LMFDB - Embedded newform 2736.2.cg.b.1151.8

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			1151.8
Character	$\chi$	$=$	2736.1151
Dual form			2736.2.cg.b.2591.8

$q$-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 $\left(\mathbb{Z}/2736\mathbb{Z}\right)^\times$.

$n$	$1009$	$1217$	$1711$	$2053$
$\chi(n)$	$e\left(\frac{2}{3}\right)$	$-1$	$-1$	$1$

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	0.711727	−	0.702456i	$-0.247913\pi$
$5$	1.02690	+	0.592883i	0.459245	+	0.265145i	−0.252482	+	0.967602i	$0.581247\pi$
$6$		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	$-0.689580\pi$
$11$	−3.72120			−1.12198			−0.560992	+	0.827821i	$0.689580\pi$
$12$		0			0
$13$	−0.163969	−	0.284003i	−0.0454768	−	0.0787681i	0.842391	−	0.538867i	$-0.181147\pi$
$13$	−0.163969	−	0.284003i	−0.0454768	−	0.0787681i	−0.887868	+	0.460099i	$0.847814\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−2.92851	−	1.69078i	−0.710268	−	0.410073i	0.100892	−	0.994897i	$-0.467830\pi$
$17$	−2.92851	−	1.69078i	−0.710268	−	0.410073i	−0.811160	+	0.584824i	$0.801164\pi$
$18$		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	$-0.247519\pi$
$23$	−1.20510	−	2.08730i	−0.251282	−	0.435232i	−0.963879	+	0.266341i	$0.914185\pi$
$24$		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	−0.415609	−	0.909543i	$-0.636432\pi$
$29$	0.884644	−	0.510749i	0.164274	−	0.0948438i	0.579883	+	0.814700i	$0.303098\pi$
$30$		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	−0.0527193	−	0.998609i	$-0.516789\pi$
$41$	−6.04391	−	3.48945i	−0.943900	−	0.544961i	−0.891181	+	0.453648i	$0.850122\pi$
$42$		0			0
$43$	−0.904236	−	0.522061i	−0.137895	−	0.0796135i	0.429466	−	0.903083i	$-0.358702\pi$
$43$	−0.904236	−	0.522061i	−0.137895	−	0.0796135i	−0.567360	+	0.823470i	$0.692035\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−3.48518	−	6.03651i	−0.508365	−	0.880515i	−0.999953	−	0.00968668i	$-0.996917\pi$
$47$	−3.48518	−	6.03651i	−0.508365	−	0.880515i	0.491588	−	0.870828i	$-0.336417\pi$
$48$		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	0.485359	−	0.874315i	$-0.338689\pi$
$53$	10.8125	−	6.24262i	1.48522	−	0.857490i	0.999858	+	0.0168248i	$0.00535574\pi$
$54$		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	$-0.857453\pi$
$59$	−0.581471	+	1.00714i	−0.0757010	+	0.131118i	0.825690	+	0.564124i	$0.190786\pi$
$60$		0			0
$61$	−4.11586	−	7.12888i	−0.526982	−	0.912759i	−0.999506	−	0.0314412i	$-0.989990\pi$
$61$	−4.11586	−	7.12888i	−0.526982	−	0.912759i	0.472524	−	0.881318i	$-0.343343\pi$
$62$		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	$-0.471284\pi$
$67$	8.16604	−	4.71466i	0.997640	−	0.575988i	0.907549	+	0.419946i	$0.137951\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−1.12392	+	1.94669i	−0.133385	+	0.231030i	−0.924979	−	0.380017i	$-0.875918\pi$
$71$	−1.12392	+	1.94669i	−0.133385	+	0.231030i	0.791594	+	0.611047i	$0.209251\pi$
$72$		0			0
$73$	−1.05266	+	1.82325i	−0.123204	+	0.213395i	−0.921029	−	0.389493i	$-0.872650\pi$
$73$	−1.05266	+	1.82325i	−0.123204	+	0.213395i	0.797826	+	0.602888i	$0.205984\pi$
$74$		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	$-0.215698\pi$
$79$	5.56073	+	3.21049i	0.625631	+	0.361208i	−0.153427	+	0.988160i	$0.549031\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	5.14422			0.564651			0.282326	−	0.959319i	$-0.408894\pi$
$83$	5.14422			0.564651			0.282326	+	0.959319i	$0.408894\pi$
$84$		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	−0.281043	−	0.959695i	$-0.590680\pi$
$89$	3.86374	−	2.23073i	0.409556	−	0.236457i	0.690599	+	0.723238i	$0.257347\pi$
$90$		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	−0.995846	−	0.0910556i	$-0.970976\pi$
$97$	−4.12733	+	7.14874i	−0.419066	+	0.725844i	0.576779	+	0.816900i	$0.304309\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	12.2411	−	7.06739i	1.21803	−	0.703232i	0.253537	−	0.967326i	$-0.418406\pi$
$101$	12.2411	−	7.06739i	1.21803	−	0.703232i	0.964497	+	0.264094i	$0.0850728\pi$
$102$		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	$-0.486701\pi$
$107$	0.864112			0.0835368			0.0417684	+	0.999127i	$0.486701\pi$
$108$		0			0
$109$	−9.36177	+	16.2151i	−0.896695	+	1.55312i	−0.0650023	+	0.997885i	$0.520705\pi$
$109$	−9.36177	+	16.2151i	−0.896695	+	1.55312i	−0.831693	+	0.555236i	$0.812628\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	15.4626i		−	1.45460i	−0.686320	−	0.727300i	$-0.740775\pi$
$113$		−	15.4626i		−	1.45460i	0.686320	−	0.727300i	$-0.259225\pi$
$114$		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	$-0.899208\pi$
$127$	−13.0248	+	7.51987i	−1.15576	+	0.667281i	−0.205479	+	0.978662i	$0.565875\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−6.07702	+	10.5257i	−0.530952	+	0.919636i	0.468396	+	0.883519i	$0.344832\pi$
$131$	−6.07702	+	10.5257i	−0.530952	+	0.919636i	−0.999348	+	0.0361169i	$0.988501\pi$
$132$		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	−0.755539	−	0.655103i	$-0.772625\pi$
$137$	−6.62454	+	3.82468i	−0.565973	+	0.326765i	0.189567	+	0.981868i	$0.439292\pi$
$138$		0			0
$139$	−6.50872	+	3.75781i	−0.552062	+	0.318733i	−0.749953	−	0.661491i	$-0.769924\pi$
$139$	−6.50872	+	3.75781i	−0.552062	+	0.318733i	0.197891	+	0.980224i	$0.436591\pi$
$140$		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	0.604881	−	0.796316i	$-0.293221\pi$
$149$	2.65727	+	1.53418i	0.217692	+	0.125685i	−0.387189	+	0.922000i	$0.626554\pi$
$150$		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	−0.665746	−	0.746178i	$-0.731887\pi$
$157$	3.92609	−	6.80019i	0.313336	−	0.542714i	0.979082	+	0.203464i	$0.0652200\pi$
$158$		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	−0.895274	−	0.445516i	$-0.853020\pi$
$167$	−10.7707	−	18.6555i	−0.833465	−	1.44360i	0.0618093	−	0.998088i	$-0.480313\pi$
$168$		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	−0.0233997	−	0.999726i	$-0.507449\pi$
$173$	−11.8493	−	6.84121i	−0.900888	−	0.520128i	−0.877488	+	0.479598i	$0.840782\pi$
$174$		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	$-0.657698\pi$
$179$	−12.7209			−0.950806			−0.475403	+	0.879768i	$0.657698\pi$
$180$		0			0
$181$	2.80425	+	4.85711i	0.208438	+	0.361026i	0.951223	−	0.308505i	$-0.0998286\pi$
$181$	2.80425	+	4.85711i	0.208438	+	0.361026i	−0.742784	+	0.669531i	$0.766495\pi$
$182$		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	$-0.687746\pi$
$191$	−15.3741			−1.11243			−0.556214	+	0.831039i	$0.687746\pi$
$192$		0			0
$193$	−12.9663	+	22.4582i	−0.933332	+	1.61658i	−0.155749	+	0.987797i	$0.549779\pi$
$193$	−12.9663	+	22.4582i	−0.933332	+	1.61658i	−0.777583	+	0.628781i	$0.783554\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			20.7134i			1.47577i	0.674929	+	0.737883i	$0.264174\pi$
$197$			20.7134i			1.47577i	−0.674929	+	0.737883i	$0.735826\pi$
$198$		0			0
$199$	−22.0917	+	12.7546i	−1.56604	+	0.904151i	−0.569412	+	0.822052i	$0.692829\pi$
$199$	−22.0917	+	12.7546i	−1.56604	+	0.904151i	−0.996624	+	0.0820992i	$0.973838\pi$
$200$		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	$-0.139611\pi$
$211$	14.3841	+	8.30467i	0.990243	+	0.571717i	0.0848964	+	0.996390i	$0.472944\pi$
$212$		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	$-0.461597\pi$
$223$	−10.1426	−	5.85583i	−0.679199	−	0.392136i	−0.799553	+	0.600595i	$0.794930\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−19.4717			−1.29238			−0.646191	−	0.763176i	$-0.723639\pi$
$227$	−19.4717			−1.29238			−0.646191	+	0.763176i	$0.723639\pi$
$228$		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	0.857785	−	0.514008i	$-0.171840\pi$
$233$	12.8455	+	7.41633i	0.841534	+	0.485860i	−0.0162514	+	0.999868i	$0.505173\pi$
$234$		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	$-0.825933\pi$
$239$	−26.4103			−1.70834			−0.854169	+	0.519996i	$0.825933\pi$
$240$		0			0
$241$	−9.04260	−	15.6622i	−0.582485	−	1.00889i	−0.995184	−	0.0980259i	$-0.968747\pi$
$241$	−9.04260	−	15.6622i	−0.582485	−	1.00889i	0.412699	−	0.910867i	$-0.364586\pi$
$242$		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	$-0.0150222\pi$
$251$	7.26539	+	12.5840i	0.458588	+	0.794297i	−0.540299	+	0.841473i	$0.681689\pi$
$252$		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	−0.936993	−	0.349349i	$-0.886403\pi$
$257$	−17.6815	+	10.2084i	−1.10294	+	0.636785i	−0.165951	+	0.986134i	$0.553069\pi$
$258$		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	$-0.825753\pi$
$263$	0.386150	−	0.668831i	0.0238110	−	0.0412419i	0.877685	+	0.479237i	$0.159087\pi$
$264$		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	−0.207608	−	0.978212i	$-0.566568\pi$
$269$	−19.0019	−	10.9708i	−1.15857	−	0.668900i	−0.950960	+	0.309313i	$0.899901\pi$
$270$		0			0
$271$	−2.24497	−	1.29614i	−0.136372	−	0.0787346i	0.430262	−	0.902704i	$-0.358421\pi$
$271$	−2.24497	−	1.29614i	−0.136372	−	0.0787346i	−0.566634	+	0.823970i	$0.691755\pi$
$272$		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	0.0796995	−	0.996819i	$-0.474604\pi$
$281$	16.4751	−	9.51188i	0.982820	−	0.567431i	0.903120	+	0.429388i	$0.141271\pi$
$282$		0			0
$283$	−7.56151	−	4.36564i	−0.449485	−	0.259510i	0.258128	−	0.966111i	$-0.416895\pi$
$283$	−7.56151	−	4.36564i	−0.449485	−	0.259510i	−0.707613	+	0.706600i	$0.750228\pi$
$284$		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	$0.197573\pi$
$293$			19.9108i			1.16320i	−0.813475	+	0.581600i	$0.802427\pi$
$294$		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	$-0.226476\pi$
$307$	9.99756	+	5.77209i	0.570591	+	0.329431i	−0.186794	+	0.982399i	$0.559810\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	28.9643			1.64241			0.821207	−	0.570630i	$-0.193301\pi$
$311$	28.9643			1.64241			0.821207	+	0.570630i	$0.193301\pi$
$312$		0			0
$313$	−9.67318	−	16.7544i	−0.546760	−	0.947016i	−0.998494	−	0.0548636i	$-0.982528\pi$
$313$	−9.67318	−	16.7544i	−0.546760	−	0.947016i	0.451734	−	0.892153i	$-0.350806\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−18.5107	+	10.6871i	−1.03966	+	0.600249i	−0.919738	−	0.392534i	$-0.871599\pi$
$317$	−18.5107	+	10.6871i	−1.03966	+	0.600249i	−0.119925	+	0.992783i	$0.538265\pi$
$318$		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	0.355465	+	0.934689i	$0.384322\pi$
$337$	−11.5971	+	20.0867i	−0.631732	+	1.09419i	−0.987197	+	0.159503i	$0.949011\pi$
$338$		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	$-0.726094\pi$
$347$	6.15774	−	10.6655i	0.330565	−	0.572555i	0.982623	+	0.185614i	$0.0594274\pi$
$348$		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	$0.0419635\pi$
$353$			4.93946i			0.262901i	−0.991323	+	0.131451i	$0.958037\pi$
$354$		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	$-0.850047\pi$
$359$	−0.994398	+	1.72235i	−0.0524823	+	0.0909020i	0.838591	+	0.544762i	$0.183380\pi$
$360$		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	$-0.765103\pi$
$367$	−10.0985	+	5.83036i	−0.527136	+	0.304342i	0.212713	+	0.977115i	$0.431770\pi$
$368$		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	$-0.908761\pi$
$383$	−4.59417	+	7.95733i	−0.234751	+	0.406601i	0.724449	+	0.689328i	$0.242094\pi$
$384$		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	−0.646492	−	0.762921i	$-0.723764\pi$
$389$	−6.09499	+	3.51894i	−0.309028	+	0.178418i	0.337463	+	0.941339i	$0.390431\pi$
$390$		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	0.486695	+	0.873572i	$0.338202\pi$
$397$	−10.2252	+	17.7106i	−0.513188	+	0.888868i	−0.999883	+	0.0152962i	$0.995131\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−15.8453	−	9.14827i	−0.791275	−	0.456843i	0.0491363	−	0.998792i	$-0.484353\pi$
$401$	−15.8453	−	9.14827i	−0.791275	−	0.456843i	−0.840411	+	0.541949i	$0.817686\pi$
$402$		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	0.921125	−	0.389267i	$-0.127272\pi$
$409$	2.49657	+	4.32418i	0.123447	+	0.213817i	−0.797678	+	0.603084i	$0.793938\pi$
$410$		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	$-0.203785\pi$
$419$	32.8318			1.60394			0.801970	+	0.597364i	$0.203785\pi$
$420$		0			0
$421$	−3.85236	+	6.67249i	−0.187753	+	0.325197i	−0.944501	−	0.328510i	$-0.893454\pi$
$421$	−3.85236	+	6.67249i	−0.187753	+	0.325197i	0.756748	+	0.653707i	$0.226787\pi$
$422$		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	−0.993274	−	0.115791i	$-0.963060\pi$
$431$	−12.3923	−	21.4640i	−0.596915	−	1.03389i	0.396359	−	0.918096i	$-0.370274\pi$
$432$		0			0
$433$	2.32776	+	4.03180i	0.111865	+	0.193756i	0.916522	−	0.399984i	$-0.130984\pi$
$433$	2.32776	+	4.03180i	0.111865	+	0.193756i	−0.804657	+	0.593740i	$0.797651\pi$
$434$		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	$-0.537818\pi$
$439$	−21.7426	−	12.5531i	−1.03772	−	0.599125i	−0.919185	+	0.393825i	$0.871152\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	5.10222	+	8.83731i	0.242414	+	0.419873i	0.961401	−	0.275150i	$-0.0887275\pi$
$443$	5.10222	+	8.83731i	0.242414	+	0.419873i	−0.718987	+	0.695023i	$0.755394\pi$
$444$		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	$0.279251\pi$
$449$			32.5901i			1.53802i	−0.639236	+	0.769011i	$0.720749\pi$
$450$		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	0.948344	−	0.317242i	$-0.102757\pi$
$461$	24.6438	+	14.2281i	1.14778	+	0.662669i	0.199432	+	0.979912i	$0.436090\pi$
$462$		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	$-0.637657\pi$
$467$	−18.1140			−0.838217			−0.419109	+	0.907936i	$0.637657\pi$
$468$		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	$-0.269575\pi$
$479$	−6.95305	−	12.0430i	−0.317693	−	0.550260i	−0.980006	+	0.198967i	$0.936241\pi$
$480$		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	$-0.987741\pi$
$491$	−10.3322	+	17.8958i	−0.466284	+	0.807627i	0.532975	+	0.846131i	$0.321074\pi$
$492$		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	$-0.383144\pi$
$499$	−6.02993	−	3.48138i	−0.269937	−	0.155848i	−0.628859	+	0.777519i	$0.716478\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−18.7543	−	32.4835i	−0.836215	−	1.44837i	−0.893038	−	0.449982i	$-0.851430\pi$
$503$	−18.7543	−	32.4835i	−0.836215	−	1.44837i	0.0568230	−	0.998384i	$-0.481903\pi$
$504$		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	−0.0462946	−	0.998928i	$-0.514741\pi$
$509$	17.9508	−	10.3639i	0.795655	−	0.459372i	0.841950	+	0.539556i	$0.181408\pi$
$510$		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	$0.323494\pi$
$521$			38.8104i			1.70032i	−0.526528	+	0.850158i	$0.676506\pi$
$522$		0			0
$523$	30.8774	−	17.8271i	1.35017	−	0.779524i	0.361901	−	0.932217i	$-0.382128\pi$
$523$	30.8774	−	17.8271i	1.35017	−	0.779524i	0.988274	+	0.152693i	$0.0487945\pi$
$524$		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	0.900593	−	0.434663i	$-0.143132\pi$
$541$	1.71812	+	2.97586i	0.0738676	+	0.127942i	−0.826726	+	0.562605i	$0.809799\pi$
$542$		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	$-0.864878\pi$
$547$	−23.6262	+	13.6406i	−1.01018	+	0.583229i	−0.0989378	+	0.995094i	$0.531544\pi$
$548$		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	−0.893463	−	0.449137i	$-0.851732\pi$
$557$	−22.4498	+	12.9614i	−0.951230	+	0.549193i	−0.0577675	+	0.998330i	$0.518398\pi$
$558$		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	$-0.567850\pi$
$563$	−10.0390			−0.423092			−0.211546	+	0.977368i	$0.567850\pi$
$564$		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	$0.378805\pi$
$569$			44.2910i			1.85678i	−0.371613	+	0.928388i	$0.621195\pi$
$570$		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	−0.246477	−	0.969149i	$-0.579273\pi$
$587$	17.3490	−	30.0493i	0.716069	−	1.24027i	0.962546	−	0.271119i	$-0.0873935\pi$
$588$		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	−0.617573	−	0.786513i	$-0.711884\pi$
$593$	−5.97148	+	3.44763i	−0.245219	+	0.141577i	0.372354	+	0.928091i	$0.378551\pi$
$594$		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	$-0.296412\pi$
$599$	−9.70202	−	16.8044i	−0.396414	−	0.686609i	−0.993281	+	0.115731i	$0.963079\pi$
$600$		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	−0.628628	−	0.777706i	$-0.716383\pi$
$613$	8.89335	−	15.4037i	0.359199	−	0.622151i	0.987827	+	0.155555i	$0.0497166\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−20.1208	+	11.6168i	−0.810034	+	0.467673i	−0.846968	−	0.531645i	$-0.821574\pi$
$617$	−20.1208	+	11.6168i	−0.810034	+	0.467673i	0.0369339	+	0.999318i	$0.488241\pi$
$618$		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	$-0.496591\pi$
$631$	22.1565	−	12.7921i	0.882038	−	0.509245i	0.871330	+	0.490698i	$0.163258\pi$
$632$		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	0.479364	−	0.877616i	$-0.340868\pi$
$641$	−1.03785	−	0.599206i	−0.0409928	−	0.0236672i	−0.520356	+	0.853949i	$0.674201\pi$
$642$		0			0
$643$	28.2815	+	16.3283i	1.11531	+	0.643926i	0.940200	−	0.340623i	$-0.110638\pi$
$643$	28.2815	+	16.3283i	1.11531	+	0.643926i	0.175112	+	0.984548i	$0.443971\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	7.82161			0.307499			0.153750	−	0.988110i	$-0.450865\pi$
$647$	7.82161			0.307499			0.153750	+	0.988110i	$0.450865\pi$
$648$		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	$0.182343\pi$
$653$			27.7018i			1.08405i	−0.840361	+	0.542027i	$0.817657\pi$
$654$		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	$-0.178166\pi$
$659$	−0.927195	−	1.60595i	−0.0361184	−	0.0625589i	−0.883520	+	0.468394i	$0.844833\pi$
$660$		0			0
$661$	9.79648	+	16.9680i	0.381039	+	0.659979i	0.991211	−	0.132290i	$-0.0422331\pi$
$661$	9.79648	+	16.9680i	0.381039	+	0.659979i	−0.610172	+	0.792269i	$0.708900\pi$
$662$		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	$-0.988886\pi$
$677$		−	1.81662i		−	0.0698184i	0.999390	−	0.0349092i	$-0.0111142\pi$
$678$		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	$-0.634955\pi$
$683$	−21.5025			−0.822772			−0.411386	+	0.911461i	$0.634955\pi$
$684$		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	0.695501	−	0.718525i	$-0.255183\pi$
$701$	11.1463	+	6.43532i	0.420990	+	0.243059i	−0.274511	+	0.961584i	$0.588516\pi$
$702$		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	−0.888716	−	0.458458i	$-0.848402\pi$
$709$	−22.4038	−	38.8046i	−0.841394	−	1.45734i	0.0473221	−	0.998880i	$-0.484931\pi$
$710$		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	$-0.865763\pi$
$719$	−2.72709	+	4.72347i	−0.101703	+	0.176156i	0.810683	+	0.585485i	$0.199096\pi$
$720$		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	$-0.356825\pi$
$727$	−3.44349	−	1.98810i	−0.127712	−	0.0737346i	−0.562495	+	0.826801i	$0.690158\pi$
$728$		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	$-0.118940\pi$
$739$	29.3695	+	16.9565i	1.08038	+	0.623755i	0.149378	+	0.988780i	$0.452273\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	21.8637	−	37.8690i	0.802100	−	1.38928i	−0.116131	−	0.993234i	$-0.537049\pi$
$743$	21.8637	−	37.8690i	0.802100	−	1.38928i	0.918231	−	0.396045i	$-0.129617\pi$
$744$		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	$-0.516098\pi$
$751$	21.6246	−	12.4850i	0.789092	−	0.455583i	0.839643	+	0.543139i	$0.182764\pi$
$752$		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	−0.468792	−	0.883309i	$-0.655311\pi$
$757$	14.5980	−	25.2844i	0.530572	−	0.918977i	0.999364	−	0.0356686i	$-0.0113561\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			2.01476i			0.0730351i	0.999333	+	0.0365176i	$0.0116265\pi$
$761$			2.01476i			0.0730351i	−0.999333	+	0.0365176i	$0.988374\pi$
$762$		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	0.934601	+	0.355697i	$0.115757\pi$
$769$	21.5009	+	37.2407i	0.775343	+	1.34293i	−0.159258	+	0.987237i	$0.550910\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	15.8967	−	9.17794i	0.571763	−	0.330107i	−0.186090	−	0.982533i	$-0.559582\pi$
$773$	15.8967	−	9.17794i	0.571763	−	0.330107i	0.757853	+	0.652425i	$0.226248\pi$
$774$		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	$0.0719493\pi$
$797$			12.6541i			0.448231i	−0.974563	+	0.224115i	$0.928051\pi$
$798$		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	$0.0422051\pi$
$809$			7.52049i			0.264406i	−0.991223	+	0.132203i	$0.957795\pi$
$810$		0			0
$811$	20.3422	−	11.7446i	0.714313	−	0.412409i	−0.0983431	−	0.995153i	$-0.531354\pi$
$811$	20.3422	−	11.7446i	0.714313	−	0.412409i	0.812656	+	0.582744i	$0.198021\pi$
$812$		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	−0.404122	−	0.914705i	$-0.632423\pi$
$821$	5.32878	−	3.07657i	0.185976	−	0.107373i	0.590097	+	0.807332i	$0.299089\pi$
$822$		0			0
$823$	−0.640832	+	0.369984i	−0.0223380	+	0.0128968i	−0.511127	−	0.859505i	$-0.670772\pi$
$823$	−0.640832	+	0.369984i	−0.0223380	+	0.0128968i	0.488789	+	0.872402i	$0.337439\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	25.3235	+	43.8616i	0.880584	+	1.52522i	0.850692	+	0.525664i	$0.176183\pi$
$827$	25.3235	+	43.8616i	0.880584	+	1.52522i	0.0298920	+	0.999553i	$0.490484\pi$
$828$		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	$-0.684418\pi$
$839$	13.0620	−	22.6241i	0.450952	−	0.781071i	0.998445	+	0.0557387i	$0.0177514\pi$
$840$		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	−0.667954	−	0.744203i	$-0.732830\pi$
$853$	9.06915	−	15.7082i	0.310522	−	0.537840i	0.978475	+	0.206363i	$0.0661629\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−45.3247	−	26.1683i	−1.54826	−	0.893890i	−0.998275	−	0.0587185i	$-0.981299\pi$
$857$	−45.3247	−	26.1683i	−1.54826	−	0.893890i	−0.549989	−	0.835172i	$-0.685368\pi$
$858$		0			0
$859$	−1.74572	+	1.00789i	−0.0595633	+	0.0343889i	−0.529486	−	0.848319i	$-0.677615\pi$
$859$	−1.74572	+	1.00789i	−0.0595633	+	0.0343889i	0.469923	+	0.882708i	$0.344282\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−28.2094			−0.960260			−0.480130	−	0.877197i	$-0.659410\pi$
$863$	−28.2094			−0.960260			−0.480130	+	0.877197i	$0.659410\pi$
$864$		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	−0.833540	−	0.552459i	$-0.813689\pi$
$877$	1.82642	−	3.16346i	0.0616739	−	0.106822i	0.895214	+	0.445637i	$0.147023\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	42.7712i		−	1.44100i	−0.693456	−	0.720499i	$-0.743913\pi$
$881$		−	42.7712i		−	1.44100i	0.693456	−	0.720499i	$-0.256087\pi$
$882$		0			0
$883$	22.5134	−	12.9981i	0.757636	−	0.437421i	−0.0708104	−	0.997490i	$-0.522559\pi$
$883$	22.5134	−	12.9981i	0.757636	−	0.437421i	0.828446	+	0.560068i	$0.189225\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−21.7353	−	37.6467i	−0.729801	−	1.26405i	−0.956967	−	0.290197i	$-0.906279\pi$
$887$	−21.7353	−	37.6467i	−0.729801	−	1.26405i	0.227166	−	0.973856i	$-0.427054\pi$
$888$		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	$-0.522041\pi$
$907$	−29.1447	−	16.8267i	−0.967735	−	0.558722i	−0.898545	+	0.438881i	$0.855375\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−0.923995			−0.0306133			−0.0153067	−	0.999883i	$-0.504872\pi$
$911$	−0.923995			−0.0306133			−0.0153067	+	0.999883i	$0.504872\pi$
$912$		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	0.642957	−	0.765903i	$-0.277708\pi$
$929$	9.17871	+	5.29933i	0.301144	+	0.173865i	−0.341813	+	0.939768i	$0.611041\pi$
$930$		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	0.939912	+	0.341418i	$0.110907\pi$
$937$	23.4364	+	40.5930i	0.765632	+	1.32611i	−0.174279	+	0.984696i	$0.555760\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	27.0848	−	15.6374i	0.882941	−	0.509766i	0.0113138	−	0.999936i	$-0.496399\pi$
$941$	27.0848	−	15.6374i	0.882941	−	0.509766i	0.871627	+	0.490170i	$0.163065\pi$
$942$		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	0.0681937	+	0.997672i	$0.478276\pi$
$947$	−25.5392	+	44.2352i	−0.829913	+	1.43745i	−0.898106	+	0.439779i	$0.855057\pi$
$948$		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	0.215275	−	0.976553i	$-0.430935\pi$
$953$	−16.1394	−	9.31810i	−0.522807	−	0.301843i	−0.738082	+	0.674711i	$0.764268\pi$
$954$		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	$-0.271580\pi$
$967$	10.3838	+	5.99509i	0.333920	+	0.192789i	−0.323660	+	0.946173i	$0.604913\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	16.1961	−	28.0524i	0.519757	−	0.900245i	−0.479980	−	0.877280i	$-0.659356\pi$
$971$	16.1961	−	28.0524i	0.519757	−	0.900245i	0.999736	−	0.0229653i	$-0.00731072\pi$
$972$		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	$0.0446043\pi$
$977$			8.73136i			0.279341i	−0.990198	+	0.139671i	$0.955396\pi$
$978$		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	$-0.701988\pi$
$983$	12.5732	−	21.7774i	0.401022	−	0.694591i	0.993850	+	0.110739i	$0.0353217\pi$
$984$		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	$-0.198181\pi$
$991$	22.4616	+	12.9682i	0.713518	+	0.411950i	−0.0988446	+	0.995103i	$0.531515\pi$
$992$		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	0.787073	−	0.616860i	$-0.211595\pi$
$997$	−4.44201	−	7.69378i	−0.140680	−	0.243664i	−0.927753	+	0.373195i	$0.878262\pi$
$998$		0			0
$999$		0			0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2736.2.cg.a.1151.5	✓	24	12.11	even	2
2736.2.cg.a.1151.8	yes	24	4.3	odd	2
2736.2.cg.a.2591.5	yes	24	19.7	even	3
2736.2.cg.a.2591.8	yes	24	57.26	odd	6
2736.2.cg.b.1151.5	yes	24	3.2	odd	2	inner
2736.2.cg.b.1151.8	yes	24	1.1	even	1	trivial
2736.2.cg.b.2591.5	yes	24	76.7	odd	6	inner
2736.2.cg.b.2591.8	yes	24	228.83	even	6	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:	\( N \)	\(=\)	\( 2736 = 2^{4} \cdot 3^{2} \cdot 19 \)
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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