LMFDB - Embedded newform 507.2.x.a.197.1

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			197.1
Character	$\chi$	$=$	507.197
Dual form			507.2.x.a.332.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.742517 - 1.56482i) q^{3} +(-0.633336 - 1.89707i) q^{4} +(-0.0345359 + 0.0318602i) q^{7} +(-1.89734 - 2.32381i) q^{9} +O(q^{10})$	\(q+(0.742517 - 1.56482i) q^{3} +(-0.633336 - 1.89707i) q^{4} +(-0.0345359 + 0.0318602i) q^{7} +(-1.89734 - 2.32381i) q^{9} +(-3.43884 - 0.417551i) q^{12} +(2.59808 - 2.50000i) q^{13} +(-3.19777 + 2.40297i) q^{16} +(-1.66737 - 6.22270i) q^{19} +(0.0242121 + 0.0776993i) q^{21} +(-1.19658 + 4.85471i) q^{25} +(-5.04516 + 1.24352i) q^{27} +(0.0823140 + 0.0453389i) q^{28} +(-3.34425 + 5.53206i) q^{31} +(-3.20679 + 5.07114i) q^{36} +(0.233758 - 11.6060i) q^{37} +(-1.98294 - 5.92182i) q^{39} +(5.46315 + 5.24743i) q^{43} +(1.38582 + 6.78819i) q^{48} +(-0.563088 + 6.97510i) q^{49} +(-6.38814 - 3.34540i) q^{52} +(-10.9755 - 2.01133i) q^{57} +(12.7648 - 8.07198i) q^{61} +(0.139563 + 0.0198055i) q^{63} +(6.58387 + 4.54452i) q^{64} +(14.0332 - 7.00847i) q^{67} +(4.86243 - 10.8039i) q^{73} +(6.70828 + 5.47714i) q^{75} +(-10.7489 + 7.10418i) q^{76} +(13.1635 - 11.6618i) q^{79} +(-1.80023 + 8.81812i) q^{81} +(0.132067 - 0.0951418i) q^{84} +(-0.0100763 + 0.169115i) q^{91} +(6.17353 + 9.34080i) q^{93} +(9.21810 - 3.70997i) q^{97} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/507\mathbb{Z}\right)^\times$.

$n$	$170$	$340$
$\chi(n)$	$-1$	$e\left(\frac{109}{156}\right)$

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		0.584522	−	0.811378i	$-0.301282\pi$
$2$		0			0		−0.584522	+	0.811378i	$0.698718\pi$
$3$	0.742517	−	1.56482i	0.428693	−	0.903450i
$3$	0.742517	−	1.56482i	0.428693	−	0.903450i
$4$	−0.633336	−	1.89707i	−0.316668	−	0.948536i
$5$		0			0		−0.616719	−	0.787183i	$-0.711538\pi$
$5$		0			0		0.616719	+	0.787183i	$0.288462\pi$
$6$		0			0
$7$	−0.0345359	+	0.0318602i	−0.0130533	+	0.0120420i	−0.684561	−	0.728956i	$-0.740006\pi$
$7$	−0.0345359	+	0.0318602i	−0.0130533	+	0.0120420i	0.671508	+	0.740998i	$0.265647\pi$
$8$		0			0
$9$	−1.89734	−	2.32381i	−0.632445	−	0.774605i
$10$		0			0
$11$		0			0		0.994935	−	0.100522i	$-0.0320513\pi$
$11$		0			0		−0.994935	+	0.100522i	$0.967949\pi$
$12$	−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
$16$	−3.19777	+	2.40297i	−0.799443	+	0.600742i
$17$		0			0		−0.999189	−	0.0402659i	$-0.987179\pi$
$17$		0			0		0.999189	+	0.0402659i	$0.0128205\pi$
$18$		0			0
$19$	−1.66737	−	6.22270i	−0.382521	−	1.42759i	−0.842038	−	0.539417i	$-0.818644\pi$
$19$	−1.66737	−	6.22270i	−0.382521	−	1.42759i	0.459518	−	0.888168i	$-0.348022\pi$
$20$		0			0
$21$	0.0242121	+	0.0776993i	0.00528351	+	0.0169554i
$22$		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
$25$	−1.19658	+	4.85471i	−0.239316	+	0.970942i
$26$		0			0
$27$	−5.04516	+	1.24352i	−0.970942	+	0.239316i
$28$	0.0823140	+	0.0453389i	0.0155559	+	0.00856824i
$29$		0			0		0.987050	−	0.160411i	$-0.0512821\pi$
$29$		0			0		−0.987050	+	0.160411i	$0.948718\pi$
$30$		0			0
$31$	−3.34425	+	5.53206i	−0.600644	+	0.993587i	0.396498	+	0.918036i	$0.370225\pi$
$31$	−3.34425	+	5.53206i	−0.600644	+	0.993587i	−0.997142	+	0.0755512i	$0.975928\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	−3.20679	+	5.07114i	−0.534466	+	0.845190i
$37$	0.233758	−	11.6060i	0.0384296	−	1.90802i	−0.279887	−	0.960033i	$-0.590297\pi$
$37$	0.233758	−	11.6060i	0.0384296	−	1.90802i	0.318317	−	0.947984i	$-0.396882\pi$
$38$		0			0
$39$	−1.98294	−	5.92182i	−0.317524	−	0.948250i
$40$		0			0
$41$		0			0		−0.335705	−	0.941967i	$-0.608974\pi$
$41$		0			0		0.335705	+	0.941967i	$0.391026\pi$
$42$		0			0
$43$	5.46315	+	5.24743i	0.833122	+	0.800225i	0.982050	−	0.188623i	$-0.0604025\pi$
$43$	5.46315	+	5.24743i	0.833122	+	0.800225i	−0.148927	+	0.988848i	$0.547582\pi$
$44$		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	$0.0192308\pi$
$48$	1.38582	+	6.78819i	0.200026	+	0.979791i
$49$	−0.563088	+	6.97510i	−0.0804412	+	0.996443i
$50$		0			0
$51$		0			0
$52$	−6.38814	−	3.34540i	−0.885875	−	0.463924i
$53$		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
$57$	−10.9755	−	2.01133i	−1.45374	−	0.266407i
$58$		0			0
$59$		0			0		−0.140502	−	0.990080i	$-0.544872\pi$
$59$		0			0		0.140502	+	0.990080i	$0.455128\pi$
$60$		0			0
$61$	12.7648	−	8.07198i	1.63437	−	1.03351i	0.680737	−	0.732528i	$-0.261660\pi$
$61$	12.7648	−	8.07198i	1.63437	−	1.03351i	0.953630	−	0.300983i	$-0.0973147\pi$
$62$		0			0
$63$	0.139563	+	0.0198055i	0.0175833	+	0.00249526i
$64$	6.58387	+	4.54452i	0.822984	+	0.568065i
$65$		0			0
$66$		0			0
$67$	14.0332	−	7.00847i	1.71443	−	0.856221i	0.729348	−	0.684143i	$-0.239824\pi$
$67$	14.0332	−	7.00847i	1.71443	−	0.856221i	0.985083	−	0.172078i	$-0.0550481\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		0.647915	−	0.761712i	$-0.275641\pi$
$71$		0			0		−0.647915	+	0.761712i	$0.724359\pi$
$72$		0			0
$73$	4.86243	−	10.8039i	0.569105	−	1.26450i	−0.372579	−	0.928000i	$-0.621527\pi$
$73$	4.86243	−	10.8039i	0.569105	−	1.26450i	0.941684	−	0.336498i	$-0.109243\pi$
$74$		0			0
$75$	6.70828	+	5.47714i	0.774605	+	0.632445i
$76$	−10.7489	+	7.10418i	−1.23299	+	0.814905i
$77$		0			0
$78$		0			0
$79$	13.1635	−	11.6618i	1.48101	−	1.31206i	0.643763	−	0.765225i	$-0.277372\pi$
$79$	13.1635	−	11.6618i	1.48101	−	1.31206i	0.837242	−	0.546832i	$-0.184166\pi$
$80$		0			0
$81$	−1.80023	+	8.81812i	−0.200026	+	0.979791i
$82$		0			0
$83$		0			0		−0.180255	−	0.983620i	$-0.557692\pi$
$83$		0			0		0.180255	+	0.983620i	$0.442308\pi$
$84$	0.132067	−	0.0951418i	0.0144097	−	0.0103808i
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$89$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$90$		0			0
$91$	−0.0100763	+	0.169115i	−0.00105629	+	0.0177281i
$92$		0			0
$93$	6.17353	+	9.34080i	0.640165	+	0.968596i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	9.21810	−	3.70997i	0.935956	−	0.376690i	0.145604	−	0.989343i	$-0.453487\pi$
$97$	9.21810	−	3.70997i	0.935956	−	0.376690i	0.790352	+	0.612653i	$0.209898\pi$
$98$		0			0
$99$		0			0
$100$	9.96757	−	0.804666i	0.996757	−	0.0804666i
$101$		0			0		0.960518	−	0.278217i	$-0.0897436\pi$
$101$		0			0		−0.960518	+	0.278217i	$0.910256\pi$
$102$		0			0
$103$	−12.1672	+	8.39838i	−1.19887	+	0.827517i	−0.988665	−	0.150141i	$-0.952027\pi$
$103$	−12.1672	+	8.39838i	−1.19887	+	0.827517i	−0.210201	+	0.977658i	$0.567412\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		0.919979	−	0.391967i	$-0.128205\pi$
$107$		0			0		−0.919979	+	0.391967i	$0.871795\pi$
$108$	5.55433	+	8.78347i	0.534466	+	0.845190i
$109$	−8.22880	+	4.97448i	−0.788176	+	0.476469i	−0.853308	−	0.521407i	$-0.825407\pi$
$109$	−8.22880	+	4.97448i	−0.788176	+	0.476469i	0.0651315	+	0.997877i	$0.479253\pi$
$110$		0			0
$111$	−17.9878	−	8.98346i	−1.70732	−	0.852672i
$112$	0.0338788	−	0.184870i	0.00320124	−	0.0174686i
$113$		0			0		−0.428693	−	0.903450i	$-0.641026\pi$
$113$		0			0		0.428693	+	0.903450i	$0.358974\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−10.7390	−	1.29411i	−0.992817	−	0.119640i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	10.7777	−	2.20028i	0.979791	−	0.200026i
$122$		0			0
$123$		0			0
$124$	12.6127	+	2.84063i	1.13266	+	0.255096i
$125$		0			0
$126$		0			0
$127$	−7.11649	+	21.3165i	−0.631486	+	1.89153i	−0.243300	+	0.969951i	$0.578230\pi$
$127$	−7.11649	+	21.3165i	−0.631486	+	1.89153i	−0.388186	+	0.921581i	$0.626898\pi$
$128$		0			0
$129$	12.2678	−	4.65255i	1.08012	−	0.409634i
$130$		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
$133$	0.255841	+	0.161784i	0.0221842	+	0.0140284i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		0.999797	−	0.0201371i	$-0.00641026\pi$
$137$		0			0		−0.999797	+	0.0201371i	$0.993590\pi$
$138$		0			0
$139$	−1.29533	−	0.551890i	−0.109869	−	0.0468107i	0.336333	−	0.941743i	$-0.390813\pi$
$139$	−1.29533	−	0.551890i	−0.109869	−	0.0468107i	−0.446202	+	0.894932i	$0.647224\pi$
$140$		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
$147$	10.4967	+	6.06027i	0.865752	+	0.499842i
$148$	−22.1655	+	6.90705i	−1.82199	+	0.567756i
$149$		0			0		−0.834256	−	0.551377i	$-0.814103\pi$
$149$		0			0		0.834256	+	0.551377i	$0.185897\pi$
$150$		0			0
$151$	1.12801	+	18.6482i	0.0917958	+	1.51757i	0.694437	+	0.719553i	$0.255653\pi$
$151$	1.12801	+	18.6482i	0.0917958	+	1.51757i	−0.602641	+	0.798012i	$0.705885\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	−9.97826	+	7.51228i	−0.798900	+	0.601464i
$157$	−0.807317	+	6.64886i	−0.0644309	+	0.530636i	0.924381	+	0.381470i	$0.124582\pi$
$157$	−0.807317	+	6.64886i	−0.0644309	+	0.530636i	−0.988812	+	0.149166i	$0.952341\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	7.43363	−	4.09448i	0.582247	−	0.320704i	−0.163909	−	0.986476i	$-0.552410\pi$
$163$	7.43363	−	4.09448i	0.582247	−	0.320704i	0.746156	+	0.665771i	$0.231897\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		−0.811378	−	0.584522i	$-0.801282\pi$
$167$		0			0		0.811378	+	0.584522i	$0.198718\pi$
$168$		0			0
$169$	0.500000	−	12.9904i	0.0384615	−	0.999260i
$170$		0			0
$171$	−11.2969	+	15.6812i	−0.863892	+	1.19917i
$172$	6.49474	−	13.6874i	0.495219	−	1.04365i
$173$		0			0		−0.316668	−	0.948536i	$-0.602564\pi$
$173$		0			0		0.316668	+	0.948536i	$0.397436\pi$
$174$		0			0
$175$	−0.113347	−	0.205785i	−0.00856824	−	0.0155559i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		0.999189	−	0.0402659i	$-0.0128205\pi$
$179$		0			0		−0.999189	+	0.0402659i	$0.987179\pi$
$180$		0			0
$181$	12.8547	+	1.56084i	0.955481	+	0.116016i	0.583401	−	0.812184i	$-0.301722\pi$
$181$	12.8547	+	1.56084i	0.955481	+	0.116016i	0.372080	+	0.928201i	$0.378645\pi$
$182$		0			0
$183$	−3.15312	−	25.9682i	−0.233085	−	1.91963i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	0.134620	−	0.203686i	0.00979219	−	0.0148160i
$190$		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
$193$	5.45165	−	5.90949i	0.392418	−	0.425374i	−0.507351	−	0.861739i	$-0.669375\pi$
$193$	5.45165	−	5.90949i	0.392418	−	0.425374i	0.899770	+	0.436365i	$0.143734\pi$
$194$		0			0
$195$		0			0
$196$	13.5889	−	3.34936i	0.970636	−	0.239240i
$197$		0			0		−0.875918	−	0.482459i	$-0.839744\pi$
$197$		0			0		0.875918	+	0.482459i	$0.160256\pi$
$198$		0			0
$199$	−5.88936	+	13.8228i	−0.417486	+	0.979875i	0.569925	+	0.821697i	$0.306972\pi$
$199$	−5.88936	+	13.8228i	−0.417486	+	0.979875i	−0.987411	+	0.158178i	$0.949438\pi$
$200$		0			0
$201$	−0.547102	−	27.1634i	−0.0385896	−	1.91596i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	−2.30063	+	14.2375i	−0.159520	+	0.987195i
$209$		0			0
$210$		0			0
$211$	−26.8786	−	8.97339i	−1.85040	−	0.617754i	−0.995011	−	0.0997656i	$-0.968191\pi$
$211$	−26.8786	−	8.97339i	−1.85040	−	0.617754i	−0.855388	−	0.517988i	$-0.826681\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−0.0607561	−	0.297603i	−0.00412439	−	0.0202026i
$218$		0			0
$219$	−13.2957	−	15.6309i	−0.898441	−	1.05624i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−10.1358	+	25.1843i	−0.678744	+	1.68647i	0.0473562	+	0.998878i	$0.484920\pi$
$223$	−10.1358	+	25.1843i	−0.678744	+	1.68647i	−0.726101	+	0.687589i	$0.758669\pi$
$224$		0			0
$225$	13.5518	−	6.43039i	0.903450	−	0.428693i
$226$		0			0
$227$		0			0		0.446798	−	0.894635i	$-0.352564\pi$
$227$		0			0		−0.446798	+	0.894635i	$0.647436\pi$
$228$	3.13552	+	22.0951i	0.207655	+	1.46329i
$229$	−15.5906	−	25.7901i	−1.03026	−	1.70426i	−0.594118	−	0.804378i	$-0.702499\pi$
$229$	−15.5906	−	25.7901i	−1.03026	−	1.70426i	−0.436141	−	0.899878i	$-0.643655\pi$
$230$		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
$237$	−8.47456	−	29.2576i	−0.550482	−	1.90048i
$238$		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
$241$	10.6436	+	26.4461i	0.685616	+	1.70354i	0.710352	+	0.703846i	$0.248536\pi$
$241$	10.6436	+	26.4461i	0.685616	+	1.70354i	−0.0247366	+	0.999694i	$0.507875\pi$
$242$		0			0
$243$	12.4621	+	9.36465i	0.799443	+	0.600742i
$244$	−23.3975	−	19.1035i	−1.49787	−	1.22298i
$245$		0			0
$246$		0			0
$247$	−19.8887	−	11.9986i	−1.26549	−	0.763455i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		−0.960518	−	0.278217i	$-0.910256\pi$
$251$		0			0		0.960518	+	0.278217i	$0.0897436\pi$
$252$	−0.0508181	−	0.277306i	−0.00320124	−	0.0174686i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	4.45148	−	15.3683i	0.278217	−	0.960518i
$257$		0			0		−0.979791	−	0.200026i	$-0.935897\pi$
$257$		0			0		0.979791	+	0.200026i	$0.0641026\pi$
$258$		0			0
$259$	0.361697	+	0.408272i	0.0224748	+	0.0253688i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		0.632445	−	0.774605i	$-0.282051\pi$
$263$		0			0		−0.632445	+	0.774605i	$0.717949\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	−22.1833	−	22.1833i	−1.35506	−	1.35506i
$269$		0			0		0.996757	−	0.0804666i	$-0.0256410\pi$
$269$		0			0		−0.996757	+	0.0804666i	$0.974359\pi$
$270$		0			0
$271$	−7.73453	−	15.4870i	−0.469839	−	0.940769i	−0.995912	−	0.0903297i	$-0.971208\pi$
$271$	−7.73453	−	15.4870i	−0.469839	−	0.940769i	0.526073	−	0.850439i	$-0.323664\pi$
$272$		0			0
$273$	0.257153	+	0.141339i	0.0155636	+	0.00855420i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−5.12584	−	8.10586i	−0.307982	−	0.487034i	0.655358	−	0.755318i	$-0.272518\pi$
$277$	−5.12584	−	8.10586i	−0.307982	−	0.487034i	−0.963340	+	0.268284i	$0.913543\pi$
$278$		0			0
$279$	19.2006	−	2.72477i	1.14951	−	0.163127i
$280$		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
$283$	−13.9474	+	13.3966i	−0.829086	+	0.796348i	−0.981398	−	0.191985i	$-0.938508\pi$
$283$	−13.9474	+	13.3966i	−0.829086	+	0.796348i	0.152312	+	0.988332i	$0.451328\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	16.9449	+	1.36793i	0.996757	+	0.0804666i
$290$		0			0
$291$	1.03916	−	17.1794i	0.0609168	−	1.00707i
$292$	−23.5753	−	2.38190i	−1.37964	−	0.139390i
$293$		0			0		−0.975564	−	0.219715i	$-0.929487\pi$
$293$		0			0		0.975564	+	0.219715i	$0.0705128\pi$
$294$		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
$301$	−0.355859	−	0.00716740i	−0.0205114	−	0.000413122i
$302$		0			0
$303$		0			0
$304$	20.2848	+	15.8921i	1.16341	+	0.911477i
$305$		0			0
$306$		0			0
$307$	−10.3261	−	6.24232i	−0.589339	−	0.356268i	0.191033	−	0.981584i	$-0.438816\pi$
$307$	−10.3261	−	6.24232i	−0.589339	−	0.356268i	−0.780372	+	0.625316i	$0.784970\pi$
$308$		0			0
$309$	4.10765	+	25.2754i	0.233676	+	1.43787i
$310$		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
$313$	−7.76329	−	1.91348i	−0.438807	−	0.108156i	0.0137219	−	0.999906i	$-0.495632\pi$
$313$	−7.76329	−	1.91348i	−0.438807	−	0.108156i	−0.452529	+	0.891750i	$0.649478\pi$
$314$		0			0
$315$		0			0
$316$	−30.4602	−	17.5862i	−1.71352	−	0.989302i
$317$		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
$325$	9.02797	+	15.6043i	0.500782	+	0.865574i
$326$		0			0
$327$	1.67415	+	16.5703i	0.0925810	+	0.916337i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−11.3050	−	12.2545i	−0.621382	−	0.673566i	0.342617	−	0.939475i	$-0.388687\pi$
$331$	−11.3050	−	12.2545i	−0.621382	−	0.673566i	−0.963998	+	0.265909i	$0.914328\pi$
$332$		0			0
$333$	−27.4138	+	21.4773i	−1.50226	+	1.17695i
$334$		0			0
$335$		0			0
$336$	−0.264134	−	0.190284i	−0.0144097	−	0.0103808i
$337$			36.1422i			1.96879i	0.175964	+	0.984397i	$0.443696\pi$
$337$			36.1422i			1.96879i	−0.175964	+	0.984397i	$0.556304\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−0.405627	−	0.517745i	−0.0219018	−	0.0279556i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		−0.632445	−	0.774605i	$-0.717949\pi$
$347$		0			0		0.632445	+	0.774605i	$0.282051\pi$
$348$		0			0
$349$	36.8963	−	3.72777i	1.97501	−	0.199543i	0.975246	−	0.221125i	$-0.0709727\pi$
$349$	36.8963	−	3.72777i	1.97501	−	0.199543i	0.999767	+	0.0215816i	$0.00687016\pi$
$350$		0			0
$351$	−9.99891	+	15.8437i	−0.533702	+	0.845672i
$352$		0			0
$353$		0			0		0.975564	−	0.219715i	$-0.0705128\pi$
$353$		0			0		−0.975564	+	0.219715i	$0.929487\pi$
$354$		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
$361$	−19.4874	+	11.2511i	−1.02565	+	0.592162i
$362$		0			0
$363$	4.55958	−	18.4989i	0.239316	−	0.970942i
$364$	0.327205	−	0.0879911i	0.0171502	−	0.00461199i
$365$		0			0
$366$		0			0
$367$	−0.908917	+	0.147713i	−0.0474451	+	0.00771057i	−0.184087	−	0.982910i	$-0.558933\pi$
$367$	−0.908917	+	0.147713i	−0.0474451	+	0.00771057i	0.136642	+	0.990620i	$0.456369\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	13.8103	−	17.6275i	0.716028	−	0.913943i
$373$	18.3641	+	2.98446i	0.950857	+	0.154529i	0.616006	−	0.787742i	$-0.288750\pi$
$373$	18.3641	+	2.98446i	0.950857	+	0.154529i	0.334852	+	0.942271i	$0.391314\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−9.65631	−	27.0950i	−0.496011	−	1.39178i	−0.882860	−	0.469635i	$-0.844385\pi$
$379$	−9.65631	−	27.0950i	−0.496011	−	1.39178i	0.386849	−	0.922143i	$-0.373563\pi$
$380$		0			0
$381$	28.0724	+	26.9639i	1.43819	+	1.38140i
$382$		0			0
$383$		0			0		0.219715	−	0.975564i	$-0.429487\pi$
$383$		0			0		−0.219715	+	0.975564i	$0.570513\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	1.82862	−	22.6515i	0.0929538	−	1.15144i
$388$	−12.8762	−	15.1378i	−0.653692	−	0.768503i
$389$		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
$397$	5.57853	+	39.3103i	0.279978	+	1.97293i	0.224550	+	0.974463i	$0.427909\pi$
$397$	5.57853	+	39.3103i	0.279978	+	1.97293i	0.0554283	+	0.998463i	$0.482348\pi$
$398$		0			0
$399$	0.443129	−	0.280218i	0.0221842	−	0.0140284i
$400$	−7.83933	−	18.3996i	−0.391967	−	0.919979i
$401$		0			0		−0.990080	−	0.140502i	$-0.955128\pi$
$401$		0			0		0.990080	+	0.140502i	$0.0448718\pi$
$402$		0			0
$403$	5.14154	+	22.7333i	0.256118	+	1.13243i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	11.4072	−	13.4108i	0.564052	−	0.663119i	−0.403739	−	0.914874i	$-0.632290\pi$
$409$	11.4072	−	13.4108i	0.564052	−	0.663119i	0.967791	+	0.251755i	$0.0810077\pi$
$410$		0			0
$411$		0			0
$412$	23.6382	+	17.7630i	1.16457	+	0.875119i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−1.82542	+	1.61718i	−0.0893911	+	0.0791936i
$418$		0			0
$419$		0			0		0.200026	−	0.979791i	$-0.435897\pi$
$419$		0			0		−0.200026	+	0.979791i	$0.564103\pi$
$420$		0			0
$421$	7.27328	+	39.6890i	0.354478	+	1.93432i	0.356421	+	0.934325i	$0.383997\pi$
$421$	7.27328	+	39.6890i	0.354478	+	1.93432i	−0.00194321	+	0.999998i	$0.500619\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−0.183669	+	0.685463i	−0.00888837	+	0.0331719i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		−0.551377	−	0.834256i	$-0.685897\pi$
$431$		0			0		0.551377	+	0.834256i	$0.314103\pi$
$432$	13.1451	−	16.0999i	0.632445	−	0.774605i
$433$	20.1036	−	26.7530i	0.966118	−	1.28567i	0.00779854	−	0.999970i	$-0.497518\pi$
$433$	20.1036	−	26.7530i	0.966118	−	1.28567i	0.958319	−	0.285700i	$-0.0922260\pi$
$434$		0			0
$435$		0			0
$436$	14.6486	+	12.4601i	0.701539	+	0.596731i
$437$		0			0
$438$		0			0
$439$	−1.89342	+	0.548434i	−0.0903678	+	0.0261753i	−0.323093	−	0.946367i	$-0.604723\pi$
$439$	−1.89342	+	0.548434i	−0.0903678	+	0.0261753i	0.232726	+	0.972542i	$0.425236\pi$
$440$		0			0
$441$	17.2772	−	11.9256i	0.822724	−	0.567886i
$442$		0			0
$443$		0			0		0.568065	−	0.822984i	$-0.307692\pi$
$443$		0			0		−0.568065	+	0.822984i	$0.692308\pi$
$444$	−5.64997	+	39.8137i	−0.268136	+	1.88947i
$445$		0			0
$446$		0			0
$447$		0			0
$448$	−0.372169	+	0.0528146i	−0.0175833	+	0.00249526i
$449$		0			0		−0.894635	−	0.446798i	$-0.852564\pi$
$449$		0			0		0.894635	+	0.446798i	$0.147436\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	30.0186	+	12.0815i	1.41040	+	0.567636i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−30.2795	+	25.7558i	−1.41641	+	1.20481i	−0.469660	+	0.882847i	$0.655624\pi$
$457$	−30.2795	+	25.7558i	−1.41641	+	1.20481i	−0.946753	+	0.321960i	$0.895658\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		−0.994935	−	0.100522i	$-0.967949\pi$
$461$		0			0		0.994935	+	0.100522i	$0.0320513\pi$
$462$		0			0
$463$	−36.4054	+	16.3847i	−1.69190	+	0.761463i	−0.692711	+	0.721215i	$0.743584\pi$
$463$	−36.4054	+	16.3847i	−1.69190	+	0.761463i	−0.999190	+	0.0402476i	$0.987185\pi$
$464$		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	$0.884615\pi$
$468$	4.34635	+	21.1922i	0.200910	+	0.979610i
$469$	−0.261358	+	0.689146i	−0.0120684	+	0.0318218i
$470$		0			0
$471$	9.80483	+	6.20020i	0.451783	+	0.285690i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	32.2045	−	0.648636i	1.47765	−	0.0297615i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.482459	−	0.875918i	$-0.339744\pi$
$479$		0			0		−0.482459	+	0.875918i	$0.660256\pi$
$480$		0			0
$481$	−28.4077	−	30.7377i	−1.29528	−	1.40152i
$482$		0			0
$483$		0			0
$484$	−11.0000	−	19.0526i	−0.500000	−	0.866025i
$485$		0			0
$486$		0			0
$487$	5.10101	+	3.37136i	0.231149	+	0.152771i	0.661635	−	0.749826i	$-0.269863\pi$
$487$	5.10101	+	3.37136i	0.231149	+	0.152771i	−0.430486	+	0.902597i	$0.641658\pi$
$488$		0			0
$489$	−0.887525	−	14.6725i	−0.0401353	−	0.663515i
$490$		0			0
$491$		0			0		−0.600742	−	0.799443i	$-0.705128\pi$
$491$		0			0		0.600742	+	0.799443i	$0.294872\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	−2.59923	−	25.7264i	−0.116709	−	1.15515i
$497$		0			0
$498$		0			0
$499$	1.43008	−	4.58929i	0.0640193	−	0.205445i	−0.917169	−	0.398499i	$-0.869531\pi$
$499$	1.43008	−	4.58929i	0.0640193	−	0.205445i	0.981188	+	0.193054i	$0.0618391\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		0.948536	−	0.316668i	$-0.102564\pi$
$503$		0			0		−0.948536	+	0.316668i	$0.897436\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−19.9564	−	10.4280i	−0.886294	−	0.463123i
$508$	44.9461			1.99416
$509$		0			0		0.584522	−	0.811378i	$-0.301282\pi$
$509$		0			0		−0.584522	+	0.811378i	$0.698718\pi$
$510$		0			0
$511$	0.176286	+	0.528040i	0.00779842	+	0.0233591i
$512$		0			0
$513$	16.1502	+	29.3211i	0.713049	+	1.29456i
$514$		0			0
$515$		0			0
$516$	−16.5958	−	20.3262i	−0.730592	−	0.894812i
$517$		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
$523$	−31.3942	+	23.5912i	−1.37277	+	1.03157i	−0.378397	+	0.925644i	$0.623524\pi$
$523$	−31.3942	+	23.5912i	−1.37277	+	1.03157i	−0.994374	+	0.105927i	$0.966219\pi$
$524$		0			0
$525$	−0.406179	+	0.0245693i	−0.0177271	+	0.00107229i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	11.5000	−	19.9186i	0.500000	−	0.866025i
$530$		0			0
$531$		0			0
$532$	0.144883	−	0.587812i	0.00628146	−	0.0254849i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−28.2773	+	36.0933i	−1.21574	+	1.55177i	−0.474979	+	0.879997i	$0.657544\pi$
$541$	−28.2773	+	36.0933i	−1.21574	+	1.55177i	−0.740756	+	0.671774i	$0.765533\pi$
$542$		0			0
$543$	11.9873	−	18.9563i	0.514422	−	0.813494i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	16.1772	+	42.6557i	0.691686	+	1.82382i	0.552413	+	0.833571i	$0.313707\pi$
$547$	16.1772	+	42.6557i	0.691686	+	1.82382i	0.139273	+	0.990254i	$0.455523\pi$
$548$		0			0
$549$	−42.9769	−	14.3478i	−1.83421	−	0.612349i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−0.0830641	+	0.822142i	−0.00353225	+	0.0349610i
$554$		0			0
$555$		0			0
$556$	−0.226594	+	2.80687i	−0.00960974	+	0.119038i
$557$		0			0		−0.647915	−	0.761712i	$-0.724359\pi$
$557$		0			0		0.647915	+	0.761712i	$0.275641\pi$
$558$		0			0
$559$	27.3122	−	0.0246603i	1.15518	−	0.00104302i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		0.903450	−	0.428693i	$-0.141026\pi$
$563$		0			0		−0.903450	+	0.428693i	$0.858974\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−0.218775	−	0.361897i	−0.00918767	−	0.0151983i
$568$		0			0
$569$		0			0		−0.391967	−	0.919979i	$-0.628205\pi$
$569$		0			0		0.391967	+	0.919979i	$0.371795\pi$
$570$		0			0
$571$	37.9813	+	26.2166i	1.58947	+	1.09713i	0.942293	+	0.334790i	$0.108665\pi$
$571$	37.9813	+	26.2166i	1.58947	+	1.09713i	0.647175	+	0.762341i	$0.275950\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−1.93120	−	23.9222i	−0.0804666	−	0.996757i
$577$	25.2123	−	25.2123i	1.04960	−	1.04960i	0.0508984	−	0.998704i	$-0.483792\pi$
$577$	25.2123	−	25.2123i	1.04960	−	1.04960i	0.998704	−	0.0508984i	$-0.0162085\pi$
$578$		0			0
$579$	−5.19935	−	12.9188i	−0.216078	−	0.536885i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$587$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$588$	4.84884	−	23.7512i	0.199963	−	0.979482i
$589$	40.0004	+	11.5863i	1.64819	+	0.477404i
$590$		0			0
$591$		0			0
$592$	27.1414	+	37.6751i	1.11550	+	1.54844i
$593$		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
$597$	17.2573	+	19.4795i	0.706296	+	0.797243i
$598$		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
$601$	15.6372	−	19.1520i	0.637853	−	0.781228i	−0.349978	−	0.936758i	$-0.613811\pi$
$601$	15.6372	−	19.1520i	0.637853	−	0.781228i	0.987831	+	0.155530i	$0.0497085\pi$
$602$		0			0
$603$	−42.9121	−	19.3132i	−1.74752	−	0.786494i
$604$	34.6625	−	13.9505i	1.41040	−	0.567636i
$605$		0			0
$606$		0			0
$607$	−3.20847	+	0.259014i	−0.130228	+	0.0105131i	−0.145409	−	0.989372i	$-0.546450\pi$
$607$	−3.20847	+	0.259014i	−0.130228	+	0.0105131i	0.0151808	+	0.999885i	$0.495168\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−3.58761	+	25.2809i	−0.144902	+	1.02108i	0.775845	+	0.630924i	$0.217324\pi$
$613$	−3.58761	+	25.2809i	−0.144902	+	1.02108i	−0.920747	+	0.390160i	$0.872420\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		0.990080	−	0.140502i	$-0.0448718\pi$
$617$		0			0		−0.990080	+	0.140502i	$0.955128\pi$
$618$		0			0
$619$	0.417726	−	2.27945i	0.0167898	−	0.0916190i	−0.974158	−	0.225869i	$-0.927478\pi$
$619$	0.417726	−	2.27945i	0.0167898	−	0.0916190i	0.990948	+	0.134250i	$0.0428625\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$	20.5709	+	14.1717i	0.823496	+	0.567321i
$625$	−22.1364	−	11.6181i	−0.885456	−	0.464723i
$626$		0			0
$627$		0			0
$628$	13.1247	−	2.67942i	0.523731	−	0.106920i
$629$		0			0
$630$		0			0
$631$	−47.0735	−	10.6018i	−1.87397	−	0.422052i	−0.875806	−	0.482663i	$-0.839670\pi$
$631$	−47.0735	−	10.6018i	−1.87397	−	0.422052i	−0.998161	+	0.0606105i	$0.980695\pi$
$632$		0			0
$633$	−33.9996	+	35.3973i	−1.35136	+	1.40692i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	15.9748	+	19.5296i	0.632945	+	0.773789i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		0.160411	−	0.987050i	$-0.448718\pi$
$641$		0			0		−0.160411	+	0.987050i	$0.551282\pi$
$642$		0			0
$643$	−30.3055	−	10.8005i	−1.19513	−	0.425929i	−0.338059	−	0.941125i	$-0.609770\pi$
$643$	−30.3055	−	10.8005i	−1.19513	−	0.425929i	−0.857072	+	0.515196i	$0.827719\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.160411	−	0.987050i	$-0.551282\pi$
$647$		0			0		0.160411	+	0.987050i	$0.448718\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	−0.510808	−	0.125903i	−0.0200201	−	0.00493452i
$652$	−12.4755	−	11.5090i	−0.488579	−	0.450726i
$653$		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
$657$	−34.3319	+	9.19920i	−1.33941	+	0.358895i
$658$		0			0
$659$		0			0		0.0402659	−	0.999189i	$-0.487179\pi$
$659$		0			0		−0.0402659	+	0.999189i	$0.512821\pi$
$660$		0			0
$661$	−11.2232	−	49.8326i	−0.436533	−	1.93826i	−0.330112	−	0.943942i	$-0.607086\pi$
$661$	−11.2232	−	49.8326i	−0.436533	−	1.93826i	−0.106421	−	0.994321i	$-0.533939\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	31.8830	+	34.5605i	1.23267	+	1.33619i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−27.0735	−	12.8466i	−1.04361	−	0.495198i	−0.171844	−	0.985124i	$-0.554973\pi$
$673$	−27.0735	−	12.8466i	−1.04361	−	0.495198i	−0.871764	+	0.489926i	$0.837024\pi$
$674$		0			0
$675$		−	25.9808i		−	1.00000i
$676$	−24.9604	+	7.27874i	−0.960014	+	0.279952i
$677$		0			0		1.00000			$0$
$677$		0			0		−1.00000			$\pi$
$678$		0			0
$679$	−0.200155	+	0.421818i	−0.00768124	+	0.0161879i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		0.735006	−	0.678061i	$-0.237179\pi$
$683$		0			0		−0.735006	+	0.678061i	$0.762821\pi$
$684$	36.9031	+	11.4995i	1.41103	+	0.439693i
$685$		0			0
$686$		0			0
$687$	−51.9332	+	5.24701i	−1.98138	+	0.200186i
$688$	−30.0793	−	3.65229i	−1.14676	−	0.139242i
$689$		0			0
$690$		0			0
$691$	41.8636	−	9.42846i	1.59257	−	0.358676i	0.669347	−	0.742950i	$-0.266574\pi$
$691$	41.8636	−	9.42846i	1.59257	−	0.358676i	0.923219	+	0.384274i	$0.125548\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	−0.318602	+	0.345359i	−0.0120420	+	0.0130533i
$701$		0			0		0.239316	−	0.970942i	$-0.423077\pi$
$701$		0			0		−0.239316	+	0.970942i	$0.576923\pi$
$702$		0			0
$703$	−72.6106	+	17.8969i	−2.73856	+	0.674994i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	16.1288	−	45.2564i	0.605729	−	1.69964i	−0.103470	−	0.994633i	$-0.532995\pi$
$709$	16.1288	−	45.2564i	0.605729	−	1.69964i	0.709199	−	0.705008i	$-0.249057\pi$
$710$		0			0
$711$	−52.0754	−	8.46308i	−1.95298	−	0.317390i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		−0.721202	−	0.692724i	$-0.756410\pi$
$719$		0			0		0.721202	+	0.692724i	$0.243590\pi$
$720$		0			0
$721$	0.152629	−	0.677694i	0.00568421	−	0.0252386i
$722$		0			0
$723$	49.2864	+	2.98128i	1.83298	+	0.110875i
$724$	−5.18030	−	25.3748i	−0.192524	−	0.943047i
$725$		0			0
$726$		0			0
$727$	15.3002	−	29.1522i	0.567454	−	1.08119i	−0.417548	−	0.908655i	$-0.637111\pi$
$727$	15.3002	−	29.1522i	0.567454	−	1.08119i	0.985003	−	0.172539i	$-0.0551971\pi$
$728$		0			0
$729$	23.9073	−	12.5475i	0.885456	−	0.464723i
$730$		0			0
$731$		0			0
$732$	−47.2667	+	22.4283i	−1.74703	+	0.828974i
$733$	9.13877	+	1.67474i	0.337548	+	0.0618580i	0.346356	−	0.938103i	$-0.387419\pi$
$733$	9.13877	+	1.67474i	0.337548	+	0.0618580i	−0.00880766	+	0.999961i	$0.502804\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	50.3957	+	7.15167i	1.85384	+	0.263078i	0.976299	−	0.216427i	$-0.0694404\pi$
$739$	50.3957	+	7.15167i	1.85384	+	0.263078i	0.877539	+	0.479506i	$0.159184\pi$
$740$		0			0
$741$	−33.5434	+	22.2131i	−1.23225	+	0.816018i
$742$		0			0
$743$		0			0		0.894635	−	0.446798i	$-0.147436\pi$
$743$		0			0		−0.894635	+	0.446798i	$0.852564\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	41.6030	+	33.9678i	1.51811	+	1.23950i	0.885137	+	0.465331i	$0.154065\pi$
$751$	41.6030	+	33.9678i	1.51811	+	1.23950i	0.632977	+	0.774171i	$0.281833\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$	−0.471667	−	0.126383i	−0.0171544	−	0.00459650i
$757$	−11.0064	+	53.9131i	−0.400036	+	1.95951i	−0.191400	+	0.981512i	$0.561303\pi$
$757$	−11.0064	+	53.9131i	−0.400036	+	1.95951i	−0.208635	+	0.977993i	$0.566902\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		−0.584522	−	0.811378i	$-0.698718\pi$
$761$		0			0		0.584522	+	0.811378i	$0.301282\pi$
$762$		0			0
$763$	0.125701	−	0.433970i	0.00455068	−	0.0157108i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	−20.7433	−	18.3770i	−0.748511	−	0.663123i
$769$	−16.7538	−	25.3492i	−0.604157	−	0.914114i	0.395834	−	0.918322i	$-0.370455\pi$
$769$	−16.7538	−	25.3492i	−0.604157	−	0.914114i	−0.999991	+	0.00420807i	$0.998661\pi$
$770$		0			0
$771$		0			0
$772$	−14.6635	−	6.59948i	−0.527749	−	0.237521i
$773$		0			0		0.927686	−	0.373361i	$-0.121795\pi$
$773$		0			0		−0.927686	+	0.373361i	$0.878205\pi$
$774$		0			0
$775$	−22.8549	−	22.8549i	−0.820971	−	0.820971i
$776$		0			0
$777$	0.907439	−	0.262843i	0.0325542	−	0.00942944i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−14.9603	−	23.6579i	−0.534297	−	0.844924i
$785$		0			0
$786$		0			0
$787$	−45.3196	−	22.6335i	−1.61547	−	0.806797i	−0.999824	−	0.0187723i	$-0.994024\pi$
$787$	−45.3196	−	22.6335i	−1.61547	−	0.806797i	−0.615644	−	0.788025i	$-0.711104\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	12.9840	−	52.8836i	0.461076	−	1.87795i
$794$		0			0
$795$		0			0
$796$	29.9529	+	2.41805i	1.06165	+	0.0857054i
$797$		0			0		0.979791	−	0.200026i	$-0.0641026\pi$
$797$		0			0		−0.979791	+	0.200026i	$0.935897\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$	−51.1845	+	18.2415i	−1.80514	+	0.643327i
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		−0.845190	−	0.534466i	$-0.820513\pi$
$809$		0			0		0.845190	+	0.534466i	$0.179487\pi$
$810$		0			0
$811$	3.70798	+	2.90501i	0.130205	+	0.102009i	0.679708	−	0.733482i	$-0.262106\pi$
$811$	3.70798	+	2.90501i	0.130205	+	0.102009i	−0.549504	+	0.835491i	$0.685183\pi$
$812$		0			0
$813$	−29.9774	+	0.603780i	−1.05135	+	0.0211755i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	23.5441	−	42.7449i	0.823703	−	1.49546i
$818$		0			0
$819$	0.412110	−	0.297453i	0.0144003	−	0.0103938i
$820$		0			0
$821$		0			0		−0.735006	−	0.678061i	$-0.762821\pi$
$821$		0			0		0.735006	+	0.678061i	$0.237179\pi$
$822$		0			0
$823$	−34.6602	−	20.0111i	−1.20818	−	0.697542i	−0.245817	−	0.969316i	$-0.579056\pi$
$823$	−34.6602	−	20.0111i	−1.20818	−	0.697542i	−0.962361	+	0.271774i	$0.912390\pi$
$824$		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
$829$	17.2659	+	22.9767i	0.599669	+	0.798015i	0.992584	−	0.121560i	$-0.0387897\pi$
$829$	17.2659	+	22.9767i	0.599669	+	0.798015i	−0.392915	+	0.919575i	$0.628533\pi$
$830$		0			0
$831$	−16.4903	+	2.00228i	−0.572040	+	0.0694583i
$832$	28.4667	−	4.65267i	0.986905	−	0.161302i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	9.99303	−	32.0688i	0.345410	−	1.10846i
$838$		0			0
$839$		0			0		0.875918	−	0.482459i	$-0.160256\pi$
$839$		0			0		−0.875918	+	0.482459i	$0.839744\pi$
$840$		0			0
$841$	27.5076	−	9.18337i	0.948536	−	0.316668i
$842$		0			0
$843$		0			0
$844$			56.6738i			1.95079i
$845$		0			0
$846$		0			0
$847$	−0.302116	+	0.419369i	−0.0103808	+	0.0144097i
$848$		0			0
$849$	10.6072	+	31.7724i	0.364038	+	1.09043i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−1.23127	−	0.383679i	−0.0421579	−	0.0131369i	0.276352	−	0.961057i	$-0.410875\pi$
$853$	−1.23127	−	0.383679i	−0.0421579	−	0.0131369i	−0.318509	+	0.947920i	$0.603182\pi$
$854$		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
$859$	−5.20333	−	42.8533i	−0.177535	−	1.46214i	−0.760341	−	0.649524i	$-0.774968\pi$
$859$	−5.20333	−	42.8533i	−0.177535	−	1.46214i	0.582805	−	0.812612i	$-0.301955\pi$
$860$		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
$868$	−0.526096	+	0.303741i	−0.0178568	+	0.0103097i
$869$		0			0
$870$		0			0
$871$	18.9382	−	53.2916i	0.641696	−	1.80572i
$872$		0			0
$873$	−26.1111	−	14.3821i	−0.883727	−	0.486761i
$874$		0			0
$875$		0			0
$876$	−21.2323	+	35.1226i	−0.717374	+	1.18668i
$877$	−0.728687	−	36.1791i	−0.0246060	−	1.22168i	−0.803801	−	0.594898i	$-0.797192\pi$
$877$	−0.728687	−	36.1791i	−0.0246060	−	1.22168i	0.779195	−	0.626782i	$-0.215628\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		0.534466	−	0.845190i	$-0.320513\pi$
$881$		0			0		−0.534466	+	0.845190i	$0.679487\pi$
$882$		0			0
$883$	−55.3194	−	20.9799i	−1.86165	−	0.706029i	−0.964369	−	0.264560i	$-0.914773\pi$
$883$	−55.3194	−	20.9799i	−1.86165	−	0.706029i	−0.897276	−	0.441469i	$-0.854458\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		−0.948536	−	0.316668i	$-0.897436\pi$
$887$		0			0		0.948536	+	0.316668i	$0.102564\pi$
$888$		0			0
$889$	−0.433374	−	0.962917i	−0.0145349	−	0.0322952i
$890$		0			0
$891$		0			0
$892$	54.1959	+	3.27825i	1.81461	+	0.109764i
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	−20.7817	−	21.6361i	−0.692724	−	0.721202i
$901$		0			0
$902$		0			0
$903$	−0.275447	+	0.551534i	−0.00916631	+	0.0183539i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	10.9988	+	25.8151i	0.365208	+	0.857176i	0.996653	+	0.0817542i	$0.0260522\pi$
$907$	10.9988	+	25.8151i	0.365208	+	0.857176i	−0.631444	+	0.775421i	$0.717537\pi$
$908$		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	$0.307692\pi$
$912$	39.9302	−	19.9419i	1.32222	−	0.660344i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	−39.0515	+	45.9104i	−1.29030	+	1.51692i
$917$		0			0
$918$		0			0
$919$	−14.2513	−	10.7092i	−0.470107	−	0.353263i	0.339135	−	0.940738i	$-0.389866\pi$
$919$	−14.2513	−	10.7092i	−0.470107	−	0.353263i	−0.809242	+	0.587475i	$0.800122\pi$
$920$		0			0
$921$	−17.4354	+	11.5234i	−0.574516	+	0.379709i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	56.0641	+	15.0223i	1.84338	+	0.493931i
$926$		0			0
$927$	42.6015	+	12.3397i	1.39922	+	0.405288i
$928$		0			0
$929$		0			0		0.811378	−	0.584522i	$-0.198718\pi$
$929$		0			0		−0.811378	+	0.584522i	$0.801282\pi$
$930$		0			0
$931$	44.3429	−	8.12613i	1.45328	−	0.266323i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	32.9080	+	29.1539i	1.07506	+	0.952418i	0.999001	−	0.0446860i	$-0.0142287\pi$
$937$	32.9080	+	29.1539i	1.07506	+	0.952418i	0.0760561	+	0.997104i	$0.475767\pi$
$938$		0			0
$939$	−8.75864	+	10.7274i	−0.285827	+	0.350075i
$940$		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		−0.446798	−	0.894635i	$-0.647436\pi$
$947$		0			0		0.446798	+	0.894635i	$0.352564\pi$
$948$	−50.1365	+	34.6067i	−1.62836	+	1.12397i
$949$	−14.3767	−	40.2254i	−0.466689	−	1.30577i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		−0.534466	−	0.845190i	$-0.679487\pi$
$953$		0			0		0.534466	+	0.845190i	$0.320513\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−5.01326	−	9.55198i	−0.161718	−	0.308128i
$962$		0			0
$963$		0			0
$964$	43.4291	−	36.9410i	1.39876	−	1.18979i
$965$		0			0
$966$		0			0
$967$	−1.42005	+	23.4763i	−0.0456658	+	0.754946i	0.901217	+	0.433368i	$0.142675\pi$
$967$	−1.42005	+	23.4763i	−0.0456658	+	0.754946i	−0.946883	+	0.321578i	$0.895787\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		0.692724	−	0.721202i	$-0.256410\pi$
$971$		0			0		−0.692724	+	0.721202i	$0.743590\pi$
$972$	9.87273	−	29.5724i	0.316668	−	0.948536i
$973$	0.0623189	−	0.0222096i	0.00199785	−	0.000712007i
$974$		0			0
$975$	31.1215	−	2.54067i	0.996684	−	0.0813665i
$976$	−21.4222	+	56.4858i	−0.685709	+	1.80807i
$977$		0			0		−0.999797	−	0.0201371i	$-0.993590\pi$
$977$		0			0		0.999797	+	0.0201371i	$0.00641026\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	27.1726	+	9.68394i	0.867554	+	0.309185i
$982$		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
$988$	−10.1661	+	45.3295i	−0.323425	+	1.44212i
$989$		0			0
$990$		0			0
$991$	28.8548	+	49.9779i	0.916601	+	1.58760i	0.804540	+	0.593899i	$0.202412\pi$
$991$	28.8548	+	49.9779i	0.916601	+	1.58760i	0.112062	+	0.993701i	$0.464255\pi$
$992$		0			0
$993$	−27.5702	+	8.59124i	−0.874915	+	0.272635i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−2.08968	+	51.8549i	−0.0661808	+	1.64226i	0.536228	+	0.844073i	$0.319849\pi$
$997$	−2.08968	+	51.8549i	−0.0661808	+	1.64226i	−0.602409	+	0.798188i	$0.705792\pi$
$998$		0			0
$999$	13.2530	+	58.8449i	0.419305	+	1.86177i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	507.2.x.a.197.1	✓	48
3.2	odd	2	CM	507.2.x.a.197.1	✓	48
169.163	odd	156	inner	507.2.x.a.332.1	yes	48
507.332	even	156	inner	507.2.x.a.332.1	yes	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
507.2.x.a.197.1	✓	48	1.1	even	1	trivial
507.2.x.a.197.1	✓	48	3.2	odd	2	CM
507.2.x.a.332.1	yes	48	169.163	odd	156	inner
507.2.x.a.332.1	yes	48	507.332	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)

\(f(q)\)	\(=\)	\(q+(0.742517 - 1.56482i) q^{3} +(-0.633336 - 1.89707i) q^{4} +(-0.0345359 + 0.0318602i) q^{7} +(-1.89734 - 2.32381i) q^{9} +O(q^{10})\)	\(q+(0.742517 - 1.56482i) q^{3} +(-0.633336 - 1.89707i) q^{4} +(-0.0345359 + 0.0318602i) q^{7} +(-1.89734 - 2.32381i) q^{9} +(-3.43884 - 0.417551i) q^{12} +(2.59808 - 2.50000i) q^{13} +(-3.19777 + 2.40297i) q^{16} +(-1.66737 - 6.22270i) q^{19} +(0.0242121 + 0.0776993i) q^{21} +(-1.19658 + 4.85471i) q^{25} +(-5.04516 + 1.24352i) q^{27} +(0.0823140 + 0.0453389i) q^{28} +(-3.34425 + 5.53206i) q^{31} +(-3.20679 + 5.07114i) q^{36} +(0.233758 - 11.6060i) q^{37} +(-1.98294 - 5.92182i) q^{39} +(5.46315 + 5.24743i) q^{43} +(1.38582 + 6.78819i) q^{48} +(-0.563088 + 6.97510i) q^{49} +(-6.38814 - 3.34540i) q^{52} +(-10.9755 - 2.01133i) q^{57} +(12.7648 - 8.07198i) q^{61} +(0.139563 + 0.0198055i) q^{63} +(6.58387 + 4.54452i) q^{64} +(14.0332 - 7.00847i) q^{67} +(4.86243 - 10.8039i) q^{73} +(6.70828 + 5.47714i) q^{75} +(-10.7489 + 7.10418i) q^{76} +(13.1635 - 11.6618i) q^{79} +(-1.80023 + 8.81812i) q^{81} +(0.132067 - 0.0951418i) q^{84} +(-0.0100763 + 0.169115i) q^{91} +(6.17353 + 9.34080i) q^{93} +(9.21810 - 3.70997i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 10 q^{7} + 6 q^{9}+O(q^{10}) \) 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

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