LMFDB - Embedded newform 507.2.b.e.337.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [507,2,Mod(337,507)] 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("507.337"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(507, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 507 = 3 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	507.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,4,-4,0,0,0,0,4,16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(10)] == traces)

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

Self dual:	no
Analytic conductor:	$4.04841538248$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 39)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.4
Root			$0.707107 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	507.337
Dual form			507.2.b.e.337.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.41421i q^{2} +1.00000 q^{3} -3.82843 q^{4} -2.82843i q^{5} +2.41421i q^{6} -2.82843i q^{7} -4.41421i q^{8} +1.00000 q^{9} +6.82843 q^{10} -2.00000i q^{11} -3.82843 q^{12} +6.82843 q^{14} -2.82843i q^{15} +3.00000 q^{16} +3.65685 q^{17} +2.41421i q^{18} -2.82843i q^{19} +10.8284i q^{20} -2.82843i q^{21} +4.82843 q^{22} +4.00000 q^{23} -4.41421i q^{24} -3.00000 q^{25} +1.00000 q^{27} +10.8284i q^{28} +2.00000 q^{29} +6.82843 q^{30} +6.82843i q^{31} -1.58579i q^{32} -2.00000i q^{33} +8.82843i q^{34} -8.00000 q^{35} -3.82843 q^{36} +3.65685i q^{37} +6.82843 q^{38} -12.4853 q^{40} -10.8284i q^{41} +6.82843 q^{42} -9.65685 q^{43} +7.65685i q^{44} -2.82843i q^{45} +9.65685i q^{46} -0.343146i q^{47} +3.00000 q^{48} -1.00000 q^{49} -7.24264i q^{50} +3.65685 q^{51} -2.00000 q^{53} +2.41421i q^{54} -5.65685 q^{55} -12.4853 q^{56} -2.82843i q^{57} +4.82843i q^{58} -3.65685i q^{59} +10.8284i q^{60} -9.31371 q^{61} -16.4853 q^{62} -2.82843i q^{63} +9.82843 q^{64} +4.82843 q^{66} -1.17157i q^{67} -14.0000 q^{68} +4.00000 q^{69} -19.3137i q^{70} -2.00000i q^{71} -4.41421i q^{72} +11.6569i q^{73} -8.82843 q^{74} -3.00000 q^{75} +10.8284i q^{76} -5.65685 q^{77} +11.3137 q^{79} -8.48528i q^{80} +1.00000 q^{81} +26.1421 q^{82} +7.65685i q^{83} +10.8284i q^{84} -10.3431i q^{85} -23.3137i q^{86} +2.00000 q^{87} -8.82843 q^{88} +9.17157i q^{89} +6.82843 q^{90} -15.3137 q^{92} +6.82843i q^{93} +0.828427 q^{94} -8.00000 q^{95} -1.58579i q^{96} +7.65685i q^{97} -2.41421i q^{98} -2.00000i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} - 4 q^{4} + 4 q^{9} + 16 q^{10} - 4 q^{12} + 16 q^{14} + 12 q^{16} - 8 q^{17} + 8 q^{22} + 16 q^{23} - 12 q^{25} + 4 q^{27} + 8 q^{29} + 16 q^{30} - 32 q^{35} - 4 q^{36} + 16 q^{38} - 16 q^{40}+ \cdots - 32 q^{95}+O(q^{100}) $ 4 * q + 4 * q^3 - 4 * q^4 + 4 * q^9 + 16 * q^10 - 4 * q^12 + 16 * q^14 + 12 * q^16 - 8 * q^17 + 8 * q^22 + 16 * q^23 - 12 * q^25 + 4 * q^27 + 8 * q^29 + 16 * q^30 - 32 * q^35 - 4 * q^36 + 16 * q^38 - 16 * q^40 + 16 * q^42 - 16 * q^43 + 12 * q^48 - 4 * q^49 - 8 * q^51 - 8 * q^53 - 16 * q^56 + 8 * q^61 - 32 * q^62 + 28 * q^64 + 8 * q^66 - 56 * q^68 + 16 * q^69 - 24 * q^74 - 12 * q^75 + 4 * q^81 + 48 * q^82 + 8 * q^87 - 24 * q^88 + 16 * q^90 - 16 * q^92 - 8 * q^94 - 32 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/507\mathbb{Z}\right)^\times$.

$n$	$170$	$340$
$\chi(n)$	$1$	$-1$

Coefficient data

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

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

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

$2$	2.41421i	1.70711i	0.521005	+	0.853553i	$0.325557\pi$
$2$	2.41421i	1.70711i	−0.521005	+	0.853553i	$0.674443\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−3.82843	−1.91421
$5$	− 2.82843i	− 1.26491i	−0.774597	−	0.632456i	$-0.782047\pi$
$5$	− 2.82843i	− 1.26491i	0.774597	−	0.632456i	$-0.217953\pi$
$6$	2.41421i	0.985599i
$7$	− 2.82843i	− 1.06904i	−0.845154	−	0.534522i	$-0.820491\pi$
$7$	− 2.82843i	− 1.06904i	0.845154	−	0.534522i	$-0.179509\pi$
$8$	− 4.41421i	− 1.56066i
$9$	1.00000	0.333333
$10$	6.82843	2.15934
$11$	− 2.00000i	− 0.603023i	−0.953463	−	0.301511i	$-0.902509\pi$
$11$	− 2.00000i	− 0.603023i	0.953463	−	0.301511i	$-0.0974911\pi$
$12$	−3.82843	−1.10517
$13$	0	0
$13$	0	0
$14$	6.82843	1.82497
$15$	− 2.82843i	− 0.730297i
$16$	3.00000	0.750000
$17$	3.65685	0.886917	0.443459	−	0.896295i	$-0.353751\pi$
$17$	3.65685	0.886917	0.443459	+	0.896295i	$0.353751\pi$
$18$	2.41421i	0.569036i
$19$	− 2.82843i	− 0.648886i	−0.945905	−	0.324443i	$-0.894823\pi$
$19$	− 2.82843i	− 0.648886i	0.945905	−	0.324443i	$-0.105177\pi$
$20$	10.8284i	2.42131i
$21$	− 2.82843i	− 0.617213i
$22$	4.82843	1.02942
$23$	4.00000	0.834058	0.417029	−	0.908893i	$-0.363071\pi$
$23$	4.00000	0.834058	0.417029	+	0.908893i	$0.363071\pi$
$24$	− 4.41421i	− 0.901048i
$25$	−3.00000	−0.600000
$26$	0	0
$27$	1.00000	0.192450
$28$	10.8284i	2.04638i
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	6.82843	1.24669
$31$	6.82843i	1.22642i	0.789919	+	0.613211i	$0.210122\pi$
$31$	6.82843i	1.22642i	−0.789919	+	0.613211i	$0.789878\pi$
$32$	− 1.58579i	− 0.280330i
$33$	− 2.00000i	− 0.348155i
$34$	8.82843i	1.51406i
$35$	−8.00000	−1.35225
$36$	−3.82843	−0.638071
$37$	3.65685i	0.601183i	0.953753	+	0.300592i	$0.0971841\pi$
$37$	3.65685i	0.601183i	−0.953753	+	0.300592i	$0.902816\pi$
$38$	6.82843	1.10772
$39$	0	0
$40$	−12.4853	−1.97410
$41$	− 10.8284i	− 1.69112i	−0.533883	−	0.845558i	$-0.679268\pi$
$41$	− 10.8284i	− 1.69112i	0.533883	−	0.845558i	$-0.320732\pi$
$42$	6.82843	1.05365
$43$	−9.65685	−1.47266	−0.736328	−	0.676625i	$-0.763442\pi$
$43$	−9.65685	−1.47266	−0.736328	+	0.676625i	$0.763442\pi$
$44$	7.65685i	1.15431i
$45$	− 2.82843i	− 0.421637i
$46$	9.65685i	1.42383i
$47$	− 0.343146i	− 0.0500530i	−0.999687	−	0.0250265i	$-0.992033\pi$
$47$	− 0.343146i	− 0.0500530i	0.999687	−	0.0250265i	$-0.00796701\pi$
$48$	3.00000	0.433013
$49$	−1.00000	−0.142857
$50$	− 7.24264i	− 1.02426i
$51$	3.65685	0.512062
$52$	0	0
$53$	−2.00000	−0.274721	−0.137361	−	0.990521i	$-0.543862\pi$
$53$	−2.00000	−0.274721	−0.137361	+	0.990521i	$0.543862\pi$
$54$	2.41421i	0.328533i
$55$	−5.65685	−0.762770
$56$	−12.4853	−1.66842
$57$	− 2.82843i	− 0.374634i
$58$	4.82843i	0.634004i
$59$	− 3.65685i	− 0.476082i	−0.971255	−	0.238041i	$-0.923495\pi$
$59$	− 3.65685i	− 0.476082i	0.971255	−	0.238041i	$-0.0765052\pi$
$60$	10.8284i	1.39794i
$61$	−9.31371	−1.19250	−0.596249	−	0.802799i	$-0.703343\pi$
$61$	−9.31371	−1.19250	−0.596249	+	0.802799i	$0.703343\pi$
$62$	−16.4853	−2.09363
$63$	− 2.82843i	− 0.356348i
$64$	9.82843	1.22855
$65$	0	0
$66$	4.82843	0.594338
$67$	− 1.17157i	− 0.143130i	−0.997436	−	0.0715652i	$-0.977201\pi$
$67$	− 1.17157i	− 0.143130i	0.997436	−	0.0715652i	$-0.0227994\pi$
$68$	−14.0000	−1.69775
$69$	4.00000	0.481543
$70$	− 19.3137i	− 2.30843i
$71$	− 2.00000i	− 0.237356i	−0.992933	−	0.118678i	$-0.962134\pi$
$71$	− 2.00000i	− 0.237356i	0.992933	−	0.118678i	$-0.0378657\pi$
$72$	− 4.41421i	− 0.520220i
$73$	11.6569i	1.36433i	0.731198	+	0.682166i	$0.238962\pi$
$73$	11.6569i	1.36433i	−0.731198	+	0.682166i	$0.761038\pi$
$74$	−8.82843	−1.02628
$75$	−3.00000	−0.346410
$76$	10.8284i	1.24211i
$77$	−5.65685	−0.644658
$78$	0	0
$79$	11.3137	1.27289	0.636446	−	0.771321i	$-0.280404\pi$
$79$	11.3137	1.27289	0.636446	+	0.771321i	$0.280404\pi$
$80$	− 8.48528i	− 0.948683i
$81$	1.00000	0.111111
$82$	26.1421	2.88692
$83$	7.65685i	0.840449i	0.907420	+	0.420224i	$0.138049\pi$
$83$	7.65685i	0.840449i	−0.907420	+	0.420224i	$0.861951\pi$
$84$	10.8284i	1.18148i
$85$	− 10.3431i	− 1.12187i
$86$	− 23.3137i	− 2.51398i
$87$	2.00000	0.214423
$88$	−8.82843	−0.941113
$89$	9.17157i	0.972185i	0.873907	+	0.486092i	$0.161578\pi$
$89$	9.17157i	0.972185i	−0.873907	+	0.486092i	$0.838422\pi$
$90$	6.82843	0.719779
$91$	0	0
$92$	−15.3137	−1.59656
$93$	6.82843i	0.708075i
$94$	0.828427	0.0854457
$95$	−8.00000	−0.820783
$96$	− 1.58579i	− 0.161849i
$97$	7.65685i	0.777436i	0.921357	+	0.388718i	$0.127082\pi$
$97$	7.65685i	0.777436i	−0.921357	+	0.388718i	$0.872918\pi$
$98$	− 2.41421i	− 0.243872i
$99$	− 2.00000i	− 0.201008i
$100$	11.4853	1.14853
$101$	3.65685	0.363871	0.181935	−	0.983311i	$-0.441764\pi$
$101$	3.65685	0.363871	0.181935	+	0.983311i	$0.441764\pi$
$102$	8.82843i	0.874145i
$103$	−13.6569	−1.34565	−0.672825	−	0.739802i	$-0.734919\pi$
$103$	−13.6569	−1.34565	−0.672825	+	0.739802i	$0.734919\pi$
$104$	0	0
$105$	−8.00000	−0.780720
$106$	− 4.82843i	− 0.468978i
$107$	11.3137	1.09374	0.546869	−	0.837218i	$-0.315820\pi$
$107$	11.3137	1.09374	0.546869	+	0.837218i	$0.315820\pi$
$108$	−3.82843	−0.368391
$109$	17.3137i	1.65835i	0.558987	+	0.829176i	$0.311190\pi$
$109$	17.3137i	1.65835i	−0.558987	+	0.829176i	$0.688810\pi$
$110$	− 13.6569i	− 1.30213i
$111$	3.65685i	0.347093i
$112$	− 8.48528i	− 0.801784i
$113$	17.3137	1.62874	0.814368	−	0.580348i	$-0.197084\pi$
$113$	17.3137	1.62874	0.814368	+	0.580348i	$0.197084\pi$
$114$	6.82843	0.639541
$115$	− 11.3137i	− 1.05501i
$116$	−7.65685	−0.710921
$117$	0	0
$118$	8.82843	0.812723
$119$	− 10.3431i	− 0.948155i
$120$	−12.4853	−1.13975
$121$	7.00000	0.636364
$122$	− 22.4853i	− 2.03572i
$123$	− 10.8284i	− 0.976366i
$124$	− 26.1421i	− 2.34763i
$125$	− 5.65685i	− 0.505964i
$126$	6.82843	0.608325
$127$	5.65685	0.501965	0.250982	−	0.967992i	$-0.419246\pi$
$127$	5.65685	0.501965	0.250982	+	0.967992i	$0.419246\pi$
$128$	20.5563i	1.81694i
$129$	−9.65685	−0.850239
$130$	0	0
$131$	−8.00000	−0.698963	−0.349482	−	0.936943i	$-0.613642\pi$
$131$	−8.00000	−0.698963	−0.349482	+	0.936943i	$0.613642\pi$
$132$	7.65685i	0.666444i
$133$	−8.00000	−0.693688
$134$	2.82843	0.244339
$135$	− 2.82843i	− 0.243432i
$136$	− 16.1421i	− 1.38418i
$137$	− 5.17157i	− 0.441837i	−0.975292	−	0.220919i	$-0.929094\pi$
$137$	− 5.17157i	− 0.441837i	0.975292	−	0.220919i	$-0.0709055\pi$
$138$	9.65685i	0.822046i
$139$	15.3137	1.29889	0.649446	−	0.760408i	$-0.275001\pi$
$139$	15.3137	1.29889	0.649446	+	0.760408i	$0.275001\pi$
$140$	30.6274	2.58849
$141$	− 0.343146i	− 0.0288981i
$142$	4.82843	0.405193
$143$	0	0
$144$	3.00000	0.250000
$145$	− 5.65685i	− 0.469776i
$146$	−28.1421	−2.32906
$147$	−1.00000	−0.0824786
$148$	− 14.0000i	− 1.15079i
$149$	14.8284i	1.21479i	0.794399	+	0.607396i	$0.207786\pi$
$149$	14.8284i	1.21479i	−0.794399	+	0.607396i	$0.792214\pi$
$150$	− 7.24264i	− 0.591359i
$151$	− 20.4853i	− 1.66707i	−0.552468	−	0.833534i	$-0.686314\pi$
$151$	− 20.4853i	− 1.66707i	0.552468	−	0.833534i	$-0.313686\pi$
$152$	−12.4853	−1.01269
$153$	3.65685	0.295639
$154$	− 13.6569i	− 1.10050i
$155$	19.3137	1.55131
$156$	0	0
$157$	−10.0000	−0.798087	−0.399043	−	0.916932i	$-0.630658\pi$
$157$	−10.0000	−0.798087	−0.399043	+	0.916932i	$0.630658\pi$
$158$	27.3137i	2.17296i
$159$	−2.00000	−0.158610
$160$	−4.48528	−0.354593
$161$	− 11.3137i	− 0.891645i
$162$	2.41421i	0.189679i
$163$	13.1716i	1.03168i	0.856686	+	0.515839i	$0.172520\pi$
$163$	13.1716i	1.03168i	−0.856686	+	0.515839i	$0.827480\pi$
$164$	41.4558i	3.23716i
$165$	−5.65685	−0.440386
$166$	−18.4853	−1.43474
$167$	7.65685i	0.592505i	0.955110	+	0.296253i	$0.0957370\pi$
$167$	7.65685i	0.592505i	−0.955110	+	0.296253i	$0.904263\pi$
$168$	−12.4853	−0.963260
$169$	0	0
$170$	24.9706	1.91515
$171$	− 2.82843i	− 0.216295i
$172$	36.9706	2.81898
$173$	0.343146	0.0260889	0.0130444	−	0.999915i	$-0.495848\pi$
$173$	0.343146	0.0260889	0.0130444	+	0.999915i	$0.495848\pi$
$174$	4.82843i	0.366042i
$175$	8.48528i	0.641427i
$176$	− 6.00000i	− 0.452267i
$177$	− 3.65685i	− 0.274866i
$178$	−22.1421	−1.65962
$179$	0.686292	0.0512958	0.0256479	−	0.999671i	$-0.491835\pi$
$179$	0.686292	0.0512958	0.0256479	+	0.999671i	$0.491835\pi$
$180$	10.8284i	0.807103i
$181$	−14.0000	−1.04061	−0.520306	−	0.853980i	$-0.674182\pi$
$181$	−14.0000	−1.04061	−0.520306	+	0.853980i	$0.674182\pi$
$182$	0	0
$183$	−9.31371	−0.688489
$184$	− 17.6569i	− 1.30168i
$185$	10.3431	0.760443
$186$	−16.4853	−1.20876
$187$	− 7.31371i	− 0.534831i
$188$	1.31371i	0.0958120i
$189$	− 2.82843i	− 0.205738i
$190$	− 19.3137i	− 1.40116i
$191$	−19.3137	−1.39749	−0.698745	−	0.715370i	$-0.746258\pi$
$191$	−19.3137	−1.39749	−0.698745	+	0.715370i	$0.746258\pi$
$192$	9.82843	0.709306
$193$	− 17.3137i	− 1.24627i	−0.782115	−	0.623134i	$-0.785859\pi$
$193$	− 17.3137i	− 1.24627i	0.782115	−	0.623134i	$-0.214141\pi$
$194$	−18.4853	−1.32717
$195$	0	0
$196$	3.82843	0.273459
$197$	16.4853i	1.17453i	0.809396	+	0.587264i	$0.199795\pi$
$197$	16.4853i	1.17453i	−0.809396	+	0.587264i	$0.800205\pi$
$198$	4.82843	0.343141
$199$	−10.3431	−0.733206	−0.366603	−	0.930377i	$-0.619479\pi$
$199$	−10.3431	−0.733206	−0.366603	+	0.930377i	$0.619479\pi$
$200$	13.2426i	0.936396i
$201$	− 1.17157i	− 0.0826364i
$202$	8.82843i	0.621166i
$203$	− 5.65685i	− 0.397033i
$204$	−14.0000	−0.980196
$205$	−30.6274	−2.13911
$206$	− 32.9706i	− 2.29717i
$207$	4.00000	0.278019
$208$	0	0
$209$	−5.65685	−0.391293
$210$	− 19.3137i	− 1.33277i
$211$	−12.0000	−0.826114	−0.413057	−	0.910705i	$-0.635539\pi$
$211$	−12.0000	−0.826114	−0.413057	+	0.910705i	$0.635539\pi$
$212$	7.65685	0.525875
$213$	− 2.00000i	− 0.137038i
$214$	27.3137i	1.86713i
$215$	27.3137i	1.86278i
$216$	− 4.41421i	− 0.300349i
$217$	19.3137	1.31110
$218$	−41.7990	−2.83098
$219$	11.6569i	0.787697i
$220$	21.6569	1.46010
$221$	0	0
$222$	−8.82843	−0.592525
$223$	− 4.48528i	− 0.300357i	−0.988659	−	0.150178i	$-0.952015\pi$
$223$	− 4.48528i	− 0.300357i	0.988659	−	0.150178i	$-0.0479848\pi$
$224$	−4.48528	−0.299685
$225$	−3.00000	−0.200000
$226$	41.7990i	2.78043i
$227$	− 5.31371i	− 0.352683i	−0.984329	−	0.176342i	$-0.943574\pi$
$227$	− 5.31371i	− 0.352683i	0.984329	−	0.176342i	$-0.0564263\pi$
$228$	10.8284i	0.717130i
$229$	21.3137i	1.40845i	0.709977	+	0.704225i	$0.248705\pi$
$229$	21.3137i	1.40845i	−0.709977	+	0.704225i	$0.751295\pi$
$230$	27.3137	1.80101
$231$	−5.65685	−0.372194
$232$	− 8.82843i	− 0.579615i
$233$	26.9706	1.76690	0.883450	−	0.468525i	$-0.155214\pi$
$233$	26.9706	1.76690	0.883450	+	0.468525i	$0.155214\pi$
$234$	0	0
$235$	−0.970563	−0.0633125
$236$	14.0000i	0.911322i
$237$	11.3137	0.734904
$238$	24.9706	1.61860
$239$	− 2.00000i	− 0.129369i	−0.997906	−	0.0646846i	$-0.979396\pi$
$239$	− 2.00000i	− 0.129369i	0.997906	−	0.0646846i	$-0.0206041\pi$
$240$	− 8.48528i	− 0.547723i
$241$	11.6569i	0.750884i	0.926846	+	0.375442i	$0.122509\pi$
$241$	11.6569i	0.750884i	−0.926846	+	0.375442i	$0.877491\pi$
$242$	16.8995i	1.08634i
$243$	1.00000	0.0641500
$244$	35.6569	2.28270
$245$	2.82843i	0.180702i
$246$	26.1421	1.66676
$247$	0	0
$248$	30.1421	1.91403
$249$	7.65685i	0.485233i
$250$	13.6569	0.863735
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	10.8284i	0.682127i
$253$	− 8.00000i	− 0.502956i
$254$	13.6569i	0.856907i
$255$	− 10.3431i	− 0.647713i
$256$	−29.9706	−1.87316
$257$	15.6569	0.976648	0.488324	−	0.872662i	$-0.337608\pi$
$257$	15.6569	0.976648	0.488324	+	0.872662i	$0.337608\pi$
$258$	− 23.3137i	− 1.45145i
$259$	10.3431	0.642692
$260$	0	0
$261$	2.00000	0.123797
$262$	− 19.3137i	− 1.19320i
$263$	12.0000	0.739952	0.369976	−	0.929041i	$-0.379366\pi$
$263$	12.0000	0.739952	0.369976	+	0.929041i	$0.379366\pi$
$264$	−8.82843	−0.543352
$265$	5.65685i	0.347498i
$266$	− 19.3137i	− 1.18420i
$267$	9.17157i	0.561291i
$268$	4.48528i	0.273982i
$269$	18.0000	1.09748	0.548740	−	0.835993i	$-0.315108\pi$
$269$	18.0000	1.09748	0.548740	+	0.835993i	$0.315108\pi$
$270$	6.82843	0.415565
$271$	11.7990i	0.716738i	0.933580	+	0.358369i	$0.116667\pi$
$271$	11.7990i	0.716738i	−0.933580	+	0.358369i	$0.883333\pi$
$272$	10.9706	0.665188
$273$	0	0
$274$	12.4853	0.754263
$275$	6.00000i	0.361814i
$276$	−15.3137	−0.921777
$277$	2.00000	0.120168	0.0600842	−	0.998193i	$-0.480863\pi$
$277$	2.00000	0.120168	0.0600842	+	0.998193i	$0.480863\pi$
$278$	36.9706i	2.21735i
$279$	6.82843i	0.408807i
$280$	35.3137i	2.11040i
$281$	26.8284i	1.60045i	0.599700	+	0.800225i	$0.295287\pi$
$281$	26.8284i	1.60045i	−0.599700	+	0.800225i	$0.704713\pi$
$282$	0.828427	0.0493321
$283$	4.97056	0.295469	0.147735	−	0.989027i	$-0.452802\pi$
$283$	4.97056	0.295469	0.147735	+	0.989027i	$0.452802\pi$
$284$	7.65685i	0.454351i
$285$	−8.00000	−0.473879
$286$	0	0
$287$	−30.6274	−1.80788
$288$	− 1.58579i	− 0.0934434i
$289$	−3.62742	−0.213377
$290$	13.6569	0.801958
$291$	7.65685i	0.448853i
$292$	− 44.6274i	− 2.61162i
$293$	− 26.1421i	− 1.52724i	−0.645666	−	0.763620i	$-0.723420\pi$
$293$	− 26.1421i	− 1.52724i	0.645666	−	0.763620i	$-0.276580\pi$
$294$	− 2.41421i	− 0.140800i
$295$	−10.3431	−0.602201
$296$	16.1421	0.938243
$297$	− 2.00000i	− 0.116052i
$298$	−35.7990	−2.07378
$299$	0	0
$300$	11.4853	0.663103
$301$	27.3137i	1.57434i
$302$	49.4558	2.84586
$303$	3.65685	0.210081
$304$	− 8.48528i	− 0.486664i
$305$	26.3431i	1.50840i
$306$	8.82843i	0.504688i
$307$	− 17.1716i	− 0.980033i	−0.871713	−	0.490017i	$-0.836991\pi$
$307$	− 17.1716i	− 0.980033i	0.871713	−	0.490017i	$-0.163009\pi$
$308$	21.6569	1.23401
$309$	−13.6569	−0.776911
$310$	46.6274i	2.64826i
$311$	−34.6274	−1.96354	−0.981770	−	0.190071i	$-0.939128\pi$
$311$	−34.6274	−1.96354	−0.981770	+	0.190071i	$0.939128\pi$
$312$	0	0
$313$	6.00000	0.339140	0.169570	−	0.985518i	$-0.445762\pi$
$313$	6.00000	0.339140	0.169570	+	0.985518i	$0.445762\pi$
$314$	− 24.1421i	− 1.36242i
$315$	−8.00000	−0.450749
$316$	−43.3137	−2.43659
$317$	8.48528i	0.476581i	0.971194	+	0.238290i	$0.0765870\pi$
$317$	8.48528i	0.476581i	−0.971194	+	0.238290i	$0.923413\pi$
$318$	− 4.82843i	− 0.270765i
$319$	− 4.00000i	− 0.223957i
$320$	− 27.7990i	− 1.55401i
$321$	11.3137	0.631470
$322$	27.3137	1.52213
$323$	− 10.3431i	− 0.575508i
$324$	−3.82843	−0.212690
$325$	0	0
$326$	−31.7990	−1.76118
$327$	17.3137i	0.957450i
$328$	−47.7990	−2.63926
$329$	−0.970563	−0.0535089
$330$	− 13.6569i	− 0.751785i
$331$	2.14214i	0.117742i	0.998266	+	0.0588712i	$0.0187501\pi$
$331$	2.14214i	0.117742i	−0.998266	+	0.0588712i	$0.981250\pi$
$332$	− 29.3137i	− 1.60880i
$333$	3.65685i	0.200394i
$334$	−18.4853	−1.01147
$335$	−3.31371	−0.181047
$336$	− 8.48528i	− 0.462910i
$337$	13.3137	0.725244	0.362622	−	0.931936i	$-0.381882\pi$
$337$	13.3137	0.725244	0.362622	+	0.931936i	$0.381882\pi$
$338$	0	0
$339$	17.3137	0.940352
$340$	39.5980i	2.14750i
$341$	13.6569	0.739560
$342$	6.82843	0.369239
$343$	− 16.9706i	− 0.916324i
$344$	42.6274i	2.29832i
$345$	− 11.3137i	− 0.609110i
$346$	0.828427i	0.0445365i
$347$	−31.3137	−1.68101	−0.840504	−	0.541805i	$-0.817741\pi$
$347$	−31.3137	−1.68101	−0.840504	+	0.541805i	$0.817741\pi$
$348$	−7.65685	−0.410450
$349$	− 7.65685i	− 0.409862i	−0.978776	−	0.204931i	$-0.934303\pi$
$349$	− 7.65685i	− 0.409862i	0.978776	−	0.204931i	$-0.0656970\pi$
$350$	−20.4853	−1.09498
$351$	0	0
$352$	−3.17157	−0.169045
$353$	− 17.4558i	− 0.929081i	−0.885552	−	0.464540i	$-0.846220\pi$
$353$	− 17.4558i	− 0.929081i	0.885552	−	0.464540i	$-0.153780\pi$
$354$	8.82843	0.469226
$355$	−5.65685	−0.300235
$356$	− 35.1127i	− 1.86097i
$357$	− 10.3431i	− 0.547417i
$358$	1.65685i	0.0875675i
$359$	1.02944i	0.0543316i	0.999631	+	0.0271658i	$0.00864821\pi$
$359$	1.02944i	0.0543316i	−0.999631	+	0.0271658i	$0.991352\pi$
$360$	−12.4853	−0.658032
$361$	11.0000	0.578947
$362$	− 33.7990i	− 1.77644i
$363$	7.00000	0.367405
$364$	0	0
$365$	32.9706	1.72576
$366$	− 22.4853i	− 1.17532i
$367$	−24.0000	−1.25279	−0.626395	−	0.779506i	$-0.715470\pi$
$367$	−24.0000	−1.25279	−0.626395	+	0.779506i	$0.715470\pi$
$368$	12.0000	0.625543
$369$	− 10.8284i	− 0.563705i
$370$	24.9706i	1.29816i
$371$	5.65685i	0.293689i
$372$	− 26.1421i	− 1.35541i
$373$	10.0000	0.517780	0.258890	−	0.965907i	$-0.416643\pi$
$373$	10.0000	0.517780	0.258890	+	0.965907i	$0.416643\pi$
$374$	17.6569	0.913014
$375$	− 5.65685i	− 0.292119i
$376$	−1.51472	−0.0781156
$377$	0	0
$378$	6.82843	0.351216
$379$	16.4853i	0.846792i	0.905945	+	0.423396i	$0.139162\pi$
$379$	16.4853i	0.846792i	−0.905945	+	0.423396i	$0.860838\pi$
$380$	30.6274	1.57115
$381$	5.65685	0.289809
$382$	− 46.6274i	− 2.38567i
$383$	2.97056i	0.151789i	0.997116	+	0.0758943i	$0.0241812\pi$
$383$	2.97056i	0.151789i	−0.997116	+	0.0758943i	$0.975819\pi$
$384$	20.5563i	1.04901i
$385$	16.0000i	0.815436i
$386$	41.7990	2.12751
$387$	−9.65685	−0.490885
$388$	− 29.3137i	− 1.48818i
$389$	−6.97056	−0.353422	−0.176711	−	0.984263i	$-0.556546\pi$
$389$	−6.97056	−0.353422	−0.176711	+	0.984263i	$0.556546\pi$
$390$	0	0
$391$	14.6274	0.739740
$392$	4.41421i	0.222951i
$393$	−8.00000	−0.403547
$394$	−39.7990	−2.00504
$395$	− 32.0000i	− 1.61009i
$396$	7.65685i	0.384771i
$397$	− 2.97056i	− 0.149088i	−0.997218	−	0.0745441i	$-0.976250\pi$
$397$	− 2.97056i	− 0.149088i	0.997218	−	0.0745441i	$-0.0237502\pi$
$398$	− 24.9706i	− 1.25166i
$399$	−8.00000	−0.400501
$400$	−9.00000	−0.450000
$401$	− 2.14214i	− 0.106973i	−0.998569	−	0.0534866i	$-0.982967\pi$
$401$	− 2.14214i	− 0.106973i	0.998569	−	0.0534866i	$-0.0170334\pi$
$402$	2.82843	0.141069
$403$	0	0
$404$	−14.0000	−0.696526
$405$	− 2.82843i	− 0.140546i
$406$	13.6569	0.677778
$407$	7.31371	0.362527
$408$	− 16.1421i	− 0.799155i
$409$	1.02944i	0.0509024i	0.999676	+	0.0254512i	$0.00810224\pi$
$409$	1.02944i	0.0509024i	−0.999676	+	0.0254512i	$0.991898\pi$
$410$	− 73.9411i	− 3.65169i
$411$	− 5.17157i	− 0.255095i
$412$	52.2843	2.57586
$413$	−10.3431	−0.508953
$414$	9.65685i	0.474608i
$415$	21.6569	1.06309
$416$	0	0
$417$	15.3137	0.749916
$418$	− 13.6569i	− 0.667979i
$419$	−30.6274	−1.49625	−0.748124	−	0.663559i	$-0.769045\pi$
$419$	−30.6274	−1.49625	−0.748124	+	0.663559i	$0.769045\pi$
$420$	30.6274	1.49446
$421$	− 14.6863i	− 0.715766i	−0.933766	−	0.357883i	$-0.883499\pi$
$421$	− 14.6863i	− 0.715766i	0.933766	−	0.357883i	$-0.116501\pi$
$422$	− 28.9706i	− 1.41026i
$423$	− 0.343146i	− 0.0166843i
$424$	8.82843i	0.428746i
$425$	−10.9706	−0.532150
$426$	4.82843	0.233938
$427$	26.3431i	1.27483i
$428$	−43.3137	−2.09365
$429$	0	0
$430$	−65.9411	−3.17996
$431$	− 19.6569i	− 0.946837i	−0.880838	−	0.473419i	$-0.843020\pi$
$431$	− 19.6569i	− 0.946837i	0.880838	−	0.473419i	$-0.156980\pi$
$432$	3.00000	0.144338
$433$	−1.31371	−0.0631328	−0.0315664	−	0.999502i	$-0.510050\pi$
$433$	−1.31371	−0.0631328	−0.0315664	+	0.999502i	$0.510050\pi$
$434$	46.6274i	2.23819i
$435$	− 5.65685i	− 0.271225i
$436$	− 66.2843i	− 3.17444i
$437$	− 11.3137i	− 0.541208i
$438$	−28.1421	−1.34468
$439$	16.9706	0.809961	0.404980	−	0.914325i	$-0.367278\pi$
$439$	16.9706	0.809961	0.404980	+	0.914325i	$0.367278\pi$
$440$	24.9706i	1.19042i
$441$	−1.00000	−0.0476190
$442$	0	0
$443$	41.9411	1.99268	0.996342	−	0.0854611i	$-0.0272364\pi$
$443$	41.9411	1.99268	0.996342	+	0.0854611i	$0.0272364\pi$
$444$	− 14.0000i	− 0.664411i
$445$	25.9411	1.22973
$446$	10.8284	0.512741
$447$	14.8284i	0.701361i
$448$	− 27.7990i	− 1.31338i
$449$	7.79899i	0.368057i	0.982921	+	0.184029i	$0.0589139\pi$
$449$	7.79899i	0.368057i	−0.982921	+	0.184029i	$0.941086\pi$
$450$	− 7.24264i	− 0.341421i
$451$	−21.6569	−1.01978
$452$	−66.2843	−3.11775
$453$	− 20.4853i	− 0.962482i
$454$	12.8284	0.602068
$455$	0	0
$456$	−12.4853	−0.584677
$457$	− 3.65685i	− 0.171060i	−0.996336	−	0.0855302i	$-0.972742\pi$
$457$	− 3.65685i	− 0.171060i	0.996336	−	0.0855302i	$-0.0272584\pi$
$458$	−51.4558	−2.40437
$459$	3.65685	0.170687
$460$	43.3137i	2.01951i
$461$	− 10.8284i	− 0.504330i	−0.967684	−	0.252165i	$-0.918857\pi$
$461$	− 10.8284i	− 0.504330i	0.967684	−	0.252165i	$-0.0811426\pi$
$462$	− 13.6569i	− 0.635374i
$463$	− 7.51472i	− 0.349239i	−0.984636	−	0.174619i	$-0.944131\pi$
$463$	− 7.51472i	− 0.349239i	0.984636	−	0.174619i	$-0.0558695\pi$
$464$	6.00000	0.278543
$465$	19.3137	0.895652
$466$	65.1127i	3.01629i
$467$	8.00000	0.370196	0.185098	−	0.982720i	$-0.440740\pi$
$467$	8.00000	0.370196	0.185098	+	0.982720i	$0.440740\pi$
$468$	0	0
$469$	−3.31371	−0.153013
$470$	− 2.34315i	− 0.108081i
$471$	−10.0000	−0.460776
$472$	−16.1421	−0.743002
$473$	19.3137i	0.888045i
$474$	27.3137i	1.25456i
$475$	8.48528i	0.389331i
$476$	39.5980i	1.81497i
$477$	−2.00000	−0.0915737
$478$	4.82843	0.220847
$479$	2.68629i	0.122740i	0.998115	+	0.0613699i	$0.0195469\pi$
$479$	2.68629i	0.122740i	−0.998115	+	0.0613699i	$0.980453\pi$
$480$	−4.48528	−0.204724
$481$	0	0
$482$	−28.1421	−1.28184
$483$	− 11.3137i	− 0.514792i
$484$	−26.7990	−1.21814
$485$	21.6569	0.983387
$486$	2.41421i	0.109511i
$487$	− 31.7990i	− 1.44095i	−0.693481	−	0.720475i	$-0.743924\pi$
$487$	− 31.7990i	− 1.44095i	0.693481	−	0.720475i	$-0.256076\pi$
$488$	41.1127i	1.86108i
$489$	13.1716i	0.595639i
$490$	−6.82843	−0.308477
$491$	14.6274	0.660126	0.330063	−	0.943959i	$-0.392930\pi$
$491$	14.6274	0.660126	0.330063	+	0.943959i	$0.392930\pi$
$492$	41.4558i	1.86897i
$493$	7.31371	0.329393
$494$	0	0
$495$	−5.65685	−0.254257
$496$	20.4853i	0.919816i
$497$	−5.65685	−0.253745
$498$	−18.4853	−0.828345
$499$	2.14214i	0.0958952i	0.998850	+	0.0479476i	$0.0152680\pi$
$499$	2.14214i	0.0958952i	−0.998850	+	0.0479476i	$0.984732\pi$
$500$	21.6569i	0.968524i
$501$	7.65685i	0.342083i
$502$	0	0
$503$	15.3137	0.682805	0.341402	−	0.939917i	$-0.389098\pi$
$503$	15.3137	0.682805	0.341402	+	0.939917i	$0.389098\pi$
$504$	−12.4853	−0.556139
$505$	− 10.3431i	− 0.460264i
$506$	19.3137	0.858599
$507$	0	0
$508$	−21.6569	−0.960868
$509$	27.7990i	1.23217i	0.787680	+	0.616084i	$0.211282\pi$
$509$	27.7990i	1.23217i	−0.787680	+	0.616084i	$0.788718\pi$
$510$	24.9706	1.10572
$511$	32.9706	1.45853
$512$	− 31.2426i	− 1.38074i
$513$	− 2.82843i	− 0.124878i
$514$	37.7990i	1.66724i
$515$	38.6274i	1.70213i
$516$	36.9706	1.62754
$517$	−0.686292	−0.0301831
$518$	24.9706i	1.09714i
$519$	0.343146	0.0150624
$520$	0	0
$521$	2.68629	0.117689	0.0588443	−	0.998267i	$-0.481258\pi$
$521$	2.68629	0.117689	0.0588443	+	0.998267i	$0.481258\pi$
$522$	4.82843i	0.211335i
$523$	7.31371	0.319806	0.159903	−	0.987133i	$-0.448882\pi$
$523$	7.31371	0.319806	0.159903	+	0.987133i	$0.448882\pi$
$524$	30.6274	1.33796
$525$	8.48528i	0.370328i
$526$	28.9706i	1.26318i
$527$	24.9706i	1.08773i
$528$	− 6.00000i	− 0.261116i
$529$	−7.00000	−0.304348
$530$	−13.6569	−0.593216
$531$	− 3.65685i	− 0.158694i
$532$	30.6274	1.32787
$533$	0	0
$534$	−22.1421	−0.958184
$535$	− 32.0000i	− 1.38348i
$536$	−5.17157	−0.223378
$537$	0.686292	0.0296157
$538$	43.4558i	1.87351i
$539$	2.00000i	0.0861461i
$540$	10.8284i	0.465981i
$541$	10.0000i	0.429934i	0.976621	+	0.214967i	$0.0689643\pi$
$541$	10.0000i	0.429934i	−0.976621	+	0.214967i	$0.931036\pi$
$542$	−28.4853	−1.22355
$543$	−14.0000	−0.600798
$544$	− 5.79899i	− 0.248630i
$545$	48.9706	2.09767
$546$	0	0
$547$	0.686292	0.0293437	0.0146719	−	0.999892i	$-0.495330\pi$
$547$	0.686292	0.0293437	0.0146719	+	0.999892i	$0.495330\pi$
$548$	19.7990i	0.845771i
$549$	−9.31371	−0.397499
$550$	−14.4853	−0.617654
$551$	− 5.65685i	− 0.240990i
$552$	− 17.6569i	− 0.751526i
$553$	− 32.0000i	− 1.36078i
$554$	4.82843i	0.205140i
$555$	10.3431	0.439042
$556$	−58.6274	−2.48636
$557$	31.7990i	1.34737i	0.739020	+	0.673683i	$0.235289\pi$
$557$	31.7990i	1.34737i	−0.739020	+	0.673683i	$0.764711\pi$
$558$	−16.4853	−0.697878
$559$	0	0
$560$	−24.0000	−1.01419
$561$	− 7.31371i	− 0.308785i
$562$	−64.7696	−2.73214
$563$	−4.00000	−0.168580	−0.0842900	−	0.996441i	$-0.526862\pi$
$563$	−4.00000	−0.168580	−0.0842900	+	0.996441i	$0.526862\pi$
$564$	1.31371i	0.0553171i
$565$	− 48.9706i	− 2.06021i
$566$	12.0000i	0.504398i
$567$	− 2.82843i	− 0.118783i
$568$	−8.82843	−0.370433
$569$	9.02944	0.378534	0.189267	−	0.981926i	$-0.439389\pi$
$569$	9.02944	0.378534	0.189267	+	0.981926i	$0.439389\pi$
$570$	− 19.3137i	− 0.808962i
$571$	−20.9706	−0.877591	−0.438795	−	0.898587i	$-0.644595\pi$
$571$	−20.9706	−0.877591	−0.438795	+	0.898587i	$0.644595\pi$
$572$	0	0
$573$	−19.3137	−0.806842
$574$	− 73.9411i	− 3.08624i
$575$	−12.0000	−0.500435
$576$	9.82843	0.409518
$577$	− 35.9411i	− 1.49625i	−0.663559	−	0.748124i	$-0.730955\pi$
$577$	− 35.9411i	− 1.49625i	0.663559	−	0.748124i	$-0.269045\pi$
$578$	− 8.75736i	− 0.364258i
$579$	− 17.3137i	− 0.719533i
$580$	21.6569i	0.899252i
$581$	21.6569	0.898478
$582$	−18.4853	−0.766240
$583$	4.00000i	0.165663i
$584$	51.4558	2.12926
$585$	0	0
$586$	63.1127	2.60716
$587$	− 22.9706i	− 0.948097i	−0.880499	−	0.474048i	$-0.842792\pi$
$587$	− 22.9706i	− 0.948097i	0.880499	−	0.474048i	$-0.157208\pi$
$588$	3.82843	0.157882
$589$	19.3137	0.795807
$590$	− 24.9706i	− 1.02802i
$591$	16.4853i	0.678114i
$592$	10.9706i	0.450887i
$593$	− 3.51472i	− 0.144332i	−0.997393	−	0.0721661i	$-0.977009\pi$
$593$	− 3.51472i	− 0.144332i	0.997393	−	0.0721661i	$-0.0229912\pi$
$594$	4.82843	0.198113
$595$	−29.2548	−1.19933
$596$	− 56.7696i	− 2.32537i
$597$	−10.3431	−0.423317
$598$	0	0
$599$	−0.686292	−0.0280411	−0.0140206	−	0.999902i	$-0.504463\pi$
$599$	−0.686292	−0.0280411	−0.0140206	+	0.999902i	$0.504463\pi$
$600$	13.2426i	0.540629i
$601$	44.6274	1.82039	0.910195	−	0.414180i	$-0.135931\pi$
$601$	44.6274	1.82039	0.910195	+	0.414180i	$0.135931\pi$
$602$	−65.9411	−2.68756
$603$	− 1.17157i	− 0.0477101i
$604$	78.4264i	3.19113i
$605$	− 19.7990i	− 0.804943i
$606$	8.82843i	0.358630i
$607$	−25.9411	−1.05292	−0.526459	−	0.850201i	$-0.676481\pi$
$607$	−25.9411	−1.05292	−0.526459	+	0.850201i	$0.676481\pi$
$608$	−4.48528	−0.181902
$609$	− 5.65685i	− 0.229227i
$610$	−63.5980	−2.57501
$611$	0	0
$612$	−14.0000	−0.565916
$613$	36.3431i	1.46789i	0.679211	+	0.733943i	$0.262322\pi$
$613$	36.3431i	1.46789i	−0.679211	+	0.733943i	$0.737678\pi$
$614$	41.4558	1.67302
$615$	−30.6274	−1.23502
$616$	24.9706i	1.00609i
$617$	29.1716i	1.17440i	0.809441	+	0.587202i	$0.199770\pi$
$617$	29.1716i	1.17440i	−0.809441	+	0.587202i	$0.800230\pi$
$618$	− 32.9706i	− 1.32627i
$619$	− 15.7990i	− 0.635015i	−0.948256	−	0.317508i	$-0.897154\pi$
$619$	− 15.7990i	− 0.635015i	0.948256	−	0.317508i	$-0.102846\pi$
$620$	−73.9411	−2.96955
$621$	4.00000	0.160514
$622$	− 83.5980i	− 3.35197i
$623$	25.9411	1.03931
$624$	0	0
$625$	−31.0000	−1.24000
$626$	14.4853i	0.578948i
$627$	−5.65685	−0.225913
$628$	38.2843	1.52771
$629$	13.3726i	0.533200i
$630$	− 19.3137i	− 0.769477i
$631$	− 19.1127i	− 0.760865i	−0.924809	−	0.380432i	$-0.875775\pi$
$631$	− 19.1127i	− 0.760865i	0.924809	−	0.380432i	$-0.124225\pi$
$632$	− 49.9411i	− 1.98655i
$633$	−12.0000	−0.476957
$634$	−20.4853	−0.813574
$635$	− 16.0000i	− 0.634941i
$636$	7.65685	0.303614
$637$	0	0
$638$	9.65685	0.382319
$639$	− 2.00000i	− 0.0791188i
$640$	58.1421	2.29827
$641$	26.2843	1.03817	0.519083	−	0.854724i	$-0.326273\pi$
$641$	26.2843	1.03817	0.519083	+	0.854724i	$0.326273\pi$
$642$	27.3137i	1.07799i
$643$	− 17.1716i	− 0.677181i	−0.940934	−	0.338590i	$-0.890050\pi$
$643$	− 17.1716i	− 0.677181i	0.940934	−	0.338590i	$-0.109950\pi$
$644$	43.3137i	1.70680i
$645$	27.3137i	1.07548i
$646$	24.9706	0.982454
$647$	11.3137	0.444788	0.222394	−	0.974957i	$-0.428613\pi$
$647$	11.3137	0.444788	0.222394	+	0.974957i	$0.428613\pi$
$648$	− 4.41421i	− 0.173407i
$649$	−7.31371	−0.287088
$650$	0	0
$651$	19.3137	0.756964
$652$	− 50.4264i	− 1.97485i
$653$	−2.68629	−0.105123	−0.0525614	−	0.998618i	$-0.516739\pi$
$653$	−2.68629	−0.105123	−0.0525614	+	0.998618i	$0.516739\pi$
$654$	−41.7990	−1.63447
$655$	22.6274i	0.884126i
$656$	− 32.4853i	− 1.26834i
$657$	11.6569i	0.454777i
$658$	− 2.34315i	− 0.0913453i
$659$	−24.6863	−0.961641	−0.480821	−	0.876819i	$-0.659661\pi$
$659$	−24.6863	−0.961641	−0.480821	+	0.876819i	$0.659661\pi$
$660$	21.6569	0.842992
$661$	− 1.02944i	− 0.0400405i	−0.999800	−	0.0200202i	$-0.993627\pi$
$661$	− 1.02944i	− 0.0400405i	0.999800	−	0.0200202i	$-0.00637306\pi$
$662$	−5.17157	−0.200999
$663$	0	0
$664$	33.7990	1.31166
$665$	22.6274i	0.877454i
$666$	−8.82843	−0.342095
$667$	8.00000	0.309761
$668$	− 29.3137i	− 1.13418i
$669$	− 4.48528i	− 0.173411i
$670$	− 8.00000i	− 0.309067i
$671$	18.6274i	0.719103i
$672$	−4.48528	−0.173023
$673$	28.6274	1.10351	0.551753	−	0.834008i	$-0.313959\pi$
$673$	28.6274	1.10351	0.551753	+	0.834008i	$0.313959\pi$
$674$	32.1421i	1.23807i
$675$	−3.00000	−0.115470
$676$	0	0
$677$	49.3137	1.89528	0.947640	−	0.319341i	$-0.103462\pi$
$677$	49.3137	1.89528	0.947640	+	0.319341i	$0.103462\pi$
$678$	41.7990i	1.60528i
$679$	21.6569	0.831114
$680$	−45.6569	−1.75086
$681$	− 5.31371i	− 0.203622i
$682$	32.9706i	1.26251i
$683$	− 19.9411i	− 0.763026i	−0.924363	−	0.381513i	$-0.875403\pi$
$683$	− 19.9411i	− 0.763026i	0.924363	−	0.381513i	$-0.124597\pi$
$684$	10.8284i	0.414035i
$685$	−14.6274	−0.558885
$686$	40.9706	1.56426
$687$	21.3137i	0.813169i
$688$	−28.9706	−1.10449
$689$	0	0
$690$	27.3137	1.03982
$691$	34.1421i	1.29883i	0.760435	+	0.649414i	$0.224986\pi$
$691$	34.1421i	1.29883i	−0.760435	+	0.649414i	$0.775014\pi$
$692$	−1.31371	−0.0499397
$693$	−5.65685	−0.214886
$694$	− 75.5980i	− 2.86966i
$695$	− 43.3137i	− 1.64298i
$696$	− 8.82843i	− 0.334641i
$697$	− 39.5980i	− 1.49988i
$698$	18.4853	0.699678
$699$	26.9706	1.02012
$700$	− 32.4853i	− 1.22783i
$701$	−38.9706	−1.47190	−0.735949	−	0.677037i	$-0.763264\pi$
$701$	−38.9706	−1.47190	−0.735949	+	0.677037i	$0.763264\pi$
$702$	0	0
$703$	10.3431	0.390099
$704$	− 19.6569i	− 0.740846i
$705$	−0.970563	−0.0365535
$706$	42.1421	1.58604
$707$	− 10.3431i	− 0.388994i
$708$	14.0000i	0.526152i
$709$	40.6274i	1.52579i	0.646520	+	0.762897i	$0.276224\pi$
$709$	40.6274i	1.52579i	−0.646520	+	0.762897i	$0.723776\pi$
$710$	− 13.6569i	− 0.512533i
$711$	11.3137	0.424297
$712$	40.4853	1.51725
$713$	27.3137i	1.02291i
$714$	24.9706	0.934500
$715$	0	0
$716$	−2.62742	−0.0981912
$717$	− 2.00000i	− 0.0746914i
$718$	−2.48528	−0.0927499
$719$	−37.9411	−1.41497	−0.707483	−	0.706731i	$-0.750169\pi$
$719$	−37.9411	−1.41497	−0.707483	+	0.706731i	$0.750169\pi$
$720$	− 8.48528i	− 0.316228i
$721$	38.6274i	1.43856i
$722$	26.5563i	0.988325i
$723$	11.6569i	0.433523i
$724$	53.5980	1.99195
$725$	−6.00000	−0.222834
$726$	16.8995i	0.627199i
$727$	21.6569	0.803208	0.401604	−	0.915813i	$-0.368453\pi$
$727$	21.6569	0.803208	0.401604	+	0.915813i	$0.368453\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	79.5980i	2.94605i
$731$	−35.3137	−1.30612
$732$	35.6569	1.31792
$733$	− 8.62742i	− 0.318661i	−0.987225	−	0.159330i	$-0.949066\pi$
$733$	− 8.62742i	− 0.318661i	0.987225	−	0.159330i	$-0.0509335\pi$
$734$	− 57.9411i	− 2.13865i
$735$	2.82843i	0.104328i
$736$	− 6.34315i	− 0.233811i
$737$	−2.34315	−0.0863109
$738$	26.1421	0.962305
$739$	10.1421i	0.373084i	0.982447	+	0.186542i	$0.0597281\pi$
$739$	10.1421i	0.373084i	−0.982447	+	0.186542i	$0.940272\pi$
$740$	−39.5980	−1.45565
$741$	0	0
$742$	−13.6569	−0.501359
$743$	− 2.00000i	− 0.0733729i	−0.999327	−	0.0366864i	$-0.988320\pi$
$743$	− 2.00000i	− 0.0733729i	0.999327	−	0.0366864i	$-0.0116803\pi$
$744$	30.1421	1.10506
$745$	41.9411	1.53660
$746$	24.1421i	0.883906i
$747$	7.65685i	0.280150i
$748$	28.0000i	1.02378i
$749$	− 32.0000i	− 1.16925i
$750$	13.6569	0.498678
$751$	−32.9706	−1.20311	−0.601556	−	0.798830i	$-0.705453\pi$
$751$	−32.9706	−1.20311	−0.601556	+	0.798830i	$0.705453\pi$
$752$	− 1.02944i	− 0.0375397i
$753$	0	0
$754$	0	0
$755$	−57.9411	−2.10869
$756$	10.8284i	0.393826i
$757$	−15.9411	−0.579390	−0.289695	−	0.957119i	$-0.593554\pi$
$757$	−15.9411	−0.579390	−0.289695	+	0.957119i	$0.593554\pi$
$758$	−39.7990	−1.44556
$759$	− 8.00000i	− 0.290382i
$760$	35.3137i	1.28096i
$761$	15.5147i	0.562408i	0.959648	+	0.281204i	$0.0907338\pi$
$761$	15.5147i	0.562408i	−0.959648	+	0.281204i	$0.909266\pi$
$762$	13.6569i	0.494736i
$763$	48.9706	1.77285
$764$	73.9411	2.67510
$765$	− 10.3431i	− 0.373957i
$766$	−7.17157	−0.259119
$767$	0	0
$768$	−29.9706	−1.08147
$769$	− 42.0000i	− 1.51456i	−0.653091	−	0.757279i	$-0.726528\pi$
$769$	− 42.0000i	− 1.51456i	0.653091	−	0.757279i	$-0.273472\pi$
$770$	−38.6274	−1.39204
$771$	15.6569	0.563868
$772$	66.2843i	2.38562i
$773$	− 5.85786i	− 0.210693i	−0.994436	−	0.105346i	$-0.966405\pi$
$773$	− 5.85786i	− 0.210693i	0.994436	−	0.105346i	$-0.0335951\pi$
$774$	− 23.3137i	− 0.837994i
$775$	− 20.4853i	− 0.735853i
$776$	33.7990	1.21331
$777$	10.3431	0.371058
$778$	− 16.8284i	− 0.603328i
$779$	−30.6274	−1.09734
$780$	0	0
$781$	−4.00000	−0.143131
$782$	35.3137i	1.26282i
$783$	2.00000	0.0714742
$784$	−3.00000	−0.107143
$785$	28.2843i	1.00951i
$786$	− 19.3137i	− 0.688897i
$787$	32.7696i	1.16811i	0.811715	+	0.584054i	$0.198534\pi$
$787$	32.7696i	1.16811i	−0.811715	+	0.584054i	$0.801466\pi$
$788$	− 63.1127i	− 2.24830i
$789$	12.0000	0.427211
$790$	77.2548	2.74860
$791$	− 48.9706i	− 1.74119i
$792$	−8.82843	−0.313704
$793$	0	0
$794$	7.17157	0.254510
$795$	5.65685i	0.200628i
$796$	39.5980	1.40351
$797$	−35.6569	−1.26303	−0.631515	−	0.775363i	$-0.717567\pi$
$797$	−35.6569	−1.26303	−0.631515	+	0.775363i	$0.717567\pi$
$798$	− 19.3137i	− 0.683698i
$799$	− 1.25483i	− 0.0443928i
$800$	4.75736i	0.168198i
$801$	9.17157i	0.324062i
$802$	5.17157	0.182615
$803$	23.3137	0.822723
$804$	4.48528i	0.158184i
$805$	−32.0000	−1.12785
$806$	0	0
$807$	18.0000	0.633630
$808$	− 16.1421i	− 0.567878i
$809$	41.3137	1.45251	0.726256	−	0.687424i	$-0.241259\pi$
$809$	41.3137	1.45251	0.726256	+	0.687424i	$0.241259\pi$
$810$	6.82843	0.239926
$811$	1.85786i	0.0652384i	0.999468	+	0.0326192i	$0.0103849\pi$
$811$	1.85786i	0.0652384i	−0.999468	+	0.0326192i	$0.989615\pi$
$812$	21.6569i	0.760007i
$813$	11.7990i	0.413809i
$814$	17.6569i	0.618872i
$815$	37.2548	1.30498
$816$	10.9706	0.384047
$817$	27.3137i	0.955586i
$818$	−2.48528	−0.0868958
$819$	0	0
$820$	117.255	4.09472
$821$	− 15.7990i	− 0.551389i	−0.961245	−	0.275694i	$-0.911092\pi$
$821$	− 15.7990i	− 0.551389i	0.961245	−	0.275694i	$-0.0889077\pi$
$822$	12.4853	0.435474
$823$	−48.9706	−1.70701	−0.853503	−	0.521088i	$-0.825527\pi$
$823$	−48.9706	−1.70701	−0.853503	+	0.521088i	$0.825527\pi$
$824$	60.2843i	2.10010i
$825$	6.00000i	0.208893i
$826$	− 24.9706i	− 0.868837i
$827$	− 26.0000i	− 0.904109i	−0.891990	−	0.452054i	$-0.850691\pi$
$827$	− 26.0000i	− 0.904109i	0.891990	−	0.452054i	$-0.149309\pi$
$828$	−15.3137	−0.532188
$829$	−5.31371	−0.184553	−0.0922764	−	0.995733i	$-0.529414\pi$
$829$	−5.31371	−0.184553	−0.0922764	+	0.995733i	$0.529414\pi$
$830$	52.2843i	1.81481i
$831$	2.00000	0.0693792
$832$	0	0
$833$	−3.65685	−0.126702
$834$	36.9706i	1.28019i
$835$	21.6569	0.749466
$836$	21.6569	0.749018
$837$	6.82843i	0.236025i
$838$	− 73.9411i	− 2.55425i
$839$	− 47.2548i	− 1.63142i	−0.578462	−	0.815709i	$-0.696347\pi$
$839$	− 47.2548i	− 1.63142i	0.578462	−	0.815709i	$-0.303653\pi$
$840$	35.3137i	1.21844i
$841$	−25.0000	−0.862069
$842$	35.4558	1.22189
$843$	26.8284i	0.924020i
$844$	45.9411	1.58136
$845$	0	0
$846$	0.828427	0.0284819
$847$	− 19.7990i	− 0.680301i
$848$	−6.00000	−0.206041
$849$	4.97056	0.170589
$850$	− 26.4853i	− 0.908438i
$851$	14.6274i	0.501421i
$852$	7.65685i	0.262320i
$853$	− 7.65685i	− 0.262166i	−0.991371	−	0.131083i	$-0.958155\pi$
$853$	− 7.65685i	− 0.262166i	0.991371	−	0.131083i	$-0.0418454\pi$
$854$	−63.5980	−2.17628
$855$	−8.00000	−0.273594
$856$	− 49.9411i	− 1.70695i
$857$	−29.5980	−1.01105	−0.505524	−	0.862813i	$-0.668701\pi$
$857$	−29.5980	−1.01105	−0.505524	+	0.862813i	$0.668701\pi$
$858$	0	0
$859$	−23.3137	−0.795453	−0.397727	−	0.917504i	$-0.630201\pi$
$859$	−23.3137	−0.795453	−0.397727	+	0.917504i	$0.630201\pi$
$860$	− 104.569i	− 3.56576i
$861$	−30.6274	−1.04378
$862$	47.4558	1.61635
$863$	39.6569i	1.34994i	0.737847	+	0.674968i	$0.235842\pi$
$863$	39.6569i	1.34994i	−0.737847	+	0.674968i	$0.764158\pi$
$864$	− 1.58579i	− 0.0539496i
$865$	− 0.970563i	− 0.0330001i
$866$	− 3.17157i	− 0.107774i
$867$	−3.62742	−0.123194
$868$	−73.9411	−2.50973
$869$	− 22.6274i	− 0.767583i
$870$	13.6569	0.463011
$871$	0	0
$872$	76.4264	2.58812
$873$	7.65685i	0.259145i
$874$	27.3137	0.923900
$875$	−16.0000	−0.540899
$876$	− 44.6274i	− 1.50782i
$877$	14.2843i	0.482346i	0.970482	+	0.241173i	$0.0775321\pi$
$877$	14.2843i	0.482346i	−0.970482	+	0.241173i	$0.922468\pi$
$878$	40.9706i	1.38269i
$879$	− 26.1421i	− 0.881752i
$880$	−16.9706	−0.572078
$881$	−53.5980	−1.80576	−0.902881	−	0.429891i	$-0.858552\pi$
$881$	−53.5980	−1.80576	−0.902881	+	0.429891i	$0.858552\pi$
$882$	− 2.41421i	− 0.0812908i
$883$	51.5980	1.73641	0.868205	−	0.496205i	$-0.165274\pi$
$883$	51.5980	1.73641	0.868205	+	0.496205i	$0.165274\pi$
$884$	0	0
$885$	−10.3431	−0.347681
$886$	101.255i	3.40172i
$887$	−8.00000	−0.268614	−0.134307	−	0.990940i	$-0.542881\pi$
$887$	−8.00000	−0.268614	−0.134307	+	0.990940i	$0.542881\pi$
$888$	16.1421	0.541695
$889$	− 16.0000i	− 0.536623i
$890$	62.6274i	2.09928i
$891$	− 2.00000i	− 0.0670025i
$892$	17.1716i	0.574947i
$893$	−0.970563	−0.0324786
$894$	−35.7990	−1.19730
$895$	− 1.94113i	− 0.0648847i
$896$	58.1421	1.94239
$897$	0	0
$898$	−18.8284	−0.628313
$899$	13.6569i	0.455482i
$900$	11.4853	0.382843
$901$	−7.31371	−0.243655
$902$	− 52.2843i	− 1.74088i
$903$	27.3137i	0.908943i
$904$	− 76.4264i	− 2.54190i
$905$	39.5980i	1.31628i
$906$	49.4558	1.64306
$907$	−20.9706	−0.696316	−0.348158	−	0.937436i	$-0.613193\pi$
$907$	−20.9706	−0.696316	−0.348158	+	0.937436i	$0.613193\pi$
$908$	20.3431i	0.675111i
$909$	3.65685	0.121290
$910$	0	0
$911$	−40.0000	−1.32526	−0.662630	−	0.748947i	$-0.730560\pi$
$911$	−40.0000	−1.32526	−0.662630	+	0.748947i	$0.730560\pi$
$912$	− 8.48528i	− 0.280976i
$913$	15.3137	0.506810
$914$	8.82843	0.292018
$915$	26.3431i	0.870878i
$916$	− 81.5980i	− 2.69607i
$917$	22.6274i	0.747223i
$918$	8.82843i	0.291382i
$919$	−19.3137	−0.637100	−0.318550	−	0.947906i	$-0.603196\pi$
$919$	−19.3137	−0.637100	−0.318550	+	0.947906i	$0.603196\pi$
$920$	−49.9411	−1.64651
$921$	− 17.1716i	− 0.565823i
$922$	26.1421	0.860945
$923$	0	0
$924$	21.6569	0.712458
$925$	− 10.9706i	− 0.360710i
$926$	18.1421	0.596188
$927$	−13.6569	−0.448550
$928$	− 3.17157i	− 0.104112i
$929$	27.7990i	0.912055i	0.889966	+	0.456028i	$0.150728\pi$
$929$	27.7990i	0.912055i	−0.889966	+	0.456028i	$0.849272\pi$
$930$	46.6274i	1.52897i
$931$	2.82843i	0.0926980i
$932$	−103.255	−3.38222
$933$	−34.6274	−1.13365
$934$	19.3137i	0.631964i
$935$	−20.6863	−0.676514
$936$	0	0
$937$	1.31371	0.0429170	0.0214585	−	0.999770i	$-0.493169\pi$
$937$	1.31371	0.0429170	0.0214585	+	0.999770i	$0.493169\pi$
$938$	− 8.00000i	− 0.261209i
$939$	6.00000	0.195803
$940$	3.71573	0.121194
$941$	− 5.85786i	− 0.190961i	−0.995431	−	0.0954805i	$-0.969561\pi$
$941$	− 5.85786i	− 0.190961i	0.995431	−	0.0954805i	$-0.0304387\pi$
$942$	− 24.1421i	− 0.786593i
$943$	− 43.3137i	− 1.41049i
$944$	− 10.9706i	− 0.357061i
$945$	−8.00000	−0.260240
$946$	−46.6274	−1.51599
$947$	− 54.9706i	− 1.78630i	−0.449756	−	0.893152i	$-0.648489\pi$
$947$	− 54.9706i	− 1.78630i	0.449756	−	0.893152i	$-0.351511\pi$
$948$	−43.3137	−1.40676
$949$	0	0
$950$	−20.4853	−0.664630
$951$	8.48528i	0.275154i
$952$	−45.6569	−1.47975
$953$	−51.6569	−1.67333	−0.836665	−	0.547715i	$-0.815498\pi$
$953$	−51.6569	−1.67333	−0.836665	+	0.547715i	$0.815498\pi$
$954$	− 4.82843i	− 0.156326i
$955$	54.6274i	1.76770i
$956$	7.65685i	0.247640i
$957$	− 4.00000i	− 0.129302i
$958$	−6.48528	−0.209530
$959$	−14.6274	−0.472344
$960$	− 27.7990i	− 0.897209i
$961$	−15.6274	−0.504110
$962$	0	0
$963$	11.3137	0.364579
$964$	− 44.6274i	− 1.43735i
$965$	−48.9706	−1.57642
$966$	27.3137	0.878804
$967$	10.1421i	0.326149i	0.986614	+	0.163075i	$0.0521411\pi$
$967$	10.1421i	0.326149i	−0.986614	+	0.163075i	$0.947859\pi$
$968$	− 30.8995i	− 0.993147i
$969$	− 10.3431i	− 0.332270i
$970$	52.2843i	1.67875i
$971$	7.31371	0.234708	0.117354	−	0.993090i	$-0.462559\pi$
$971$	7.31371	0.234708	0.117354	+	0.993090i	$0.462559\pi$
$972$	−3.82843	−0.122797
$973$	− 43.3137i	− 1.38857i
$974$	76.7696	2.45986
$975$	0	0
$976$	−27.9411	−0.894374
$977$	− 13.8579i	− 0.443352i	−0.975120	−	0.221676i	$-0.928847\pi$
$977$	− 13.8579i	− 0.443352i	0.975120	−	0.221676i	$-0.0711528\pi$
$978$	−31.7990	−1.01682
$979$	18.3431	0.586249
$980$	− 10.8284i	− 0.345901i
$981$	17.3137i	0.552784i
$982$	35.3137i	1.12691i
$983$	2.68629i	0.0856794i	0.999082	+	0.0428397i	$0.0136405\pi$
$983$	2.68629i	0.0856794i	−0.999082	+	0.0428397i	$0.986360\pi$
$984$	−47.7990	−1.52378
$985$	46.6274	1.48567
$986$	17.6569i	0.562309i
$987$	−0.970563	−0.0308934
$988$	0	0
$989$	−38.6274	−1.22828
$990$	− 13.6569i	− 0.434043i
$991$	27.3137	0.867649	0.433824	−	0.900998i	$-0.357164\pi$
$991$	27.3137	0.867649	0.433824	+	0.900998i	$0.357164\pi$
$992$	10.8284	0.343803
$993$	2.14214i	0.0679786i
$994$	− 13.6569i	− 0.433169i
$995$	29.2548i	0.927441i
$996$	− 29.3137i	− 0.928840i
$997$	51.2548	1.62326	0.811628	−	0.584174i	$-0.198581\pi$
$997$	51.2548	1.62326	0.811628	+	0.584174i	$0.198581\pi$
$998$	−5.17157	−0.163703
$999$	3.65685i	0.115698i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	507.2.b.e.337.4		4
3.2	odd	2		1521.2.b.j.1351.1		4
13.2	odd	12		507.2.e.d.22.1		4
13.3	even	3		507.2.j.f.316.1		8
13.4	even	6		507.2.j.f.361.1		8
13.5	odd	4		507.2.a.h.1.2		2
13.6	odd	12		507.2.e.d.484.1		4
13.7	odd	12		507.2.e.h.484.2		4
13.8	odd	4		39.2.a.b.1.1	✓	2
13.9	even	3		507.2.j.f.361.4		8
13.10	even	6		507.2.j.f.316.4		8
13.11	odd	12		507.2.e.h.22.2		4
13.12	even	2	inner	507.2.b.e.337.1		4
39.5	even	4		1521.2.a.f.1.1		2
39.8	even	4		117.2.a.c.1.2		2
39.38	odd	2		1521.2.b.j.1351.4		4
52.31	even	4		8112.2.a.bm.1.1		2
52.47	even	4		624.2.a.k.1.2		2
65.8	even	4		975.2.c.h.274.4		4
65.34	odd	4		975.2.a.l.1.2		2
65.47	even	4		975.2.c.h.274.1		4
91.34	even	4		1911.2.a.h.1.1		2
104.21	odd	4		2496.2.a.bf.1.1		2
104.99	even	4		2496.2.a.bi.1.1		2
117.34	odd	12		1053.2.e.m.352.2		4
117.47	even	12		1053.2.e.e.352.1		4
117.86	even	12		1053.2.e.e.703.1		4
117.112	odd	12		1053.2.e.m.703.2		4
143.21	even	4		4719.2.a.p.1.2		2
156.47	odd	4		1872.2.a.w.1.1		2
195.8	odd	4		2925.2.c.u.2224.1		4
195.47	odd	4		2925.2.c.u.2224.4		4
195.164	even	4		2925.2.a.v.1.1		2
273.125	odd	4		5733.2.a.u.1.2		2
312.125	even	4		7488.2.a.cl.1.2		2
312.203	odd	4		7488.2.a.co.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.2.a.b.1.1	✓	2	13.8	odd	4
117.2.a.c.1.2		2	39.8	even	4
507.2.a.h.1.2		2	13.5	odd	4
507.2.b.e.337.1		4	13.12	even	2	inner
507.2.b.e.337.4		4	1.1	even	1	trivial
507.2.e.d.22.1		4	13.2	odd	12
507.2.e.d.484.1		4	13.6	odd	12
507.2.e.h.22.2		4	13.11	odd	12
507.2.e.h.484.2		4	13.7	odd	12
507.2.j.f.316.1		8	13.3	even	3
507.2.j.f.316.4		8	13.10	even	6
507.2.j.f.361.1		8	13.4	even	6
507.2.j.f.361.4		8	13.9	even	3
624.2.a.k.1.2		2	52.47	even	4
975.2.a.l.1.2		2	65.34	odd	4
975.2.c.h.274.1		4	65.47	even	4
975.2.c.h.274.4		4	65.8	even	4
1053.2.e.e.352.1		4	117.47	even	12
1053.2.e.e.703.1		4	117.86	even	12
1053.2.e.m.352.2		4	117.34	odd	12
1053.2.e.m.703.2		4	117.112	odd	12
1521.2.a.f.1.1		2	39.5	even	4
1521.2.b.j.1351.1		4	3.2	odd	2
1521.2.b.j.1351.4		4	39.38	odd	2
1872.2.a.w.1.1		2	156.47	odd	4
1911.2.a.h.1.1		2	91.34	even	4
2496.2.a.bf.1.1		2	104.21	odd	4
2496.2.a.bi.1.1		2	104.99	even	4
2925.2.a.v.1.1		2	195.164	even	4
2925.2.c.u.2224.1		4	195.8	odd	4
2925.2.c.u.2224.4		4	195.47	odd	4
4719.2.a.p.1.2		2	143.21	even	4
5733.2.a.u.1.2		2	273.125	odd	4
7488.2.a.cl.1.2		2	312.125	even	4
7488.2.a.co.1.2		2	312.203	odd	4
8112.2.a.bm.1.1		2	52.31	even	4

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:	\( N \)	\(=\)	\( 507 = 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	507.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.41421i q^{2} +1.00000 q^{3} -3.82843 q^{4} -2.82843i q^{5} +2.41421i q^{6} -2.82843i q^{7} -4.41421i q^{8} +1.00000 q^{9} +6.82843 q^{10} -2.00000i q^{11} -3.82843 q^{12} +6.82843 q^{14} -2.82843i q^{15} +3.00000 q^{16} +3.65685 q^{17} +2.41421i q^{18} -2.82843i q^{19} +10.8284i q^{20} -2.82843i q^{21} +4.82843 q^{22} +4.00000 q^{23} -4.41421i q^{24} -3.00000 q^{25} +1.00000 q^{27} +10.8284i q^{28} +2.00000 q^{29} +6.82843 q^{30} +6.82843i q^{31} -1.58579i q^{32} -2.00000i q^{33} +8.82843i q^{34} -8.00000 q^{35} -3.82843 q^{36} +3.65685i q^{37} +6.82843 q^{38} -12.4853 q^{40} -10.8284i q^{41} +6.82843 q^{42} -9.65685 q^{43} +7.65685i q^{44} -2.82843i q^{45} +9.65685i q^{46} -0.343146i q^{47} +3.00000 q^{48} -1.00000 q^{49} -7.24264i q^{50} +3.65685 q^{51} -2.00000 q^{53} +2.41421i q^{54} -5.65685 q^{55} -12.4853 q^{56} -2.82843i q^{57} +4.82843i q^{58} -3.65685i q^{59} +10.8284i q^{60} -9.31371 q^{61} -16.4853 q^{62} -2.82843i q^{63} +9.82843 q^{64} +4.82843 q^{66} -1.17157i q^{67} -14.0000 q^{68} +4.00000 q^{69} -19.3137i q^{70} -2.00000i q^{71} -4.41421i q^{72} +11.6569i q^{73} -8.82843 q^{74} -3.00000 q^{75} +10.8284i q^{76} -5.65685 q^{77} +11.3137 q^{79} -8.48528i q^{80} +1.00000 q^{81} +26.1421 q^{82} +7.65685i q^{83} +10.8284i q^{84} -10.3431i q^{85} -23.3137i q^{86} +2.00000 q^{87} -8.82843 q^{88} +9.17157i q^{89} +6.82843 q^{90} -15.3137 q^{92} +6.82843i q^{93} +0.828427 q^{94} -8.00000 q^{95} -1.58579i q^{96} +7.65685i q^{97} -2.41421i q^{98} -2.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} - 4 q^{4} + 4 q^{9} + 16 q^{10} - 4 q^{12} + 16 q^{14} + 12 q^{16} - 8 q^{17} + 8 q^{22} + 16 q^{23} - 12 q^{25} + 4 q^{27} + 8 q^{29} + 16 q^{30} - 32 q^{35} - 4 q^{36} + 16 q^{38} - 16 q^{40}+ \cdots - 32 q^{95}+O(q^{100}) \) 4 * q + 4 * q^3 - 4 * q^4 + 4 * q^9 + 16 * q^10 - 4 * q^12 + 16 * q^14 + 12 * q^16 - 8 * q^17 + 8 * q^22 + 16 * q^23 - 12 * q^25 + 4 * q^27 + 8 * q^29 + 16 * q^30 - 32 * q^35 - 4 * q^36 + 16 * q^38 - 16 * q^40 + 16 * q^42 - 16 * q^43 + 12 * q^48 - 4 * q^49 - 8 * q^51 - 8 * q^53 - 16 * q^56 + 8 * q^61 - 32 * q^62 + 28 * q^64 + 8 * q^66 - 56 * q^68 + 16 * q^69 - 24 * q^74 - 12 * q^75 + 4 * q^81 + 48 * q^82 + 8 * q^87 - 24 * q^88 + 16 * q^90 - 16 * q^92 - 8 * q^94 - 32 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more