LMFDB - Embedded newform 507.2.x.a.149.1

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			149.1
Character	$\chi$	$=$	507.149
Dual form			507.2.x.a.245.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.19983 + 1.24916i) q^{3} +(-1.80690 - 0.857385i) q^{4} +(1.81421 + 0.257455i) q^{7} +(-0.120798 - 2.99757i) q^{9} +O(q^{10})$	$q+(-1.19983 + 1.24916i) q^{3} +(-1.80690 - 0.857385i) q^{4} +(1.81421 + 0.257455i) q^{7} +(-0.120798 - 2.99757i) q^{9} +(3.23899 - 1.22839i) q^{12} +(-2.59808 - 2.50000i) q^{13} +(2.52978 + 3.09842i) q^{16} +(8.37298 + 2.24353i) q^{19} +(-2.49835 + 1.95733i) q^{21} +(3.31561 + 3.74255i) q^{25} +(3.88938 + 3.44569i) q^{27} +(-3.05736 - 2.02067i) q^{28} +(5.31457 - 0.321472i) q^{31} +(-2.35180 + 5.51988i) q^{36} +(10.3965 + 5.19223i) q^{37} +(6.24016 - 0.245827i) q^{39} +(4.15278 - 12.4391i) q^{43} +(-6.90574 - 0.557489i) q^{48} +(-3.49856 - 1.01337i) q^{49} +(2.55100 + 6.74480i) q^{52} +(-12.8487 + 7.76732i) q^{57} +(-5.31675 + 2.26526i) q^{61} +(0.552585 - 5.46931i) q^{63} +(-1.91453 - 7.76753i) q^{64} +(-10.7743 - 3.83981i) q^{67} +(14.1217 + 4.40050i) q^{73} +(-8.65323 - 0.348713i) q^{75} +(-13.2056 - 11.2327i) q^{76} +(-4.91997 + 7.12781i) q^{79} +(-8.97082 + 0.724199i) q^{81} +(6.19246 - 1.39466i) q^{84} +(-4.06981 - 5.20441i) q^{91} +(-5.97503 + 7.02446i) q^{93} +(9.94139 - 13.7997i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/507\mathbb{Z}\right)^\times$.

$n$	$170$	$340$
$\chi(n)$	$-1$	$e\left(\frac{89}{156}\right)$

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		0.219715	−	0.975564i	$-0.429487\pi$
$2$		0			0		−0.219715	+	0.975564i	$0.570513\pi$
$3$	−1.19983	+	1.24916i	−0.692724	+	0.721202i
$3$	−1.19983	+	1.24916i	−0.692724	+	0.721202i
$4$	−1.80690	−	0.857385i	−0.903450	−	0.428693i
$5$		0			0		−0.911900	−	0.410413i	$-0.865385\pi$
$5$		0			0		0.911900	+	0.410413i	$0.134615\pi$
$6$		0			0
$7$	1.81421	+	0.257455i	0.685706	+	0.0973087i	0.474662	−	0.880168i	$-0.342570\pi$
$7$	1.81421	+	0.257455i	0.685706	+	0.0973087i	0.211044	+	0.977477i	$0.432314\pi$
$8$		0			0
$9$	−0.120798	−	2.99757i	−0.0402659	−	0.999189i
$10$		0			0
$11$		0			0		0.735006	−	0.678061i	$-0.237179\pi$
$11$		0			0		−0.735006	+	0.678061i	$0.762821\pi$
$12$	3.23899	−	1.22839i	0.935016	−	0.354605i
$13$	−2.59808	−	2.50000i	−0.720577	−	0.693375i
$13$	−2.59808	−	2.50000i	−0.720577	−	0.693375i
$14$		0			0
$15$		0			0
$16$	2.52978	+	3.09842i	0.632445	+	0.774605i
$17$		0			0		−0.600742	−	0.799443i	$-0.705128\pi$
$17$		0			0		0.600742	+	0.799443i	$0.294872\pi$
$18$		0			0
$19$	8.37298	+	2.24353i	1.92089	+	0.514702i	0.987975	+	0.154615i	$0.0494137\pi$
$19$	8.37298	+	2.24353i	1.92089	+	0.514702i	0.932919	+	0.360087i	$0.117253\pi$
$20$		0			0
$21$	−2.49835	+	1.95733i	−0.545185	+	0.427125i
$22$		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
$25$	3.31561	+	3.74255i	0.663123	+	0.748511i
$26$		0			0
$27$	3.88938	+	3.44569i	0.748511	+	0.663123i
$28$	−3.05736	−	2.02067i	−0.577786	−	0.381871i
$29$		0			0		0.845190	−	0.534466i	$-0.179487\pi$
$29$		0			0		−0.845190	+	0.534466i	$0.820513\pi$
$30$		0			0
$31$	5.31457	−	0.321472i	0.954525	−	0.0577381i	0.424233	−	0.905553i	$-0.360544\pi$
$31$	5.31457	−	0.321472i	0.954525	−	0.0577381i	0.530292	+	0.847815i	$0.322082\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	−2.35180	+	5.51988i	−0.391967	+	0.919979i
$37$	10.3965	+	5.19223i	1.70918	+	0.853597i	0.986770	+	0.162127i	$0.0518355\pi$
$37$	10.3965	+	5.19223i	1.70918	+	0.853597i	0.722406	+	0.691469i	$0.243036\pi$
$38$		0			0
$39$	6.24016	−	0.245827i	0.999225	−	0.0393638i
$40$		0			0
$41$		0			0		−0.0201371	−	0.999797i	$-0.506410\pi$
$41$		0			0		0.0201371	+	0.999797i	$0.493590\pi$
$42$		0			0
$43$	4.15278	−	12.4391i	0.633293	−	1.89694i	0.305621	−	0.952153i	$-0.401136\pi$
$43$	4.15278	−	12.4391i	0.633293	−	1.89694i	0.327672	−	0.944791i	$-0.393736\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		0.983620	−	0.180255i	$-0.0576923\pi$
$47$		0			0		−0.983620	+	0.180255i	$0.942308\pi$
$48$	−6.90574	−	0.557489i	−0.996757	−	0.0804666i
$49$	−3.49856	−	1.01337i	−0.499794	−	0.144767i
$50$		0			0
$51$		0			0
$52$	2.55100	+	6.74480i	0.353761	+	0.935336i
$53$		0			0		−0.120537	−	0.992709i	$-0.538462\pi$
$53$		0			0		0.120537	+	0.992709i	$0.461538\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−12.8487	+	7.76732i	−1.70185	+	1.02881i
$58$		0			0
$59$		0			0		0.994935	−	0.100522i	$-0.0320513\pi$
$59$		0			0		−0.994935	+	0.100522i	$0.967949\pi$
$60$		0			0
$61$	−5.31675	+	2.26526i	−0.680740	+	0.290036i	−0.704523	−	0.709681i	$-0.748839\pi$
$61$	−5.31675	+	2.26526i	−0.680740	+	0.290036i	0.0237832	+	0.999717i	$0.492429\pi$
$62$		0			0
$63$	0.552585	−	5.46931i	0.0696192	−	0.689068i
$64$	−1.91453	−	7.76753i	−0.239316	−	0.970942i
$65$		0			0
$66$		0			0
$67$	−10.7743	−	3.83981i	−1.31629	−	0.469108i	−0.417974	−	0.908459i	$-0.637260\pi$
$67$	−10.7743	−	3.83981i	−1.31629	−	0.469108i	−0.898315	+	0.439351i	$0.855208\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		−0.875918	−	0.482459i	$-0.839744\pi$
$71$		0			0		0.875918	+	0.482459i	$0.160256\pi$
$72$		0			0
$73$	14.1217	+	4.40050i	1.65282	+	0.515039i	0.975379	−	0.220537i	$-0.0707811\pi$
$73$	14.1217	+	4.40050i	1.65282	+	0.515039i	0.677441	+	0.735577i	$0.263089\pi$
$74$		0			0
$75$	−8.65323	−	0.348713i	−0.999189	−	0.0402659i
$76$	−13.2056	−	11.2327i	−1.51478	−	1.28848i
$77$		0			0
$78$		0			0
$79$	−4.91997	+	7.12781i	−0.553540	+	0.801942i	−0.995467	−	0.0951096i	$-0.969680\pi$
$79$	−4.91997	+	7.12781i	−0.553540	+	0.801942i	0.441926	+	0.897051i	$0.354295\pi$
$80$		0			0
$81$	−8.97082	+	0.724199i	−0.996757	+	0.0804666i
$82$		0			0
$83$		0			0		0.517338	−	0.855781i	$-0.326923\pi$
$83$		0			0		−0.517338	+	0.855781i	$0.673077\pi$
$84$	6.19246	−	1.39466i	0.675653	−	0.152170i
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$89$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$90$		0			0
$91$	−4.06981	−	5.20441i	−0.426633	−	0.545570i
$92$		0			0
$93$	−5.97503	+	7.02446i	−0.619582	+	0.728402i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	9.94139	−	13.7997i	1.00940	−	1.40115i	0.0953921	−	0.995440i	$-0.469590\pi$
$97$	9.94139	−	13.7997i	1.00940	−	1.40115i	0.914003	−	0.405707i	$-0.132975\pi$
$98$		0			0
$99$		0			0
$100$	−2.78217	−	9.60518i	−0.278217	−	0.960518i
$101$		0			0		0.979791	−	0.200026i	$-0.0641026\pi$
$101$		0			0		−0.979791	+	0.200026i	$0.935897\pi$
$102$		0			0
$103$	4.23181	−	17.1691i	0.416972	−	1.69172i	−0.268217	−	0.963359i	$-0.586434\pi$
$103$	4.23181	−	17.1691i	0.416972	−	1.69172i	0.685189	−	0.728365i	$-0.259719\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		−0.987050	−	0.160411i	$-0.948718\pi$
$107$		0			0		0.987050	+	0.160411i	$0.0512821\pi$
$108$	−4.07344	−	9.56071i	−0.391967	−	0.919979i
$109$	−0.947648	+	15.6665i	−0.0907682	+	1.50058i	0.612963	+	0.790112i	$0.289977\pi$
$109$	−0.947648	+	15.6665i	−0.0907682	+	1.50058i	−0.703731	+	0.710466i	$0.748484\pi$
$110$		0			0
$111$	−18.9600	+	6.75709i	−1.79960	+	0.641355i
$112$	3.79185	+	6.27248i	0.358296	+	0.592694i
$113$		0			0		−0.692724	−	0.721202i	$-0.743590\pi$
$113$		0			0		0.692724	+	0.721202i	$0.256410\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−7.18008	+	8.08990i	−0.663798	+	0.747912i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	0.885132	−	10.9643i	0.0804666	−	0.996757i
$122$		0			0
$123$		0			0
$124$	−9.87853	−	3.97577i	−0.887118	−	0.357034i
$125$		0			0
$126$		0			0
$127$	19.3411	−	9.17747i	1.71625	−	0.814369i	0.724040	−	0.689758i	$-0.242283\pi$
$127$	19.3411	−	9.17747i	1.71625	−	0.814369i	0.992206	−	0.124611i	$-0.0397684\pi$
$128$		0			0
$129$	10.5558	+	20.1123i	0.929384	+	1.77079i
$130$		0			0
$131$		0			0		−0.885456	−	0.464723i	$-0.846154\pi$
$131$		0			0		0.885456	+	0.464723i	$0.153846\pi$
$132$		0			0
$133$	14.6127	+	6.22590i	1.26708	+	0.539854i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		−0.446798	−	0.894635i	$-0.647436\pi$
$137$		0			0		0.446798	+	0.894635i	$0.352564\pi$
$138$		0			0
$139$	−9.56623	+	1.55466i	−0.811397	+	0.131865i	−0.551933	−	0.833888i	$-0.686110\pi$
$139$	−9.56623	+	1.55466i	−0.811397	+	0.131865i	−0.259463	+	0.965753i	$0.583546\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	8.98213	−	7.95747i	0.748511	−	0.663123i
$145$		0			0
$146$		0			0
$147$	5.46355	−	3.15438i	0.450626	−	0.260169i
$148$	−14.3337	−	18.2956i	−1.17822	−	1.50389i
$149$		0			0		0.761712	−	0.647915i	$-0.224359\pi$
$149$		0			0		−0.761712	+	0.647915i	$0.775641\pi$
$150$		0			0
$151$	−0.983616	+	5.36742i	−0.0800456	+	0.436794i	0.919045	+	0.394153i	$0.128962\pi$
$151$	−0.983616	+	5.36742i	−0.0800456	+	0.436794i	−0.999090	+	0.0426416i	$0.986423\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	−11.4861	−	4.90603i	−0.919625	−	0.392797i
$157$	8.78695	+	23.1693i	0.701275	+	1.84911i	0.490111	+	0.871660i	$0.336956\pi$
$157$	8.78695	+	23.1693i	0.701275	+	1.84911i	0.211165	+	0.977451i	$0.432274\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−21.0801	+	13.9323i	−1.65112	+	1.09126i	−0.746156	+	0.665771i	$0.768103\pi$
$163$	−21.0801	+	13.9323i	−1.65112	+	1.09126i	−0.904964	+	0.425488i	$0.860103\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		−0.975564	−	0.219715i	$-0.929487\pi$
$167$		0			0		0.975564	+	0.219715i	$0.0705128\pi$
$168$		0			0
$169$	0.500000	+	12.9904i	0.0384615	+	0.999260i
$170$		0			0
$171$	5.71370	−	25.3696i	0.436938	−	1.94006i
$172$	−18.1688	+	18.9157i	−1.38536	+	1.44231i
$173$		0			0		−0.903450	−	0.428693i	$-0.858974\pi$
$173$		0			0		0.903450	+	0.428693i	$0.141026\pi$
$174$		0			0
$175$	5.05168	+	7.64339i	0.381871	+	0.577786i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		0.600742	−	0.799443i	$-0.294872\pi$
$179$		0			0		−0.600742	+	0.799443i	$0.705128\pi$
$180$		0			0
$181$	−20.9696	+	7.95273i	−1.55866	+	0.591122i	−0.975260	−	0.221062i	$-0.929048\pi$
$181$	−20.9696	+	7.95273i	−1.55866	+	0.591122i	−0.583401	+	0.812184i	$0.698278\pi$
$182$		0			0
$183$	3.54955	−	9.35940i	0.262390	−	0.691867i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	6.16903	+	7.25253i	0.448731	+	0.527544i
$190$		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
$193$	−1.19308	−	8.40729i	−0.0858798	−	0.605170i	−0.985649	−	0.168805i	$-0.946009\pi$
$193$	−1.19308	−	8.40729i	−0.0858798	−	0.605170i	0.899770	−	0.436365i	$-0.143734\pi$
$194$		0			0
$195$		0			0
$196$	5.45270	+	4.83067i	0.389479	+	0.345048i
$197$		0			0		−0.834256	−	0.551377i	$-0.814103\pi$
$197$		0			0		0.834256	+	0.551377i	$0.185897\pi$
$198$		0			0
$199$	−0.736085	−	4.52932i	−0.0521797	−	0.321075i	−0.999997	−	0.00226195i	$-0.999280\pi$
$199$	−0.736085	−	4.52932i	−0.0521797	−	0.321075i	0.947818	−	0.318813i	$-0.103284\pi$
$200$		0			0
$201$	17.7239	−	8.85167i	1.25015	−	0.624349i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	1.17348	−	14.3744i	0.0813665	−	0.996684i
$209$		0			0
$210$		0			0
$211$	12.0323	+	25.3576i	0.828339	+	1.74569i	0.647336	+	0.762205i	$0.275883\pi$
$211$	12.0323	+	25.3576i	0.828339	+	1.74569i	0.181003	+	0.983483i	$0.442066\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	9.72450	+	0.785043i	0.660142	+	0.0532922i
$218$		0			0
$219$	−22.4406	+	12.3604i	−1.51640	+	0.835237i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	10.1021	−	7.27760i	0.676485	−	0.487344i	−0.193067	−	0.981186i	$-0.561843\pi$
$223$	10.1021	−	7.27760i	0.676485	−	0.487344i	0.869552	+	0.493841i	$0.164408\pi$
$224$		0			0
$225$	10.8180	−	10.3909i	0.721202	−	0.692724i
$226$		0			0
$227$		0			0		−0.335705	−	0.941967i	$-0.608974\pi$
$227$		0			0		0.335705	+	0.941967i	$0.391026\pi$
$228$	29.8759	−	3.01848i	1.97858	−	0.199904i
$229$	9.90629	+	0.599220i	0.654626	+	0.0395976i	0.384353	−	0.923186i	$-0.374424\pi$
$229$	9.90629	+	0.599220i	0.654626	+	0.0395976i	0.270273	+	0.962784i	$0.412886\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		−0.239316	−	0.970942i	$-0.576923\pi$
$233$		0			0		0.239316	+	0.970942i	$0.423077\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−3.00062	−	14.6980i	−0.194911	−	0.954739i
$238$		0			0
$239$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$239$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$240$		0			0
$241$	−24.1648	−	17.4085i	−1.55659	−	1.12138i	−0.943502	−	0.331367i	$-0.892490\pi$
$241$	−24.1648	−	17.4085i	−1.55659	−	1.12138i	−0.613091	−	0.790012i	$-0.710074\pi$
$242$		0			0
$243$	9.85885	−	12.0749i	0.632445	−	0.774605i
$244$	11.5490	+	0.465410i	0.739351	+	0.0297948i
$245$		0			0
$246$		0			0
$247$	−16.1448	−	26.7613i	−1.02727	−	1.70278i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		−0.979791	−	0.200026i	$-0.935897\pi$
$251$		0			0		0.979791	+	0.200026i	$0.0641026\pi$
$252$	−5.68777	+	9.40872i	−0.358296	+	0.592694i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−3.20041	+	15.6767i	−0.200026	+	0.979791i
$257$		0			0		−0.0804666	−	0.996757i	$-0.525641\pi$
$257$		0			0		0.0804666	+	0.996757i	$0.474359\pi$
$258$		0			0
$259$	17.5247	+	12.0964i	1.08893	+	0.751634i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		0.0402659	−	0.999189i	$-0.487179\pi$
$263$		0			0		−0.0402659	+	0.999189i	$0.512821\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	16.1759	+	16.1759i	0.988099	+	0.988099i
$269$		0			0		−0.278217	−	0.960518i	$-0.589744\pi$
$269$		0			0		0.278217	+	0.960518i	$0.410256\pi$
$270$		0			0
$271$	6.90226	−	19.3673i	0.419282	−	1.17648i	−0.526073	−	0.850439i	$-0.676336\pi$
$271$	6.90226	−	19.3673i	0.419282	−	1.17648i	0.945355	−	0.326042i	$-0.105715\pi$
$272$		0			0
$273$	11.3842	+	1.16058i	0.689005	+	0.0702413i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	10.3490	+	24.2901i	0.621813	+	1.45945i	0.871142	+	0.491031i	$0.163380\pi$
$277$	10.3490	+	24.2901i	0.621813	+	1.45945i	−0.249329	+	0.968419i	$0.580210\pi$
$278$		0			0
$279$	−1.60562	−	15.8919i	−0.0961262	−	0.951426i
$280$		0			0
$281$		0			0		−0.517338	−	0.855781i	$-0.673077\pi$
$281$		0			0		0.517338	+	0.855781i	$0.326923\pi$
$282$		0			0
$283$	−3.85965	−	11.5611i	−0.229432	−	0.687234i	−0.998825	−	0.0484581i	$-0.984569\pi$
$283$	−3.85965	−	11.5611i	−0.229432	−	0.687234i	0.769393	−	0.638776i	$-0.220559\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−4.72970	+	16.3288i	−0.278217	+	0.960518i
$290$		0			0
$291$	5.31000	+	28.9757i	0.311278	+	1.69859i
$292$	−21.7436	−	20.0590i	−1.27245	−	1.17386i
$293$		0			0		−0.927686	−	0.373361i	$-0.878205\pi$
$293$		0			0		0.927686	+	0.373361i	$0.121795\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$	15.3365	+	8.04924i	0.885456	+	0.464723i
$301$	10.7365	−	21.4980i	0.618842	−	1.23912i
$302$		0			0
$303$		0			0
$304$	14.2304	+	31.6187i	0.816170	+	1.81345i
$305$		0			0
$306$		0			0
$307$	−1.72801	−	28.5674i	−0.0986227	−	1.63043i	−0.625316	−	0.780372i	$-0.715030\pi$
$307$	−1.72801	−	28.5674i	−0.0986227	−	1.63043i	0.526693	−	0.850056i	$-0.323432\pi$
$308$		0			0
$309$	16.3695	+	25.8863i	0.931228	+	1.47262i
$310$		0			0
$311$		0			0		0.663123	−	0.748511i	$-0.269231\pi$
$311$		0			0		−0.663123	+	0.748511i	$0.730769\pi$
$312$		0			0
$313$	−26.2456	+	23.2516i	−1.48349	+	1.31426i	−0.652789	+	0.757540i	$0.726401\pi$
$313$	−26.2456	+	23.2516i	−1.48349	+	1.31426i	−0.830701	+	0.556718i	$0.812060\pi$
$314$		0			0
$315$		0			0
$316$	15.0012	−	8.66094i	0.843883	−	0.487216i
$317$		0			0		−0.616719	−	0.787183i	$-0.711538\pi$
$317$		0			0		0.616719	+	0.787183i	$0.288462\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	16.8303	+	6.38289i	0.935016	+	0.354605i
$325$	0.742168	−	18.0125i	0.0411681	−	0.999152i
$326$		0			0
$327$	−18.4329	−	19.9810i	−1.01934	−	1.10495i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−2.93503	+	20.6823i	−0.161324	+	1.13680i	0.730004	+	0.683443i	$0.239518\pi$
$331$	−2.93503	+	20.6823i	−0.161324	+	1.13680i	−0.891328	+	0.453360i	$0.850225\pi$
$332$		0			0
$333$	14.3082	−	31.7914i	0.784083	−	1.74216i
$334$		0			0
$335$		0			0
$336$	−12.3849	−	2.78931i	−0.675653	−	0.152170i
$337$			25.8481i			1.40803i	0.710184	+	0.704017i	$0.248612\pi$
$337$			25.8481i			1.40803i	−0.710184	+	0.704017i	$0.751388\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−17.7829	−	8.00342i	−0.960185	−	0.432144i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		−0.0402659	−	0.999189i	$-0.512821\pi$
$347$		0			0		0.0402659	+	0.999189i	$0.487179\pi$
$348$		0			0
$349$	−3.34957	+	3.09006i	−0.179299	+	0.165407i	−0.762648	−	0.646814i	$-0.776101\pi$
$349$	−3.34957	+	3.09006i	−0.179299	+	0.165407i	0.583349	+	0.812222i	$0.301742\pi$
$350$		0			0
$351$	−1.49068	−	18.6756i	−0.0795666	−	0.996830i
$352$		0			0
$353$		0			0		0.927686	−	0.373361i	$-0.121795\pi$
$353$		0			0		−0.927686	+	0.373361i	$0.878205\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		0.787183	−	0.616719i	$-0.211538\pi$
$359$		0			0		−0.787183	+	0.616719i	$0.788462\pi$
$360$		0			0
$361$	48.6189	+	28.0701i	2.55889	+	1.47738i
$362$		0			0
$363$	12.6342	+	14.2610i	0.663123	+	0.748511i
$364$	2.89157	+	12.8932i	0.151559	+	0.675790i
$365$		0			0
$366$		0			0
$367$	29.9939	−	18.9670i	1.56567	−	0.990070i	0.581355	−	0.813650i	$-0.302523\pi$
$367$	29.9939	−	18.9670i	1.56567	−	0.990070i	0.984315	−	0.176420i	$-0.0564516\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	16.8190	−	7.56960i	0.872022	−	0.392465i
$373$	−26.5060	−	16.7614i	−1.37243	−	0.867873i	−0.374201	−	0.927348i	$-0.622083\pi$
$373$	−26.5060	−	16.7614i	−1.37243	−	0.867873i	−0.998230	+	0.0594747i	$0.981057\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−0.0114331	−	0.567647i	−0.000587277	−	0.0291581i	0.999397	−	0.0347140i	$-0.0110520\pi$
$379$	−0.0114331	−	0.567647i	−0.000587277	−	0.0291581i	−0.999985	+	0.00555587i	$0.998232\pi$
$380$		0			0
$381$	−11.7420	+	35.1716i	−0.601561	+	1.80189i
$382$		0			0
$383$		0			0		0.373361	−	0.927686i	$-0.378205\pi$
$383$		0			0		−0.373361	+	0.927686i	$0.621795\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−37.7887	−	10.9456i	−1.92091	−	0.556397i
$388$	−29.7948	+	16.4111i	−1.51260	+	0.833146i
$389$		0			0		−0.992709	−	0.120537i	$-0.961538\pi$
$389$		0			0		0.992709	+	0.120537i	$0.0384615\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−38.8068	+	3.92080i	−1.94766	+	0.196779i	−0.994424	−	0.105454i	$-0.966370\pi$
$397$	−38.8068	+	3.92080i	−1.94766	+	0.196779i	−0.953234	+	0.302233i	$0.902268\pi$
$398$		0			0
$399$	−25.3100	+	10.7836i	−1.26708	+	0.539854i
$400$	−3.20823	+	19.7410i	−0.160411	+	0.987050i
$401$		0			0		0.100522	−	0.994935i	$-0.467949\pi$
$401$		0			0		−0.100522	+	0.994935i	$0.532051\pi$
$402$		0			0
$403$	−14.6113	−	12.4512i	−0.727843	−	0.620239i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−13.6484	−	7.51758i	−0.674868	−	0.371720i	0.107788	−	0.994174i	$-0.465623\pi$
$409$	−13.6484	−	7.51758i	−0.674868	−	0.371720i	−0.782657	+	0.622454i	$0.786136\pi$
$410$		0			0
$411$		0			0
$412$	−22.3670	+	27.3946i	−1.10194	+	1.34964i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	9.53586	−	13.8151i	0.466973	−	0.676527i
$418$		0			0
$419$		0			0		0.996757	−	0.0804666i	$-0.0256410\pi$
$419$		0			0		−0.996757	+	0.0804666i	$0.974359\pi$
$420$		0			0
$421$	1.24064	−	2.05227i	0.0604651	−	0.100022i	−0.824086	−	0.566464i	$-0.808311\pi$
$421$	1.24064	−	2.05227i	0.0604651	−	0.100022i	0.884551	+	0.466443i	$0.154465\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−10.2289	+	2.74082i	−0.495011	+	0.132638i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.647915	−	0.761712i	$-0.275641\pi$
$431$		0			0		−0.647915	+	0.761712i	$0.724359\pi$
$432$	−0.836912	+	20.7678i	−0.0402659	+	0.999189i
$433$	−32.1112	−	26.2180i	−1.54317	−	1.25996i	−0.827389	−	0.561629i	$-0.810175\pi$
$433$	−32.1112	−	26.2180i	−1.54317	−	1.25996i	−0.715778	−	0.698328i	$-0.753928\pi$
$434$		0			0
$435$		0			0
$436$	15.1445	−	27.4953i	0.725291	−	1.31679i
$437$		0			0
$438$		0			0
$439$	28.6419	−	5.84728i	1.36700	−	0.279076i	0.540185	−	0.841546i	$-0.318354\pi$
$439$	28.6419	−	5.84728i	1.36700	−	0.279076i	0.826817	+	0.562470i	$0.190149\pi$
$440$		0			0
$441$	−2.61503	+	10.6096i	−0.124525	+	0.505218i
$442$		0			0
$443$		0			0		0.970942	−	0.239316i	$-0.0769231\pi$
$443$		0			0		−0.970942	+	0.239316i	$0.923077\pi$
$444$	40.0523	+	4.04663i	1.90080	+	0.192045i
$445$		0			0
$446$		0			0
$447$		0			0
$448$	−1.47356	−	14.5848i	−0.0696192	−	0.689068i
$449$		0			0		0.941967	−	0.335705i	$-0.108974\pi$
$449$		0			0		−0.941967	+	0.335705i	$0.891026\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	−5.52459	−	7.66871i	−0.259568	−	0.360307i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−2.78414	−	5.05469i	−0.130237	−	0.236448i	0.802785	−	0.596269i	$-0.203351\pi$
$457$	−2.78414	−	5.05469i	−0.130237	−	0.236448i	−0.933022	+	0.359821i	$0.882838\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		−0.735006	−	0.678061i	$-0.762821\pi$
$461$		0			0		0.735006	+	0.678061i	$0.237179\pi$
$462$		0			0
$463$	−4.29781	−	13.7921i	−0.199736	−	0.640975i	−0.999190	−	0.0402476i	$-0.987185\pi$
$463$	−4.29781	−	13.7921i	−0.199736	−	0.640975i	0.799454	−	0.600728i	$-0.205122\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		−0.464723	−	0.885456i	$-0.653846\pi$
$467$		0			0		0.464723	+	0.885456i	$0.346154\pi$
$468$	19.9098	−	8.46156i	0.920333	−	0.391136i
$469$	−18.5582	−	9.74011i	−0.856939	−	0.449756i
$470$		0			0
$471$	−39.4850	−	16.8230i	−1.81937	−	0.775163i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	19.3650	+	38.7750i	0.888528	+	1.77912i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.551377	−	0.834256i	$-0.314103\pi$
$479$		0			0		−0.551377	+	0.834256i	$0.685897\pi$
$480$		0			0
$481$	−14.0304	−	39.4811i	−0.639729	−	1.80018i
$482$		0			0
$483$		0			0
$484$	−11.0000	+	19.0526i	−0.500000	+	0.866025i
$485$		0			0
$486$		0			0
$487$	8.63674	−	7.34644i	0.391368	−	0.332899i	−0.430486	−	0.902597i	$-0.641658\pi$
$487$	8.63674	−	7.34644i	0.391368	−	0.332899i	0.821854	+	0.569698i	$0.192940\pi$
$488$		0			0
$489$	7.88898	−	43.0488i	0.356752	−	1.94673i
$490$		0			0
$491$		0			0		0.774605	−	0.632445i	$-0.217949\pi$
$491$		0			0		−0.774605	+	0.632445i	$0.782051\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	14.4408	+	15.6535i	0.648409	+	0.702864i
$497$		0			0
$498$		0			0
$499$	−33.0430	−	25.8875i	−1.47921	−	1.15888i	−0.950764	−	0.309915i	$-0.899699\pi$
$499$	−33.0430	−	25.8875i	−1.47921	−	1.15888i	−0.528442	−	0.848969i	$-0.677224\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		0.428693	−	0.903450i	$-0.358974\pi$
$503$		0			0		−0.428693	+	0.903450i	$0.641026\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−16.8270	−	14.9617i	−0.747312	−	0.664473i
$508$	−42.8161			−1.89966
$509$		0			0		0.219715	−	0.975564i	$-0.429487\pi$
$509$		0			0		−0.219715	+	0.975564i	$0.570513\pi$
$510$		0			0
$511$	24.4868	+	11.6191i	1.08323	+	0.513999i
$512$		0			0
$513$	24.8352	+	37.5766i	1.09650	+	1.65905i
$514$		0			0
$515$		0			0
$516$	−1.82921	−	45.3914i	−0.0805264	−	1.99824i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		0.354605	−	0.935016i	$-0.384615\pi$
$521$		0			0		−0.354605	+	0.935016i	$0.615385\pi$
$522$		0			0
$523$	−8.75581	−	10.7239i	−0.382865	−	0.468925i	0.546836	−	0.837240i	$-0.315832\pi$
$523$	−8.75581	−	10.7239i	−0.382865	−	0.468925i	−0.929701	+	0.368315i	$0.879935\pi$
$524$		0			0
$525$	−15.6090	−	2.86045i	−0.681232	−	0.124840i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	11.5000	+	19.9186i	0.500000	+	0.866025i
$530$		0			0
$531$		0			0
$532$	−21.0657	−	23.7783i	−0.913316	−	1.03092i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−13.0899	+	5.89128i	−0.562778	+	0.253286i	−0.671774	−	0.740756i	$-0.734467\pi$
$541$	−13.0899	+	5.89128i	−0.562778	+	0.253286i	0.108996	+	0.994042i	$0.465237\pi$
$542$		0			0
$543$	15.2258	−	35.7364i	0.653403	−	1.53359i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	4.21460	−	2.21199i	0.180203	−	0.0945780i	−0.372209	−	0.928149i	$-0.621400\pi$
$547$	4.21460	−	2.21199i	0.180203	−	0.0945780i	0.552413	+	0.833571i	$0.313707\pi$
$548$		0			0
$549$	7.43251	+	15.6637i	0.317212	+	0.668509i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−10.7609	+	11.6647i	−0.457602	+	0.496032i
$554$		0			0
$555$		0			0
$556$	18.6182	+	5.39282i	0.789586	+	0.228706i
$557$		0			0		0.875918	−	0.482459i	$-0.160256\pi$
$557$		0			0		−0.875918	+	0.482459i	$0.839744\pi$
$558$		0			0
$559$	−41.8870	+	21.9358i	−1.77163	+	0.927784i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		0.721202	−	0.692724i	$-0.243590\pi$
$563$		0			0		−0.721202	+	0.692724i	$0.756410\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−16.4614	−	0.995730i	−0.691313	−	0.0418167i
$568$		0			0
$569$		0			0		0.160411	−	0.987050i	$-0.448718\pi$
$569$		0			0		−0.160411	+	0.987050i	$0.551282\pi$
$570$		0			0
$571$	1.13864	+	4.61963i	0.0476505	+	0.193326i	0.989943	−	0.141464i	$-0.0451809\pi$
$571$	1.13864	+	4.61963i	0.0476505	+	0.193326i	−0.942293	+	0.334790i	$0.891335\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−23.0524	+	6.67722i	−0.960518	+	0.278217i
$577$	31.8760	−	31.8760i	1.32701	−	1.32701i	0.419053	−	0.907962i	$-0.362362\pi$
$577$	31.8760	−	31.8760i	1.32701	−	1.32701i	0.907962	−	0.419053i	$-0.137638\pi$
$578$		0			0
$579$	11.9335	+	8.59700i	0.495941	+	0.357279i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$587$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$588$	−12.5766	+	1.01529i	−0.518651	+	0.0418698i
$589$	45.2200	+	9.23174i	1.86326	+	0.380387i
$590$		0			0
$591$		0			0
$592$	10.2132	+	45.3479i	0.419760	+	1.86379i
$593$		0			0		−0.855781	−	0.517338i	$-0.826923\pi$
$593$		0			0		0.855781	+	0.517338i	$0.173077\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	6.54102	+	4.51494i	0.267706	+	0.184784i
$598$		0			0
$599$		0			0		−0.568065	−	0.822984i	$-0.692308\pi$
$599$		0			0		0.568065	+	0.822984i	$0.307692\pi$
$600$		0			0
$601$	0.837524	−	20.7829i	0.0341633	−	0.847754i	−0.887743	−	0.460340i	$-0.847727\pi$
$601$	0.837524	−	20.7829i	0.0341633	−	0.847754i	0.921906	−	0.387414i	$-0.126632\pi$
$602$		0			0
$603$	−10.2086	+	32.7605i	−0.415726	+	1.33411i
$604$	6.37924	−	8.85506i	0.259568	−	0.360307i
$605$		0			0
$606$		0			0
$607$	11.7670	+	40.6244i	0.477608	+	1.64889i	0.731635	+	0.681696i	$0.238757\pi$
$607$	11.7670	+	40.6244i	0.477608	+	1.64889i	−0.254028	+	0.967197i	$0.581755\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	3.07116	+	0.310291i	0.124043	+	0.0125325i	0.162348	−	0.986734i	$-0.448093\pi$
$613$	3.07116	+	0.310291i	0.124043	+	0.0125325i	−0.0383050	+	0.999266i	$0.512196\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		−0.100522	−	0.994935i	$-0.532051\pi$
$617$		0			0		0.100522	+	0.994935i	$0.467949\pi$
$618$		0			0
$619$	13.6208	+	22.5316i	0.547468	+	0.905622i	0.999904	+	0.0138256i	$0.00440096\pi$
$619$	13.6208	+	22.5316i	0.547468	+	0.905622i	−0.452437	+	0.891796i	$0.649445\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$	16.5479	+	18.7127i	0.662447	+	0.749109i
$625$	−3.01342	+	24.8177i	−0.120537	+	0.992709i
$626$		0			0
$627$		0			0
$628$	3.98785	−	49.3984i	0.159133	−	1.97121i
$629$		0			0
$630$		0			0
$631$	−29.4676	−	11.8597i	−1.17309	−	0.472127i	−0.297281	−	0.954790i	$-0.596080\pi$
$631$	−29.4676	−	11.8597i	−1.17309	−	0.472127i	−0.875806	+	0.482663i	$0.839670\pi$
$632$		0			0
$633$	−46.1124	−	15.3946i	−1.83280	−	0.611880i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	6.55610	+	11.3792i	0.259762	+	0.450861i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		0.534466	−	0.845190i	$-0.320513\pi$
$641$		0			0		−0.534466	+	0.845190i	$0.679487\pi$
$642$		0			0
$643$	−48.7378	−	0.981635i	−1.92203	−	0.0387119i	−0.955462	−	0.295115i	$-0.904642\pi$
$643$	−48.7378	−	0.981635i	−1.92203	−	0.0387119i	−0.966570	+	0.256403i	$0.917463\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.534466	−	0.845190i	$-0.679487\pi$
$647$		0			0		0.534466	+	0.845190i	$0.320513\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	−12.6484	+	11.2055i	−0.495731	+	0.439179i
$652$	50.0349	−	7.10047i	1.95952	−	0.278076i
$653$		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
$657$	11.4849	−	42.8623i	0.448069	−	1.67222i
$658$		0			0
$659$		0			0		0.799443	−	0.600742i	$-0.205128\pi$
$659$		0			0		−0.799443	+	0.600742i	$0.794872\pi$
$660$		0			0
$661$	10.9993	+	27.3299i	0.427824	+	1.06301i	0.974244	+	0.225498i	$0.0724008\pi$
$661$	10.9993	+	27.3299i	0.427824	+	1.06301i	−0.546419	+	0.837512i	$0.684009\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	−3.02993	+	21.3510i	−0.117144	+	0.825478i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−36.4333	−	34.9946i	−1.40440	−	1.34894i	−0.860170	−	0.510007i	$-0.829643\pi$
$673$	−36.4333	−	34.9946i	−1.40440	−	1.34894i	−0.544229	−	0.838937i	$-0.683178\pi$
$674$		0			0
$675$			25.9808i			1.00000i
$676$	10.2343	−	23.9010i	0.393627	−	0.919270i
$677$		0			0		1.00000			$0$
$677$		0			0		−1.00000			$\pi$
$678$		0			0
$679$	21.5885	−	22.4761i	0.828492	−	0.862552i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		−0.990080	−	0.140502i	$-0.955128\pi$
$683$		0			0		0.990080	+	0.140502i	$0.0448718\pi$
$684$	−32.0756	+	40.9415i	−1.22644	+	1.56544i
$685$		0			0
$686$		0			0
$687$	−12.6344	+	11.6556i	−0.482033	+	0.444688i
$688$	49.0472	−	18.6011i	1.86991	−	0.709162i
$689$		0			0
$690$		0			0
$691$	−42.9151	+	17.2719i	−1.63257	+	0.657052i	−0.993680	−	0.112247i	$-0.964195\pi$
$691$	−42.9151	+	17.2719i	−1.63257	+	0.657052i	−0.638889	+	0.769299i	$0.720606\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	−2.57455	−	18.1421i	−0.0973087	−	0.685706i
$701$		0			0		−0.663123	−	0.748511i	$-0.730769\pi$
$701$		0			0		0.663123	+	0.748511i	$0.269231\pi$
$702$		0			0
$703$	75.4008	+	66.7993i	2.84380	+	2.51938i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−0.0400045	+	1.98621i	−0.00150240	+	0.0745936i	0.998350	+	0.0574199i	$0.0182874\pi$
$709$	−0.0400045	+	1.98621i	−0.00150240	+	0.0745936i	−0.999853	+	0.0171738i	$0.994533\pi$
$710$		0			0
$711$	21.9604	+	13.8869i	0.823580	+	0.520801i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		0.316668	−	0.948536i	$-0.397436\pi$
$719$		0			0		−0.316668	+	0.948536i	$0.602564\pi$
$720$		0			0
$721$	12.0976	−	30.0589i	0.450540	−	1.11945i
$722$		0			0
$723$	50.7397	−	9.29840i	1.88703	−	0.345811i
$724$	44.7086	+	3.60925i	1.66158	+	0.134137i
$725$		0			0
$726$		0			0
$727$	16.3263	+	1.98237i	0.605508	+	0.0735220i	0.417548	−	0.908655i	$-0.362889\pi$
$727$	16.3263	+	1.98237i	0.605508	+	0.0735220i	0.187960	+	0.982177i	$0.439813\pi$
$728$		0			0
$729$	3.25449	+	26.8031i	0.120537	+	0.992709i
$730$		0			0
$731$		0			0
$732$	−14.4383	+	13.8682i	−0.533655	+	0.512582i
$733$	32.4766	−	19.6328i	1.19955	−	0.725154i	0.230755	−	0.973012i	$-0.425881\pi$
$733$	32.4766	−	19.6328i	1.19955	−	0.725154i	0.968796	+	0.247858i	$0.0797267\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	4.84781	−	47.9820i	0.178329	−	1.76505i	−0.370234	−	0.928939i	$-0.620722\pi$
$739$	4.84781	−	47.9820i	0.178329	−	1.76505i	0.548563	−	0.836109i	$-0.315175\pi$
$740$		0			0
$741$	52.8002	+	11.9417i	1.93967	+	0.438689i
$742$		0			0
$743$		0			0		−0.941967	−	0.335705i	$-0.891026\pi$
$743$		0			0		0.941967	+	0.335705i	$0.108974\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	54.7184	+	2.20507i	1.99670	+	0.0804643i	1.00000		0.000686069i	$-0.000218382\pi$
$751$	54.7184	+	2.20507i	1.99670	+	0.0804643i	0.996702	+	0.0811504i	$0.0258594\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$	−4.92861	−	18.3938i	−0.179252	−	0.668977i
$757$	31.5535	−	2.54726i	1.14683	−	0.0925818i	0.507598	−	0.861594i	$-0.330534\pi$
$757$	31.5535	−	2.54726i	1.14683	−	0.0925818i	0.639234	+	0.769012i	$0.279252\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		−0.219715	−	0.975564i	$-0.570513\pi$
$761$		0			0		0.219715	+	0.975564i	$0.429487\pi$
$762$		0			0
$763$	−5.75264	+	28.1783i	−0.208260	+	1.02012i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	−15.7427	−	22.8072i	−0.568065	−	0.822984i
$769$	−35.7674	+	42.0494i	−1.28980	+	1.51634i	−0.571523	+	0.820586i	$0.693647\pi$
$769$	−35.7674	+	42.0494i	−1.28980	+	1.51634i	−0.718281	+	0.695753i	$0.755071\pi$
$770$		0			0
$771$		0			0
$772$	−5.05251	+	16.2141i	−0.181844	+	0.583557i
$773$		0			0		0.584522	−	0.811378i	$-0.301282\pi$
$773$		0			0		−0.584522	+	0.811378i	$0.698718\pi$
$774$		0			0
$775$	18.8242	+	18.8242i	0.676185	+	0.676185i
$776$		0			0
$777$	−36.1370	+	7.37743i	−1.29641	+	0.264664i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−5.71074	−	13.4036i	−0.203955	−	0.478700i
$785$		0			0
$786$		0			0
$787$	33.4812	−	11.9323i	1.19348	−	0.425339i	0.336990	−	0.941508i	$-0.390591\pi$
$787$	33.4812	−	11.9323i	1.19348	−	0.425339i	0.856487	+	0.516169i	$0.172642\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	19.4765	+	7.40657i	0.691629	+	0.263015i
$794$		0			0
$795$		0			0
$796$	−2.55333	+	8.81513i	−0.0905005	+	0.312444i
$797$		0			0		0.0804666	−	0.996757i	$-0.474359\pi$
$797$		0			0		−0.0804666	+	0.996757i	$0.525641\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$	−39.6146	+	0.797883i	−1.39710	+	0.0281392i
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		−0.919979	−	0.391967i	$-0.871795\pi$
$809$		0			0		0.919979	+	0.391967i	$0.128205\pi$
$810$		0			0
$811$	−18.7189	−	41.5916i	−0.657308	−	1.46048i	−0.874834	−	0.484424i	$-0.839029\pi$
$811$	−18.7189	−	41.5916i	−0.657308	−	1.46048i	0.217526	−	0.976055i	$-0.430201\pi$
$812$		0			0
$813$	15.9113	+	31.8596i	0.558034	+	1.11736i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	62.6787	−	94.8354i	2.19285	−	3.31787i
$818$		0			0
$819$	−15.1089	+	12.8282i	−0.527949	+	0.448254i
$820$		0			0
$821$		0			0		0.990080	−	0.140502i	$-0.0448718\pi$
$821$		0			0		−0.990080	+	0.140502i	$0.955128\pi$
$822$		0			0
$823$	2.33345	−	1.34722i	0.0813390	−	0.0469611i	−0.458779	−	0.888550i	$-0.651713\pi$
$823$	2.33345	−	1.34722i	0.0813390	−	0.0469611i	0.540118	+	0.841589i	$0.318380\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.180255	−	0.983620i	$-0.442308\pi$
$827$		0			0		−0.180255	+	0.983620i	$0.557692\pi$
$828$		0			0
$829$	−30.8661	+	25.2014i	−1.07202	+	0.875280i	−0.992584	−	0.121560i	$-0.961210\pi$
$829$	−30.8661	+	25.2014i	−1.07202	+	0.875280i	−0.0794389	+	0.996840i	$0.525313\pi$
$830$		0			0
$831$	−42.7593	−	16.2165i	−1.48330	−	0.562543i
$832$	−14.4448	+	24.9670i	−0.500782	+	0.865574i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	21.7781	+	17.0620i	0.752760	+	0.589750i
$838$		0			0
$839$		0			0		0.834256	−	0.551377i	$-0.185897\pi$
$839$		0			0		−0.834256	+	0.551377i	$0.814103\pi$
$840$		0			0
$841$	12.4321	−	26.2001i	0.428693	−	0.903450i
$842$		0			0
$843$		0			0
$844$		−	56.1349i		−	1.93224i
$845$		0			0
$846$		0			0
$847$	4.42863	−	19.6637i	0.152170	−	0.675653i
$848$		0			0
$849$	19.0726	+	9.05004i	0.654568	+	0.310597i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−35.1590	+	44.8772i	−1.20382	+	1.53656i	−0.430446	+	0.902616i	$0.641644\pi$
$853$	−35.1590	+	44.8772i	−1.20382	+	1.53656i	−0.773375	+	0.633948i	$0.781433\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		0.935016	−	0.354605i	$-0.115385\pi$
$857$		0			0		−0.935016	+	0.354605i	$0.884615\pi$
$858$		0			0
$859$	18.5762	−	48.9815i	0.633812	−	1.67123i	−0.102625	−	0.994720i	$-0.532724\pi$
$859$	18.5762	−	48.9815i	0.633812	−	1.67123i	0.736437	−	0.676506i	$-0.236507\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		−0.983620	−	0.180255i	$-0.942308\pi$
$863$		0			0		0.983620	+	0.180255i	$0.0576923\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−14.7224	−	25.5000i	−0.500000	−	0.866025i
$868$	−16.8981	−	9.75614i	−0.573560	−	0.331145i
$869$		0			0
$870$		0			0
$871$	18.3929	+	36.9119i	0.623220	+	1.25071i
$872$		0			0
$873$	−42.5664	−	28.1330i	−1.44065	−	0.952158i
$874$		0			0
$875$		0			0
$876$	51.1456	−	3.09374i	1.72805	−	0.104528i
$877$	2.81366	−	1.40520i	0.0950105	−	0.0474502i	−0.398663	−	0.917098i	$-0.630526\pi$
$877$	2.81366	−	1.40520i	0.0950105	−	0.0474502i	0.493673	+	0.869647i	$0.335654\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		0.391967	−	0.919979i	$-0.371795\pi$
$881$		0			0		−0.391967	+	0.919979i	$0.628205\pi$
$882$		0			0
$883$	−13.4686	+	25.6623i	−0.453255	+	0.863605i	0.546405	+	0.837521i	$0.315996\pi$
$883$	−13.4686	+	25.6623i	−0.453255	+	0.863605i	−0.999660	+	0.0260838i	$0.991696\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		−0.428693	−	0.903450i	$-0.641026\pi$
$887$		0			0		0.428693	+	0.903450i	$0.358974\pi$
$888$		0			0
$889$	37.4516	−	11.6704i	1.25609	−	0.391412i
$890$		0			0
$891$		0			0
$892$	−24.4932	+	4.48854i	−0.820092	+	0.150287i
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	−28.4561	+	9.50004i	−0.948536	+	0.316668i
$901$		0			0
$902$		0			0
$903$	13.9723	+	39.2056i	0.464971	+	1.30468i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	0.0124823	−	0.0768067i	0.000414468	−	0.00255033i	−0.986842	−	0.161686i	$-0.948307\pi$
$907$	0.0124823	−	0.0768067i	0.000414468	−	0.00255033i	0.987257	+	0.159136i	$0.0508708\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		−0.970942	−	0.239316i	$-0.923077\pi$
$911$		0			0		0.970942	+	0.239316i	$0.0769231\pi$
$912$	−56.5709	−	20.1611i	−1.87325	−	0.667601i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	−17.3859	−	9.57624i	−0.574447	−	0.316408i
$917$		0			0
$918$		0			0
$919$	7.04931	−	8.63384i	0.232535	−	0.284804i	−0.645135	−	0.764068i	$-0.723199\pi$
$919$	7.04931	−	8.63384i	0.232535	−	0.284804i	0.877671	+	0.479264i	$0.159096\pi$
$920$		0			0
$921$	37.7586	+	32.1176i	1.24419	+	1.05831i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	15.0386	+	56.1249i	0.494467	+	1.84538i
$926$		0			0
$927$	−51.9768	−	10.6111i	−1.70714	−	0.348515i
$928$		0			0
$929$		0			0		0.975564	−	0.219715i	$-0.0705128\pi$
$929$		0			0		−0.975564	+	0.219715i	$0.929487\pi$
$930$		0			0
$931$	−27.0198	−	16.3341i	−0.885539	−	0.535327i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−34.7427	−	50.3335i	−1.13499	−	1.64432i	−0.604273	−	0.796777i	$-0.706536\pi$
$937$	−34.7427	−	50.3335i	−1.13499	−	1.64432i	−0.530721	−	0.847546i	$-0.678079\pi$
$938$		0			0
$939$	2.44544	−	60.6830i	0.0798040	−	1.98032i
$940$		0			0
$941$		0			0		0.297503	−	0.954721i	$-0.403846\pi$
$941$		0			0		−0.297503	+	0.954721i	$0.596154\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.335705	−	0.941967i	$-0.391026\pi$
$947$		0			0		−0.335705	+	0.941967i	$0.608974\pi$
$948$	−7.18004	+	29.1306i	−0.233197	+	0.946117i
$949$	−25.6880	−	46.7371i	−0.833868	−	1.51715i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		−0.391967	−	0.919979i	$-0.628205\pi$
$953$		0			0		0.391967	+	0.919979i	$0.371795\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−2.63265	+	0.319662i	−0.0849243	+	0.0103117i
$962$		0			0
$963$		0			0
$964$	28.7376	+	52.1740i	0.925577	+	1.68041i
$965$		0			0
$966$		0			0
$967$	5.45949	+	29.7915i	0.175565	+	0.958029i	0.946883	+	0.321578i	$0.104213\pi$
$967$	5.45949	+	29.7915i	0.175565	+	0.958029i	−0.771318	+	0.636450i	$0.780402\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		−0.948536	−	0.316668i	$-0.897436\pi$
$971$		0			0		0.948536	+	0.316668i	$0.102564\pi$
$972$	−28.1668	+	13.3653i	−0.903450	+	0.428693i
$973$	−17.7554	+	0.357614i	−0.569211	+	0.0114646i
$974$		0			0
$975$	21.6100	+	22.5391i	0.692073	+	0.721828i
$976$	−20.4689	−	10.7429i	−0.655194	−	0.343873i
$977$		0			0		0.446798	−	0.894635i	$-0.352564\pi$
$977$		0			0		−0.446798	+	0.894635i	$0.647436\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	47.0758	+	0.948161i	1.50302	+	0.0302724i
$982$		0			0
$983$		0			0		−0.0603785	−	0.998176i	$-0.519231\pi$
$983$		0			0		0.0603785	+	0.998176i	$0.480769\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	6.22731	+	62.1974i	0.198117	+	1.97876i
$989$		0			0
$990$		0			0
$991$	−29.9440	+	51.8646i	−0.951204	+	1.64753i	−0.208379	+	0.978048i	$0.566819\pi$
$991$	−29.9440	+	51.8646i	−0.951204	+	1.64753i	−0.742825	+	0.669485i	$0.766515\pi$
$992$		0			0
$993$	−22.3140	−	28.4817i	−0.708112	−	0.903838i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	23.3684	−	17.5602i	0.740084	−	0.556137i	−0.162470	−	0.986713i	$-0.551946\pi$
$997$	23.3684	−	17.5602i	0.740084	−	0.556137i	0.902554	+	0.430577i	$0.141690\pi$
$998$		0			0
$999$	22.5452	+	56.0176i	0.713297	+	1.77232i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	507.2.x.a.149.1	✓	48
3.2	odd	2	CM	507.2.x.a.149.1	✓	48
169.76	odd	156	inner	507.2.x.a.245.1	yes	48
507.245	even	156	inner	507.2.x.a.245.1	yes	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
507.2.x.a.149.1	✓	48	1.1	even	1	trivial
507.2.x.a.149.1	✓	48	3.2	odd	2	CM
507.2.x.a.245.1	yes	48	169.76	odd	156	inner
507.2.x.a.245.1	yes	48	507.245	even	156	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+(-1.19983 + 1.24916i) q^{3} +(-1.80690 - 0.857385i) q^{4} +(1.81421 + 0.257455i) q^{7} +(-0.120798 - 2.99757i) q^{9} +O(q^{10})\)	\(q+(-1.19983 + 1.24916i) q^{3} +(-1.80690 - 0.857385i) q^{4} +(1.81421 + 0.257455i) q^{7} +(-0.120798 - 2.99757i) q^{9} +(3.23899 - 1.22839i) q^{12} +(-2.59808 - 2.50000i) q^{13} +(2.52978 + 3.09842i) q^{16} +(8.37298 + 2.24353i) q^{19} +(-2.49835 + 1.95733i) q^{21} +(3.31561 + 3.74255i) q^{25} +(3.88938 + 3.44569i) q^{27} +(-3.05736 - 2.02067i) q^{28} +(5.31457 - 0.321472i) q^{31} +(-2.35180 + 5.51988i) q^{36} +(10.3965 + 5.19223i) q^{37} +(6.24016 - 0.245827i) q^{39} +(4.15278 - 12.4391i) q^{43} +(-6.90574 - 0.557489i) q^{48} +(-3.49856 - 1.01337i) q^{49} +(2.55100 + 6.74480i) q^{52} +(-12.8487 + 7.76732i) q^{57} +(-5.31675 + 2.26526i) q^{61} +(0.552585 - 5.46931i) q^{63} +(-1.91453 - 7.76753i) q^{64} +(-10.7743 - 3.83981i) q^{67} +(14.1217 + 4.40050i) q^{73} +(-8.65323 - 0.348713i) q^{75} +(-13.2056 - 11.2327i) q^{76} +(-4.91997 + 7.12781i) q^{79} +(-8.97082 + 0.724199i) q^{81} +(6.19246 - 1.39466i) q^{84} +(-4.06981 - 5.20441i) q^{91} +(-5.97503 + 7.02446i) q^{93} +(9.94139 - 13.7997i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 10 q^{7} + 6 q^{9}+O(q^{10}) \) 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

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