LMFDB - Embedded newform 2736.2.cg.a.2591.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2736,2,Mod(1151,2736)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2736, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 0, 3, 4]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2736.1151");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2736 = 2^{4} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2736.cg (of order $6$, degree $2$, 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:	$21.8470699930$
Analytic rank:	$0$
Dimension:	$24$
Relative dimension:	$12$ over $\Q(\zeta_{6})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			2591.8
Character	$\chi$	$=$	2736.2591
Dual form			2736.2.cg.a.1151.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.02690 - 0.592883i) q^{5} +4.00663i q^{7} +O(q^{10})$	$q+(1.02690 - 0.592883i) q^{5} +4.00663i q^{7} +3.72120 q^{11} +(-0.163969 + 0.284003i) q^{13} +(-2.92851 + 1.69078i) q^{17} +(3.68810 + 2.32335i) q^{19} +(1.20510 - 2.08730i) q^{23} +(-1.79698 + 3.11246i) q^{25} +(0.884644 + 0.510749i) q^{29} +0.252014i q^{31} +(2.37546 + 4.11443i) q^{35} -2.26602 q^{37} +(-6.04391 + 3.48945i) q^{41} +(0.904236 - 0.522061i) q^{43} +(3.48518 - 6.03651i) q^{47} -9.05312 q^{49} +(10.8125 + 6.24262i) q^{53} +(3.82131 - 2.20623i) q^{55} +(0.581471 + 1.00714i) q^{59} +(-4.11586 + 7.12888i) q^{61} +0.388857i q^{65} +(-8.16604 - 4.71466i) q^{67} +(1.12392 + 1.94669i) q^{71} +(-1.05266 - 1.82325i) q^{73} +14.9095i q^{77} +(-5.56073 + 3.21049i) q^{79} -5.14422 q^{83} +(-2.00486 + 3.47253i) q^{85} +(3.86374 + 2.23073i) q^{89} +(-1.13789 - 0.656964i) q^{91} +(5.16479 + 0.199241i) q^{95} +(-4.12733 - 7.14874i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q+O(q^{10}) $ 24 * q	$ 24 q + 12 q^{13} - 12 q^{19} + 8 q^{25} + 16 q^{37} - 12 q^{43} + 16 q^{49} + 12 q^{55} - 60 q^{67} + 8 q^{73} + 12 q^{79} + 16 q^{85} - 12 q^{91} - 32 q^{97}+O(q^{100}) $ 24 * q + 12 * q^13 - 12 * q^19 + 8 * q^25 + 16 * q^37 - 12 * q^43 + 16 * q^49 + 12 * q^55 - 60 * q^67 + 8 * q^73 + 12 * q^79 + 16 * q^85 - 12 * q^91 - 32 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2736\mathbb{Z}\right)^\times$.

$n$	$1009$	$1217$	$1711$	$2053$
$\chi(n)$	$e\left(\frac{1}{3}\right)$	$-1$	$-1$	$1$

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
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	1.02690	−	0.592883i	0.459245	−	0.265145i	−0.252482	−	0.967602i	$-0.581247\pi$
$5$	1.02690	−	0.592883i	0.459245	−	0.265145i	0.711727	+	0.702456i	$0.247913\pi$
$6$		0			0
$7$			4.00663i			1.51437i	0.653203	+	0.757183i	$0.273425\pi$
$7$			4.00663i			1.51437i	−0.653203	+	0.757183i	$0.726575\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	3.72120			1.12198			0.560992	−	0.827821i	$-0.310420\pi$
$11$	3.72120			1.12198			0.560992	+	0.827821i	$0.310420\pi$
$12$		0			0
$13$	−0.163969	+	0.284003i	−0.0454768	+	0.0787681i	−0.887868	−	0.460099i	$-0.847814\pi$
$13$	−0.163969	+	0.284003i	−0.0454768	+	0.0787681i	0.842391	+	0.538867i	$0.181147\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−2.92851	+	1.69078i	−0.710268	+	0.410073i	−0.811160	−	0.584824i	$-0.801164\pi$
$17$	−2.92851	+	1.69078i	−0.710268	+	0.410073i	0.100892	+	0.994897i	$0.467830\pi$
$18$		0			0
$19$	3.68810	+	2.32335i	0.846108	+	0.533012i
$19$	3.68810	+	2.32335i	0.846108	+	0.533012i
$20$		0			0
$21$		0			0
$22$		0			0
$23$	1.20510	−	2.08730i	0.251282	−	0.435232i	−0.712597	−	0.701573i	$-0.752481\pi$
$23$	1.20510	−	2.08730i	0.251282	−	0.435232i	0.963879	+	0.266341i	$0.0858147\pi$
$24$		0			0
$25$	−1.79698	+	3.11246i	−0.359396	+	0.622492i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	0.884644	+	0.510749i	0.164274	+	0.0948438i	0.579883	−	0.814700i	$-0.303098\pi$
$29$	0.884644	+	0.510749i	0.164274	+	0.0948438i	−0.415609	+	0.909543i	$0.636432\pi$
$30$		0			0
$31$			0.252014i			0.0452630i	0.999744	+	0.0226315i	$0.00720445\pi$
$31$			0.252014i			0.0452630i	−0.999744	+	0.0226315i	$0.992796\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	2.37546	+	4.11443i	0.401527	+	0.695465i
$36$		0			0
$37$	−2.26602			−0.372532			−0.186266	−	0.982499i	$-0.559639\pi$
$37$	−2.26602			−0.372532			−0.186266	+	0.982499i	$0.559639\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−6.04391	+	3.48945i	−0.943900	+	0.544961i	−0.891181	−	0.453648i	$-0.850122\pi$
$41$	−6.04391	+	3.48945i	−0.943900	+	0.544961i	−0.0527193	+	0.998609i	$0.516789\pi$
$42$		0			0
$43$	0.904236	−	0.522061i	0.137895	−	0.0796135i	−0.429466	−	0.903083i	$-0.641298\pi$
$43$	0.904236	−	0.522061i	0.137895	−	0.0796135i	0.567360	+	0.823470i	$0.307965\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	3.48518	−	6.03651i	0.508365	−	0.880515i	−0.491588	−	0.870828i	$-0.663583\pi$
$47$	3.48518	−	6.03651i	0.508365	−	0.880515i	0.999953	−	0.00968668i	$-0.00308341\pi$
$48$		0			0
$49$	−9.05312			−1.29330
$50$		0			0
$51$		0			0
$52$		0			0
$53$	10.8125	+	6.24262i	1.48522	+	0.857490i	0.999858	−	0.0168248i	$-0.00535574\pi$
$53$	10.8125	+	6.24262i	1.48522	+	0.857490i	0.485359	+	0.874315i	$0.338689\pi$
$54$		0			0
$55$	3.82131	−	2.20623i	0.515265	−	0.297489i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	0.581471	+	1.00714i	0.0757010	+	0.131118i	0.901391	−	0.433006i	$-0.142547\pi$
$59$	0.581471	+	1.00714i	0.0757010	+	0.131118i	−0.825690	+	0.564124i	$0.809214\pi$
$60$		0			0
$61$	−4.11586	+	7.12888i	−0.526982	+	0.912759i	0.472524	+	0.881318i	$0.343343\pi$
$61$	−4.11586	+	7.12888i	−0.526982	+	0.912759i	−0.999506	+	0.0314412i	$0.989990\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$			0.388857i			0.0482318i
$66$		0			0
$67$	−8.16604	−	4.71466i	−0.997640	−	0.575988i	−0.0900911	−	0.995934i	$-0.528716\pi$
$67$	−8.16604	−	4.71466i	−0.997640	−	0.575988i	−0.907549	+	0.419946i	$0.862049\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	1.12392	+	1.94669i	0.133385	+	0.231030i	0.924979	−	0.380017i	$-0.124082\pi$
$71$	1.12392	+	1.94669i	0.133385	+	0.231030i	−0.791594	+	0.611047i	$0.790749\pi$
$72$		0			0
$73$	−1.05266	−	1.82325i	−0.123204	−	0.213395i	0.797826	−	0.602888i	$-0.205984\pi$
$73$	−1.05266	−	1.82325i	−0.123204	−	0.213395i	−0.921029	+	0.389493i	$0.872650\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$			14.9095i			1.69909i
$78$		0			0
$79$	−5.56073	+	3.21049i	−0.625631	+	0.361208i	−0.779058	−	0.626952i	$-0.784302\pi$
$79$	−5.56073	+	3.21049i	−0.625631	+	0.361208i	0.153427	+	0.988160i	$0.450969\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−5.14422			−0.564651			−0.282326	−	0.959319i	$-0.591106\pi$
$83$	−5.14422			−0.564651			−0.282326	+	0.959319i	$0.591106\pi$
$84$		0			0
$85$	−2.00486	+	3.47253i	−0.217458	+	0.376648i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	3.86374	+	2.23073i	0.409556	+	0.236457i	0.690599	−	0.723238i	$-0.257347\pi$
$89$	3.86374	+	2.23073i	0.409556	+	0.236457i	−0.281043	+	0.959695i	$0.590680\pi$
$90$		0			0
$91$	−1.13789	−	0.656964i	−0.119284	−	0.0688685i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	5.16479	+	0.199241i	0.529896	+	0.0204417i
$96$		0			0
$97$	−4.12733	−	7.14874i	−0.419066	−	0.725844i	0.576779	−	0.816900i	$-0.304309\pi$
$97$	−4.12733	−	7.14874i	−0.419066	−	0.725844i	−0.995846	+	0.0910556i	$0.970976\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	12.2411	+	7.06739i	1.21803	+	0.703232i	0.964497	−	0.264094i	$-0.0850728\pi$
$101$	12.2411	+	7.06739i	1.21803	+	0.703232i	0.253537	+	0.967326i	$0.418406\pi$
$102$		0			0
$103$			5.07024i			0.499586i	0.968299	+	0.249793i	$0.0803626\pi$
$103$			5.07024i			0.499586i	−0.968299	+	0.249793i	$0.919637\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−0.864112			−0.0835368			−0.0417684	−	0.999127i	$-0.513299\pi$
$107$	−0.864112			−0.0835368			−0.0417684	+	0.999127i	$0.513299\pi$
$108$		0			0
$109$	−9.36177	−	16.2151i	−0.896695	−	1.55312i	−0.831693	−	0.555236i	$-0.812628\pi$
$109$	−9.36177	−	16.2151i	−0.896695	−	1.55312i	−0.0650023	−	0.997885i	$-0.520705\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			15.4626i			1.45460i	0.686320	+	0.727300i	$0.259225\pi$
$113$			15.4626i			1.45460i	−0.686320	+	0.727300i	$0.740775\pi$
$114$		0			0
$115$		−	2.85794i		−	0.266504i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−6.77432	−	11.7335i	−0.621001	−	1.07561i
$120$		0			0
$121$	2.84732			0.258847
$122$		0			0
$123$		0			0
$124$		0			0
$125$			10.1904i			0.911459i
$126$		0			0
$127$	13.0248	+	7.51987i	1.15576	+	0.667281i	0.950285	−	0.311381i	$-0.100792\pi$
$127$	13.0248	+	7.51987i	1.15576	+	0.667281i	0.205479	+	0.978662i	$0.434125\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	6.07702	+	10.5257i	0.530952	+	0.919636i	0.999348	+	0.0361169i	$0.0114989\pi$
$131$	6.07702	+	10.5257i	0.530952	+	0.919636i	−0.468396	+	0.883519i	$0.655168\pi$
$132$		0			0
$133$	−9.30880	+	14.7769i	−0.807175	+	1.28132i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−6.62454	−	3.82468i	−0.565973	−	0.326765i	0.189567	−	0.981868i	$-0.439292\pi$
$137$	−6.62454	−	3.82468i	−0.565973	−	0.326765i	−0.755539	+	0.655103i	$0.772625\pi$
$138$		0			0
$139$	6.50872	+	3.75781i	0.552062	+	0.318733i	0.749953	−	0.661491i	$-0.230076\pi$
$139$	6.50872	+	3.75781i	0.552062	+	0.318733i	−0.197891	+	0.980224i	$0.563409\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−0.610161	+	1.05683i	−0.0510242	+	0.0883766i
$144$		0			0
$145$	1.21126			0.100590
$146$		0			0
$147$		0			0
$148$		0			0
$149$	2.65727	−	1.53418i	0.217692	−	0.125685i	−0.387189	−	0.922000i	$-0.626554\pi$
$149$	2.65727	−	1.53418i	0.217692	−	0.125685i	0.604881	+	0.796316i	$0.293221\pi$
$150$		0			0
$151$			5.37855i			0.437701i	0.975758	+	0.218850i	$0.0702306\pi$
$151$			5.37855i			0.437701i	−0.975758	+	0.218850i	$0.929769\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	0.149415	+	0.258794i	0.0120013	+	0.0207868i
$156$		0			0
$157$	3.92609	+	6.80019i	0.313336	+	0.542714i	0.979082	−	0.203464i	$-0.0652200\pi$
$157$	3.92609	+	6.80019i	0.313336	+	0.542714i	−0.665746	+	0.746178i	$0.731887\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	8.36305	+	4.82841i	0.659101	+	0.380532i
$162$		0			0
$163$			11.0089i			0.862286i	0.902284	+	0.431143i	$0.141890\pi$
$163$			11.0089i			0.862286i	−0.902284	+	0.431143i	$0.858110\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	10.7707	−	18.6555i	0.833465	−	1.44360i	−0.0618093	−	0.998088i	$-0.519687\pi$
$167$	10.7707	−	18.6555i	0.833465	−	1.44360i	0.895274	−	0.445516i	$-0.146980\pi$
$168$		0			0
$169$	6.44623	+	11.1652i	0.495864	+	0.858861i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−11.8493	+	6.84121i	−0.900888	+	0.520128i	−0.877488	−	0.479598i	$-0.840782\pi$
$173$	−11.8493	+	6.84121i	−0.900888	+	0.520128i	−0.0233997	+	0.999726i	$0.507449\pi$
$174$		0			0
$175$	−12.4705	−	7.19984i	−0.942681	−	0.544257i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	12.7209			0.950806			0.475403	−	0.879768i	$-0.342302\pi$
$179$	12.7209			0.950806			0.475403	+	0.879768i	$0.342302\pi$
$180$		0			0
$181$	2.80425	−	4.85711i	0.208438	−	0.361026i	−0.742784	−	0.669531i	$-0.766495\pi$
$181$	2.80425	−	4.85711i	0.208438	−	0.361026i	0.951223	+	0.308505i	$0.0998286\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−2.32699	+	1.34349i	−0.171083	+	0.0987750i
$186$		0			0
$187$	−10.8976	+	6.29171i	−0.796909	+	0.460096i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	15.3741			1.11243			0.556214	−	0.831039i	$-0.312254\pi$
$191$	15.3741			1.11243			0.556214	+	0.831039i	$0.312254\pi$
$192$		0			0
$193$	−12.9663	−	22.4582i	−0.933332	−	1.61658i	−0.777583	−	0.628781i	$-0.783554\pi$
$193$	−12.9663	−	22.4582i	−0.933332	−	1.61658i	−0.155749	−	0.987797i	$-0.549779\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	20.7134i		−	1.47577i	−0.674929	−	0.737883i	$-0.735826\pi$
$197$		−	20.7134i		−	1.47577i	0.674929	−	0.737883i	$-0.264174\pi$
$198$		0			0
$199$	22.0917	+	12.7546i	1.56604	+	0.904151i	0.996624	+	0.0820992i	$0.0261624\pi$
$199$	22.0917	+	12.7546i	1.56604	+	0.904151i	0.569412	+	0.822052i	$0.307171\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−2.04639	+	3.54445i	−0.143628	+	0.248771i
$204$		0			0
$205$	−4.13767	+	7.16666i	−0.288988	+	0.500541i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	13.7241	+	8.64563i	0.949319	+	0.598031i
$210$		0			0
$211$	−14.3841	+	8.30467i	−0.990243	+	0.571717i	−0.905347	−	0.424672i	$-0.860389\pi$
$211$	−14.3841	+	8.30467i	−0.990243	+	0.571717i	−0.0848964	+	0.996390i	$0.527056\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	0.619042	−	1.07221i	0.0422183	−	0.0731242i
$216$		0			0
$217$	−1.00973			−0.0685448
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	1.10894i		−	0.0745953i
$222$		0			0
$223$	10.1426	−	5.85583i	0.679199	−	0.392136i	−0.120354	−	0.992731i	$-0.538403\pi$
$223$	10.1426	−	5.85583i	0.679199	−	0.392136i	0.799553	+	0.600595i	$0.205070\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	19.4717			1.29238			0.646191	−	0.763176i	$-0.276361\pi$
$227$	19.4717			1.29238			0.646191	+	0.763176i	$0.276361\pi$
$228$		0			0
$229$	−1.42187			−0.0939598			−0.0469799	−	0.998896i	$-0.514960\pi$
$229$	−1.42187			−0.0939598			−0.0469799	+	0.998896i	$0.514960\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	12.8455	−	7.41633i	0.841534	−	0.485860i	−0.0162514	−	0.999868i	$-0.505173\pi$
$233$	12.8455	−	7.41633i	0.841534	−	0.485860i	0.857785	+	0.514008i	$0.171840\pi$
$234$		0			0
$235$		−	8.26521i		−	0.539163i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	26.4103			1.70834			0.854169	−	0.519996i	$-0.174067\pi$
$239$	26.4103			1.70834			0.854169	+	0.519996i	$0.174067\pi$
$240$		0			0
$241$	−9.04260	+	15.6622i	−0.582485	+	1.00889i	0.412699	+	0.910867i	$0.364586\pi$
$241$	−9.04260	+	15.6622i	−0.582485	+	1.00889i	−0.995184	+	0.0980259i	$0.968747\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−9.29668	+	5.36744i	−0.593943	+	0.342913i
$246$		0			0
$247$	−1.26457	+	0.666473i	−0.0804626	+	0.0424066i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−7.26539	+	12.5840i	−0.458588	+	0.794297i	−0.998887	−	0.0471760i	$-0.984978\pi$
$251$	−7.26539	+	12.5840i	−0.458588	+	0.794297i	0.540299	+	0.841473i	$0.318311\pi$
$252$		0			0
$253$	4.48443	−	7.76726i	0.281934	−	0.488324i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−17.6815	−	10.2084i	−1.10294	−	0.636785i	−0.165951	−	0.986134i	$-0.553069\pi$
$257$	−17.6815	−	10.2084i	−1.10294	−	0.636785i	−0.936993	+	0.349349i	$0.886403\pi$
$258$		0			0
$259$		−	9.07912i		−	0.564149i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−0.386150	−	0.668831i	−0.0238110	−	0.0412419i	0.853874	−	0.520479i	$-0.174247\pi$
$263$	−0.386150	−	0.668831i	−0.0238110	−	0.0412419i	−0.877685	+	0.479237i	$0.840913\pi$
$264$		0			0
$265$	14.8046			0.909438
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−19.0019	+	10.9708i	−1.15857	+	0.668900i	−0.950960	−	0.309313i	$-0.899901\pi$
$269$	−19.0019	+	10.9708i	−1.15857	+	0.668900i	−0.207608	+	0.978212i	$0.566568\pi$
$270$		0			0
$271$	2.24497	−	1.29614i	0.136372	−	0.0787346i	−0.430262	−	0.902704i	$-0.641579\pi$
$271$	2.24497	−	1.29614i	0.136372	−	0.0787346i	0.566634	+	0.823970i	$0.308245\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−6.68692	+	11.5821i	−0.403236	+	0.698426i
$276$		0			0
$277$	−8.39074			−0.504151			−0.252075	−	0.967708i	$-0.581113\pi$
$277$	−8.39074			−0.504151			−0.252075	+	0.967708i	$0.581113\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	16.4751	+	9.51188i	0.982820	+	0.567431i	0.903120	−	0.429388i	$-0.141271\pi$
$281$	16.4751	+	9.51188i	0.982820	+	0.567431i	0.0796995	+	0.996819i	$0.474604\pi$
$282$		0			0
$283$	7.56151	−	4.36564i	0.449485	−	0.259510i	−0.258128	−	0.966111i	$-0.583105\pi$
$283$	7.56151	−	4.36564i	0.449485	−	0.259510i	0.707613	+	0.706600i	$0.249772\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−13.9810	−	24.2157i	−0.825270	−	1.42941i
$288$		0			0
$289$	−2.78255	+	4.81953i	−0.163680	+	0.283502i
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	19.9108i		−	1.16320i	−0.813475	−	0.581600i	$-0.802427\pi$
$293$		−	19.9108i		−	1.16320i	0.813475	−	0.581600i	$-0.197573\pi$
$294$		0			0
$295$	1.19423	+	0.689488i	0.0695306	+	0.0401435i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	0.395199	+	0.684505i	0.0228550	+	0.0395860i
$300$		0			0
$301$	2.09171	+	3.62294i	0.120564	+	0.208823i
$302$		0			0
$303$		0			0
$304$		0			0
$305$			9.76089i			0.558907i
$306$		0			0
$307$	−9.99756	+	5.77209i	−0.570591	+	0.329431i	−0.757385	−	0.652968i	$-0.773524\pi$
$307$	−9.99756	+	5.77209i	−0.570591	+	0.329431i	0.186794	+	0.982399i	$0.440190\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−28.9643			−1.64241			−0.821207	−	0.570630i	$-0.806699\pi$
$311$	−28.9643			−1.64241			−0.821207	+	0.570630i	$0.806699\pi$
$312$		0			0
$313$	−9.67318	+	16.7544i	−0.546760	+	0.947016i	0.451734	+	0.892153i	$0.350806\pi$
$313$	−9.67318	+	16.7544i	−0.546760	+	0.947016i	−0.998494	+	0.0548636i	$0.982528\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−18.5107	−	10.6871i	−1.03966	−	0.600249i	−0.119925	−	0.992783i	$-0.538265\pi$
$317$	−18.5107	−	10.6871i	−1.03966	−	0.600249i	−0.919738	+	0.392534i	$0.871599\pi$
$318$		0			0
$319$	3.29194	+	1.90060i	0.184313	+	0.106413i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−14.7289	−	0.568193i	−0.819537	−	0.0316151i
$324$		0			0
$325$	−0.589298	−	1.02069i	−0.0326884	−	0.0566179i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	24.1861	+	13.9638i	1.33342	+	0.769851i
$330$		0			0
$331$		−	18.2194i		−	1.00143i	−0.865612	−	0.500715i	$-0.833070\pi$
$331$		−	18.2194i		−	1.00143i	0.865612	−	0.500715i	$-0.166930\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−11.1810			−0.610882
$336$		0			0
$337$	−11.5971	−	20.0867i	−0.631732	−	1.09419i	−0.987197	−	0.159503i	$-0.949011\pi$
$337$	−11.5971	−	20.0867i	−0.631732	−	1.09419i	0.355465	−	0.934689i	$-0.384322\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			0.937794i			0.0507844i
$342$		0			0
$343$		−	8.22611i		−	0.444168i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−6.15774	−	10.6655i	−0.330565	−	0.572555i	0.652058	−	0.758169i	$-0.273906\pi$
$347$	−6.15774	−	10.6655i	−0.330565	−	0.572555i	−0.982623	+	0.185614i	$0.940573\pi$
$348$		0			0
$349$	9.44133			0.505383			0.252691	−	0.967547i	$-0.418684\pi$
$349$	9.44133			0.505383			0.252691	+	0.967547i	$0.418684\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	4.93946i		−	0.262901i	−0.991323	−	0.131451i	$-0.958037\pi$
$353$		−	4.93946i		−	0.262901i	0.991323	−	0.131451i	$-0.0419635\pi$
$354$		0			0
$355$	2.30832	+	1.33271i	0.122513	+	0.0707329i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	0.994398	+	1.72235i	0.0524823	+	0.0909020i	0.891073	−	0.453860i	$-0.149953\pi$
$359$	0.994398	+	1.72235i	0.0524823	+	0.0909020i	−0.838591	+	0.544762i	$0.816620\pi$
$360$		0			0
$361$	8.20413	+	17.1375i	0.431796	+	0.901971i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−2.16195	−	1.24820i	−0.113162	−	0.0653339i
$366$		0			0
$367$	10.0985	+	5.83036i	0.527136	+	0.304342i	0.739850	−	0.672772i	$-0.234897\pi$
$367$	10.0985	+	5.83036i	0.527136	+	0.304342i	−0.212713	+	0.977115i	$0.568230\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−25.0119	+	43.3219i	−1.29855	+	2.24916i
$372$		0			0
$373$	27.9319			1.44626			0.723129	−	0.690713i	$-0.242703\pi$
$373$	27.9319			1.44626			0.723129	+	0.690713i	$0.242703\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−0.290108	+	0.167494i	−0.0149413	+	0.00862638i
$378$		0			0
$379$		−	5.72714i		−	0.294183i	−0.989123	−	0.147092i	$-0.953009\pi$
$379$		−	5.72714i		−	0.294183i	0.989123	−	0.147092i	$-0.0469913\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	4.59417	+	7.95733i	0.234751	+	0.406601i	0.959200	−	0.282728i	$-0.0912392\pi$
$383$	4.59417	+	7.95733i	0.234751	+	0.406601i	−0.724449	+	0.689328i	$0.757906\pi$
$384$		0			0
$385$	8.83958	+	15.3106i	0.450507	+	0.780300i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−6.09499	−	3.51894i	−0.309028	−	0.178418i	0.337463	−	0.941339i	$-0.390431\pi$
$389$	−6.09499	−	3.51894i	−0.309028	−	0.178418i	−0.646492	+	0.762921i	$0.723764\pi$
$390$		0			0
$391$			8.15024i			0.412175i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−3.80689	+	6.59373i	−0.191545	+	0.331766i
$396$		0			0
$397$	−10.2252	−	17.7106i	−0.513188	−	0.888868i	−0.999883	−	0.0152962i	$-0.995131\pi$
$397$	−10.2252	−	17.7106i	−0.513188	−	0.888868i	0.486695	−	0.873572i	$-0.338202\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−15.8453	+	9.14827i	−0.791275	+	0.456843i	−0.840411	−	0.541949i	$-0.817686\pi$
$401$	−15.8453	+	9.14827i	−0.791275	+	0.456843i	0.0491363	+	0.998792i	$0.484353\pi$
$402$		0			0
$403$	−0.0715726	−	0.0413224i	−0.00356528	−	0.00205842i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−8.43232			−0.417974
$408$		0			0
$409$	2.49657	−	4.32418i	0.123447	−	0.213817i	−0.797678	−	0.603084i	$-0.793938\pi$
$409$	2.49657	−	4.32418i	0.123447	−	0.213817i	0.921125	+	0.389267i	$0.127272\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−4.03523	+	2.32974i	−0.198561	+	0.114639i
$414$		0			0
$415$	−5.28261	+	3.04992i	−0.259313	+	0.149715i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−32.8318			−1.60394			−0.801970	−	0.597364i	$-0.796215\pi$
$419$	−32.8318			−1.60394			−0.801970	+	0.597364i	$0.796215\pi$
$420$		0			0
$421$	−3.85236	−	6.67249i	−0.187753	−	0.325197i	0.756748	−	0.653707i	$-0.226787\pi$
$421$	−3.85236	−	6.67249i	−0.187753	−	0.325197i	−0.944501	+	0.328510i	$0.893454\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		−	12.1532i		−	0.589515i
$426$		0			0
$427$	−28.5628	−	16.4907i	−1.38225	−	0.798043i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	12.3923	−	21.4640i	0.596915	−	1.03389i	−0.396359	−	0.918096i	$-0.629726\pi$
$431$	12.3923	−	21.4640i	0.596915	−	1.03389i	0.993274	−	0.115791i	$-0.0369403\pi$
$432$		0			0
$433$	2.32776	−	4.03180i	0.111865	−	0.193756i	−0.804657	−	0.593740i	$-0.797651\pi$
$433$	2.32776	−	4.03180i	0.111865	−	0.193756i	0.916522	+	0.399984i	$0.130984\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	9.29406	−	4.89830i	0.444595	−	0.234317i
$438$		0			0
$439$	21.7426	−	12.5531i	1.03772	−	0.599125i	0.118530	−	0.992950i	$-0.462182\pi$
$439$	21.7426	−	12.5531i	1.03772	−	0.599125i	0.919185	+	0.393825i	$0.128848\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−5.10222	+	8.83731i	−0.242414	+	0.419873i	−0.961401	−	0.275150i	$-0.911273\pi$
$443$	−5.10222	+	8.83731i	−0.242414	+	0.419873i	0.718987	+	0.695023i	$0.244606\pi$
$444$		0			0
$445$	5.29025			0.250782
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	32.5901i		−	1.53802i	−0.639236	−	0.769011i	$-0.720749\pi$
$449$		−	32.5901i		−	1.53802i	0.639236	−	0.769011i	$-0.279251\pi$
$450$		0			0
$451$	−22.4906	+	12.9849i	−1.05904	+	0.611437i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−1.55801			−0.0730406
$456$		0			0
$457$	2.39243			0.111913			0.0559566	−	0.998433i	$-0.482179\pi$
$457$	2.39243			0.111913			0.0559566	+	0.998433i	$0.482179\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	24.6438	−	14.2281i	1.14778	−	0.662669i	0.199432	−	0.979912i	$-0.436090\pi$
$461$	24.6438	−	14.2281i	1.14778	−	0.662669i	0.948344	+	0.317242i	$0.102757\pi$
$462$		0			0
$463$			2.85291i			0.132586i	0.997800	+	0.0662929i	$0.0211172\pi$
$463$			2.85291i			0.132586i	−0.997800	+	0.0662929i	$0.978883\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	18.1140			0.838217			0.419109	−	0.907936i	$-0.362343\pi$
$467$	18.1140			0.838217			0.419109	+	0.907936i	$0.362343\pi$
$468$		0			0
$469$	18.8899	−	32.7183i	0.872256	−	1.51079i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	3.36484	−	1.94269i	0.154715	−	0.0893250i
$474$		0			0
$475$	−13.8588	+	7.30405i	−0.635883	+	0.335133i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	6.95305	−	12.0430i	0.317693	−	0.550260i	−0.662313	−	0.749227i	$-0.730425\pi$
$479$	6.95305	−	12.0430i	0.317693	−	0.550260i	0.980006	+	0.198967i	$0.0637586\pi$
$480$		0			0
$481$	0.371557	−	0.643556i	0.0169416	−	0.0293436i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−8.47673	−	4.89404i	−0.384908	−	0.222227i
$486$		0			0
$487$		−	39.0698i		−	1.77042i	−0.465189	−	0.885211i	$-0.654014\pi$
$487$		−	39.0698i		−	1.77042i	0.465189	−	0.885211i	$-0.345986\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	10.3322	+	17.8958i	0.466284	+	0.807627i	0.999258	−	0.0385040i	$-0.0122592\pi$
$491$	10.3322	+	17.8958i	0.466284	+	0.807627i	−0.532975	+	0.846131i	$0.678926\pi$
$492$		0			0
$493$	−3.45425			−0.155572
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−7.79969	+	4.50315i	−0.349864	+	0.201994i
$498$		0			0
$499$	6.02993	−	3.48138i	0.269937	−	0.155848i	−0.358922	−	0.933367i	$-0.616856\pi$
$499$	6.02993	−	3.48138i	0.269937	−	0.155848i	0.628859	+	0.777519i	$0.283522\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	18.7543	−	32.4835i	0.836215	−	1.44837i	−0.0568230	−	0.998384i	$-0.518097\pi$
$503$	18.7543	−	32.4835i	0.836215	−	1.44837i	0.893038	−	0.449982i	$-0.148570\pi$
$504$		0			0
$505$	16.7605			0.745835
$506$		0			0
$507$		0			0
$508$		0			0
$509$	17.9508	+	10.3639i	0.795655	+	0.459372i	0.841950	−	0.539556i	$-0.181408\pi$
$509$	17.9508	+	10.3639i	0.795655	+	0.459372i	−0.0462946	+	0.998928i	$0.514741\pi$
$510$		0			0
$511$	7.30510	−	4.21760i	0.323159	−	0.186576i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	3.00606	+	5.20665i	0.132463	+	0.229432i
$516$		0			0
$517$	12.9690	−	22.4630i	0.570378	−	0.987923i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	38.8104i		−	1.70032i	−0.526528	−	0.850158i	$-0.676506\pi$
$521$		−	38.8104i		−	1.70032i	0.526528	−	0.850158i	$-0.323494\pi$
$522$		0			0
$523$	−30.8774	−	17.8271i	−1.35017	−	0.779524i	−0.361901	−	0.932217i	$-0.617872\pi$
$523$	−30.8774	−	17.8271i	−1.35017	−	0.779524i	−0.988274	+	0.152693i	$0.951206\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−0.426099	−	0.738025i	−0.0185612	−	0.0321489i
$528$		0			0
$529$	8.59545	+	14.8878i	0.373715	+	0.647294i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	2.28865i		−	0.0991323i
$534$		0			0
$535$	−0.887359	+	0.512317i	−0.0383639	+	0.0221494i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−33.6885			−1.45106
$540$		0			0
$541$	1.71812	−	2.97586i	0.0738676	−	0.127942i	−0.826726	−	0.562605i	$-0.809799\pi$
$541$	1.71812	−	2.97586i	0.0738676	−	0.127942i	0.900593	+	0.434663i	$0.143132\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−19.2273	−	11.1009i	−0.823606	−	0.475509i
$546$		0			0
$547$	23.6262	+	13.6406i	1.01018	+	0.583229i	0.911245	−	0.411864i	$-0.135122\pi$
$547$	23.6262	+	13.6406i	1.01018	+	0.583229i	0.0989378	+	0.995094i	$0.468456\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	2.07601	+	3.93903i	0.0884408	+	0.167808i
$552$		0			0
$553$	−12.8633	−	22.2798i	−0.547002	−	0.947434i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−22.4498	−	12.9614i	−0.951230	−	0.549193i	−0.0577675	−	0.998330i	$-0.518398\pi$
$557$	−22.4498	−	12.9614i	−0.951230	−	0.549193i	−0.893463	+	0.449137i	$0.851732\pi$
$558$		0			0
$559$			0.342407i			0.0144823i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	10.0390			0.423092			0.211546	−	0.977368i	$-0.432150\pi$
$563$	10.0390			0.423092			0.211546	+	0.977368i	$0.432150\pi$
$564$		0			0
$565$	9.16751	+	15.8786i	0.385680	+	0.668018i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	44.2910i		−	1.85678i	−0.371613	−	0.928388i	$-0.621195\pi$
$569$		−	44.2910i		−	1.85678i	0.371613	−	0.928388i	$-0.378805\pi$
$570$		0			0
$571$		−	37.2489i		−	1.55882i	−0.626516	−	0.779409i	$-0.715520\pi$
$571$		−	37.2489i		−	1.55882i	0.626516	−	0.779409i	$-0.284480\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	4.33109	+	7.50168i	0.180619	+	0.312842i
$576$		0			0
$577$	38.2192			1.59109			0.795544	−	0.605896i	$-0.207185\pi$
$577$	38.2192			1.59109			0.795544	+	0.605896i	$0.207185\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		−	20.6110i		−	0.855088i
$582$		0			0
$583$	40.2356	+	23.2300i	1.66639	+	0.962090i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−17.3490	−	30.0493i	−0.716069	−	1.24027i	−0.962546	−	0.271119i	$-0.912606\pi$
$587$	−17.3490	−	30.0493i	−0.716069	−	1.24027i	0.246477	−	0.969149i	$-0.420727\pi$
$588$		0			0
$589$	−0.585515	+	0.929452i	−0.0241257	+	0.0382974i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−5.97148	−	3.44763i	−0.245219	−	0.141577i	0.372354	−	0.928091i	$-0.378551\pi$
$593$	−5.97148	−	3.44763i	−0.245219	−	0.141577i	−0.617573	+	0.786513i	$0.711884\pi$
$594$		0			0
$595$	−13.9131	−	8.03276i	−0.570383	−	0.329311i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	9.70202	−	16.8044i	0.396414	−	0.686609i	−0.596867	−	0.802340i	$-0.703588\pi$
$599$	9.70202	−	16.8044i	0.396414	−	0.686609i	0.993281	+	0.115731i	$0.0369212\pi$
$600$		0			0
$601$	1.22714			0.0500559			0.0250279	−	0.999687i	$-0.492033\pi$
$601$	1.22714			0.0500559			0.0250279	+	0.999687i	$0.492033\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	2.92392	−	1.68813i	0.118874	−	0.0686321i
$606$		0			0
$607$			2.35663i			0.0956528i	0.998856	+	0.0478264i	$0.0152294\pi$
$607$			2.35663i			0.0956528i	−0.998856	+	0.0478264i	$0.984771\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	1.14292	+	1.97960i	0.0462377	+	0.0800860i
$612$		0			0
$613$	8.89335	+	15.4037i	0.359199	+	0.622151i	0.987827	−	0.155555i	$-0.0497166\pi$
$613$	8.89335	+	15.4037i	0.359199	+	0.622151i	−0.628628	+	0.777706i	$0.716383\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−20.1208	−	11.6168i	−0.810034	−	0.467673i	0.0369339	−	0.999318i	$-0.488241\pi$
$617$	−20.1208	−	11.6168i	−0.810034	−	0.467673i	−0.846968	+	0.531645i	$0.821574\pi$
$618$		0			0
$619$			36.7083i			1.47543i	0.675112	+	0.737715i	$0.264095\pi$
$619$			36.7083i			1.47543i	−0.675112	+	0.737715i	$0.735905\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−8.93773	+	15.4806i	−0.358083	+	0.620217i
$624$		0			0
$625$	−2.94317	−	5.09773i	−0.117727	−	0.203909i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	6.63607	−	3.83133i	0.264597	−	0.152765i
$630$		0			0
$631$	−22.1565	−	12.7921i	−0.882038	−	0.509245i	−0.0107083	−	0.999943i	$-0.503409\pi$
$631$	−22.1565	−	12.7921i	−0.882038	−	0.509245i	−0.871330	+	0.490698i	$0.836742\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	17.8336			0.707705
$636$		0			0
$637$	1.48443	−	2.57111i	0.0588153	−	0.101871i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−1.03785	+	0.599206i	−0.0409928	+	0.0236672i	−0.520356	−	0.853949i	$-0.674201\pi$
$641$	−1.03785	+	0.599206i	−0.0409928	+	0.0236672i	0.479364	+	0.877616i	$0.340868\pi$
$642$		0			0
$643$	−28.2815	+	16.3283i	−1.11531	+	0.643926i	−0.940200	−	0.340623i	$-0.889362\pi$
$643$	−28.2815	+	16.3283i	−1.11531	+	0.643926i	−0.175112	+	0.984548i	$0.556029\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−7.82161			−0.307499			−0.153750	−	0.988110i	$-0.549135\pi$
$647$	−7.82161			−0.307499			−0.153750	+	0.988110i	$0.549135\pi$
$648$		0			0
$649$	2.16377	+	3.74775i	0.0849353	+	0.147112i
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	27.7018i		−	1.08405i	−0.840361	−	0.542027i	$-0.817657\pi$
$653$		−	27.7018i		−	1.08405i	0.840361	−	0.542027i	$-0.182343\pi$
$654$		0			0
$655$	12.4810	+	7.20592i	0.487674	+	0.281559i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	0.927195	−	1.60595i	0.0361184	−	0.0625589i	−0.847401	−	0.530953i	$-0.821834\pi$
$659$	0.927195	−	1.60595i	0.0361184	−	0.0625589i	0.883520	+	0.468394i	$0.155167\pi$
$660$		0			0
$661$	9.79648	−	16.9680i	0.381039	−	0.659979i	−0.610172	−	0.792269i	$-0.708900\pi$
$661$	9.79648	−	16.9680i	0.381039	−	0.659979i	0.991211	+	0.132290i	$0.0422331\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−0.798286	+	20.6934i	−0.0309562	+	0.802457i
$666$		0			0
$667$	2.13218	−	1.23101i	0.0825582	−	0.0476650i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−15.3159	+	26.5280i	−0.591265	+	1.02410i
$672$		0			0
$673$	29.6231			1.14189			0.570944	−	0.820989i	$-0.306577\pi$
$673$	29.6231			1.14189			0.570944	+	0.820989i	$0.306577\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			1.81662i			0.0698184i	0.999390	+	0.0349092i	$0.0111142\pi$
$677$			1.81662i			0.0698184i	−0.999390	+	0.0349092i	$0.988886\pi$
$678$		0			0
$679$	28.6424	−	16.5367i	1.09919	−	0.634620i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	21.5025			0.822772			0.411386	−	0.911461i	$-0.365045\pi$
$683$	21.5025			0.822772			0.411386	+	0.911461i	$0.365045\pi$
$684$		0			0
$685$	−9.07035			−0.346560
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−3.54584	+	2.04719i	−0.135086	+	0.0779918i
$690$		0			0
$691$			7.92674i			0.301547i	0.988568	+	0.150774i	$0.0481765\pi$
$691$			7.92674i			0.301547i	−0.988568	+	0.150774i	$0.951824\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	8.91176			0.338042
$696$		0			0
$697$	11.7998	−	20.4378i	0.446948	−	0.774136i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	11.1463	−	6.43532i	0.420990	−	0.243059i	−0.274511	−	0.961584i	$-0.588516\pi$
$701$	11.1463	−	6.43532i	0.420990	−	0.243059i	0.695501	+	0.718525i	$0.255183\pi$
$702$		0			0
$703$	−8.35731	−	5.26475i	−0.315202	−	0.198564i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−28.3165	+	49.0456i	−1.06495	+	1.84455i
$708$		0			0
$709$	−22.4038	+	38.8046i	−0.841394	+	1.45734i	0.0473221	+	0.998880i	$0.484931\pi$
$709$	−22.4038	+	38.8046i	−0.841394	+	1.45734i	−0.888716	+	0.458458i	$0.848402\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	0.526029	+	0.303703i	0.0196999	+	0.0113738i
$714$		0			0
$715$			1.44702i			0.0541153i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	2.72709	+	4.72347i	0.101703	+	0.176156i	0.912387	−	0.409330i	$-0.134237\pi$
$719$	2.72709	+	4.72347i	0.101703	+	0.176156i	−0.810683	+	0.585485i	$0.800904\pi$
$720$		0			0
$721$	−20.3146			−0.756556
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−3.17937	+	1.83561i	−0.118079	+	0.0681730i
$726$		0			0
$727$	3.44349	−	1.98810i	0.127712	−	0.0737346i	−0.434783	−	0.900535i	$-0.643175\pi$
$727$	3.44349	−	1.98810i	0.127712	−	0.0737346i	0.562495	+	0.826801i	$0.309842\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−1.76537	+	3.05772i	−0.0652947	+	0.113094i
$732$		0			0
$733$	29.8710			1.10331			0.551654	−	0.834073i	$-0.313997\pi$
$733$	29.8710			1.10331			0.551654	+	0.834073i	$0.313997\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−30.3874	−	17.5442i	−1.11934	−	0.646249i
$738$		0			0
$739$	−29.3695	+	16.9565i	−1.08038	+	0.623755i	−0.930998	−	0.365025i	$-0.881060\pi$
$739$	−29.3695	+	16.9565i	−1.08038	+	0.623755i	−0.149378	+	0.988780i	$0.547727\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−21.8637	−	37.8690i	−0.802100	−	1.38928i	−0.918231	−	0.396045i	$-0.870383\pi$
$743$	−21.8637	−	37.8690i	−0.802100	−	1.38928i	0.116131	−	0.993234i	$-0.462951\pi$
$744$		0			0
$745$	1.81917	−	3.15090i	0.0666494	−	0.115440i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		−	3.46218i		−	0.126505i
$750$		0			0
$751$	−21.6246	−	12.4850i	−0.789092	−	0.455583i	0.0505507	−	0.998721i	$-0.483902\pi$
$751$	−21.6246	−	12.4850i	−0.789092	−	0.455583i	−0.839643	+	0.543139i	$0.817236\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	3.18885	+	5.52325i	0.116054	+	0.201012i
$756$		0			0
$757$	14.5980	+	25.2844i	0.530572	+	0.918977i	0.999364	+	0.0356686i	$0.0113561\pi$
$757$	14.5980	+	25.2844i	0.530572	+	0.918977i	−0.468792	+	0.883309i	$0.655311\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	2.01476i		−	0.0730351i	−0.999333	−	0.0365176i	$-0.988374\pi$
$761$		−	2.01476i		−	0.0730351i	0.999333	−	0.0365176i	$-0.0116265\pi$
$762$		0			0
$763$	64.9678	−	37.5092i	2.35199	−	1.35792i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−0.381372			−0.0137706
$768$		0			0
$769$	21.5009	−	37.2407i	0.775343	−	1.34293i	−0.159258	−	0.987237i	$-0.550910\pi$
$769$	21.5009	−	37.2407i	0.775343	−	1.34293i	0.934601	−	0.355697i	$-0.115757\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	15.8967	+	9.17794i	0.571763	+	0.330107i	0.757853	−	0.652425i	$-0.226248\pi$
$773$	15.8967	+	9.17794i	0.571763	+	0.330107i	−0.186090	+	0.982533i	$0.559582\pi$
$774$		0			0
$775$	−0.784383	−	0.452864i	−0.0281759	−	0.0162674i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−30.3977	−	1.17265i	−1.08911	−	0.0420144i
$780$		0			0
$781$	4.18234	+	7.24403i	0.149656	+	0.259212i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	8.06343	+	4.65542i	0.287796	+	0.166159i
$786$		0			0
$787$			5.84314i			0.208286i	0.994562	+	0.104143i	$0.0332099\pi$
$787$			5.84314i			0.208286i	−0.994562	+	0.104143i	$0.966790\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−61.9530			−2.20280
$792$		0			0
$793$	−1.34975	−	2.33783i	−0.0479309	−	0.0830187i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	12.6541i		−	0.448231i	−0.974563	−	0.224115i	$-0.928051\pi$
$797$		−	12.6541i		−	0.448231i	0.974563	−	0.224115i	$-0.0719493\pi$
$798$		0			0
$799$			23.5706i			0.833868i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−3.91714	−	6.78468i	−0.138233	−	0.239426i
$804$		0			0
$805$	11.4507			0.403585
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	7.52049i		−	0.264406i	−0.991223	−	0.132203i	$-0.957795\pi$
$809$		−	7.52049i		−	0.264406i	0.991223	−	0.132203i	$-0.0422051\pi$
$810$		0			0
$811$	−20.3422	−	11.7446i	−0.714313	−	0.412409i	0.0983431	−	0.995153i	$-0.468646\pi$
$811$	−20.3422	−	11.7446i	−0.714313	−	0.412409i	−0.812656	+	0.582744i	$0.801979\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	6.52701	+	11.3051i	0.228631	+	0.396001i
$816$		0			0
$817$	4.54784	+	0.175441i	0.159109	+	0.00613790i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	5.32878	+	3.07657i	0.185976	+	0.107373i	0.590097	−	0.807332i	$-0.299089\pi$
$821$	5.32878	+	3.07657i	0.185976	+	0.107373i	−0.404122	+	0.914705i	$0.632423\pi$
$822$		0			0
$823$	0.640832	+	0.369984i	0.0223380	+	0.0128968i	0.511127	−	0.859505i	$-0.329228\pi$
$823$	0.640832	+	0.369984i	0.0223380	+	0.0128968i	−0.488789	+	0.872402i	$0.662561\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−25.3235	+	43.8616i	−0.880584	+	1.52522i	−0.0298920	+	0.999553i	$0.509516\pi$
$827$	−25.3235	+	43.8616i	−0.880584	+	1.52522i	−0.850692	+	0.525664i	$0.823817\pi$
$828$		0			0
$829$	−39.2083			−1.36176			−0.680880	−	0.732395i	$-0.738402\pi$
$829$	−39.2083			−1.36176			−0.680880	+	0.732395i	$0.738402\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	26.5122	−	15.3068i	0.918592	−	0.530349i
$834$		0			0
$835$		−	25.5431i		−	0.883957i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−13.0620	−	22.6241i	−0.450952	−	0.781071i	0.547494	−	0.836810i	$-0.315582\pi$
$839$	−13.0620	−	22.6241i	−0.450952	−	0.781071i	−0.998445	+	0.0557387i	$0.982249\pi$
$840$		0			0
$841$	−13.9783	−	24.2111i	−0.482009	−	0.834865i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	13.2393	+	7.64372i	0.455446	+	0.262952i
$846$		0			0
$847$			11.4082i			0.391989i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−2.73079	+	4.72987i	−0.0936103	+	0.162138i
$852$		0			0
$853$	9.06915	+	15.7082i	0.310522	+	0.537840i	0.978475	−	0.206363i	$-0.0661629\pi$
$853$	9.06915	+	15.7082i	0.310522	+	0.537840i	−0.667954	+	0.744203i	$0.732830\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−45.3247	+	26.1683i	−1.54826	+	0.893890i	−0.549989	+	0.835172i	$0.685368\pi$
$857$	−45.3247	+	26.1683i	−1.54826	+	0.893890i	−0.998275	+	0.0587185i	$0.981299\pi$
$858$		0			0
$859$	1.74572	+	1.00789i	0.0595633	+	0.0343889i	0.529486	−	0.848319i	$-0.322385\pi$
$859$	1.74572	+	1.00789i	0.0595633	+	0.0343889i	−0.469923	+	0.882708i	$0.655718\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	28.2094			0.960260			0.480130	−	0.877197i	$-0.340590\pi$
$863$	28.2094			0.960260			0.480130	+	0.877197i	$0.340590\pi$
$864$		0			0
$865$	−8.11208	+	14.0505i	−0.275819	+	0.477732i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−20.6926	+	11.9469i	−0.701948	+	0.405270i
$870$		0			0
$871$	2.67795	−	1.54612i	0.0907390	−	0.0523882i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−40.8293			−1.38028
$876$		0			0
$877$	1.82642	+	3.16346i	0.0616739	+	0.106822i	0.895214	−	0.445637i	$-0.147023\pi$
$877$	1.82642	+	3.16346i	0.0616739	+	0.106822i	−0.833540	+	0.552459i	$0.813689\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			42.7712i			1.44100i	0.693456	+	0.720499i	$0.256087\pi$
$881$			42.7712i			1.44100i	−0.693456	+	0.720499i	$0.743913\pi$
$882$		0			0
$883$	−22.5134	−	12.9981i	−0.757636	−	0.437421i	0.0708104	−	0.997490i	$-0.477441\pi$
$883$	−22.5134	−	12.9981i	−0.757636	−	0.437421i	−0.828446	+	0.560068i	$0.810775\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	21.7353	−	37.6467i	0.729801	−	1.26405i	−0.227166	−	0.973856i	$-0.572946\pi$
$887$	21.7353	−	37.6467i	0.729801	−	1.26405i	0.956967	−	0.290197i	$-0.0937208\pi$
$888$		0			0
$889$	−30.1294	+	52.1856i	−1.01051	+	1.75025i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	26.8786	−	14.1659i	0.899457	−	0.474045i
$894$		0			0
$895$	13.0632	−	7.54201i	0.436653	−	0.252102i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−0.128716	+	0.222943i	−0.00429292	+	0.00743555i
$900$		0			0
$901$	−42.2195			−1.40654
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	6.65037i		−	0.221066i
$906$		0			0
$907$	29.1447	−	16.8267i	0.967735	−	0.558722i	0.0691900	−	0.997603i	$-0.477959\pi$
$907$	29.1447	−	16.8267i	0.967735	−	0.558722i	0.898545	+	0.438881i	$0.144625\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	0.923995			0.0306133			0.0153067	−	0.999883i	$-0.495128\pi$
$911$	0.923995			0.0306133			0.0153067	+	0.999883i	$0.495128\pi$
$912$		0			0
$913$	−19.1426			−0.633529
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−42.1727	+	24.3484i	−1.39266	+	0.804055i
$918$		0			0
$919$		−	43.4679i		−	1.43387i	−0.697138	−	0.716937i	$-0.745544\pi$
$919$		−	43.4679i		−	1.43387i	0.697138	−	0.716937i	$-0.254456\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−0.737154			−0.0242637
$924$		0			0
$925$	4.07200	−	7.05290i	0.133886	−	0.231898i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	9.17871	−	5.29933i	0.301144	−	0.173865i	−0.341813	−	0.939768i	$-0.611041\pi$
$929$	9.17871	−	5.29933i	0.301144	−	0.173865i	0.642957	+	0.765903i	$0.277708\pi$
$930$		0			0
$931$	−33.3888	−	21.0335i	−1.09427	−	0.689346i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−7.46050	+	12.9220i	−0.243984	+	0.422593i
$936$		0			0
$937$	23.4364	−	40.5930i	0.765632	−	1.32611i	−0.174279	−	0.984696i	$-0.555760\pi$
$937$	23.4364	−	40.5930i	0.765632	−	1.32611i	0.939912	−	0.341418i	$-0.110907\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	27.0848	+	15.6374i	0.882941	+	0.509766i	0.871627	−	0.490170i	$-0.163065\pi$
$941$	27.0848	+	15.6374i	0.882941	+	0.509766i	0.0113138	+	0.999936i	$0.496399\pi$
$942$		0			0
$943$			16.8206i			0.547754i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	25.5392	+	44.2352i	0.829913	+	1.43745i	0.898106	+	0.439779i	$0.144943\pi$
$947$	25.5392	+	44.2352i	0.829913	+	1.43745i	−0.0681937	+	0.997672i	$0.521724\pi$
$948$		0			0
$949$	0.690411			0.0224117
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−16.1394	+	9.31810i	−0.522807	+	0.301843i	−0.738082	−	0.674711i	$-0.764268\pi$
$953$	−16.1394	+	9.31810i	−0.522807	+	0.301843i	0.215275	+	0.976553i	$0.430935\pi$
$954$		0			0
$955$	15.7877	−	9.11501i	0.510877	−	0.294955i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	15.3241	−	26.5421i	0.494841	−	0.857090i
$960$		0			0
$961$	30.9365			0.997951
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−26.6302	−	15.3749i	−0.857256	−	0.494937i
$966$		0			0
$967$	−10.3838	+	5.99509i	−0.333920	+	0.192789i	−0.657580	−	0.753384i	$-0.728420\pi$
$967$	−10.3838	+	5.99509i	−0.333920	+	0.192789i	0.323660	+	0.946173i	$0.395087\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−16.1961	−	28.0524i	−0.519757	−	0.900245i	−0.999736	−	0.0229653i	$-0.992689\pi$
$971$	−16.1961	−	28.0524i	−0.519757	−	0.900245i	0.479980	−	0.877280i	$-0.340644\pi$
$972$		0			0
$973$	−15.0562	+	26.0781i	−0.482679	+	0.836024i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	8.73136i		−	0.279341i	−0.990198	−	0.139671i	$-0.955396\pi$
$977$		−	8.73136i		−	0.279341i	0.990198	−	0.139671i	$-0.0446043\pi$
$978$		0			0
$979$	14.3777	+	8.30100i	0.459515	+	0.265301i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−12.5732	−	21.7774i	−0.401022	−	0.694591i	0.592827	−	0.805329i	$-0.298012\pi$
$983$	−12.5732	−	21.7774i	−0.401022	−	0.694591i	−0.993850	+	0.110739i	$0.964678\pi$
$984$		0			0
$985$	−12.2806	−	21.2706i	−0.391292	−	0.677738i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	2.51655i		−	0.0800216i
$990$		0			0
$991$	−22.4616	+	12.9682i	−0.713518	+	0.411950i	−0.812362	−	0.583153i	$-0.801819\pi$
$991$	−22.4616	+	12.9682i	−0.713518	+	0.411950i	0.0988446	+	0.995103i	$0.468485\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	30.2480			0.958926
$996$		0			0
$997$	−4.44201	+	7.69378i	−0.140680	+	0.243664i	−0.927753	−	0.373195i	$-0.878262\pi$
$997$	−4.44201	+	7.69378i	−0.140680	+	0.243664i	0.787073	+	0.616860i	$0.211595\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2736.2.cg.a.2591.8	yes	24
3.2	odd	2	inner	2736.2.cg.a.2591.5	yes	24
4.3	odd	2		2736.2.cg.b.2591.8	yes	24
12.11	even	2		2736.2.cg.b.2591.5	yes	24
19.11	even	3		2736.2.cg.b.1151.5	yes	24
57.11	odd	6		2736.2.cg.b.1151.8	yes	24
76.11	odd	6	inner	2736.2.cg.a.1151.5	✓	24
228.11	even	6	inner	2736.2.cg.a.1151.8	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2736.2.cg.a.1151.5	✓	24	76.11	odd	6	inner
2736.2.cg.a.1151.8	yes	24	228.11	even	6	inner
2736.2.cg.a.2591.5	yes	24	3.2	odd	2	inner
2736.2.cg.a.2591.8	yes	24	1.1	even	1	trivial
2736.2.cg.b.1151.5	yes	24	19.11	even	3
2736.2.cg.b.1151.8	yes	24	57.11	odd	6
2736.2.cg.b.2591.5	yes	24	12.11	even	2
2736.2.cg.b.2591.8	yes	24	4.3	odd	2

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:	\( N \)	\(=\)	\( 2736 = 2^{4} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2736.cg (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(1.02690 - 0.592883i) q^{5} +4.00663i q^{7} +O(q^{10})\)	\(q+(1.02690 - 0.592883i) q^{5} +4.00663i q^{7} +3.72120 q^{11} +(-0.163969 + 0.284003i) q^{13} +(-2.92851 + 1.69078i) q^{17} +(3.68810 + 2.32335i) q^{19} +(1.20510 - 2.08730i) q^{23} +(-1.79698 + 3.11246i) q^{25} +(0.884644 + 0.510749i) q^{29} +0.252014i q^{31} +(2.37546 + 4.11443i) q^{35} -2.26602 q^{37} +(-6.04391 + 3.48945i) q^{41} +(0.904236 - 0.522061i) q^{43} +(3.48518 - 6.03651i) q^{47} -9.05312 q^{49} +(10.8125 + 6.24262i) q^{53} +(3.82131 - 2.20623i) q^{55} +(0.581471 + 1.00714i) q^{59} +(-4.11586 + 7.12888i) q^{61} +0.388857i q^{65} +(-8.16604 - 4.71466i) q^{67} +(1.12392 + 1.94669i) q^{71} +(-1.05266 - 1.82325i) q^{73} +14.9095i q^{77} +(-5.56073 + 3.21049i) q^{79} -5.14422 q^{83} +(-2.00486 + 3.47253i) q^{85} +(3.86374 + 2.23073i) q^{89} +(-1.13789 - 0.656964i) q^{91} +(5.16479 + 0.199241i) q^{95} +(-4.12733 - 7.14874i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q+O(q^{10}) \) 24 * q	\( 24 q + 12 q^{13} - 12 q^{19} + 8 q^{25} + 16 q^{37} - 12 q^{43} + 16 q^{49} + 12 q^{55} - 60 q^{67} + 8 q^{73} + 12 q^{79} + 16 q^{85} - 12 q^{91} - 32 q^{97}+O(q^{100}) \) 24 * q + 12 * q^13 - 12 * q^19 + 8 * q^25 + 16 * q^37 - 12 * q^43 + 16 * q^49 + 12 * q^55 - 60 * q^67 + 8 * q^73 + 12 * q^79 + 16 * q^85 - 12 * q^91 - 32 * q^97

Introduction

L-functions

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