LMFDB - Embedded newform 507.2.x.a.119.1

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			119.1
Character	$\chi$	$=$	507.119
Dual form			507.2.x.a.98.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.64291 + 0.548485i) q^{3} +(-1.44240 - 1.38545i) q^{4} +(-0.527444 + 5.22047i) q^{7} +(2.39833 + 1.80223i) q^{9} +O(q^{10})$	$q+(1.64291 + 0.548485i) q^{3} +(-1.44240 - 1.38545i) q^{4} +(-0.527444 + 5.22047i) q^{7} +(2.39833 + 1.80223i) q^{9} +(-1.60985 - 3.06731i) q^{12} +(2.59808 - 2.50000i) q^{13} +(0.161064 + 3.99676i) q^{16} +(1.98272 + 7.39960i) q^{19} +(-3.72990 + 8.28749i) q^{21} +(-4.11492 - 2.84032i) q^{25} +(2.95175 + 4.27635i) q^{27} +(7.99349 - 6.79929i) q^{28} +(1.48545 + 0.272220i) q^{31} +(-0.962468 - 5.92230i) q^{36} +(5.91139 - 2.10674i) q^{37} +(5.63963 - 2.68228i) q^{39} +(11.5365 - 5.47415i) q^{43} +(-1.92755 + 6.65466i) q^{48} +(-20.1166 - 4.10684i) q^{49} +(-7.21110 + 0.00651091i) q^{52} +(-0.801139 + 13.2444i) q^{57} +(-11.3705 - 1.84789i) q^{61} +(-10.6735 + 11.5698i) q^{63} +(5.30498 - 5.98809i) q^{64} +(-9.94094 - 0.200222i) q^{67} +(2.09927 - 2.67951i) q^{73} +(-5.20258 - 6.92338i) q^{75} +(7.39189 - 13.4202i) q^{76} +(-0.441406 + 0.108797i) q^{79} +(2.50396 + 8.64466i) q^{81} +(16.8619 - 6.78634i) q^{84} +(11.6808 + 14.8818i) q^{91} +(2.29117 + 1.26198i) q^{93} +(2.18324 - 9.69385i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/507\mathbb{Z}\right)^\times$.

$n$	$170$	$340$
$\chi(n)$	$-1$	$e\left(\frac{97}{156}\right)$

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		0.373361	−	0.927686i	$-0.378205\pi$
$2$		0			0		−0.373361	+	0.927686i	$0.621795\pi$
$3$	1.64291	+	0.548485i	0.948536	+	0.316668i
$3$	1.64291	+	0.548485i	0.948536	+	0.316668i
$4$	−1.44240	−	1.38545i	−0.721202	−	0.692724i
$5$		0			0		0.297503	−	0.954721i	$-0.403846\pi$
$5$		0			0		−0.297503	+	0.954721i	$0.596154\pi$
$6$		0			0
$7$	−0.527444	+	5.22047i	−0.199355	+	1.97315i	0.0290142	+	0.999579i	$0.490763\pi$
$7$	−0.527444	+	5.22047i	−0.199355	+	1.97315i	−0.228369	+	0.973575i	$0.573339\pi$
$8$		0			0
$9$	2.39833	+	1.80223i	0.799443	+	0.600742i
$10$		0			0
$11$		0			0		−0.990080	−	0.140502i	$-0.955128\pi$
$11$		0			0		0.990080	+	0.140502i	$0.0448718\pi$
$12$	−1.60985	−	3.06731i	−0.464723	−	0.885456i
$13$	2.59808	−	2.50000i	0.720577	−	0.693375i
$13$	2.59808	−	2.50000i	0.720577	−	0.693375i
$14$		0			0
$15$		0			0
$16$	0.161064	+	3.99676i	0.0402659	+	0.999189i
$17$		0			0		0.774605	−	0.632445i	$-0.217949\pi$
$17$		0			0		−0.774605	+	0.632445i	$0.782051\pi$
$18$		0			0
$19$	1.98272	+	7.39960i	0.454867	+	1.69759i	0.688479	+	0.725256i	$0.258279\pi$
$19$	1.98272	+	7.39960i	0.454867	+	1.69759i	−0.233613	+	0.972330i	$0.575055\pi$
$20$		0			0
$21$	−3.72990	+	8.28749i	−0.813930	+	1.80848i
$22$		0			0
$23$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$23$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$24$		0			0
$25$	−4.11492	−	2.84032i	−0.822984	−	0.568065i
$26$		0			0
$27$	2.95175	+	4.27635i	0.568065	+	0.822984i
$28$	7.99349	−	6.79929i	1.51063	−	1.28495i
$29$		0			0		0.919979	−	0.391967i	$-0.128205\pi$
$29$		0			0		−0.919979	+	0.391967i	$0.871795\pi$
$30$		0			0
$31$	1.48545	+	0.272220i	0.266796	+	0.0488921i	0.311987	−	0.950086i	$-0.399005\pi$
$31$	1.48545	+	0.272220i	0.266796	+	0.0488921i	−0.0451919	+	0.998978i	$0.514390\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	−0.962468	−	5.92230i	−0.160411	−	0.987050i
$37$	5.91139	−	2.10674i	0.971826	−	0.346346i	0.198322	−	0.980137i	$-0.436451\pi$
$37$	5.91139	−	2.10674i	0.971826	−	0.346346i	0.773504	+	0.633791i	$0.218502\pi$
$38$		0			0
$39$	5.63963	−	2.68228i	0.903063	−	0.429508i
$40$		0			0
$41$		0			0		0.894635	−	0.446798i	$-0.147436\pi$
$41$		0			0		−0.894635	+	0.446798i	$0.852564\pi$
$42$		0			0
$43$	11.5365	−	5.47415i	1.75930	−	0.834800i	0.781904	−	0.623399i	$-0.214249\pi$
$43$	11.5365	−	5.47415i	1.75930	−	0.834800i	0.977399	−	0.211401i	$-0.0678026\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		−0.855781	−	0.517338i	$-0.826923\pi$
$47$		0			0		0.855781	+	0.517338i	$0.173077\pi$
$48$	−1.92755	+	6.65466i	−0.278217	+	0.960518i
$49$	−20.1166	−	4.10684i	−2.87380	−	0.586691i
$50$		0			0
$51$		0			0
$52$	−7.21110	+	0.00651091i	−1.00000	+	0.000902901i
$53$		0			0		0.354605	−	0.935016i	$-0.384615\pi$
$53$		0			0		−0.354605	+	0.935016i	$0.615385\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−0.801139	+	13.2444i	−0.106113	+	1.75426i
$58$		0			0
$59$		0			0		0.735006	−	0.678061i	$-0.237179\pi$
$59$		0			0		−0.735006	+	0.678061i	$0.762821\pi$
$60$		0			0
$61$	−11.3705	−	1.84789i	−1.45585	−	0.236598i	−0.619586	−	0.784929i	$-0.712699\pi$
$61$	−11.3705	−	1.84789i	−1.45585	−	0.236598i	−0.836261	+	0.548331i	$0.815263\pi$
$62$		0			0
$63$	−10.6735	+	11.5698i	−1.34473	+	1.45766i
$64$	5.30498	−	5.98809i	0.663123	−	0.748511i
$65$		0			0
$66$		0			0
$67$	−9.94094	−	0.200222i	−1.21448	−	0.0244610i	−0.591242	−	0.806494i	$-0.701362\pi$
$67$	−9.94094	−	0.200222i	−1.21448	−	0.0244610i	−0.623237	+	0.782033i	$0.714183\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		−0.834256	−	0.551377i	$-0.814103\pi$
$71$		0			0		0.834256	+	0.551377i	$0.185897\pi$
$72$		0			0
$73$	2.09927	−	2.67951i	0.245700	−	0.313614i	−0.648557	−	0.761166i	$-0.724627\pi$
$73$	2.09927	−	2.67951i	0.245700	−	0.313614i	0.894258	+	0.447552i	$0.147704\pi$
$74$		0			0
$75$	−5.20258	−	6.92338i	−0.600742	−	0.799443i
$76$	7.39189	−	13.4202i	0.847908	−	1.53940i
$77$		0			0
$78$		0			0
$79$	−0.441406	+	0.108797i	−0.0496620	+	0.0122406i	−0.264068	−	0.964504i	$-0.585064\pi$
$79$	−0.441406	+	0.108797i	−0.0496620	+	0.0122406i	0.214406	+	0.976745i	$0.431218\pi$
$80$		0			0
$81$	2.50396	+	8.64466i	0.278217	+	0.960518i
$82$		0			0
$83$		0			0		0.998176	−	0.0603785i	$-0.0192308\pi$
$83$		0			0		−0.998176	+	0.0603785i	$0.980769\pi$
$84$	16.8619	−	6.78634i	1.83979	−	0.740450i
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$89$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$90$		0			0
$91$	11.6808	+	14.8818i	1.22449	+	1.56004i
$92$		0			0
$93$	2.29117	+	1.26198i	0.237583	+	0.130862i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	2.18324	−	9.69385i	0.221674	−	0.984261i	−0.731501	−	0.681841i	$-0.761180\pi$
$97$	2.18324	−	9.69385i	0.221674	−	0.984261i	0.953175	−	0.302420i	$-0.0977946\pi$
$98$		0			0
$99$		0			0
$100$	2.00026	+	9.79791i	0.200026	+	0.979791i
$101$		0			0		0.0804666	−	0.996757i	$-0.474359\pi$
$101$		0			0		−0.0804666	+	0.996757i	$0.525641\pi$
$102$		0			0
$103$	−13.1592	−	14.8537i	−1.29661	−	1.46358i	−0.805645	−	0.592399i	$-0.798181\pi$
$103$	−13.1592	−	14.8537i	−1.29661	−	1.46358i	−0.490970	−	0.871177i	$-0.663357\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		−0.845190	−	0.534466i	$-0.820513\pi$
$107$		0			0		0.845190	+	0.534466i	$0.179487\pi$
$108$	1.66704	−	10.2577i	0.160411	−	0.987050i
$109$	−3.73419	−	20.3768i	−0.357671	−	1.95175i	−0.302046	−	0.953293i	$-0.597670\pi$
$109$	−3.73419	−	20.3768i	−0.357671	−	1.95175i	−0.0556241	−	0.998452i	$-0.517715\pi$
$110$		0			0
$111$	10.8674	−	0.218882i	1.03149	−	0.0207754i
$112$	−20.9499	−	1.26724i	−1.97958	−	0.119743i
$113$		0			0		0.948536	−	0.316668i	$-0.102564\pi$
$113$		0			0		−0.948536	+	0.316668i	$0.897436\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	10.7366	−	1.31350i	0.992600	−	0.121433i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	10.5657	+	3.06039i	0.960518	+	0.278217i
$122$		0			0
$123$		0			0
$124$	−1.76548	−	2.45067i	−0.158545	−	0.220077i
$125$		0			0
$126$		0			0
$127$	−8.18527	+	7.86205i	−0.726325	+	0.697644i	−0.961652	−	0.274272i	$-0.911563\pi$
$127$	−8.18527	+	7.86205i	−0.726325	+	0.697644i	0.235327	+	0.971916i	$0.424384\pi$
$128$		0			0
$129$	21.9560	−	2.66594i	1.93312	−	0.234723i
$130$		0			0
$131$		0			0		0.120537	−	0.992709i	$-0.461538\pi$
$131$		0			0		−0.120537	+	0.992709i	$0.538462\pi$
$132$		0			0
$133$	−39.6752	+	6.44785i	−3.44028	+	0.559099i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		0.335705	−	0.941967i	$-0.391026\pi$
$137$		0			0		−0.335705	+	0.941967i	$0.608974\pi$
$138$		0			0
$139$	19.0229	−	12.0294i	1.61351	−	1.02032i	0.647407	−	0.762144i	$-0.275853\pi$
$139$	19.0229	−	12.0294i	1.61351	−	1.02032i	0.966098	−	0.258175i	$-0.0831210\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	−6.81678	+	9.87581i	−0.568065	+	0.822984i
$145$		0			0
$146$		0			0
$147$	−30.7973	−	17.7808i	−2.54012	−	1.46654i
$148$	−11.4454	−	5.15115i	−0.940805	−	0.423422i
$149$		0			0		−0.482459	−	0.875918i	$-0.660256\pi$
$149$		0			0		0.482459	+	0.875918i	$0.339744\pi$
$150$		0			0
$151$	4.29671	+	7.10763i	0.349662	+	0.578411i	0.980263	−	0.197696i	$-0.0633457\pi$
$151$	4.29671	+	7.10763i	0.349662	+	0.578411i	−0.630602	+	0.776107i	$0.717192\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	−11.8508	−	3.94448i	−0.948822	−	0.315811i
$157$	4.68563	−	2.45921i	0.373954	−	0.196266i	−0.267267	−	0.963623i	$-0.586121\pi$
$157$	4.68563	−	2.45921i	0.373954	−	0.196266i	0.641221	+	0.767356i	$0.278428\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	19.4443	+	16.5394i	1.52300	+	1.29547i	0.776842	+	0.629696i	$0.216821\pi$
$163$	19.4443	+	16.5394i	1.52300	+	1.29547i	0.746156	+	0.665771i	$0.231897\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		−0.927686	−	0.373361i	$-0.878205\pi$
$167$		0			0		0.927686	+	0.373361i	$0.121795\pi$
$168$		0			0
$169$	0.500000	−	12.9904i	0.0384615	−	0.999260i
$170$		0			0
$171$	−8.58056	+	21.3200i	−0.656172	+	1.63038i
$172$	−24.2245	−	8.08732i	−1.84710	−	0.616653i
$173$		0			0		−0.721202	−	0.692724i	$-0.756410\pi$
$173$		0			0		0.721202	+	0.692724i	$0.243590\pi$
$174$		0			0
$175$	16.9982	−	19.9837i	1.28495	−	1.51063i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		−0.774605	−	0.632445i	$-0.782051\pi$
$179$		0			0		0.774605	+	0.632445i	$0.217949\pi$
$180$		0			0
$181$	8.11496	+	15.4618i	0.603180	+	1.14926i	0.975260	+	0.221062i	$0.0709525\pi$
$181$	8.11496	+	15.4618i	0.603180	+	1.14926i	−0.372080	+	0.928201i	$0.621355\pi$
$182$		0			0
$183$	−17.6673	−	9.27249i	−1.30600	−	0.685442i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	−23.8815	+	13.1540i	−1.73712	+	0.956813i
$190$		0			0
$191$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$191$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$192$	12.0000	−	6.92820i	0.866025	−	0.500000i
$193$	23.5348	−	2.37781i	1.69407	−	0.171158i	0.794302	−	0.607524i	$-0.207837\pi$
$193$	23.5348	−	2.37781i	1.69407	−	0.171158i	0.899770	+	0.436365i	$0.143734\pi$
$194$		0			0
$195$		0			0
$196$	23.3265	+	33.7943i	1.66618	+	2.41388i
$197$		0			0		0.761712	−	0.647915i	$-0.224359\pi$
$197$		0			0		−0.761712	+	0.647915i	$0.775641\pi$
$198$		0			0
$199$	−10.8515	−	17.1602i	−0.769239	−	1.21645i	−0.971481	−	0.237119i	$-0.923797\pi$
$199$	−10.8515	−	17.1602i	−0.769239	−	1.21645i	0.202241	−	0.979336i	$-0.435177\pi$
$200$		0			0
$201$	−16.2223	−	5.78140i	−1.14423	−	0.407789i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	10.4103	+	9.98122i	0.721828	+	0.692073i
$209$		0			0
$210$		0			0
$211$	2.81244	+	2.92806i	0.193616	+	0.201576i	0.810933	−	0.585139i	$-0.198960\pi$
$211$	2.81244	+	2.92806i	0.193616	+	0.201576i	−0.617317	+	0.786714i	$0.711780\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−2.20461	+	7.61120i	−0.149659	+	0.516682i
$218$		0			0
$219$	4.91859	−	3.25079i	0.332367	−	0.219668i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−26.4841	+	5.96471i	−1.77351	+	0.399427i	−0.978346	−	0.206974i	$-0.933638\pi$
$223$	−26.4841	+	5.96471i	−1.77351	+	0.399427i	−0.795159	+	0.606401i	$0.792613\pi$
$224$		0			0
$225$	−4.75002	−	14.2280i	−0.316668	−	0.948536i
$226$		0			0
$227$		0			0		−0.0201371	−	0.999797i	$-0.506410\pi$
$227$		0			0		0.0201371	+	0.999797i	$0.493590\pi$
$228$	19.5050	−	17.9939i	1.29175	−	1.19167i
$229$	14.5776	−	2.67145i	0.963316	−	0.176534i	0.324493	−	0.945888i	$-0.394806\pi$
$229$	14.5776	−	2.67145i	0.963316	−	0.176534i	0.638823	+	0.769354i	$0.279422\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		0.663123	−	0.748511i	$-0.269231\pi$
$233$		0			0		−0.663123	+	0.748511i	$0.730769\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−0.784865	−	0.0633608i	−0.0509824	−	0.00411573i
$238$		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
$241$	30.2023	+	6.80213i	1.94550	+	0.438164i	0.989423	+	0.145056i	$0.0463362\pi$
$241$	30.2023	+	6.80213i	1.94550	+	0.438164i	0.956079	+	0.293108i	$0.0946894\pi$
$242$		0			0
$243$	−0.627684	+	15.5758i	−0.0402659	+	0.999189i
$244$	13.8407	+	18.4187i	0.886063	+	1.17914i
$245$		0			0
$246$		0			0
$247$	23.6503	+	14.2679i	1.50483	+	0.907847i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		−0.0804666	−	0.996757i	$-0.525641\pi$
$251$		0			0		0.0804666	+	0.996757i	$0.474359\pi$
$252$	31.4249	−	1.90085i	1.97958	−	0.119743i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−15.9481	+	1.28747i	−0.996757	+	0.0804666i
$257$		0			0		0.960518	−	0.278217i	$-0.0897436\pi$
$257$		0			0		−0.960518	+	0.278217i	$0.910256\pi$
$258$		0			0
$259$	7.88025	+	31.9714i	0.489655	+	1.98661i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		0.799443	−	0.600742i	$-0.205128\pi$
$263$		0			0		−0.799443	+	0.600742i	$0.794872\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	14.0615	+	14.0615i	0.858940	+	0.858940i
$269$		0			0		−0.200026	−	0.979791i	$-0.564103\pi$
$269$		0			0		0.200026	+	0.979791i	$0.435897\pi$
$270$		0			0
$271$	0.570747	−	28.3374i	0.0346704	−	1.72137i	−0.491403	−	0.870933i	$-0.663516\pi$
$271$	0.570747	−	28.3374i	0.0346704	−	1.72137i	0.526073	−	0.850439i	$-0.323664\pi$
$272$		0			0
$273$	11.0282	+	30.8563i	0.667455	+	1.86751i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−5.26128	+	32.3740i	−0.316120	+	1.94516i	0.0103260	+	0.999947i	$0.496713\pi$
$277$	−5.26128	+	32.3740i	−0.316120	+	1.94516i	−0.326446	+	0.945216i	$0.605851\pi$
$278$		0			0
$279$	3.07201	+	3.33000i	0.183916	+	0.199362i
$280$		0			0
$281$		0			0		−0.998176	−	0.0603785i	$-0.980769\pi$
$281$		0			0		0.998176	+	0.0603785i	$0.0192308\pi$
$282$		0			0
$283$	−10.5454	−	5.00387i	−0.626860	−	0.297449i	0.0886374	−	0.996064i	$-0.471749\pi$
$283$	−10.5454	−	5.00387i	−0.626860	−	0.297449i	−0.715498	+	0.698615i	$0.753800\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	3.40044	−	16.6564i	0.200026	−	0.979791i
$290$		0			0
$291$	8.90380	−	14.7287i	0.521950	−	0.863411i
$292$	−6.74032	+	0.956520i	−0.394448	+	0.0559761i
$293$		0			0		−0.584522	−	0.811378i	$-0.698718\pi$
$293$		0			0		0.584522	+	0.811378i	$0.301282\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$	−2.08776	+	17.1942i	−0.120537	+	0.992709i
$301$	22.4928	+	63.1134i	1.29646	+	3.63780i
$302$		0			0
$303$		0			0
$304$	−29.2551	+	9.11625i	−1.67789	+	0.522853i
$305$		0			0
$306$		0			0
$307$	−2.30456	+	12.5756i	−0.131528	+	0.717726i	0.850056	+	0.526693i	$0.176568\pi$
$307$	−2.30456	+	12.5756i	−0.131528	+	0.717726i	−0.981584	+	0.191033i	$0.938816\pi$
$308$		0			0
$309$	−13.4724	−	31.6209i	−0.766419	−	1.79885i
$310$		0			0
$311$		0			0		0.822984	−	0.568065i	$-0.192308\pi$
$311$		0			0		−0.822984	+	0.568065i	$0.807692\pi$
$312$		0			0
$313$	−18.8900	+	27.3669i	−1.06773	+	1.54687i	−0.252616	+	0.967567i	$0.581291\pi$
$313$	−18.8900	+	27.3669i	−1.06773	+	1.54687i	−0.815111	+	0.579304i	$0.803324\pi$
$314$		0			0
$315$		0			0
$316$	0.787418	+	0.454616i	0.0442957	+	0.0255741i
$317$		0			0		−0.911900	−	0.410413i	$-0.865385\pi$
$317$		0			0		0.911900	+	0.410413i	$0.134615\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	8.36502	−	15.9382i	0.464723	−	0.885456i
$325$	−17.7917	+	2.90792i	−0.986905	+	0.161302i
$326$		0			0
$327$	5.04143	−	35.5255i	0.278792	−	1.96456i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	24.2052	+	2.44554i	1.33044	+	0.134419i	0.739968	−	0.672642i	$-0.234841\pi$
$331$	24.2052	+	2.44554i	1.33044	+	0.134419i	0.590468	+	0.807061i	$0.298943\pi$
$332$		0			0
$333$	17.9743	+	5.60101i	0.984984	+	0.306933i
$334$		0			0
$335$		0			0
$336$	−33.7238	−	13.5727i	−1.83979	−	0.740450i
$337$		−	31.3370i		−	1.70703i	−0.521065	−	0.853517i	$-0.674465\pi$
$337$		−	31.3370i		−	1.70703i	0.521065	−	0.853517i	$-0.325535\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	21.1229	−	67.7858i	1.14053	−	3.66009i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		−0.799443	−	0.600742i	$-0.794872\pi$
$347$		0			0		0.799443	+	0.600742i	$0.205128\pi$
$348$		0			0
$349$	−36.7163	−	5.21041i	−1.96538	−	0.278907i	−0.999825	−	0.0186925i	$-0.994050\pi$
$349$	−36.7163	−	5.21041i	−1.96538	−	0.278907i	−0.965551	−	0.260214i	$-0.916207\pi$
$350$		0			0
$351$	18.3597	+	3.73090i	0.979971	+	0.199141i
$352$		0			0
$353$		0			0		0.584522	−	0.811378i	$-0.301282\pi$
$353$		0			0		−0.584522	+	0.811378i	$0.698718\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		0.410413	−	0.911900i	$-0.365385\pi$
$359$		0			0		−0.410413	+	0.911900i	$0.634615\pi$
$360$		0			0
$361$	−34.3685	+	19.8427i	−1.80887	+	1.04435i
$362$		0			0
$363$	15.6799	+	10.8231i	0.822984	+	0.568065i
$364$	3.76946	−	37.6488i	0.197573	−	1.97333i
$365$		0			0
$366$		0			0
$367$	35.0835	−	14.9477i	1.83135	−	0.780264i	0.877792	−	0.479042i	$-0.159016\pi$
$367$	35.0835	−	14.9477i	1.83135	−	0.780264i	0.953554	−	0.301222i	$-0.0973945\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	−1.55637	−	4.99458i	−0.0806943	−	0.258957i
$373$	−29.6280	−	12.6233i	−1.53408	−	0.653609i	−0.550622	−	0.834755i	$-0.685609\pi$
$373$	−29.6280	−	12.6233i	−1.53408	−	0.653609i	−0.983456	+	0.181146i	$0.942019\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−19.3675	+	9.67250i	−0.994840	+	0.496843i	−0.868790	−	0.495181i	$-0.835102\pi$
$379$	−19.3675	+	9.67250i	−0.994840	+	0.496843i	−0.126050	+	0.992024i	$0.540230\pi$
$380$		0			0
$381$	−17.7599	+	8.42718i	−0.909867	+	0.431737i
$382$		0			0
$383$		0			0		0.811378	−	0.584522i	$-0.198718\pi$
$383$		0			0		−0.811378	+	0.584522i	$0.801282\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	37.5340	+	7.66263i	1.90796	+	0.389513i
$388$	−16.5794	+	10.9577i	−0.841693	+	0.556293i
$389$		0			0		0.935016	−	0.354605i	$-0.115385\pi$
$389$		0			0		−0.935016	+	0.354605i	$0.884615\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−24.6775	+	22.7656i	−1.23853	+	1.14257i	−0.254026	+	0.967197i	$0.581755\pi$
$397$	−24.6775	+	22.7656i	−1.23853	+	1.14257i	−0.984502	+	0.175376i	$0.943886\pi$
$398$		0			0
$399$	−68.7195	−	11.1680i	−3.44028	−	0.559099i
$400$	10.6893	−	16.9038i	0.534466	−	0.845190i
$401$		0			0		0.678061	−	0.735006i	$-0.262821\pi$
$401$		0			0		−0.678061	+	0.735006i	$0.737179\pi$
$402$		0			0
$403$	4.53987	−	3.00639i	0.226147	−	0.149759i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−6.99095	−	4.62046i	−0.345680	−	0.228467i	0.366574	−	0.930389i	$-0.380531\pi$
$409$	−6.99095	−	4.62046i	−0.345680	−	0.228467i	−0.712254	+	0.701922i	$0.752326\pi$
$410$		0			0
$411$		0			0
$412$	−1.59810	+	39.6564i	−0.0787327	+	1.95373i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	37.8510	−	9.32943i	1.85357	−	0.456864i
$418$		0			0
$419$		0			0		−0.278217	−	0.960518i	$-0.589744\pi$
$419$		0			0		0.278217	+	0.960518i	$0.410256\pi$
$420$		0			0
$421$	29.0205	−	1.75542i	1.41437	−	0.0855539i	0.664576	−	0.747221i	$-0.268612\pi$
$421$	29.0205	−	1.75542i	1.41437	−	0.0855539i	0.749798	+	0.661667i	$0.230151\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	15.6442	−	58.3849i	0.757075	−	2.82544i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		−0.875918	−	0.482459i	$-0.839744\pi$
$431$		0			0		0.875918	+	0.482459i	$0.160256\pi$
$432$	−16.6161	+	12.4862i	−0.799443	+	0.600742i
$433$	−26.5497	−	1.06992i	−1.27590	−	0.0514169i	−0.606958	−	0.794734i	$-0.707611\pi$
$433$	−26.5497	−	1.06992i	−1.27590	−	0.0514169i	−0.668940	+	0.743317i	$0.733252\pi$
$434$		0			0
$435$		0			0
$436$	−22.8448	+	34.5651i	−1.09407	+	1.65537i
$437$		0			0
$438$		0			0
$439$	0.652051	−	8.07710i	0.0311207	−	0.385499i	−0.962383	−	0.271697i	$-0.912415\pi$
$439$	0.652051	−	8.07710i	0.0311207	−	0.385499i	0.993503	−	0.113802i	$-0.0363029\pi$
$440$		0			0
$441$	−40.8448	−	46.1042i	−1.94499	−	2.19544i
$442$		0			0
$443$		0			0		−0.748511	−	0.663123i	$-0.769231\pi$
$443$		0			0		0.748511	+	0.663123i	$0.230769\pi$
$444$	−15.9785	−	14.7405i	−0.758304	−	0.699554i
$445$		0			0
$446$		0			0
$447$		0			0
$448$	28.4626	+	30.8529i	1.34473	+	1.45766i
$449$		0			0		0.999797	−	0.0201371i	$-0.00641026\pi$
$449$		0			0		−0.999797	+	0.0201371i	$0.993590\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	3.16070	+	14.0339i	0.148503	+	0.659371i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	19.5940	+	29.6466i	0.916570	+	1.38681i	0.922154	+	0.386823i	$0.126428\pi$
$457$	19.5940	+	29.6466i	0.916570	+	1.38681i	−0.00558406	+	0.999984i	$0.501777\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		0.990080	−	0.140502i	$-0.0448718\pi$
$461$		0			0		−0.990080	+	0.140502i	$0.955128\pi$
$462$		0			0
$463$	−27.4861	+	21.5340i	−1.27739	+	1.00077i	−0.278200	+	0.960523i	$0.589738\pi$
$463$	−27.4861	+	21.5340i	−1.27739	+	1.00077i	−0.999190	+	0.0402476i	$0.987185\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		0.992709	−	0.120537i	$-0.0384615\pi$
$467$		0			0		−0.992709	+	0.120537i	$0.961538\pi$
$468$	−17.3063	−	12.9804i	−0.799985	−	0.600020i
$469$	6.28854	−	51.7908i	0.290378	−	2.39148i
$470$		0			0
$471$	9.04692	−	1.47027i	0.416860	−	0.0677464i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	12.8585	−	36.0803i	0.589991	−	1.65548i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.647915	−	0.761712i	$-0.724359\pi$
$479$		0			0		0.647915	+	0.761712i	$0.275641\pi$
$480$		0			0
$481$	10.0914	−	20.2519i	0.460128	−	0.923409i
$482$		0			0
$483$		0			0
$484$	−11.0000	−	19.0526i	−0.500000	−	0.866025i
$485$		0			0
$486$		0			0
$487$	12.4124	+	22.5351i	0.562460	+	1.02116i	0.992946	+	0.118565i	$0.0378293\pi$
$487$	12.4124	+	22.5351i	0.562460	+	1.02116i	−0.430486	+	0.902597i	$0.641658\pi$
$488$		0			0
$489$	22.8737	+	37.8378i	1.03439	+	1.71108i
$490$		0			0
$491$		0			0		0.999189	−	0.0402659i	$-0.0128205\pi$
$491$		0			0		−0.999189	+	0.0402659i	$0.987179\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	−0.848743	+	5.98085i	−0.0381097	+	0.268548i
$497$		0			0
$498$		0			0
$499$	16.1232	+	35.8243i	0.721774	+	1.60372i	0.795151	+	0.606412i	$0.207392\pi$
$499$	16.1232	+	35.8243i	0.721774	+	1.60372i	−0.0733768	+	0.997304i	$0.523378\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		0.692724	−	0.721202i	$-0.256410\pi$
$503$		0			0		−0.692724	+	0.721202i	$0.743590\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	7.94649	−	21.0678i	0.352916	−	0.935655i
$508$	22.6989			1.00710
$509$		0			0		0.373361	−	0.927686i	$-0.378205\pi$
$509$		0			0		−0.373361	+	0.927686i	$0.621795\pi$
$510$		0			0
$511$	12.8811	+	12.3725i	0.569826	+	0.547325i
$512$		0			0
$513$	−25.7908	+	30.3206i	−1.13869	+	1.33869i
$514$		0			0
$515$		0			0
$516$	−35.3630	−	26.5735i	−1.55677	−	1.16984i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		−0.885456	−	0.464723i	$-0.846154\pi$
$521$		0			0		0.885456	+	0.464723i	$0.153846\pi$
$522$		0			0
$523$	−1.76126	−	43.7052i	−0.0770145	−	1.91110i	−0.330581	−	0.943778i	$-0.607245\pi$
$523$	−1.76126	−	43.7052i	−0.0770145	−	1.91110i	0.253567	−	0.967318i	$-0.418396\pi$
$524$		0			0
$525$	38.8874	−	23.5082i	1.69718	−	1.02598i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	11.5000	−	19.9186i	0.500000	−	0.866025i
$530$		0			0
$531$		0			0
$532$	66.1609	+	45.6676i	2.86844	+	1.97994i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−2.65262	−	8.51255i	−0.114045	−	0.365983i	0.879997	−	0.474979i	$-0.157544\pi$
$541$	−2.65262	−	8.51255i	−0.114045	−	0.365983i	−0.994042	+	0.108996i	$0.965237\pi$
$542$		0			0
$543$	4.85163	+	29.8533i	0.208203	+	1.28113i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−5.44792	−	44.8677i	−0.232936	−	1.91840i	−0.372209	−	0.928149i	$-0.621400\pi$
$547$	−5.44792	−	44.8677i	−0.232936	−	1.91840i	0.139273	−	0.990254i	$-0.455523\pi$
$548$		0			0
$549$	−23.9399	−	24.9241i	−1.02173	−	1.06374i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−0.335154	−	2.36173i	−0.0142522	−	0.100431i
$554$		0			0
$555$		0			0
$556$	−44.1049	−	9.00408i	−1.87046	−	0.381858i
$557$		0			0		0.834256	−	0.551377i	$-0.185897\pi$
$557$		0			0		−0.834256	+	0.551377i	$0.814103\pi$
$558$		0			0
$559$	16.2874	−	43.0636i	0.688884	−	1.82139i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		−0.316668	−	0.948536i	$-0.602564\pi$
$563$		0			0		0.316668	+	0.948536i	$0.397436\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−46.4499	+	8.51227i	−1.95071	+	0.357482i
$568$		0			0
$569$		0			0		0.534466	−	0.845190i	$-0.320513\pi$
$569$		0			0		−0.534466	+	0.845190i	$0.679487\pi$
$570$		0			0
$571$	11.8609	−	13.3882i	0.496363	−	0.560279i	−0.445929	−	0.895068i	$-0.647127\pi$
$571$	11.8609	−	13.3882i	0.496363	−	0.560279i	0.942293	+	0.334790i	$0.108665\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	23.5150	−	4.80062i	0.979791	−	0.200026i
$577$	−30.2280	+	30.2280i	−1.25841	+	1.25841i	−0.306554	+	0.951853i	$0.599176\pi$
$577$	−30.2280	+	30.2280i	−1.25841	+	1.25841i	−0.951853	+	0.306554i	$0.900824\pi$
$578$		0			0
$579$	39.9698	+	9.00195i	1.66109	+	0.374108i
$580$		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	$-0.916667\pi$
$587$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$588$	19.7878	+	68.3153i	0.816033	+	2.81727i
$589$	0.930920	+	11.5315i	0.0383579	+	0.475148i
$590$		0			0
$591$		0			0
$592$	9.37223	+	23.2871i	0.385196	+	0.957092i
$593$		0			0		−0.0603785	−	0.998176i	$-0.519231\pi$
$593$		0			0		0.0603785	+	0.998176i	$0.480769\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−8.41588	−	34.1446i	−0.344439	−	1.39744i
$598$		0			0
$599$		0			0		−0.970942	−	0.239316i	$-0.923077\pi$
$599$		0			0		0.970942	+	0.239316i	$0.0769231\pi$
$600$		0			0
$601$	−7.65049	+	5.74897i	−0.312070	+	0.234505i	−0.745223	−	0.666815i	$-0.767657\pi$
$601$	−7.65049	+	5.74897i	−0.312070	+	0.234505i	0.433154	+	0.901320i	$0.357401\pi$
$602$		0			0
$603$	−23.4808	−	18.3960i	−0.956211	−	0.749144i
$604$	3.64966	−	16.2050i	0.148503	−	0.659371i
$605$		0			0
$606$		0			0
$607$	7.00386	+	34.3072i	0.284278	+	1.39248i	0.831513	+	0.555506i	$0.187475\pi$
$607$	7.00386	+	34.3072i	0.284278	+	1.39248i	−0.547235	+	0.836979i	$0.684320\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−34.7265	−	32.0360i	−1.40259	−	1.29392i	−0.904290	−	0.426919i	$-0.859599\pi$
$613$	−34.7265	−	32.0360i	−1.40259	−	1.29392i	−0.498300	−	0.867005i	$-0.666042\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		−0.678061	−	0.735006i	$-0.737179\pi$
$617$		0			0		0.678061	+	0.735006i	$0.262821\pi$
$618$		0			0
$619$	−34.6321	−	2.09486i	−1.39198	−	0.0841994i	−0.652711	−	0.757607i	$-0.726368\pi$
$619$	−34.6321	−	2.09486i	−1.39198	−	0.0841994i	−0.739271	+	0.673408i	$0.764830\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$	11.6287	+	22.1082i	0.465522	+	0.885036i
$625$	8.86512	+	23.3754i	0.354605	+	0.935016i
$626$		0			0
$627$		0			0
$628$	−10.1657	−	2.94453i	−0.405655	−	0.117499i
$629$		0			0
$630$		0			0
$631$	−26.5337	−	36.8316i	−1.05629	−	1.46624i	−0.875806	−	0.482663i	$-0.839670\pi$
$631$	−26.5337	−	36.8316i	−1.05629	−	1.46624i	−0.180484	−	0.983578i	$-0.557766\pi$
$632$		0			0
$633$	3.01459	+	6.35312i	0.119819	+	0.252514i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−62.5316	+	39.6217i	−2.47759	+	1.56987i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		0.391967	−	0.919979i	$-0.371795\pi$
$641$		0			0		−0.391967	+	0.919979i	$0.628205\pi$
$642$		0			0
$643$	−0.855787	+	1.71356i	−0.0337490	+	0.0675763i	−0.910871	−	0.412692i	$-0.864589\pi$
$643$	−0.855787	+	1.71356i	−0.0337490	+	0.0675763i	0.877122	+	0.480268i	$0.159461\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.391967	−	0.919979i	$-0.628205\pi$
$647$		0			0		0.391967	+	0.919979i	$0.371795\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	−7.79661	+	11.2953i	−0.305573	+	0.442699i
$652$	−5.13208	−	50.7957i	−0.200988	−	1.98931i
$653$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$653$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	9.86382	−	2.64300i	0.384824	−	0.103113i
$658$		0			0
$659$		0			0		−0.632445	−	0.774605i	$-0.717949\pi$
$659$		0			0		0.632445	+	0.774605i	$0.282051\pi$
$660$		0			0
$661$	−39.0968	−	28.1656i	−1.52069	−	1.09551i	−0.964374	−	0.264543i	$-0.914779\pi$
$661$	−39.0968	−	28.1656i	−1.52069	−	1.09551i	−0.556315	−	0.830971i	$-0.687785\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	−46.7826	−	4.72662i	−1.80872	−	0.182742i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−6.00393	+	17.9840i	−0.231435	+	0.693231i	0.767220	+	0.641384i	$0.221639\pi$
$673$	−6.00393	+	17.9840i	−0.231435	+	0.693231i	−0.998655	+	0.0518477i	$0.983489\pi$
$674$		0			0
$675$		−	25.9808i		−	1.00000i
$676$	−18.7187	+	18.0447i	−0.719950	+	0.694026i
$677$		0			0		1.00000			$0$
$677$		0			0		−1.00000			$\pi$
$678$		0			0
$679$	49.4549	+	16.5105i	1.89791	+	0.633614i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		0.100522	−	0.994935i	$-0.467949\pi$
$683$		0			0		−0.100522	+	0.994935i	$0.532051\pi$
$684$	41.9144	−	18.8641i	1.60264	−	0.721288i
$685$		0			0
$686$		0			0
$687$	25.4150	+	3.60665i	0.969643	+	0.137602i
$688$	23.7370	+	45.2270i	0.904963	+	1.72426i
$689$		0			0
$690$		0			0
$691$	−30.4607	+	42.2826i	−1.15878	+	1.60851i	−0.472098	+	0.881546i	$0.656503\pi$
$691$	−30.4607	+	42.2826i	−1.15878	+	1.60851i	−0.686681	+	0.726959i	$0.740933\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	−52.2047	+	5.27444i	−1.97315	+	0.199355i
$701$		0			0		−0.822984	−	0.568065i	$-0.807692\pi$
$701$		0			0		0.822984	+	0.568065i	$0.192308\pi$
$702$		0			0
$703$	27.3096	+	39.5649i	1.03000	+	1.49222i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−3.97439	−	1.98489i	−0.149262	−	0.0745442i	0.370610	−	0.928788i	$-0.379149\pi$
$709$	−3.97439	−	1.98489i	−0.149262	−	0.0745442i	−0.519872	+	0.854244i	$0.674020\pi$
$710$		0			0
$711$	−1.25471	−	0.534583i	−0.0470554	−	0.0200484i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		0.903450	−	0.428693i	$-0.141026\pi$
$719$		0			0		−0.903450	+	0.428693i	$0.858974\pi$
$720$		0			0
$721$	84.4839	−	60.8628i	3.14635	−	2.26665i
$722$		0			0
$723$	45.8889	+	27.7408i	1.70663	+	1.03169i
$724$	9.71642	−	33.5450i	0.361108	−	1.24669i
$725$		0			0
$726$		0			0
$727$	−3.43881	+	1.30417i	−0.127538	+	0.0483689i	−0.417548	−	0.908655i	$-0.637111\pi$
$727$	−3.43881	+	1.30417i	−0.127538	+	0.0483689i	0.290010	+	0.957024i	$0.406341\pi$
$728$		0			0
$729$	−9.57433	+	25.2454i	−0.354605	+	0.935016i
$730$		0			0
$731$		0			0
$732$	12.6368	+	37.8518i	0.467069	+	1.39904i
$733$	−1.71595	+	28.3680i	−0.0633799	+	1.04780i	0.817949	+	0.575291i	$0.195111\pi$
$733$	−1.71595	+	28.3680i	−0.0633799	+	1.04780i	−0.881329	+	0.472504i	$0.843350\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	24.5392	−	26.6001i	0.902690	−	0.978499i	−0.0971693	−	0.995268i	$-0.530979\pi$
$739$	24.5392	−	26.6001i	0.902690	−	0.978499i	0.999859	+	0.0167685i	$0.00533782\pi$
$740$		0			0
$741$	31.0296	+	36.4128i	1.13990	+	1.33766i
$742$		0			0
$743$		0			0		−0.999797	−	0.0201371i	$-0.993590\pi$
$743$		0			0		0.999797	+	0.0201371i	$0.00641026\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−23.7306	−	31.5797i	−0.865942	−	1.15236i	−0.987160	−	0.159734i	$-0.948936\pi$
$751$	−23.7306	−	31.5797i	−0.865942	−	1.15236i	0.121218	−	0.992626i	$-0.461320\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$	52.6709	+	14.1131i	1.91562	+	0.513290i
$757$	−11.5482	−	39.8690i	−0.419727	−	1.44906i	−0.840457	−	0.541879i	$-0.817713\pi$
$757$	−11.5482	−	39.8690i	−0.419727	−	1.44906i	0.420730	−	0.907186i	$-0.361774\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		−0.373361	−	0.927686i	$-0.621795\pi$
$761$		0			0		0.373361	+	0.927686i	$0.378205\pi$
$762$		0			0
$763$	108.346	−	8.74661i	3.92240	−	0.316649i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	−26.9075	−	6.63211i	−0.970942	−	0.239316i
$769$	47.4373	+	26.1287i	1.71063	+	0.942224i	0.959339	+	0.282257i	$0.0910831\pi$
$769$	47.4373	+	26.1287i	1.71063	+	0.942224i	0.751295	+	0.659967i	$0.229430\pi$
$770$		0			0
$771$		0			0
$772$	−37.2410	−	29.1765i	−1.34033	−	1.05008i
$773$		0			0		0.219715	−	0.975564i	$-0.429487\pi$
$773$		0			0		−0.219715	+	0.975564i	$0.570513\pi$
$774$		0			0
$775$	−5.33934	−	5.33934i	−0.191795	−	0.191795i
$776$		0			0
$777$	−4.58928	+	56.8485i	−0.164640	+	2.03943i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	13.1740	−	81.0627i	0.470499	−	2.89509i
$785$		0			0
$786$		0			0
$787$	−8.92279	+	0.179715i	−0.318063	+	0.00640615i	−0.178912	−	0.983865i	$-0.557258\pi$
$787$	−8.92279	+	0.179715i	−0.318063	+	0.00640615i	−0.139151	+	0.990271i	$0.544437\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−34.1612	+	23.6254i	−1.21310	+	0.838961i
$794$		0			0
$795$		0			0
$796$	−8.12239	+	39.7861i	−0.287890	+	1.41018i
$797$		0			0		−0.960518	−	0.278217i	$-0.910256\pi$
$797$		0			0		0.960518	+	0.278217i	$0.0897436\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$	15.3893	+	30.8143i	0.542737	+	1.08674i
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		0.987050	−	0.160411i	$-0.0512821\pi$
$809$		0			0		−0.987050	+	0.160411i	$0.948718\pi$
$810$		0			0
$811$	−17.3352	+	5.40187i	−0.608721	+	0.189685i	−0.586341	−	0.810064i	$-0.699432\pi$
$811$	−17.3352	+	5.40187i	−0.608721	+	0.189685i	−0.0223803	+	0.999750i	$0.507124\pi$
$812$		0			0
$813$	16.4803	−	46.2428i	0.577990	−	1.62181i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	63.3802	+	74.5120i	2.21739	+	2.60685i
$818$		0			0
$819$	1.19412	+	56.7430i	0.0417260	+	1.98276i
$820$		0			0
$821$		0			0		−0.100522	−	0.994935i	$-0.532051\pi$
$821$		0			0		0.100522	+	0.994935i	$0.467949\pi$
$822$		0			0
$823$	42.1737	+	24.3490i	1.47008	+	0.848753i	0.999436	−	0.0335690i	$-0.0106873\pi$
$823$	42.1737	+	24.3490i	1.47008	+	0.848753i	0.470647	+	0.882322i	$0.344021\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		−0.517338	−	0.855781i	$-0.673077\pi$
$827$		0			0		0.517338	+	0.855781i	$0.326923\pi$
$828$		0			0
$829$	57.3466	−	2.31099i	1.99173	−	0.0802640i	0.992584	−	0.121560i	$-0.0387897\pi$
$829$	57.3466	−	2.31099i	1.99173	−	0.0802640i	0.999147	+	0.0412960i	$0.0131487\pi$
$830$		0			0
$831$	−26.4005	+	50.3019i	−0.915822	+	1.74495i
$832$	−1.18747	−	28.8200i	−0.0411681	−	0.999152i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	3.22059	+	7.15585i	0.111320	+	0.247342i
$838$		0			0
$839$		0			0		−0.761712	−	0.647915i	$-0.775641\pi$
$839$		0			0		0.761712	+	0.647915i	$0.224359\pi$
$840$		0			0
$841$	20.0890	−	20.9149i	0.692724	−	0.721202i
$842$		0			0
$843$		0			0
$844$		−	8.11993i		−	0.279500i
$845$		0			0
$846$		0			0
$847$	−21.5495	+	53.5438i	−0.740450	+	1.83979i
$848$		0			0
$849$	−14.5807	−	14.0049i	−0.500407	−	0.480648i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−43.1585	+	19.4240i	−1.47772	+	0.665067i	−0.979400	−	0.201928i	$-0.935279\pi$
$853$	−43.1585	+	19.4240i	−1.47772	+	0.665067i	−0.498317	+	0.866995i	$0.666048\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		−0.464723	−	0.885456i	$-0.653846\pi$
$857$		0			0		0.464723	+	0.885456i	$0.346154\pi$
$858$		0			0
$859$	30.2494	+	15.8761i	1.03210	+	0.541686i	0.893688	−	0.448689i	$-0.148109\pi$
$859$	30.2494	+	15.8761i	1.03210	+	0.541686i	0.138409	+	0.990375i	$0.455801\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.855781	−	0.517338i	$-0.173077\pi$
$863$		0			0		−0.855781	+	0.517338i	$0.826923\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	14.7224	−	25.5000i	0.500000	−	0.866025i
$868$	13.7249	−	7.92405i	0.465852	−	0.268960i
$869$		0			0
$870$		0			0
$871$	−26.3279	+	24.3321i	−0.892086	+	0.824463i
$872$		0			0
$873$	22.7066	−	19.3143i	0.768503	−	0.653692i
$874$		0			0
$875$		0			0
$876$	−11.5984	−	2.12549i	−0.391874	−	0.0718135i
$877$	−54.8026	−	19.5309i	−1.85055	−	0.659511i	−0.988193	−	0.153211i	$-0.951038\pi$
$877$	−54.8026	−	19.5309i	−1.85055	−	0.659511i	−0.862357	−	0.506300i	$-0.831013\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		−0.160411	−	0.987050i	$-0.551282\pi$
$881$		0			0		0.160411	+	0.987050i	$0.448718\pi$
$882$		0			0
$883$	−40.2610	−	4.88857i	−1.35489	−	0.164514i	−0.589338	−	0.807887i	$-0.700611\pi$
$883$	−40.2610	−	4.88857i	−1.35489	−	0.164514i	−0.765553	+	0.643373i	$0.777534\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		−0.692724	−	0.721202i	$-0.743590\pi$
$887$		0			0		0.692724	+	0.721202i	$0.256410\pi$
$888$		0			0
$889$	−36.7264	−	46.8778i	−1.23176	−	1.57223i
$890$		0			0
$891$		0			0
$892$	46.4646	+	28.0888i	1.55575	+	0.940483i
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	−12.8608	+	27.1035i	−0.428693	+	0.903450i
$901$		0			0
$902$		0			0
$903$	2.33691	+	116.027i	0.0777676	+	3.86113i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	31.9527	−	50.5291i	1.06097	−	1.67779i	0.427525	−	0.904004i	$-0.359386\pi$
$907$	31.9527	−	50.5291i	1.06097	−	1.67779i	0.633446	−	0.773787i	$-0.281640\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		0.748511	−	0.663123i	$-0.230769\pi$
$911$		0			0		−0.748511	+	0.663123i	$0.769231\pi$
$912$	−53.0637	−	1.06876i	−1.75711	−	0.0353903i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	−24.7280	−	16.3432i	−0.817035	−	0.539996i
$917$		0			0
$918$		0			0
$919$	−2.43632	+	60.4566i	−0.0803667	+	1.99428i	0.0237804	+	0.999717i	$0.492430\pi$
$919$	−2.43632	+	60.4566i	−0.0803667	+	1.99428i	−0.104147	+	0.994562i	$0.533211\pi$
$920$		0			0
$921$	−10.6837	+	19.3966i	−0.352040	+	0.639138i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−30.3087	−	8.12119i	−0.996544	−	0.267023i
$926$		0			0
$927$	−4.79041	−	59.3398i	−0.157338	−	1.94898i
$928$		0			0
$929$		0			0		0.927686	−	0.373361i	$-0.121795\pi$
$929$		0			0		−0.927686	+	0.373361i	$0.878205\pi$
$930$		0			0
$931$	−9.49661	−	156.998i	−0.311239	−	5.14539i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−31.5471	−	7.77567i	−1.03060	−	0.254020i	−0.312469	−	0.949928i	$-0.601156\pi$
$937$	−31.5471	−	7.77567i	−1.03060	−	0.254020i	−0.718131	+	0.695908i	$0.755002\pi$
$938$		0			0
$939$	−46.0450	+	34.6006i	−1.50262	+	1.12915i
$940$		0			0
$941$		0			0		−0.787183	−	0.616719i	$-0.788462\pi$
$941$		0			0		0.787183	+	0.616719i	$0.211538\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.0201371	−	0.999797i	$-0.493590\pi$
$947$		0			0		−0.0201371	+	0.999797i	$0.506410\pi$
$948$	1.04431	+	1.17878i	0.0339176	+	0.0382850i
$949$	−1.24473	−	12.2097i	−0.0404058	−	0.396345i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		0.160411	−	0.987050i	$-0.448718\pi$
$953$		0			0		−0.160411	+	0.987050i	$0.551282\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−26.8530	−	10.1840i	−0.866227	−	0.328516i
$962$		0			0
$963$		0			0
$964$	−34.1400	−	51.6552i	−1.09957	−	1.66370i
$965$		0			0
$966$		0			0
$967$	−24.6157	+	40.7194i	−0.791588	+	1.30945i	0.155295	+	0.987868i	$0.450367\pi$
$967$	−24.6157	+	40.7194i	−0.791588	+	1.30945i	−0.946883	+	0.321578i	$0.895787\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		−0.428693	−	0.903450i	$-0.641026\pi$
$971$		0			0		0.428693	+	0.903450i	$0.358974\pi$
$972$	22.4849	−	21.5970i	0.721202	−	0.692724i
$973$	52.7655	+	105.654i	1.69159	+	3.38710i
$974$		0			0
$975$	−30.8251	−	4.98101i	−0.987195	−	0.159520i
$976$	5.55419	−	45.7429i	0.177785	−	1.46419i
$977$		0			0		−0.335705	−	0.941967i	$-0.608974\pi$
$977$		0			0		0.335705	+	0.941967i	$0.391026\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	27.7678	−	55.6002i	0.886559	−	1.77518i
$982$		0			0
$983$		0			0		0.180255	−	0.983620i	$-0.442308\pi$
$983$		0			0		−0.180255	+	0.983620i	$0.557692\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	−14.3458	−	53.3464i	−0.456399	−	1.69717i
$989$		0			0
$990$		0			0
$991$	−25.0043	−	43.3087i	−0.794288	−	1.37575i	−0.923291	−	0.384102i	$-0.874511\pi$
$991$	−25.0043	−	43.3087i	−0.794288	−	1.37575i	0.129003	−	0.991644i	$-0.458822\pi$
$992$		0			0
$993$	38.4256	+	17.2940i	1.21940	+	0.548808i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	32.9167	+	40.3156i	1.04248	+	1.27681i	0.959936	+	0.280221i	$0.0904077\pi$
$997$	32.9167	+	40.3156i	1.04248	+	1.27681i	0.0825460	+	0.996587i	$0.473695\pi$
$998$		0			0
$999$	26.4581	+	19.0606i	0.837097	+	0.603050i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	507.2.x.a.119.1	yes	48
3.2	odd	2	CM	507.2.x.a.119.1	yes	48
169.98	odd	156	inner	507.2.x.a.98.1	✓	48
507.98	even	156	inner	507.2.x.a.98.1	✓	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
507.2.x.a.98.1	✓	48	169.98	odd	156	inner
507.2.x.a.98.1	✓	48	507.98	even	156	inner
507.2.x.a.119.1	yes	48	1.1	even	1	trivial
507.2.x.a.119.1	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)

\(f(q)\)	\(=\)	\(q+(1.64291 + 0.548485i) q^{3} +(-1.44240 - 1.38545i) q^{4} +(-0.527444 + 5.22047i) q^{7} +(2.39833 + 1.80223i) q^{9} +O(q^{10})\)	\(q+(1.64291 + 0.548485i) q^{3} +(-1.44240 - 1.38545i) q^{4} +(-0.527444 + 5.22047i) q^{7} +(2.39833 + 1.80223i) q^{9} +(-1.60985 - 3.06731i) q^{12} +(2.59808 - 2.50000i) q^{13} +(0.161064 + 3.99676i) q^{16} +(1.98272 + 7.39960i) q^{19} +(-3.72990 + 8.28749i) q^{21} +(-4.11492 - 2.84032i) q^{25} +(2.95175 + 4.27635i) q^{27} +(7.99349 - 6.79929i) q^{28} +(1.48545 + 0.272220i) q^{31} +(-0.962468 - 5.92230i) q^{36} +(5.91139 - 2.10674i) q^{37} +(5.63963 - 2.68228i) q^{39} +(11.5365 - 5.47415i) q^{43} +(-1.92755 + 6.65466i) q^{48} +(-20.1166 - 4.10684i) q^{49} +(-7.21110 + 0.00651091i) q^{52} +(-0.801139 + 13.2444i) q^{57} +(-11.3705 - 1.84789i) q^{61} +(-10.6735 + 11.5698i) q^{63} +(5.30498 - 5.98809i) q^{64} +(-9.94094 - 0.200222i) q^{67} +(2.09927 - 2.67951i) q^{73} +(-5.20258 - 6.92338i) q^{75} +(7.39189 - 13.4202i) q^{76} +(-0.441406 + 0.108797i) q^{79} +(2.50396 + 8.64466i) q^{81} +(16.8619 - 6.78634i) q^{84} +(11.6808 + 14.8818i) q^{91} +(2.29117 + 1.26198i) q^{93} +(2.18324 - 9.69385i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 10 q^{7} + 6 q^{9}+O(q^{10}) \) 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

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Representations

Groups

Database

Properties

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