0.104542\pi$
$648$			11826.1i			0.716932i
$649$			14138.8i			0.855158i
$650$		−	34936.8i		−	2.10821i
$651$			2618.67i			0.157655i
$652$		−	2511.38i		−	0.150849i
$653$	−28168.6			−1.68809			−0.844044	−	0.536274i	$-0.819831\pi$
$653$	−28168.6			−1.68809			−0.844044	+	0.536274i	$0.819831\pi$
$654$		−	3402.64i		−	0.203446i
$655$	−10775.1			−0.642778
$656$	−4208.38			−0.250472
$657$	−7534.63			−0.447418
$658$	46670.7			2.76506
$659$	−26569.3			−1.57055			−0.785274	−	0.619148i	$-0.787478\pi$
$659$	−26569.3			−1.57055			−0.785274	+	0.619148i	$0.787478\pi$
$660$			2891.24i			0.170517i
$661$		−	8667.44i		−	0.510022i	−0.966938	−	0.255011i	$-0.917921\pi$
$661$		−	8667.44i		−	0.510022i	0.966938	−	0.255011i	$-0.0820790\pi$
$662$			49836.1i			2.92588i
$663$	−3196.31			−0.187231
$664$	−20780.1			−1.21449
$665$	−35663.6			−2.07966
$666$		−	46160.0i		−	2.68568i
$667$		−	4554.55i		−	0.264397i
$668$			38238.0i			2.21478i
$669$	−3235.23			−0.186967
$670$		−	30093.3i		−	1.73524i
$671$	10723.4			0.616951
$672$		−	3268.95i		−	0.187652i
$673$		−	12938.6i		−	0.741079i	−0.928817	−	0.370540i	$-0.879173\pi$
$673$		−	12938.6i		−	0.741079i	0.928817	−	0.370540i	$-0.120827\pi$
$674$	6012.71			0.343621
$675$			5816.88i			0.331692i
$676$	−8300.77			−0.472279
$677$			30000.7i			1.70313i	0.524249	+	0.851565i	$0.324346\pi$
$677$			30000.7i			1.70313i	−0.524249	+	0.851565i	$0.675654\pi$
$678$			7440.54i			0.421463i
$679$		−	14784.6i		−	0.835610i
$680$	22598.8			1.27445
$681$	3074.49			0.173003
$682$	−15434.4			−0.866586
$683$		−	3237.67i		−	0.181385i	−0.995879	−	0.0906925i	$-0.971092\pi$
$683$		−	3237.67i		−	0.181385i	0.995879	−	0.0906925i	$-0.0289080\pi$
$684$			33986.9i			1.89989i
$685$		−	16500.8i		−	0.920383i
$686$	25426.2			1.41512
$687$	−2021.92			−0.112287
$688$	−8222.60			−0.455645
$689$	−12656.0			−0.699789
$690$	7692.46			0.424416
$691$			25012.5i			1.37702i	0.725228	+	0.688508i	$0.241734\pi$
$691$			25012.5i			1.37702i	−0.725228	+	0.688508i	$0.758266\pi$
$692$	−17601.5			−0.966918
$693$			10485.9i			0.574784i
$694$		−	17494.5i		−	0.956889i
$695$		−	24475.3i		−	1.33583i
$696$		−	419.462i		−	0.0228443i
$697$			18637.3i			1.01282i
$698$			2579.01i			0.139852i
$699$			1407.06i			0.0761371i
$700$		−	34648.9i		−	1.87087i
$701$		−	24548.6i		−	1.32266i	−0.750093	−	0.661332i	$-0.769992\pi$
$701$		−	24548.6i		−	1.32266i	0.750093	−	0.661332i	$-0.230008\pi$
$702$	9584.05			0.515280
$703$		−	42374.1i		−	2.27336i
$704$	16425.5			0.879346
$705$		−	6425.30i		−	0.343249i
$706$			16825.0i			0.896910i
$707$	−16389.4			−0.871833
$708$	6341.18			0.336605
$709$		−	298.567i		−	0.0158151i	−0.999969	−	0.00790756i	$-0.997483\pi$
$709$		−	298.567i		−	0.0158151i	0.999969	−	0.00790756i	$-0.00251708\pi$
$710$	76293.1			4.03272
$711$	28203.2			1.48763
$712$		−	15297.6i		−	0.805199i
$713$			24537.8i			1.28885i
$714$	−5305.06			−0.278063
$715$	−17524.3			−0.916604
$716$		−	8128.09i		−	0.424247i
$717$			1551.70i			0.0808218i
$718$	−15957.4			−0.829423
$719$		−	21978.7i		−	1.14001i	−0.821641	−	0.570005i	$-0.806941\pi$
$719$		−	21978.7i		−	1.14001i	0.821641	−	0.570005i	$-0.193059\pi$
$720$	7804.61			0.403973
$721$			3197.15i			0.165143i
$722$			21632.9i			1.11509i
$723$	−5307.67			−0.273021
$724$			33336.3i			1.71123i
$725$			4726.65i			0.242129i
$726$	−3129.73			−0.159993
$727$	−8906.26			−0.454353			−0.227177	−	0.973854i	$-0.572949\pi$
$727$	−8906.26			−0.454353			−0.227177	+	0.973854i	$0.572949\pi$
$728$	−18637.4			−0.948830
$729$	17281.1			0.877970
$730$		−	20899.7i		−	1.05963i
$731$			36414.7i			1.84247i
$732$		−	4809.41i		−	0.242843i
$733$			12614.5i			0.635646i	0.948150	+	0.317823i	$0.102952\pi$
$733$			12614.5i			0.635646i	−0.948150	+	0.317823i	$0.897048\pi$
$734$	−45722.6			−2.29925
$735$			717.177i			0.0359911i
$736$		−	30631.1i		−	1.53407i
$737$	−8122.42			−0.405961
$738$		−	27649.6i		−	1.37913i
$739$		−	9217.46i		−	0.458823i	−0.973330	−	0.229411i	$-0.926320\pi$
$739$		−	9217.46i		−	0.458823i	0.973330	−	0.229411i	$-0.0736801\pi$
$740$	76508.3			3.80068
$741$	4353.01			0.215805
$742$	−21005.7			−1.03928
$743$	21880.3			1.08037			0.540183	−	0.841547i	$-0.318355\pi$
$743$	21880.3			1.08037			0.540183	+	0.841547i	$0.318355\pi$
$744$			2259.87i			0.111358i
$745$	22733.9			1.11799
$746$	34433.5			1.68995
$747$			31781.9i			1.55668i
$748$		−	18683.7i		−	0.913295i
$749$	11250.8			0.548859
$750$	−1130.28			−0.0550294
$751$	−29522.4			−1.43447			−0.717235	−	0.696832i	$-0.754592\pi$
$751$	−29522.4			−1.43447			−0.717235	+	0.696832i	$0.754592\pi$
$752$	−9375.73			−0.454651
$753$	−317.057			−0.0153442
$754$	7787.76			0.376145
$755$	−10404.3			−0.501523
$756$	9505.08			0.457271
$757$			25257.7i			1.21269i	0.795201	+	0.606346i	$0.207365\pi$
$757$			25257.7i			1.21269i	−0.795201	+	0.606346i	$0.792635\pi$
$758$	−5953.72			−0.285289
$759$		−	2076.25i		−	0.0992927i
$760$	−30777.0			−1.46895
$761$			6128.16i			0.291913i	0.989291	+	0.145956i	$0.0466259\pi$
$761$			6128.16i			0.291913i	−0.989291	+	0.145956i	$0.953374\pi$
$762$	1080.53			0.0513694
$763$	20454.1			0.970494
$764$	24881.0			1.17822
$765$		−	34563.6i		−	1.63353i
$766$	−28146.3			−1.32763
$767$			38435.0i			1.80940i
$768$		−	1546.62i		−	0.0726679i
$769$	38812.4			1.82004			0.910021	−	0.414562i	$-0.136065\pi$
$769$	38812.4			1.82004			0.910021	+	0.414562i	$0.136065\pi$
$770$	−29085.9			−1.36128
$771$	1020.37			0.0476623
$772$			43891.3i			2.04622i
$773$	−9178.07			−0.427053			−0.213527	−	0.976937i	$-0.568495\pi$
$773$	−9178.07			−0.427053			−0.213527	+	0.976937i	$0.568495\pi$
$774$		−	54023.5i		−	2.50883i
$775$		−	25465.0i		−	1.18030i
$776$		−	12758.8i		−	0.590225i
$777$	−5863.41			−0.270719
$778$			55185.2i			2.54304i
$779$		−	25381.9i		−	1.16739i
$780$			7859.55i			0.360791i
$781$		−	20592.1i		−	0.943459i
$782$	−49710.1			−2.27319
$783$	−1296.64			−0.0591803
$784$	1046.50			0.0476721
$785$		−	9136.63i		−	0.415414i
$786$	2182.88			0.0990594
$787$	29667.7			1.34376			0.671879	−	0.740661i	$-0.265487\pi$
$787$	29667.7			1.34376			0.671879	+	0.740661i	$0.265487\pi$
$788$			29154.5i			1.31800i
$789$			2174.61i			0.0981217i
$790$			78230.4i			3.52318i
$791$	−44726.8			−2.01050
$792$			9049.12i			0.405993i
$793$	29150.7			1.30538
$794$	58479.1			2.61378
$795$			2891.92i			0.129014i
$796$	−11892.3			−0.529539
$797$		−	9650.21i		−	0.428893i	−0.976736	−	0.214447i	$-0.931205\pi$
$797$		−	9650.21i		−	0.428893i	0.976736	−	0.214447i	$-0.0687948\pi$
$798$	7224.89			0.320499
$799$			41521.5i			1.83845i
$800$			31788.6i			1.40487i
$801$	−23396.8			−1.03207
$802$	6730.80			0.296350
$803$	−5640.97			−0.247902
$804$			3642.86i			0.159793i
$805$			46241.2i			2.02458i
$806$	−41956.8			−1.83358
$807$	−3249.04	+	565.229i	−0.141725	+	0.0246555i
$808$	−14143.7			−0.615811
$809$		−	6170.70i		−	0.268171i	−0.990970	−	0.134086i	$-0.957190\pi$
$809$		−	6170.70i		−	0.268171i	0.990970	−	0.134086i	$-0.0428097\pi$
$810$			50170.8i			2.17632i
$811$	27379.0			1.18546			0.592729	−	0.805402i	$-0.298050\pi$
$811$	27379.0			1.18546			0.592729	+	0.805402i	$0.298050\pi$
$812$	7723.59			0.333799
$813$	3491.35			0.150611
$814$		−	34558.7i		−	1.48806i
$815$		−	3478.25i		−	0.149494i
$816$	1065.74			0.0457211
$817$		−	49592.7i		−	2.12366i
$818$	−43408.8			−1.85545
$819$			28504.8i			1.21617i
$820$	45828.0			1.95169
$821$	−20109.0			−0.854821			−0.427410	−	0.904058i	$-0.640574\pi$
$821$	−20109.0			−0.854821			−0.427410	+	0.904058i	$0.640574\pi$
$822$			3342.80i			0.141841i
$823$	−17328.3			−0.733931			−0.366965	−	0.930235i	$-0.619603\pi$
$823$	−17328.3			−0.733931			−0.366965	+	0.930235i	$0.619603\pi$
$824$			2759.09i			0.116647i
$825$			2154.71i			0.0909300i
$826$			63792.4i			2.68719i
$827$	11441.6			0.481094			0.240547	−	0.970637i	$-0.422673\pi$
$827$	11441.6			0.481094			0.240547	+	0.970637i	$0.422673\pi$
$828$	44067.3			1.84957
$829$			18350.4i			0.768799i	0.923167	+	0.384400i	$0.125591\pi$
$829$			18350.4i			0.768799i	−0.923167	+	0.384400i	$0.874409\pi$
$830$	−88157.1			−3.68672
$831$	1806.83			0.0754251
$832$	44651.2			1.86058
$833$		−	4634.53i		−	0.192770i
$834$			4958.32i			0.205866i
$835$			52959.4i			2.19489i
$836$			25445.1i			1.05268i
$837$	6985.70			0.288484
$838$		−	6914.54i		−	0.285034i
$839$			7854.35i			0.323197i	0.986857	+	0.161599i	$0.0516650\pi$
$839$			7854.35i			0.323197i	−0.986857	+	0.161599i	$0.948335\pi$
$840$			4258.69i			0.174927i
$841$	23335.4			0.956799
$842$		−	146.242i		−	0.00598554i
$843$	−2150.62			−0.0878662
$844$	34371.2			1.40178
$845$	−11496.5			−0.468038
$846$		−	61599.7i		−	2.50336i
$847$		−	18813.5i		−	0.763212i
$848$	4219.87			0.170885
$849$			4224.94i			0.170788i
$850$	51588.5			2.08173
$851$	−54942.1			−2.21315
$852$	−9235.42			−0.371362
$853$		−	15100.2i		−	0.606119i	−0.952972	−	0.303059i	$-0.901992\pi$
$853$		−	15100.2i		−	0.606119i	0.952972	−	0.303059i	$-0.0980081\pi$
$854$	48382.7			1.93867
$855$			47071.6i			1.88283i
$856$	9709.24			0.387681
$857$			10317.9i			0.411264i	0.978629	+	0.205632i	$0.0659249\pi$
$857$			10317.9i			0.411264i	−0.978629	+	0.205632i	$0.934075\pi$
$858$	3550.15			0.141259
$859$	−1666.85			−0.0662073			−0.0331037	−	0.999452i	$-0.510539\pi$
$859$	−1666.85			−0.0662073			−0.0331037	+	0.999452i	$0.510539\pi$
$860$	89541.8			3.55041
$861$	−3512.15			−0.139017
$862$	−23244.7			−0.918464
$863$	7426.25			0.292923			0.146462	−	0.989216i	$-0.453212\pi$
$863$	7426.25			0.292923			0.146462	+	0.989216i	$0.453212\pi$
$864$	−8720.41			−0.343373
$865$	−24377.9			−0.958235
$866$			5154.56i			0.202262i
$867$		−	1047.36i		−	0.0410269i
$868$	−41611.1			−1.62716
$869$	21115.0			0.824253
$870$		−	1779.52i		−	0.0693464i
$871$	−22080.0			−0.858958
$872$	17651.5			0.685499
$873$	−19513.9			−0.756522
$874$	67699.6			2.62011
$875$		−	6794.39i		−	0.262506i
$876$			2529.94i			0.0975787i
$877$	28401.2			1.09355			0.546773	−	0.837281i	$-0.315856\pi$
$877$	28401.2			1.09355			0.546773	+	0.837281i	$0.315856\pi$
$878$		−	1734.25i		−	0.0666606i
$879$			3563.59i			0.136743i
$880$	5843.10			0.223831
$881$		−	38213.7i		−	1.46135i	−0.682724	−	0.730676i	$-0.739205\pi$
$881$		−	38213.7i		−	1.46135i	0.682724	−	0.730676i	$-0.260795\pi$
$882$			6875.62i			0.262488i
$883$			1882.98i			0.0717637i	0.999356	+	0.0358819i	$0.0114240\pi$
$883$			1882.98i			0.0717637i	−0.999356	+	0.0358819i	$0.988576\pi$
$884$		−	50789.9i		−	1.93241i
$885$	8782.49			0.333582
$886$	9806.88			0.371861
$887$	−8735.76			−0.330686			−0.165343	−	0.986236i	$-0.552873\pi$
$887$	−8735.76			−0.330686			−0.165343	+	0.986236i	$0.552873\pi$
$888$	−5060.02			−0.191220
$889$			6495.32i			0.245046i
$890$		−	64898.4i		−	2.44427i
$891$	13541.5			0.509154
$892$		−	51408.3i		−	1.92968i
$893$		−	56547.5i		−	2.11903i
$894$	−4605.53			−0.172295
$895$		−	11257.3i		−	0.420437i
$896$	39123.4			1.45873
$897$		−	5644.09i		−	0.210090i
$898$			46739.3i			1.73687i
$899$	5676.40			0.210588
$900$	−45732.4			−1.69379
$901$		−	18688.2i		−	0.691002i
$902$		−	20700.5i		−	0.764136i
$903$	−6862.26			−0.252892
$904$	−38598.4			−1.42009
$905$			46170.5i			1.69587i
$906$	2107.74			0.0772903
$907$	−19862.0			−0.727131			−0.363565	−	0.931569i	$-0.618441\pi$
$907$	−19862.0			−0.727131			−0.363565	+	0.931569i	$0.618441\pi$
$908$			48854.2i			1.78556i
$909$			21632.0i			0.789317i
$910$	−79067.1			−2.88027
$911$			5577.05i			0.202828i	0.994844	+	0.101414i	$0.0323366\pi$
$911$			5577.05i			0.202828i	−0.994844	+	0.101414i	$0.967663\pi$
$912$	−1451.42			−0.0526987
$913$			23794.2i			0.862513i
$914$			20866.8i			0.755154i
$915$		−	6660.99i		−	0.240662i
$916$		−	32128.6i		−	1.15891i
$917$			13121.8i			0.472541i
$918$			14152.0i			0.508810i
$919$		−	13312.8i		−	0.477857i	−0.971037	−	0.238928i	$-0.923204\pi$
$919$		−	13312.8i		−	0.477857i	0.971037	−	0.238928i	$-0.0767961\pi$
$920$			39905.3i			1.43004i
$921$			5387.09i			0.192737i
$922$	38549.9			1.37698
$923$		−	55977.5i		−	1.99623i
$924$	3520.90			0.125356
$925$	57018.1			2.02675
$926$	9348.20			0.331751
$927$	4219.86			0.149513
$928$	−7085.99			−0.250656
$929$		−	48945.0i		−	1.72856i	−0.503011	−	0.864280i	$-0.667775\pi$
$929$		−	48945.0i		−	1.72856i	0.503011	−	0.864280i	$-0.332225\pi$
$930$			9587.23i			0.338040i
$931$			6311.71i			0.222189i
$932$	−22358.4			−0.785809
$933$	354.340			0.0124336
$934$	−43425.4			−1.52133
$935$		−	25876.8i		−	0.905093i
$936$			24599.2i			0.859026i
$937$		−	1925.82i		−	0.0671439i	−0.999436	−	0.0335720i	$-0.989312\pi$
$937$		−	1925.82i		−	0.0671439i	0.999436	−	0.0335720i	$-0.0106883\pi$
$938$	−36647.2			−1.27566
$939$		−	7056.22i		−	0.245230i
$940$	102099.			3.54266
$941$		−	6411.21i		−	0.222104i	−0.993815	−	0.111052i	$-0.964578\pi$
$941$		−	6411.21i		−	0.222104i	0.993815	−	0.111052i	$-0.0354220\pi$
$942$			1850.94i			0.0640200i
$943$	−32910.0			−1.13648
$944$		−	12815.3i		−	0.441847i
$945$	13164.5			0.453164
$946$		−	40446.0i		−	1.39008i
$947$		−	49838.0i		−	1.71015i	−0.518501	−	0.855077i	$-0.673510\pi$
$947$		−	49838.0i		−	1.71015i	0.518501	−	0.855077i	$-0.326490\pi$
$948$		−	9469.94i		−	0.324440i
$949$	−15334.4			−0.524528
$950$	−70257.7			−2.39943
$951$	6002.98			0.204690
$952$		−	27520.5i		−	0.936915i
$953$		−	1248.12i		−	0.0424244i	−0.999775	−	0.0212122i	$-0.993247\pi$
$953$		−	1248.12i		−	0.0424244i	0.999775	−	0.0212122i	$-0.00675256\pi$
$954$			27725.1i			0.940914i
$955$	34460.0			1.16764
$956$	−24656.8			−0.834160
$957$	−480.306			−0.0162237
$958$	−22693.9			−0.765352
$959$	−20094.4			−0.676623
$960$		−	10202.9i		−	0.343018i
$961$	−790.797			−0.0265448
$962$		−	93944.6i		−	3.14854i
$963$		−	14849.7i		−	0.496911i
$964$		−	84339.8i		−	2.81784i
$965$			60789.1i			2.02784i
$966$		−	9367.75i		−	0.312011i
$967$		−	1878.73i		−	0.0624775i	−0.999512	−	0.0312388i	$-0.990055\pi$
$967$		−	1878.73i		−	0.0624775i	0.999512	−	0.0312388i	$-0.00994523\pi$
$968$		−	16235.7i		−	0.539087i
$969$			6427.76i			0.213095i
$970$		−	54127.9i		−	1.79169i
$971$	1174.83			0.0388281			0.0194141	−	0.999812i	$-0.493820\pi$
$971$	1174.83			0.0388281			0.0194141	+	0.999812i	$0.493820\pi$
$972$		−	18883.9i		−	0.623151i
$973$	−29805.6			−0.982040
$974$		−	26036.7i		−	0.856541i
$975$			5857.36i			0.192396i
$976$	−9719.66			−0.318769
$977$	−18087.9			−0.592307			−0.296154	−	0.955140i	$-0.595704\pi$
$977$	−18087.9			−0.592307			−0.296154	+	0.955140i	$0.595704\pi$
$978$			704.640i			0.0230387i
$979$	−17516.5			−0.571840
$980$	−11396.1			−0.371463
$981$		−	26996.9i		−	0.878640i
$982$			9864.68i			0.320565i
$983$	32703.1			1.06111			0.530553	−	0.847652i	$-0.321984\pi$
$983$	32703.1			1.06111			0.530553	+	0.847652i	$0.321984\pi$
$984$	−3030.92			−0.0981934
$985$			40378.7i			1.30617i
$986$			11499.6i			0.371422i
$987$	−7824.62			−0.252341
$988$			69170.1i			2.22732i
$989$	−64301.7			−2.06742
$990$			38389.9i			1.23243i
$991$			25266.5i			0.809906i	0.914337	+	0.404953i	$0.132712\pi$
$991$			25266.5i			0.809906i	−0.914337	+	0.404953i	$0.867288\pi$
$992$	38176.0			1.22186
$993$		−	8355.32i		−	0.267017i
$994$		−	92908.4i		−	2.96466i
$995$	−16470.8			−0.524784
$996$	10671.6			0.339500
$997$	−14310.5			−0.454583			−0.227291	−	0.973827i	$-0.572987\pi$
$997$	−14310.5			−0.454583			−0.227291	+	0.973827i	$0.572987\pi$
$998$	−59324.3			−1.88164
$999$			15641.5i			0.495371i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	269.4.b.a.268.9	✓	66
269.268	even	2	inner	269.4.b.a.268.58	yes	66

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
269.4.b.a.268.9	✓	66	1.1	even	1	trivial
269.4.b.a.268.58	yes	66	269.268	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 269 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	269.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-4.45843i q^{2} +0.747483i q^{3} -11.8776 q^{4} -16.4504 q^{5} +3.33261 q^{6} +20.0331i q^{7} +17.2882i q^{8} +26.4413 q^{9} +73.3432i q^{10} +19.7959 q^{11} -8.87834i q^{12} +53.8132 q^{13} +89.3161 q^{14} -12.2964i q^{15} -17.9428 q^{16} +79.4618i q^{17} -117.887i q^{18} -108.218i q^{19} +195.392 q^{20} -14.9744 q^{21} -88.2586i q^{22} -140.315 q^{23} -12.9226 q^{24} +145.617 q^{25} -239.922i q^{26} +39.9465i q^{27} -237.946i q^{28} +32.4595i q^{29} -54.8228 q^{30} -174.877i q^{31} +218.302i q^{32} +14.7971i q^{33} +354.275 q^{34} -329.553i q^{35} -314.060 q^{36} +391.562 q^{37} -482.483 q^{38} +40.2244i q^{39} -284.398i q^{40} +234.544 q^{41} +66.7623i q^{42} +458.267 q^{43} -235.128 q^{44} -434.971 q^{45} +625.585i q^{46} +522.533 q^{47} -13.4120i q^{48} -58.3240 q^{49} -649.224i q^{50} -59.3964 q^{51} -639.173 q^{52} -235.184 q^{53} +178.099 q^{54} -325.651 q^{55} -346.336 q^{56} +80.8912 q^{57} +144.718 q^{58} +714.231i q^{59} +146.053i q^{60} +541.701 q^{61} -779.675 q^{62} +529.700i q^{63} +829.745 q^{64} -885.250 q^{65} +65.9718 q^{66} -410.309 q^{67} -943.819i q^{68} -104.883i q^{69} -1469.29 q^{70} -1040.22i q^{71} +457.122i q^{72} -284.957 q^{73} -1745.75i q^{74} +108.846i q^{75} +1285.37i q^{76} +396.572i q^{77} +179.338 q^{78} +1066.64 q^{79} +295.168 q^{80} +684.055 q^{81} -1045.70i q^{82} +1201.98i q^{83} +177.860 q^{84} -1307.18i q^{85} -2043.15i q^{86} -24.2629 q^{87} +342.235i q^{88} -884.858 q^{89} +1939.29i q^{90} +1078.04i q^{91} +1666.61 q^{92} +130.717 q^{93} -2329.68i q^{94} +1780.23i q^{95} -163.177 q^{96} -738.008 q^{97} +260.034i q^{98} +523.428 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 66 q - 258 q^{4} - 34 q^{5} + 48 q^{6} - 564 q^{9} + 22 q^{11} + 118 q^{13} + 60 q^{14} + 1030 q^{16} + 144 q^{20} - 64 q^{21} - 80 q^{23} - 778 q^{24} + 1676 q^{25} - 1146 q^{30} + 308 q^{34} + 2030 q^{36}+ \cdots - 3982 q^{99}+O(q^{100}) \) 66 * q - 258 * q^4 - 34 * q^5 + 48 * q^6 - 564 * q^9 + 22 * q^11 + 118 * q^13 + 60 * q^14 + 1030 * q^16 + 144 * q^20 - 64 * q^21 - 80 * q^23 - 778 * q^24 + 1676 * q^25 - 1146 * q^30 + 308 * q^34 + 2030 * q^36 + 470 * q^37 + 696 * q^38 + 76 * q^41 + 426 * q^43 + 698 * q^44 + 1254 * q^45 - 1280 * q^47 - 3502 * q^49 - 640 * q^51 - 1924 * q^52 - 618 * q^53 - 3766 * q^54 + 176 * q^55 + 3164 * q^56 + 1668 * q^57 - 2170 * q^58 - 1026 * q^61 + 516 * q^62 - 2936 * q^64 - 900 * q^65 - 5296 * q^66 + 862 * q^67 - 1776 * q^70 + 3560 * q^73 + 8818 * q^78 + 2064 * q^79 - 3966 * q^80 + 7446 * q^81 - 3304 * q^84 - 776 * q^87 - 892 * q^89 - 6402 * q^92 + 7388 * q^93 + 8632 * q^96 + 4552 * q^97 - 3982 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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