LMFDB - Embedded newform 2736.2.cg.c.2591.2

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:	$32$
Relative dimension:	$16$ over $\Q(\zeta_{6})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			2591.2
Character	$\chi$	$=$	2736.2591
Dual form			2736.2.cg.c.1151.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-3.37787 + 1.95021i) q^{5} +3.27561i q^{7} +O(q^{10})$	$q+(-3.37787 + 1.95021i) q^{5} +3.27561i q^{7} -5.67850 q^{11} +(-2.34864 + 4.06796i) q^{13} +(-1.20495 + 0.695678i) q^{17} +(3.88569 - 1.97520i) q^{19} +(-3.59477 + 6.22632i) q^{23} +(5.10665 - 8.84498i) q^{25} +(4.52822 + 2.61437i) q^{29} +9.36294i q^{31} +(-6.38813 - 11.0646i) q^{35} -8.48371 q^{37} +(-3.32327 + 1.91869i) q^{41} +(0.584394 - 0.337400i) q^{43} +(3.54888 - 6.14684i) q^{47} -3.72960 q^{49} +(4.58282 + 2.64589i) q^{53} +(19.1812 - 11.0743i) q^{55} +(-5.63261 - 9.75597i) q^{59} +(0.574458 - 0.994990i) q^{61} -18.3214i q^{65} +(-5.96859 - 3.44597i) q^{67} +(-4.92298 - 8.52686i) q^{71} +(5.95529 + 10.3149i) q^{73} -18.6005i q^{77} +(6.25791 - 3.61301i) q^{79} +4.05126 q^{83} +(2.71344 - 4.69981i) q^{85} +(13.8078 + 7.97196i) q^{89} +(-13.3250 - 7.69321i) q^{91} +(-9.27327 + 14.2499i) q^{95} +(4.66740 + 8.08417i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q+O(q^{10}) $ 32 * q	$ 32 q - 16 q^{13} + 24 q^{25} - 16 q^{37} - 96 q^{49} - 8 q^{61} - 8 q^{73} + 16 q^{85} + 48 q^{97}+O(q^{100}) $ 32 * q - 16 * q^13 + 24 * q^25 - 16 * q^37 - 96 * q^49 - 8 * q^61 - 8 * q^73 + 16 * q^85 + 48 * 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
$5$	−3.37787	+	1.95021i	−1.51063	+	0.872161i	−0.510705	+	0.859756i	$0.670615\pi$
$5$	−3.37787	+	1.95021i	−1.51063	+	0.872161i	−0.999923	+	0.0124051i	$0.996051\pi$
$6$		0			0
$7$			3.27561i			1.23806i	0.785366	+	0.619032i	$0.212475\pi$
$7$			3.27561i			1.23806i	−0.785366	+	0.619032i	$0.787525\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−5.67850			−1.71213			−0.856066	−	0.516866i	$-0.827098\pi$
$11$	−5.67850			−1.71213			−0.856066	+	0.516866i	$0.827098\pi$
$12$		0			0
$13$	−2.34864	+	4.06796i	−0.651395	+	1.12825i	0.331390	+	0.943494i	$0.392482\pi$
$13$	−2.34864	+	4.06796i	−0.651395	+	1.12825i	−0.982785	+	0.184755i	$0.940851\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−1.20495	+	0.695678i	−0.292243	+	0.168727i	−0.638953	−	0.769246i	$-0.720632\pi$
$17$	−1.20495	+	0.695678i	−0.292243	+	0.168727i	0.346710	+	0.937972i	$0.387299\pi$
$18$		0			0
$19$	3.88569	−	1.97520i	0.891438	−	0.453143i
$19$	3.88569	−	1.97520i	0.891438	−	0.453143i
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−3.59477	+	6.22632i	−0.749561	+	1.29828i	0.198472	+	0.980107i	$0.436402\pi$
$23$	−3.59477	+	6.22632i	−0.749561	+	1.29828i	−0.948033	+	0.318171i	$0.896931\pi$
$24$		0			0
$25$	5.10665	−	8.84498i	1.02133	−	1.76900i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	4.52822	+	2.61437i	0.840869	+	0.485476i	0.857560	−	0.514385i	$-0.171980\pi$
$29$	4.52822	+	2.61437i	0.840869	+	0.485476i	−0.0166905	+	0.999861i	$0.505313\pi$
$30$		0			0
$31$			9.36294i			1.68163i	0.541319	+	0.840817i	$0.317925\pi$
$31$			9.36294i			1.68163i	−0.541319	+	0.840817i	$0.682075\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−6.38813	−	11.0646i	−1.07979	−	1.87025i
$36$		0			0
$37$	−8.48371			−1.39471			−0.697356	−	0.716725i	$-0.745640\pi$
$37$	−8.48371			−1.39471			−0.697356	+	0.716725i	$0.745640\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−3.32327	+	1.91869i	−0.519007	+	0.299649i	−0.736528	−	0.676407i	$-0.763536\pi$
$41$	−3.32327	+	1.91869i	−0.519007	+	0.299649i	0.217521	+	0.976056i	$0.430203\pi$
$42$		0			0
$43$	0.584394	−	0.337400i	0.0891192	−	0.0514530i	−0.454778	−	0.890605i	$-0.650281\pi$
$43$	0.584394	−	0.337400i	0.0891192	−	0.0514530i	0.543897	+	0.839152i	$0.316948\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	3.54888	−	6.14684i	0.517657	−	0.896608i	−0.482133	−	0.876098i	$-0.660138\pi$
$47$	3.54888	−	6.14684i	0.517657	−	0.896608i	0.999790	−	0.0205100i	$-0.00652899\pi$
$48$		0			0
$49$	−3.72960			−0.532800
$50$		0			0
$51$		0			0
$52$		0			0
$53$	4.58282	+	2.64589i	0.629498	+	0.363441i	0.780558	−	0.625084i	$-0.214935\pi$
$53$	4.58282	+	2.64589i	0.629498	+	0.363441i	−0.151060	+	0.988525i	$0.548269\pi$
$54$		0			0
$55$	19.1812	−	11.0743i	2.58639	−	1.49326i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−5.63261	−	9.75597i	−0.733303	−	1.27012i	−0.955464	−	0.295108i	$-0.904644\pi$
$59$	−5.63261	−	9.75597i	−0.733303	−	1.27012i	0.222160	−	0.975010i	$-0.428689\pi$
$60$		0			0
$61$	0.574458	−	0.994990i	0.0735518	−	0.127395i	−0.826904	−	0.562343i	$-0.809900\pi$
$61$	0.574458	−	0.994990i	0.0735518	−	0.127395i	0.900456	+	0.434948i	$0.143233\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	18.3214i		−	2.27249i
$66$		0			0
$67$	−5.96859	−	3.44597i	−0.729179	−	0.420992i	0.0889425	−	0.996037i	$-0.471651\pi$
$67$	−5.96859	−	3.44597i	−0.729179	−	0.420992i	−0.818122	+	0.575045i	$0.804985\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−4.92298	−	8.52686i	−0.584251	−	1.01195i	−0.994968	−	0.100189i	$-0.968055\pi$
$71$	−4.92298	−	8.52686i	−0.584251	−	1.01195i	0.410718	−	0.911762i	$-0.365278\pi$
$72$		0			0
$73$	5.95529	+	10.3149i	0.697014	+	1.20726i	0.969497	+	0.245103i	$0.0788219\pi$
$73$	5.95529	+	10.3149i	0.697014	+	1.20726i	−0.272483	+	0.962161i	$0.587845\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		−	18.6005i		−	2.11973i
$78$		0			0
$79$	6.25791	−	3.61301i	0.704070	−	0.406495i	−0.104791	−	0.994494i	$-0.533418\pi$
$79$	6.25791	−	3.61301i	0.704070	−	0.406495i	0.808862	+	0.587999i	$0.200084\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	4.05126			0.444683			0.222342	−	0.974969i	$-0.428630\pi$
$83$	4.05126			0.444683			0.222342	+	0.974969i	$0.428630\pi$
$84$		0			0
$85$	2.71344	−	4.69981i	0.294314	−	0.509766i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	13.8078	+	7.97196i	1.46363	+	0.845026i	0.999177	−	0.0405732i	$-0.0129184\pi$
$89$	13.8078	+	7.97196i	1.46363	+	0.845026i	0.464451	+	0.885599i	$0.346252\pi$
$90$		0			0
$91$	−13.3250	−	7.69321i	−1.39684	−	0.806468i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−9.27327	+	14.2499i	−0.951417	+	1.46201i
$96$		0			0
$97$	4.66740	+	8.08417i	0.473902	+	0.820823i	0.999554	−	0.0298774i	$-0.00951170\pi$
$97$	4.66740	+	8.08417i	0.473902	+	0.820823i	−0.525651	+	0.850700i	$0.676178\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−2.65167	−	1.53094i	−0.263851	−	0.152334i	0.362239	−	0.932085i	$-0.382012\pi$
$101$	−2.65167	−	1.53094i	−0.263851	−	0.152334i	−0.626090	+	0.779751i	$0.715346\pi$
$102$		0			0
$103$		−	6.89193i		−	0.679083i	−0.940591	−	0.339541i	$-0.889728\pi$
$103$		−	6.89193i		−	0.679083i	0.940591	−	0.339541i	$-0.110272\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	2.54022			0.245573			0.122786	−	0.992433i	$-0.460817\pi$
$107$	2.54022			0.245573			0.122786	+	0.992433i	$0.460817\pi$
$108$		0			0
$109$	−4.44157	−	7.69303i	−0.425426	−	0.736859i	0.571034	−	0.820926i	$-0.306542\pi$
$109$	−4.44157	−	7.69303i	−0.425426	−	0.736859i	−0.996460	+	0.0840672i	$0.973209\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			17.2092i			1.61890i	0.587186	+	0.809452i	$0.300236\pi$
$113$			17.2092i			1.61890i	−0.587186	+	0.809452i	$0.699764\pi$
$114$		0			0
$115$		−	28.0422i		−	2.61495i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−2.27877	−	3.94694i	−0.208894	−	0.361815i
$120$		0			0
$121$	21.2454			1.93140
$122$		0			0
$123$		0			0
$124$		0			0
$125$			20.3341i			1.81874i
$126$		0			0
$127$	5.10408	+	2.94684i	0.452914	+	0.261490i	0.709060	−	0.705148i	$-0.249120\pi$
$127$	5.10408	+	2.94684i	0.452914	+	0.261490i	−0.256146	+	0.966638i	$0.582453\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	3.64066	+	6.30581i	0.318086	+	0.550941i	0.980089	−	0.198560i	$-0.0636266\pi$
$131$	3.64066	+	6.30581i	0.318086	+	0.550941i	−0.662003	+	0.749501i	$0.730293\pi$
$132$		0			0
$133$	6.46999	+	12.7280i	0.561019	+	1.10366i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−8.02007	−	4.63039i	−0.685201	−	0.395601i	0.116611	−	0.993178i	$-0.462797\pi$
$137$	−8.02007	−	4.63039i	−0.685201	−	0.395601i	−0.801812	+	0.597577i	$0.796130\pi$
$138$		0			0
$139$	−6.37032	−	3.67791i	−0.540324	−	0.311956i	0.204886	−	0.978786i	$-0.434317\pi$
$139$	−6.37032	−	3.67791i	−0.540324	−	0.311956i	−0.745210	+	0.666830i	$0.767651\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	13.3367	−	23.0999i	1.11527	−	1.93171i
$144$		0			0
$145$	−20.3943			−1.69365
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−3.72883	+	2.15284i	−0.305478	+	0.176368i	−0.644901	−	0.764266i	$-0.723101\pi$
$149$	−3.72883	+	2.15284i	−0.305478	+	0.176368i	0.339423	+	0.940634i	$0.389768\pi$
$150$		0			0
$151$		−	24.0918i		−	1.96057i	−0.197599	−	0.980283i	$-0.563314\pi$
$151$		−	24.0918i		−	1.96057i	0.197599	−	0.980283i	$-0.436686\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−18.2597	−	31.6268i	−1.46666	−	2.54032i
$156$		0			0
$157$	−0.483837	−	0.838030i	−0.0386144	−	0.0668821i	0.846072	−	0.533068i	$-0.178961\pi$
$157$	−0.483837	−	0.838030i	−0.0386144	−	0.0668821i	−0.884687	+	0.466186i	$0.845628\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−20.3950	−	11.7751i	−1.60735	−	0.928004i
$162$		0			0
$163$		−	1.13868i		−	0.0891882i	−0.999005	−	0.0445941i	$-0.985801\pi$
$163$		−	1.13868i		−	0.0891882i	0.999005	−	0.0445941i	$-0.0141994\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	5.11802	−	8.86467i	0.396044	−	0.685968i	−0.597190	−	0.802100i	$-0.703716\pi$
$167$	5.11802	−	8.86467i	0.396044	−	0.685968i	0.993234	+	0.116132i	$0.0370494\pi$
$168$		0			0
$169$	−4.53220	−	7.84999i	−0.348630	−	0.603846i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−6.16300	+	3.55821i	−0.468564	+	0.270525i	−0.715638	−	0.698471i	$-0.753864\pi$
$173$	−6.16300	+	3.55821i	−0.468564	+	0.270525i	0.247074	+	0.968996i	$0.420531\pi$
$174$		0			0
$175$	28.9727	+	16.7274i	2.19013	+	1.26447i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−8.72500			−0.652137			−0.326068	−	0.945346i	$-0.605724\pi$
$179$	−8.72500			−0.652137			−0.326068	+	0.945346i	$0.605724\pi$
$180$		0			0
$181$	−9.07811	+	15.7237i	−0.674770	+	1.16874i	0.301765	+	0.953382i	$0.402424\pi$
$181$	−9.07811	+	15.7237i	−0.674770	+	1.16874i	−0.976536	+	0.215355i	$0.930909\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	28.6568	−	16.5450i	2.10689	−	1.21641i
$186$		0			0
$187$	6.84230	−	3.95041i	0.500359	−	0.288882i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	4.16746			0.301547			0.150774	−	0.988568i	$-0.451824\pi$
$191$	4.16746			0.301547			0.150774	+	0.988568i	$0.451824\pi$
$192$		0			0
$193$	−3.31631	−	5.74402i	−0.238713	−	0.413464i	0.721632	−	0.692277i	$-0.243392\pi$
$193$	−3.31631	−	5.74402i	−0.238713	−	0.413464i	−0.960345	+	0.278813i	$0.910059\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			15.2595i			1.08719i	0.839347	+	0.543596i	$0.182938\pi$
$197$			15.2595i			1.08719i	−0.839347	+	0.543596i	$0.817062\pi$
$198$		0			0
$199$	7.53911	+	4.35271i	0.534433	+	0.308555i	0.742820	−	0.669491i	$-0.233488\pi$
$199$	7.53911	+	4.35271i	0.534433	+	0.308555i	−0.208387	+	0.978047i	$0.566821\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−8.56364	+	14.8327i	−0.601050	+	1.04105i
$204$		0			0
$205$	7.48371	−	12.9622i	0.522685	−	0.905316i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−22.0649	+	11.2162i	−1.52626	+	0.775840i
$210$		0			0
$211$	8.10855	−	4.68147i	0.558215	−	0.322286i	−0.194214	−	0.980959i	$-0.562215\pi$
$211$	8.10855	−	4.68147i	0.558215	−	0.322286i	0.752429	+	0.658674i	$0.228882\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−1.31600	+	2.27938i	−0.0897506	+	0.155453i
$216$		0			0
$217$	−30.6693			−2.08197
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	6.53558i		−	0.439631i
$222$		0			0
$223$	−14.6062	+	8.43291i	−0.978106	+	0.564710i	−0.901698	−	0.432367i	$-0.857678\pi$
$223$	−14.6062	+	8.43291i	−0.978106	+	0.564710i	−0.0764080	+	0.997077i	$0.524345\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−12.7763			−0.847990			−0.423995	−	0.905665i	$-0.639373\pi$
$227$	−12.7763			−0.847990			−0.423995	+	0.905665i	$0.639373\pi$
$228$		0			0
$229$	18.0918			1.19554			0.597771	−	0.801667i	$-0.296053\pi$
$229$	18.0918			1.19554			0.597771	+	0.801667i	$0.296053\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	7.17989	−	4.14531i	0.470370	−	0.271568i	−0.246025	−	0.969264i	$-0.579124\pi$
$233$	7.17989	−	4.14531i	0.470370	−	0.271568i	0.716395	+	0.697695i	$0.245791\pi$
$234$		0			0
$235$			27.6843i			1.80592i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−19.5757			−1.26625			−0.633124	−	0.774050i	$-0.718228\pi$
$239$	−19.5757			−1.26625			−0.633124	+	0.774050i	$0.718228\pi$
$240$		0			0
$241$	13.0035	−	22.5228i	0.837630	−	1.45082i	−0.0542404	−	0.998528i	$-0.517274\pi$
$241$	13.0035	−	22.5228i	0.837630	−	1.45082i	0.891871	−	0.452290i	$-0.149393\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	12.5981	−	7.27351i	0.804863	−	0.464688i
$246$		0			0
$247$	−1.09103	+	20.4459i	−0.0694203	+	1.30094i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	11.2530	−	19.4908i	0.710284	−	1.23025i	−0.254467	−	0.967081i	$-0.581900\pi$
$251$	11.2530	−	19.4908i	0.710284	−	1.23025i	0.964751	−	0.263166i	$-0.0847666\pi$
$252$		0			0
$253$	20.4129	−	35.3562i	1.28335	−	2.22282i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−4.99317	−	2.88281i	−0.311465	−	0.179825i	0.336117	−	0.941820i	$-0.390886\pi$
$257$	−4.99317	−	2.88281i	−0.311465	−	0.179825i	−0.647582	+	0.761996i	$0.724220\pi$
$258$		0			0
$259$		−	27.7893i		−	1.72674i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	7.85327	+	13.6023i	0.484254	+	0.838752i	0.999836	−	0.0180879i	$-0.00575788\pi$
$263$	7.85327	+	13.6023i	0.484254	+	0.838752i	−0.515583	+	0.856840i	$0.672425\pi$
$264$		0			0
$265$	−20.6402			−1.26792
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−5.43680	+	3.13894i	−0.331488	+	0.191384i	−0.656501	−	0.754325i	$-0.727964\pi$
$269$	−5.43680	+	3.13894i	−0.331488	+	0.191384i	0.325014	+	0.945709i	$0.394631\pi$
$270$		0			0
$271$	−25.7135	+	14.8457i	−1.56198	+	0.901812i	−0.564927	+	0.825141i	$0.691096\pi$
$271$	−25.7135	+	14.8457i	−1.56198	+	0.901812i	−0.997056	+	0.0766706i	$0.975571\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−28.9981	+	50.2262i	−1.74865	+	3.02876i
$276$		0			0
$277$	−30.6970			−1.84440			−0.922202	−	0.386708i	$-0.873612\pi$
$277$	−30.6970			−1.84440			−0.922202	+	0.386708i	$0.873612\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	7.72370	+	4.45928i	0.460757	+	0.266018i	0.712363	−	0.701812i	$-0.247625\pi$
$281$	7.72370	+	4.45928i	0.460757	+	0.266018i	−0.251605	+	0.967830i	$0.580958\pi$
$282$		0			0
$283$	3.36586	−	1.94328i	0.200079	−	0.115516i	−0.396613	−	0.917986i	$-0.629815\pi$
$283$	3.36586	−	1.94328i	0.200079	−	0.115516i	0.596692	+	0.802470i	$0.296481\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−6.28488	−	10.8857i	−0.370984	−	0.642564i
$288$		0			0
$289$	−7.53207	+	13.0459i	−0.443063	+	0.767407i
$290$		0			0
$291$		0			0
$292$		0			0
$293$			7.70978i			0.450410i	0.974311	+	0.225205i	$0.0723053\pi$
$293$			7.70978i			0.450410i	−0.974311	+	0.225205i	$0.927695\pi$
$294$		0			0
$295$	38.0524	+	21.9696i	2.21550	+	1.27912i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−16.8856	−	29.2467i	−0.976520	−	1.69138i
$300$		0			0
$301$	1.10519	+	1.91424i	0.0637020	+	0.110335i
$302$		0			0
$303$		0			0
$304$		0			0
$305$			4.48126i			0.256596i
$306$		0			0
$307$	−0.310029	+	0.178995i	−0.0176943	+	0.0102158i	−0.508821	−	0.860872i	$-0.669919\pi$
$307$	−0.310029	+	0.178995i	−0.0176943	+	0.0102158i	0.491127	+	0.871088i	$0.336585\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	3.68561			0.208992			0.104496	−	0.994525i	$-0.466677\pi$
$311$	3.68561			0.208992			0.104496	+	0.994525i	$0.466677\pi$
$312$		0			0
$313$	−13.8361	+	23.9649i	−0.782064	+	1.35457i	0.148673	+	0.988886i	$0.452500\pi$
$313$	−13.8361	+	23.9649i	−0.782064	+	1.35457i	−0.930737	+	0.365688i	$0.880834\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	9.52139	+	5.49717i	0.534774	+	0.308752i	0.742958	−	0.669338i	$-0.233422\pi$
$317$	9.52139	+	5.49717i	0.534774	+	0.308752i	−0.208184	+	0.978090i	$0.566755\pi$
$318$		0			0
$319$	−25.7135	−	14.8457i	−1.43968	−	0.831199i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−3.30795	+	5.08321i	−0.184059	+	0.282837i
$324$		0			0
$325$	23.9874	+	41.5473i	1.33058	+	2.30463i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	20.1346	+	11.6247i	1.11006	+	0.640892i
$330$		0			0
$331$			32.7627i			1.80080i	0.435061	+	0.900401i	$0.356727\pi$
$331$			32.7627i			1.80080i	−0.435061	+	0.900401i	$0.643273\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	26.8815			1.46869
$336$		0			0
$337$	2.77186	+	4.80101i	0.150993	+	0.261528i	0.931593	−	0.363504i	$-0.118420\pi$
$337$	2.77186	+	4.80101i	0.150993	+	0.261528i	−0.780600	+	0.625031i	$0.785086\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	53.1675i		−	2.87918i
$342$		0			0
$343$			10.7125i			0.578423i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	14.0923	+	24.4085i	0.756512	+	1.31032i	0.944619	+	0.328169i	$0.106431\pi$
$347$	14.0923	+	24.4085i	0.756512	+	1.31032i	−0.188107	+	0.982149i	$0.560235\pi$
$348$		0			0
$349$	32.4312			1.73600			0.868002	−	0.496560i	$-0.165404\pi$
$349$	32.4312			1.73600			0.868002	+	0.496560i	$0.165404\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			9.23276i			0.491410i	0.969345	+	0.245705i	$0.0790195\pi$
$353$			9.23276i			0.491410i	−0.969345	+	0.245705i	$0.920981\pi$
$354$		0			0
$355$	33.2584	+	19.2017i	1.76517	+	1.01912i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	1.28232	+	2.22105i	0.0676784	+	0.117222i	0.897879	−	0.440242i	$-0.145107\pi$
$359$	1.28232	+	2.22105i	0.0676784	+	0.117222i	−0.830201	+	0.557465i	$0.811774\pi$
$360$		0			0
$361$	11.1971	−	15.3500i	0.589323	−	0.807897i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−40.2324	−	23.2282i	−2.10586	−	1.21582i
$366$		0			0
$367$	−24.5597	−	14.1795i	−1.28200	−	0.740165i	−0.304789	−	0.952420i	$-0.598586\pi$
$367$	−24.5597	−	14.1795i	−1.28200	−	0.740165i	−0.977214	+	0.212254i	$0.931919\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−8.66689	+	15.0115i	−0.449963	+	0.779358i
$372$		0			0
$373$	5.99511			0.310415			0.155207	−	0.987882i	$-0.450395\pi$
$373$	5.99511			0.310415			0.155207	+	0.987882i	$0.450395\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−21.2703	+	12.2804i	−1.09548	+	0.632473i
$378$		0			0
$379$			0.804599i			0.0413295i	0.999786	+	0.0206647i	$0.00657826\pi$
$379$			0.804599i			0.0413295i	−0.999786	+	0.0206647i	$0.993422\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−2.07152	−	3.58798i	−0.105850	−	0.183337i	0.808235	−	0.588860i	$-0.200423\pi$
$383$	−2.07152	−	3.58798i	−0.105850	−	0.183337i	−0.914085	+	0.405522i	$0.867090\pi$
$384$		0			0
$385$	36.2750	+	62.8301i	1.84874	+	3.20212i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	26.5385	+	15.3220i	1.34556	+	0.776857i	0.987616	−	0.156888i	$-0.0501462\pi$
$389$	26.5385	+	15.3220i	1.34556	+	0.776857i	0.357939	+	0.933745i	$0.383480\pi$
$390$		0			0
$391$		−	10.0032i		−	0.505884i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−14.0923	+	24.4085i	−0.709058	+	1.22813i
$396$		0			0
$397$	−2.33247	−	4.03996i	−0.117064	−	0.202760i	0.801539	−	0.597942i	$-0.204015\pi$
$397$	−2.33247	−	4.03996i	−0.117064	−	0.202760i	−0.918603	+	0.395182i	$0.870681\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−8.16733	+	4.71541i	−0.407857	+	0.235476i	−0.689869	−	0.723935i	$-0.742332\pi$
$401$	−8.16733	+	4.71541i	−0.407857	+	0.235476i	0.282011	+	0.959411i	$0.408998\pi$
$402$		0			0
$403$	−38.0881	−	21.9902i	−1.89730	−	1.09541i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	48.1747			2.38793
$408$		0			0
$409$	10.6228	−	18.3993i	0.525264	−	0.909785i	−0.474303	−	0.880362i	$-0.657300\pi$
$409$	10.6228	−	18.3993i	0.525264	−	0.909785i	0.999567	−	0.0294229i	$-0.00936695\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	31.9567	−	18.4502i	1.57249	−	0.907876i
$414$		0			0
$415$	−13.6846	+	7.90081i	−0.671751	+	0.387836i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	2.08467			0.101843			0.0509214	−	0.998703i	$-0.483784\pi$
$419$	2.08467			0.101843			0.0509214	+	0.998703i	$0.483784\pi$
$420$		0			0
$421$	−2.33625	−	4.04651i	−0.113862	−	0.197215i	0.803462	−	0.595356i	$-0.202989\pi$
$421$	−2.33625	−	4.04651i	−0.113862	−	0.197215i	−0.917324	+	0.398141i	$0.869656\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$			14.2103i			0.689303i
$426$		0			0
$427$	3.25920	+	1.88170i	0.157724	+	0.0910618i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	9.31916	−	16.1413i	0.448888	−	0.777497i	−0.549426	−	0.835543i	$-0.685153\pi$
$431$	9.31916	−	16.1413i	0.448888	−	0.777497i	0.998314	+	0.0580454i	$0.0184868\pi$
$432$		0			0
$433$	13.1872	−	22.8409i	0.633737	−	1.09766i	−0.353044	−	0.935607i	$-0.614854\pi$
$433$	13.1872	−	22.8409i	0.633737	−	1.09766i	0.986781	−	0.162058i	$-0.0518131\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−1.66990	+	31.2939i	−0.0798821	+	1.49699i
$438$		0			0
$439$	−10.3816	+	5.99383i	−0.495488	+	0.286070i	−0.726848	−	0.686798i	$-0.759016\pi$
$439$	−10.3816	+	5.99383i	−0.495488	+	0.286070i	0.231361	+	0.972868i	$0.425682\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−2.79336	+	4.83824i	−0.132717	+	0.229872i	−0.924723	−	0.380641i	$-0.875703\pi$
$443$	−2.79336	+	4.83824i	−0.132717	+	0.229872i	0.792006	+	0.610513i	$0.209037\pi$
$444$		0			0
$445$	−62.1880			−2.94799
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	18.9428i		−	0.893964i	−0.894543	−	0.446982i	$-0.852499\pi$
$449$		−	18.9428i		−	0.893964i	0.894543	−	0.446982i	$-0.147501\pi$
$450$		0			0
$451$	18.8712	−	10.8953i	0.888610	−	0.513039i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	60.0136			2.81348
$456$		0			0
$457$	−26.2451			−1.22769			−0.613847	−	0.789425i	$-0.710379\pi$
$457$	−26.2451			−1.22769			−0.613847	+	0.789425i	$0.710379\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	8.07467	−	4.66191i	0.376075	−	0.217127i	−0.300034	−	0.953928i	$-0.596998\pi$
$461$	8.07467	−	4.66191i	0.376075	−	0.217127i	0.676109	+	0.736802i	$0.263665\pi$
$462$		0			0
$463$			18.6793i			0.868101i	0.900889	+	0.434050i	$0.142916\pi$
$463$			18.6793i			0.868101i	−0.900889	+	0.434050i	$0.857084\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−9.63798			−0.445992			−0.222996	−	0.974819i	$-0.571584\pi$
$467$	−9.63798			−0.445992			−0.222996	+	0.974819i	$0.571584\pi$
$468$		0			0
$469$	11.2876	−	19.5508i	0.521215	−	0.902770i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−3.31848	+	1.91592i	−0.152584	+	0.0880943i
$474$		0			0
$475$	2.37222	−	44.4555i	0.108845	−	2.03976i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−2.03784	+	3.52964i	−0.0931113	+	0.161274i	−0.908819	−	0.417191i	$-0.863015\pi$
$479$	−2.03784	+	3.52964i	−0.0931113	+	0.161274i	0.815707	+	0.578465i	$0.196348\pi$
$480$		0			0
$481$	19.9252	−	34.5114i	0.908509	−	1.57358i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−31.5317	−	18.2048i	−1.43178	−	0.826638i
$486$		0			0
$487$		−	7.99607i		−	0.362337i	−0.983452	−	0.181168i	$-0.942012\pi$
$487$		−	7.99607i		−	0.362337i	0.983452	−	0.181168i	$-0.0579879\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	4.92298	+	8.52686i	0.222171	+	0.384812i	0.955467	−	0.295098i	$-0.0953523\pi$
$491$	4.92298	+	8.52686i	0.222171	+	0.384812i	−0.733296	+	0.679910i	$0.762019\pi$
$492$		0			0
$493$	−7.27503			−0.327651
$494$		0			0
$495$		0			0
$496$		0			0
$497$	27.9306	−	16.1258i	1.25286	−	0.723339i
$498$		0			0
$499$	−0.416687	+	0.240574i	−0.0186535	+	0.0107696i	−0.509298	−	0.860590i	$-0.670095\pi$
$499$	−0.416687	+	0.240574i	−0.0186535	+	0.0107696i	0.490644	+	0.871360i	$0.336761\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	19.9206	−	34.5036i	0.888218	−	1.53844i	0.0462369	−	0.998931i	$-0.485277\pi$
$503$	19.9206	−	34.5036i	0.888218	−	1.53844i	0.841981	−	0.539508i	$-0.181390\pi$
$504$		0			0
$505$	11.9426			0.531441
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−19.3040	−	11.1452i	−0.855636	−	0.494001i	0.00691272	−	0.999976i	$-0.497800\pi$
$509$	−19.3040	−	11.1452i	−0.855636	−	0.494001i	−0.862548	+	0.505975i	$0.831133\pi$
$510$		0			0
$511$	−33.7874	+	19.5072i	−1.49467	+	0.862947i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	13.4407	+	23.2800i	0.592269	+	1.02584i
$516$		0			0
$517$	−20.1523	+	34.9048i	−0.886297	+	1.53511i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			7.73125i			0.338712i	0.985555	+	0.169356i	$0.0541688\pi$
$521$			7.73125i			0.338712i	−0.985555	+	0.169356i	$0.945831\pi$
$522$		0			0
$523$	−0.986124	−	0.569339i	−0.0431202	−	0.0248955i	0.478285	−	0.878205i	$-0.341259\pi$
$523$	−0.986124	−	0.569339i	−0.0431202	−	0.0248955i	−0.521405	+	0.853309i	$0.674592\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−6.51359	−	11.2819i	−0.283736	−	0.491446i
$528$		0			0
$529$	−14.3447	−	24.8458i	−0.623684	−	1.08025i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	18.0252i		−	0.780760i
$534$		0			0
$535$	−8.58053	+	4.95397i	−0.370969	+	0.214179i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	21.1785			0.912224
$540$		0			0
$541$	−8.22960	+	14.2541i	−0.353818	+	0.612831i	−0.986915	−	0.161242i	$-0.948450\pi$
$541$	−8.22960	+	14.2541i	−0.353818	+	0.612831i	0.633097	+	0.774073i	$0.281784\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	30.0061	+	17.3240i	1.28532	+	0.742080i
$546$		0			0
$547$	−10.7926	−	6.23108i	−0.461456	−	0.266422i	0.251200	−	0.967935i	$-0.419175\pi$
$547$	−10.7926	−	6.23108i	−0.461456	−	0.266422i	−0.712656	+	0.701513i	$0.752508\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	22.7592	+	1.21447i	0.969573	+	0.0517381i
$552$		0			0
$553$	11.8348	+	20.4985i	0.503266	+	0.871683i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	15.9642	+	9.21695i	0.676426	+	0.390535i	0.798507	−	0.601985i	$-0.205624\pi$
$557$	15.9642	+	9.21695i	0.676426	+	0.390535i	−0.122081	+	0.992520i	$0.538957\pi$
$558$		0			0
$559$			3.16972i			0.134065i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	34.0466			1.43489			0.717446	−	0.696614i	$-0.245311\pi$
$563$	34.0466			1.43489			0.717446	+	0.696614i	$0.245311\pi$
$564$		0			0
$565$	−33.5616	−	58.1303i	−1.41195	−	2.44556i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			25.8995i			1.08576i	0.839809	+	0.542882i	$0.182667\pi$
$569$			25.8995i			1.08576i	−0.839809	+	0.542882i	$0.817333\pi$
$570$		0			0
$571$		−	4.59067i		−	0.192113i	−0.995376	−	0.0960567i	$-0.969377\pi$
$571$		−	4.59067i		−	0.192113i	0.995376	−	0.0960567i	$-0.0306230\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	36.7145	+	63.5913i	1.53110	+	2.65194i
$576$		0			0
$577$	−20.4295			−0.850493			−0.425247	−	0.905078i	$-0.639813\pi$
$577$	−20.4295			−0.850493			−0.425247	+	0.905078i	$0.639813\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$			13.2703i			0.550546i
$582$		0			0
$583$	−26.0235	−	15.0247i	−1.07778	−	0.622259i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−4.95666	−	8.58519i	−0.204583	−	0.354349i	0.745417	−	0.666599i	$-0.232251\pi$
$587$	−4.95666	−	8.58519i	−0.204583	−	0.354349i	−0.950000	+	0.312250i	$0.898917\pi$
$588$		0			0
$589$	18.4937	+	36.3815i	0.762020	+	1.49907i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−12.3914	−	7.15420i	−0.508856	−	0.293788i	0.223508	−	0.974702i	$-0.428249\pi$
$593$	−12.3914	−	7.15420i	−0.508856	−	0.293788i	−0.732363	+	0.680914i	$0.761583\pi$
$594$		0			0
$595$	15.3947	+	8.88816i	0.631123	+	0.364379i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	3.35384	−	5.80903i	0.137034	−	0.237350i	−0.789338	−	0.613958i	$-0.789576\pi$
$599$	3.35384	−	5.80903i	0.137034	−	0.237350i	0.926373	+	0.376608i	$0.122910\pi$
$600$		0			0
$601$	−3.81850			−0.155760			−0.0778799	−	0.996963i	$-0.524815\pi$
$601$	−3.81850			−0.155760			−0.0778799	+	0.996963i	$0.524815\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−71.7640	+	41.4330i	−2.91762	+	1.68449i
$606$		0			0
$607$			19.5650i			0.794120i	0.917793	+	0.397060i	$0.129969\pi$
$607$			19.5650i			0.794120i	−0.917793	+	0.397060i	$0.870031\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	16.6701	+	28.8734i	0.674398	+	1.16809i
$612$		0			0
$613$	−21.2753	−	36.8498i	−0.859299	−	1.48835i	−0.872598	−	0.488439i	$-0.837567\pi$
$613$	−21.2753	−	36.8498i	−0.859299	−	1.48835i	0.0132989	−	0.999912i	$-0.495767\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	13.5604	+	7.82909i	0.545921	+	0.315187i	0.747475	−	0.664290i	$-0.231266\pi$
$617$	13.5604	+	7.82909i	0.545921	+	0.315187i	−0.201554	+	0.979477i	$0.564599\pi$
$618$		0			0
$619$		−	34.6107i		−	1.39112i	−0.718467	−	0.695561i	$-0.755156\pi$
$619$		−	34.6107i		−	1.39112i	0.718467	−	0.695561i	$-0.244844\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−26.1130	+	45.2290i	−1.04620	+	1.81206i
$624$		0			0
$625$	−14.1226	−	24.4610i	−0.564902	−	0.978439i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	10.2224	−	5.90193i	0.407595	−	0.235325i
$630$		0			0
$631$	6.11091	+	3.52814i	0.243272	+	0.140453i	0.616679	−	0.787214i	$-0.288477\pi$
$631$	6.11091	+	3.52814i	0.243272	+	0.140453i	−0.373408	+	0.927667i	$0.621811\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−22.9879			−0.912246
$636$		0			0
$637$	8.75948	−	15.1719i	0.347063	−	0.601131i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	16.9743	−	9.80009i	0.670443	−	0.387080i	−0.125802	−	0.992055i	$-0.540150\pi$
$641$	16.9743	−	9.80009i	0.670443	−	0.387080i	0.796244	+	0.604975i	$0.206817\pi$
$642$		0			0
$643$	−12.8109	+	7.39637i	−0.505212	+	0.291685i	−0.730863	−	0.682524i	$-0.760882\pi$
$643$	−12.8109	+	7.39637i	−0.505212	+	0.291685i	0.225651	+	0.974208i	$0.427549\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	30.8654			1.21344			0.606721	−	0.794915i	$-0.292484\pi$
$647$	30.8654			1.21344			0.606721	+	0.794915i	$0.292484\pi$
$648$		0			0
$649$	31.9848	+	55.3993i	1.25551	+	2.17461i
$650$		0			0
$651$		0			0
$652$		0			0
$653$			27.6677i			1.08272i	0.840791	+	0.541360i	$0.182090\pi$
$653$			27.6677i			1.08272i	−0.840791	+	0.541360i	$0.817910\pi$
$654$		0			0
$655$	−24.5953	−	14.2001i	−0.961019	−	0.554845i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−18.4089	+	31.8851i	−0.717108	+	1.24207i	0.245033	+	0.969515i	$0.421201\pi$
$659$	−18.4089	+	31.8851i	−0.717108	+	1.24207i	−0.962141	+	0.272552i	$0.912132\pi$
$660$		0			0
$661$	−5.48371	+	9.49806i	−0.213291	+	0.369432i	−0.952743	−	0.303778i	$-0.901752\pi$
$661$	−5.48371	+	9.49806i	−0.213291	+	0.369432i	0.739451	+	0.673210i	$0.235085\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−46.6770	−	30.3756i	−1.81006	−	1.17791i
$666$		0			0
$667$	−32.5558	+	18.7961i	−1.26057	+	0.727788i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−3.26206	+	5.65005i	−0.125930	+	0.218118i
$672$		0			0
$673$	−17.0122			−0.655772			−0.327886	−	0.944717i	$-0.606336\pi$
$673$	−17.0122			−0.655772			−0.327886	+	0.944717i	$0.606336\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	3.69536i		−	0.142024i	−0.997475	−	0.0710120i	$-0.977377\pi$
$677$		−	3.69536i		−	0.142024i	0.997475	−	0.0710120i	$-0.0226229\pi$
$678$		0			0
$679$	−26.4805	+	15.2886i	−1.01623	+	0.586721i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−49.8693			−1.90820			−0.954099	−	0.299493i	$-0.903183\pi$
$683$	−49.8693			−1.90820			−0.954099	+	0.299493i	$0.903183\pi$
$684$		0			0
$685$	36.1210			1.38011
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−21.5267	+	12.4285i	−0.820104	+	0.473487i
$690$		0			0
$691$		−	19.6882i		−	0.748974i	−0.927232	−	0.374487i	$-0.877819\pi$
$691$		−	19.6882i		−	0.748974i	0.927232	−	0.374487i	$-0.122181\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	28.6908			1.08830
$696$		0			0
$697$	2.66958	−	4.62385i	0.101118	−	0.175141i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	19.0808	−	11.0163i	0.720673	−	0.416081i	−0.0943271	−	0.995541i	$-0.530070\pi$
$701$	19.0808	−	11.0163i	0.720673	−	0.416081i	0.815000	+	0.579460i	$0.196737\pi$
$702$		0			0
$703$	−32.9650	+	16.7570i	−1.24330	+	0.632004i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	5.01476	−	8.68583i	0.188600	−	0.326664i
$708$		0			0
$709$	3.54702	−	6.14363i	0.133211	−	0.230729i	−0.791701	−	0.610908i	$-0.790804\pi$
$709$	3.54702	−	6.14363i	0.133211	−	0.230729i	0.924913	+	0.380179i	$0.124138\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−58.2967	−	33.6576i	−2.18323	−	1.26049i
$714$		0			0
$715$			104.038i			3.89080i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−8.41376	−	14.5731i	−0.313780	−	0.543483i	0.665397	−	0.746489i	$-0.268262\pi$
$719$	−8.41376	−	14.5731i	−0.313780	−	0.543483i	−0.979177	+	0.203006i	$0.934929\pi$
$720$		0			0
$721$	22.5753			0.840747
$722$		0			0
$723$		0			0
$724$		0			0
$725$	46.2481	−	26.7013i	1.71761	−	0.991663i
$726$		0			0
$727$	22.0180	−	12.7121i	0.816602	−	0.471465i	−0.0326416	−	0.999467i	$-0.510392\pi$
$727$	22.0180	−	12.7121i	0.816602	−	0.471465i	0.849243	+	0.528002i	$0.177059\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−0.469443	+	0.813099i	−0.0173630	+	0.0300736i
$732$		0			0
$733$	−17.8756			−0.660250			−0.330125	−	0.943937i	$-0.607091\pi$
$733$	−17.8756			−0.660250			−0.330125	+	0.943937i	$0.607091\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	33.8926	+	19.5679i	1.24845	+	0.720794i
$738$		0			0
$739$	3.86322	−	2.23043i	0.142111	−	0.0820478i	−0.427259	−	0.904129i	$-0.640521\pi$
$739$	3.86322	−	2.23043i	0.142111	−	0.0820478i	0.569370	+	0.822082i	$0.307187\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−16.3588	−	28.3343i	−0.600147	−	1.03948i	−0.992798	−	0.119797i	$-0.961776\pi$
$743$	−16.3588	−	28.3343i	−0.600147	−	1.03948i	0.392652	−	0.919687i	$-0.371558\pi$
$744$		0			0
$745$	8.39700	−	14.5440i	0.307642	−	0.532852i
$746$		0			0
$747$		0			0
$748$		0			0
$749$			8.32077i			0.304034i
$750$		0			0
$751$	2.49214	+	1.43884i	0.0909395	+	0.0525040i	0.544780	−	0.838579i	$-0.316613\pi$
$751$	2.49214	+	1.43884i	0.0909395	+	0.0525040i	−0.453841	+	0.891083i	$0.649946\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	46.9842	+	81.3790i	1.70993	+	2.96169i
$756$		0			0
$757$	12.8360	+	22.2326i	0.466532	+	0.808057i	0.999269	−	0.0382235i	$-0.0121699\pi$
$757$	12.8360	+	22.2326i	0.466532	+	0.808057i	−0.532737	+	0.846281i	$0.678837\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	17.9352i		−	0.650150i	−0.945688	−	0.325075i	$-0.894610\pi$
$761$		−	17.9352i		−	0.650150i	0.945688	−	0.325075i	$-0.105390\pi$
$762$		0			0
$763$	25.1994	−	14.5489i	0.912278	−	0.526704i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	52.9158			1.91068
$768$		0			0
$769$	−9.92395	+	17.1888i	−0.357867	+	0.619843i	−0.987604	−	0.156965i	$-0.949829\pi$
$769$	−9.92395	+	17.1888i	−0.357867	+	0.619843i	0.629738	+	0.776808i	$0.283162\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	26.4410	+	15.2657i	0.951018	+	0.549071i	0.893397	−	0.449268i	$-0.148315\pi$
$773$	26.4410	+	15.2657i	0.951018	+	0.549071i	0.0576213	+	0.998339i	$0.481648\pi$
$774$		0			0
$775$	82.8151	+	47.8133i	2.97481	+	1.71750i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−9.12338	+	14.0196i	−0.326879	+	0.502303i
$780$		0			0
$781$	27.9552	+	48.4198i	1.00031	+	1.73260i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	3.26867	+	1.88717i	0.116664	+	0.0673559i
$786$		0			0
$787$			25.1007i			0.894744i	0.894348	+	0.447372i	$0.147640\pi$
$787$			25.1007i			0.894744i	−0.894348	+	0.447372i	$0.852360\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−56.3705			−2.00430
$792$		0			0
$793$	2.69839	+	4.67374i	0.0958225	+	0.165970i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			24.4922i			0.867560i	0.901019	+	0.433780i	$0.142820\pi$
$797$			24.4922i			0.867560i	−0.901019	+	0.433780i	$0.857180\pi$
$798$		0			0
$799$			9.87550i			0.349370i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−33.8171	−	58.5730i	−1.19338	−	2.06700i
$804$		0			0
$805$	91.8554			3.23748
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	4.71435i		−	0.165748i	−0.996560	−	0.0828738i	$-0.973590\pi$
$809$		−	4.71435i		−	0.165748i	0.996560	−	0.0828738i	$-0.0264098\pi$
$810$		0			0
$811$	−40.1007	−	23.1521i	−1.40812	−	0.812981i	−0.412917	−	0.910769i	$-0.635490\pi$
$811$	−40.1007	−	23.1521i	−1.40812	−	0.812981i	−0.995207	+	0.0977875i	$0.968823\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	2.22066	+	3.84630i	0.0777865	+	0.134730i
$816$		0			0
$817$	1.60434	−	2.46533i	0.0561287	−	0.0862508i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	6.52776	+	3.76881i	0.227821	+	0.131532i	0.609566	−	0.792735i	$-0.291344\pi$
$821$	6.52776	+	3.76881i	0.227821	+	0.131532i	−0.381746	+	0.924267i	$0.624677\pi$
$822$		0			0
$823$	36.9895	+	21.3559i	1.28937	+	0.744420i	0.978543	−	0.206045i	$-0.0660592\pi$
$823$	36.9895	+	21.3559i	1.28937	+	0.744420i	0.310831	+	0.950465i	$0.399393\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	13.1212	−	22.7265i	0.456268	−	0.790279i	−0.542492	−	0.840061i	$-0.682519\pi$
$827$	13.1212	−	22.7265i	0.456268	−	0.790279i	0.998760	+	0.0497815i	$0.0158525\pi$
$828$		0			0
$829$	−45.4613			−1.57894			−0.789469	−	0.613791i	$-0.789644\pi$
$829$	−45.4613			−1.57894			−0.789469	+	0.613791i	$0.789644\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	4.49398	−	2.59460i	0.155707	−	0.0898976i
$834$		0			0
$835$			39.9249i			1.38166i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	10.3025	+	17.8444i	0.355680	+	0.616056i	0.987234	−	0.159276i	$-0.0509159\pi$
$839$	10.3025	+	17.8444i	0.355680	+	0.616056i	−0.631554	+	0.775332i	$0.717583\pi$
$840$		0			0
$841$	−0.830158	−	1.43788i	−0.0286262	−	0.0495820i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	30.6183	+	17.6775i	1.05330	+	0.608124i
$846$		0			0
$847$			69.5915i			2.39119i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	30.4970	−	52.8223i	1.04542	−	1.81072i
$852$		0			0
$853$	−3.52842	−	6.11139i	−0.120811	−	0.209250i	0.799277	−	0.600963i	$-0.205216\pi$
$853$	−3.52842	−	6.11139i	−0.120811	−	0.209250i	−0.920088	+	0.391713i	$0.871883\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	16.4352	−	9.48889i	0.561417	−	0.324134i	−0.192297	−	0.981337i	$-0.561594\pi$
$857$	16.4352	−	9.48889i	0.561417	−	0.324134i	0.753714	+	0.657203i	$0.228260\pi$
$858$		0			0
$859$	6.42562	+	3.70983i	0.219239	+	0.126578i	0.605598	−	0.795771i	$-0.292934\pi$
$859$	6.42562	+	3.70983i	0.219239	+	0.126578i	−0.386359	+	0.922349i	$0.626267\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−37.7307			−1.28437			−0.642184	−	0.766550i	$-0.721972\pi$
$863$	−37.7307			−1.28437			−0.642184	+	0.766550i	$0.721972\pi$
$864$		0			0
$865$	13.8785	−	24.0383i	0.471884	−	0.817327i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−35.5356	+	20.5165i	−1.20546	+	0.695973i
$870$		0			0
$871$	28.0361	−	16.1867i	0.949967	−	0.548464i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−66.6066			−2.25171
$876$		0			0
$877$	−0.927858	−	1.60710i	−0.0313315	−	0.0542678i	0.849934	−	0.526888i	$-0.176641\pi$
$877$	−0.927858	−	1.60710i	−0.0313315	−	0.0542678i	−0.881266	+	0.472621i	$0.843308\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	13.9367i		−	0.469539i	−0.972051	−	0.234770i	$-0.924566\pi$
$881$		−	13.9367i		−	0.469539i	0.972051	−	0.234770i	$-0.0754336\pi$
$882$		0			0
$883$	−31.5144	−	18.1948i	−1.06054	−	0.612305i	−0.134961	−	0.990851i	$-0.543091\pi$
$883$	−31.5144	−	18.1948i	−1.06054	−	0.612305i	−0.925582	+	0.378546i	$0.876424\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	18.2719	−	31.6479i	0.613511	−	1.06263i	−0.377132	−	0.926159i	$-0.623090\pi$
$887$	18.2719	−	31.6479i	0.613511	−	1.06263i	0.990644	−	0.136474i	$-0.0435769\pi$
$888$		0			0
$889$	−9.65270	+	16.7190i	−0.323741	+	0.560736i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	1.64858	−	30.8944i	0.0551676	−	1.03384i
$894$		0			0
$895$	29.4719	−	17.0156i	0.985136	−	0.568769i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−24.4782	+	42.3974i	−0.816393	+	1.41403i
$900$		0			0
$901$	−7.36275			−0.245289
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	70.8169i		−	2.35403i
$906$		0			0
$907$	29.6994	−	17.1470i	0.986153	−	0.569355i	0.0820306	−	0.996630i	$-0.473859\pi$
$907$	29.6994	−	17.1470i	0.986153	−	0.569355i	0.904122	+	0.427274i	$0.140526\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−49.5696			−1.64231			−0.821156	−	0.570703i	$-0.806671\pi$
$911$	−49.5696			−1.64231			−0.821156	+	0.570703i	$0.806671\pi$
$912$		0			0
$913$	−23.0051			−0.761357
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−20.6553	+	11.9254i	−0.682100	+	0.393810i
$918$		0			0
$919$			12.3617i			0.407774i	0.978994	+	0.203887i	$0.0653575\pi$
$919$			12.3617i			0.407774i	−0.978994	+	0.203887i	$0.934642\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	46.2492			1.52231
$924$		0			0
$925$	−43.3233	+	75.0382i	−1.42446	+	2.46724i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	22.5934	−	13.0443i	0.741267	−	0.427971i	−0.0812629	−	0.996693i	$-0.525895\pi$
$929$	22.5934	−	13.0443i	0.741267	−	0.427971i	0.822530	+	0.568722i	$0.192562\pi$
$930$		0			0
$931$	−14.4921	+	7.36672i	−0.474958	+	0.241435i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−15.4083	+	26.6879i	−0.503904	+	0.872787i
$936$		0			0
$937$	13.5296	−	23.4340i	0.441993	−	0.765555i	−0.555844	−	0.831287i	$-0.687605\pi$
$937$	13.5296	−	23.4340i	0.441993	−	0.765555i	0.997837	+	0.0657317i	$0.0209381\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−28.6188	−	16.5230i	−0.932945	−	0.538636i	−0.0452031	−	0.998978i	$-0.514393\pi$
$941$	−28.6188	−	16.5230i	−0.932945	−	0.538636i	−0.887742	+	0.460342i	$0.847727\pi$
$942$		0			0
$943$		−	27.5890i		−	0.898421i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	19.0733	+	33.0360i	0.619800	+	1.07353i	0.989522	+	0.144383i	$0.0461198\pi$
$947$	19.0733	+	33.0360i	0.619800	+	1.07353i	−0.369721	+	0.929143i	$0.620547\pi$
$948$		0			0
$949$	−55.9473			−1.81613
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−25.6143	+	14.7884i	−0.829728	+	0.479044i	−0.853759	−	0.520668i	$-0.825683\pi$
$953$	−25.6143	+	14.7884i	−0.829728	+	0.479044i	0.0240316	+	0.999711i	$0.492350\pi$
$954$		0			0
$955$	−14.0771	+	8.12744i	−0.455525	+	0.262998i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	15.1673	−	26.2706i	0.489779	−	0.848322i
$960$		0			0
$961$	−56.6647			−1.82789
$962$		0			0
$963$		0			0
$964$		0			0
$965$	22.4041	+	12.9350i	0.721214	+	0.416393i
$966$		0			0
$967$	−50.8725	+	29.3713i	−1.63595	+	0.944516i	−0.653742	+	0.756718i	$0.726802\pi$
$967$	−50.8725	+	29.3713i	−1.63595	+	0.944516i	−0.982208	+	0.187798i	$0.939865\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	11.9963	+	20.7782i	0.384980	+	0.666805i	0.991766	−	0.128060i	$-0.0408750\pi$
$971$	11.9963	+	20.7782i	0.384980	+	0.666805i	−0.606786	+	0.794865i	$0.707542\pi$
$972$		0			0
$973$	12.0474	−	20.8667i	0.386221	−	0.668955i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	51.3764i		−	1.64368i	−0.569721	−	0.821838i	$-0.692949\pi$
$977$		−	51.3764i		−	1.64368i	0.569721	−	0.821838i	$-0.307051\pi$
$978$		0			0
$979$	−78.4078	−	45.2688i	−2.50592	−	1.44680i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	22.6437	+	39.2200i	0.722222	+	1.25092i	0.960107	+	0.279631i	$0.0902122\pi$
$983$	22.6437	+	39.2200i	0.722222	+	1.25092i	−0.237886	+	0.971293i	$0.576454\pi$
$984$		0			0
$985$	−29.7592	−	51.5445i	−0.948208	−	1.64234i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			4.85150i			0.154269i
$990$		0			0
$991$	35.9481	−	20.7547i	1.14193	−	0.659293i	0.195022	−	0.980799i	$-0.437522\pi$
$991$	35.9481	−	20.7547i	1.14193	−	0.659293i	0.946908	+	0.321506i	$0.104189\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−33.9548			−1.07644
$996$		0			0
$997$	13.1534	−	22.7824i	0.416573	−	0.721525i	−0.579019	−	0.815314i	$-0.696564\pi$
$997$	13.1534	−	22.7824i	0.416573	−	0.721525i	0.995592	+	0.0937884i	$0.0298977\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2736.2.cg.c.2591.2	yes	32
3.2	odd	2	inner	2736.2.cg.c.2591.16	yes	32
4.3	odd	2	inner	2736.2.cg.c.2591.1	yes	32
12.11	even	2	inner	2736.2.cg.c.2591.15	yes	32
19.11	even	3	inner	2736.2.cg.c.1151.15	yes	32
57.11	odd	6	inner	2736.2.cg.c.1151.1	✓	32
76.11	odd	6	inner	2736.2.cg.c.1151.16	yes	32
228.11	even	6	inner	2736.2.cg.c.1151.2	yes	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2736.2.cg.c.1151.1	✓	32	57.11	odd	6	inner
2736.2.cg.c.1151.2	yes	32	228.11	even	6	inner
2736.2.cg.c.1151.15	yes	32	19.11	even	3	inner
2736.2.cg.c.1151.16	yes	32	76.11	odd	6	inner
2736.2.cg.c.2591.1	yes	32	4.3	odd	2	inner
2736.2.cg.c.2591.2	yes	32	1.1	even	1	trivial
2736.2.cg.c.2591.15	yes	32	12.11	even	2	inner
2736.2.cg.c.2591.16	yes	32	3.2	odd	2	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)

\(f(q)\)	\(=\)	\(q+(-3.37787 + 1.95021i) q^{5} +3.27561i q^{7} +O(q^{10})\)	\(q+(-3.37787 + 1.95021i) q^{5} +3.27561i q^{7} -5.67850 q^{11} +(-2.34864 + 4.06796i) q^{13} +(-1.20495 + 0.695678i) q^{17} +(3.88569 - 1.97520i) q^{19} +(-3.59477 + 6.22632i) q^{23} +(5.10665 - 8.84498i) q^{25} +(4.52822 + 2.61437i) q^{29} +9.36294i q^{31} +(-6.38813 - 11.0646i) q^{35} -8.48371 q^{37} +(-3.32327 + 1.91869i) q^{41} +(0.584394 - 0.337400i) q^{43} +(3.54888 - 6.14684i) q^{47} -3.72960 q^{49} +(4.58282 + 2.64589i) q^{53} +(19.1812 - 11.0743i) q^{55} +(-5.63261 - 9.75597i) q^{59} +(0.574458 - 0.994990i) q^{61} -18.3214i q^{65} +(-5.96859 - 3.44597i) q^{67} +(-4.92298 - 8.52686i) q^{71} +(5.95529 + 10.3149i) q^{73} -18.6005i q^{77} +(6.25791 - 3.61301i) q^{79} +4.05126 q^{83} +(2.71344 - 4.69981i) q^{85} +(13.8078 + 7.97196i) q^{89} +(-13.3250 - 7.69321i) q^{91} +(-9.27327 + 14.2499i) q^{95} +(4.66740 + 8.08417i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q+O(q^{10}) \) 32 * q	\( 32 q - 16 q^{13} + 24 q^{25} - 16 q^{37} - 96 q^{49} - 8 q^{61} - 8 q^{73} + 16 q^{85} + 48 q^{97}+O(q^{100}) \) 32 * q - 16 * q^13 + 24 * q^25 - 16 * q^37 - 96 * q^49 - 8 * q^61 - 8 * q^73 + 16 * q^85 + 48 * q^97

Introduction

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