LMFDB - Embedded newform 507.2.x.a.20.1

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			20.1
Character	$\chi$	$=$	507.20
Dual form			507.2.x.a.431.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.19983 - 1.24916i) q^{3} +(1.80690 + 0.857385i) q^{4} +(-0.697469 + 4.91487i) q^{7} +(-0.120798 - 2.99757i) q^{9} +O(q^{10})$	$q+(1.19983 - 1.24916i) q^{3} +(1.80690 + 0.857385i) q^{4} +(-0.697469 + 4.91487i) q^{7} +(-0.120798 - 2.99757i) q^{9} +(3.23899 - 1.22839i) q^{12} +(2.59808 + 2.50000i) q^{13} +(2.52978 + 3.09842i) q^{16} +(0.239984 - 0.895631i) q^{19} +(5.30260 + 6.76827i) q^{21} +(-3.31561 - 3.74255i) q^{25} +(-3.88938 - 3.44569i) q^{27} +(-5.47419 + 8.28267i) q^{28} +(-0.590513 - 9.76234i) q^{31} +(2.35180 - 5.51988i) q^{36} +(-1.60806 + 3.21986i) q^{37} +(6.24016 - 0.245827i) q^{39} +(4.15278 - 12.4391i) q^{43} +(6.90574 + 0.557489i) q^{48} +(-16.9458 - 4.90841i) q^{49} +(2.55100 + 6.74480i) q^{52} +(-0.830846 - 1.37439i) q^{57} +(5.31675 - 2.26526i) q^{61} +(14.8169 + 1.49701i) q^{63} +(1.91453 + 7.76753i) q^{64} +(-3.93177 + 11.0323i) q^{67} +(-2.54558 + 8.16904i) q^{73} +(-8.65323 - 0.348713i) q^{75} +(1.20153 - 1.41256i) q^{76} +(4.91997 - 7.12781i) q^{79} +(-8.97082 + 0.724199i) q^{81} +(3.77826 + 16.7760i) q^{84} +(-14.0992 + 11.0255i) q^{91} +(-12.9032 - 10.9755i) q^{93} +(-8.06239 - 5.80820i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/507\mathbb{Z}\right)^\times$.

$n$	$170$	$340$
$\chi(n)$	$-1$	$e\left(\frac{11}{156}\right)$

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		−0.975564	−	0.219715i	$-0.929487\pi$
$2$		0			0		0.975564	+	0.219715i	$0.0705128\pi$
$3$	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		0.410413	−	0.911900i	$-0.365385\pi$
$5$		0			0		−0.410413	+	0.911900i	$0.634615\pi$
$6$		0			0
$7$	−0.697469	+	4.91487i	−0.263619	+	1.85764i	0.211044	+	0.977477i	$0.432314\pi$
$7$	−0.697469	+	4.91487i	−0.263619	+	1.85764i	−0.474662	+	0.880168i	$0.657430\pi$
$8$		0			0
$9$	−0.120798	−	2.99757i	−0.0402659	−	0.999189i
$10$		0			0
$11$		0			0		−0.678061	−	0.735006i	$-0.737179\pi$
$11$		0			0		0.678061	+	0.735006i	$0.262821\pi$
$12$	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
$19$	0.239984	−	0.895631i	0.0550560	−	0.205472i	−0.932919	−	0.360087i	$-0.882747\pi$
$19$	0.239984	−	0.895631i	0.0550560	−	0.205472i	0.987975	+	0.154615i	$0.0494137\pi$
$20$		0			0
$21$	5.30260	+	6.76827i	1.15712	+	1.47696i
$22$		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
$28$	−5.47419	+	8.28267i	−1.03452	+	1.56528i
$29$		0			0		0.845190	−	0.534466i	$-0.179487\pi$
$29$		0			0		−0.845190	+	0.534466i	$0.820513\pi$
$30$		0			0
$31$	−0.590513	−	9.76234i	−0.106059	−	1.75337i	−0.530292	−	0.847815i	$-0.677918\pi$
$31$	−0.590513	−	9.76234i	−0.106059	−	1.75337i	0.424233	−	0.905553i	$-0.360544\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	2.35180	−	5.51988i	0.391967	−	0.919979i
$37$	−1.60806	+	3.21986i	−0.264364	+	0.529342i	−0.986770	−	0.162127i	$-0.948165\pi$
$37$	−1.60806	+	3.21986i	−0.264364	+	0.529342i	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		0.999797	−	0.0201371i	$-0.00641026\pi$
$41$		0			0		−0.999797	+	0.0201371i	$0.993590\pi$
$42$		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		−0.180255	−	0.983620i	$-0.557692\pi$
$47$		0			0		0.180255	+	0.983620i	$0.442308\pi$
$48$	6.90574	+	0.557489i	0.996757	+	0.0804666i
$49$	−16.9458	−	4.90841i	−2.42083	−	0.701202i
$50$		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
$57$	−0.830846	−	1.37439i	−0.110048	−	0.182042i
$58$		0			0
$59$		0			0		−0.100522	−	0.994935i	$-0.532051\pi$
$59$		0			0		0.100522	+	0.994935i	$0.467949\pi$
$60$		0			0
$61$	5.31675	−	2.26526i	0.680740	−	0.290036i	−0.0237832	−	0.999717i	$-0.507571\pi$
$61$	5.31675	−	2.26526i	0.680740	−	0.290036i	0.704523	+	0.709681i	$0.251161\pi$
$62$		0			0
$63$	14.8169	+	1.49701i	1.86675	+	0.188605i
$64$	1.91453	+	7.76753i	0.239316	+	0.970942i
$65$		0			0
$66$		0			0
$67$	−3.93177	+	11.0323i	−0.480342	+	1.34781i	0.417974	+	0.908459i	$0.362740\pi$
$67$	−3.93177	+	11.0323i	−0.480342	+	1.34781i	−0.898315	+	0.439351i	$0.855208\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		0.482459	−	0.875918i	$-0.339744\pi$
$71$		0			0		−0.482459	+	0.875918i	$0.660256\pi$
$72$		0			0
$73$	−2.54558	+	8.16904i	−0.297937	+	0.956114i	0.677441	+	0.735577i	$0.263089\pi$
$73$	−2.54558	+	8.16904i	−0.297937	+	0.956114i	−0.975379	+	0.220537i	$0.929219\pi$
$74$		0			0
$75$	−8.65323	−	0.348713i	−0.999189	−	0.0402659i
$76$	1.20153	−	1.41256i	0.137825	−	0.162032i
$77$		0			0
$78$		0			0
$79$	4.91997	−	7.12781i	0.553540	−	0.801942i	−0.441926	−	0.897051i	$-0.645705\pi$
$79$	4.91997	−	7.12781i	0.553540	−	0.801942i	0.995467	+	0.0951096i	$0.0303201\pi$
$80$		0			0
$81$	−8.97082	+	0.724199i	−0.996757	+	0.0804666i
$82$		0			0
$83$		0			0		−0.855781	−	0.517338i	$-0.826923\pi$
$83$		0			0		0.855781	+	0.517338i	$0.173077\pi$
$84$	3.77826	+	16.7760i	0.412242	+	1.83041i
$85$		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
$91$	−14.0992	+	11.0255i	−1.47800	+	1.15579i
$92$		0			0
$93$	−12.9032	−	10.9755i	−1.33800	−	1.13811i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−8.06239	−	5.80820i	−0.818611	−	0.589733i	0.0953921	−	0.995440i	$-0.469590\pi$
$97$	−8.06239	−	5.80820i	−0.818611	−	0.589733i	−0.914003	+	0.405707i	$0.867025\pi$
$98$		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
$109$	13.7467	+	0.831522i	1.31669	+	0.0796453i	0.703731	−	0.710466i	$-0.251516\pi$
$109$	13.7467	+	0.831522i	1.31669	+	0.0796453i	0.612963	+	0.790112i	$0.289977\pi$
$110$		0			0
$111$	2.09271	+	5.87203i	0.198631	+	0.557348i
$112$	−16.9928	+	10.2725i	−1.60567	+	0.970658i
$113$		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
$124$	7.30309	−	18.1459i	0.655837	−	1.62955i
$125$		0			0
$126$		0			0
$127$	−19.3411	+	9.17747i	−1.71625	+	0.814369i	−0.724040	+	0.689758i	$0.757717\pi$
$127$	−19.3411	+	9.17747i	−1.71625	+	0.814369i	−0.992206	+	0.124611i	$0.960232\pi$
$128$		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
$133$	4.23453	+	1.80416i	0.367180	+	0.156441i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		0.894635	−	0.446798i	$-0.147436\pi$
$137$		0			0		−0.894635	+	0.446798i	$0.852564\pi$
$138$		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
$147$	−26.4635	+	15.2787i	−2.18268	+	1.26017i
$148$	−5.66627	+	4.43924i	−0.465765	+	0.364903i
$149$		0			0		−0.647915	−	0.761712i	$-0.724359\pi$
$149$		0			0		0.647915	+	0.761712i	$0.275641\pi$
$150$		0			0
$151$	−23.5704	−	4.31944i	−1.91814	−	0.351511i	−0.919045	−	0.394153i	$-0.871038\pi$
$151$	−23.5704	−	4.31944i	−1.91814	−	0.351511i	−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
$163$	−2.02753	−	3.06773i	−0.158808	−	0.240284i	0.746156	−	0.665771i	$-0.231897\pi$
$163$	−2.02753	−	3.06773i	−0.158808	−	0.240284i	−0.904964	+	0.425488i	$0.860103\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.219715	−	0.975564i	$-0.429487\pi$
$167$		0			0		−0.219715	+	0.975564i	$0.570513\pi$
$168$		0			0
$169$	0.500000	+	12.9904i	0.0384615	+	0.999260i
$170$		0			0
$171$	−2.71370	−	0.611177i	−0.207522	−	0.0467379i
$172$	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
$175$	20.7067	−	13.6855i	1.56528	−	1.03452i
$176$		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
$189$	19.6478	−	16.7125i	1.42917	−	1.21566i
$190$		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
$193$	26.1931	−	3.71706i	1.88542	−	0.267560i	0.899770	−	0.436365i	$-0.143734\pi$
$193$	26.1931	−	3.71706i	1.88542	−	0.267560i	0.985649	+	0.168805i	$0.0539908\pi$
$194$		0			0
$195$		0			0
$196$	−26.4110	−	23.3981i	−1.88650	−	1.67129i
$197$		0			0		0.551377	−	0.834256i	$-0.314103\pi$
$197$		0			0		−0.551377	+	0.834256i	$0.685897\pi$
$198$		0			0
$199$	0.736085	+	4.52932i	0.0521797	+	0.321075i	0.999997	+	0.00226195i	$0.000720001\pi$
$199$	0.736085	+	4.52932i	0.0521797	+	0.321075i	−0.947818	+	0.318813i	$0.896716\pi$
$200$		0			0
$201$	9.06364	+	18.1483i	0.639300	+	1.28008i
$202$		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	$-0.724117\pi$
$211$	−12.0323	−	25.3576i	−0.828339	−	1.74569i	−0.181003	−	0.983483i	$-0.557934\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	48.3925	+	3.90664i	3.28509	+	0.265200i
$218$		0			0
$219$	7.15017	+	12.9813i	0.483164	+	0.877197i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	15.8683	+	22.0268i	1.06262	+	1.47503i	0.869552	+	0.493841i	$0.164408\pi$
$223$	15.8683	+	22.0268i	1.06262	+	1.47503i	0.193067	+	0.981186i	$0.438157\pi$
$224$		0			0
$225$	−10.8180	+	10.3909i	−0.721202	+	0.692724i
$226$		0			0
$227$		0			0		0.941967	−	0.335705i	$-0.108974\pi$
$227$		0			0		−0.941967	+	0.335705i	$0.891026\pi$
$228$	−0.322877	−	3.19573i	−0.0213831	−	0.211643i
$229$	1.72635	−	28.5399i	0.114080	−	1.88597i	−0.270273	−	0.962784i	$-0.587114\pi$
$229$	1.72635	−	28.5399i	0.114080	−	1.88597i	0.384353	−	0.923186i	$-0.374424\pi$
$230$		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	$-0.750000\pi$
$239$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$240$		0			0
$241$	5.12936	−	7.12009i	0.330411	−	0.458645i	−0.613091	−	0.790012i	$-0.710074\pi$
$241$	5.12936	−	7.12009i	0.330411	−	0.458645i	0.943502	+	0.331367i	$0.107510\pi$
$242$		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
$247$	2.86257	−	1.72696i	0.182141	−	0.109884i
$248$		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	$0.0641026\pi$
$252$	25.4891	+	15.4087i	1.60567	+	0.970658i
$253$		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
$259$	−14.7036	−	10.1492i	−0.913638	−	0.630639i
$260$		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
$268$	−16.5632	+	16.5632i	−1.01176	+	1.01176i
$269$		0			0		−0.278217	−	0.960518i	$-0.589744\pi$
$269$		0			0		0.278217	+	0.960518i	$0.410256\pi$
$270$		0			0
$271$	24.2228	+	8.63267i	1.47143	+	0.524397i	0.945355	−	0.326042i	$-0.105715\pi$
$271$	24.2228	+	8.63267i	1.47143	+	0.524397i	0.526073	+	0.850439i	$0.323664\pi$
$272$		0			0
$273$	−3.14411	+	30.8410i	−0.190290	+	1.86658i
$274$		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
$279$	−29.1919	+	2.94937i	−1.74768	+	0.176574i
$280$		0			0
$281$		0			0		0.855781	−	0.517338i	$-0.173077\pi$
$281$		0			0		−0.855781	+	0.517338i	$0.826923\pi$
$282$		0			0
$283$	3.85965	+	11.5611i	0.229432	+	0.687234i	0.998825	+	0.0484581i	$0.0154307\pi$
$283$	3.85965	+	11.5611i	0.229432	+	0.687234i	−0.769393	+	0.638776i	$0.779441\pi$
$284$		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
$291$	−16.9289	+	3.10233i	−0.992389	+	0.181862i
$292$	−11.6036	+	12.5781i	−0.679051	+	0.736078i
$293$		0			0		0.373361	−	0.927686i	$-0.378205\pi$
$293$		0			0		−0.373361	+	0.927686i	$0.621795\pi$
$294$		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
$301$	58.2401	+	29.0863i	3.35690	+	1.67650i
$302$		0			0
$303$		0			0
$304$	3.38215	−	1.52218i	0.193979	−	0.0873031i
$305$		0			0
$306$		0			0
$307$	20.1848	−	1.22096i	1.15201	−	0.0696837i	0.526693	−	0.850056i	$-0.323432\pi$
$307$	20.1848	−	1.22096i	1.15201	−	0.0696837i	0.625316	+	0.780372i	$0.284970\pi$
$308$		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	$-0.273599\pi$
$313$	26.2456	−	23.2516i	1.48349	−	1.31426i	0.830701	−	0.556718i	$-0.187940\pi$
$314$		0			0
$315$		0			0
$316$	15.0012	−	8.66094i	0.843883	−	0.487216i
$317$		0			0		0.787183	−	0.616719i	$-0.211538\pi$
$317$		0			0		−0.787183	+	0.616719i	$0.788462\pi$
$318$		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
$327$	17.5324	−	16.1741i	0.969547	−	0.894431i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−29.4975	−	4.18600i	−1.62133	−	0.230083i	−0.730004	−	0.683443i	$-0.760482\pi$
$331$	−29.4975	−	4.18600i	−1.62133	−	0.230083i	−0.891328	+	0.453360i	$0.850225\pi$
$332$		0			0
$333$	9.84600	+	4.43133i	0.539558	+	0.242835i
$334$		0			0
$335$		0			0
$336$	−7.55652	+	33.5519i	−0.412242	+	1.83041i
$337$		−	25.8481i		−	1.40803i	−0.710184	−	0.704017i	$-0.751388\pi$
$337$		−	25.8481i		−	1.40803i	0.710184	−	0.704017i	$-0.248612\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	21.6820	−	48.1755i	1.17072	−	2.60123i
$344$		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
$349$	−25.1453	−	27.2570i	−1.34600	−	1.45904i	−0.762648	−	0.646814i	$-0.776101\pi$
$349$	−25.1453	−	27.2570i	−1.34600	−	1.45904i	−0.583349	−	0.812222i	$-0.698258\pi$
$350$		0			0
$351$	−1.49068	−	18.6756i	−0.0795666	−	0.996830i
$352$		0			0
$353$		0			0		−0.373361	−	0.927686i	$-0.621795\pi$
$353$		0			0		0.373361	+	0.927686i	$0.378205\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		−0.616719	−	0.787183i	$-0.711538\pi$
$359$		0			0		0.616719	+	0.787183i	$0.288462\pi$
$360$		0			0
$361$	15.7099	+	9.07013i	0.826838	+	0.477375i
$362$		0			0
$363$	12.6342	+	14.2610i	0.663123	+	0.748511i
$364$	−34.9291	+	7.83354i	−1.83078	+	0.410589i
$365$		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
$372$	−13.9046	−	30.8948i	−0.720920	−	1.60182i
$373$	26.5060	+	16.7614i	1.37243	+	0.867873i	0.998230	−	0.0594747i	$-0.0189426\pi$
$373$	26.5060	+	16.7614i	1.37243	+	0.867873i	0.374201	+	0.927348i	$0.377917\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−38.9238	+	0.783970i	−1.99938	+	0.0402698i	−0.999397	+	0.0347140i	$0.988948\pi$
$379$	−38.9238	+	0.783970i	−1.99938	+	0.0402698i	−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		−0.927686	−	0.373361i	$-0.878205\pi$
$383$		0			0		0.927686	+	0.373361i	$0.121795\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−37.7887	−	10.9456i	−1.92091	−	0.556397i
$388$	−9.58807	−	17.4074i	−0.486761	−	0.883727i
$389$		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
$397$	0.820709	+	8.12312i	0.0411902	+	0.407687i	0.994424	+	0.105454i	$0.0336296\pi$
$397$	0.820709	+	8.12312i	0.0411902	+	0.407687i	−0.953234	+	0.302233i	$0.902268\pi$
$398$		0			0
$399$	7.33441	−	3.12490i	0.367180	−	0.156441i
$400$	3.20823	−	19.7410i	0.160411	−	0.987050i
$401$		0			0		−0.994935	−	0.100522i	$-0.967949\pi$
$401$		0			0		0.994935	+	0.100522i	$0.0320513\pi$
$402$		0			0
$403$	22.8717	−	26.8396i	1.13932	−	1.33697i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−18.0081	+	32.6943i	−0.890445	+	1.61663i	−0.107788	+	0.994174i	$0.534377\pi$
$409$	−18.0081	+	32.6943i	−0.890445	+	1.61663i	−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
$421$	−35.0583	−	21.1935i	−1.70864	−	1.03291i	−0.884551	−	0.466443i	$-0.845535\pi$
$421$	−35.0583	−	21.1935i	−1.70864	−	1.03291i	−0.824086	−	0.566464i	$-0.808311\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	7.42516	+	27.7111i	0.359328	+	1.34103i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		−0.761712	−	0.647915i	$-0.775641\pi$
$431$		0			0		0.761712	+	0.647915i	$0.224359\pi$
$432$	0.836912	−	20.7678i	0.0402659	−	0.999189i
$433$	32.1112	+	26.2180i	1.54317	+	1.25996i	0.827389	+	0.561629i	$0.189825\pi$
$433$	32.1112	+	26.2180i	1.54317	+	1.25996i	0.715778	+	0.698328i	$0.246072\pi$
$434$		0			0
$435$		0			0
$436$	24.1260	+	13.2887i	1.15542	+	0.636413i
$437$		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
$441$	−12.6663	+	51.3891i	−0.603156	+	2.44710i
$442$		0			0
$443$		0			0		0.970942	−	0.239316i	$-0.0769231\pi$
$443$		0			0		−0.970942	+	0.239316i	$0.923077\pi$
$444$	−1.25327	+	12.4044i	−0.0594774	+	0.588688i
$445$		0			0
$446$		0			0
$447$		0			0
$448$	−39.5117	+	3.99202i	−1.86675	+	0.188605i
$449$		0			0		−0.335705	−	0.941967i	$-0.608974\pi$
$449$		0			0		0.335705	+	0.941967i	$0.391026\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	−33.6763	+	24.2606i	−1.58225	+	1.13986i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−37.1073	+	20.4389i	−1.73581	+	0.956089i	−0.802785	+	0.596269i	$0.796649\pi$
$457$	−37.1073	+	20.4389i	−1.73581	+	0.956089i	−0.933022	+	0.359821i	$0.882838\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		0.678061	−	0.735006i	$-0.262821\pi$
$461$		0			0		−0.678061	+	0.735006i	$0.737179\pi$
$462$		0			0
$463$	−38.7022	+	12.0601i	−1.79864	+	0.560480i	−0.999190	−	0.0402476i	$-0.987185\pi$
$463$	−38.7022	+	12.0601i	−1.79864	+	0.560480i	−0.799454	+	0.600728i	$0.794878\pi$
$464$		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
$469$	−51.4800	−	27.0188i	−2.37713	−	1.24761i
$470$		0			0
$471$	39.4850	+	16.8230i	1.81937	+	0.775163i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−4.14764	+	2.07142i	−0.190307	+	0.0950430i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.834256	−	0.551377i	$-0.814103\pi$
$479$		0			0		0.834256	+	0.551377i	$0.185897\pi$
$480$		0			0
$481$	−12.2275	+	4.34529i	−0.557527	+	0.198128i
$482$		0			0
$483$		0			0
$484$	−11.0000	+	19.0526i	−0.500000	+	0.866025i
$485$		0			0
$486$		0			0
$487$	−27.6367	−	32.4907i	−1.25234	−	1.47230i	−0.821854	−	0.569698i	$-0.807060\pi$
$487$	−27.6367	−	32.4907i	−1.25234	−	1.47230i	−0.430486	−	0.902597i	$-0.641658\pi$
$488$		0			0
$489$	−6.26479	−	1.14806i	−0.283303	−	0.0519173i
$490$		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
$496$	28.7540	−	26.5262i	1.29109	−	1.19106i
$497$		0			0
$498$		0			0
$499$	9.43396	−	12.0416i	0.422322	−	0.539054i	−0.528442	−	0.848969i	$-0.677224\pi$
$499$	9.43396	−	12.0416i	0.422322	−	0.539054i	0.950764	+	0.309915i	$0.100301\pi$
$500$		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		−0.975564	−	0.219715i	$-0.929487\pi$
$509$		0			0		0.975564	+	0.219715i	$0.0705128\pi$
$510$		0			0
$511$	−38.3743	−	18.2088i	−1.69758	−	0.805511i
$512$		0			0
$513$	−4.01945	+	2.65654i	−0.177463	+	0.117289i
$514$		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
$525$	7.74924	−	42.2862i	0.338205	−	1.84552i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	11.5000	+	19.9186i	0.500000	+	0.866025i
$530$		0			0
$531$		0			0
$532$	6.10451	+	6.89056i	0.264664	+	0.298744i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	18.1602	+	40.3504i	0.780769	+	1.73480i	0.671774	+	0.740756i	$0.265533\pi$
$541$	18.1602	+	40.3504i	0.780769	+	1.73480i	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
$553$	31.6007	+	29.1524i	1.34380	+	1.23969i
$554$		0			0
$555$		0			0
$556$	−18.6182	−	5.39282i	−0.789586	−	0.228706i
$557$		0			0		−0.482459	−	0.875918i	$-0.660256\pi$
$557$		0			0		0.482459	+	0.875918i	$0.339744\pi$
$558$		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
$567$	2.69753	−	44.5955i	0.113286	−	1.87283i
$568$		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	0.942293	−	0.334790i	$-0.108665\pi$
$571$	−1.13864	−	4.61963i	−0.0476505	−	0.193326i	−0.989943	+	0.141464i	$0.954819\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	23.0524	−	6.67722i	0.960518	−	0.278217i
$577$	11.7440	+	11.7440i	0.488909	+	0.488909i	0.907962	−	0.419053i	$-0.137638\pi$
$577$	11.7440	+	11.7440i	0.488909	+	0.488909i	−0.419053	+	0.907962i	$0.637638\pi$
$578$		0			0
$579$	26.7841	−	37.1792i	1.11311	−	1.54511i
$580$		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	$0.916667\pi$
$588$	−60.9168	+	4.91771i	−2.51217	+	0.202803i
$589$	−8.88517	−	1.81392i	−0.366107	−	0.0747413i
$590$		0			0
$591$		0			0
$592$	−14.0445	+	3.16309i	−0.577227	+	0.130002i
$593$		0			0		0.517338	−	0.855781i	$-0.326923\pi$
$593$		0			0		−0.517338	+	0.855781i	$0.673077\pi$
$594$		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	$0.152273\pi$
$601$	−0.837524	+	20.7829i	−0.0341633	+	0.847754i	−0.921906	+	0.387414i	$0.873368\pi$
$602$		0			0
$603$	33.5450	+	10.4531i	1.36606	+	0.425681i
$604$	−38.8860	−	28.0138i	−1.58225	−	1.13986i
$605$		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
$613$	−4.96794	+	49.1710i	−0.200653	+	1.98600i	−0.0383050	+	0.999266i	$0.512196\pi$
$613$	−4.96794	+	49.1710i	−0.200653	+	1.98600i	−0.162348	+	0.986734i	$0.551907\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		0.994935	−	0.100522i	$-0.0320513\pi$
$617$		0			0		−0.994935	+	0.100522i	$0.967949\pi$
$618$		0			0
$619$	36.1338	−	21.8437i	1.45234	−	0.877971i	0.452437	−	0.891796i	$-0.350555\pi$
$619$	36.1338	−	21.8437i	1.45234	−	0.877971i	0.999904	+	0.0138256i	$0.00440096\pi$
$620$		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
$631$	−14.5324	+	36.1084i	−0.578525	+	1.43745i	0.297281	+	0.954790i	$0.403920\pi$
$631$	−14.5324	+	36.1084i	−0.578525	+	1.43745i	−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
$637$	−31.7555	−	55.1170i	−1.25820	−	2.18381i
$638$		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
$643$	−0.281676	+	13.9851i	−0.0111082	+	0.551518i	0.955462	+	0.295115i	$0.0953580\pi$
$643$	−0.281676	+	13.9851i	−0.0111082	+	0.551518i	−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
$651$	62.9429	−	55.7626i	2.46693	−	2.18551i
$652$	−1.03331	−	7.28147i	−0.0404677	−	0.285164i
$653$		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
$657$	24.7948	+	6.64374i	0.967336	+	0.259197i
$658$		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
$661$	39.0961	−	15.7348i	1.52066	−	0.612014i	0.546419	−	0.837512i	$-0.315991\pi$
$661$	39.0961	−	15.7348i	1.52066	−	0.612014i	0.974244	+	0.225498i	$0.0724008\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	46.5543	+	6.60654i	1.79990	+	0.255423i
$670$		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
$679$	34.1698	−	35.5745i	1.31132	−	1.36522i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		0.140502	−	0.990080i	$-0.455128\pi$
$683$		0			0		−0.140502	+	0.990080i	$0.544872\pi$
$684$	−4.37938	−	3.43103i	−0.167450	−	0.131189i
$685$		0			0
$686$		0			0
$687$	−33.5796	−	36.3996i	−1.28114	−	1.38873i
$688$	49.0472	−	18.6011i	1.86991	−	0.709162i
$689$		0			0
$690$		0			0
$691$	−9.32637	−	23.1731i	−0.354792	−	0.881546i	−0.993680	−	0.112247i	$-0.964195\pi$
$691$	−9.32637	−	23.1731i	−0.354792	−	0.881546i	0.638889	−	0.769299i	$-0.279394\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	49.1487	−	6.97469i	1.85764	−	0.263619i
$701$		0			0		−0.663123	−	0.748511i	$-0.730769\pi$
$701$		0			0		0.663123	+	0.748511i	$0.269231\pi$
$702$		0			0
$703$	2.49790	+	2.21295i	0.0942101	+	0.0834629i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−53.2062	−	1.07163i	−1.99820	−	0.0402461i	−0.998350	−	0.0574199i	$-0.981713\pi$
$709$	−53.2062	−	1.07163i	−1.99820	−	0.0402461i	−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
$721$	81.4324	+	32.7737i	3.03270	+	1.22056i
$722$		0			0
$723$	−2.73975	−	14.9503i	−0.101892	−	0.556008i
$724$	−44.7086	−	3.60925i	−1.66158	−	0.134137i
$725$		0			0
$726$		0			0
$727$	−16.3263	−	1.98237i	−0.605508	−	0.0735220i	−0.187960	−	0.982177i	$-0.560187\pi$
$727$	−16.3263	−	1.98237i	−0.605508	−	0.0735220i	−0.417548	+	0.908655i	$0.637111\pi$
$728$		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
$733$	19.9817	+	33.0538i	0.738042	+	1.22087i	0.968796	+	0.247858i	$0.0797267\pi$
$733$	19.9817	+	33.0538i	0.738042	+	1.22087i	−0.230755	+	0.973012i	$0.574119\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−24.9771	−	2.52353i	−0.918797	−	0.0928295i	−0.370234	−	0.928939i	$-0.620722\pi$
$739$	−24.9771	−	2.52353i	−0.918797	−	0.0928295i	−0.548563	+	0.836109i	$0.684825\pi$
$740$		0			0
$741$	1.27737	−	5.64787i	0.0469252	−	0.207480i
$742$		0			0
$743$		0			0		0.335705	−	0.941967i	$-0.391026\pi$
$743$		0			0		−0.335705	+	0.941967i	$0.608974\pi$
$744$		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
$756$	49.8307	−	13.3521i	1.81232	−	0.485611i
$757$	−31.5535	+	2.54726i	−1.14683	+	0.0925818i	−0.639234	−	0.769012i	$-0.720748\pi$
$757$	−31.5535	+	2.54726i	−1.14683	+	0.0925818i	−0.507598	+	0.861594i	$0.669466\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		0.975564	−	0.219715i	$-0.0705128\pi$
$761$		0			0		−0.975564	+	0.219715i	$0.929487\pi$
$762$		0			0
$763$	−13.6747	+	66.9832i	−0.495058	+	2.42495i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	15.7427	+	22.8072i	0.568065	+	0.822984i
$769$	4.06973	+	3.46173i	0.146758	+	0.124833i	0.718281	−	0.695753i	$-0.244929\pi$
$769$	4.06973	+	3.46173i	0.146758	+	0.124833i	−0.571523	+	0.820586i	$0.693647\pi$
$770$		0			0
$771$		0			0
$772$	50.5153	+	15.7412i	1.81808	+	0.566538i
$773$		0			0		−0.811378	−	0.584522i	$-0.801282\pi$
$773$		0			0		0.811378	+	0.584522i	$0.198718\pi$
$774$		0			0
$775$	−34.5782	+	34.5782i	−1.24208	+	1.24208i
$776$		0			0
$777$	−30.3198	+	6.18984i	−1.08772	+	0.222059i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−27.6609	−	64.9225i	−0.987888	−	2.31866i
$785$		0			0
$786$		0			0
$787$	−14.5737	−	40.8930i	−0.519497	−	1.45768i	−0.856487	−	0.516169i	$-0.827358\pi$
$787$	−14.5737	−	40.8930i	−0.519497	−	1.45768i	0.336990	−	0.941508i	$-0.390591\pi$
$788$		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
$804$	0.816990	+	40.5633i	0.0288130	+	1.43056i
$805$		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
$811$	−31.1083	+	14.0007i	−1.09236	+	0.491631i	−0.874834	−	0.484424i	$-0.839029\pi$
$811$	−31.1083	+	14.0007i	−1.09236	+	0.491631i	−0.217526	+	0.976055i	$0.569799\pi$
$812$		0			0
$813$	39.8469	−	19.9003i	1.39749	−	0.697935i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−10.1442	−	6.70454i	−0.354902	−	0.234562i
$818$		0			0
$819$	34.7529	+	40.9316i	1.21436	+	1.43026i
$820$		0			0
$821$		0			0		−0.140502	−	0.990080i	$-0.544872\pi$
$821$		0			0		0.140502	+	0.990080i	$0.455128\pi$
$822$		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		−0.983620	−	0.180255i	$-0.942308\pi$
$827$		0			0		0.983620	+	0.180255i	$0.0576923\pi$
$828$		0			0
$829$	30.8661	−	25.2014i	1.07202	−	0.875280i	0.0794389	−	0.996840i	$-0.474687\pi$
$829$	30.8661	−	25.2014i	1.07202	−	0.875280i	0.992584	+	0.121560i	$0.0387897\pi$
$830$		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
$837$	−31.3412	+	40.0041i	−1.08331	+	1.38275i
$838$		0			0
$839$		0			0		−0.551377	−	0.834256i	$-0.685897\pi$
$839$		0			0		0.551377	+	0.834256i	$0.314103\pi$
$840$		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
$847$	−53.2709	−	11.9976i	−1.83041	−	0.412242i
$848$		0			0
$849$	19.0726	+	9.05004i	0.654568	+	0.310597i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−10.0156	−	7.84675i	−0.342929	−	0.268668i	0.430446	−	0.902616i	$-0.358356\pi$
$853$	−10.0156	−	7.84675i	−0.342929	−	0.268668i	−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	$0.467276\pi$
$859$	−18.5762	+	48.9815i	−0.633812	+	1.67123i	−0.736437	+	0.676506i	$0.763493\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.180255	−	0.983620i	$-0.442308\pi$
$863$		0			0		−0.180255	+	0.983620i	$0.557692\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	14.7224	+	25.5000i	0.500000	+	0.866025i
$868$	84.0909	+	48.5499i	2.85423	+	1.64789i
$869$		0			0
$870$		0			0
$871$	−37.7958	+	18.8334i	−1.28066	+	0.638144i
$872$		0			0
$873$	−16.4365	+	24.8692i	−0.556293	+	0.841693i
$874$		0			0
$875$		0			0
$876$	1.78965	+	29.5864i	0.0604666	+	0.999632i
$877$	26.4258	+	52.9130i	0.892336	+	1.78674i	0.493673	+	0.869647i	$0.335654\pi$
$877$	26.4258	+	52.9130i	0.892336	+	1.78674i	0.398663	+	0.917098i	$0.369474\pi$
$878$		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	$-0.684004\pi$
$883$	13.4686	−	25.6623i	0.453255	−	0.863605i	0.999660	−	0.0260838i	$-0.00830366\pi$
$884$		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
$889$	−31.6162	−	101.460i	−1.06037	−	3.40286i
$890$		0			0
$891$		0			0
$892$	9.78693	+	53.4056i	0.327691	+	1.78815i
$893$		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
$903$	106.212	−	37.8524i	3.53451	−	1.25965i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−0.0124823	+	0.0768067i	−0.000414468	+	0.00255033i	−0.987257	−	0.159136i	$-0.949129\pi$
$907$	−0.0124823	+	0.0768067i	−0.000414468	+	0.00255033i	0.986842	+	0.161686i	$0.0516933\pi$
$908$		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	$0.0769231\pi$
$912$	2.15657	−	6.05121i	0.0714111	−	0.200375i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	27.5890	−	50.0886i	0.911567	−	1.65497i
$917$		0			0
$918$		0			0
$919$	−7.04931	+	8.63384i	−0.232535	+	0.284804i	−0.877671	−	0.479264i	$-0.840904\pi$
$919$	−7.04931	+	8.63384i	−0.232535	+	0.284804i	0.645135	+	0.764068i	$0.276801\pi$
$920$		0			0
$921$	22.6933	−	26.6790i	0.747769	−	0.879103i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	17.3822	−	4.65755i	0.571524	−	0.153139i
$926$		0			0
$927$	−51.9768	−	10.6111i	−1.70714	−	0.348515i
$928$		0			0
$929$		0			0		−0.219715	−	0.975564i	$-0.570513\pi$
$929$		0			0		0.219715	+	0.975564i	$0.429487\pi$
$930$		0			0
$931$	−8.46285	+	13.9993i	−0.277359	+	0.458807i
$932$		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	$0.293464\pi$
$937$	34.7427	+	50.3335i	1.13499	+	1.64432i	0.530721	+	0.847546i	$0.321921\pi$
$938$		0			0
$939$	2.44544	−	60.6830i	0.0798040	−	1.98032i
$940$		0			0
$941$		0			0		−0.954721	−	0.297503i	$-0.903846\pi$
$941$		0			0		0.954721	+	0.297503i	$0.0961538\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		−0.941967	−	0.335705i	$-0.891026\pi$
$947$		0			0		0.941967	+	0.335705i	$0.108974\pi$
$948$	7.18004	−	29.1306i	0.233197	−	0.946117i
$949$	−27.0362	+	14.8599i	−0.877633	+	0.482371i
$950$		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
$961$	−64.1806	+	7.79294i	−2.07034	+	0.251385i
$962$		0			0
$963$		0			0
$964$	15.3729	−	8.46746i	0.495128	−	0.272718i
$965$		0			0
$966$		0			0
$967$	−53.4302	+	9.79146i	−1.71820	+	0.314872i	−0.946883	−	0.321578i	$-0.895787\pi$
$967$	−53.4302	+	9.79146i	−1.71820	+	0.314872i	−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
$973$	−0.968810	−	48.1010i	−0.0310586	−	1.54205i
$974$		0			0
$975$	−21.6100	−	22.5391i	−0.692073	−	0.721828i
$976$	20.4689	+	10.7429i	0.655194	+	0.343873i
$977$		0			0		−0.894635	−	0.446798i	$-0.852564\pi$
$977$		0			0		0.894635	+	0.446798i	$0.147436\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	0.831971	−	41.3071i	0.0265628	−	1.31883i
$982$		0			0
$983$		0			0		0.998176	−	0.0603785i	$-0.0192308\pi$
$983$		0			0		−0.998176	+	0.0603785i	$0.980769\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	6.65306	−	0.666115i	0.211662	−	0.0211920i
$989$		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
$993$	−40.6211	+	31.8246i	−1.28907	+	1.00992i
$994$		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
$999$	17.3490	−	6.98237i	0.548898	−	0.220912i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	507.2.x.a.20.1	✓	48
3.2	odd	2	CM	507.2.x.a.20.1	✓	48
169.93	odd	156	inner	507.2.x.a.431.1	yes	48
507.431	even	156	inner	507.2.x.a.431.1	yes	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
507.2.x.a.20.1	✓	48	1.1	even	1	trivial
507.2.x.a.20.1	✓	48	3.2	odd	2	CM
507.2.x.a.431.1	yes	48	169.93	odd	156	inner
507.2.x.a.431.1	yes	48	507.431	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)

\(f(q)\)	\(=\)	\(q+(1.19983 - 1.24916i) q^{3} +(1.80690 + 0.857385i) q^{4} +(-0.697469 + 4.91487i) q^{7} +(-0.120798 - 2.99757i) q^{9} +O(q^{10})\)	\(q+(1.19983 - 1.24916i) q^{3} +(1.80690 + 0.857385i) q^{4} +(-0.697469 + 4.91487i) q^{7} +(-0.120798 - 2.99757i) q^{9} +(3.23899 - 1.22839i) q^{12} +(2.59808 + 2.50000i) q^{13} +(2.52978 + 3.09842i) q^{16} +(0.239984 - 0.895631i) q^{19} +(5.30260 + 6.76827i) q^{21} +(-3.31561 - 3.74255i) q^{25} +(-3.88938 - 3.44569i) q^{27} +(-5.47419 + 8.28267i) q^{28} +(-0.590513 - 9.76234i) q^{31} +(2.35180 - 5.51988i) q^{36} +(-1.60806 + 3.21986i) q^{37} +(6.24016 - 0.245827i) q^{39} +(4.15278 - 12.4391i) q^{43} +(6.90574 + 0.557489i) q^{48} +(-16.9458 - 4.90841i) q^{49} +(2.55100 + 6.74480i) q^{52} +(-0.830846 - 1.37439i) q^{57} +(5.31675 - 2.26526i) q^{61} +(14.8169 + 1.49701i) q^{63} +(1.91453 + 7.76753i) q^{64} +(-3.93177 + 11.0323i) q^{67} +(-2.54558 + 8.16904i) q^{73} +(-8.65323 - 0.348713i) q^{75} +(1.20153 - 1.41256i) q^{76} +(4.91997 - 7.12781i) q^{79} +(-8.97082 + 0.724199i) q^{81} +(3.77826 + 16.7760i) q^{84} +(-14.0992 + 11.0255i) q^{91} +(-12.9032 - 10.9755i) q^{93} +(-8.06239 - 5.80820i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 10 q^{7} + 6 q^{9}+O(q^{10}) \) 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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