LMFDB - Embedded newform 507.2.x.a.32.1

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			32.1
Character	$\chi$	$=$	507.32
Dual form			507.2.x.a.206.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.72643 - 0.139372i) q^{3} +(1.95958 + 0.400051i) q^{4} +(-0.421909 - 1.87333i) q^{7} +(2.96115 - 0.481234i) q^{9} +O(q^{10})$	\(q+(1.72643 - 0.139372i) q^{3} +(1.95958 + 0.400051i) q^{4} +(-0.421909 - 1.87333i) q^{7} +(2.96115 - 0.481234i) q^{9} +(3.43884 + 0.417551i) q^{12} +(-2.59808 - 2.50000i) q^{13} +(3.67992 + 1.56787i) q^{16} +(0.120150 + 0.0321942i) q^{19} +(-0.989488 - 3.17538i) q^{21} +(-1.19658 + 4.85471i) q^{25} +(5.04516 - 1.24352i) q^{27} +(-0.0773365 - 3.83973i) q^{28} +(-3.88527 + 6.42702i) q^{31} +(5.99513 + 0.241596i) q^{36} +(-6.55717 + 3.61171i) q^{37} +(-4.83384 - 3.95399i) q^{39} +(-4.32977 + 1.25413i) q^{43} +(6.57165 + 2.19394i) q^{48} +(2.99280 - 1.42010i) q^{49} +(-4.09101 - 5.93832i) q^{52} +(0.211919 + 0.0388355i) q^{57} +(0.0861007 + 2.13657i) q^{61} +(-2.15084 - 5.34417i) q^{63} +(6.58387 + 4.54452i) q^{64} +(-2.62057 - 1.73199i) q^{67} +(3.74328 - 8.31724i) q^{73} +(-1.38920 + 8.54811i) q^{75} +(0.222565 + 0.111153i) q^{76} +(2.85285 - 2.52741i) q^{79} +(8.53683 - 2.85001i) q^{81} +(-0.668667 - 6.61826i) q^{84} +(-3.58717 + 5.92182i) q^{91} +(-5.81192 + 11.6373i) q^{93} +(2.38963 - 16.8390i) q^{97} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/507\mathbb{Z}\right)^\times$.

$n$	$170$	$340$
$\chi(n)$	$-1$	$e\left(\frac{5}{156}\right)$

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		−0.994935	−	0.100522i	$-0.967949\pi$
$2$		0			0		0.994935	+	0.100522i	$0.0320513\pi$
$3$	1.72643	−	0.139372i	0.996757	−	0.0804666i
$3$	1.72643	−	0.139372i	0.996757	−	0.0804666i
$4$	1.95958	+	0.400051i	0.979791	+	0.200026i
$5$		0			0		−0.616719	−	0.787183i	$-0.711538\pi$
$5$		0			0		0.616719	+	0.787183i	$0.288462\pi$
$6$		0			0
$7$	−0.421909	−	1.87333i	−0.159467	−	0.708052i	−0.988794	−	0.149289i	$-0.952302\pi$
$7$	−0.421909	−	1.87333i	−0.159467	−	0.708052i	0.829327	−	0.558763i	$-0.188724\pi$
$8$		0			0
$9$	2.96115	−	0.481234i	0.987050	−	0.160411i
$10$		0			0
$11$		0			0		−0.584522	−	0.811378i	$-0.698718\pi$
$11$		0			0		0.584522	+	0.811378i	$0.301282\pi$
$12$	3.43884	+	0.417551i	0.992709	+	0.120537i
$13$	−2.59808	−	2.50000i	−0.720577	−	0.693375i
$13$	−2.59808	−	2.50000i	−0.720577	−	0.693375i
$14$		0			0
$15$		0			0
$16$	3.67992	+	1.56787i	0.919979	+	0.391967i
$17$		0			0		0.534466	−	0.845190i	$-0.320513\pi$
$17$		0			0		−0.534466	+	0.845190i	$0.679487\pi$
$18$		0			0
$19$	0.120150	+	0.0321942i	0.0275644	+	0.00738585i	0.272575	−	0.962135i	$-0.412125\pi$
$19$	0.120150	+	0.0321942i	0.0275644	+	0.00738585i	−0.245011	+	0.969520i	$0.578791\pi$
$20$		0			0
$21$	−0.989488	−	3.17538i	−0.215924	−	0.692924i
$22$		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
$25$	−1.19658	+	4.85471i	−0.239316	+	0.970942i
$26$		0			0
$27$	5.04516	−	1.24352i	0.970942	−	0.239316i
$28$	−0.0773365	−	3.83973i	−0.0146152	−	0.725640i
$29$		0			0		−0.632445	−	0.774605i	$-0.717949\pi$
$29$		0			0		0.632445	+	0.774605i	$0.282051\pi$
$30$		0			0
$31$	−3.88527	+	6.42702i	−0.697815	+	1.15433i	0.282950	+	0.959135i	$0.408687\pi$
$31$	−3.88527	+	6.42702i	−0.697815	+	1.15433i	−0.980765	+	0.195192i	$0.937467\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	5.99513	+	0.241596i	0.999189	+	0.0402659i
$37$	−6.55717	+	3.61171i	−1.07799	+	0.593762i	−0.919297	−	0.393566i	$-0.871241\pi$
$37$	−6.55717	+	3.61171i	−1.07799	+	0.593762i	−0.158695	+	0.987328i	$0.550729\pi$
$38$		0			0
$39$	−4.83384	−	3.95399i	−0.774034	−	0.633145i
$40$		0			0
$41$		0			0		0.647915	−	0.761712i	$-0.275641\pi$
$41$		0			0		−0.647915	+	0.761712i	$0.724359\pi$
$42$		0			0
$43$	−4.32977	+	1.25413i	−0.660284	+	0.191254i	−0.591409	−	0.806372i	$-0.701428\pi$
$43$	−4.32977	+	1.25413i	−0.660284	+	0.191254i	−0.0688750	+	0.997625i	$0.521941\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		−0.998176	−	0.0603785i	$-0.980769\pi$
$47$		0			0		0.998176	+	0.0603785i	$0.0192308\pi$
$48$	6.57165	+	2.19394i	0.948536	+	0.316668i
$49$	2.99280	−	1.42010i	0.427542	−	0.202871i
$50$		0			0
$51$		0			0
$52$	−4.09101	−	5.93832i	−0.567321	−	0.823496i
$53$		0			0		0.885456	−	0.464723i	$-0.153846\pi$
$53$		0			0		−0.885456	+	0.464723i	$0.846154\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	0.211919	+	0.0388355i	0.0280693	+	0.00514389i
$58$		0			0
$59$		0			0		−0.927686	−	0.373361i	$-0.878205\pi$
$59$		0			0		0.927686	+	0.373361i	$0.121795\pi$
$60$		0			0
$61$	0.0861007	+	2.13657i	0.0110241	+	0.273559i	0.995295	+	0.0968928i	$0.0308904\pi$
$61$	0.0861007	+	2.13657i	0.0110241	+	0.273559i	−0.984271	+	0.176667i	$0.943469\pi$
$62$		0			0
$63$	−2.15084	−	5.34417i	−0.270981	−	0.673303i
$64$	6.58387	+	4.54452i	0.822984	+	0.568065i
$65$		0			0
$66$		0			0
$67$	−2.62057	−	1.73199i	−0.320154	−	0.211596i	0.381055	−	0.924552i	$-0.375561\pi$
$67$	−2.62057	−	1.73199i	−0.320154	−	0.211596i	−0.701208	+	0.712956i	$0.747356\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		−0.335705	−	0.941967i	$-0.608974\pi$
$71$		0			0		0.335705	+	0.941967i	$0.391026\pi$
$72$		0			0
$73$	3.74328	−	8.31724i	0.438118	−	0.973459i	−0.552080	−	0.833791i	$-0.686166\pi$
$73$	3.74328	−	8.31724i	0.438118	−	0.973459i	0.990198	−	0.139668i	$-0.0446035\pi$
$74$		0			0
$75$	−1.38920	+	8.54811i	−0.160411	+	0.987050i
$76$	0.222565	+	0.111153i	0.0255299	+	0.0127502i
$77$		0			0
$78$		0			0
$79$	2.85285	−	2.52741i	0.320971	−	0.284355i	−0.487216	−	0.873282i	$-0.661988\pi$
$79$	2.85285	−	2.52741i	0.320971	−	0.284355i	0.808187	+	0.588926i	$0.200449\pi$
$80$		0			0
$81$	8.53683	−	2.85001i	0.948536	−	0.316668i
$82$		0			0
$83$		0			0		−0.180255	−	0.983620i	$-0.557692\pi$
$83$		0			0		0.180255	+	0.983620i	$0.442308\pi$
$84$	−0.668667	−	6.61826i	−0.0729576	−	0.722111i
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$89$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$90$		0			0
$91$	−3.58717	+	5.92182i	−0.376038	+	0.620776i
$92$		0			0
$93$	−5.81192	+	11.6373i	−0.602668	+	1.20673i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	2.38963	−	16.8390i	0.242630	−	1.70974i	−0.378138	−	0.925749i	$-0.623436\pi$
$97$	2.38963	−	16.8390i	0.242630	−	1.70974i	0.620768	−	0.783994i	$-0.286821\pi$
$98$		0			0
$99$		0			0
$100$	−4.28693	+	9.03450i	−0.428693	+	0.903450i
$101$		0			0		−0.721202	−	0.692724i	$-0.756410\pi$
$101$		0			0		0.721202	+	0.692724i	$0.243590\pi$
$102$		0			0
$103$	−9.07441	+	6.26362i	−0.894128	+	0.617172i	−0.924005	−	0.382381i	$-0.875104\pi$
$103$	−9.07441	+	6.26362i	−0.894128	+	0.617172i	0.0298761	+	0.999554i	$0.490489\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		−0.799443	−	0.600742i	$-0.794872\pi$
$107$		0			0		0.799443	+	0.600742i	$0.205128\pi$
$108$	10.3839	−	0.418456i	0.999189	−	0.0402659i
$109$	−17.3702	+	10.5006i	−1.66376	+	1.00578i	−0.710466	+	0.703731i	$0.751516\pi$
$109$	−17.3702	+	10.5006i	−1.66376	+	1.00578i	−0.953293	+	0.302046i	$0.902330\pi$
$110$		0			0
$111$	−10.8171	+	7.14927i	−1.02672	+	0.678579i
$112$	1.38454	−	7.55519i	0.130827	−	0.713899i
$113$		0			0		−0.996757	−	0.0804666i	$-0.974359\pi$
$113$		0			0		0.996757	+	0.0804666i	$0.0256410\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−8.89638	−	6.15259i	−0.822471	−	0.568808i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−3.48335	+	10.4339i	−0.316668	+	0.948536i
$122$		0			0
$123$		0			0
$124$	−10.1846	+	11.0400i	−0.914608	+	0.991418i
$125$		0			0
$126$		0			0
$127$	−21.7827	+	4.44696i	−1.93290	+	0.394604i	−0.933553	+	0.358441i	$0.883309\pi$
$127$	−21.7827	+	4.44696i	−1.93290	+	0.394604i	−0.999346	+	0.0361634i	$0.988486\pi$
$128$		0			0
$129$	−7.30028	+	2.76863i	−0.642753	+	0.243764i
$130$		0			0
$131$		0			0		0.354605	−	0.935016i	$-0.384615\pi$
$131$		0			0		−0.354605	+	0.935016i	$0.615385\pi$
$132$		0			0
$133$	0.00961783	−	0.238664i	0.000833972	−	0.0206948i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		0.482459	−	0.875918i	$-0.339744\pi$
$137$		0			0		−0.482459	+	0.875918i	$0.660256\pi$
$138$		0			0
$139$	9.74136	−	7.32015i	0.826251	−	0.620887i	−0.101186	−	0.994868i	$-0.532264\pi$
$139$	9.74136	−	7.32015i	0.826251	−	0.620887i	0.927437	+	0.373980i	$0.122007\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	11.6513	+	2.87179i	0.970942	+	0.239316i
$145$		0			0
$146$		0			0
$147$	4.96895	−	2.86882i	0.409832	−	0.236616i
$148$	−14.2942	+	4.45424i	−1.17497	+	0.366137i
$149$		0			0		0.894635	−	0.446798i	$-0.147436\pi$
$149$		0			0		−0.894635	+	0.446798i	$0.852564\pi$
$150$		0			0
$151$	−1.37789	−	22.7793i	−0.112131	−	1.85375i	−0.426543	−	0.904467i	$-0.640269\pi$
$151$	−1.37789	−	22.7793i	−0.112131	−	1.85375i	0.314412	−	0.949287i	$-0.398193\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	−7.89050	−	9.68194i	−0.631746	−	0.775176i
$157$	2.79222	−	22.9960i	0.222844	−	1.83528i	−0.267267	−	0.963623i	$-0.586121\pi$
$157$	2.79222	−	22.9960i	0.222844	−	1.83528i	0.490111	−	0.871660i	$-0.336956\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−0.349986	+	17.3767i	−0.0274130	+	1.36105i	0.718743	+	0.695276i	$0.244718\pi$
$163$	−0.349986	+	17.3767i	−0.0274130	+	1.36105i	−0.746156	+	0.665771i	$0.768103\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.100522	−	0.994935i	$-0.467949\pi$
$167$		0			0		−0.100522	+	0.994935i	$0.532051\pi$
$168$		0			0
$169$	0.500000	+	12.9904i	0.0384615	+	0.999260i
$170$		0			0
$171$	0.371276	+	0.0375114i	0.0283922	+	0.00286857i
$172$	−8.98626	+	0.725446i	−0.685196	+	0.0553147i
$173$		0			0		−0.979791	−	0.200026i	$-0.935897\pi$
$173$		0			0		0.979791	+	0.200026i	$0.0641026\pi$
$174$		0			0
$175$	9.59932	+	0.193341i	0.725640	+	0.0146152i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		−0.534466	−	0.845190i	$-0.679487\pi$
$179$		0			0		0.534466	+	0.845190i	$0.320513\pi$
$180$		0			0
$181$	−0.500995	−	0.0608318i	−0.0372387	−	0.00452160i	0.101896	−	0.994795i	$-0.467509\pi$
$181$	−0.500995	−	0.0608318i	−0.0372387	−	0.00452160i	−0.139135	+	0.990273i	$0.544432\pi$
$182$		0			0
$183$	0.446425	+	3.67664i	0.0330007	+	0.271785i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	−4.45812	−	8.92660i	−0.324281	−	0.649314i
$190$		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
$193$	21.1943	+	4.77336i	1.52560	+	0.343594i	0.899770	−	0.436365i	$-0.143734\pi$
$193$	21.1943	+	4.77336i	1.52560	+	0.343594i	0.625831	+	0.779959i	$0.284760\pi$
$194$		0			0
$195$		0			0
$196$	6.43274	−	1.58553i	0.459482	−	0.113252i
$197$		0			0		−0.0201371	−	0.999797i	$-0.506410\pi$
$197$		0			0		0.0201371	+	0.999797i	$0.493590\pi$
$198$		0			0
$199$	16.2691	+	21.6503i	1.15329	+	1.53475i	0.800800	+	0.598932i	$0.204408\pi$
$199$	16.2691	+	21.6503i	1.15329	+	1.53475i	0.352489	+	0.935816i	$0.385335\pi$
$200$		0			0
$201$	−4.76563	−	2.62493i	−0.336142	−	0.185148i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	−5.64104	−	13.2732i	−0.391136	−	0.920333i
$209$		0			0
$210$		0			0
$211$	−4.42351	−	21.6678i	−0.304527	−	1.49167i	−0.787652	−	0.616120i	$-0.788704\pi$
$211$	−4.42351	−	21.6678i	−0.304527	−	1.49167i	0.483125	−	0.875551i	$-0.339502\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	13.6792	+	4.56677i	0.928602	+	0.310013i
$218$		0			0
$219$	5.30334	−	14.8809i	0.358367	−	1.00556i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	12.3270	−	1.74933i	0.825478	−	0.117144i	0.285027	−	0.958519i	$-0.407997\pi$
$223$	12.3270	−	1.74933i	0.825478	−	0.117144i	0.540451	+	0.841376i	$0.318254\pi$
$224$		0			0
$225$	−1.20700	+	14.9514i	−0.0804666	+	0.996757i
$226$		0			0
$227$		0			0		−0.551377	−	0.834256i	$-0.685897\pi$
$227$		0			0		0.551377	+	0.834256i	$0.314103\pi$
$228$	0.399735	+	0.160880i	0.0264731	+	0.0106545i
$229$	−2.32790	−	3.85082i	−0.153832	−	0.254470i	0.769354	−	0.638823i	$-0.220578\pi$
$229$	−2.32790	−	3.85082i	−0.153832	−	0.254470i	−0.923186	+	0.384353i	$0.874424\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		−0.822984	−	0.568065i	$-0.807692\pi$
$233$		0			0		0.822984	+	0.568065i	$0.192308\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	4.57301	−	4.76101i	0.297049	−	0.309261i
$238$		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
$241$	26.5509	+	3.76785i	1.71030	+	0.242708i	0.925958	−	0.377625i	$-0.123259\pi$
$241$	26.5509	+	3.76785i	1.71030	+	0.242708i	0.784338	+	0.620334i	$0.213003\pi$
$242$		0			0
$243$	14.3411	−	6.11015i	0.919979	−	0.391967i
$244$	−0.686015	+	4.22122i	−0.0439176	+	0.270236i
$245$		0			0
$246$		0			0
$247$	−0.231674	−	0.384019i	−0.0147411	−	0.0244345i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		0.721202	−	0.692724i	$-0.243590\pi$
$251$		0			0		−0.721202	+	0.692724i	$0.756410\pi$
$252$	−2.07681	−	11.3328i	−0.130827	−	0.713899i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	11.0836	+	11.5392i	0.692724	+	0.721202i
$257$		0			0		−0.316668	−	0.948536i	$-0.602564\pi$
$257$		0			0		0.316668	+	0.948536i	$0.397436\pi$
$258$		0			0
$259$	9.53246	+	10.7599i	0.592318	+	0.668589i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		−0.987050	−	0.160411i	$-0.948718\pi$
$263$		0			0		0.987050	+	0.160411i	$0.0512821\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	−4.44234	−	4.44234i	−0.271359	−	0.271359i
$269$		0			0		0.428693	−	0.903450i	$-0.358974\pi$
$269$		0			0		−0.428693	+	0.903450i	$0.641026\pi$
$270$		0			0
$271$	7.61399	−	11.5203i	0.462517	−	0.699807i	−0.526073	−	0.850439i	$-0.676336\pi$
$271$	7.61399	−	11.5203i	0.462517	−	0.699807i	0.988590	+	0.150632i	$0.0481310\pi$
$272$		0			0
$273$	−5.36768	+	10.7236i	−0.324867	+	0.649021i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	33.1777	−	1.33701i	1.99345	−	0.0803334i	0.993901	−	0.110280i	$-0.0351746\pi$
$277$	33.1777	−	1.33701i	1.99345	−	0.0803334i	0.999552	+	0.0299462i	$0.00953358\pi$
$278$		0			0
$279$	−8.41197	+	20.9011i	−0.503611	+	1.25132i
$280$		0			0
$281$		0			0		0.180255	−	0.983620i	$-0.442308\pi$
$281$		0			0		−0.180255	+	0.983620i	$0.557692\pi$
$282$		0			0
$283$	−24.3644	−	7.05724i	−1.44831	−	0.419509i	−0.541379	−	0.840779i	$-0.682097\pi$
$283$	−24.3644	−	7.05724i	−1.44831	−	0.419509i	−0.906935	+	0.421270i	$0.861585\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−7.28777	−	15.3587i	−0.428693	−	0.903450i
$290$		0			0
$291$	1.77864	−	29.4045i	0.104266	−	1.72372i
$292$	10.6626	−	14.8008i	0.623981	−	0.866151i
$293$		0			0		0.678061	−	0.735006i	$-0.262821\pi$
$293$		0			0		−0.678061	+	0.735006i	$0.737179\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$	−6.14194	+	16.1950i	−0.354605	+	0.935016i
$301$	4.17617	+	7.58196i	0.240711	+	0.437017i
$302$		0			0
$303$		0			0
$304$	0.391667	+	0.306851i	0.0224636	+	0.0175991i
$305$		0			0
$306$		0			0
$307$	−7.49577	−	4.53135i	−0.427806	−	0.258618i	0.287014	−	0.957927i	$-0.407338\pi$
$307$	−7.49577	−	4.53135i	−0.427806	−	0.258618i	−0.714820	+	0.699309i	$0.753491\pi$
$308$		0			0
$309$	−14.7934	+	12.0784i	−0.841567	+	0.687118i
$310$		0			0
$311$		0			0		−0.239316	−	0.970942i	$-0.576923\pi$
$311$		0			0		0.239316	+	0.970942i	$0.423077\pi$
$312$		0			0
$313$	28.5391	+	7.03426i	1.61313	+	0.397600i	0.939794	−	0.341741i	$-0.111017\pi$
$313$	28.5391	+	7.03426i	1.61313	+	0.397600i	0.673331	+	0.739341i	$0.264863\pi$
$314$		0			0
$315$		0			0
$316$	6.60149	−	3.81137i	0.371363	−	0.214406i
$317$		0			0		0.954721	−	0.297503i	$-0.0961538\pi$
$317$		0			0		−0.954721	+	0.297503i	$0.903846\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	17.8688	−	2.16966i	0.992709	−	0.120537i
$325$	15.2456	−	9.62146i	0.845672	−	0.533702i
$326$		0			0
$327$	−28.5249	+	20.5496i	−1.57743	+	1.13639i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	35.4967	−	7.99452i	1.95108	−	0.439419i	0.973923	−	0.226877i	$-0.0728516\pi$
$331$	35.4967	−	7.99452i	1.95108	−	0.439419i	0.977152	−	0.212541i	$-0.0681740\pi$
$332$		0			0
$333$	−17.6787	+	13.8504i	−0.968786	+	0.758995i
$334$		0			0
$335$		0			0
$336$	1.33733	−	13.2365i	0.0729576	−	0.722111i
$337$			15.2142i			0.828771i	0.910101	+	0.414385i	$0.136003\pi$
$337$			15.2142i			0.828771i	−0.910101	+	0.414385i	$0.863997\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−12.2128	−	15.5885i	−0.659429	−	0.841699i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		0.987050	−	0.160411i	$-0.0512821\pi$
$347$		0			0		−0.987050	+	0.160411i	$0.948718\pi$
$348$		0			0
$349$	2.66379	+	3.69762i	0.142589	+	0.197929i	0.876614	−	0.481193i	$-0.159797\pi$
$349$	2.66379	+	3.69762i	0.142589	+	0.197929i	−0.734025	+	0.679122i	$0.762361\pi$
$350$		0			0
$351$	−16.2165	−	9.38214i	−0.865574	−	0.500782i
$352$		0			0
$353$		0			0		−0.678061	−	0.735006i	$-0.737179\pi$
$353$		0			0		0.678061	+	0.735006i	$0.262821\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		−0.297503	−	0.954721i	$-0.596154\pi$
$359$		0			0		0.297503	+	0.954721i	$0.403846\pi$
$360$		0			0
$361$	−16.4411	−	9.49226i	−0.865320	−	0.499593i
$362$		0			0
$363$	−4.55958	+	18.4989i	−0.239316	+	0.970942i
$364$	−9.39839	+	10.1692i	−0.492609	+	0.533013i
$365$		0			0
$366$		0			0
$367$	−22.8571	−	27.9949i	−1.19313	−	1.46132i	−0.853758	−	0.520669i	$-0.825682\pi$
$367$	−22.8571	−	27.9949i	−1.19313	−	1.46132i	−0.339373	−	0.940652i	$-0.610215\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	−16.0445	+	20.4792i	−0.831866	+	1.06180i
$373$	22.6708	−	27.7667i	1.17385	−	1.43770i	0.298367	−	0.954451i	$-0.403558\pi$
$373$	22.6708	−	27.7667i	1.17385	−	1.43770i	0.875481	−	0.483252i	$-0.160545\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−25.0849	+	29.4907i	−1.28852	+	1.51483i	−0.563484	+	0.826127i	$0.690539\pi$
$379$	−25.0849	+	29.4907i	−1.28852	+	1.51483i	−0.725040	+	0.688707i	$0.758179\pi$
$380$		0			0
$381$	−36.9865	+	10.7133i	−1.89488	+	0.548858i
$382$		0			0
$383$		0			0		−0.735006	−	0.678061i	$-0.762821\pi$
$383$		0			0		0.735006	+	0.678061i	$0.237179\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−12.2176	+	5.79731i	−0.621054	+	0.294694i
$388$	11.4191	−	32.0414i	0.579719	−	1.62666i
$389$		0			0		0.464723	−	0.885456i	$-0.346154\pi$
$389$		0			0		−0.464723	+	0.885456i	$0.653846\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	36.9450	+	14.8691i	1.85421	+	0.746257i	0.940291	+	0.340371i	$0.110553\pi$
$397$	36.9450	+	14.8691i	1.85421	+	0.746257i	0.913924	+	0.405886i	$0.133037\pi$
$398$		0			0
$399$	−0.0166586	−	0.413378i	−0.000833972	−	0.0206948i
$400$	−12.0148	+	15.9889i	−0.600742	+	0.799443i
$401$		0			0		−0.373361	−	0.927686i	$-0.621795\pi$
$401$		0			0		0.373361	+	0.927686i	$0.378205\pi$
$402$		0			0
$403$	26.1618	−	6.98472i	1.30321	−	0.347934i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	13.0696	+	36.6724i	0.646248	+	1.81333i	0.578578	+	0.815627i	$0.303608\pi$
$409$	13.0696	+	36.6724i	0.646248	+	1.81333i	0.0676696	+	0.997708i	$0.478444\pi$
$410$		0			0
$411$		0			0
$412$	−20.2878	+	8.64383i	−0.999509	+	0.425851i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	15.7976	−	13.9954i	0.773611	−	0.685360i
$418$		0			0
$419$		0			0		0.948536	−	0.316668i	$-0.102564\pi$
$419$		0			0		−0.948536	+	0.316668i	$0.897436\pi$
$420$		0			0
$421$	−3.04895	−	16.6376i	−0.148596	−	0.810865i	−0.970475	−	0.241202i	$-0.922458\pi$
$421$	−3.04895	−	16.6376i	−0.148596	−	0.810865i	0.821878	−	0.569663i	$-0.192926\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	3.96617	−	1.06273i	0.191936	−	0.0514292i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.446798	−	0.894635i	$-0.352564\pi$
$431$		0			0		−0.446798	+	0.894635i	$0.647436\pi$
$432$	20.5155	+	3.33409i	0.987050	+	0.160411i
$433$	−3.38748	−	7.95071i	−0.162792	−	0.382087i	0.818529	−	0.574466i	$-0.194790\pi$
$433$	−3.38748	−	7.95071i	−0.162792	−	0.382087i	−0.981321	+	0.192379i	$0.938380\pi$
$434$		0			0
$435$		0			0
$436$	−38.2390	+	13.6279i	−1.83132	+	0.652657i
$437$		0			0
$438$		0			0
$439$	−18.9544	−	18.2059i	−0.904644	−	0.868923i	0.0872244	−	0.996189i	$-0.472200\pi$
$439$	−18.9544	−	18.2059i	−0.904644	−	0.868923i	−0.991869	+	0.127266i	$0.959380\pi$
$440$		0			0
$441$	8.17872	−	5.64537i	0.389463	−	0.268827i
$442$		0			0
$443$		0			0		0.568065	−	0.822984i	$-0.307692\pi$
$443$		0			0		−0.568065	+	0.822984i	$0.692308\pi$
$444$	−24.0571	+	9.68217i	−1.14170	+	0.459495i
$445$		0			0
$446$		0			0
$447$		0			0
$448$	5.73559	−	14.2511i	0.270981	−	0.673303i
$449$		0			0		0.834256	−	0.551377i	$-0.185897\pi$
$449$		0			0		−0.834256	+	0.551377i	$0.814103\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	−5.55365	−	39.1349i	−0.260933	−	1.83872i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	23.3582	+	8.32453i	1.09265	+	0.389405i	0.819799	−	0.572651i	$-0.194085\pi$
$457$	23.3582	+	8.32453i	1.09265	+	0.389405i	0.272850	+	0.962057i	$0.412034\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		0.584522	−	0.811378i	$-0.301282\pi$
$461$		0			0		−0.584522	+	0.811378i	$0.698718\pi$
$462$		0			0
$463$	−35.1089	+	15.8012i	−1.63165	+	0.734346i	−0.999190	−	0.0402476i	$-0.987185\pi$
$463$	−35.1089	+	15.8012i	−1.63165	+	0.734346i	−0.632460	+	0.774593i	$0.717955\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		0.935016	−	0.354605i	$-0.115385\pi$
$467$		0			0		−0.935016	+	0.354605i	$0.884615\pi$
$468$	−14.9718	−	15.6155i	−0.692073	−	0.721828i
$469$	−2.13894	+	5.63993i	−0.0987673	+	0.260428i
$470$		0			0
$471$	1.61558	−	40.0903i	0.0744422	−	1.84726i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−0.300063	+	0.544772i	−0.0137678	+	0.0249958i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.999797	−	0.0201371i	$-0.00641026\pi$
$479$		0			0		−0.999797	+	0.0201371i	$0.993590\pi$
$480$		0			0
$481$	26.0653	+	7.00941i	1.18848	+	0.319601i
$482$		0			0
$483$		0			0
$484$	−11.0000	+	19.0526i	−0.500000	+	0.866025i
$485$		0			0
$486$		0			0
$487$	0.716717	−	0.357943i	0.0324775	−	0.0162199i	−0.430486	−	0.902597i	$-0.641658\pi$
$487$	0.716717	−	0.357943i	0.0324775	−	0.0162199i	0.462964	+	0.886377i	$0.346786\pi$
$488$		0			0
$489$	1.81760	+	30.0485i	0.0821947	+	1.35884i
$490$		0			0
$491$		0			0		0.391967	−	0.919979i	$-0.371795\pi$
$491$		0			0		−0.391967	+	0.919979i	$0.628205\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	−24.3742	+	17.5593i	−1.09443	+	0.788437i
$497$		0			0
$498$		0			0
$499$	−13.1725	+	42.2721i	−0.589683	+	1.89236i	−0.167426	+	0.985885i	$0.553546\pi$
$499$	−13.1725	+	42.2721i	−0.589683	+	1.89236i	−0.422257	+	0.906476i	$0.638762\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		0.200026	−	0.979791i	$-0.435897\pi$
$503$		0			0		−0.200026	+	0.979791i	$0.564103\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	2.67371	+	22.3574i	0.118744	+	0.992925i
$508$	−44.4639			−1.97277
$509$		0			0		−0.994935	−	0.100522i	$-0.967949\pi$
$509$		0			0		0.994935	+	0.100522i	$0.0320513\pi$
$510$		0			0
$511$	−17.1602	−	3.50329i	−0.759125	−	0.154976i
$512$		0			0
$513$	0.646212	+	0.0130154i	0.0285309	+	0.000574646i
$514$		0			0
$515$		0			0
$516$	−15.4131	+	2.50487i	−0.678523	+	0.110271i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		−0.120537	−	0.992709i	$-0.538462\pi$
$521$		0			0		0.120537	+	0.992709i	$0.461538\pi$
$522$		0			0
$523$	−41.3473	−	17.6164i	−1.80799	−	0.770313i	−0.976743	−	0.214411i	$-0.931217\pi$
$523$	−41.3473	−	17.6164i	−1.80799	−	0.770313i	−0.831248	−	0.555902i	$-0.812373\pi$
$524$		0			0
$525$	16.5995	−	1.00409i	0.724463	−	0.0438219i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	11.5000	+	19.9186i	0.500000	+	0.866025i
$530$		0			0
$531$		0			0
$532$	0.114325	−	0.463834i	0.00495661	−	0.0201098i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	17.4295	−	22.2471i	0.749353	−	0.956478i	−0.250579	−	0.968096i	$-0.580621\pi$
$541$	17.4295	−	22.2471i	0.749353	−	0.956478i	0.999933	+	0.0116175i	$0.00369806\pi$
$542$		0			0
$543$	−0.873414	−	0.0351974i	−0.0374818	−	0.00151046i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	5.58773	+	14.7336i	0.238914	+	0.629965i	0.999821	−	0.0188956i	$-0.00601502\pi$
$547$	5.58773	+	14.7336i	0.238914	+	0.629965i	−0.760907	+	0.648861i	$0.775246\pi$
$548$		0			0
$549$	1.28315	+	6.28526i	0.0547633	+	0.268248i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−5.93831	−	4.27800i	−0.252523	−	0.181919i
$554$		0			0
$555$		0			0
$556$	22.0174	−	10.4474i	0.933746	−	0.443068i
$557$		0			0		0.335705	−	0.941967i	$-0.391026\pi$
$557$		0			0		−0.335705	+	0.941967i	$0.608974\pi$
$558$		0			0
$559$	14.3844	+	7.56610i	0.608396	+	0.320012i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		0.0804666	−	0.996757i	$-0.474359\pi$
$563$		0			0		−0.0804666	+	0.996757i	$0.525641\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−8.94077	−	14.7898i	−0.375477	−	0.621115i
$568$		0			0
$569$		0			0		0.600742	−	0.799443i	$-0.294872\pi$
$569$		0			0		−0.600742	+	0.799443i	$0.705128\pi$
$570$		0			0
$571$	−23.0210	−	15.8903i	−0.963401	−	0.664988i	−0.0211080	−	0.999777i	$-0.506719\pi$
$571$	−23.0210	−	15.8903i	−0.963401	−	0.664988i	−0.942293	+	0.334790i	$0.891335\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	21.6828	+	10.2886i	0.903450	+	0.428693i
$577$	8.44249	−	8.44249i	0.351466	−	0.351466i	−0.509189	−	0.860655i	$-0.670055\pi$
$577$	8.44249	−	8.44249i	0.351466	−	0.351466i	0.860655	+	0.509189i	$0.170055\pi$
$578$		0			0
$579$	37.2559	+	5.28699i	1.54830	+	0.219720i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$587$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$588$	10.8847	−	3.63386i	0.448879	−	0.149858i
$589$	−0.673729	+	0.647125i	−0.0277605	+	0.0266643i
$590$		0			0
$591$		0			0
$592$	−29.7925	+	3.01005i	−1.22446	+	0.123712i
$593$		0			0		0.983620	−	0.180255i	$-0.0576923\pi$
$593$		0			0		−0.983620	+	0.180255i	$0.942308\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	31.1050	+	35.1103i	1.27304	+	1.43697i
$598$		0			0
$599$		0			0		−0.748511	−	0.663123i	$-0.769231\pi$
$599$		0			0		0.748511	+	0.663123i	$0.230769\pi$
$600$		0			0
$601$	48.2574	+	7.84259i	1.96846	+	0.319906i	0.996347	+	0.0853931i	$0.0272146\pi$
$601$	48.2574	+	7.84259i	1.96846	+	0.319906i	0.972113	+	0.234513i	$0.0753495\pi$
$602$		0			0
$603$	−8.59340	−	3.86757i	−0.349950	−	0.157500i
$604$	6.41280	−	45.1891i	0.260933	−	1.83872i
$605$		0			0
$606$		0			0
$607$	−16.5635	+	34.9067i	−0.672290	+	1.41682i	0.224548	+	0.974463i	$0.427909\pi$
$607$	−16.5635	+	34.9067i	−0.672290	+	1.41682i	−0.896838	+	0.442358i	$0.854142\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	41.8527	−	16.8443i	1.69042	−	0.680334i	0.691308	−	0.722560i	$-0.257035\pi$
$613$	41.8527	−	16.8443i	1.69042	−	0.680334i	0.999108	+	0.0422267i	$0.0134452\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		0.373361	−	0.927686i	$-0.378205\pi$
$617$		0			0		−0.373361	+	0.927686i	$0.621795\pi$
$618$		0			0
$619$	5.43344	−	29.6493i	0.218389	−	1.19171i	−0.673408	−	0.739271i	$-0.735170\pi$
$619$	5.43344	−	29.6493i	0.218389	−	1.19171i	0.891796	−	0.452437i	$-0.149445\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$	−11.5888	−	22.1292i	−0.463924	−	0.885875i
$625$	−22.1364	−	11.6181i	−0.885456	−	0.464723i
$626$		0			0
$627$		0			0
$628$	14.6712	−	43.9456i	0.585444	−	1.75362i
$629$		0			0
$630$		0			0
$631$	−32.3148	+	35.0286i	−1.28643	+	1.39447i	−0.410625	+	0.911804i	$0.634689\pi$
$631$	−32.3148	+	35.0286i	−1.28643	+	1.39447i	−0.875806	+	0.482663i	$0.839670\pi$
$632$		0			0
$633$	−10.6568	−	36.7915i	−0.423569	−	1.46233i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−11.3258	−	3.79247i	−0.448743	−	0.150263i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		−0.774605	−	0.632445i	$-0.782051\pi$
$641$		0			0		0.774605	+	0.632445i	$0.217949\pi$
$642$		0			0
$643$	−3.20470	+	2.72593i	−0.126381	+	0.107500i	−0.708873	−	0.705337i	$-0.750796\pi$
$643$	−3.20470	+	2.72593i	−0.126381	+	0.107500i	0.582492	+	0.812837i	$0.302078\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.774605	−	0.632445i	$-0.217949\pi$
$647$		0			0		−0.774605	+	0.632445i	$0.782051\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	24.2526	+	5.97774i	0.950536	+	0.234286i
$652$	−7.63740	+	33.9110i	−0.299104	+	1.32806i
$653$		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
$657$	7.08189	−	26.4300i	0.276291	−	1.03113i
$658$		0			0
$659$		0			0		−0.845190	−	0.534466i	$-0.820513\pi$
$659$		0			0		0.845190	+	0.534466i	$0.179487\pi$
$660$		0			0
$661$	−28.1400	+	25.9598i	−1.09452	+	1.00972i	−0.0946192	+	0.995514i	$0.530163\pi$
$661$	−28.1400	+	25.9598i	−1.09452	+	1.00972i	−0.999899	+	0.0142068i	$0.995478\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	21.0380	−	4.73814i	0.813375	−	0.183187i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−0.455152	−	5.63806i	−0.0175448	−	0.217331i	−0.999589	−	0.0286787i	$-0.990870\pi$
$673$	−0.455152	−	5.63806i	−0.0175448	−	0.217331i	0.982044	−	0.188653i	$-0.0604120\pi$
$674$		0			0
$675$			25.9808i			1.00000i
$676$	−4.21703	+	25.6557i	−0.162193	+	0.986759i
$677$		0			0		1.00000			$0$
$677$		0			0		−1.00000			$\pi$
$678$		0			0
$679$	−32.5532	+	2.62797i	−1.24928	+	0.100852i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		−0.219715	−	0.975564i	$-0.570513\pi$
$683$		0			0		0.219715	+	0.975564i	$0.429487\pi$
$684$	0.712539	+	0.222036i	0.0272446	+	0.00848976i
$685$		0			0
$686$		0			0
$687$	−4.55567	−	6.32375i	−0.173810	−	0.241266i
$688$	−17.8995	−	2.17340i	−0.682413	−	0.0828599i
$689$		0			0
$690$		0			0
$691$	−26.4025	−	28.6198i	−1.00440	−	1.08875i	−0.996051	−	0.0887833i	$-0.971702\pi$
$691$	−26.4025	−	28.6198i	−1.00440	−	1.08875i	−0.00834661	−	0.999965i	$-0.502657\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	18.7333	+	4.21909i	0.708052	+	0.159467i
$701$		0			0		0.239316	−	0.970942i	$-0.423077\pi$
$701$		0			0		−0.239316	+	0.970942i	$0.576923\pi$
$702$		0			0
$703$	−0.904121	+	0.222846i	−0.0340996	+	0.00840479i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	27.6454	+	32.5009i	1.03825	+	1.22060i	0.974909	+	0.222606i	$0.0714562\pi$
$709$	27.6454	+	32.5009i	1.03825	+	1.22060i	0.0633366	+	0.997992i	$0.479826\pi$
$710$		0			0
$711$	7.23145	−	8.85692i	0.271201	−	0.332160i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		0.960518	−	0.278217i	$-0.0897436\pi$
$719$		0			0		−0.960518	+	0.278217i	$0.910256\pi$
$720$		0			0
$721$	15.5624	+	14.3567i	0.579574	+	0.534671i
$722$		0			0
$723$	46.3636	+	2.80448i	1.72428	+	0.104300i
$724$	−0.957405	−	0.319629i	−0.0355817	−	0.0118789i
$725$		0			0
$726$		0			0
$727$	25.0260	−	47.6830i	0.928162	−	1.76846i	0.417548	−	0.908655i	$-0.362889\pi$
$727$	25.0260	−	47.6830i	0.928162	−	1.76846i	0.510614	−	0.859810i	$-0.329418\pi$
$728$		0			0
$729$	23.9073	−	12.5475i	0.885456	−	0.464723i
$730$		0			0
$731$		0			0
$732$	−0.596039	+	7.38327i	−0.0220303	+	0.272894i
$733$	−22.2859	−	4.08405i	−0.823149	−	0.150848i	−0.247858	−	0.968796i	$-0.579727\pi$
$733$	−22.2859	−	4.08405i	−0.823149	−	0.150848i	−0.575291	+	0.817949i	$0.695111\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	19.7257	+	49.0122i	0.725622	+	1.80294i	0.581785	+	0.813343i	$0.302354\pi$
$739$	19.7257	+	49.0122i	0.725622	+	1.80294i	0.143837	+	0.989601i	$0.454056\pi$
$740$		0			0
$741$	−0.453492	−	0.630694i	−0.0166594	−	0.0231691i
$742$		0			0
$743$		0			0		−0.834256	−	0.551377i	$-0.814103\pi$
$743$		0			0		0.834256	+	0.551377i	$0.185897\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	7.94567	−	48.8917i	0.289942	−	1.78408i	−0.277558	−	0.960709i	$-0.589525\pi$
$751$	7.94567	−	48.8917i	0.289942	−	1.78408i	0.567500	−	0.823373i	$-0.307911\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$	−5.16495	−	19.2759i	−0.187848	−	0.701057i
$757$	−29.2714	+	9.77223i	−1.06389	+	0.355178i	−0.794128	−	0.607750i	$-0.792072\pi$
$757$	−29.2714	+	9.77223i	−1.06389	+	0.355178i	−0.269759	+	0.962928i	$0.586944\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		0.994935	−	0.100522i	$-0.0320513\pi$
$761$		0			0		−0.994935	+	0.100522i	$0.967949\pi$
$762$		0			0
$763$	26.9998	+	28.1097i	0.977457	+	1.01764i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	20.7433	+	18.3770i	0.748511	+	0.663123i
$769$	12.0467	−	24.1213i	0.434414	−	0.869837i	−0.564597	−	0.825367i	$-0.690968\pi$
$769$	12.0467	−	24.1213i	0.434414	−	0.869837i	0.999011	−	0.0444702i	$-0.0141600\pi$
$770$		0			0
$771$		0			0
$772$	39.6224	+	17.8326i	1.42604	+	0.641809i
$773$		0			0		0.140502	−	0.990080i	$-0.455128\pi$
$773$		0			0		−0.140502	+	0.990080i	$0.544872\pi$
$774$		0			0
$775$	−26.5523	−	26.5523i	−0.953786	−	0.953786i
$776$		0			0
$777$	17.9568	+	17.2477i	0.644196	+	0.618759i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	13.2398	−	0.533545i	0.472849	−	0.0190552i
$785$		0			0
$786$		0			0
$787$	−9.42532	+	6.22939i	−0.335976	+	0.222054i	−0.708071	−	0.706141i	$-0.750434\pi$
$787$	−9.42532	+	6.22939i	−0.335976	+	0.222054i	0.372095	+	0.928195i	$0.378640\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	5.11772	−	5.76622i	0.181736	−	0.204764i
$794$		0			0
$795$		0			0
$796$	23.2195	+	48.9340i	0.822992	+	1.73442i
$797$		0			0		0.316668	−	0.948536i	$-0.397436\pi$
$797$		0			0		−0.316668	+	0.948536i	$0.602564\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$	−8.28854	−	7.05026i	−0.292314	−	0.248644i
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		0.0402659	−	0.999189i	$-0.487179\pi$
$809$		0			0		−0.0402659	+	0.999189i	$0.512821\pi$
$810$		0			0
$811$	−30.9856	−	24.2757i	−1.08805	−	0.852434i	−0.0982894	−	0.995158i	$-0.531337\pi$
$811$	−30.9856	−	24.2757i	−1.08805	−	0.852434i	−0.989763	+	0.142724i	$0.954414\pi$
$812$		0			0
$813$	11.5394	−	20.9502i	0.404706	−	0.734755i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−0.560599	+	0.0112911i	−0.0196129	+	0.000395026i
$818$		0			0
$819$	−7.77238	+	19.2617i	−0.271589	+	0.673058i
$820$		0			0
$821$		0			0		0.219715	−	0.975564i	$-0.429487\pi$
$821$		0			0		−0.219715	+	0.975564i	$0.570513\pi$
$822$		0			0
$823$	49.5536	−	28.6098i	1.72733	−	0.997274i	0.826772	−	0.562537i	$-0.190175\pi$
$823$	49.5536	−	28.6098i	1.72733	−	0.997274i	0.900557	−	0.434737i	$-0.143159\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		−0.0603785	−	0.998176i	$-0.519231\pi$
$827$		0			0		0.0603785	+	0.998176i	$0.480769\pi$
$828$		0			0
$829$	−6.25737	+	14.6866i	−0.217328	+	0.510086i	−0.992584	−	0.121560i	$-0.961210\pi$
$829$	−6.25737	+	14.6866i	−0.217328	+	0.510086i	0.775257	+	0.631646i	$0.217621\pi$
$830$		0			0
$831$	57.0927	−	6.93231i	1.98052	−	0.240479i
$832$	−5.74410	−	28.2667i	−0.199141	−	0.979971i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−11.6097	+	37.2568i	−0.401289	+	1.28778i
$838$		0			0
$839$		0			0		0.0201371	−	0.999797i	$-0.493590\pi$
$839$		0			0		−0.0201371	+	0.999797i	$0.506410\pi$
$840$		0			0
$841$	−5.80075	+	28.4139i	−0.200026	+	0.979791i
$842$		0			0
$843$		0			0
$844$		−	44.2294i		−	1.52244i
$845$		0			0
$846$		0			0
$847$	21.0158	+	2.12330i	0.722111	+	0.0729576i
$848$		0			0
$849$	−43.0471	−	8.78814i	−1.47737	−	0.301608i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	53.8391	+	16.7769i	1.84342	+	0.574432i	0.999265	+	0.0383254i	$0.0122023\pi$
$853$	53.8391	+	16.7769i	1.84342	+	0.574432i	0.844151	+	0.536106i	$0.180105\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		−0.992709	−	0.120537i	$-0.961538\pi$
$857$		0			0		0.992709	+	0.120537i	$0.0384615\pi$
$858$		0			0
$859$	2.62414	+	21.6117i	0.0895345	+	0.737383i	0.966471	+	0.256776i	$0.0826604\pi$
$859$	2.62414	+	21.6117i	0.0895345	+	0.737383i	−0.876936	+	0.480607i	$0.840417\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.998176	−	0.0603785i	$-0.0192308\pi$
$863$		0			0		−0.998176	+	0.0603785i	$0.980769\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−14.7224	−	25.5000i	−0.500000	−	0.866025i
$868$	24.9785	+	14.4213i	0.847825	+	0.489492i
$869$		0			0
$870$		0			0
$871$	2.47847	+	11.0513i	0.0839797	+	0.374458i
$872$		0			0
$873$	−1.02746	−	51.0128i	−0.0347741	−	1.72652i
$874$		0			0
$875$		0			0
$876$	16.3454	−	27.0387i	0.552261	−	0.913552i
$877$	−49.4163	−	27.2187i	−1.66867	−	0.919111i	−0.981223	−	0.192878i	$-0.938218\pi$
$877$	−49.4163	−	27.2187i	−1.66867	−	0.919111i	−0.687448	−	0.726233i	$-0.741269\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		−0.999189	−	0.0402659i	$-0.987179\pi$
$881$		0			0		0.999189	+	0.0402659i	$0.0128205\pi$
$882$		0			0
$883$	−35.7513	−	13.5587i	−1.20313	−	0.456286i	−0.330095	−	0.943948i	$-0.607081\pi$
$883$	−35.7513	−	13.5587i	−1.20313	−	0.456286i	−0.873033	+	0.487661i	$0.837850\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		−0.200026	−	0.979791i	$-0.564103\pi$
$887$		0			0		0.200026	+	0.979791i	$0.435897\pi$
$888$		0			0
$889$	17.5209	+	38.9299i	0.587633	+	1.30567i
$890$		0			0
$891$		0			0
$892$	24.8556	+	1.50349i	0.832227	+	0.0503405i
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	−8.34652	+	28.8155i	−0.278217	+	0.960518i
$901$		0			0
$902$		0			0
$903$	8.26660	+	12.5077i	0.275095	+	0.416231i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	29.8057	−	39.6642i	0.989683	−	1.31703i	0.0415568	−	0.999136i	$-0.486768\pi$
$907$	29.8057	−	39.6642i	0.989683	−	1.31703i	0.948127	−	0.317893i	$-0.102975\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		−0.568065	−	0.822984i	$-0.692308\pi$
$911$		0			0		0.568065	+	0.822984i	$0.307692\pi$
$912$	0.718954	+	0.475171i	0.0238069	+	0.0157345i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	−3.02119	−	8.47728i	−0.0998229	−	0.280097i
$917$		0			0
$918$		0			0
$919$	−39.2973	+	16.7430i	−1.29630	+	0.552301i	−0.926319	−	0.376740i	$-0.877045\pi$
$919$	−39.2973	+	16.7430i	−1.29630	+	0.552301i	−0.369978	+	0.929040i	$0.620635\pi$
$920$		0			0
$921$	−13.5725	−	6.77838i	−0.447229	−	0.223355i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−9.68766	−	36.1548i	−0.318528	−	1.18876i
$926$		0			0
$927$	−23.8564	+	22.9144i	−0.783548	+	0.752608i
$928$		0			0
$929$		0			0		−0.100522	−	0.994935i	$-0.532051\pi$
$929$		0			0		0.100522	+	0.994935i	$0.467949\pi$
$930$		0			0
$931$	0.405304	−	0.0742748i	0.0132833	−	0.00243426i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	3.48523	+	3.08765i	0.113858	+	0.100869i	0.718131	−	0.695908i	$-0.244998\pi$
$937$	3.48523	+	3.08765i	0.113858	+	0.100869i	−0.604273	+	0.796777i	$0.706536\pi$
$938$		0			0
$939$	50.2513	+	8.16662i	1.63989	+	0.266508i
$940$		0			0
$941$		0			0		−0.911900	−	0.410413i	$-0.865385\pi$
$941$		0			0		0.911900	+	0.410413i	$0.134615\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.551377	−	0.834256i	$-0.314103\pi$
$947$		0			0		−0.551377	+	0.834256i	$0.685897\pi$
$948$	10.8658	−	7.50015i	0.352906	−	0.243593i
$949$	−30.5184	+	12.2506i	−0.990670	+	0.397671i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		0.999189	−	0.0402659i	$-0.0128205\pi$
$953$		0			0		−0.999189	+	0.0402659i	$0.987179\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−11.8049	−	22.4923i	−0.380802	−	0.725558i
$962$		0			0
$963$		0			0
$964$	50.5214	+	18.0051i	1.62718	+	0.579907i
$965$		0			0
$966$		0			0
$967$	−0.990681	+	16.3779i	−0.0318581	+	0.526678i	0.946883	+	0.321578i	$0.104213\pi$
$967$	−0.990681	+	16.3779i	−0.0318581	+	0.526678i	−0.978741	+	0.205100i	$0.934248\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		−0.278217	−	0.960518i	$-0.589744\pi$
$971$		0			0		0.278217	+	0.960518i	$0.410256\pi$
$972$	30.5468	−	6.23618i	0.979791	−	0.200026i
$973$	−17.8230	−	15.1603i	−0.571380	−	0.486018i
$974$		0			0
$975$	24.9795	−	18.7356i	0.799985	−	0.600020i
$976$	−3.03301	+	7.99739i	−0.0970842	+	0.255990i
$977$		0			0		−0.482459	−	0.875918i	$-0.660256\pi$
$977$		0			0		0.482459	+	0.875918i	$0.339744\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−46.3824	+	39.4530i	−1.48088	+	1.25964i
$982$		0			0
$983$		0			0		−0.855781	−	0.517338i	$-0.826923\pi$
$983$		0			0		0.855781	+	0.517338i	$0.173077\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	−0.300357	−	0.845197i	−0.00955563	−	0.0268893i
$989$		0			0
$990$		0			0
$991$	−13.2521	+	22.9534i	−0.420968	+	0.729138i	−0.996034	−	0.0889688i	$-0.971643\pi$
$991$	−13.2521	+	22.9534i	−0.420968	+	0.729138i	0.575066	+	0.818107i	$0.304976\pi$
$992$		0			0
$993$	60.1685	−	18.7493i	1.90939	−	0.594990i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−52.9716	−	33.4972i	−1.67763	−	1.06087i	−0.904343	−	0.426807i	$-0.859638\pi$
$997$	−52.9716	−	33.4972i	−1.67763	−	1.06087i	−0.773284	−	0.634060i	$-0.781387\pi$
$998$		0			0
$999$	−28.5907	+	26.3757i	−0.904570	+	0.834489i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	507.2.x.a.32.1	✓	48
3.2	odd	2	CM	507.2.x.a.32.1	✓	48
169.37	odd	156	inner	507.2.x.a.206.1	yes	48
507.206	even	156	inner	507.2.x.a.206.1	yes	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
507.2.x.a.32.1	✓	48	1.1	even	1	trivial
507.2.x.a.32.1	✓	48	3.2	odd	2	CM
507.2.x.a.206.1	yes	48	169.37	odd	156	inner
507.2.x.a.206.1	yes	48	507.206	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)

\(f(q)\)	\(=\)	\(q+(1.72643 - 0.139372i) q^{3} +(1.95958 + 0.400051i) q^{4} +(-0.421909 - 1.87333i) q^{7} +(2.96115 - 0.481234i) q^{9} +O(q^{10})\)	\(q+(1.72643 - 0.139372i) q^{3} +(1.95958 + 0.400051i) q^{4} +(-0.421909 - 1.87333i) q^{7} +(2.96115 - 0.481234i) q^{9} +(3.43884 + 0.417551i) q^{12} +(-2.59808 - 2.50000i) q^{13} +(3.67992 + 1.56787i) q^{16} +(0.120150 + 0.0321942i) q^{19} +(-0.989488 - 3.17538i) q^{21} +(-1.19658 + 4.85471i) q^{25} +(5.04516 - 1.24352i) q^{27} +(-0.0773365 - 3.83973i) q^{28} +(-3.88527 + 6.42702i) q^{31} +(5.99513 + 0.241596i) q^{36} +(-6.55717 + 3.61171i) q^{37} +(-4.83384 - 3.95399i) q^{39} +(-4.32977 + 1.25413i) q^{43} +(6.57165 + 2.19394i) q^{48} +(2.99280 - 1.42010i) q^{49} +(-4.09101 - 5.93832i) q^{52} +(0.211919 + 0.0388355i) q^{57} +(0.0861007 + 2.13657i) q^{61} +(-2.15084 - 5.34417i) q^{63} +(6.58387 + 4.54452i) q^{64} +(-2.62057 - 1.73199i) q^{67} +(3.74328 - 8.31724i) q^{73} +(-1.38920 + 8.54811i) q^{75} +(0.222565 + 0.111153i) q^{76} +(2.85285 - 2.52741i) q^{79} +(8.53683 - 2.85001i) q^{81} +(-0.668667 - 6.61826i) q^{84} +(-3.58717 + 5.92182i) q^{91} +(-5.81192 + 11.6373i) q^{93} +(2.38963 - 16.8390i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 10 q^{7} + 6 q^{9}+O(q^{10}) \) 48 * q + 10 * q^7 + 6 * q^9	\( 48 q + 10 q^{7} + 6 q^{9} - 8 q^{16} - 14 q^{19} - 18 q^{21} + 20 q^{28} + 14 q^{31} + 2 q^{37} + 24 q^{39} + 6 q^{43} - 18 q^{49} - 28 q^{52} - 12 q^{57} - 24 q^{63} - 32 q^{67} + 34 q^{73} + 30 q^{75} + 28 q^{76} + 18 q^{81} + 12 q^{84} - 2 q^{91} - 6 q^{93} + 38 q^{97}+O(q^{100}) \) 48 * q + 10 * q^7 + 6 * q^9 - 8 * q^16 - 14 * q^19 - 18 * q^21 + 20 * q^28 + 14 * q^31 + 2 * q^37 + 24 * q^39 + 6 * q^43 - 18 * q^49 - 28 * q^52 - 12 * q^57 - 24 * q^63 - 32 * q^67 + 34 * q^73 + 30 * q^75 + 28 * q^76 + 18 * q^81 + 12 * q^84 - 2 * q^91 - 6 * q^93 + 38 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

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