LMFDB - Embedded newform 507.2.x.a.50.1

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			50.1
Character	$\chi$	$=$	507.50
Dual form			507.2.x.a.71.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.64291 - 0.548485i) q^{3} +(1.44240 + 1.38545i) q^{4} +(0.680973 + 0.0688013i) q^{7} +(2.39833 + 1.80223i) q^{9} +O(q^{10})$	\(q+(-1.64291 - 0.548485i) q^{3} +(1.44240 + 1.38545i) q^{4} +(0.680973 + 0.0688013i) q^{7} +(2.39833 + 1.80223i) q^{9} +(-1.60985 - 3.06731i) q^{12} +(-2.59808 + 2.50000i) q^{13} +(0.161064 + 3.99676i) q^{16} +(4.01931 - 1.07697i) q^{19} +(-1.08104 - 0.486538i) q^{21} +(4.11492 + 2.84032i) q^{25} +(-2.95175 - 4.27635i) q^{27} +(0.886918 + 1.04269i) q^{28} +(-1.98869 + 10.8519i) q^{31} +(0.962468 + 5.92230i) q^{36} +(3.49870 + 9.81714i) q^{37} +(5.63963 - 2.68228i) q^{39} +(11.5365 - 5.47415i) q^{43} +(1.92755 - 6.65466i) q^{48} +(-6.39954 - 1.30648i) q^{49} +(-7.21110 + 0.00651091i) q^{52} +(-7.19407 - 0.435161i) q^{57} +(11.3705 + 1.84789i) q^{61} +(1.50920 + 1.39228i) q^{63} +(-5.30498 + 5.98809i) q^{64} +(0.261888 - 13.0027i) q^{67} +(-13.1818 - 10.3273i) q^{73} +(-5.20258 - 6.92338i) q^{75} +(7.28955 + 4.01512i) q^{76} +(0.441406 - 0.108797i) q^{79} +(2.50396 + 8.64466i) q^{81} +(-0.885228 - 2.19951i) q^{84} +(-1.94122 + 1.52368i) q^{91} +(9.21937 - 16.7380i) q^{93} +(-16.5921 - 3.73686i) q^{97} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/507\mathbb{Z}\right)^\times$.

$n$	$170$	$340$
$\chi(n)$	$-1$	$e\left(\frac{19}{156}\right)$

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		−0.927686	−	0.373361i	$-0.878205\pi$
$2$		0			0		0.927686	+	0.373361i	$0.121795\pi$
$3$	−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		−0.954721	−	0.297503i	$-0.903846\pi$
$5$		0			0		0.954721	+	0.297503i	$0.0961538\pi$
$6$		0			0
$7$	0.680973	+	0.0688013i	0.257384	+	0.0260044i	0.228369	−	0.973575i	$-0.426661\pi$
$7$	0.680973	+	0.0688013i	0.257384	+	0.0260044i	0.0290142	+	0.999579i	$0.490763\pi$
$8$		0			0
$9$	2.39833	+	1.80223i	0.799443	+	0.600742i
$10$		0			0
$11$		0			0		0.140502	−	0.990080i	$-0.455128\pi$
$11$		0			0		−0.140502	+	0.990080i	$0.544872\pi$
$12$	−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
$19$	4.01931	−	1.07697i	0.922092	−	0.247074i	0.233613	−	0.972330i	$-0.424945\pi$
$19$	4.01931	−	1.07697i	0.922092	−	0.247074i	0.688479	+	0.725256i	$0.258279\pi$
$20$		0			0
$21$	−1.08104	−	0.486538i	−0.235903	−	0.106171i
$22$		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
$28$	0.886918	+	1.04269i	0.167612	+	0.197050i
$29$		0			0		0.919979	−	0.391967i	$-0.128205\pi$
$29$		0			0		−0.919979	+	0.391967i	$0.871795\pi$
$30$		0			0
$31$	−1.98869	+	10.8519i	−0.357179	+	1.94906i	−0.0451919	+	0.998978i	$0.514390\pi$
$31$	−1.98869	+	10.8519i	−0.357179	+	1.94906i	−0.311987	+	0.950086i	$0.600995\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	0.962468	+	5.92230i	0.160411	+	0.987050i
$37$	3.49870	+	9.81714i	0.575182	+	1.61393i	0.773504	+	0.633791i	$0.218502\pi$
$37$	3.49870	+	9.81714i	0.575182	+	1.61393i	−0.198322	+	0.980137i	$0.563549\pi$
$38$		0			0
$39$	5.63963	−	2.68228i	0.903063	−	0.429508i
$40$		0			0
$41$		0			0		−0.446798	−	0.894635i	$-0.647436\pi$
$41$		0			0		0.446798	+	0.894635i	$0.352564\pi$
$42$		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		0.517338	−	0.855781i	$-0.326923\pi$
$47$		0			0		−0.517338	+	0.855781i	$0.673077\pi$
$48$	1.92755	−	6.65466i	0.278217	−	0.960518i
$49$	−6.39954	−	1.30648i	−0.914221	−	0.186639i
$50$		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
$57$	−7.19407	−	0.435161i	−0.952878	−	0.0576385i
$58$		0			0
$59$		0			0		−0.678061	−	0.735006i	$-0.737179\pi$
$59$		0			0		0.678061	+	0.735006i	$0.262821\pi$
$60$		0			0
$61$	11.3705	+	1.84789i	1.45585	+	0.236598i	0.836261	−	0.548331i	$-0.184737\pi$
$61$	11.3705	+	1.84789i	1.45585	+	0.236598i	0.619586	+	0.784929i	$0.287301\pi$
$62$		0			0
$63$	1.50920	+	1.39228i	0.190141	+	0.175410i
$64$	−5.30498	+	5.98809i	−0.663123	+	0.748511i
$65$		0			0
$66$		0			0
$67$	0.261888	−	13.0027i	0.0319948	−	1.58853i	−0.591242	−	0.806494i	$-0.701362\pi$
$67$	0.261888	−	13.0027i	0.0319948	−	1.58853i	0.623237	−	0.782033i	$-0.285817\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		0.551377	−	0.834256i	$-0.314103\pi$
$71$		0			0		−0.551377	+	0.834256i	$0.685897\pi$
$72$		0			0
$73$	−13.1818	−	10.3273i	−1.54281	−	1.20872i	−0.894258	−	0.447552i	$-0.852296\pi$
$73$	−13.1818	−	10.3273i	−1.54281	−	1.20872i	−0.648557	−	0.761166i	$-0.724627\pi$
$74$		0			0
$75$	−5.20258	−	6.92338i	−0.600742	−	0.799443i
$76$	7.28955	+	4.01512i	0.836169	+	0.460565i
$77$		0			0
$78$		0			0
$79$	0.441406	−	0.108797i	0.0496620	−	0.0122406i	−0.214406	−	0.976745i	$-0.568782\pi$
$79$	0.441406	−	0.108797i	0.0496620	−	0.0122406i	0.264068	+	0.964504i	$0.414936\pi$
$80$		0			0
$81$	2.50396	+	8.64466i	0.278217	+	0.960518i
$82$		0			0
$83$		0			0		−0.0603785	−	0.998176i	$-0.519231\pi$
$83$		0			0		0.0603785	+	0.998176i	$0.480769\pi$
$84$	−0.885228	−	2.19951i	−0.0965863	−	0.239987i
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$89$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$90$		0			0
$91$	−1.94122	+	1.52368i	−0.203495	+	0.159725i
$92$		0			0
$93$	9.21937	−	16.7380i	0.956004	−	1.73565i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−16.5921	−	3.73686i	−1.68468	−	0.379420i	−0.731501	−	0.681841i	$-0.761180\pi$
$97$	−16.5921	−	3.73686i	−1.68468	−	0.379420i	−0.953175	+	0.302420i	$0.902205\pi$
$98$		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
$109$	−2.57272	+	0.471469i	−0.246422	+	0.0451586i	−0.302046	−	0.953293i	$-0.597670\pi$
$109$	−2.57272	+	0.471469i	−0.246422	+	0.0451586i	0.0556241	+	0.998452i	$0.482285\pi$
$110$		0			0
$111$	−0.363501	−	18.0477i	−0.0345020	−	1.71301i
$112$	−0.165302	+	2.73276i	−0.0156195	+	0.258222i
$113$		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
$124$	−17.9033	+	12.8977i	−1.60776	+	1.15824i
$125$		0			0
$126$		0			0
$127$	8.18527	−	7.86205i	0.726325	−	0.697644i	−0.235327	−	0.971916i	$-0.575616\pi$
$127$	8.18527	−	7.86205i	0.726325	−	0.697644i	0.961652	+	0.274272i	$0.0884367\pi$
$128$		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
$133$	2.81113	−	0.456854i	0.243756	−	0.0396143i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		−0.941967	−	0.335705i	$-0.891026\pi$
$137$		0			0		0.941967	+	0.335705i	$0.108974\pi$
$138$		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
$147$	9.79731	+	5.65648i	0.808069	+	0.466539i
$148$	−8.55461	+	19.0076i	−0.703184	+	1.56241i
$149$		0			0		0.875918	−	0.482459i	$-0.160256\pi$
$149$		0			0		−0.875918	+	0.482459i	$0.839744\pi$
$150$		0			0
$151$	19.7946	−	11.9663i	1.61087	−	0.973802i	0.630602	−	0.776107i	$-0.282808\pi$
$151$	19.7946	−	11.9663i	1.61087	−	0.973802i	0.980263	−	0.197696i	$-0.0633457\pi$
$152$		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
$163$	0.391770	−	0.460579i	0.0306858	−	0.0360754i	−0.746156	−	0.665771i	$-0.768103\pi$
$163$	0.391770	−	0.460579i	0.0306858	−	0.0360754i	0.776842	+	0.629696i	$0.216821\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.373361	−	0.927686i	$-0.378205\pi$
$167$		0			0		−0.373361	+	0.927686i	$0.621795\pi$
$168$		0			0
$169$	0.500000	−	12.9904i	0.0384615	−	0.999260i
$170$		0			0
$171$	11.5806	+	4.66077i	0.885587	+	0.356418i
$172$	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
$175$	2.60673	+	2.21730i	0.197050	+	0.167612i
$176$		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
$189$	−1.71584	−	3.11516i	−0.124809	−	0.226595i
$190$		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
$193$	1.46521	+	14.5022i	0.105468	+	1.04389i	0.899770	+	0.436365i	$0.143734\pi$
$193$	1.46521	+	14.5022i	0.105468	+	1.04389i	−0.794302	+	0.607524i	$0.792163\pi$
$194$		0			0
$195$		0			0
$196$	−7.42068	−	10.7507i	−0.530048	−	0.767908i
$197$		0			0		−0.647915	−	0.761712i	$-0.724359\pi$
$197$		0			0		0.647915	+	0.761712i	$0.275641\pi$
$198$		0			0
$199$	10.8515	+	17.1602i	0.769239	+	1.21645i	0.971481	+	0.237119i	$0.0762031\pi$
$199$	10.8515	+	17.1602i	0.769239	+	1.21645i	−0.202241	+	0.979336i	$0.564823\pi$
$200$		0			0
$201$	−7.56202	+	21.2186i	−0.533384	+	1.49664i
$202$		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	0.617317	−	0.786714i	$-0.288220\pi$
$211$	−2.81244	−	2.92806i	−0.193616	−	0.201576i	−0.810933	+	0.585139i	$0.801040\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−2.10087	+	7.25305i	−0.142616	+	0.492369i
$218$		0			0
$219$	15.9922	+	24.1969i	1.08065	+	1.63507i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−2.73557	−	12.1463i	−0.183187	−	0.813375i	−0.978346	−	0.206974i	$-0.933638\pi$
$223$	−2.73557	−	12.1463i	−0.183187	−	0.813375i	0.795159	−	0.606401i	$-0.207387\pi$
$224$		0			0
$225$	4.75002	+	14.2280i	0.316668	+	0.948536i
$226$		0			0
$227$		0			0		0.999797	−	0.0201371i	$-0.00641026\pi$
$227$		0			0		−0.999797	+	0.0201371i	$0.993590\pi$
$228$	−9.77387	−	10.5947i	−0.647290	−	0.701651i
$229$	−4.75667	−	25.9563i	−0.314330	−	1.71524i	−0.638823	−	0.769354i	$-0.720578\pi$
$229$	−4.75667	−	25.9563i	−0.314330	−	1.71524i	0.324493	−	0.945888i	$-0.394806\pi$
$230$		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	$-0.750000\pi$
$239$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$240$		0			0
$241$	−0.517638	+	2.29838i	−0.0333440	+	0.148052i	−0.989423	−	0.145056i	$-0.953664\pi$
$241$	−0.517638	+	2.29838i	−0.0333440	+	0.148052i	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
$247$	−7.75004	+	12.8463i	−0.493123	+	0.817391i
$248$		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	$0.474359\pi$
$252$	0.247953	+	4.09915i	0.0156195	+	0.258222i
$253$		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
$259$	1.70709	+	6.92592i	0.106073	+	0.430356i
$260$		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
$268$	18.3923	−	18.3923i	1.12349	−	1.12349i
$269$		0			0		−0.200026	−	0.979791i	$-0.564103\pi$
$269$		0			0		0.200026	+	0.979791i	$0.435897\pi$
$270$		0			0
$271$	−16.7498	−	0.337359i	−1.01748	−	0.0204931i	−0.491403	−	0.870933i	$-0.663516\pi$
$271$	−16.7498	−	0.337359i	−1.01748	−	0.0204931i	−0.526073	+	0.850439i	$0.676336\pi$
$272$		0			0
$273$	4.02498	−	1.43854i	0.243603	−	0.0870647i
$274$		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
$279$	−24.3272	+	22.4424i	−1.45643	+	1.34359i
$280$		0			0
$281$		0			0		0.0603785	−	0.998176i	$-0.480769\pi$
$281$		0			0		−0.0603785	+	0.998176i	$0.519231\pi$
$282$		0			0
$283$	10.5454	+	5.00387i	0.626860	+	0.297449i	0.715498	−	0.698615i	$-0.246200\pi$
$283$	10.5454	+	5.00387i	0.626860	+	0.297449i	−0.0886374	+	0.996064i	$0.528251\pi$
$284$		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
$291$	25.2098	+	15.2399i	1.47783	+	0.893377i
$292$	−4.70558	−	33.1589i	−0.275373	−	1.94048i
$293$		0			0		0.811378	−	0.584522i	$-0.198718\pi$
$293$		0			0		−0.811378	+	0.584522i	$0.801282\pi$
$294$		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
$301$	8.23269	−	2.93402i	0.474524	−	0.169114i
$302$		0			0
$303$		0			0
$304$	4.95175	+	15.8907i	0.284002	+	0.911395i
$305$		0			0
$306$		0			0
$307$	−32.0929	−	5.88124i	−1.83164	−	0.335660i	−0.850056	−	0.526693i	$-0.823432\pi$
$307$	−32.0929	−	5.88124i	−1.83164	−	0.335660i	−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	$-0.418709\pi$
$313$	18.8900	−	27.3669i	1.06773	−	1.54687i	0.815111	−	0.579304i	$-0.196676\pi$
$314$		0			0
$315$		0			0
$316$	0.787418	+	0.454616i	0.0442957	+	0.0255741i
$317$		0			0		0.410413	−	0.911900i	$-0.365385\pi$
$317$		0			0		−0.410413	+	0.911900i	$0.634615\pi$
$318$		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
$327$	4.48536	+	0.636518i	0.248041	+	0.0351995i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	2.71991	−	26.9208i	0.149500	−	1.47970i	−0.590468	−	0.807061i	$-0.701057\pi$
$331$	2.71991	−	26.9208i	0.149500	−	1.47970i	0.739968	−	0.672642i	$-0.234841\pi$
$332$		0			0
$333$	−9.30169	+	29.8502i	−0.509730	+	1.63578i
$334$		0			0
$335$		0			0
$336$	1.77046	−	4.39903i	0.0965863	−	0.239987i
$337$			31.3370i			1.70703i	0.521065	+	0.853517i	$0.325535\pi$
$337$			31.3370i			1.70703i	−0.521065	+	0.853517i	$0.674465\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−8.84217	−	2.75533i	−0.477432	−	0.148774i
$344$		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
$349$	0.640299	−	4.51200i	0.0342744	−	0.241522i	−0.965551	−	0.260214i	$-0.916207\pi$
$349$	0.640299	−	4.51200i	0.0342744	−	0.241522i	0.999825	+	0.0186925i	$0.00595035\pi$
$350$		0			0
$351$	18.3597	+	3.73090i	0.979971	+	0.199141i
$352$		0			0
$353$		0			0		−0.811378	−	0.584522i	$-0.801282\pi$
$353$		0			0		0.811378	+	0.584522i	$0.198718\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		−0.911900	−	0.410413i	$-0.865385\pi$
$359$		0			0		0.911900	+	0.410413i	$0.134615\pi$
$360$		0			0
$361$	−1.45953	+	0.842661i	−0.0768174	+	0.0443506i
$362$		0			0
$363$	15.6799	+	10.8231i	0.822984	+	0.568065i
$364$	−4.91101	−	0.491699i	−0.257407	−	0.0257720i
$365$		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
$372$	36.4877	−	11.3700i	1.89180	−	0.589509i
$373$	29.6280	+	12.6233i	1.53408	+	0.653609i	0.983456	−	0.181146i	$-0.0579805\pi$
$373$	29.6280	+	12.6233i	1.53408	+	0.653609i	0.550622	+	0.834755i	$0.314391\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	14.4596	+	28.9528i	0.742740	+	1.48720i	0.868790	+	0.495181i	$0.164898\pi$
$379$	14.4596	+	28.9528i	0.742740	+	1.48720i	−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		−0.584522	−	0.811378i	$-0.698718\pi$
$383$		0			0		0.584522	+	0.811378i	$0.301282\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	37.5340	+	7.66263i	1.90796	+	0.389513i
$388$	−18.7553	−	28.3776i	−0.952158	−	1.44065i
$389$		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
$397$	−14.5546	−	15.7769i	−0.730475	−	0.791822i	0.254026	−	0.967197i	$-0.418245\pi$
$397$	−14.5546	−	15.7769i	−0.730475	−	0.791822i	−0.984502	+	0.175376i	$0.943886\pi$
$398$		0			0
$399$	−4.86903	−	0.791294i	−0.243756	−	0.0396143i
$400$	−10.6893	+	16.9038i	−0.534466	+	0.845190i
$401$		0			0		−0.735006	−	0.678061i	$-0.762821\pi$
$401$		0			0		0.735006	+	0.678061i	$0.237179\pi$
$402$		0			0
$403$	−21.9631	−	33.1659i	−1.09406	−	1.65211i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−21.8179	+	33.0114i	−1.07883	+	1.63231i	−0.366574	+	0.930389i	$0.619469\pi$
$409$	−21.8179	+	33.0114i	−1.07883	+	1.63231i	−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
$421$	−1.74861	−	28.9080i	−0.0852220	−	1.40889i	−0.749798	−	0.661667i	$-0.769849\pi$
$421$	−1.74861	−	28.9080i	−0.0852220	−	1.40889i	0.664576	−	0.747221i	$-0.268612\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	7.61588	+	2.04067i	0.368558	+	0.0987550i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.482459	−	0.875918i	$-0.339744\pi$
$431$		0			0		−0.482459	+	0.875918i	$0.660256\pi$
$432$	16.6161	−	12.4862i	0.799443	−	0.600742i
$433$	26.5497	+	1.06992i	1.27590	+	0.0514169i	0.668940	−	0.743317i	$-0.266748\pi$
$433$	26.5497	+	1.06992i	1.27590	+	0.0514169i	0.606958	+	0.794734i	$0.292389\pi$
$434$		0			0
$435$		0			0
$436$	−4.36411	−	2.88433i	−0.209003	−	0.138134i
$437$		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
$441$	−12.9936	−	14.6668i	−0.618745	−	0.698419i
$442$		0			0
$443$		0			0		−0.748511	−	0.663123i	$-0.769231\pi$
$443$		0			0		0.748511	+	0.663123i	$0.230769\pi$
$444$	24.4798	−	26.5357i	1.16176	−	1.25933i
$445$		0			0
$446$		0			0
$447$		0			0
$448$	−4.02454	+	3.71273i	−0.190141	+	0.175410i
$449$		0			0		−0.0201371	−	0.999797i	$-0.506410\pi$
$449$		0			0		0.0201371	+	0.999797i	$0.493590\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	−39.0842	+	8.80249i	−1.83634	+	0.413577i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−19.8328	+	13.1079i	−0.927738	+	0.613161i	−0.922154	−	0.386823i	$-0.873572\pi$
$457$	−19.8328	+	13.1079i	−0.927738	+	0.613161i	−0.00558406	+	0.999984i	$0.501777\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		−0.140502	−	0.990080i	$-0.544872\pi$
$461$		0			0		0.140502	+	0.990080i	$0.455128\pi$
$462$		0			0
$463$	−15.5139	−	19.8020i	−0.720990	−	0.920276i	0.278200	−	0.960523i	$-0.410262\pi$
$463$	−15.5139	−	19.8020i	−0.720990	−	0.920276i	−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
$469$	1.07294	−	8.83644i	0.0495437	−	0.408029i
$470$		0			0
$471$	−9.04692	+	1.47027i	−0.416860	+	0.0677464i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	19.5981	+	6.98449i	0.899221	+	0.320470i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.761712	−	0.647915i	$-0.224359\pi$
$479$		0			0		−0.761712	+	0.647915i	$0.775641\pi$
$480$		0			0
$481$	−33.6327	−	16.7589i	−1.53352	−	0.764142i
$482$		0			0
$483$		0			0
$484$	−11.0000	−	19.0526i	−0.500000	−	0.866025i
$485$		0			0
$486$		0			0
$487$	−31.4124	+	17.3021i	−1.42343	+	0.784032i	−0.992946	−	0.118565i	$-0.962171\pi$
$487$	−31.4124	+	17.3021i	−1.42343	+	0.784032i	−0.430486	+	0.902597i	$0.641658\pi$
$488$		0			0
$489$	−0.896266	+	0.541812i	−0.0405305	+	0.0245016i
$490$		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
$496$	−43.6928	−	6.20046i	−1.96187	−	0.278409i
$497$		0			0
$498$		0			0
$499$	−19.4014	+	8.73188i	−0.868528	+	0.390893i	−0.795151	−	0.606412i	$-0.792608\pi$
$499$	−19.4014	+	8.73188i	−0.868528	+	0.390893i	−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		−0.927686	−	0.373361i	$-0.878205\pi$
$509$		0			0		0.927686	+	0.373361i	$0.121795\pi$
$510$		0			0
$511$	−8.26593	−	7.93953i	−0.365663	−	0.351224i
$512$		0			0
$513$	−16.4695	−	14.0090i	−0.727146	−	0.618513i
$514$		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
$525$	−3.06648	−	5.07258i	−0.133832	−	0.221385i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	11.5000	−	19.9186i	0.500000	−	0.866025i
$530$		0			0
$531$		0			0
$532$	4.68774	+	3.23571i	0.203239	+	0.140286i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	43.5890	−	13.5829i	1.87404	−	0.583974i	0.879997	−	0.474979i	$-0.157544\pi$
$541$	43.5890	−	13.5829i	1.87404	−	0.583974i	0.994042	−	0.108996i	$-0.0347635\pi$
$542$		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
$553$	0.308071	−	0.0437184i	0.0131005	−	0.00185909i
$554$		0			0
$555$		0			0
$556$	44.1049	+	9.00408i	1.87046	+	0.381858i
$557$		0			0		−0.551377	−	0.834256i	$-0.685897\pi$
$557$		0			0		0.551377	+	0.834256i	$0.314103\pi$
$558$		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
$567$	1.11036	+	6.05906i	0.0466309	+	0.254456i
$568$		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	−0.942293	−	0.334790i	$-0.891335\pi$
$571$	−11.8609	+	13.3882i	−0.496363	+	0.560279i	0.445929	+	0.895068i	$0.352873\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−23.5150	+	4.80062i	−0.979791	+	0.200026i
$577$	15.5006	+	15.5006i	0.645299	+	0.645299i	0.951853	−	0.306554i	$-0.0991761\pi$
$577$	15.5006	+	15.5006i	0.645299	+	0.645299i	−0.306554	+	0.951853i	$0.599176\pi$
$578$		0			0
$579$	5.54701	−	24.6294i	0.230526	−	1.02356i
$580$		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	$-0.416667\pi$
$587$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$588$	6.29493	+	21.7326i	0.259598	+	0.896238i
$589$	3.69405	+	45.7590i	0.152211	+	1.88547i
$590$		0			0
$591$		0			0
$592$	−38.6732	+	15.5646i	−1.58946	+	0.639702i
$593$		0			0		0.998176	−	0.0603785i	$-0.0192308\pi$
$593$		0			0		−0.998176	+	0.0603785i	$0.980769\pi$
$594$		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	−0.433154	−	0.901320i	$-0.642599\pi$
$601$	7.65049	−	5.74897i	0.312070	−	0.234505i	0.745223	+	0.666815i	$0.232343\pi$
$602$		0			0
$603$	24.0618	−	30.7127i	0.979874	−	1.25072i
$604$	45.1306	+	10.1642i	1.83634	+	0.413577i
$605$		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
$613$	10.0519	−	10.8960i	0.405990	−	0.440086i	−0.498300	−	0.867005i	$-0.666042\pi$
$613$	10.0519	−	10.8960i	0.405990	−	0.440086i	0.904290	+	0.426919i	$0.140401\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		0.735006	−	0.678061i	$-0.237179\pi$
$617$		0			0		−0.735006	+	0.678061i	$0.762821\pi$
$618$		0			0
$619$	−2.15360	+	35.6032i	−0.0865605	+	1.43102i	0.652711	+	0.757607i	$0.273632\pi$
$619$	−2.15360	+	35.6032i	−0.0865605	+	1.43102i	−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
$631$	−17.4663	+	12.5828i	−0.695323	+	0.500915i	−0.875806	−	0.482663i	$-0.839670\pi$
$631$	−17.4663	+	12.5828i	−0.695323	+	0.500915i	0.180484	+	0.983578i	$0.442234\pi$
$632$		0			0
$633$	3.01459	+	6.35312i	0.119819	+	0.252514i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	19.8927	−	12.6045i	0.788177	−	0.499410i
$638$		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
$643$	45.3389	+	22.6432i	1.78799	+	0.892959i	0.910871	+	0.412692i	$0.135411\pi$
$643$	45.3389	+	22.6432i	1.78799	+	0.892959i	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
$651$	7.42974	−	10.7638i	0.291194	−	0.421868i
$652$	1.20320	−	0.121564i	0.0471210	−	0.00476081i
$653$		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
$657$	−13.0022	−	48.5249i	−0.507264	−	1.89314i
$658$		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
$661$	10.4911	−	14.5628i	0.408058	−	0.566428i	−0.556315	−	0.830971i	$-0.687785\pi$
$661$	10.4911	−	14.5628i	0.408058	−	0.566428i	0.964374	+	0.264543i	$0.0852213\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	−2.16775	+	21.4557i	−0.0838101	+	0.829525i
$670$		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
$679$	−11.0417	−	3.68626i	−0.423741	−	0.141466i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		−0.994935	−	0.100522i	$-0.967949\pi$
$683$		0			0		0.994935	+	0.100522i	$0.0320513\pi$
$684$	10.2466	+	22.7670i	0.391788	+	0.870518i
$685$		0			0
$686$		0			0
$687$	−6.42185	+	45.2529i	−0.245009	+	1.72651i
$688$	23.7370	+	45.2270i	0.904963	+	1.72426i
$689$		0			0
$690$		0			0
$691$	5.64072	+	4.06361i	0.214583	+	0.154587i	0.686681	−	0.726959i	$-0.259067\pi$
$691$	5.64072	+	4.06361i	0.214583	+	0.154587i	−0.472098	+	0.881546i	$0.656503\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	0.688013	+	6.80973i	0.0260044	+	0.257384i
$701$		0			0		−0.822984	−	0.568065i	$-0.807692\pi$
$701$		0			0		0.822984	+	0.568065i	$0.192308\pi$
$702$		0			0
$703$	24.6351	+	35.6901i	0.929130	+	1.34608i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	23.7109	−	47.4769i	0.890482	−	1.78303i	0.370610	−	0.928788i	$-0.379149\pi$
$709$	23.7109	−	47.4769i	0.890482	−	1.78303i	0.519872	−	0.854244i	$-0.325980\pi$
$710$		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
$721$	−7.93911	−	11.0203i	−0.295668	−	0.410418i
$722$		0			0
$723$	2.11106	−	3.49212i	0.0785112	−	0.129873i
$724$	−9.71642	+	33.5450i	−0.361108	+	1.24669i
$725$		0			0
$726$		0			0
$727$	3.43881	−	1.30417i	0.127538	−	0.0483689i	−0.290010	−	0.957024i	$-0.593659\pi$
$727$	3.43881	−	1.30417i	0.127538	−	0.0483689i	0.417548	+	0.908655i	$0.362889\pi$
$728$		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
$733$	−46.0062	−	2.78286i	−1.69928	−	0.102787i	−0.817949	−	0.575291i	$-0.804889\pi$
$733$	−46.0062	−	2.78286i	−1.69928	−	0.102787i	−0.881329	+	0.472504i	$0.843350\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	29.8222	+	27.5118i	1.09703	+	1.01204i	0.999859	+	0.0167685i	$0.00533782\pi$
$739$	29.8222	+	27.5118i	1.09703	+	1.01204i	0.0971693	+	0.995268i	$0.469021\pi$
$740$		0			0
$741$	19.7786	−	16.8546i	0.726587	−	0.619169i
$742$		0			0
$743$		0			0		0.0201371	−	0.999797i	$-0.493590\pi$
$743$		0			0		−0.0201371	+	0.999797i	$0.506410\pi$
$744$		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
$756$	1.84096	−	6.87054i	0.0669549	−	0.249879i
$757$	11.5482	+	39.8690i	0.419727	+	1.44906i	0.840457	+	0.541879i	$0.182287\pi$
$757$	11.5482	+	39.8690i	0.419727	+	1.44906i	−0.420730	+	0.907186i	$0.638226\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		0.927686	−	0.373361i	$-0.121795\pi$
$761$		0			0		−0.927686	+	0.373361i	$0.878205\pi$
$762$		0			0
$763$	−1.78439	+	0.144051i	−0.0645994	+	0.00521500i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	26.9075	+	6.63211i	0.970942	+	0.239316i
$769$	−5.76924	+	10.4742i	−0.208044	+	0.377710i	−0.959339	−	0.282257i	$-0.908917\pi$
$769$	−5.76924	+	10.4742i	−0.208044	+	0.377710i	0.751295	+	0.659967i	$0.229430\pi$
$770$		0			0
$771$		0			0
$772$	−17.9786	+	22.9480i	−0.647063	+	0.825915i
$773$		0			0		−0.975564	−	0.219715i	$-0.929487\pi$
$773$		0			0		0.975564	+	0.219715i	$0.0705128\pi$
$774$		0			0
$775$	−39.0063	+	39.0063i	−1.40115	+	1.40115i
$776$		0			0
$777$	0.994170	−	12.3150i	0.0356656	−	0.441798i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	4.19093	−	25.7878i	0.149676	−	0.920994i
$785$		0			0
$786$		0			0
$787$	1.11545	+	55.3815i	0.0397613	+	1.97414i	0.178912	+	0.983865i	$0.442742\pi$
$787$	1.11545	+	55.3815i	0.0397613	+	1.97414i	−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
$804$	−40.3048	+	20.1290i	−1.42144	+	0.709896i
$805$		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
$811$	−16.0605	−	51.5400i	−0.563961	−	1.80981i	−0.586341	−	0.810064i	$-0.699432\pi$
$811$	−16.0605	−	51.5400i	−0.563961	−	1.80981i	0.0223803	−	0.999750i	$-0.492876\pi$
$812$		0			0
$813$	27.3334	+	9.74125i	0.958623	+	0.341640i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	40.4733	−	34.4268i	1.41598	−	1.20444i
$818$		0			0
$819$	−7.40171	+	0.155765i	−0.258637	+	0.00544286i
$820$		0			0
$821$		0			0		0.994935	−	0.100522i	$-0.0320513\pi$
$821$		0			0		−0.994935	+	0.100522i	$0.967949\pi$
$822$		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		0.855781	−	0.517338i	$-0.173077\pi$
$827$		0			0		−0.855781	+	0.517338i	$0.826923\pi$
$828$		0			0
$829$	−57.3466	+	2.31099i	−1.99173	+	0.0802640i	−0.999147	−	0.0412960i	$-0.986851\pi$
$829$	−57.3466	+	2.31099i	−1.99173	+	0.0802640i	−0.992584	+	0.121560i	$0.961210\pi$
$830$		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
$837$	52.2768	−	23.5279i	1.80695	−	0.813242i
$838$		0			0
$839$		0			0		0.647915	−	0.761712i	$-0.275641\pi$
$839$		0			0		−0.647915	+	0.761712i	$0.724359\pi$
$840$		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
$847$	−6.98440	−	2.81098i	−0.239987	−	0.0965863i
$848$		0			0
$849$	−14.5807	−	14.0049i	−0.500407	−	0.480648i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−14.0506	−	31.2191i	−0.481083	−	1.06892i	−0.979400	−	0.201928i	$-0.935279\pi$
$853$	−14.0506	−	31.2191i	−0.481083	−	1.06892i	0.498317	−	0.866995i	$-0.333952\pi$
$854$		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	−0.138409	−	0.990375i	$-0.544199\pi$
$859$	−30.2494	−	15.8761i	−1.03210	−	0.541686i	−0.893688	+	0.448689i	$0.851891\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		−0.517338	−	0.855781i	$-0.673077\pi$
$863$		0			0		0.517338	+	0.855781i	$0.326923\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−14.7224	+	25.5000i	−0.500000	+	0.866025i
$868$	−13.0790	+	7.55118i	−0.443931	+	0.256304i
$869$		0			0
$870$		0			0
$871$	31.8262	+	34.4366i	1.07839	+	1.16684i
$872$		0			0
$873$	−33.0587	−	38.8650i	−1.11887	−	1.31538i
$874$		0			0
$875$		0			0
$876$	−10.4563	+	57.0581i	−0.353285	+	1.92781i
$877$	−3.72653	+	10.4564i	−0.125836	+	0.353089i	−0.988193	−	0.153211i	$-0.951038\pi$
$877$	−3.72653	+	10.4564i	−0.125836	+	0.353089i	0.862357	+	0.506300i	$0.168987\pi$
$878$		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	0.765553	−	0.643373i	$-0.222466\pi$
$883$	40.2610	+	4.88857i	1.35489	+	0.164514i	0.589338	+	0.807887i	$0.299389\pi$
$884$		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
$889$	6.11486	−	4.79069i	0.205086	−	0.160675i
$890$		0			0
$891$		0			0
$892$	12.8823	−	21.3098i	0.431330	−	0.713506i
$893$		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
$903$	−15.1349	+	0.304833i	−0.503657	+	0.0101442i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−31.9527	+	50.5291i	−1.06097	+	1.67779i	−0.427525	+	0.904004i	$0.640614\pi$
$907$	−31.9527	+	50.5291i	−1.06097	+	1.67779i	−0.633446	+	0.773787i	$0.718360\pi$
$908$		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	$0.769231\pi$
$912$	0.580529	−	28.8230i	0.0192232	−	0.954426i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	29.1001	−	44.0296i	0.961494	−	1.45478i
$917$		0			0
$918$		0			0
$919$	2.43632	−	60.4566i	0.0803667	−	1.99428i	−0.0237804	−	0.999717i	$-0.507570\pi$
$919$	2.43632	−	60.4566i	0.0803667	−	1.99428i	0.104147	−	0.994562i	$-0.466789\pi$
$920$		0			0
$921$	49.5001	+	27.2649i	1.63108	+	0.898408i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−13.4870	+	50.3342i	−0.443450	+	1.65498i
$926$		0			0
$927$	−4.79041	−	59.3398i	−0.157338	−	1.94898i
$928$		0			0
$929$		0			0		−0.373361	−	0.927686i	$-0.621795\pi$
$929$		0			0		0.373361	+	0.927686i	$0.378205\pi$
$930$		0			0
$931$	−27.1288	+	1.64099i	−0.889109	+	0.0537812i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	31.5471	+	7.77567i	1.03060	+	0.254020i	0.718131	−	0.695908i	$-0.244998\pi$
$937$	31.5471	+	7.77567i	1.03060	+	0.254020i	0.312469	+	0.949928i	$0.398844\pi$
$938$		0			0
$939$	−46.0450	+	34.6006i	−1.50262	+	1.12915i
$940$		0			0
$941$		0			0		0.616719	−	0.787183i	$-0.288462\pi$
$941$		0			0		−0.616719	+	0.787183i	$0.711538\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		−0.999797	−	0.0201371i	$-0.993590\pi$
$947$		0			0		0.999797	+	0.0201371i	$0.00641026\pi$
$948$	−1.04431	−	1.17878i	−0.0339176	−	0.0382850i
$949$	60.0656	−	6.12345i	1.94981	−	0.198775i
$950$		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
$961$	−84.8240	−	32.1695i	−2.73626	−	1.03773i
$962$		0			0
$963$		0			0
$964$	−3.93093	+	2.59803i	−0.126607	+	0.0836770i
$965$		0			0
$966$		0			0
$967$	34.2740	+	20.7194i	1.10218	+	0.666290i	0.946883	−	0.321578i	$-0.104213\pi$
$967$	34.2740	+	20.7194i	1.10218	+	0.666290i	0.155295	+	0.987868i	$0.450367\pi$
$968$		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
$973$	13.7817	−	6.88288i	0.441823	−	0.220655i
$974$		0			0
$975$	30.8251	+	4.98101i	0.987195	+	0.159520i
$976$	−5.55419	+	45.7429i	−0.177785	+	1.46419i
$977$		0			0		0.941967	−	0.335705i	$-0.108974\pi$
$977$		0			0		−0.941967	+	0.335705i	$0.891026\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−7.01993	−	3.50589i	−0.224129	−	0.111935i
$982$		0			0
$983$		0			0		−0.983620	−	0.180255i	$-0.942308\pi$
$983$		0			0		0.983620	+	0.180255i	$0.0576923\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	−28.9766	+	7.79230i	−0.921868	+	0.247906i
$989$		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
$993$	−19.2343	+	42.7368i	−0.610381	+	1.35621i
$994$		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
$999$	31.6542	−	43.9394i	1.00150	−	1.39018i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	507.2.x.a.50.1	✓	48
3.2	odd	2	CM	507.2.x.a.50.1	✓	48
169.71	odd	156	inner	507.2.x.a.71.1	yes	48
507.71	even	156	inner	507.2.x.a.71.1	yes	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
507.2.x.a.50.1	✓	48	1.1	even	1	trivial
507.2.x.a.50.1	✓	48	3.2	odd	2	CM
507.2.x.a.71.1	yes	48	169.71	odd	156	inner
507.2.x.a.71.1	yes	48	507.71	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)

\(f(q)\)	\(=\)	\(q+(-1.64291 - 0.548485i) q^{3} +(1.44240 + 1.38545i) q^{4} +(0.680973 + 0.0688013i) q^{7} +(2.39833 + 1.80223i) q^{9} +O(q^{10})\)	\(q+(-1.64291 - 0.548485i) q^{3} +(1.44240 + 1.38545i) q^{4} +(0.680973 + 0.0688013i) q^{7} +(2.39833 + 1.80223i) q^{9} +(-1.60985 - 3.06731i) q^{12} +(-2.59808 + 2.50000i) q^{13} +(0.161064 + 3.99676i) q^{16} +(4.01931 - 1.07697i) q^{19} +(-1.08104 - 0.486538i) q^{21} +(4.11492 + 2.84032i) q^{25} +(-2.95175 - 4.27635i) q^{27} +(0.886918 + 1.04269i) q^{28} +(-1.98869 + 10.8519i) q^{31} +(0.962468 + 5.92230i) q^{36} +(3.49870 + 9.81714i) q^{37} +(5.63963 - 2.68228i) q^{39} +(11.5365 - 5.47415i) q^{43} +(1.92755 - 6.65466i) q^{48} +(-6.39954 - 1.30648i) q^{49} +(-7.21110 + 0.00651091i) q^{52} +(-7.19407 - 0.435161i) q^{57} +(11.3705 + 1.84789i) q^{61} +(1.50920 + 1.39228i) q^{63} +(-5.30498 + 5.98809i) q^{64} +(0.261888 - 13.0027i) q^{67} +(-13.1818 - 10.3273i) q^{73} +(-5.20258 - 6.92338i) q^{75} +(7.28955 + 4.01512i) q^{76} +(0.441406 - 0.108797i) q^{79} +(2.50396 + 8.64466i) q^{81} +(-0.885228 - 2.19951i) q^{84} +(-1.94122 + 1.52368i) q^{91} +(9.21937 - 16.7380i) q^{93} +(-16.5921 - 3.73686i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 10 q^{7} + 6 q^{9}+O(q^{10}) \) 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

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