LMFDB - Embedded newform 507.2.x.a.215.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [507,2,Mod(2,507)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(507, base_ring=CyclotomicField(156))

chi = DirichletCharacter(H, H._module([78, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("507.2");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 507 = 3 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	507.x (of order $156$, degree $48$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$4.04841538248$
Analytic rank:	$0$
Dimension:	$48$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{156}]$

Embedding invariants

Embedding label			215.1
Character	$\chi$	$=$	507.215
Dual form			507.2.x.a.158.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.59345 - 0.678906i) q^{3} +(1.06893 - 1.69038i) q^{4} +(3.10761 - 1.55200i) q^{7} +(2.07817 + 2.16361i) q^{9} +O(q^{10})$	$q+(-1.59345 - 0.678906i) q^{3} +(1.06893 - 1.69038i) q^{4} +(3.10761 - 1.55200i) q^{7} +(2.07817 + 2.16361i) q^{9} +(-2.85090 + 1.96783i) q^{12} +(2.59808 + 2.50000i) q^{13} +(-1.71477 - 3.61380i) q^{16} +(1.30811 - 4.88192i) q^{19} +(-6.00549 + 0.363265i) q^{21} +(-4.67508 - 1.77302i) q^{25} +(-1.84258 - 4.85849i) q^{27} +(0.698348 - 6.91202i) q^{28} +(-1.68540 + 3.74480i) q^{31} +(5.87874 - 1.20015i) q^{36} +(6.10103 - 4.39522i) q^{37} +(-2.44264 - 5.74748i) q^{39} +(-1.17487 - 7.22929i) q^{43} +(0.278971 + 6.92258i) q^{48} +(3.04333 - 4.04993i) q^{49} +(7.00312 - 1.71941i) q^{52} +(-5.39877 + 6.89102i) q^{57} +(-2.74229 + 13.4326i) q^{61} +(9.81607 + 3.49832i) q^{63} +(-7.94167 - 0.964293i) q^{64} +(-4.93139 - 1.11064i) q^{67} +(-4.08038 + 2.46667i) q^{73} +(6.24580 + 5.99917i) q^{75} +(-6.85403 - 7.42964i) q^{76} +(10.1330 + 5.31819i) q^{79} +(-0.362393 + 8.99270i) q^{81} +(-5.80540 + 10.5399i) q^{84} +(11.9538 + 3.73680i) q^{91} +(5.22797 - 4.82293i) q^{93} +(6.43811 + 7.56888i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q + 10 q^{7} + 6 q^{9}+O(q^{10}) $ 48 * q + 10 * q^7 + 6 * q^9	$ 48 q + 10 q^{7} + 6 q^{9} - 8 q^{16} - 14 q^{19} - 18 q^{21} + 20 q^{28} + 14 q^{31} + 2 q^{37} + 24 q^{39} + 6 q^{43} - 18 q^{49} - 28 q^{52} - 12 q^{57} - 24 q^{63} - 32 q^{67} + 34 q^{73} + 30 q^{75} + 28 q^{76} + 18 q^{81} + 12 q^{84} - 2 q^{91} - 6 q^{93} + 38 q^{97}+O(q^{100}) $ 48 * q + 10 * q^7 + 6 * q^9 - 8 * q^16 - 14 * q^19 - 18 * q^21 + 20 * q^28 + 14 * q^31 + 2 * q^37 + 24 * q^39 + 6 * q^43 - 18 * q^49 - 28 * q^52 - 12 * q^57 - 24 * q^63 - 32 * q^67 + 34 * q^73 + 30 * q^75 + 28 * q^76 + 18 * q^81 + 12 * q^84 - 2 * q^91 - 6 * q^93 + 38 * q^97

Display 100 coefficients 10 coefficients

Character values

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

$n$	$170$	$340$
$\chi(n)$	$-1$	$e\left(\frac{131}{156}\right)$

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 $

$2$		0			0		0.875918	−	0.482459i	$-0.160256\pi$
$2$		0			0		−0.875918	+	0.482459i	$0.839744\pi$
$3$	−1.59345	−	0.678906i	−0.919979	−	0.391967i
$3$	−1.59345	−	0.678906i	−0.919979	−	0.391967i
$4$	1.06893	−	1.69038i	0.534466	−	0.845190i
$5$		0			0		0.180255	−	0.983620i	$-0.442308\pi$
$5$		0			0		−0.180255	+	0.983620i	$0.557692\pi$
$6$		0			0
$7$	3.10761	−	1.55200i	1.17457	−	0.586602i	0.250232	−	0.968186i	$-0.419493\pi$
$7$	3.10761	−	1.55200i	1.17457	−	0.586602i	0.924334	+	0.381584i	$0.124621\pi$
$8$		0			0
$9$	2.07817	+	2.16361i	0.692724	+	0.721202i
$10$		0			0
$11$		0			0		−0.0201371	−	0.999797i	$-0.506410\pi$
$11$		0			0		0.0201371	+	0.999797i	$0.493590\pi$
$12$	−2.85090	+	1.96783i	−0.822984	+	0.568065i
$13$	2.59808	+	2.50000i	0.720577	+	0.693375i
$13$	2.59808	+	2.50000i	0.720577	+	0.693375i
$14$		0			0
$15$		0			0
$16$	−1.71477	−	3.61380i	−0.428693	−	0.903450i
$17$		0			0		0.316668	−	0.948536i	$-0.397436\pi$
$17$		0			0		−0.316668	+	0.948536i	$0.602564\pi$
$18$		0			0
$19$	1.30811	−	4.88192i	0.300100	−	1.11999i	−0.636981	−	0.770879i	$-0.719817\pi$
$19$	1.30811	−	4.88192i	0.300100	−	1.11999i	0.937082	−	0.349111i	$-0.113516\pi$
$20$		0			0
$21$	−6.00549	+	0.363265i	−1.31050	+	0.0792709i
$22$		0			0
$23$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$23$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$24$		0			0
$25$	−4.67508	−	1.77302i	−0.935016	−	0.354605i
$26$		0			0
$27$	−1.84258	−	4.85849i	−0.354605	−	0.935016i
$28$	0.698348	−	6.91202i	0.131975	−	1.30625i
$29$		0			0		0.278217	−	0.960518i	$-0.410256\pi$
$29$		0			0		−0.278217	+	0.960518i	$0.589744\pi$
$30$		0			0
$31$	−1.68540	+	3.74480i	−0.302707	+	0.672586i	−0.998978	−	0.0451919i	$-0.985610\pi$
$31$	−1.68540	+	3.74480i	−0.302707	+	0.672586i	0.696272	+	0.717778i	$0.254841\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	5.87874	−	1.20015i	0.979791	−	0.200026i
$37$	6.10103	−	4.39522i	1.00300	−	0.722570i	0.0420032	−	0.999117i	$-0.486626\pi$
$37$	6.10103	−	4.39522i	1.00300	−	0.722570i	0.961000	+	0.276547i	$0.0891901\pi$
$38$		0			0
$39$	−2.44264	−	5.74748i	−0.391136	−	0.920333i
$40$		0			0
$41$		0			0		−0.373361	−	0.927686i	$-0.621795\pi$
$41$		0			0		0.373361	+	0.927686i	$0.378205\pi$
$42$		0			0
$43$	−1.17487	−	7.22929i	−0.179167	−	1.10246i	−0.908372	−	0.418162i	$-0.862674\pi$
$43$	−1.17487	−	7.22929i	−0.179167	−	1.10246i	0.729206	−	0.684294i	$-0.239890\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		0.954721	−	0.297503i	$-0.0961538\pi$
$47$		0			0		−0.954721	+	0.297503i	$0.903846\pi$
$48$	0.278971	+	6.92258i	0.0402659	+	0.999189i
$49$	3.04333	−	4.04993i	0.434761	−	0.578562i
$50$		0			0
$51$		0			0
$52$	7.00312	−	1.71941i	0.971157	−	0.238439i
$53$		0			0		0.748511	−	0.663123i	$-0.230769\pi$
$53$		0			0		−0.748511	+	0.663123i	$0.769231\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−5.39877	+	6.89102i	−0.715085	+	0.912739i
$58$		0			0
$59$		0			0		−0.335705	−	0.941967i	$-0.608974\pi$
$59$		0			0		0.335705	+	0.941967i	$0.391026\pi$
$60$		0			0
$61$	−2.74229	+	13.4326i	−0.351114	+	1.71987i	0.294019	+	0.955800i	$0.405007\pi$
$61$	−2.74229	+	13.4326i	−0.351114	+	1.71987i	−0.645133	+	0.764070i	$0.723198\pi$
$62$		0			0
$63$	9.81607	+	3.49832i	1.23671	+	0.440746i
$64$	−7.94167	−	0.964293i	−0.992709	−	0.120537i
$65$		0			0
$66$		0			0
$67$	−4.93139	−	1.11064i	−0.602465	−	0.135686i	−0.0922537	−	0.995736i	$-0.529407\pi$
$67$	−4.93139	−	1.11064i	−0.602465	−	0.135686i	−0.510211	+	0.860049i	$0.670433\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		−0.990080	−	0.140502i	$-0.955128\pi$
$71$		0			0		0.990080	+	0.140502i	$0.0448718\pi$
$72$		0			0
$73$	−4.08038	+	2.46667i	−0.477572	+	0.288702i	−0.735577	−	0.677441i	$-0.763089\pi$
$73$	−4.08038	+	2.46667i	−0.477572	+	0.288702i	0.258005	+	0.966144i	$0.416935\pi$
$74$		0			0
$75$	6.24580	+	5.99917i	0.721202	+	0.692724i
$76$	−6.85403	−	7.42964i	−0.786211	−	0.852238i
$77$		0			0
$78$		0			0
$79$	10.1330	+	5.31819i	1.14005	+	0.598343i	0.925642	−	0.378401i	$-0.123526\pi$
$79$	10.1330	+	5.31819i	1.14005	+	0.598343i	0.214406	+	0.976745i	$0.431218\pi$
$80$		0			0
$81$	−0.362393	+	8.99270i	−0.0402659	+	0.999189i
$82$		0			0
$83$		0			0		0.787183	−	0.616719i	$-0.211538\pi$
$83$		0			0		−0.787183	+	0.616719i	$0.788462\pi$
$84$	−5.80540	+	10.5399i	−0.633421	+	1.14999i
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$89$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$90$		0			0
$91$	11.9538	+	3.73680i	1.25310	+	0.391723i
$92$		0			0
$93$	5.22797	−	4.82293i	0.542115	−	0.500115i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	6.43811	+	7.56888i	0.653692	+	0.768503i	0.984536	−	0.175182i	$-0.0560515\pi$
$97$	6.43811	+	7.56888i	0.653692	+	0.768503i	−0.330844	+	0.943685i	$0.607334\pi$
$98$		0			0
$99$		0			0
$100$	−7.99443	+	6.00742i	−0.799443	+	0.600742i
$101$		0			0		0.774605	−	0.632445i	$-0.217949\pi$
$101$		0			0		−0.774605	+	0.632445i	$0.782051\pi$
$102$		0			0
$103$	19.4115	−	2.35698i	1.91267	−	0.232240i	0.924005	−	0.382381i	$-0.124896\pi$
$103$	19.4115	−	2.35698i	1.91267	−	0.232240i	0.988665	+	0.150141i	$0.0479728\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		0.996757	−	0.0804666i	$-0.0256410\pi$
$107$		0			0		−0.996757	+	0.0804666i	$0.974359\pi$
$108$	−10.1823	−	2.07873i	−0.979791	−	0.200026i
$109$	−15.6188	+	7.02944i	−1.49601	+	0.673298i	−0.982750	−	0.184937i	$-0.940792\pi$
$109$	−15.6188	+	7.02944i	−1.49601	+	0.673298i	−0.513256	+	0.858235i	$0.671561\pi$
$110$		0			0
$111$	−12.7056	+	2.86155i	−1.20597	+	0.271606i
$112$	−10.9375	−	8.56895i	−1.03349	−	0.809690i
$113$		0			0		0.919979	−	0.391967i	$-0.128205\pi$
$113$		0			0		−0.919979	+	0.391967i	$0.871795\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−0.00976636	+	10.8166i	−0.000902901	+	1.00000i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−10.9911	+	0.442925i	−0.999189	+	0.0402659i
$122$		0			0
$123$		0			0
$124$	4.52856	+	6.85190i	0.406677	+	0.615319i
$125$		0			0
$126$		0			0
$127$	10.2077	+	16.1422i	0.905788	+	1.43239i	0.901683	+	0.432398i	$0.142332\pi$
$127$	10.2077	+	16.1422i	0.905788	+	1.43239i	0.00410555	+	0.999992i	$0.498693\pi$
$128$		0			0
$129$	−3.03590	+	12.3171i	−0.267297	+	1.08446i
$130$		0			0
$131$		0			0		0.970942	−	0.239316i	$-0.0769231\pi$
$131$		0			0		−0.970942	+	0.239316i	$0.923077\pi$
$132$		0			0
$133$	−3.51167	−	17.2013i	−0.304500	−	1.49154i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		0.584522	−	0.811378i	$-0.301282\pi$
$137$		0			0		−0.584522	+	0.811378i	$0.698718\pi$
$138$		0			0
$139$	18.5110	+	1.49436i	1.57008	+	0.126750i	0.834624	−	0.550820i	$-0.185685\pi$
$139$	18.5110	+	1.49436i	1.57008	+	0.126750i	0.735458	+	0.677571i	$0.236967\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	4.25526	−	11.2202i	0.354605	−	0.935016i
$145$		0			0
$146$		0			0
$147$	−7.59891	+	4.38723i	−0.626748	+	0.361853i
$148$	−0.908014	−	15.0113i	−0.0746383	−	1.23392i
$149$		0			0		0.678061	−	0.735006i	$-0.262821\pi$
$149$		0			0		−0.678061	+	0.735006i	$0.737179\pi$
$150$		0			0
$151$	−7.06690	+	22.6785i	−0.575096	+	1.84555i	−0.0426416	+	0.999090i	$0.513577\pi$
$151$	−7.06690	+	22.6785i	−0.575096	+	1.84555i	−0.532455	+	0.846458i	$0.678730\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	−12.3264	−	2.01467i	−0.986905	−	0.161302i
$157$	0.411278	+	0.595838i	0.0328235	+	0.0475531i	0.839052	−	0.544051i	$-0.183110\pi$
$157$	0.411278	+	0.595838i	0.0328235	+	0.0475531i	−0.806229	+	0.591604i	$0.798495\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−1.50770	−	14.9227i	−0.118092	−	1.16884i	−0.864248	−	0.503066i	$-0.832205\pi$
$163$	−1.50770	−	14.9227i	−0.118092	−	1.16884i	0.746156	−	0.665771i	$-0.231897\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		−0.482459	−	0.875918i	$-0.660256\pi$
$167$		0			0		0.482459	+	0.875918i	$0.339744\pi$
$168$		0			0
$169$	0.500000	+	12.9904i	0.0384615	+	0.999260i
$170$		0			0
$171$	13.2810	−	7.31525i	1.01563	−	0.559411i
$172$	−13.4761	−	5.74163i	−1.02754	−	0.437795i
$173$		0			0		0.534466	−	0.845190i	$-0.320513\pi$
$173$		0			0		−0.534466	+	0.845190i	$0.679487\pi$
$174$		0			0
$175$	−17.2801	+	1.74587i	−1.30625	+	0.131975i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		−0.316668	−	0.948536i	$-0.602564\pi$
$179$		0			0		0.316668	+	0.948536i	$0.397436\pi$
$180$		0			0
$181$	22.0290	−	15.2055i	1.63740	−	1.13022i	0.760817	−	0.648967i	$-0.224799\pi$
$181$	22.0290	−	15.2055i	1.63740	−	1.13022i	0.876584	−	0.481249i	$-0.159817\pi$
$182$		0			0
$183$	13.4892	−	19.5425i	0.997149	−	1.44462i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	−13.2664	−	12.2386i	−0.964989	−	0.890226i
$190$		0			0
$191$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$191$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$192$	12.0000	+	6.92820i	0.866025	+	0.500000i
$193$	9.83709	−	19.6970i	0.708089	−	1.41782i	−0.191681	−	0.981457i	$-0.561394\pi$
$193$	9.83709	−	19.6970i	0.708089	−	1.41782i	0.899770	−	0.436365i	$-0.143734\pi$
$194$		0			0
$195$		0			0
$196$	−3.59282	−	9.47348i	−0.256630	−	0.676677i
$197$		0			0		0.100522	−	0.994935i	$-0.467949\pi$
$197$		0			0		−0.100522	+	0.994935i	$0.532051\pi$
$198$		0			0
$199$	−1.76178	+	21.8236i	−0.124890	+	1.54703i	0.566202	+	0.824267i	$0.308412\pi$
$199$	−1.76178	+	21.8236i	−0.124890	+	1.54703i	−0.691091	+	0.722768i	$0.742870\pi$
$200$		0			0
$201$	7.10390	+	5.11770i	0.501071	+	0.360975i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	4.57940	−	13.6759i	0.317524	−	0.948250i
$209$		0			0
$210$		0			0
$211$	18.0601	−	11.4205i	1.24331	−	0.786222i	0.259553	−	0.965729i	$-0.416425\pi$
$211$	18.0601	−	11.4205i	1.24331	−	0.786222i	0.983757	+	0.179507i	$0.0574503\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	0.574381	+	14.2531i	0.0389915	+	0.967565i
$218$		0			0
$219$	8.17652	−	1.16033i	0.552518	−	0.0784080i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−20.6786	−	17.5893i	−1.38474	−	1.17786i	−0.962468	−	0.271394i	$-0.912515\pi$
$223$	−20.6786	−	17.5893i	−1.38474	−	1.17786i	−0.422270	−	0.906470i	$-0.638767\pi$
$224$		0			0
$225$	−5.87950	−	13.7997i	−0.391967	−	0.919979i
$226$		0			0
$227$		0			0		−0.219715	−	0.975564i	$-0.570513\pi$
$227$		0			0		0.219715	+	0.975564i	$0.429487\pi$
$228$	5.87753	+	16.4920i	0.389249	+	1.09221i
$229$	10.4188	+	23.1496i	0.688494	+	1.52977i	0.840746	+	0.541430i	$0.182117\pi$
$229$	10.4188	+	23.1496i	0.688494	+	1.52977i	−0.152252	+	0.988342i	$0.548652\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		−0.992709	−	0.120537i	$-0.961538\pi$
$233$		0			0		0.992709	+	0.120537i	$0.0384615\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−12.5358	−	15.3536i	−0.814290	−	0.997324i
$238$		0			0
$239$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$239$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$240$		0			0
$241$	−23.3259	+	19.8411i	−1.50256	+	1.27808i	−0.644404	+	0.764685i	$0.722895\pi$
$241$	−23.3259	+	19.8411i	−1.50256	+	1.27808i	−0.858152	+	0.513395i	$0.828388\pi$
$242$		0			0
$243$	6.68266	−	14.0834i	0.428693	−	0.903450i
$244$	19.7749	+	18.9941i	1.26596	+	1.21597i
$245$		0			0
$246$		0			0
$247$	15.6034	−	9.41334i	0.992819	−	0.598957i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		−0.774605	−	0.632445i	$-0.782051\pi$
$251$		0			0		0.774605	+	0.632445i	$0.217949\pi$
$252$	16.4062	−	12.8534i	1.03349	−	0.809690i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−10.1191	+	12.3937i	−0.632445	+	0.774605i
$257$		0			0		−0.999189	−	0.0402659i	$-0.987179\pi$
$257$		0			0		0.999189	+	0.0402659i	$0.0128205\pi$
$258$		0			0
$259$	12.1382	−	23.1275i	0.754233	−	1.43707i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		0.692724	−	0.721202i	$-0.256410\pi$
$263$		0			0		−0.692724	+	0.721202i	$0.743590\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	−7.14872	+	7.14872i	−0.436677	+	0.436677i
$269$		0			0		0.799443	−	0.600742i	$-0.205128\pi$
$269$		0			0		−0.799443	+	0.600742i	$0.794872\pi$
$270$		0			0
$271$	−5.16555	+	22.9357i	−0.313785	+	1.39325i	0.526073	+	0.850439i	$0.323664\pi$
$271$	−5.16555	+	22.9357i	−0.313785	+	1.39325i	−0.839858	+	0.542806i	$0.817362\pi$
$272$		0			0
$273$	−16.5109	−	14.0699i	−0.999284	−	0.851551i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−27.7435	−	5.66388i	−1.66695	−	0.340309i	−0.728317	−	0.685240i	$-0.759697\pi$
$277$	−27.7435	−	5.66388i	−1.66695	−	0.340309i	−0.938628	+	0.344931i	$0.887902\pi$
$278$		0			0
$279$	−11.6048	+	4.13581i	−0.694763	+	0.247604i
$280$		0			0
$281$		0			0		−0.787183	−	0.616719i	$-0.788462\pi$
$281$		0			0		0.787183	+	0.616719i	$0.211538\pi$
$282$		0			0
$283$	0.694457	−	4.27317i	0.0412812	−	0.254013i	−0.958204	−	0.286085i	$-0.907646\pi$
$283$	0.694457	−	4.27317i	0.0412812	−	0.254013i	0.999486	+	0.0320711i	$0.0102103\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−13.5905	−	10.2126i	−0.799443	−	0.600742i
$290$		0			0
$291$	−5.12027	−	16.4315i	−0.300155	−	0.963232i
$292$	−0.192028	+	9.53410i	−0.0112376	+	0.557941i
$293$		0			0		−0.551377	−	0.834256i	$-0.685897\pi$
$293$		0			0		0.551377	+	0.834256i	$0.314103\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$	16.8172	−	4.14507i	0.970942	−	0.239316i
$301$	−14.8709	−	20.6424i	−0.857146	−	1.18981i
$302$		0			0
$303$		0			0
$304$	−19.8854	+	3.64414i	−1.14051	+	0.209006i
$305$		0			0
$306$		0			0
$307$	31.9536	+	14.3812i	1.82369	+	0.820776i	0.916346	+	0.400386i	$0.131124\pi$
$307$	31.9536	+	14.3812i	1.82369	+	0.820776i	0.907344	+	0.420390i	$0.138107\pi$
$308$		0			0
$309$	−32.5314	−	9.42284i	−1.85065	−	0.536046i
$310$		0			0
$311$		0			0		0.935016	−	0.354605i	$-0.115385\pi$
$311$		0			0		−0.935016	+	0.354605i	$0.884615\pi$
$312$		0			0
$313$	−5.98288	+	15.7755i	−0.338172	+	0.891687i	0.652789	+	0.757540i	$0.273599\pi$
$313$	−5.98288	+	15.7755i	−0.338172	+	0.891687i	−0.990961	+	0.134147i	$0.957171\pi$
$314$		0			0
$315$		0			0
$316$	19.8212	−	11.4438i	1.11503	−	0.643763i
$317$		0			0		−0.0603785	−	0.998176i	$-0.519231\pi$
$317$		0			0		0.0603785	+	0.998176i	$0.480769\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	14.8137	+	10.2252i	0.822984	+	0.568065i
$325$	−7.71366	−	16.2942i	−0.427877	−	0.903837i
$326$		0			0
$327$	29.6601	−	0.597388i	1.64021	−	0.0330356i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−16.1242	−	32.2859i	−0.886266	−	1.77459i	−0.557492	−	0.830182i	$-0.688236\pi$
$331$	−16.1242	−	32.2859i	−0.886266	−	1.77459i	−0.328774	−	0.944408i	$-0.606636\pi$
$332$		0			0
$333$	22.1885	+	4.06620i	1.21592	+	0.222826i
$334$		0			0
$335$		0			0
$336$	11.6108	+	21.0797i	0.633421	+	1.14999i
$337$			6.77550i			0.369085i	0.982825	+	0.184543i	$0.0590804\pi$
$337$			6.77550i			0.369085i	−0.982825	+	0.184543i	$0.940920\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−1.21098	+	6.60811i	−0.0653869	+	0.356805i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		−0.692724	−	0.721202i	$-0.743590\pi$
$347$		0			0		0.692724	+	0.721202i	$0.256410\pi$
$348$		0			0
$349$	0.746765	+	37.0766i	0.0399734	+	1.98466i	0.141933	+	0.989876i	$0.454668\pi$
$349$	0.746765	+	37.0766i	0.0399734	+	1.98466i	−0.101959	+	0.994789i	$0.532511\pi$
$350$		0			0
$351$	7.35905	−	17.2292i	0.392797	−	0.919625i
$352$		0			0
$353$		0			0		0.551377	−	0.834256i	$-0.314103\pi$
$353$		0			0		−0.551377	+	0.834256i	$0.685897\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		0.998176	−	0.0603785i	$-0.0192308\pi$
$359$		0			0		−0.998176	+	0.0603785i	$0.980769\pi$
$360$		0			0
$361$	−5.66755	−	3.27216i	−0.298292	−	0.172219i
$362$		0			0
$363$	17.8145	+	6.75613i	0.935016	+	0.354605i
$364$	19.0944	−	16.2121i	1.00082	−	0.849745i
$365$		0			0
$366$		0			0
$367$	7.25845	−	25.0591i	0.378888	−	1.30807i	−0.513998	−	0.857791i	$-0.671836\pi$
$367$	7.25845	−	25.0591i	0.378888	−	1.30807i	0.892886	−	0.450282i	$-0.148677\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	−2.56425	−	13.9926i	−0.132950	−	0.725485i
$373$	−7.27966	−	25.1323i	−0.376926	−	1.30130i	−0.895039	−	0.445988i	$-0.852852\pi$
$373$	−7.27966	−	25.1323i	−0.376926	−	1.30130i	0.518112	−	0.855313i	$-0.326635\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−11.0206	−	27.3828i	−0.566091	−	1.40656i	−0.888024	−	0.459797i	$-0.847922\pi$
$379$	−11.0206	−	27.3828i	−0.566091	−	1.40656i	0.321933	−	0.946762i	$-0.395667\pi$
$380$		0			0
$381$	−5.30645	−	32.6519i	−0.271858	−	1.67281i
$382$		0			0
$383$		0			0		0.834256	−	0.551377i	$-0.185897\pi$
$383$		0			0		−0.834256	+	0.551377i	$0.814103\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	13.1998	−	17.5657i	0.670981	−	0.892914i
$388$	19.6762	−	2.79225i	0.998907	−	0.141755i
$389$		0			0		0.663123	−	0.748511i	$-0.269231\pi$
$389$		0			0		−0.663123	+	0.748511i	$0.730769\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	12.6681	+	35.5460i	0.635794	+	1.78400i	0.620615	+	0.784116i	$0.286883\pi$
$397$	12.6681	+	35.5460i	0.635794	+	1.78400i	0.0151793	+	0.999885i	$0.495168\pi$
$398$		0			0
$399$	−6.08239	+	29.7935i	−0.304500	+	1.49154i
$400$	1.60933	+	19.9351i	0.0804666	+	0.996757i
$401$		0			0		−0.941967	−	0.335705i	$-0.891026\pi$
$401$		0			0		0.941967	+	0.335705i	$0.108974\pi$
$402$		0			0
$403$	−13.7408	+	5.51579i	−0.684478	+	0.274761i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−37.8917	−	5.37722i	−1.87362	−	0.265886i	−0.891346	−	0.453323i	$-0.850238\pi$
$409$	−37.8917	−	5.37722i	−1.87362	−	0.265886i	−0.982277	+	0.187437i	$0.939982\pi$
$410$		0			0
$411$		0			0
$412$	16.7653	−	35.3322i	0.825969	−	1.74069i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−28.4818	−	14.9484i	−1.39476	−	0.732027i
$418$		0			0
$419$		0			0		0.0402659	−	0.999189i	$-0.487179\pi$
$419$		0			0		−0.0402659	+	0.999189i	$0.512821\pi$
$420$		0			0
$421$	−30.8593	+	24.1767i	−1.50399	+	1.17830i	−0.569663	+	0.821878i	$0.692926\pi$
$421$	−30.8593	+	24.1767i	−1.50399	+	1.17830i	−0.934325	+	0.356421i	$0.883997\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	12.3255	+	45.9993i	0.596472	+	2.22606i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.735006	−	0.678061i	$-0.237179\pi$
$431$		0			0		−0.735006	+	0.678061i	$0.762821\pi$
$432$	−14.3980	+	14.9899i	−0.692724	+	0.721202i
$433$	25.8336	+	12.2582i	1.24148	+	0.589091i	0.932222	−	0.361886i	$-0.117867\pi$
$433$	25.8336	+	12.2582i	1.24148	+	0.589091i	0.309261	+	0.950977i	$0.399918\pi$
$434$		0			0
$435$		0			0
$436$	−4.81298	+	33.9156i	−0.230500	+	1.62426i
$437$		0			0
$438$		0			0
$439$	−32.3713	+	26.4303i	−1.54500	+	1.26145i	−0.725884	+	0.687817i	$0.758569\pi$
$439$	−32.3713	+	26.4303i	−1.54500	+	1.26145i	−0.819113	+	0.573633i	$0.805534\pi$
$440$		0			0
$441$	15.0870	−	1.83190i	0.718429	−	0.0872331i
$442$		0			0
$443$		0			0		0.120537	−	0.992709i	$-0.461538\pi$
$443$		0			0		−0.120537	+	0.992709i	$0.538462\pi$
$444$	−8.74436	+	24.5362i	−0.414989	+	1.16443i
$445$		0			0
$446$		0			0
$447$		0			0
$448$	−26.1762	+	9.32884i	−1.23671	+	0.440746i
$449$		0			0		0.975564	−	0.219715i	$-0.0705128\pi$
$449$		0			0		−0.975564	+	0.219715i	$0.929487\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	26.6573	−	31.3393i	1.25247	−	1.47245i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	5.97331	+	42.0922i	0.279420	+	1.96899i	0.244734	+	0.969590i	$0.421299\pi$
$457$	5.97331	+	42.0922i	0.279420	+	1.96899i	0.0346858	+	0.999398i	$0.488957\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		0.0201371	−	0.999797i	$-0.493590\pi$
$461$		0			0		−0.0201371	+	0.999797i	$0.506410\pi$
$462$		0			0
$463$	−10.7416	+	17.7688i	−0.499206	+	0.825787i	−0.999190	−	0.0402476i	$-0.987185\pi$
$463$	−10.7416	+	17.7688i	−0.499206	+	0.825787i	0.499984	+	0.866035i	$0.333339\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		0.239316	−	0.970942i	$-0.423077\pi$
$467$		0			0		−0.239316	+	0.970942i	$0.576923\pi$
$468$	18.2738	+	11.5788i	0.844707	+	0.535229i
$469$	−17.0485	+	4.20209i	−0.787228	+	0.194034i
$470$		0			0
$471$	−0.250832	−	1.22866i	−0.0115577	−	0.0566136i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−14.7713	+	20.5041i	−0.677753	+	0.940792i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.994935	−	0.100522i	$-0.967949\pi$
$479$		0			0		0.994935	+	0.100522i	$0.0320513\pi$
$480$		0			0
$481$	26.8390	+	3.83345i	1.22375	+	0.174790i
$482$		0			0
$483$		0			0
$484$	−11.0000	+	19.0526i	−0.500000	+	0.866025i
$485$		0			0
$486$		0			0
$487$	11.1184	−	12.0522i	0.503824	−	0.546136i	−0.430486	−	0.902597i	$-0.641658\pi$
$487$	11.1184	−	12.0522i	0.503824	−	0.546136i	0.934310	+	0.356461i	$0.116017\pi$
$488$		0			0
$489$	−7.72868	+	24.8022i	−0.349503	+	1.12159i
$490$		0			0
$491$		0			0		0.903450	−	0.428693i	$-0.141026\pi$
$491$		0			0		−0.903450	+	0.428693i	$0.858974\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	16.4230	−	0.330779i	0.737417	−	0.0148524i
$497$		0			0
$498$		0			0
$499$	−24.8526	−	1.50331i	−1.11256	−	0.0672973i	−0.506146	−	0.862448i	$-0.668930\pi$
$499$	−24.8526	−	1.50331i	−1.11256	−	0.0672973i	−0.606412	+	0.795151i	$0.707392\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		−0.845190	−	0.534466i	$-0.820513\pi$
$503$		0			0		0.845190	+	0.534466i	$0.179487\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	8.02252	−	21.0390i	0.356293	−	0.934374i
$508$	38.1978			1.69475
$509$		0			0		0.875918	−	0.482459i	$-0.160256\pi$
$509$		0			0		−0.875918	+	0.482459i	$0.839744\pi$
$510$		0			0
$511$	−8.85193	+	13.9982i	−0.391587	+	0.619245i
$512$		0			0
$513$	−26.1291	+	2.63992i	−1.15363	+	0.116555i
$514$		0			0
$515$		0			0
$516$	17.5755	+	18.2980i	0.773718	+	0.805526i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		0.568065	−	0.822984i	$-0.307692\pi$
$521$		0			0		−0.568065	+	0.822984i	$0.692308\pi$
$522$		0			0
$523$	7.94998	+	16.7542i	0.347628	+	0.732611i	0.999673	−	0.0255692i	$-0.00813983\pi$
$523$	7.94998	+	16.7542i	0.347628	+	0.732611i	−0.652045	+	0.758181i	$0.726089\pi$
$524$		0			0
$525$	28.7202	+	8.94958i	1.25345	+	0.390592i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	11.5000	+	19.9186i	0.500000	+	0.866025i
$530$		0			0
$531$		0			0
$532$	−32.8305	−	12.4510i	−1.42338	−	0.539817i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−5.85998	−	31.9769i	−0.251940	−	1.37479i	−0.829528	−	0.558465i	$-0.811390\pi$
$541$	−5.85998	−	31.9769i	−0.251940	−	1.37479i	0.577587	−	0.816329i	$-0.303994\pi$
$542$		0			0
$543$	−45.4252	+	9.27362i	−1.94938	+	0.397969i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−45.4087	−	11.1922i	−1.94154	−	0.478546i	−0.975290	−	0.220926i	$-0.929092\pi$
$547$	−45.4087	−	11.1922i	−1.94154	−	0.478546i	−0.966246	−	0.257619i	$-0.917062\pi$
$548$		0			0
$549$	−34.7618	+	21.9821i	−1.48360	+	0.938171i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	39.7431	+	0.800472i	1.69005	+	0.0340395i
$554$		0			0
$555$		0			0
$556$	22.3130	−	29.6932i	0.946283	−	1.25927i
$557$		0			0		0.990080	−	0.140502i	$-0.0448718\pi$
$557$		0			0		−0.990080	+	0.140502i	$0.955128\pi$
$558$		0			0
$559$	15.0208	−	21.7194i	0.635313	−	0.918634i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		−0.391967	−	0.919979i	$-0.628205\pi$
$563$		0			0		0.391967	+	0.919979i	$0.371795\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	12.8305	+	28.5082i	0.538831	+	1.19723i
$568$		0			0
$569$		0			0		−0.0804666	−	0.996757i	$-0.525641\pi$
$569$		0			0		0.0804666	+	0.996757i	$0.474359\pi$
$570$		0			0
$571$	42.4645	+	5.15612i	1.77708	+	0.215777i	0.942293	−	0.334790i	$-0.108665\pi$
$571$	42.4645	+	5.15612i	1.77708	+	0.215777i	0.834791	+	0.550567i	$0.185588\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−14.4178	−	19.1866i	−0.600742	−	0.799443i
$577$	−4.41470	−	4.41470i	−0.183787	−	0.183787i	0.609217	−	0.793004i	$-0.291484\pi$
$577$	−4.41470	−	4.41470i	−0.183787	−	0.183787i	−0.793004	+	0.609217i	$0.791484\pi$
$578$		0			0
$579$	−29.0473	+	24.7078i	−1.20717	+	1.02682i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$587$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$588$	−0.706624	+	17.5347i	−0.0291407	+	0.723119i
$589$	16.0772	+	13.1266i	0.662448	+	0.540872i
$590$		0			0
$591$		0			0
$592$	−26.3453	−	14.5111i	−1.08279	−	0.596403i
$593$		0			0		−0.616719	−	0.787183i	$-0.711538\pi$
$593$		0			0		0.616719	+	0.787183i	$0.288462\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	17.6235	−	33.5788i	0.721282	−	1.37429i
$598$		0			0
$599$		0			0		0.885456	−	0.464723i	$-0.153846\pi$
$599$		0			0		−0.885456	+	0.464723i	$0.846154\pi$
$600$		0			0
$601$	33.4975	−	34.8746i	1.36639	−	1.42256i	0.563989	−	0.825782i	$-0.309266\pi$
$601$	33.4975	−	34.8746i	1.36639	−	1.42256i	0.802403	−	0.596783i	$-0.203555\pi$
$602$		0			0
$603$	−7.84528	−	12.9777i	−0.319485	−	0.528492i
$604$	30.7812	+	36.1875i	1.25247	+	1.47245i
$605$		0			0
$606$		0			0
$607$	35.8413	−	26.9330i	1.45475	−	1.09318i	0.478112	−	0.878299i	$-0.341321\pi$
$607$	35.8413	−	26.9330i	1.45475	−	1.09318i	0.976641	−	0.214877i	$-0.0689350\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	10.2776	−	28.8383i	0.415107	−	1.16477i	−0.532806	−	0.846237i	$-0.678863\pi$
$613$	10.2776	−	28.8383i	0.415107	−	1.16477i	0.947913	−	0.318529i	$-0.103189\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		0.941967	−	0.335705i	$-0.108974\pi$
$617$		0			0		−0.941967	+	0.335705i	$0.891026\pi$
$618$		0			0
$619$	−35.9378	−	28.1554i	−1.44446	−	1.13166i	−0.967540	−	0.252717i	$-0.918676\pi$
$619$	−35.9378	−	28.1554i	−1.44446	−	1.13166i	−0.476921	−	0.878946i	$-0.658247\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$	−16.5817	+	18.6828i	−0.663798	+	0.747912i
$625$	18.7128	+	16.5781i	0.748511	+	0.663123i
$626$		0			0
$627$		0			0
$628$	1.44682	−	0.0583049i	0.0577344	−	0.00232662i
$629$		0			0
$630$		0			0
$631$	−2.22258	−	3.36285i	−0.0884794	−	0.133873i	0.787327	−	0.616536i	$-0.211464\pi$
$631$	−2.22258	−	3.36285i	−0.0884794	−	0.133873i	−0.875806	+	0.482663i	$0.839670\pi$
$632$		0			0
$633$	−36.5314	+	5.93693i	−1.45199	+	0.235972i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	18.0316	−	2.91371i	0.714439	−	0.115446i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		0.960518	−	0.278217i	$-0.0897436\pi$
$641$		0			0		−0.960518	+	0.278217i	$0.910256\pi$
$642$		0			0
$643$	−44.5889	−	17.9455i	−1.75841	−	0.707700i	−0.998323	−	0.0578826i	$-0.981565\pi$
$643$	−44.5889	−	17.9455i	−1.75841	−	0.707700i	−0.760090	−	0.649818i	$-0.774845\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.960518	−	0.278217i	$-0.910256\pi$
$647$		0			0		0.960518	+	0.278217i	$0.0897436\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	8.76128	−	23.1016i	0.343382	−	0.905423i
$652$	−26.8367	−	13.4028i	−1.05101	−	0.524894i
$653$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$653$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−13.8166	−	3.70216i	−0.539039	−	0.144435i
$658$		0			0
$659$		0			0		−0.948536	−	0.316668i	$-0.897436\pi$
$659$		0			0		0.948536	+	0.316668i	$0.102564\pi$
$660$		0			0
$661$	−10.0188	−	6.62162i	−0.389686	−	0.257551i	0.341286	−	0.939960i	$-0.389138\pi$
$661$	−10.0188	−	6.62162i	−0.389686	−	0.257551i	−0.730971	+	0.682408i	$0.760933\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	21.0088	+	42.0664i	0.812248	+	1.62638i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	17.8967	−	42.0052i	0.689868	−	1.61918i	−0.0920175	−	0.995757i	$-0.529332\pi$
$673$	17.8967	−	42.0052i	0.689868	−	1.61918i	0.781885	−	0.623422i	$-0.214258\pi$
$674$		0			0
$675$			25.9808i			1.00000i
$676$	22.4931	+	13.0406i	0.865121	+	0.501563i
$677$		0			0		1.00000			$0$
$677$		0			0		−1.00000			$\pi$
$678$		0			0
$679$	31.7541	+	13.5291i	1.21861	+	0.519201i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		0.894635	−	0.446798i	$-0.147436\pi$
$683$		0			0		−0.894635	+	0.446798i	$0.852564\pi$
$684$	1.83097	−	30.2695i	0.0700088	−	1.15738i
$685$		0			0
$686$		0			0
$687$	−0.885429	−	43.9612i	−0.0337812	−	1.67723i
$688$	−24.1106	+	16.6423i	−0.919207	+	0.634483i
$689$		0			0
$690$		0			0
$691$	26.9811	−	40.8236i	1.02641	−	1.55300i	0.208197	−	0.978087i	$-0.433241\pi$
$691$	26.9811	−	40.8236i	1.02641	−	1.55300i	0.818214	−	0.574914i	$-0.194965\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	−15.5200	+	31.0761i	−0.586602	+	1.17457i
$701$		0			0		−0.935016	−	0.354605i	$-0.884615\pi$
$701$		0			0		0.935016	+	0.354605i	$0.115385\pi$
$702$		0			0
$703$	−13.4763	−	35.5342i	−0.508270	−	1.34020i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	17.2486	−	42.8574i	0.647785	−	1.60954i	−0.137567	−	0.990492i	$-0.543928\pi$
$709$	17.2486	−	42.8574i	0.647785	−	1.60954i	0.785352	−	0.619049i	$-0.212482\pi$
$710$		0			0
$711$	9.55158	+	32.9759i	0.358212	+	1.23669i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		−0.160411	−	0.987050i	$-0.551282\pi$
$719$		0			0		0.160411	+	0.987050i	$0.448718\pi$
$720$		0			0
$721$	56.6652	−	37.4512i	2.11032	−	1.39476i
$722$		0			0
$723$	50.6390	−	15.7798i	1.88329	−	0.586856i
$724$	−2.15561	−	53.4910i	−0.0801127	−	1.98798i
$725$		0			0
$726$		0			0
$727$	−34.2227	+	38.6294i	−1.26925	+	1.43268i	−0.417548	+	0.908655i	$0.637111\pi$
$727$	−34.2227	+	38.6294i	−1.26925	+	1.43268i	−0.851700	+	0.524030i	$0.824428\pi$
$728$		0			0
$729$	−20.2098	+	17.9043i	−0.748511	+	0.663123i
$730$		0			0
$731$		0			0
$732$	−18.6152	−	43.6914i	−0.688036	−	1.61488i
$733$	32.7928	−	41.8569i	1.21123	−	1.54602i	0.456906	−	0.889515i	$-0.348957\pi$
$733$	32.7928	−	41.8569i	1.21123	−	1.54602i	0.754322	−	0.656504i	$-0.227966\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−42.2460	−	15.0559i	−1.55404	−	0.553841i	−0.587252	−	0.809404i	$-0.699790\pi$
$739$	−42.2460	−	15.0559i	−1.55404	−	0.553841i	−0.966792	+	0.255563i	$0.917739\pi$
$740$		0			0
$741$	−31.2540	+	4.40647i	−1.14814	+	0.161876i
$742$		0			0
$743$		0			0		−0.975564	−	0.219715i	$-0.929487\pi$
$743$		0			0		0.975564	+	0.219715i	$0.0705128\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	37.9600	+	36.4611i	1.38518	+	1.33048i	0.885775	+	0.464116i	$0.153628\pi$
$751$	37.9600	+	36.4611i	1.38518	+	1.33048i	0.499406	+	0.866368i	$0.333552\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$	−34.8687	+	9.34305i	−1.26816	+	0.339803i
$757$	0.767455	−	19.0442i	0.0278936	−	0.692173i	−0.923391	−	0.383860i	$-0.874595\pi$
$757$	0.767455	−	19.0442i	0.0278936	−	0.692173i	0.951285	−	0.308313i	$-0.0997644\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		−0.875918	−	0.482459i	$-0.839744\pi$
$761$		0			0		0.875918	+	0.482459i	$0.160256\pi$
$762$		0			0
$763$	−37.6273	+	46.0851i	−1.36220	+	1.66839i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	24.5385	−	12.8788i	0.885456	−	0.464723i
$769$	−15.3790	+	14.1875i	−0.554579	+	0.511613i	−0.905246	−	0.424887i	$-0.860314\pi$
$769$	−15.3790	+	14.1875i	−0.554579	+	0.511613i	0.350667	+	0.936500i	$0.385955\pi$
$770$		0			0
$771$		0			0
$772$	−22.7803	−	37.6832i	−0.819880	−	1.35625i
$773$		0			0		−0.647915	−	0.761712i	$-0.724359\pi$
$773$		0			0		0.647915	+	0.761712i	$0.275641\pi$
$774$		0			0
$775$	14.5190	−	14.5190i	0.521538	−	0.521538i
$776$		0			0
$777$	−35.0430	+	28.6118i	−1.25716	+	1.02644i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−19.8543	−	4.05327i	−0.709080	−	0.144760i
$785$		0			0
$786$		0			0
$787$	32.8225	−	7.39223i	1.16999	−	0.263505i	0.409168	−	0.912459i	$-0.365819\pi$
$787$	32.8225	−	7.39223i	1.16999	−	0.263505i	0.760827	+	0.648955i	$0.224793\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−40.7062	+	28.0432i	−1.44552	+	0.995844i
$794$		0			0
$795$		0			0
$796$	35.0070	+	26.3060i	1.24079	+	0.932392i
$797$		0			0		0.999189	−	0.0402659i	$-0.0128205\pi$
$797$		0			0		−0.999189	+	0.0402659i	$0.987179\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$	16.2444	−	6.53783i	0.572897	−	0.230571i
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		−0.200026	−	0.979791i	$-0.564103\pi$
$809$		0			0		0.200026	+	0.979791i	$0.435897\pi$
$810$		0			0
$811$	−54.8657	+	10.0545i	−1.92660	+	0.353062i	−0.999750	−	0.0223803i	$-0.992876\pi$
$811$	−54.8657	+	10.0545i	−1.92660	+	0.353062i	−0.926846	+	0.375442i	$0.877491\pi$
$812$		0			0
$813$	23.8023	−	33.0400i	0.834782	−	1.15876i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−36.8297	−	3.72104i	−1.28851	−	0.130183i
$818$		0			0
$819$	16.7571	+	33.6291i	0.585541	+	1.17509i
$820$		0			0
$821$		0			0		−0.894635	−	0.446798i	$-0.852564\pi$
$821$		0			0		0.894635	+	0.446798i	$0.147436\pi$
$822$		0			0
$823$	31.1670	−	17.9943i	1.08641	−	0.627240i	0.153793	−	0.988103i	$-0.450851\pi$
$823$	31.1670	−	17.9943i	1.08641	−	0.627240i	0.932619	+	0.360863i	$0.117518\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.297503	−	0.954721i	$-0.403846\pi$
$827$		0			0		−0.297503	+	0.954721i	$0.596154\pi$
$828$		0			0
$829$	43.9423	−	20.8509i	1.52618	−	0.724181i	0.533594	−	0.845741i	$-0.320841\pi$
$829$	43.9423	−	20.8509i	1.52618	−	0.724181i	0.992584	+	0.121560i	$0.0387897\pi$
$830$		0			0
$831$	40.3627	+	27.8603i	1.40017	+	0.966464i
$832$	−18.2223	−	22.3595i	−0.631746	−	0.775176i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	21.2996	+	1.28839i	0.736220	+	0.0445331i
$838$		0			0
$839$		0			0		−0.100522	−	0.994935i	$-0.532051\pi$
$839$		0			0		0.100522	+	0.994935i	$0.467949\pi$
$840$		0			0
$841$	−24.5105	−	15.4995i	−0.845190	−	0.534466i
$842$		0			0
$843$		0			0
$844$		−	42.7362i		−	1.47104i
$845$		0			0
$846$		0			0
$847$	−33.4686	+	18.4346i	−1.14999	+	0.633421i
$848$		0			0
$849$	−4.00766	+	6.33761i	−0.137543	+	0.217506i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−1.56966	+	25.9496i	−0.0537441	+	0.888496i	0.866995	+	0.498317i	$0.166048\pi$
$853$	−1.56966	+	25.9496i	−0.0537441	+	0.888496i	−0.920739	+	0.390179i	$0.872413\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		0.822984	−	0.568065i	$-0.192308\pi$
$857$		0			0		−0.822984	+	0.568065i	$0.807692\pi$
$858$		0			0
$859$	3.41726	−	4.95076i	0.116595	−	0.168918i	−0.760341	−	0.649524i	$-0.774968\pi$
$859$	3.41726	−	4.95076i	0.116595	−	0.168918i	0.876936	+	0.480607i	$0.159583\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		−0.954721	−	0.297503i	$-0.903846\pi$
$863$		0			0		0.954721	+	0.297503i	$0.0961538\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	14.7224	+	25.5000i	0.500000	+	0.866025i
$868$	24.7072	+	14.2647i	0.838616	+	0.484175i
$869$		0			0
$870$		0			0
$871$	−10.0355	−	15.2140i	−0.340040	−	0.515506i
$872$		0			0
$873$	−2.99656	+	29.6590i	−0.101418	+	1.00380i
$874$		0			0
$875$		0			0
$876$	6.77874	−	15.0617i	0.229032	−	0.508889i
$877$	25.1606	+	18.1259i	0.849614	+	0.612068i	0.922812	−	0.385250i	$-0.125885\pi$
$877$	25.1606	+	18.1259i	0.849614	+	0.612068i	−0.0731981	+	0.997317i	$0.523321\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		0.979791	−	0.200026i	$-0.0641026\pi$
$881$		0			0		−0.979791	+	0.200026i	$0.935897\pi$
$882$		0			0
$883$	−6.27887	−	25.4744i	−0.211301	−	0.857281i	−0.976854	−	0.213908i	$-0.931381\pi$
$883$	−6.27887	−	25.4744i	−0.211301	−	0.857281i	0.765553	−	0.643373i	$-0.222466\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		0.845190	−	0.534466i	$-0.179487\pi$
$887$		0			0		−0.845190	+	0.534466i	$0.820513\pi$
$888$		0			0
$889$	56.7743	+	34.3213i	1.90415	+	1.15110i
$890$		0			0
$891$		0			0
$892$	−51.8365	+	16.1529i	−1.73561	+	0.540839i
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	−29.6115	−	4.81234i	−0.987050	−	0.160411i
$901$		0			0
$902$		0			0
$903$	9.68184	+	42.9886i	0.322191	+	1.43057i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	2.24684	+	27.8321i	0.0746052	+	0.924151i	0.920485	+	0.390778i	$0.127794\pi$
$907$	2.24684	+	27.8321i	0.0746052	+	0.924151i	−0.845880	+	0.533373i	$0.820924\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		−0.120537	−	0.992709i	$-0.538462\pi$
$911$		0			0		0.120537	+	0.992709i	$0.461538\pi$
$912$	34.1605	+	7.69357i	1.13117	+	0.254760i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	50.2687	+	7.13364i	1.66092	+	0.235702i
$917$		0			0
$918$		0			0
$919$	16.1040	−	33.9384i	0.531222	−	1.11953i	−0.443535	−	0.896257i	$-0.646276\pi$
$919$	16.1040	−	33.9384i	0.531222	−	1.11953i	0.974757	−	0.223269i	$-0.0716728\pi$
$920$		0			0
$921$	−41.1531	−	44.6092i	−1.35604	−	1.46992i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−36.3157	+	9.73075i	−1.19405	+	0.319945i
$926$		0			0
$927$	45.4400	+	37.1006i	1.49244	+	1.21854i
$928$		0			0
$929$		0			0		0.482459	−	0.875918i	$-0.339744\pi$
$929$		0			0		−0.482459	+	0.875918i	$0.660256\pi$
$930$		0			0
$931$	−15.7905	−	20.1550i	−0.517511	−	0.660554i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−42.1414	+	22.1175i	−1.37670	+	0.722547i	−0.980666	−	0.195689i	$-0.937306\pi$
$937$	−42.1414	+	22.1175i	−1.37670	+	0.722547i	−0.396033	+	0.918236i	$0.629613\pi$
$938$		0			0
$939$	20.2435	−	21.0758i	0.660623	−	0.687781i
$940$		0			0
$941$		0			0		−0.517338	−	0.855781i	$-0.673077\pi$
$941$		0			0		0.517338	+	0.855781i	$0.326923\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.219715	−	0.975564i	$-0.429487\pi$
$947$		0			0		−0.219715	+	0.975564i	$0.570513\pi$
$948$	−39.3534	+	4.77837i	−1.27814	+	0.155194i
$949$	−16.7678	−	3.79234i	−0.544306	−	0.123104i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		−0.979791	−	0.200026i	$-0.935897\pi$
$953$		0			0		0.979791	+	0.200026i	$0.0641026\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	9.37383	+	10.5809i	0.302381	+	0.341318i
$962$		0			0
$963$		0			0
$964$	8.60523	+	60.6385i	0.277156	+	1.95304i
$965$		0			0
$966$		0			0
$967$	−10.8929	−	34.9564i	−0.350291	−	1.12412i	−0.946883	−	0.321578i	$-0.895787\pi$
$967$	−10.8929	−	34.9564i	−0.350291	−	1.12412i	0.596592	−	0.802545i	$-0.296521\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		0.987050	−	0.160411i	$-0.0512821\pi$
$971$		0			0		−0.987050	+	0.160411i	$0.948718\pi$
$972$	−16.6630	−	26.3504i	−0.534466	−	0.845190i
$973$	59.8442	−	24.0852i	1.91852	−	0.772136i
$974$		0			0
$975$	1.22913	+	31.2008i	0.0393638	+	0.999225i
$976$	53.2452	−	13.1238i	1.70434	−	0.420081i
$977$		0			0		−0.584522	−	0.811378i	$-0.698718\pi$
$977$		0			0		0.584522	+	0.811378i	$0.301282\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−47.6674	−	19.1845i	−1.52190	−	0.612514i
$982$		0			0
$983$		0			0		−0.911900	−	0.410413i	$-0.865385\pi$
$983$		0			0		0.911900	+	0.410413i	$0.134615\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	0.766813	−	36.4378i	0.0243956	−	1.15924i
$989$		0			0
$990$		0			0
$991$	15.9721	−	27.6645i	0.507371	−	0.878793i	−0.492592	−	0.870260i	$-0.663951\pi$
$991$	15.9721	−	27.6645i	0.507371	−	0.878793i	0.999964	−	0.00853283i	$-0.00271612\pi$
$992$		0			0
$993$	3.77407	+	62.3927i	0.119766	+	1.97997i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−0.124974	−	0.0417226i	−0.00395798	−	0.00132137i	0.314688	−	0.949195i	$-0.398100\pi$
$997$	−0.124974	−	0.0417226i	−0.00395798	−	0.00132137i	−0.318646	+	0.947874i	$0.603228\pi$
$998$		0			0
$999$	−32.5958	−	21.5432i	−1.03129	−	0.681598i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	507.2.x.a.215.1	yes	48
3.2	odd	2	CM	507.2.x.a.215.1	yes	48
169.158	odd	156	inner	507.2.x.a.158.1	✓	48
507.158	even	156	inner	507.2.x.a.158.1	✓	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
507.2.x.a.158.1	✓	48	169.158	odd	156	inner
507.2.x.a.158.1	✓	48	507.158	even	156	inner
507.2.x.a.215.1	yes	48	1.1	even	1	trivial
507.2.x.a.215.1	yes	48	3.2	odd	2	CM

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 507 = 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	507.x (of order \(156\), degree \(48\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.59345 - 0.678906i) q^{3} +(1.06893 - 1.69038i) q^{4} +(3.10761 - 1.55200i) q^{7} +(2.07817 + 2.16361i) q^{9} +O(q^{10})\)	\(q+(-1.59345 - 0.678906i) q^{3} +(1.06893 - 1.69038i) q^{4} +(3.10761 - 1.55200i) q^{7} +(2.07817 + 2.16361i) q^{9} +(-2.85090 + 1.96783i) q^{12} +(2.59808 + 2.50000i) q^{13} +(-1.71477 - 3.61380i) q^{16} +(1.30811 - 4.88192i) q^{19} +(-6.00549 + 0.363265i) q^{21} +(-4.67508 - 1.77302i) q^{25} +(-1.84258 - 4.85849i) q^{27} +(0.698348 - 6.91202i) q^{28} +(-1.68540 + 3.74480i) q^{31} +(5.87874 - 1.20015i) q^{36} +(6.10103 - 4.39522i) q^{37} +(-2.44264 - 5.74748i) q^{39} +(-1.17487 - 7.22929i) q^{43} +(0.278971 + 6.92258i) q^{48} +(3.04333 - 4.04993i) q^{49} +(7.00312 - 1.71941i) q^{52} +(-5.39877 + 6.89102i) q^{57} +(-2.74229 + 13.4326i) q^{61} +(9.81607 + 3.49832i) q^{63} +(-7.94167 - 0.964293i) q^{64} +(-4.93139 - 1.11064i) q^{67} +(-4.08038 + 2.46667i) q^{73} +(6.24580 + 5.99917i) q^{75} +(-6.85403 - 7.42964i) q^{76} +(10.1330 + 5.31819i) q^{79} +(-0.362393 + 8.99270i) q^{81} +(-5.80540 + 10.5399i) q^{84} +(11.9538 + 3.73680i) q^{91} +(5.22797 - 4.82293i) q^{93} +(6.43811 + 7.56888i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 10 q^{7} + 6 q^{9}+O(q^{10}) \) 48 * q + 10 * q^7 + 6 * q^9	\( 48 q + 10 q^{7} + 6 q^{9} - 8 q^{16} - 14 q^{19} - 18 q^{21} + 20 q^{28} + 14 q^{31} + 2 q^{37} + 24 q^{39} + 6 q^{43} - 18 q^{49} - 28 q^{52} - 12 q^{57} - 24 q^{63} - 32 q^{67} + 34 q^{73} + 30 q^{75} + 28 q^{76} + 18 q^{81} + 12 q^{84} - 2 q^{91} - 6 q^{93} + 38 q^{97}+O(q^{100}) \) 48 * q + 10 * q^7 + 6 * q^9 - 8 * q^16 - 14 * q^19 - 18 * q^21 + 20 * q^28 + 14 * q^31 + 2 * q^37 + 24 * q^39 + 6 * q^43 - 18 * q^49 - 28 * q^52 - 12 * q^57 - 24 * q^63 - 32 * q^67 + 34 * q^73 + 30 * q^75 + 28 * q^76 + 18 * q^81 + 12 * q^84 - 2 * q^91 - 6 * q^93 + 38 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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