LMFDB - Embedded newform 390.2.a.h.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [390,2,Mod(1,390)] 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("390.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(390, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 390 = 2 \cdot 3 \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	390.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,2,2,2,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$3.11416567883$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	390.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} +2.82843 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -5.65685 q^{11} +1.00000 q^{12} -1.00000 q^{13} +2.82843 q^{14} +1.00000 q^{15} +1.00000 q^{16} -4.82843 q^{17} +1.00000 q^{18} -2.82843 q^{19} +1.00000 q^{20} +2.82843 q^{21} -5.65685 q^{22} +8.48528 q^{23} +1.00000 q^{24} +1.00000 q^{25} -1.00000 q^{26} +1.00000 q^{27} +2.82843 q^{28} -3.17157 q^{29} +1.00000 q^{30} +4.00000 q^{31} +1.00000 q^{32} -5.65685 q^{33} -4.82843 q^{34} +2.82843 q^{35} +1.00000 q^{36} -0.343146 q^{37} -2.82843 q^{38} -1.00000 q^{39} +1.00000 q^{40} +3.65685 q^{41} +2.82843 q^{42} -1.65685 q^{43} -5.65685 q^{44} +1.00000 q^{45} +8.48528 q^{46} -8.00000 q^{47} +1.00000 q^{48} +1.00000 q^{49} +1.00000 q^{50} -4.82843 q^{51} -1.00000 q^{52} -9.31371 q^{53} +1.00000 q^{54} -5.65685 q^{55} +2.82843 q^{56} -2.82843 q^{57} -3.17157 q^{58} -13.6569 q^{59} +1.00000 q^{60} +6.00000 q^{61} +4.00000 q^{62} +2.82843 q^{63} +1.00000 q^{64} -1.00000 q^{65} -5.65685 q^{66} -5.65685 q^{67} -4.82843 q^{68} +8.48528 q^{69} +2.82843 q^{70} +5.65685 q^{71} +1.00000 q^{72} +2.48528 q^{73} -0.343146 q^{74} +1.00000 q^{75} -2.82843 q^{76} -16.0000 q^{77} -1.00000 q^{78} +13.6569 q^{79} +1.00000 q^{80} +1.00000 q^{81} +3.65685 q^{82} +17.6569 q^{83} +2.82843 q^{84} -4.82843 q^{85} -1.65685 q^{86} -3.17157 q^{87} -5.65685 q^{88} -4.34315 q^{89} +1.00000 q^{90} -2.82843 q^{91} +8.48528 q^{92} +4.00000 q^{93} -8.00000 q^{94} -2.82843 q^{95} +1.00000 q^{96} -8.82843 q^{97} +1.00000 q^{98} -5.65685 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{5} + 2 q^{6} + 2 q^{8} + 2 q^{9} + 2 q^{10} + 2 q^{12} - 2 q^{13} + 2 q^{15} + 2 q^{16} - 4 q^{17} + 2 q^{18} + 2 q^{20} + 2 q^{24} + 2 q^{25} - 2 q^{26} + 2 q^{27}+ \cdots + 2 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 + 2 * q^5 + 2 * q^6 + 2 * q^8 + 2 * q^9 + 2 * q^10 + 2 * q^12 - 2 * q^13 + 2 * q^15 + 2 * q^16 - 4 * q^17 + 2 * q^18 + 2 * q^20 + 2 * q^24 + 2 * q^25 - 2 * q^26 + 2 * q^27 - 12 * q^29 + 2 * q^30 + 8 * q^31 + 2 * q^32 - 4 * q^34 + 2 * q^36 - 12 * q^37 - 2 * q^39 + 2 * q^40 - 4 * q^41 + 8 * q^43 + 2 * q^45 - 16 * q^47 + 2 * q^48 + 2 * q^49 + 2 * q^50 - 4 * q^51 - 2 * q^52 + 4 * q^53 + 2 * q^54 - 12 * q^58 - 16 * q^59 + 2 * q^60 + 12 * q^61 + 8 * q^62 + 2 * q^64 - 2 * q^65 - 4 * q^68 + 2 * q^72 - 12 * q^73 - 12 * q^74 + 2 * q^75 - 32 * q^77 - 2 * q^78 + 16 * q^79 + 2 * q^80 + 2 * q^81 - 4 * q^82 + 24 * q^83 - 4 * q^85 + 8 * q^86 - 12 * q^87 - 20 * q^89 + 2 * q^90 + 8 * q^93 - 16 * q^94 + 2 * q^96 - 12 * q^97 + 2 * q^98

Coefficient data

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

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

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

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	1.00000	0.408248
$7$	2.82843	1.06904	0.534522	−	0.845154i	$-0.320491\pi$
$7$	2.82843	1.06904	0.534522	+	0.845154i	$0.320491\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	1.00000	0.316228
$11$	−5.65685	−1.70561	−0.852803	−	0.522233i	$-0.825099\pi$
$11$	−5.65685	−1.70561	−0.852803	+	0.522233i	$0.825099\pi$
$12$	1.00000	0.288675
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	2.82843	0.755929
$15$	1.00000	0.258199
$16$	1.00000	0.250000
$17$	−4.82843	−1.17107	−0.585533	−	0.810649i	$-0.699115\pi$
$17$	−4.82843	−1.17107	−0.585533	+	0.810649i	$0.699115\pi$
$18$	1.00000	0.235702
$19$	−2.82843	−0.648886	−0.324443	−	0.945905i	$-0.605177\pi$
$19$	−2.82843	−0.648886	−0.324443	+	0.945905i	$0.605177\pi$
$20$	1.00000	0.223607
$21$	2.82843	0.617213
$22$	−5.65685	−1.20605
$23$	8.48528	1.76930	0.884652	−	0.466252i	$-0.154396\pi$
$23$	8.48528	1.76930	0.884652	+	0.466252i	$0.154396\pi$
$24$	1.00000	0.204124
$25$	1.00000	0.200000
$26$	−1.00000	−0.196116
$27$	1.00000	0.192450
$28$	2.82843	0.534522
$29$	−3.17157	−0.588946	−0.294473	−	0.955660i	$-0.595144\pi$
$29$	−3.17157	−0.588946	−0.294473	+	0.955660i	$0.595144\pi$
$30$	1.00000	0.182574
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	1.00000	0.176777
$33$	−5.65685	−0.984732
$34$	−4.82843	−0.828068
$35$	2.82843	0.478091
$36$	1.00000	0.166667
$37$	−0.343146	−0.0564128	−0.0282064	−	0.999602i	$-0.508980\pi$
$37$	−0.343146	−0.0564128	−0.0282064	+	0.999602i	$0.508980\pi$
$38$	−2.82843	−0.458831
$39$	−1.00000	−0.160128
$40$	1.00000	0.158114
$41$	3.65685	0.571105	0.285552	−	0.958363i	$-0.407823\pi$
$41$	3.65685	0.571105	0.285552	+	0.958363i	$0.407823\pi$
$42$	2.82843	0.436436
$43$	−1.65685	−0.252668	−0.126334	−	0.991988i	$-0.540321\pi$
$43$	−1.65685	−0.252668	−0.126334	+	0.991988i	$0.540321\pi$
$44$	−5.65685	−0.852803
$45$	1.00000	0.149071
$46$	8.48528	1.25109
$47$	−8.00000	−1.16692	−0.583460	−	0.812142i	$-0.698301\pi$
$47$	−8.00000	−1.16692	−0.583460	+	0.812142i	$0.698301\pi$
$48$	1.00000	0.144338
$49$	1.00000	0.142857
$50$	1.00000	0.141421
$51$	−4.82843	−0.676115
$52$	−1.00000	−0.138675
$53$	−9.31371	−1.27934	−0.639668	−	0.768651i	$-0.720928\pi$
$53$	−9.31371	−1.27934	−0.639668	+	0.768651i	$0.720928\pi$
$54$	1.00000	0.136083
$55$	−5.65685	−0.762770
$56$	2.82843	0.377964
$57$	−2.82843	−0.374634
$58$	−3.17157	−0.416448
$59$	−13.6569	−1.77797	−0.888985	−	0.457935i	$-0.848589\pi$
$59$	−13.6569	−1.77797	−0.888985	+	0.457935i	$0.848589\pi$
$60$	1.00000	0.129099
$61$	6.00000	0.768221	0.384111	−	0.923287i	$-0.374508\pi$
$61$	6.00000	0.768221	0.384111	+	0.923287i	$0.374508\pi$
$62$	4.00000	0.508001
$63$	2.82843	0.356348
$64$	1.00000	0.125000
$65$	−1.00000	−0.124035
$66$	−5.65685	−0.696311
$67$	−5.65685	−0.691095	−0.345547	−	0.938401i	$-0.612307\pi$
$67$	−5.65685	−0.691095	−0.345547	+	0.938401i	$0.612307\pi$
$68$	−4.82843	−0.585533
$69$	8.48528	1.02151
$70$	2.82843	0.338062
$71$	5.65685	0.671345	0.335673	−	0.941979i	$-0.391036\pi$
$71$	5.65685	0.671345	0.335673	+	0.941979i	$0.391036\pi$
$72$	1.00000	0.117851
$73$	2.48528	0.290880	0.145440	−	0.989367i	$-0.453540\pi$
$73$	2.48528	0.290880	0.145440	+	0.989367i	$0.453540\pi$
$74$	−0.343146	−0.0398899
$75$	1.00000	0.115470
$76$	−2.82843	−0.324443
$77$	−16.0000	−1.82337
$78$	−1.00000	−0.113228
$79$	13.6569	1.53652	0.768258	−	0.640140i	$-0.221124\pi$
$79$	13.6569	1.53652	0.768258	+	0.640140i	$0.221124\pi$
$80$	1.00000	0.111803
$81$	1.00000	0.111111
$82$	3.65685	0.403832
$83$	17.6569	1.93809	0.969046	−	0.246881i	$-0.0794057\pi$
$83$	17.6569	1.93809	0.969046	+	0.246881i	$0.0794057\pi$
$84$	2.82843	0.308607
$85$	−4.82843	−0.523716
$86$	−1.65685	−0.178663
$87$	−3.17157	−0.340028
$88$	−5.65685	−0.603023
$89$	−4.34315	−0.460373	−0.230186	−	0.973147i	$-0.573934\pi$
$89$	−4.34315	−0.460373	−0.230186	+	0.973147i	$0.573934\pi$
$90$	1.00000	0.105409
$91$	−2.82843	−0.296500
$92$	8.48528	0.884652
$93$	4.00000	0.414781
$94$	−8.00000	−0.825137
$95$	−2.82843	−0.290191
$96$	1.00000	0.102062
$97$	−8.82843	−0.896391	−0.448195	−	0.893936i	$-0.647933\pi$
$97$	−8.82843	−0.896391	−0.448195	+	0.893936i	$0.647933\pi$
$98$	1.00000	0.101015
$99$	−5.65685	−0.568535
$100$	1.00000	0.100000
$101$	−12.1421	−1.20819	−0.604094	−	0.796913i	$-0.706465\pi$
$101$	−12.1421	−1.20819	−0.604094	+	0.796913i	$0.706465\pi$
$102$	−4.82843	−0.478086
$103$	9.65685	0.951518	0.475759	−	0.879576i	$-0.342173\pi$
$103$	9.65685	0.951518	0.475759	+	0.879576i	$0.342173\pi$
$104$	−1.00000	−0.0980581
$105$	2.82843	0.276026
$106$	−9.31371	−0.904627
$107$	−4.00000	−0.386695	−0.193347	−	0.981130i	$-0.561934\pi$
$107$	−4.00000	−0.386695	−0.193347	+	0.981130i	$0.561934\pi$
$108$	1.00000	0.0962250
$109$	3.17157	0.303782	0.151891	−	0.988397i	$-0.451464\pi$
$109$	3.17157	0.303782	0.151891	+	0.988397i	$0.451464\pi$
$110$	−5.65685	−0.539360
$111$	−0.343146	−0.0325700
$112$	2.82843	0.267261
$113$	−10.4853	−0.986372	−0.493186	−	0.869924i	$-0.664168\pi$
$113$	−10.4853	−0.986372	−0.493186	+	0.869924i	$0.664168\pi$
$114$	−2.82843	−0.264906
$115$	8.48528	0.791257
$116$	−3.17157	−0.294473
$117$	−1.00000	−0.0924500
$118$	−13.6569	−1.25722
$119$	−13.6569	−1.25192
$120$	1.00000	0.0912871
$121$	21.0000	1.90909
$122$	6.00000	0.543214
$123$	3.65685	0.329727
$124$	4.00000	0.359211
$125$	1.00000	0.0894427
$126$	2.82843	0.251976
$127$	−1.65685	−0.147022	−0.0735110	−	0.997294i	$-0.523420\pi$
$127$	−1.65685	−0.147022	−0.0735110	+	0.997294i	$0.523420\pi$
$128$	1.00000	0.0883883
$129$	−1.65685	−0.145878
$130$	−1.00000	−0.0877058
$131$	22.1421	1.93457	0.967284	−	0.253697i	$-0.0816467\pi$
$131$	22.1421	1.93457	0.967284	+	0.253697i	$0.0816467\pi$
$132$	−5.65685	−0.492366
$133$	−8.00000	−0.693688
$134$	−5.65685	−0.488678
$135$	1.00000	0.0860663
$136$	−4.82843	−0.414034
$137$	5.31371	0.453981	0.226990	−	0.973897i	$-0.427111\pi$
$137$	5.31371	0.453981	0.226990	+	0.973897i	$0.427111\pi$
$138$	8.48528	0.722315
$139$	17.6569	1.49763	0.748817	−	0.662776i	$-0.230622\pi$
$139$	17.6569	1.49763	0.748817	+	0.662776i	$0.230622\pi$
$140$	2.82843	0.239046
$141$	−8.00000	−0.673722
$142$	5.65685	0.474713
$143$	5.65685	0.473050
$144$	1.00000	0.0833333
$145$	−3.17157	−0.263385
$146$	2.48528	0.205683
$147$	1.00000	0.0824786
$148$	−0.343146	−0.0282064
$149$	−7.65685	−0.627274	−0.313637	−	0.949543i	$-0.601547\pi$
$149$	−7.65685	−0.627274	−0.313637	+	0.949543i	$0.601547\pi$
$150$	1.00000	0.0816497
$151$	12.0000	0.976546	0.488273	−	0.872691i	$-0.337627\pi$
$151$	12.0000	0.976546	0.488273	+	0.872691i	$0.337627\pi$
$152$	−2.82843	−0.229416
$153$	−4.82843	−0.390355
$154$	−16.0000	−1.28932
$155$	4.00000	0.321288
$156$	−1.00000	−0.0800641
$157$	−17.3137	−1.38178	−0.690892	−	0.722958i	$-0.742782\pi$
$157$	−17.3137	−1.38178	−0.690892	+	0.722958i	$0.742782\pi$
$158$	13.6569	1.08648
$159$	−9.31371	−0.738625
$160$	1.00000	0.0790569
$161$	24.0000	1.89146
$162$	1.00000	0.0785674
$163$	−11.3137	−0.886158	−0.443079	−	0.896483i	$-0.646114\pi$
$163$	−11.3137	−0.886158	−0.443079	+	0.896483i	$0.646114\pi$
$164$	3.65685	0.285552
$165$	−5.65685	−0.440386
$166$	17.6569	1.37044
$167$	−24.9706	−1.93228	−0.966140	−	0.258018i	$-0.916931\pi$
$167$	−24.9706	−1.93228	−0.966140	+	0.258018i	$0.916931\pi$
$168$	2.82843	0.218218
$169$	1.00000	0.0769231
$170$	−4.82843	−0.370323
$171$	−2.82843	−0.216295
$172$	−1.65685	−0.126334
$173$	13.3137	1.01222	0.506111	−	0.862468i	$-0.331083\pi$
$173$	13.3137	1.01222	0.506111	+	0.862468i	$0.331083\pi$
$174$	−3.17157	−0.240436
$175$	2.82843	0.213809
$176$	−5.65685	−0.426401
$177$	−13.6569	−1.02651
$178$	−4.34315	−0.325533
$179$	24.4853	1.83012	0.915058	−	0.403322i	$-0.132145\pi$
$179$	24.4853	1.83012	0.915058	+	0.403322i	$0.132145\pi$
$180$	1.00000	0.0745356
$181$	3.65685	0.271812	0.135906	−	0.990722i	$-0.456606\pi$
$181$	3.65685	0.271812	0.135906	+	0.990722i	$0.456606\pi$
$182$	−2.82843	−0.209657
$183$	6.00000	0.443533
$184$	8.48528	0.625543
$185$	−0.343146	−0.0252286
$186$	4.00000	0.293294
$187$	27.3137	1.99738
$188$	−8.00000	−0.583460
$189$	2.82843	0.205738
$190$	−2.82843	−0.205196
$191$	−11.3137	−0.818631	−0.409316	−	0.912393i	$-0.634232\pi$
$191$	−11.3137	−0.818631	−0.409316	+	0.912393i	$0.634232\pi$
$192$	1.00000	0.0721688
$193$	−14.4853	−1.04267	−0.521337	−	0.853351i	$-0.674566\pi$
$193$	−14.4853	−1.04267	−0.521337	+	0.853351i	$0.674566\pi$
$194$	−8.82843	−0.633844
$195$	−1.00000	−0.0716115
$196$	1.00000	0.0714286
$197$	9.31371	0.663574	0.331787	−	0.943354i	$-0.392348\pi$
$197$	9.31371	0.663574	0.331787	+	0.943354i	$0.392348\pi$
$198$	−5.65685	−0.402015
$199$	21.6569	1.53521	0.767607	−	0.640921i	$-0.221447\pi$
$199$	21.6569	1.53521	0.767607	+	0.640921i	$0.221447\pi$
$200$	1.00000	0.0707107
$201$	−5.65685	−0.399004
$202$	−12.1421	−0.854318
$203$	−8.97056	−0.629610
$204$	−4.82843	−0.338058
$205$	3.65685	0.255406
$206$	9.65685	0.672825
$207$	8.48528	0.589768
$208$	−1.00000	−0.0693375
$209$	16.0000	1.10674
$210$	2.82843	0.195180
$211$	23.3137	1.60498	0.802491	−	0.596664i	$-0.203508\pi$
$211$	23.3137	1.60498	0.802491	+	0.596664i	$0.203508\pi$
$212$	−9.31371	−0.639668
$213$	5.65685	0.387601
$214$	−4.00000	−0.273434
$215$	−1.65685	−0.112997
$216$	1.00000	0.0680414
$217$	11.3137	0.768025
$218$	3.17157	0.214806
$219$	2.48528	0.167940
$220$	−5.65685	−0.381385
$221$	4.82843	0.324795
$222$	−0.343146	−0.0230304
$223$	5.17157	0.346314	0.173157	−	0.984894i	$-0.444603\pi$
$223$	5.17157	0.346314	0.173157	+	0.984894i	$0.444603\pi$
$224$	2.82843	0.188982
$225$	1.00000	0.0666667
$226$	−10.4853	−0.697471
$227$	4.00000	0.265489	0.132745	−	0.991150i	$-0.457621\pi$
$227$	4.00000	0.265489	0.132745	+	0.991150i	$0.457621\pi$
$228$	−2.82843	−0.187317
$229$	−24.1421	−1.59536	−0.797679	−	0.603083i	$-0.793939\pi$
$229$	−24.1421	−1.59536	−0.797679	+	0.603083i	$0.793939\pi$
$230$	8.48528	0.559503
$231$	−16.0000	−1.05272
$232$	−3.17157	−0.208224
$233$	22.4853	1.47306	0.736530	−	0.676405i	$-0.236463\pi$
$233$	22.4853	1.47306	0.736530	+	0.676405i	$0.236463\pi$
$234$	−1.00000	−0.0653720
$235$	−8.00000	−0.521862
$236$	−13.6569	−0.888985
$237$	13.6569	0.887108
$238$	−13.6569	−0.885242
$239$	16.0000	1.03495	0.517477	−	0.855697i	$-0.326871\pi$
$239$	16.0000	1.03495	0.517477	+	0.855697i	$0.326871\pi$
$240$	1.00000	0.0645497
$241$	−17.3137	−1.11527	−0.557637	−	0.830085i	$-0.688292\pi$
$241$	−17.3137	−1.11527	−0.557637	+	0.830085i	$0.688292\pi$
$242$	21.0000	1.34993
$243$	1.00000	0.0641500
$244$	6.00000	0.384111
$245$	1.00000	0.0638877
$246$	3.65685	0.233153
$247$	2.82843	0.179969
$248$	4.00000	0.254000
$249$	17.6569	1.11896
$250$	1.00000	0.0632456
$251$	5.17157	0.326427	0.163213	−	0.986591i	$-0.447814\pi$
$251$	5.17157	0.326427	0.163213	+	0.986591i	$0.447814\pi$
$252$	2.82843	0.178174
$253$	−48.0000	−3.01773
$254$	−1.65685	−0.103960
$255$	−4.82843	−0.302368
$256$	1.00000	0.0625000
$257$	0.828427	0.0516759	0.0258379	−	0.999666i	$-0.491775\pi$
$257$	0.828427	0.0516759	0.0258379	+	0.999666i	$0.491775\pi$
$258$	−1.65685	−0.103151
$259$	−0.970563	−0.0603078
$260$	−1.00000	−0.0620174
$261$	−3.17157	−0.196315
$262$	22.1421	1.36795
$263$	−0.485281	−0.0299237	−0.0149619	−	0.999888i	$-0.504763\pi$
$263$	−0.485281	−0.0299237	−0.0149619	+	0.999888i	$0.504763\pi$
$264$	−5.65685	−0.348155
$265$	−9.31371	−0.572137
$266$	−8.00000	−0.490511
$267$	−4.34315	−0.265796
$268$	−5.65685	−0.345547
$269$	2.48528	0.151530	0.0757651	−	0.997126i	$-0.475860\pi$
$269$	2.48528	0.151530	0.0757651	+	0.997126i	$0.475860\pi$
$270$	1.00000	0.0608581
$271$	−15.3137	−0.930242	−0.465121	−	0.885247i	$-0.653989\pi$
$271$	−15.3137	−0.930242	−0.465121	+	0.885247i	$0.653989\pi$
$272$	−4.82843	−0.292766
$273$	−2.82843	−0.171184
$274$	5.31371	0.321013
$275$	−5.65685	−0.341121
$276$	8.48528	0.510754
$277$	26.0000	1.56219	0.781094	−	0.624413i	$-0.214662\pi$
$277$	26.0000	1.56219	0.781094	+	0.624413i	$0.214662\pi$
$278$	17.6569	1.05899
$279$	4.00000	0.239474
$280$	2.82843	0.169031
$281$	19.6569	1.17263	0.586315	−	0.810083i	$-0.300578\pi$
$281$	19.6569	1.17263	0.586315	+	0.810083i	$0.300578\pi$
$282$	−8.00000	−0.476393
$283$	−6.34315	−0.377061	−0.188530	−	0.982067i	$-0.560372\pi$
$283$	−6.34315	−0.377061	−0.188530	+	0.982067i	$0.560372\pi$
$284$	5.65685	0.335673
$285$	−2.82843	−0.167542
$286$	5.65685	0.334497
$287$	10.3431	0.610537
$288$	1.00000	0.0589256
$289$	6.31371	0.371395
$290$	−3.17157	−0.186241
$291$	−8.82843	−0.517532
$292$	2.48528	0.145440
$293$	28.6274	1.67243	0.836216	−	0.548401i	$-0.184763\pi$
$293$	28.6274	1.67243	0.836216	+	0.548401i	$0.184763\pi$
$294$	1.00000	0.0583212
$295$	−13.6569	−0.795133
$296$	−0.343146	−0.0199449
$297$	−5.65685	−0.328244
$298$	−7.65685	−0.443550
$299$	−8.48528	−0.490716
$300$	1.00000	0.0577350
$301$	−4.68629	−0.270113
$302$	12.0000	0.690522
$303$	−12.1421	−0.697547
$304$	−2.82843	−0.162221
$305$	6.00000	0.343559
$306$	−4.82843	−0.276023
$307$	−10.3431	−0.590315	−0.295157	−	0.955449i	$-0.595372\pi$
$307$	−10.3431	−0.590315	−0.295157	+	0.955449i	$0.595372\pi$
$308$	−16.0000	−0.911685
$309$	9.65685	0.549359
$310$	4.00000	0.227185
$311$	−24.0000	−1.36092	−0.680458	−	0.732787i	$-0.738219\pi$
$311$	−24.0000	−1.36092	−0.680458	+	0.732787i	$0.738219\pi$
$312$	−1.00000	−0.0566139
$313$	2.97056	0.167906	0.0839531	−	0.996470i	$-0.473245\pi$
$313$	2.97056	0.167906	0.0839531	+	0.996470i	$0.473245\pi$
$314$	−17.3137	−0.977069
$315$	2.82843	0.159364
$316$	13.6569	0.768258
$317$	2.68629	0.150877	0.0754386	−	0.997150i	$-0.475964\pi$
$317$	2.68629	0.150877	0.0754386	+	0.997150i	$0.475964\pi$
$318$	−9.31371	−0.522287
$319$	17.9411	1.00451
$320$	1.00000	0.0559017
$321$	−4.00000	−0.223258
$322$	24.0000	1.33747
$323$	13.6569	0.759888
$324$	1.00000	0.0555556
$325$	−1.00000	−0.0554700
$326$	−11.3137	−0.626608
$327$	3.17157	0.175388
$328$	3.65685	0.201916
$329$	−22.6274	−1.24749
$330$	−5.65685	−0.311400
$331$	8.48528	0.466393	0.233197	−	0.972430i	$-0.425081\pi$
$331$	8.48528	0.466393	0.233197	+	0.972430i	$0.425081\pi$
$332$	17.6569	0.969046
$333$	−0.343146	−0.0188043
$334$	−24.9706	−1.36633
$335$	−5.65685	−0.309067
$336$	2.82843	0.154303
$337$	−22.9706	−1.25129	−0.625643	−	0.780109i	$-0.715163\pi$
$337$	−22.9706	−1.25129	−0.625643	+	0.780109i	$0.715163\pi$
$338$	1.00000	0.0543928
$339$	−10.4853	−0.569482
$340$	−4.82843	−0.261858
$341$	−22.6274	−1.22534
$342$	−2.82843	−0.152944
$343$	−16.9706	−0.916324
$344$	−1.65685	−0.0893316
$345$	8.48528	0.456832
$346$	13.3137	0.715749
$347$	1.65685	0.0889446	0.0444723	−	0.999011i	$-0.485839\pi$
$347$	1.65685	0.0889446	0.0444723	+	0.999011i	$0.485839\pi$
$348$	−3.17157	−0.170014
$349$	−16.1421	−0.864069	−0.432034	−	0.901857i	$-0.642204\pi$
$349$	−16.1421	−0.864069	−0.432034	+	0.901857i	$0.642204\pi$
$350$	2.82843	0.151186
$351$	−1.00000	−0.0533761
$352$	−5.65685	−0.301511
$353$	−17.3137	−0.921516	−0.460758	−	0.887526i	$-0.652422\pi$
$353$	−17.3137	−0.921516	−0.460758	+	0.887526i	$0.652422\pi$
$354$	−13.6569	−0.725854
$355$	5.65685	0.300235
$356$	−4.34315	−0.230186
$357$	−13.6569	−0.722797
$358$	24.4853	1.29409
$359$	28.2843	1.49279	0.746393	−	0.665505i	$-0.231784\pi$
$359$	28.2843	1.49279	0.746393	+	0.665505i	$0.231784\pi$
$360$	1.00000	0.0527046
$361$	−11.0000	−0.578947
$362$	3.65685	0.192200
$363$	21.0000	1.10221
$364$	−2.82843	−0.148250
$365$	2.48528	0.130086
$366$	6.00000	0.313625
$367$	14.3431	0.748706	0.374353	−	0.927286i	$-0.377865\pi$
$367$	14.3431	0.748706	0.374353	+	0.927286i	$0.377865\pi$
$368$	8.48528	0.442326
$369$	3.65685	0.190368
$370$	−0.343146	−0.0178393
$371$	−26.3431	−1.36767
$372$	4.00000	0.207390
$373$	−25.3137	−1.31069	−0.655347	−	0.755328i	$-0.727478\pi$
$373$	−25.3137	−1.31069	−0.655347	+	0.755328i	$0.727478\pi$
$374$	27.3137	1.41236
$375$	1.00000	0.0516398
$376$	−8.00000	−0.412568
$377$	3.17157	0.163344
$378$	2.82843	0.145479
$379$	−24.4853	−1.25772	−0.628862	−	0.777517i	$-0.716479\pi$
$379$	−24.4853	−1.25772	−0.628862	+	0.777517i	$0.716479\pi$
$380$	−2.82843	−0.145095
$381$	−1.65685	−0.0848832
$382$	−11.3137	−0.578860
$383$	−18.3431	−0.937291	−0.468645	−	0.883386i	$-0.655258\pi$
$383$	−18.3431	−0.937291	−0.468645	+	0.883386i	$0.655258\pi$
$384$	1.00000	0.0510310
$385$	−16.0000	−0.815436
$386$	−14.4853	−0.737281
$387$	−1.65685	−0.0842226
$388$	−8.82843	−0.448195
$389$	10.4853	0.531625	0.265812	−	0.964025i	$-0.414360\pi$
$389$	10.4853	0.531625	0.265812	+	0.964025i	$0.414360\pi$
$390$	−1.00000	−0.0506370
$391$	−40.9706	−2.07197
$392$	1.00000	0.0505076
$393$	22.1421	1.11692
$394$	9.31371	0.469218
$395$	13.6569	0.687151
$396$	−5.65685	−0.284268
$397$	−26.2843	−1.31917	−0.659585	−	0.751630i	$-0.729268\pi$
$397$	−26.2843	−1.31917	−0.659585	+	0.751630i	$0.729268\pi$
$398$	21.6569	1.08556
$399$	−8.00000	−0.400501
$400$	1.00000	0.0500000
$401$	6.97056	0.348093	0.174047	−	0.984737i	$-0.444316\pi$
$401$	6.97056	0.348093	0.174047	+	0.984737i	$0.444316\pi$
$402$	−5.65685	−0.282138
$403$	−4.00000	−0.199254
$404$	−12.1421	−0.604094
$405$	1.00000	0.0496904
$406$	−8.97056	−0.445202
$407$	1.94113	0.0962180
$408$	−4.82843	−0.239043
$409$	7.65685	0.378607	0.189304	−	0.981919i	$-0.439377\pi$
$409$	7.65685	0.378607	0.189304	+	0.981919i	$0.439377\pi$
$410$	3.65685	0.180599
$411$	5.31371	0.262106
$412$	9.65685	0.475759
$413$	−38.6274	−1.90073
$414$	8.48528	0.417029
$415$	17.6569	0.866741
$416$	−1.00000	−0.0490290
$417$	17.6569	0.864660
$418$	16.0000	0.782586
$419$	−5.17157	−0.252648	−0.126324	−	0.991989i	$-0.540318\pi$
$419$	−5.17157	−0.252648	−0.126324	+	0.991989i	$0.540318\pi$
$420$	2.82843	0.138013
$421$	4.14214	0.201875	0.100938	−	0.994893i	$-0.467816\pi$
$421$	4.14214	0.201875	0.100938	+	0.994893i	$0.467816\pi$
$422$	23.3137	1.13489
$423$	−8.00000	−0.388973
$424$	−9.31371	−0.452314
$425$	−4.82843	−0.234213
$426$	5.65685	0.274075
$427$	16.9706	0.821263
$428$	−4.00000	−0.193347
$429$	5.65685	0.273115
$430$	−1.65685	−0.0799006
$431$	−16.0000	−0.770693	−0.385346	−	0.922772i	$-0.625918\pi$
$431$	−16.0000	−0.770693	−0.385346	+	0.922772i	$0.625918\pi$
$432$	1.00000	0.0481125
$433$	10.9706	0.527212	0.263606	−	0.964630i	$-0.415088\pi$
$433$	10.9706	0.527212	0.263606	+	0.964630i	$0.415088\pi$
$434$	11.3137	0.543075
$435$	−3.17157	−0.152065
$436$	3.17157	0.151891
$437$	−24.0000	−1.14808
$438$	2.48528	0.118751
$439$	−22.6274	−1.07995	−0.539974	−	0.841682i	$-0.681566\pi$
$439$	−22.6274	−1.07995	−0.539974	+	0.841682i	$0.681566\pi$
$440$	−5.65685	−0.269680
$441$	1.00000	0.0476190
$442$	4.82843	0.229665
$443$	41.6569	1.97918	0.989588	−	0.143926i	$-0.0459728\pi$
$443$	41.6569	1.97918	0.989588	+	0.143926i	$0.0459728\pi$
$444$	−0.343146	−0.0162850
$445$	−4.34315	−0.205885
$446$	5.17157	0.244881
$447$	−7.65685	−0.362157
$448$	2.82843	0.133631
$449$	−30.2843	−1.42920	−0.714602	−	0.699532i	$-0.753392\pi$
$449$	−30.2843	−1.42920	−0.714602	+	0.699532i	$0.753392\pi$
$450$	1.00000	0.0471405
$451$	−20.6863	−0.974079
$452$	−10.4853	−0.493186
$453$	12.0000	0.563809
$454$	4.00000	0.187729
$455$	−2.82843	−0.132599
$456$	−2.82843	−0.132453
$457$	15.1716	0.709696	0.354848	−	0.934924i	$-0.384533\pi$
$457$	15.1716	0.709696	0.354848	+	0.934924i	$0.384533\pi$
$458$	−24.1421	−1.12809
$459$	−4.82843	−0.225372
$460$	8.48528	0.395628
$461$	14.0000	0.652045	0.326023	−	0.945362i	$-0.394291\pi$
$461$	14.0000	0.652045	0.326023	+	0.945362i	$0.394291\pi$
$462$	−16.0000	−0.744387
$463$	35.7990	1.66372	0.831860	−	0.554985i	$-0.187276\pi$
$463$	35.7990	1.66372	0.831860	+	0.554985i	$0.187276\pi$
$464$	−3.17157	−0.147237
$465$	4.00000	0.185496
$466$	22.4853	1.04161
$467$	−15.3137	−0.708634	−0.354317	−	0.935125i	$-0.615287\pi$
$467$	−15.3137	−0.708634	−0.354317	+	0.935125i	$0.615287\pi$
$468$	−1.00000	−0.0462250
$469$	−16.0000	−0.738811
$470$	−8.00000	−0.369012
$471$	−17.3137	−0.797774
$472$	−13.6569	−0.628608
$473$	9.37258	0.430952
$474$	13.6569	0.627280
$475$	−2.82843	−0.129777
$476$	−13.6569	−0.625961
$477$	−9.31371	−0.426445
$478$	16.0000	0.731823
$479$	11.3137	0.516937	0.258468	−	0.966020i	$-0.416782\pi$
$479$	11.3137	0.516937	0.258468	+	0.966020i	$0.416782\pi$
$480$	1.00000	0.0456435
$481$	0.343146	0.0156461
$482$	−17.3137	−0.788618
$483$	24.0000	1.09204
$484$	21.0000	0.954545
$485$	−8.82843	−0.400878
$486$	1.00000	0.0453609
$487$	−0.485281	−0.0219902	−0.0109951	−	0.999940i	$-0.503500\pi$
$487$	−0.485281	−0.0219902	−0.0109951	+	0.999940i	$0.503500\pi$
$488$	6.00000	0.271607
$489$	−11.3137	−0.511624
$490$	1.00000	0.0451754
$491$	−9.85786	−0.444879	−0.222440	−	0.974946i	$-0.571402\pi$
$491$	−9.85786	−0.444879	−0.222440	+	0.974946i	$0.571402\pi$
$492$	3.65685	0.164864
$493$	15.3137	0.689695
$494$	2.82843	0.127257
$495$	−5.65685	−0.254257
$496$	4.00000	0.179605
$497$	16.0000	0.717698
$498$	17.6569	0.791223
$499$	−16.4853	−0.737983	−0.368991	−	0.929433i	$-0.620297\pi$
$499$	−16.4853	−0.737983	−0.368991	+	0.929433i	$0.620297\pi$
$500$	1.00000	0.0447214
$501$	−24.9706	−1.11560
$502$	5.17157	0.230819
$503$	−40.4853	−1.80515	−0.902575	−	0.430533i	$-0.858326\pi$
$503$	−40.4853	−1.80515	−0.902575	+	0.430533i	$0.858326\pi$
$504$	2.82843	0.125988
$505$	−12.1421	−0.540318
$506$	−48.0000	−2.13386
$507$	1.00000	0.0444116
$508$	−1.65685	−0.0735110
$509$	−14.6863	−0.650958	−0.325479	−	0.945549i	$-0.605526\pi$
$509$	−14.6863	−0.650958	−0.325479	+	0.945549i	$0.605526\pi$
$510$	−4.82843	−0.213806
$511$	7.02944	0.310964
$512$	1.00000	0.0441942
$513$	−2.82843	−0.124878
$514$	0.828427	0.0365404
$515$	9.65685	0.425532
$516$	−1.65685	−0.0729389
$517$	45.2548	1.99031
$518$	−0.970563	−0.0426441
$519$	13.3137	0.584407
$520$	−1.00000	−0.0438529
$521$	−6.97056	−0.305386	−0.152693	−	0.988274i	$-0.548795\pi$
$521$	−6.97056	−0.305386	−0.152693	+	0.988274i	$0.548795\pi$
$522$	−3.17157	−0.138816
$523$	34.6274	1.51415	0.757076	−	0.653327i	$-0.226627\pi$
$523$	34.6274	1.51415	0.757076	+	0.653327i	$0.226627\pi$
$524$	22.1421	0.967284
$525$	2.82843	0.123443
$526$	−0.485281	−0.0211593
$527$	−19.3137	−0.841318
$528$	−5.65685	−0.246183
$529$	49.0000	2.13043
$530$	−9.31371	−0.404562
$531$	−13.6569	−0.592657
$532$	−8.00000	−0.346844
$533$	−3.65685	−0.158396
$534$	−4.34315	−0.187946
$535$	−4.00000	−0.172935
$536$	−5.65685	−0.244339
$537$	24.4853	1.05662
$538$	2.48528	0.107148
$539$	−5.65685	−0.243658
$540$	1.00000	0.0430331
$541$	−2.48528	−0.106851	−0.0534253	−	0.998572i	$-0.517014\pi$
$541$	−2.48528	−0.106851	−0.0534253	+	0.998572i	$0.517014\pi$
$542$	−15.3137	−0.657780
$543$	3.65685	0.156931
$544$	−4.82843	−0.207017
$545$	3.17157	0.135855
$546$	−2.82843	−0.121046
$547$	23.3137	0.996822	0.498411	−	0.866941i	$-0.333917\pi$
$547$	23.3137	0.996822	0.498411	+	0.866941i	$0.333917\pi$
$548$	5.31371	0.226990
$549$	6.00000	0.256074
$550$	−5.65685	−0.241209
$551$	8.97056	0.382159
$552$	8.48528	0.361158
$553$	38.6274	1.64260
$554$	26.0000	1.10463
$555$	−0.343146	−0.0145657
$556$	17.6569	0.748817
$557$	33.3137	1.41155	0.705774	−	0.708437i	$-0.250600\pi$
$557$	33.3137	1.41155	0.705774	+	0.708437i	$0.250600\pi$
$558$	4.00000	0.169334
$559$	1.65685	0.0700775
$560$	2.82843	0.119523
$561$	27.3137	1.15319
$562$	19.6569	0.829174
$563$	−41.6569	−1.75563	−0.877814	−	0.479002i	$-0.840998\pi$
$563$	−41.6569	−1.75563	−0.877814	+	0.479002i	$0.840998\pi$
$564$	−8.00000	−0.336861
$565$	−10.4853	−0.441119
$566$	−6.34315	−0.266622
$567$	2.82843	0.118783
$568$	5.65685	0.237356
$569$	20.3431	0.852829	0.426415	−	0.904528i	$-0.359776\pi$
$569$	20.3431	0.852829	0.426415	+	0.904528i	$0.359776\pi$
$570$	−2.82843	−0.118470
$571$	−12.9706	−0.542801	−0.271401	−	0.962466i	$-0.587487\pi$
$571$	−12.9706	−0.542801	−0.271401	+	0.962466i	$0.587487\pi$
$572$	5.65685	0.236525
$573$	−11.3137	−0.472637
$574$	10.3431	0.431715
$575$	8.48528	0.353861
$576$	1.00000	0.0416667
$577$	27.4558	1.14300	0.571501	−	0.820601i	$-0.306361\pi$
$577$	27.4558	1.14300	0.571501	+	0.820601i	$0.306361\pi$
$578$	6.31371	0.262616
$579$	−14.4853	−0.601988
$580$	−3.17157	−0.131692
$581$	49.9411	2.07191
$582$	−8.82843	−0.365950
$583$	52.6863	2.18204
$584$	2.48528	0.102842
$585$	−1.00000	−0.0413449
$586$	28.6274	1.18259
$587$	−42.6274	−1.75942	−0.879711	−	0.475509i	$-0.842264\pi$
$587$	−42.6274	−1.75942	−0.879711	+	0.475509i	$0.842264\pi$
$588$	1.00000	0.0412393
$589$	−11.3137	−0.466173
$590$	−13.6569	−0.562244
$591$	9.31371	0.383115
$592$	−0.343146	−0.0141032
$593$	−11.6569	−0.478690	−0.239345	−	0.970935i	$-0.576933\pi$
$593$	−11.6569	−0.478690	−0.239345	+	0.970935i	$0.576933\pi$
$594$	−5.65685	−0.232104
$595$	−13.6569	−0.559876
$596$	−7.65685	−0.313637
$597$	21.6569	0.886356
$598$	−8.48528	−0.346989
$599$	40.0000	1.63436	0.817178	−	0.576386i	$-0.195537\pi$
$599$	40.0000	1.63436	0.817178	+	0.576386i	$0.195537\pi$
$600$	1.00000	0.0408248
$601$	6.68629	0.272740	0.136370	−	0.990658i	$-0.456456\pi$
$601$	6.68629	0.272740	0.136370	+	0.990658i	$0.456456\pi$
$602$	−4.68629	−0.190999
$603$	−5.65685	−0.230365
$604$	12.0000	0.488273
$605$	21.0000	0.853771
$606$	−12.1421	−0.493241
$607$	−4.97056	−0.201749	−0.100874	−	0.994899i	$-0.532164\pi$
$607$	−4.97056	−0.201749	−0.100874	+	0.994899i	$0.532164\pi$
$608$	−2.82843	−0.114708
$609$	−8.97056	−0.363506
$610$	6.00000	0.242933
$611$	8.00000	0.323645
$612$	−4.82843	−0.195178
$613$	−34.2843	−1.38473	−0.692364	−	0.721548i	$-0.743431\pi$
$613$	−34.2843	−1.38473	−0.692364	+	0.721548i	$0.743431\pi$
$614$	−10.3431	−0.417415
$615$	3.65685	0.147459
$616$	−16.0000	−0.644658
$617$	2.00000	0.0805170	0.0402585	−	0.999189i	$-0.487182\pi$
$617$	2.00000	0.0805170	0.0402585	+	0.999189i	$0.487182\pi$
$618$	9.65685	0.388456
$619$	−29.1716	−1.17250	−0.586252	−	0.810129i	$-0.699397\pi$
$619$	−29.1716	−1.17250	−0.586252	+	0.810129i	$0.699397\pi$
$620$	4.00000	0.160644
$621$	8.48528	0.340503
$622$	−24.0000	−0.962312
$623$	−12.2843	−0.492159
$624$	−1.00000	−0.0400320
$625$	1.00000	0.0400000
$626$	2.97056	0.118728
$627$	16.0000	0.638978
$628$	−17.3137	−0.690892
$629$	1.65685	0.0660631
$630$	2.82843	0.112687
$631$	−22.3431	−0.889467	−0.444733	−	0.895663i	$-0.646702\pi$
$631$	−22.3431	−0.889467	−0.444733	+	0.895663i	$0.646702\pi$
$632$	13.6569	0.543240
$633$	23.3137	0.926637
$634$	2.68629	0.106686
$635$	−1.65685	−0.0657503
$636$	−9.31371	−0.369313
$637$	−1.00000	−0.0396214
$638$	17.9411	0.710296
$639$	5.65685	0.223782
$640$	1.00000	0.0395285
$641$	40.6274	1.60469	0.802343	−	0.596863i	$-0.203586\pi$
$641$	40.6274	1.60469	0.802343	+	0.596863i	$0.203586\pi$
$642$	−4.00000	−0.157867
$643$	−39.5980	−1.56159	−0.780796	−	0.624786i	$-0.785186\pi$
$643$	−39.5980	−1.56159	−0.780796	+	0.624786i	$0.785186\pi$
$644$	24.0000	0.945732
$645$	−1.65685	−0.0652386
$646$	13.6569	0.537322
$647$	−8.48528	−0.333591	−0.166795	−	0.985992i	$-0.553342\pi$
$647$	−8.48528	−0.333591	−0.166795	+	0.985992i	$0.553342\pi$
$648$	1.00000	0.0392837
$649$	77.2548	3.03252
$650$	−1.00000	−0.0392232
$651$	11.3137	0.443419
$652$	−11.3137	−0.443079
$653$	14.2843	0.558987	0.279493	−	0.960148i	$-0.409834\pi$
$653$	14.2843	0.558987	0.279493	+	0.960148i	$0.409834\pi$
$654$	3.17157	0.124018
$655$	22.1421	0.865165
$656$	3.65685	0.142776
$657$	2.48528	0.0969601
$658$	−22.6274	−0.882109
$659$	−24.4853	−0.953811	−0.476906	−	0.878955i	$-0.658242\pi$
$659$	−24.4853	−0.953811	−0.476906	+	0.878955i	$0.658242\pi$
$660$	−5.65685	−0.220193
$661$	20.1421	0.783438	0.391719	−	0.920085i	$-0.371880\pi$
$661$	20.1421	0.783438	0.391719	+	0.920085i	$0.371880\pi$
$662$	8.48528	0.329790
$663$	4.82843	0.187521
$664$	17.6569	0.685219
$665$	−8.00000	−0.310227
$666$	−0.343146	−0.0132966
$667$	−26.9117	−1.04202
$668$	−24.9706	−0.966140
$669$	5.17157	0.199945
$670$	−5.65685	−0.218543
$671$	−33.9411	−1.31028
$672$	2.82843	0.109109
$673$	−12.6274	−0.486751	−0.243376	−	0.969932i	$-0.578255\pi$
$673$	−12.6274	−0.486751	−0.243376	+	0.969932i	$0.578255\pi$
$674$	−22.9706	−0.884793
$675$	1.00000	0.0384900
$676$	1.00000	0.0384615
$677$	23.6569	0.909207	0.454603	−	0.890694i	$-0.349781\pi$
$677$	23.6569	0.909207	0.454603	+	0.890694i	$0.349781\pi$
$678$	−10.4853	−0.402685
$679$	−24.9706	−0.958282
$680$	−4.82843	−0.185162
$681$	4.00000	0.153280
$682$	−22.6274	−0.866449
$683$	22.3431	0.854937	0.427468	−	0.904030i	$-0.359406\pi$
$683$	22.3431	0.854937	0.427468	+	0.904030i	$0.359406\pi$
$684$	−2.82843	−0.108148
$685$	5.31371	0.203026
$686$	−16.9706	−0.647939
$687$	−24.1421	−0.921080
$688$	−1.65685	−0.0631670
$689$	9.31371	0.354824
$690$	8.48528	0.323029
$691$	11.7990	0.448855	0.224427	−	0.974491i	$-0.427949\pi$
$691$	11.7990	0.448855	0.224427	+	0.974491i	$0.427949\pi$
$692$	13.3137	0.506111
$693$	−16.0000	−0.607790
$694$	1.65685	0.0628933
$695$	17.6569	0.669763
$696$	−3.17157	−0.120218
$697$	−17.6569	−0.668801
$698$	−16.1421	−0.610989
$699$	22.4853	0.850471
$700$	2.82843	0.106904
$701$	−28.1421	−1.06291	−0.531457	−	0.847085i	$-0.678355\pi$
$701$	−28.1421	−1.06291	−0.531457	+	0.847085i	$0.678355\pi$
$702$	−1.00000	−0.0377426
$703$	0.970563	0.0366055
$704$	−5.65685	−0.213201
$705$	−8.00000	−0.301297
$706$	−17.3137	−0.651610
$707$	−34.3431	−1.29161
$708$	−13.6569	−0.513256
$709$	−12.8284	−0.481782	−0.240891	−	0.970552i	$-0.577440\pi$
$709$	−12.8284	−0.481782	−0.240891	+	0.970552i	$0.577440\pi$
$710$	5.65685	0.212298
$711$	13.6569	0.512172
$712$	−4.34315	−0.162766
$713$	33.9411	1.27111
$714$	−13.6569	−0.511095
$715$	5.65685	0.211554
$716$	24.4853	0.915058
$717$	16.0000	0.597531
$718$	28.2843	1.05556
$719$	−18.3431	−0.684084	−0.342042	−	0.939685i	$-0.611118\pi$
$719$	−18.3431	−0.684084	−0.342042	+	0.939685i	$0.611118\pi$
$720$	1.00000	0.0372678
$721$	27.3137	1.01722
$722$	−11.0000	−0.409378
$723$	−17.3137	−0.643904
$724$	3.65685	0.135906
$725$	−3.17157	−0.117789
$726$	21.0000	0.779383
$727$	21.9411	0.813751	0.406876	−	0.913484i	$-0.366618\pi$
$727$	21.9411	0.813751	0.406876	+	0.913484i	$0.366618\pi$
$728$	−2.82843	−0.104828
$729$	1.00000	0.0370370
$730$	2.48528	0.0919844
$731$	8.00000	0.295891
$732$	6.00000	0.221766
$733$	−11.6569	−0.430556	−0.215278	−	0.976553i	$-0.569066\pi$
$733$	−11.6569	−0.430556	−0.215278	+	0.976553i	$0.569066\pi$
$734$	14.3431	0.529415
$735$	1.00000	0.0368856
$736$	8.48528	0.312772
$737$	32.0000	1.17874
$738$	3.65685	0.134611
$739$	14.1421	0.520227	0.260113	−	0.965578i	$-0.416240\pi$
$739$	14.1421	0.520227	0.260113	+	0.965578i	$0.416240\pi$
$740$	−0.343146	−0.0126143
$741$	2.82843	0.103905
$742$	−26.3431	−0.967087
$743$	20.2843	0.744158	0.372079	−	0.928201i	$-0.378645\pi$
$743$	20.2843	0.744158	0.372079	+	0.928201i	$0.378645\pi$
$744$	4.00000	0.146647
$745$	−7.65685	−0.280525
$746$	−25.3137	−0.926801
$747$	17.6569	0.646031
$748$	27.3137	0.998688
$749$	−11.3137	−0.413394
$750$	1.00000	0.0365148
$751$	−11.3137	−0.412843	−0.206422	−	0.978463i	$-0.566182\pi$
$751$	−11.3137	−0.412843	−0.206422	+	0.978463i	$0.566182\pi$
$752$	−8.00000	−0.291730
$753$	5.17157	0.188463
$754$	3.17157	0.115502
$755$	12.0000	0.436725
$756$	2.82843	0.102869
$757$	−47.9411	−1.74245	−0.871225	−	0.490884i	$-0.836674\pi$
$757$	−47.9411	−1.74245	−0.871225	+	0.490884i	$0.836674\pi$
$758$	−24.4853	−0.889345
$759$	−48.0000	−1.74229
$760$	−2.82843	−0.102598
$761$	16.3431	0.592439	0.296219	−	0.955120i	$-0.404274\pi$
$761$	16.3431	0.592439	0.296219	+	0.955120i	$0.404274\pi$
$762$	−1.65685	−0.0600215
$763$	8.97056	0.324756
$764$	−11.3137	−0.409316
$765$	−4.82843	−0.174572
$766$	−18.3431	−0.662765
$767$	13.6569	0.493120
$768$	1.00000	0.0360844
$769$	−14.0000	−0.504853	−0.252426	−	0.967616i	$-0.581229\pi$
$769$	−14.0000	−0.504853	−0.252426	+	0.967616i	$0.581229\pi$
$770$	−16.0000	−0.576600
$771$	0.828427	0.0298351
$772$	−14.4853	−0.521337
$773$	−30.6863	−1.10371	−0.551855	−	0.833940i	$-0.686080\pi$
$773$	−30.6863	−1.10371	−0.551855	+	0.833940i	$0.686080\pi$
$774$	−1.65685	−0.0595544
$775$	4.00000	0.143684
$776$	−8.82843	−0.316922
$777$	−0.970563	−0.0348187
$778$	10.4853	0.375916
$779$	−10.3431	−0.370582
$780$	−1.00000	−0.0358057
$781$	−32.0000	−1.14505
$782$	−40.9706	−1.46510
$783$	−3.17157	−0.113343
$784$	1.00000	0.0357143
$785$	−17.3137	−0.617953
$786$	22.1421	0.789784
$787$	−24.0000	−0.855508	−0.427754	−	0.903895i	$-0.640695\pi$
$787$	−24.0000	−0.855508	−0.427754	+	0.903895i	$0.640695\pi$
$788$	9.31371	0.331787
$789$	−0.485281	−0.0172765
$790$	13.6569	0.485889
$791$	−29.6569	−1.05448
$792$	−5.65685	−0.201008
$793$	−6.00000	−0.213066
$794$	−26.2843	−0.932794
$795$	−9.31371	−0.330323
$796$	21.6569	0.767607
$797$	−28.6274	−1.01404	−0.507018	−	0.861936i	$-0.669252\pi$
$797$	−28.6274	−1.01404	−0.507018	+	0.861936i	$0.669252\pi$
$798$	−8.00000	−0.283197
$799$	38.6274	1.36654
$800$	1.00000	0.0353553
$801$	−4.34315	−0.153458
$802$	6.97056	0.246139
$803$	−14.0589	−0.496127
$804$	−5.65685	−0.199502
$805$	24.0000	0.845889
$806$	−4.00000	−0.140894
$807$	2.48528	0.0874860
$808$	−12.1421	−0.427159
$809$	−9.31371	−0.327453	−0.163726	−	0.986506i	$-0.552351\pi$
$809$	−9.31371	−0.327453	−0.163726	+	0.986506i	$0.552351\pi$
$810$	1.00000	0.0351364
$811$	−30.1421	−1.05843	−0.529217	−	0.848487i	$-0.677514\pi$
$811$	−30.1421	−1.05843	−0.529217	+	0.848487i	$0.677514\pi$
$812$	−8.97056	−0.314805
$813$	−15.3137	−0.537075
$814$	1.94113	0.0680364
$815$	−11.3137	−0.396302
$816$	−4.82843	−0.169029
$817$	4.68629	0.163953
$818$	7.65685	0.267716
$819$	−2.82843	−0.0988332
$820$	3.65685	0.127703
$821$	−22.2843	−0.777726	−0.388863	−	0.921296i	$-0.627132\pi$
$821$	−22.2843	−0.777726	−0.388863	+	0.921296i	$0.627132\pi$
$822$	5.31371	0.185337
$823$	−19.0294	−0.663324	−0.331662	−	0.943398i	$-0.607609\pi$
$823$	−19.0294	−0.663324	−0.331662	+	0.943398i	$0.607609\pi$
$824$	9.65685	0.336412
$825$	−5.65685	−0.196946
$826$	−38.6274	−1.34402
$827$	−1.65685	−0.0576145	−0.0288072	−	0.999585i	$-0.509171\pi$
$827$	−1.65685	−0.0576145	−0.0288072	+	0.999585i	$0.509171\pi$
$828$	8.48528	0.294884
$829$	−30.6863	−1.06578	−0.532889	−	0.846185i	$-0.678894\pi$
$829$	−30.6863	−1.06578	−0.532889	+	0.846185i	$0.678894\pi$
$830$	17.6569	0.612878
$831$	26.0000	0.901930
$832$	−1.00000	−0.0346688
$833$	−4.82843	−0.167295
$834$	17.6569	0.611407
$835$	−24.9706	−0.864142
$836$	16.0000	0.553372
$837$	4.00000	0.138260
$838$	−5.17157	−0.178649
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	2.82843	0.0975900
$841$	−18.9411	−0.653142
$842$	4.14214	0.142747
$843$	19.6569	0.677018
$844$	23.3137	0.802491
$845$	1.00000	0.0344010
$846$	−8.00000	−0.275046
$847$	59.3970	2.04090
$848$	−9.31371	−0.319834
$849$	−6.34315	−0.217696
$850$	−4.82843	−0.165614
$851$	−2.91169	−0.0998114
$852$	5.65685	0.193801
$853$	−18.2843	−0.626042	−0.313021	−	0.949746i	$-0.601341\pi$
$853$	−18.2843	−0.626042	−0.313021	+	0.949746i	$0.601341\pi$
$854$	16.9706	0.580721
$855$	−2.82843	−0.0967302
$856$	−4.00000	−0.136717
$857$	−15.1716	−0.518251	−0.259126	−	0.965844i	$-0.583434\pi$
$857$	−15.1716	−0.518251	−0.259126	+	0.965844i	$0.583434\pi$
$858$	5.65685	0.193122
$859$	−29.9411	−1.02158	−0.510789	−	0.859706i	$-0.670647\pi$
$859$	−29.9411	−1.02158	−0.510789	+	0.859706i	$0.670647\pi$
$860$	−1.65685	−0.0564983
$861$	10.3431	0.352493
$862$	−16.0000	−0.544962
$863$	28.2843	0.962808	0.481404	−	0.876499i	$-0.340127\pi$
$863$	28.2843	0.962808	0.481404	+	0.876499i	$0.340127\pi$
$864$	1.00000	0.0340207
$865$	13.3137	0.452680
$866$	10.9706	0.372795
$867$	6.31371	0.214425
$868$	11.3137	0.384012
$869$	−77.2548	−2.62069
$870$	−3.17157	−0.107526
$871$	5.65685	0.191675
$872$	3.17157	0.107403
$873$	−8.82843	−0.298797
$874$	−24.0000	−0.811812
$875$	2.82843	0.0956183
$876$	2.48528	0.0839699
$877$	39.2548	1.32554	0.662771	−	0.748822i	$-0.269380\pi$
$877$	39.2548	1.32554	0.662771	+	0.748822i	$0.269380\pi$
$878$	−22.6274	−0.763638
$879$	28.6274	0.965579
$880$	−5.65685	−0.190693
$881$	46.2843	1.55936	0.779678	−	0.626180i	$-0.215383\pi$
$881$	46.2843	1.55936	0.779678	+	0.626180i	$0.215383\pi$
$882$	1.00000	0.0336718
$883$	8.68629	0.292317	0.146158	−	0.989261i	$-0.453309\pi$
$883$	8.68629	0.292317	0.146158	+	0.989261i	$0.453309\pi$
$884$	4.82843	0.162398
$885$	−13.6569	−0.459070
$886$	41.6569	1.39949
$887$	−23.5147	−0.789547	−0.394773	−	0.918778i	$-0.629177\pi$
$887$	−23.5147	−0.789547	−0.394773	+	0.918778i	$0.629177\pi$
$888$	−0.343146	−0.0115152
$889$	−4.68629	−0.157173
$890$	−4.34315	−0.145583
$891$	−5.65685	−0.189512
$892$	5.17157	0.173157
$893$	22.6274	0.757198
$894$	−7.65685	−0.256084
$895$	24.4853	0.818453
$896$	2.82843	0.0944911
$897$	−8.48528	−0.283315
$898$	−30.2843	−1.01060
$899$	−12.6863	−0.423112
$900$	1.00000	0.0333333
$901$	44.9706	1.49819
$902$	−20.6863	−0.688778
$903$	−4.68629	−0.155950
$904$	−10.4853	−0.348735
$905$	3.65685	0.121558
$906$	12.0000	0.398673
$907$	−48.2843	−1.60325	−0.801626	−	0.597825i	$-0.796032\pi$
$907$	−48.2843	−1.60325	−0.801626	+	0.597825i	$0.796032\pi$
$908$	4.00000	0.132745
$909$	−12.1421	−0.402729
$910$	−2.82843	−0.0937614
$911$	−8.97056	−0.297208	−0.148604	−	0.988897i	$-0.547478\pi$
$911$	−8.97056	−0.297208	−0.148604	+	0.988897i	$0.547478\pi$
$912$	−2.82843	−0.0936586
$913$	−99.8823	−3.30562
$914$	15.1716	0.501831
$915$	6.00000	0.198354
$916$	−24.1421	−0.797679
$917$	62.6274	2.06814
$918$	−4.82843	−0.159362
$919$	−25.9411	−0.855719	−0.427859	−	0.903845i	$-0.640732\pi$
$919$	−25.9411	−0.855719	−0.427859	+	0.903845i	$0.640732\pi$
$920$	8.48528	0.279751
$921$	−10.3431	−0.340818
$922$	14.0000	0.461065
$923$	−5.65685	−0.186198
$924$	−16.0000	−0.526361
$925$	−0.343146	−0.0112826
$926$	35.7990	1.17643
$927$	9.65685	0.317173
$928$	−3.17157	−0.104112
$929$	45.5980	1.49602	0.748011	−	0.663687i	$-0.231009\pi$
$929$	45.5980	1.49602	0.748011	+	0.663687i	$0.231009\pi$
$930$	4.00000	0.131165
$931$	−2.82843	−0.0926980
$932$	22.4853	0.736530
$933$	−24.0000	−0.785725
$934$	−15.3137	−0.501080
$935$	27.3137	0.893254
$936$	−1.00000	−0.0326860
$937$	−28.6274	−0.935217	−0.467608	−	0.883936i	$-0.654884\pi$
$937$	−28.6274	−0.935217	−0.467608	+	0.883936i	$0.654884\pi$
$938$	−16.0000	−0.522419
$939$	2.97056	0.0969407
$940$	−8.00000	−0.260931
$941$	21.0294	0.685540	0.342770	−	0.939419i	$-0.388635\pi$
$941$	21.0294	0.685540	0.342770	+	0.939419i	$0.388635\pi$
$942$	−17.3137	−0.564111
$943$	31.0294	1.01046
$944$	−13.6569	−0.444493
$945$	2.82843	0.0920087
$946$	9.37258	0.304729
$947$	41.6569	1.35367	0.676833	−	0.736137i	$-0.263352\pi$
$947$	41.6569	1.35367	0.676833	+	0.736137i	$0.263352\pi$
$948$	13.6569	0.443554
$949$	−2.48528	−0.0806756
$950$	−2.82843	−0.0917663
$951$	2.68629	0.0871090
$952$	−13.6569	−0.442621
$953$	−56.1421	−1.81862	−0.909311	−	0.416117i	$-0.863391\pi$
$953$	−56.1421	−1.81862	−0.909311	+	0.416117i	$0.863391\pi$
$954$	−9.31371	−0.301542
$955$	−11.3137	−0.366103
$956$	16.0000	0.517477
$957$	17.9411	0.579954
$958$	11.3137	0.365529
$959$	15.0294	0.485326
$960$	1.00000	0.0322749
$961$	−15.0000	−0.483871
$962$	0.343146	0.0110635
$963$	−4.00000	−0.128898
$964$	−17.3137	−0.557637
$965$	−14.4853	−0.466298
$966$	24.0000	0.772187
$967$	−24.4853	−0.787394	−0.393697	−	0.919240i	$-0.628804\pi$
$967$	−24.4853	−0.787394	−0.393697	+	0.919240i	$0.628804\pi$
$968$	21.0000	0.674966
$969$	13.6569	0.438721
$970$	−8.82843	−0.283464
$971$	32.4853	1.04250	0.521251	−	0.853403i	$-0.325465\pi$
$971$	32.4853	1.04250	0.521251	+	0.853403i	$0.325465\pi$
$972$	1.00000	0.0320750
$973$	49.9411	1.60104
$974$	−0.485281	−0.0155494
$975$	−1.00000	−0.0320256
$976$	6.00000	0.192055
$977$	−19.6569	−0.628878	−0.314439	−	0.949278i	$-0.601816\pi$
$977$	−19.6569	−0.628878	−0.314439	+	0.949278i	$0.601816\pi$
$978$	−11.3137	−0.361773
$979$	24.5685	0.785214
$980$	1.00000	0.0319438
$981$	3.17157	0.101261
$982$	−9.85786	−0.314577
$983$	−13.6569	−0.435586	−0.217793	−	0.975995i	$-0.569886\pi$
$983$	−13.6569	−0.435586	−0.217793	+	0.975995i	$0.569886\pi$
$984$	3.65685	0.116576
$985$	9.31371	0.296759
$986$	15.3137	0.487688
$987$	−22.6274	−0.720239
$988$	2.82843	0.0899843
$989$	−14.0589	−0.447046
$990$	−5.65685	−0.179787
$991$	58.9117	1.87139	0.935696	−	0.352808i	$-0.114773\pi$
$991$	58.9117	1.87139	0.935696	+	0.352808i	$0.114773\pi$
$992$	4.00000	0.127000
$993$	8.48528	0.269272
$994$	16.0000	0.507489
$995$	21.6569	0.686568
$996$	17.6569	0.559479
$997$	38.6863	1.22521	0.612604	−	0.790390i	$-0.290122\pi$
$997$	38.6863	1.22521	0.612604	+	0.790390i	$0.290122\pi$
$998$	−16.4853	−0.521832
$999$	−0.343146	−0.0108567

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	390.2.a.h.1.2	✓	2
3.2	odd	2		1170.2.a.o.1.2		2
4.3	odd	2		3120.2.a.bc.1.1		2
5.2	odd	4		1950.2.e.o.1249.4		4
5.3	odd	4		1950.2.e.o.1249.1		4
5.4	even	2		1950.2.a.bd.1.1		2
12.11	even	2		9360.2.a.ch.1.1		2
13.5	odd	4		5070.2.b.q.1351.2		4
13.8	odd	4		5070.2.b.q.1351.3		4
13.12	even	2		5070.2.a.bc.1.1		2
15.2	even	4		5850.2.e.bk.5149.2		4
15.8	even	4		5850.2.e.bk.5149.3		4
15.14	odd	2		5850.2.a.cl.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
390.2.a.h.1.2	✓	2	1.1	even	1	trivial
1170.2.a.o.1.2		2	3.2	odd	2
1950.2.a.bd.1.1		2	5.4	even	2
1950.2.e.o.1249.1		4	5.3	odd	4
1950.2.e.o.1249.4		4	5.2	odd	4
3120.2.a.bc.1.1		2	4.3	odd	2
5070.2.a.bc.1.1		2	13.12	even	2
5070.2.b.q.1351.2		4	13.5	odd	4
5070.2.b.q.1351.3		4	13.8	odd	4
5850.2.a.cl.1.1		2	15.14	odd	2
5850.2.e.bk.5149.2		4	15.2	even	4
5850.2.e.bk.5149.3		4	15.8	even	4
9360.2.a.ch.1.1		2	12.11	even	2

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:	\( N \)	\(=\)	\( 390 = 2 \cdot 3 \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	390.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} +2.82843 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -5.65685 q^{11} +1.00000 q^{12} -1.00000 q^{13} +2.82843 q^{14} +1.00000 q^{15} +1.00000 q^{16} -4.82843 q^{17} +1.00000 q^{18} -2.82843 q^{19} +1.00000 q^{20} +2.82843 q^{21} -5.65685 q^{22} +8.48528 q^{23} +1.00000 q^{24} +1.00000 q^{25} -1.00000 q^{26} +1.00000 q^{27} +2.82843 q^{28} -3.17157 q^{29} +1.00000 q^{30} +4.00000 q^{31} +1.00000 q^{32} -5.65685 q^{33} -4.82843 q^{34} +2.82843 q^{35} +1.00000 q^{36} -0.343146 q^{37} -2.82843 q^{38} -1.00000 q^{39} +1.00000 q^{40} +3.65685 q^{41} +2.82843 q^{42} -1.65685 q^{43} -5.65685 q^{44} +1.00000 q^{45} +8.48528 q^{46} -8.00000 q^{47} +1.00000 q^{48} +1.00000 q^{49} +1.00000 q^{50} -4.82843 q^{51} -1.00000 q^{52} -9.31371 q^{53} +1.00000 q^{54} -5.65685 q^{55} +2.82843 q^{56} -2.82843 q^{57} -3.17157 q^{58} -13.6569 q^{59} +1.00000 q^{60} +6.00000 q^{61} +4.00000 q^{62} +2.82843 q^{63} +1.00000 q^{64} -1.00000 q^{65} -5.65685 q^{66} -5.65685 q^{67} -4.82843 q^{68} +8.48528 q^{69} +2.82843 q^{70} +5.65685 q^{71} +1.00000 q^{72} +2.48528 q^{73} -0.343146 q^{74} +1.00000 q^{75} -2.82843 q^{76} -16.0000 q^{77} -1.00000 q^{78} +13.6569 q^{79} +1.00000 q^{80} +1.00000 q^{81} +3.65685 q^{82} +17.6569 q^{83} +2.82843 q^{84} -4.82843 q^{85} -1.65685 q^{86} -3.17157 q^{87} -5.65685 q^{88} -4.34315 q^{89} +1.00000 q^{90} -2.82843 q^{91} +8.48528 q^{92} +4.00000 q^{93} -8.00000 q^{94} -2.82843 q^{95} +1.00000 q^{96} -8.82843 q^{97} +1.00000 q^{98} -5.65685 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{5} + 2 q^{6} + 2 q^{8} + 2 q^{9} + 2 q^{10} + 2 q^{12} - 2 q^{13} + 2 q^{15} + 2 q^{16} - 4 q^{17} + 2 q^{18} + 2 q^{20} + 2 q^{24} + 2 q^{25} - 2 q^{26} + 2 q^{27}+ \cdots + 2 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 + 2 * q^5 + 2 * q^6 + 2 * q^8 + 2 * q^9 + 2 * q^10 + 2 * q^12 - 2 * q^13 + 2 * q^15 + 2 * q^16 - 4 * q^17 + 2 * q^18 + 2 * q^20 + 2 * q^24 + 2 * q^25 - 2 * q^26 + 2 * q^27 - 12 * q^29 + 2 * q^30 + 8 * q^31 + 2 * q^32 - 4 * q^34 + 2 * q^36 - 12 * q^37 - 2 * q^39 + 2 * q^40 - 4 * q^41 + 8 * q^43 + 2 * q^45 - 16 * q^47 + 2 * q^48 + 2 * q^49 + 2 * q^50 - 4 * q^51 - 2 * q^52 + 4 * q^53 + 2 * q^54 - 12 * q^58 - 16 * q^59 + 2 * q^60 + 12 * q^61 + 8 * q^62 + 2 * q^64 - 2 * q^65 - 4 * q^68 + 2 * q^72 - 12 * q^73 - 12 * q^74 + 2 * q^75 - 32 * q^77 - 2 * q^78 + 16 * q^79 + 2 * q^80 + 2 * q^81 - 4 * q^82 + 24 * q^83 - 4 * q^85 + 8 * q^86 - 12 * q^87 - 20 * q^89 + 2 * q^90 + 8 * q^93 - 16 * q^94 + 2 * q^96 - 12 * q^97 + 2 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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