LMFDB - Embedded newform 507.2.x.a.137.1

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			137.1
Character	$\chi$	$=$	507.137
Dual form			507.2.x.a.470.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.72643 + 0.139372i) q^{3} +(-1.95958 - 0.400051i) q^{4} +(-4.81030 + 1.08337i) q^{7} +(2.96115 - 0.481234i) q^{9} +O(q^{10})$	$q+(-1.72643 + 0.139372i) q^{3} +(-1.95958 - 0.400051i) q^{4} +(-4.81030 + 1.08337i) q^{7} +(2.96115 - 0.481234i) q^{9} +(3.43884 + 0.417551i) q^{12} +(2.59808 + 2.50000i) q^{13} +(3.67992 + 1.56787i) q^{16} +(2.25610 - 8.41989i) q^{19} +(8.15367 - 2.54079i) q^{21} +(1.19658 - 4.85471i) q^{25} +(-5.04516 + 1.24352i) q^{27} +(9.85957 - 0.198583i) q^{28} +(7.03607 + 4.25345i) q^{31} +(-5.99513 - 0.241596i) q^{36} +(4.62656 + 8.39965i) q^{37} +(-4.83384 - 3.95399i) q^{39} +(-4.32977 + 1.25413i) q^{43} +(-6.57165 - 2.19394i) q^{48} +(15.6411 - 7.42180i) q^{49} +(-4.09101 - 5.93832i) q^{52} +(-2.72151 + 14.8508i) q^{57} +(-0.0861007 - 2.13657i) q^{61} +(-13.7227 + 5.52289i) q^{63} +(-6.58387 - 4.54452i) q^{64} +(8.85871 - 13.4036i) q^{67} +(13.1772 + 5.93059i) q^{73} +(-1.38920 + 8.54811i) q^{75} +(-7.78940 + 15.5969i) q^{76} +(-2.85285 + 2.52741i) q^{79} +(8.53683 - 2.85001i) q^{81} +(-16.9942 + 1.71699i) q^{84} +(-15.2059 - 9.21107i) q^{91} +(-12.7401 - 6.36267i) q^{93} +(-9.83809 - 1.39613i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/507\mathbb{Z}\right)^\times$.

$n$	$170$	$340$
$\chi(n)$	$-1$	$e\left(\frac{83}{156}\right)$

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		0.100522	−	0.994935i	$-0.467949\pi$
$2$		0			0		−0.100522	+	0.994935i	$0.532051\pi$
$3$	−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		0.787183	−	0.616719i	$-0.211538\pi$
$5$		0			0		−0.787183	+	0.616719i	$0.788462\pi$
$6$		0			0
$7$	−4.81030	+	1.08337i	−1.81812	+	0.409475i	−0.988794	−	0.149289i	$-0.952302\pi$
$7$	−4.81030	+	1.08337i	−1.81812	+	0.409475i	−0.829327	+	0.558763i	$0.811276\pi$
$8$		0			0
$9$	2.96115	−	0.481234i	0.987050	−	0.160411i
$10$		0			0
$11$		0			0		0.811378	−	0.584522i	$-0.198718\pi$
$11$		0			0		−0.811378	+	0.584522i	$0.801282\pi$
$12$	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
$19$	2.25610	−	8.41989i	0.517585	−	1.93165i	0.245011	−	0.969520i	$-0.421209\pi$
$19$	2.25610	−	8.41989i	0.517585	−	1.93165i	0.272575	−	0.962135i	$-0.412125\pi$
$20$		0			0
$21$	8.15367	−	2.54079i	1.77928	−	0.554445i
$22$		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
$28$	9.85957	−	0.198583i	1.86328	−	0.0375286i
$29$		0			0		−0.632445	−	0.774605i	$-0.717949\pi$
$29$		0			0		0.632445	+	0.774605i	$0.282051\pi$
$30$		0			0
$31$	7.03607	+	4.25345i	1.26371	+	0.763942i	0.980765	−	0.195192i	$-0.0625332\pi$
$31$	7.03607	+	4.25345i	1.26371	+	0.763942i	0.282950	+	0.959135i	$0.408687\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	−5.99513	−	0.241596i	−0.999189	−	0.0402659i
$37$	4.62656	+	8.39965i	0.760602	+	1.38089i	0.919297	+	0.393566i	$0.128759\pi$
$37$	4.62656	+	8.39965i	0.760602	+	1.38089i	−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		−0.761712	−	0.647915i	$-0.775641\pi$
$41$		0			0		0.761712	+	0.647915i	$0.224359\pi$
$42$		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		0.0603785	−	0.998176i	$-0.480769\pi$
$47$		0			0		−0.0603785	+	0.998176i	$0.519231\pi$
$48$	−6.57165	−	2.19394i	−0.948536	−	0.316668i
$49$	15.6411	−	7.42180i	2.23444	−	1.06026i
$50$		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
$57$	−2.72151	+	14.8508i	−0.360473	+	1.96704i
$58$		0			0
$59$		0			0		0.373361	−	0.927686i	$-0.378205\pi$
$59$		0			0		−0.373361	+	0.927686i	$0.621795\pi$
$60$		0			0
$61$	−0.0861007	−	2.13657i	−0.0110241	−	0.273559i	−0.995295	−	0.0968928i	$-0.969110\pi$
$61$	−0.0861007	−	2.13657i	−0.0110241	−	0.273559i	0.984271	−	0.176667i	$-0.0565314\pi$
$62$		0			0
$63$	−13.7227	+	5.52289i	−1.72889	+	0.695819i
$64$	−6.58387	−	4.54452i	−0.822984	−	0.568065i
$65$		0			0
$66$		0			0
$67$	8.85871	−	13.4036i	1.08226	−	1.63751i	0.381055	−	0.924552i	$-0.375561\pi$
$67$	8.85871	−	13.4036i	1.08226	−	1.63751i	0.701208	−	0.712956i	$-0.252644\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		0.941967	−	0.335705i	$-0.108974\pi$
$71$		0			0		−0.941967	+	0.335705i	$0.891026\pi$
$72$		0			0
$73$	13.1772	+	5.93059i	1.54228	+	0.694123i	0.990198	−	0.139668i	$-0.0446035\pi$
$73$	13.1772	+	5.93059i	1.54228	+	0.694123i	0.552080	+	0.833791i	$0.313834\pi$
$74$		0			0
$75$	−1.38920	+	8.54811i	−0.160411	+	0.987050i
$76$	−7.78940	+	15.5969i	−0.893506	+	1.78909i
$77$		0			0
$78$		0			0
$79$	−2.85285	+	2.52741i	−0.320971	+	0.284355i	−0.808187	−	0.588926i	$-0.799551\pi$
$79$	−2.85285	+	2.52741i	−0.320971	+	0.284355i	0.487216	+	0.873282i	$0.338012\pi$
$80$		0			0
$81$	8.53683	−	2.85001i	0.948536	−	0.316668i
$82$		0			0
$83$		0			0		0.983620	−	0.180255i	$-0.0576923\pi$
$83$		0			0		−0.983620	+	0.180255i	$0.942308\pi$
$84$	−16.9942	+	1.71699i	−1.85422	+	0.187339i
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$89$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$90$		0			0
$91$	−15.2059	−	9.21107i	−1.59402	−	0.965582i
$92$		0			0
$93$	−12.7401	−	6.36267i	−1.32109	−	0.659778i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−9.83809	−	1.39613i	−0.998907	−	0.141755i	−0.378138	−	0.925749i	$-0.623436\pi$
$97$	−9.83809	−	1.39613i	−0.998907	−	0.141755i	−0.620768	+	0.783994i	$0.713179\pi$
$98$		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
$109$	2.53519	+	4.19371i	0.242827	+	0.401685i	0.953293	−	0.302046i	$-0.0976698\pi$
$109$	2.53519	+	4.19371i	0.242827	+	0.401685i	−0.710466	+	0.703731i	$0.751516\pi$
$110$		0			0
$111$	−9.15813	−	13.8566i	−0.869251	−	1.31521i
$112$	−19.4001	−	3.55519i	−1.83313	−	0.335934i
$113$		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
$124$	−12.0861	−	11.1498i	−1.08537	−	1.00128i
$125$		0			0
$126$		0			0
$127$	21.7827	−	4.44696i	1.93290	−	0.394604i	0.933553	−	0.358441i	$-0.116691\pi$
$127$	21.7827	−	4.44696i	1.93290	−	0.394604i	0.999346	−	0.0361634i	$-0.0115137\pi$
$128$		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
$133$	−1.73068	+	42.9463i	−0.150069	+	3.72392i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		−0.875918	−	0.482459i	$-0.839744\pi$
$137$		0			0		0.875918	+	0.482459i	$0.160256\pi$
$138$		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
$147$	−25.9689	+	14.9932i	−2.14188	+	1.23662i
$148$	−5.70583	−	18.3107i	−0.469016	−	1.50513i
$149$		0			0		−0.446798	−	0.894635i	$-0.647436\pi$
$149$		0			0		0.446798	+	0.894635i	$0.352564\pi$
$150$		0			0
$151$	9.10501	−	0.550752i	0.740955	−	0.0448195i	0.314412	−	0.949287i	$-0.398193\pi$
$151$	9.10501	−	0.550752i	0.740955	−	0.0448195i	0.426543	+	0.904467i	$0.359731\pi$
$152$		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
$163$	18.7026	+	0.376691i	1.46490	+	0.0295047i	0.746156	−	0.665771i	$-0.231897\pi$
$163$	18.7026	+	0.376691i	1.46490	+	0.0295047i	0.718743	+	0.695276i	$0.244718\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		−0.994935	−	0.100522i	$-0.967949\pi$
$167$		0			0		0.994935	+	0.100522i	$0.0320513\pi$
$168$		0			0
$169$	0.500000	+	12.9904i	0.0384615	+	0.999260i
$170$		0			0
$171$	2.62872	−	26.0183i	0.201024	−	1.98967i
$172$	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
$175$	−0.496457	+	24.6489i	−0.0375286	+	1.86328i
$176$		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
$189$	22.9215	−	11.4475i	1.66730	−	0.832681i
$190$		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
$193$	3.80568	−	16.8977i	0.273939	−	1.21632i	−0.625831	−	0.779959i	$-0.715240\pi$
$193$	3.80568	−	16.8977i	0.273939	−	1.21632i	0.899770	−	0.436365i	$-0.143734\pi$
$194$		0			0
$195$		0			0
$196$	−33.6191	+	8.28637i	−2.40137	+	0.591883i
$197$		0			0		0.999797	−	0.0201371i	$-0.00641026\pi$
$197$		0			0		−0.999797	+	0.0201371i	$0.993590\pi$
$198$		0			0
$199$	−16.2691	−	21.6503i	−1.15329	−	1.53475i	−0.800800	−	0.598932i	$-0.795592\pi$
$199$	−16.2691	−	21.6503i	−1.15329	−	1.53475i	−0.352489	−	0.935816i	$-0.614665\pi$
$200$		0			0
$201$	−13.4259	+	24.3751i	−0.946989	+	1.71928i
$202$		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	$0.211296\pi$
$211$	4.42351	+	21.6678i	0.304527	+	1.49167i	−0.483125	+	0.875551i	$0.660498\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−38.4536	−	12.8377i	−2.61040	−	0.871480i
$218$		0			0
$219$	−23.5762	−	8.40224i	−1.59313	−	0.567771i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	3.81429	+	26.8782i	0.255423	+	1.79990i	0.540451	+	0.841376i	$0.318254\pi$
$223$	3.81429	+	26.8782i	0.255423	+	1.79990i	−0.285027	+	0.958519i	$0.592003\pi$
$224$		0			0
$225$	1.20700	−	14.9514i	0.0804666	−	0.996757i
$226$		0			0
$227$		0			0		0.834256	−	0.551377i	$-0.185897\pi$
$227$		0			0		−0.834256	+	0.551377i	$0.814103\pi$
$228$	11.2741	−	28.0126i	0.746647	−	1.85518i
$229$	25.6128	−	15.4835i	1.69254	−	1.02318i	0.769354	−	0.638823i	$-0.220578\pi$
$229$	25.6128	−	15.4835i	1.69254	−	1.02318i	0.923186	−	0.384353i	$-0.125576\pi$
$230$		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	$-0.750000\pi$
$239$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$240$		0			0
$241$	−2.19854	+	15.4925i	−0.141621	+	0.997959i	0.784338	+	0.620334i	$0.213003\pi$
$241$	−2.19854	+	15.4925i	−0.141621	+	0.997959i	−0.925958	+	0.377625i	$0.876741\pi$
$242$		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
$247$	26.9112	−	16.2353i	1.71232	−	1.03302i
$248$		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	$0.756410\pi$
$252$	29.1001	−	5.33279i	1.83313	−	0.335934i
$253$		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
$259$	−31.3550	−	35.3925i	−1.94831	−	2.19918i
$260$		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
$268$	−22.7215	+	22.7215i	−1.38794	+	1.38794i
$269$		0			0		0.428693	−	0.903450i	$-0.358974\pi$
$269$		0			0		−0.428693	+	0.903450i	$0.641026\pi$
$270$		0			0
$271$	24.9345	+	16.4797i	1.51466	+	1.00107i	0.988590	+	0.150632i	$0.0481310\pi$
$271$	24.9345	+	16.4797i	1.51466	+	1.00107i	0.526073	+	0.850439i	$0.323664\pi$
$272$		0			0
$273$	27.5358	+	13.7830i	1.66654	+	0.834186i
$274$		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
$279$	22.8818	+	9.20911i	1.36989	+	0.551335i
$280$		0			0
$281$		0			0		−0.983620	−	0.180255i	$-0.942308\pi$
$281$		0			0		0.983620	+	0.180255i	$0.0576923\pi$
$282$		0			0
$283$	24.3644	+	7.05724i	1.44831	+	0.419509i	0.906935	−	0.421270i	$-0.138415\pi$
$283$	24.3644	+	7.05724i	1.44831	+	0.419509i	0.541379	+	0.840779i	$0.317903\pi$
$284$		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
$291$	17.1794	+	1.03916i	1.00707	+	0.0609168i
$292$	−23.4493	−	16.8930i	−1.37227	−	0.988591i
$293$		0			0		−0.735006	−	0.678061i	$-0.762821\pi$
$293$		0			0		0.735006	+	0.678061i	$0.237179\pi$
$294$		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
$301$	19.4688	−	10.7235i	1.12216	−	0.618092i
$302$		0			0
$303$		0			0
$304$	21.5035	−	27.4472i	1.23331	−	1.57421i
$305$		0			0
$306$		0			0
$307$	−17.5535	+	29.0371i	−1.00183	+	1.65724i	−0.287014	+	0.957927i	$0.592662\pi$
$307$	−17.5535	+	29.0371i	−1.00183	+	1.65724i	−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	−0.673331	−	0.739341i	$-0.735137\pi$
$313$	−28.5391	−	7.03426i	−1.61313	−	0.397600i	−0.939794	+	0.341741i	$0.888983\pi$
$314$		0			0
$315$		0			0
$316$	6.60149	−	3.81137i	0.371363	−	0.214406i
$317$		0			0		−0.297503	−	0.954721i	$-0.596154\pi$
$317$		0			0		0.297503	+	0.954721i	$0.403846\pi$
$318$		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
$327$	−4.96132	−	6.88684i	−0.274362	−	0.380843i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−0.0587411	−	0.260818i	−0.00322870	−	0.0143358i	0.973923	−	0.226877i	$-0.0728516\pi$
$331$	−0.0587411	−	0.260818i	−0.00322870	−	0.0143358i	−0.977152	+	0.212541i	$0.931826\pi$
$332$		0			0
$333$	17.7421	+	22.6462i	0.972263	+	1.24100i
$334$		0			0
$335$		0			0
$336$	33.9884	+	3.43398i	1.85422	+	0.187339i
$337$		−	15.2142i		−	0.828771i	−0.910101	−	0.414385i	$-0.863997\pi$
$337$		−	15.2142i		−	0.828771i	0.910101	−	0.414385i	$-0.136003\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−40.0278	+	31.3598i	−2.16130	+	1.69327i
$344$		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
$349$	−30.0892	+	21.6765i	−1.61064	+	1.16032i	−0.734025	+	0.679122i	$0.762361\pi$
$349$	−30.0892	+	21.6765i	−1.61064	+	1.16032i	−0.876614	+	0.481193i	$0.840203\pi$
$350$		0			0
$351$	−16.2165	−	9.38214i	−0.865574	−	0.500782i
$352$		0			0
$353$		0			0		0.735006	−	0.678061i	$-0.237179\pi$
$353$		0			0		−0.735006	+	0.678061i	$0.762821\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		0.954721	−	0.297503i	$-0.0961538\pi$
$359$		0			0		−0.954721	+	0.297503i	$0.903846\pi$
$360$		0			0
$361$	−49.3500	−	28.4923i	−2.59737	−	1.49959i
$362$		0			0
$363$	−4.55958	+	18.4989i	−0.239316	+	0.970942i
$364$	26.1124	+	24.1330i	1.36866	+	1.26491i
$365$		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
$372$	22.4199	+	17.5649i	1.16242	+	0.910696i
$373$	−22.6708	+	27.7667i	−1.17385	+	1.43770i	−0.298367	+	0.954451i	$0.596442\pi$
$373$	−22.6708	+	27.7667i	−1.17385	+	1.43770i	−0.875481	+	0.483252i	$0.839455\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	3.14517	+	2.67529i	0.161556	+	0.137421i	0.725040	−	0.688707i	$-0.241821\pi$
$379$	3.14517	+	2.67529i	0.161556	+	0.137421i	−0.563484	+	0.826127i	$0.690539\pi$
$380$		0			0
$381$	−36.9865	+	10.7133i	−1.89488	+	0.548858i
$382$		0			0
$383$		0			0		0.678061	−	0.735006i	$-0.262821\pi$
$383$		0			0		−0.678061	+	0.735006i	$0.737179\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−12.2176	+	5.79731i	−0.621054	+	0.294694i
$388$	18.7200	+	6.67156i	0.950365	+	0.338697i
$389$		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
$397$	−0.525369	+	1.30538i	−0.0263675	+	0.0655150i	−0.940291	−	0.340371i	$-0.889447\pi$
$397$	−0.525369	+	1.30538i	−0.0263675	+	0.0655150i	0.913924	+	0.405886i	$0.133037\pi$
$398$		0			0
$399$	−2.99762	−	74.3852i	−0.150069	−	3.72392i
$400$	12.0148	−	15.9889i	0.600742	−	0.799443i
$401$		0			0		0.927686	−	0.373361i	$-0.121795\pi$
$401$		0			0		−0.927686	+	0.373361i	$0.878205\pi$
$402$		0			0
$403$	7.64661	+	28.6410i	0.380905	+	1.42671i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−10.3325	+	3.68236i	−0.510909	+	0.182081i	−0.578578	−	0.815627i	$-0.696392\pi$
$409$	−10.3325	+	3.68236i	−0.510909	+	0.182081i	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
$421$	36.7760	−	6.73946i	1.79235	−	0.328461i	0.821878	−	0.569663i	$-0.192926\pi$
$421$	36.7760	−	6.73946i	1.79235	−	0.328461i	0.970475	+	0.241202i	$0.0775416\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	2.72886	+	10.1842i	0.132059	+	0.492850i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		−0.894635	−	0.446798i	$-0.852564\pi$
$431$		0			0		0.894635	+	0.446798i	$0.147436\pi$
$432$	−20.5155	−	3.33409i	−0.987050	−	0.160411i
$433$	3.38748	+	7.95071i	0.162792	+	0.382087i	0.981321	−	0.192379i	$-0.0616202\pi$
$433$	3.38748	+	7.95071i	0.162792	+	0.382087i	−0.818529	+	0.574466i	$0.805210\pi$
$434$		0			0
$435$		0			0
$436$	−3.29021	−	9.23213i	−0.157572	−	0.442139i
$437$		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
$441$	42.7440	−	29.5041i	2.03543	−	1.40496i
$442$		0			0
$443$		0			0		0.568065	−	0.822984i	$-0.307692\pi$
$443$		0			0		−0.568065	+	0.822984i	$0.692308\pi$
$444$	12.4027	+	30.8169i	0.588608	+	1.46251i
$445$		0			0
$446$		0			0
$447$		0			0
$448$	36.5938	+	14.7277i	1.72889	+	0.695819i
$449$		0			0		−0.551377	−	0.834256i	$-0.685897\pi$
$449$		0			0		0.551377	+	0.834256i	$0.314103\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	−15.6424	+	2.21982i	−0.734946	+	0.104296i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	11.6924	−	32.8083i	0.546949	−	1.53471i	−0.272850	−	0.962057i	$-0.587966\pi$
$457$	11.6924	−	32.8083i	0.546949	−	1.53471i	0.819799	−	0.572651i	$-0.194085\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		−0.811378	−	0.584522i	$-0.801282\pi$
$461$		0			0		0.811378	+	0.584522i	$0.198718\pi$
$462$		0			0
$463$	−7.89109	−	17.5333i	−0.366730	−	0.814841i	−0.999190	−	0.0402476i	$-0.987185\pi$
$463$	−7.89109	−	17.5333i	−0.366730	−	0.814841i	0.632460	−	0.774593i	$-0.282045\pi$
$464$		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
$469$	−28.0920	+	74.0725i	−1.29717	+	3.42035i
$470$		0			0
$471$	−1.61558	+	40.0903i	−0.0744422	+	1.84726i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−38.1765	−	21.0278i	−1.75166	−	0.964821i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.0201371	−	0.999797i	$-0.506410\pi$
$479$		0			0		0.0201371	+	0.999797i	$0.493590\pi$
$480$		0			0
$481$	−8.97896	+	33.3893i	−0.409405	+	1.52242i
$482$		0			0
$483$		0			0
$484$	−11.0000	+	19.0526i	−0.500000	+	0.866025i
$485$		0			0
$486$		0			0
$487$	−19.7167	−	39.4792i	−0.893450	−	1.78897i	−0.462964	−	0.886377i	$-0.653214\pi$
$487$	−19.7167	−	39.4792i	−0.893450	−	1.78897i	−0.430486	−	0.902597i	$-0.641658\pi$
$488$		0			0
$489$	−32.3413	+	1.95629i	−1.46252	+	0.0884663i
$490$		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
$496$	19.2233	+	26.6840i	0.863152	+	1.19815i
$497$		0			0
$498$		0			0
$499$	5.69250	+	1.77386i	0.254831	+	0.0794087i	0.422257	−	0.906476i	$-0.361238\pi$
$499$	5.69250	+	1.77386i	0.254831	+	0.0794087i	−0.167426	+	0.985885i	$0.553546\pi$
$500$		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		0.100522	−	0.994935i	$-0.467949\pi$
$509$		0			0		−0.100522	+	0.994935i	$0.532051\pi$
$510$		0			0
$511$	−69.8114	−	14.2521i	−3.08827	−	0.630476i
$512$		0			0
$513$	−0.912096	+	45.2852i	−0.0402700	+	1.99939i
$514$		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
$525$	−2.57827	−	42.6239i	−0.112525	−	1.86026i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	11.5000	+	19.9186i	0.500000	+	0.866025i
$530$		0			0
$531$		0			0
$532$	20.5721	−	83.4645i	0.891916	−	3.61864i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	29.0862	+	22.7876i	1.25051	+	0.979714i	0.999933	+	0.0116175i	$0.00369806\pi$
$541$	29.0862	+	22.7876i	1.25051	+	0.979714i	0.250579	+	0.968096i	$0.419379\pi$
$542$		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
$553$	10.9850	−	15.2483i	0.467128	−	0.648422i
$554$		0			0
$555$		0			0
$556$	−22.0174	+	10.4474i	−0.933746	+	0.443068i
$557$		0			0		−0.941967	−	0.335705i	$-0.891026\pi$
$557$		0			0		0.941967	+	0.335705i	$0.108974\pi$
$558$		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
$567$	−37.9771	+	22.9579i	−1.59489	+	0.964142i
$568$		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	0.942293	−	0.334790i	$-0.108665\pi$
$571$	23.0210	+	15.8903i	0.963401	+	0.664988i	0.0211080	+	0.999777i	$0.493281\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−21.6828	−	10.2886i	−0.903450	−	0.428693i
$577$	32.9048	+	32.9048i	1.36984	+	1.36984i	0.860655	+	0.509189i	$0.170055\pi$
$577$	32.9048	+	32.9048i	1.36984	+	1.36984i	0.509189	+	0.860655i	$0.329945\pi$
$578$		0			0
$579$	−4.21519	+	29.7032i	−0.175177	+	1.23442i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$587$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$588$	56.8863	−	18.9914i	2.34595	−	0.783194i
$589$	51.6877	−	49.6467i	2.12975	−	2.04566i
$590$		0			0
$591$		0			0
$592$	3.85584	+	38.1638i	0.158474	+	1.56852i
$593$		0			0		−0.180255	−	0.983620i	$-0.557692\pi$
$593$		0			0		0.180255	+	0.983620i	$0.442308\pi$
$594$		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	$-0.972785\pi$
$601$	−48.2574	−	7.84259i	−1.96846	−	0.319906i	−0.972113	−	0.234513i	$-0.924651\pi$
$602$		0			0
$603$	19.7817	−	43.9532i	0.805573	−	1.78991i
$604$	−18.0623	−	2.56323i	−0.734946	−	0.104296i
$605$		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
$613$	−7.62078	−	18.9352i	−0.307800	−	0.764787i	−0.999108	−	0.0422267i	$-0.986555\pi$
$613$	−7.62078	−	18.9352i	−0.307800	−	0.764787i	0.691308	−	0.722560i	$-0.257035\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		−0.927686	−	0.373361i	$-0.878205\pi$
$617$		0			0		0.927686	+	0.373361i	$0.121795\pi$
$618$		0			0
$619$	38.9418	+	7.13636i	1.56520	+	0.286834i	0.891796	−	0.452437i	$-0.149445\pi$
$619$	38.9418	+	7.13636i	1.56520	+	0.286834i	0.673408	+	0.739271i	$0.264830\pi$
$620$		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
$631$	−11.6852	−	10.7799i	−0.465181	−	0.429141i	0.410625	−	0.911804i	$-0.365311\pi$
$631$	−11.6852	−	10.7799i	−0.465181	−	0.429141i	−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
$637$	59.1913	+	19.8204i	2.34524	+	0.785312i
$638$		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
$643$	32.7457	+	38.4970i	1.29136	+	1.51817i	0.708873	+	0.705337i	$0.249204\pi$
$643$	32.7457	+	38.4970i	1.29136	+	1.51817i	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
$651$	68.1769	+	16.8041i	2.67206	+	0.658604i
$652$	−36.4985	−	8.22015i	−1.42939	−	0.321926i
$653$		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
$657$	41.8738	+	11.2200i	1.63365	+	0.437736i
$658$		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
$661$	−23.2747	−	25.2293i	−0.905280	−	0.981307i	0.0946192	−	0.995514i	$-0.469837\pi$
$661$	−23.2747	−	25.2293i	−0.905280	−	0.981307i	−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
$669$	−10.3312	−	45.8718i	−0.399427	−	1.77351i
$670$		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
$679$	48.8366	−	3.94250i	1.87418	−	0.151299i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		0.975564	−	0.219715i	$-0.0705128\pi$
$683$		0			0		−0.975564	+	0.219715i	$0.929487\pi$
$684$	−15.5598	+	49.9333i	−0.594946	+	1.90925i
$685$		0			0
$686$		0			0
$687$	−42.0608	+	30.3009i	−1.60472	+	1.15605i
$688$	−17.8995	−	2.17340i	−0.682413	−	0.0828599i
$689$		0			0
$690$		0			0
$691$	25.9637	−	23.9521i	0.987704	−	0.911182i	−0.00834661	−	0.999965i	$-0.502657\pi$
$691$	25.9637	−	23.9521i	0.987704	−	0.911182i	0.996051	+	0.0887833i	$0.0282979\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	10.8337	−	48.1030i	0.409475	−	1.81812i
$701$		0			0		0.239316	−	0.970942i	$-0.423077\pi$
$701$		0			0		−0.239316	+	0.970942i	$0.576923\pi$
$702$		0			0
$703$	81.1621	−	20.0047i	3.06109	−	0.754490i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	24.2725	−	20.6463i	0.911572	−	0.775387i	−0.0633366	−	0.997992i	$-0.520174\pi$
$709$	24.2725	−	20.6463i	0.911572	−	0.775387i	0.974909	+	0.222606i	$0.0714562\pi$
$710$		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
$721$	36.8648	−	39.9608i	1.37292	−	1.48822i
$722$		0			0
$723$	1.63642	−	27.0532i	0.0608590	−	1.00612i
$724$	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	$0.637111\pi$
$727$	−25.0260	+	47.6830i	−0.928162	+	1.76846i	−0.510614	+	0.859810i	$0.670582\pi$
$728$		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
$733$	−8.86492	+	48.3743i	−0.327433	+	1.78674i	0.247858	+	0.968796i	$0.420273\pi$
$733$	−8.86492	+	48.3743i	−0.327433	+	1.78674i	−0.575291	+	0.817949i	$0.695111\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	11.9054	−	4.79152i	0.437948	−	0.176259i	−0.143837	−	0.989601i	$-0.545944\pi$
$739$	11.9054	−	4.79152i	0.437948	−	0.176259i	0.581785	+	0.813343i	$0.302354\pi$
$740$		0			0
$741$	−44.1978	+	31.7798i	−1.62365	+	1.16746i
$742$		0			0
$743$		0			0		0.551377	−	0.834256i	$-0.314103\pi$
$743$		0			0		−0.551377	+	0.834256i	$0.685897\pi$
$744$		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
$756$	−49.4962	+	13.2625i	−1.80016	+	0.482351i
$757$	29.2714	−	9.77223i	1.06389	−	0.355178i	0.269759	−	0.962928i	$-0.413056\pi$
$757$	29.2714	−	9.77223i	1.06389	−	0.355178i	0.794128	+	0.607750i	$0.207928\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		−0.100522	−	0.994935i	$-0.532051\pi$
$761$		0			0		0.100522	+	0.994935i	$0.467949\pi$
$762$		0			0
$763$	−16.7383	−	17.4265i	−0.605969	−	0.630880i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	−20.7433	−	18.3770i	−0.748511	−	0.663123i
$769$	−43.3602	−	21.6549i	−1.56361	−	0.780897i	−0.564597	−	0.825367i	$-0.690968\pi$
$769$	−43.3602	−	21.6549i	−1.56361	−	0.780897i	−0.999011	+	0.0444702i	$0.985840\pi$
$770$		0			0
$771$		0			0
$772$	−14.2175	+	31.5900i	−0.511699	+	1.13695i
$773$		0			0		−0.990080	−	0.140502i	$-0.955128\pi$
$773$		0			0		0.990080	+	0.140502i	$0.0448718\pi$
$774$		0			0
$775$	29.0685	−	29.0685i	1.04417	−	1.04417i
$776$		0			0
$777$	59.0651	+	56.7328i	2.11895	+	2.03528i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	69.1944	−	2.78844i	2.47123	−	0.0995870i
$785$		0			0
$786$		0			0
$787$	30.3025	+	45.8489i	1.08017	+	1.63434i	0.708071	+	0.706141i	$0.249566\pi$
$787$	30.3025	+	45.8489i	1.08017	+	1.63434i	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
$804$	36.0604	−	42.3939i	1.27175	−	1.49512i
$805$		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
$811$	25.3874	−	32.4047i	0.891473	−	1.13788i	−0.0982894	−	0.995158i	$-0.531337\pi$
$811$	25.3874	−	32.4047i	0.891473	−	1.13788i	0.989763	−	0.142724i	$-0.0455860\pi$
$812$		0			0
$813$	−45.3446	−	24.9760i	−1.59030	−	0.875946i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	0.791258	+	39.2857i	0.0276826	+	1.37443i
$818$		0			0
$819$	−49.4597	−	19.9577i	−1.72826	−	0.697380i
$820$		0			0
$821$		0			0		−0.975564	−	0.219715i	$-0.929487\pi$
$821$		0			0		0.975564	+	0.219715i	$0.0705128\pi$
$822$		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		0.998176	−	0.0603785i	$-0.0192308\pi$
$827$		0			0		−0.998176	+	0.0603785i	$0.980769\pi$
$828$		0			0
$829$	6.25737	−	14.6866i	0.217328	−	0.510086i	−0.775257	−	0.631646i	$-0.782379\pi$
$829$	6.25737	−	14.6866i	0.217328	−	0.510086i	0.992584	+	0.121560i	$0.0387897\pi$
$830$		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
$837$	−40.7873	−	12.7099i	−1.40982	−	0.439317i
$838$		0			0
$839$		0			0		−0.999797	−	0.0201371i	$-0.993590\pi$
$839$		0			0		0.999797	+	0.0201371i	$0.00641026\pi$
$840$		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
$847$	−5.45218	+	53.9639i	−0.187339	+	1.85422i
$848$		0			0
$849$	−43.0471	−	8.78814i	−1.47737	−	0.301608i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	4.53031	−	14.5383i	0.155115	−	0.497781i	−0.844151	−	0.536106i	$-0.819895\pi$
$853$	4.53031	−	14.5383i	0.155115	−	0.497781i	0.999265	+	0.0383254i	$0.0122023\pi$
$854$		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	$-0.917340\pi$
$859$	−2.62414	−	21.6117i	−0.0895345	−	0.737383i	0.876936	−	0.480607i	$-0.159583\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		−0.0603785	−	0.998176i	$-0.519231\pi$
$863$		0			0		0.0603785	+	0.998176i	$0.480769\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	14.7224	+	25.5000i	0.500000	+	0.866025i
$868$	70.2172	+	40.5399i	2.38333	+	1.37602i
$869$		0			0
$870$		0			0
$871$	56.5246	−	12.6768i	1.91526	−	0.429536i
$872$		0			0
$873$	−29.8039	+	0.600285i	−1.00871	+	0.0203166i
$874$		0			0
$875$		0			0
$876$	42.8381	+	25.8966i	1.44737	+	0.874963i
$877$	8.69989	−	15.7949i	0.293774	−	0.533356i	−0.687448	−	0.726233i	$-0.741269\pi$
$877$	8.69989	−	15.7949i	0.293774	−	0.533356i	0.981223	+	0.192878i	$0.0617821\pi$
$878$		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	0.873033	−	0.487661i	$-0.162150\pi$
$883$	35.7513	+	13.5587i	1.20313	+	0.456286i	0.330095	+	0.943948i	$0.392919\pi$
$884$		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
$889$	−99.9633	+	44.9898i	−3.35266	+	1.50891i
$890$		0			0
$891$		0			0
$892$	3.27825	−	54.1959i	0.109764	−	1.81461i
$893$		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
$903$	−32.1170	+	21.2268i	−1.06879	+	0.706384i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−29.8057	+	39.6642i	−0.989683	+	1.31703i	−0.0415568	+	0.999136i	$0.513232\pi$
$907$	−29.8057	+	39.6642i	−0.989683	+	1.31703i	−0.948127	+	0.317893i	$0.897025\pi$
$908$		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	$0.307692\pi$
$912$	−33.2991	+	50.3828i	−1.10264	+	1.66834i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	−56.3845	+	20.0947i	−1.86300	+	0.663947i
$917$		0			0
$918$		0			0
$919$	39.2973	−	16.7430i	1.29630	−	0.552301i	0.369978	−	0.929040i	$-0.379365\pi$
$919$	39.2973	−	16.7430i	1.29630	−	0.552301i	0.926319	+	0.376740i	$0.122955\pi$
$920$		0			0
$921$	26.2581	−	52.5771i	0.865233	−	1.73248i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	46.3139	−	12.4098i	1.52279	−	0.408031i
$926$		0			0
$927$	−23.8564	+	22.9144i	−0.783548	+	0.752608i
$928$		0			0
$929$		0			0		0.994935	−	0.100522i	$-0.0320513\pi$
$929$		0			0		−0.994935	+	0.100522i	$0.967949\pi$
$930$		0			0
$931$	−27.2028	−	148.441i	−0.891535	−	4.86495i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−3.48523	−	3.08765i	−0.113858	−	0.100869i	0.604273	−	0.796777i	$-0.293464\pi$
$937$	−3.48523	−	3.08765i	−0.113858	−	0.100869i	−0.718131	+	0.695908i	$0.755002\pi$
$938$		0			0
$939$	50.2513	+	8.16662i	1.63989	+	0.266508i
$940$		0			0
$941$		0			0		0.410413	−	0.911900i	$-0.365385\pi$
$941$		0			0		−0.410413	+	0.911900i	$0.634615\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		−0.834256	−	0.551377i	$-0.814103\pi$
$947$		0			0		0.834256	+	0.551377i	$0.185897\pi$
$948$	−10.8658	+	7.50015i	−0.352906	+	0.243593i
$949$	19.4090	+	48.3512i	0.630042	+	1.56955i
$950$		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
$961$	17.0080	+	32.4060i	0.548644	+	1.04535i
$962$		0			0
$963$		0			0
$964$	10.5060	−	29.4793i	0.338376	−	0.949464i
$965$		0			0
$966$		0			0
$967$	−59.8804	−	3.62210i	−1.92562	−	0.116479i	−0.946883	−	0.321578i	$-0.895787\pi$
$967$	−59.8804	−	3.62210i	−1.92562	−	0.116479i	−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
$973$	−38.9284	+	45.7656i	−1.24799	+	1.46718i
$974$		0			0
$975$	−24.9795	+	18.7356i	−0.799985	+	0.600020i
$976$	3.03301	−	7.99739i	0.0970842	−	0.255990i
$977$		0			0		0.875918	−	0.482459i	$-0.160256\pi$
$977$		0			0		−0.875918	+	0.482459i	$0.839744\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	9.52523	+	11.1982i	0.304117	+	0.357531i
$982$		0			0
$983$		0			0		0.517338	−	0.855781i	$-0.326923\pi$
$983$		0			0		−0.517338	+	0.855781i	$0.673077\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	−59.2297	+	21.0484i	−1.88435	+	0.669640i
$989$		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
$993$	0.137763	+	0.442098i	0.00437179	+	0.0140296i
$994$		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
$999$	−33.7869	−	36.6243i	−1.06897	−	1.15874i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	507.2.x.a.137.1	✓	48
3.2	odd	2	CM	507.2.x.a.137.1	✓	48
169.132	odd	156	inner	507.2.x.a.470.1	yes	48
507.470	even	156	inner	507.2.x.a.470.1	yes	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
507.2.x.a.137.1	✓	48	1.1	even	1	trivial
507.2.x.a.137.1	✓	48	3.2	odd	2	CM
507.2.x.a.470.1	yes	48	169.132	odd	156	inner
507.2.x.a.470.1	yes	48	507.470	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)

\(f(q)\)	\(=\)	\(q+(-1.72643 + 0.139372i) q^{3} +(-1.95958 - 0.400051i) q^{4} +(-4.81030 + 1.08337i) q^{7} +(2.96115 - 0.481234i) q^{9} +O(q^{10})\)	\(q+(-1.72643 + 0.139372i) q^{3} +(-1.95958 - 0.400051i) q^{4} +(-4.81030 + 1.08337i) q^{7} +(2.96115 - 0.481234i) q^{9} +(3.43884 + 0.417551i) q^{12} +(2.59808 + 2.50000i) q^{13} +(3.67992 + 1.56787i) q^{16} +(2.25610 - 8.41989i) q^{19} +(8.15367 - 2.54079i) q^{21} +(1.19658 - 4.85471i) q^{25} +(-5.04516 + 1.24352i) q^{27} +(9.85957 - 0.198583i) q^{28} +(7.03607 + 4.25345i) q^{31} +(-5.99513 - 0.241596i) q^{36} +(4.62656 + 8.39965i) q^{37} +(-4.83384 - 3.95399i) q^{39} +(-4.32977 + 1.25413i) q^{43} +(-6.57165 - 2.19394i) q^{48} +(15.6411 - 7.42180i) q^{49} +(-4.09101 - 5.93832i) q^{52} +(-2.72151 + 14.8508i) q^{57} +(-0.0861007 - 2.13657i) q^{61} +(-13.7227 + 5.52289i) q^{63} +(-6.58387 - 4.54452i) q^{64} +(8.85871 - 13.4036i) q^{67} +(13.1772 + 5.93059i) q^{73} +(-1.38920 + 8.54811i) q^{75} +(-7.78940 + 15.5969i) q^{76} +(-2.85285 + 2.52741i) q^{79} +(8.53683 - 2.85001i) q^{81} +(-16.9942 + 1.71699i) q^{84} +(-15.2059 - 9.21107i) q^{91} +(-12.7401 - 6.36267i) q^{93} +(-9.83809 - 1.39613i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 10 q^{7} + 6 q^{9}+O(q^{10}) \) 48 * q + 10 * q^7 + 6 * q^9	\( 48 q + 10 q^{7} + 6 q^{9} - 8 q^{16} - 14 q^{19} - 18 q^{21} + 20 q^{28} + 14 q^{31} + 2 q^{37} + 24 q^{39} + 6 q^{43} - 18 q^{49} - 28 q^{52} - 12 q^{57} - 24 q^{63} - 32 q^{67} + 34 q^{73} + 30 q^{75} + 28 q^{76} + 18 q^{81} + 12 q^{84} - 2 q^{91} - 6 q^{93} + 38 q^{97}+O(q^{100}) \) 48 * q + 10 * q^7 + 6 * q^9 - 8 * q^16 - 14 * q^19 - 18 * q^21 + 20 * q^28 + 14 * q^31 + 2 * q^37 + 24 * q^39 + 6 * q^43 - 18 * q^49 - 28 * q^52 - 12 * q^57 - 24 * q^63 - 32 * q^67 + 34 * q^73 + 30 * q^75 + 28 * q^76 + 18 * q^81 + 12 * q^84 - 2 * q^91 - 6 * q^93 + 38 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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