LMFDB - Embedded newform 2736.2.d.b.2015.18

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2736,2,Mod(2015,2736)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2736, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 1, 0]))

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

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

chi := DirichletCharacter("2736.2015");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2736 = 2^{4} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2736.d (of order $2$, degree $1$, 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:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2015.18
Character	$\chi$	$=$	2736.2015
Dual form			2736.2.d.b.2015.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.03314i q^{5} +3.00897i q^{7} +O(q^{10})$	$q+2.03314i q^{5} +3.00897i q^{7} +4.39868 q^{11} +4.79346 q^{13} +2.70773i q^{17} +1.00000i q^{19} +1.06706 q^{23} +0.866325 q^{25} +2.56438i q^{29} -2.28336i q^{31} -6.11768 q^{35} +4.52611 q^{37} -2.51222i q^{41} -3.98914i q^{43} -1.12272 q^{47} -2.05393 q^{49} -6.72681i q^{53} +8.94315i q^{55} -7.15876 q^{59} +12.9611 q^{61} +9.74579i q^{65} -1.75060i q^{67} -4.70842 q^{71} -0.322316 q^{73} +13.2355i q^{77} -2.71440i q^{79} -6.57724 q^{83} -5.50520 q^{85} -3.49594i q^{89} +14.4234i q^{91} -2.03314 q^{95} -8.69074 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q+O(q^{10}) $ 24 * q	$ 24 q + 8 q^{25} - 32 q^{37} - 32 q^{49} + 8 q^{73} + 40 q^{85} + 16 q^{97}+O(q^{100}) $ 24 * q + 8 * q^25 - 32 * q^37 - 32 * q^49 + 8 * q^73 + 40 * q^85 + 16 * q^97

Display 100 coefficients 10 coefficients

Character values

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

$n$	$1009$	$1217$	$1711$	$2053$
$\chi(n)$	$1$	$-1$	$-1$	$1$

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
$5$	2.03314i	0.909250i	0.890683	+	0.454625i	$0.150227\pi$
$5$	2.03314i	0.909250i	−0.890683	+	0.454625i	$0.849773\pi$
$6$	0	0
$7$	3.00897i	1.13729i	0.822585	+	0.568643i	$0.192531\pi$
$7$	3.00897i	1.13729i	−0.822585	+	0.568643i	$0.807469\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	4.39868	1.32625	0.663126	−	0.748508i	$-0.269229\pi$
$11$	4.39868	1.32625	0.663126	+	0.748508i	$0.269229\pi$
$12$	0	0
$13$	4.79346	1.32947	0.664733	−	0.747081i	$-0.268545\pi$
$13$	4.79346	1.32947	0.664733	+	0.747081i	$0.268545\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	2.70773i	0.656721i	0.944553	+	0.328360i	$0.106496\pi$
$17$	2.70773i	0.656721i	−0.944553	+	0.328360i	$0.893504\pi$
$18$	0	0
$19$	1.00000i	0.229416i
$19$	1.00000i	0.229416i
$20$	0	0
$21$	0	0
$22$	0	0
$23$	1.06706	0.222498	0.111249	−	0.993793i	$-0.464515\pi$
$23$	1.06706	0.222498	0.111249	+	0.993793i	$0.464515\pi$
$24$	0	0
$25$	0.866325	0.173265
$26$	0	0
$27$	0	0
$28$	0	0
$29$	2.56438i	0.476194i	0.971241	+	0.238097i	$0.0765236\pi$
$29$	2.56438i	0.476194i	−0.971241	+	0.238097i	$0.923476\pi$
$30$	0	0
$31$	− 2.28336i	− 0.410104i	−0.978751	−	0.205052i	$-0.934264\pi$
$31$	− 2.28336i	− 0.410104i	0.978751	−	0.205052i	$-0.0657364\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−6.11768	−1.03408
$36$	0	0
$37$	4.52611	0.744088	0.372044	−	0.928215i	$-0.378657\pi$
$37$	4.52611	0.744088	0.372044	+	0.928215i	$0.378657\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 2.51222i	− 0.392342i	−0.980570	−	0.196171i	$-0.937149\pi$
$41$	− 2.51222i	− 0.392342i	0.980570	−	0.196171i	$-0.0628508\pi$
$42$	0	0
$43$	− 3.98914i	− 0.608338i	−0.952618	−	0.304169i	$-0.901621\pi$
$43$	− 3.98914i	− 0.608338i	0.952618	−	0.304169i	$-0.0983788\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−1.12272	−0.163765	−0.0818826	−	0.996642i	$-0.526093\pi$
$47$	−1.12272	−0.163765	−0.0818826	+	0.996642i	$0.526093\pi$
$48$	0	0
$49$	−2.05393	−0.293418
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 6.72681i	− 0.923998i	−0.886881	−	0.461999i	$-0.847132\pi$
$53$	− 6.72681i	− 0.923998i	0.886881	−	0.461999i	$-0.152868\pi$
$54$	0	0
$55$	8.94315i	1.20589i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−7.15876	−0.931991	−0.465996	−	0.884787i	$-0.654304\pi$
$59$	−7.15876	−0.931991	−0.465996	+	0.884787i	$0.654304\pi$
$60$	0	0
$61$	12.9611	1.65950	0.829749	−	0.558136i	$-0.188483\pi$
$61$	12.9611	1.65950	0.829749	+	0.558136i	$0.188483\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	9.74579i	1.20882i
$66$	0	0
$67$	− 1.75060i	− 0.213870i	−0.994266	−	0.106935i	$-0.965896\pi$
$67$	− 1.75060i	− 0.213870i	0.994266	−	0.106935i	$-0.0341036\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−4.70842	−0.558787	−0.279393	−	0.960177i	$-0.590133\pi$
$71$	−4.70842	−0.558787	−0.279393	+	0.960177i	$0.590133\pi$
$72$	0	0
$73$	−0.322316	−0.0377243	−0.0188621	−	0.999822i	$-0.506004\pi$
$73$	−0.322316	−0.0377243	−0.0188621	+	0.999822i	$0.506004\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	13.2355i	1.50833i
$78$	0	0
$79$	− 2.71440i	− 0.305394i	−0.988273	−	0.152697i	$-0.951204\pi$
$79$	− 2.71440i	− 0.305394i	0.988273	−	0.152697i	$-0.0487958\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−6.57724	−0.721946	−0.360973	−	0.932576i	$-0.617555\pi$
$83$	−6.57724	−0.721946	−0.360973	+	0.932576i	$0.617555\pi$
$84$	0	0
$85$	−5.50520	−0.597123
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 3.49594i	− 0.370569i	−0.982685	−	0.185285i	$-0.940679\pi$
$89$	− 3.49594i	− 0.370569i	0.982685	−	0.185285i	$-0.0593207\pi$
$90$	0	0
$91$	14.4234i	1.51198i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−2.03314	−0.208596
$96$	0	0
$97$	−8.69074	−0.882411	−0.441206	−	0.897406i	$-0.645449\pi$
$97$	−8.69074	−0.882411	−0.441206	+	0.897406i	$0.645449\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	15.4577i	1.53810i	0.639189	+	0.769049i	$0.279270\pi$
$101$	15.4577i	1.53810i	−0.639189	+	0.769049i	$0.720730\pi$
$102$	0	0
$103$	2.44705i	0.241115i	0.992706	+	0.120557i	$0.0384682\pi$
$103$	2.44705i	0.241115i	−0.992706	+	0.120557i	$0.961532\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−15.7807	−1.52558	−0.762791	−	0.646646i	$-0.776171\pi$
$107$	−15.7807	−1.52558	−0.762791	+	0.646646i	$0.776171\pi$
$108$	0	0
$109$	15.0071	1.43742	0.718711	−	0.695309i	$-0.244733\pi$
$109$	15.0071	1.43742	0.718711	+	0.695309i	$0.244733\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	9.11357i	0.857333i	0.903463	+	0.428666i	$0.141016\pi$
$113$	9.11357i	0.857333i	−0.903463	+	0.428666i	$0.858984\pi$
$114$	0	0
$115$	2.16949i	0.202306i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−8.14749	−0.746879
$120$	0	0
$121$	8.34838	0.758943
$122$	0	0
$123$	0	0
$124$	0	0
$125$	11.9271i	1.06679i
$126$	0	0
$127$	17.7108i	1.57158i	0.618495	+	0.785789i	$0.287743\pi$
$127$	17.7108i	1.57158i	−0.618495	+	0.785789i	$0.712257\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−11.6503	−1.01790	−0.508948	−	0.860797i	$-0.669965\pi$
$131$	−11.6503	−1.01790	−0.508948	+	0.860797i	$0.669965\pi$
$132$	0	0
$133$	−3.00897	−0.260911
$134$	0	0
$135$	0	0
$136$	0	0
$137$	3.35138i	0.286327i	0.989699	+	0.143164i	$0.0457275\pi$
$137$	3.35138i	0.286327i	−0.989699	+	0.143164i	$0.954272\pi$
$138$	0	0
$139$	− 1.81965i	− 0.154340i	−0.997018	−	0.0771702i	$-0.975412\pi$
$139$	− 1.81965i	− 0.154340i	0.997018	−	0.0771702i	$-0.0245885\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	21.0849	1.76321
$144$	0	0
$145$	−5.21376	−0.432979
$146$	0	0
$147$	0	0
$148$	0	0
$149$	23.8578i	1.95451i	0.212073	+	0.977254i	$0.431979\pi$
$149$	23.8578i	1.95451i	−0.212073	+	0.977254i	$0.568021\pi$
$150$	0	0
$151$	− 14.7987i	− 1.20430i	−0.798381	−	0.602152i	$-0.794310\pi$
$151$	− 14.7987i	− 1.20430i	0.798381	−	0.602152i	$-0.205690\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	4.64241	0.372887
$156$	0	0
$157$	−13.5569	−1.08196	−0.540979	−	0.841036i	$-0.681946\pi$
$157$	−13.5569	−1.08196	−0.540979	+	0.841036i	$0.681946\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	3.21077i	0.253044i
$162$	0	0
$163$	− 4.64350i	− 0.363707i	−0.983326	−	0.181853i	$-0.941790\pi$
$163$	− 4.64350i	− 0.363707i	0.983326	−	0.181853i	$-0.0582096\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	8.54323	0.661095	0.330547	−	0.943789i	$-0.392767\pi$
$167$	8.54323	0.661095	0.330547	+	0.943789i	$0.392767\pi$
$168$	0	0
$169$	9.97724	0.767480
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 13.9082i	− 1.05742i	−0.848802	−	0.528711i	$-0.822676\pi$
$173$	− 13.9082i	− 1.05742i	0.848802	−	0.528711i	$-0.177324\pi$
$174$	0	0
$175$	2.60675i	0.197052i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−22.9649	−1.71648	−0.858238	−	0.513252i	$-0.828441\pi$
$179$	−22.9649	−1.71648	−0.858238	+	0.513252i	$0.828441\pi$
$180$	0	0
$181$	−14.5023	−1.07795	−0.538974	−	0.842322i	$-0.681188\pi$
$181$	−14.5023	−1.07795	−0.538974	+	0.842322i	$0.681188\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	9.20223i	0.676561i
$186$	0	0
$187$	11.9104i	0.870977i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	26.6862	1.93095	0.965474	−	0.260499i	$-0.0838871\pi$
$191$	26.6862	1.93095	0.965474	+	0.260499i	$0.0838871\pi$
$192$	0	0
$193$	24.0859	1.73374	0.866870	−	0.498535i	$-0.166128\pi$
$193$	24.0859	1.73374	0.866870	+	0.498535i	$0.166128\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 14.5502i	− 1.03666i	−0.855181	−	0.518329i	$-0.826554\pi$
$197$	− 14.5502i	− 1.03666i	0.855181	−	0.518329i	$-0.173446\pi$
$198$	0	0
$199$	− 3.04572i	− 0.215905i	−0.994156	−	0.107953i	$-0.965571\pi$
$199$	− 3.04572i	− 0.215905i	0.994156	−	0.107953i	$-0.0344295\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−7.71616	−0.541569
$204$	0	0
$205$	5.10769	0.356737
$206$	0	0
$207$	0	0
$208$	0	0
$209$	4.39868i	0.304263i
$210$	0	0
$211$	− 11.5472i	− 0.794945i	−0.917614	−	0.397473i	$-0.869887\pi$
$211$	− 11.5472i	− 0.794945i	0.917614	−	0.397473i	$-0.130113\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	8.11050	0.553131
$216$	0	0
$217$	6.87059	0.466406
$218$	0	0
$219$	0	0
$220$	0	0
$221$	12.9794i	0.873088i
$222$	0	0
$223$	− 29.5633i	− 1.97970i	−0.142107	−	0.989851i	$-0.545388\pi$
$223$	− 29.5633i	− 1.97970i	0.142107	−	0.989851i	$-0.454612\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−12.6363	−0.838701	−0.419350	−	0.907825i	$-0.637742\pi$
$227$	−12.6363	−0.838701	−0.419350	+	0.907825i	$0.637742\pi$
$228$	0	0
$229$	16.6361	1.09935	0.549673	−	0.835380i	$-0.314752\pi$
$229$	16.6361	1.09935	0.549673	+	0.835380i	$0.314752\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 6.92402i	− 0.453607i	−0.973940	−	0.226804i	$-0.927172\pi$
$233$	− 6.92402i	− 0.453607i	0.973940	−	0.226804i	$-0.0728276\pi$
$234$	0	0
$235$	− 2.28265i	− 0.148903i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	7.36426	0.476354	0.238177	−	0.971222i	$-0.423450\pi$
$239$	7.36426	0.476354	0.238177	+	0.971222i	$0.423450\pi$
$240$	0	0
$241$	11.9442	0.769392	0.384696	−	0.923043i	$-0.374306\pi$
$241$	11.9442	0.769392	0.384696	+	0.923043i	$0.374306\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 4.17593i	− 0.266790i
$246$	0	0
$247$	4.79346i	0.305000i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	9.53492	0.601839	0.300919	−	0.953650i	$-0.402707\pi$
$251$	9.53492	0.601839	0.300919	+	0.953650i	$0.402707\pi$
$252$	0	0
$253$	4.69367	0.295088
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 1.58370i	− 0.0987882i	−0.998779	−	0.0493941i	$-0.984271\pi$
$257$	− 1.58370i	− 0.0987882i	0.998779	−	0.0493941i	$-0.0157290\pi$
$258$	0	0
$259$	13.6189i	0.846240i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−19.4030	−1.19644	−0.598221	−	0.801331i	$-0.704126\pi$
$263$	−19.4030	−1.19644	−0.598221	+	0.801331i	$0.704126\pi$
$264$	0	0
$265$	13.6766	0.840145
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 9.26187i	− 0.564706i	−0.959311	−	0.282353i	$-0.908885\pi$
$269$	− 9.26187i	− 0.564706i	0.959311	−	0.282353i	$-0.0911150\pi$
$270$	0	0
$271$	− 13.3507i	− 0.810995i	−0.914096	−	0.405497i	$-0.867098\pi$
$271$	− 13.3507i	− 0.810995i	0.914096	−	0.405497i	$-0.132902\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	3.81069	0.229793
$276$	0	0
$277$	−13.9813	−0.840054	−0.420027	−	0.907512i	$-0.637979\pi$
$277$	−13.9813	−0.840054	−0.420027	+	0.907512i	$0.637979\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 27.4391i	− 1.63688i	−0.574593	−	0.818439i	$-0.694840\pi$
$281$	− 27.4391i	− 1.63688i	0.574593	−	0.818439i	$-0.305160\pi$
$282$	0	0
$283$	29.1181i	1.73089i	0.501001	+	0.865446i	$0.332965\pi$
$283$	29.1181i	1.73089i	−0.501001	+	0.865446i	$0.667035\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	7.55919	0.446205
$288$	0	0
$289$	9.66820	0.568718
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 5.85015i	− 0.341769i	−0.985291	−	0.170885i	$-0.945337\pi$
$293$	− 5.85015i	− 0.341769i	0.985291	−	0.170885i	$-0.0546625\pi$
$294$	0	0
$295$	− 14.5548i	− 0.847413i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	5.11492	0.295804
$300$	0	0
$301$	12.0032	0.691854
$302$	0	0
$303$	0	0
$304$	0	0
$305$	26.3518i	1.50890i
$306$	0	0
$307$	32.8827i	1.87671i	0.345668	+	0.938357i	$0.387652\pi$
$307$	32.8827i	1.87671i	−0.345668	+	0.938357i	$0.612348\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−12.4253	−0.704575	−0.352288	−	0.935892i	$-0.614596\pi$
$311$	−12.4253	−0.704575	−0.352288	+	0.935892i	$0.614596\pi$
$312$	0	0
$313$	−32.1767	−1.81874	−0.909368	−	0.415993i	$-0.863434\pi$
$313$	−32.1767	−1.81874	−0.909368	+	0.415993i	$0.863434\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	4.93547i	0.277204i	0.990348	+	0.138602i	$0.0442608\pi$
$317$	4.93547i	0.277204i	−0.990348	+	0.138602i	$0.955739\pi$
$318$	0	0
$319$	11.2799i	0.631553i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−2.70773	−0.150662
$324$	0	0
$325$	4.15269	0.230350
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 3.37823i	− 0.186248i
$330$	0	0
$331$	5.18147i	0.284799i	0.989809	+	0.142400i	$0.0454818\pi$
$331$	5.18147i	0.284799i	−0.989809	+	0.142400i	$0.954518\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	3.55922	0.194461
$336$	0	0
$337$	20.6071	1.12254	0.561270	−	0.827633i	$-0.310313\pi$
$337$	20.6071	1.12254	0.561270	+	0.827633i	$0.310313\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 10.0438i	− 0.543902i
$342$	0	0
$343$	14.8826i	0.803585i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−25.6179	−1.37524	−0.687621	−	0.726070i	$-0.741345\pi$
$347$	−25.6179	−1.37524	−0.687621	+	0.726070i	$0.741345\pi$
$348$	0	0
$349$	−21.9243	−1.17358	−0.586789	−	0.809740i	$-0.699608\pi$
$349$	−21.9243	−1.17358	−0.586789	+	0.809740i	$0.699608\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	17.5098i	0.931950i	0.884798	+	0.465975i	$0.154296\pi$
$353$	17.5098i	0.931950i	−0.884798	+	0.465975i	$0.845704\pi$
$354$	0	0
$355$	− 9.57290i	− 0.508077i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−7.33437	−0.387094	−0.193547	−	0.981091i	$-0.561999\pi$
$359$	−7.33437	−0.387094	−0.193547	+	0.981091i	$0.561999\pi$
$360$	0	0
$361$	−1.00000	−0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 0.655315i	− 0.0343008i
$366$	0	0
$367$	− 9.78728i	− 0.510892i	−0.966823	−	0.255446i	$-0.917778\pi$
$367$	− 9.78728i	− 0.510892i	0.966823	−	0.255446i	$-0.0822223\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	20.2408	1.05085
$372$	0	0
$373$	−15.1684	−0.785390	−0.392695	−	0.919669i	$-0.628457\pi$
$373$	−15.1684	−0.785390	−0.392695	+	0.919669i	$0.628457\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	12.2923i	0.633084i
$378$	0	0
$379$	3.14950i	0.161779i	0.996723	+	0.0808896i	$0.0257761\pi$
$379$	3.14950i	0.161779i	−0.996723	+	0.0808896i	$0.974224\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	12.9070	0.659517	0.329759	−	0.944065i	$-0.393033\pi$
$383$	12.9070	0.659517	0.329759	+	0.944065i	$0.393033\pi$
$384$	0	0
$385$	−26.9097	−1.37145
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 29.7262i	− 1.50718i	−0.657347	−	0.753588i	$-0.728321\pi$
$389$	− 29.7262i	− 1.50718i	0.657347	−	0.753588i	$-0.271679\pi$
$390$	0	0
$391$	2.88932i	0.146119i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	5.51876	0.277679
$396$	0	0
$397$	0.0678479	0.00340519	0.00170259	−	0.999999i	$-0.499458\pi$
$397$	0.0678479	0.00340519	0.00170259	+	0.999999i	$0.499458\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 28.2624i	− 1.41135i	−0.708533	−	0.705677i	$-0.750643\pi$
$401$	− 28.2624i	− 1.41135i	0.708533	−	0.705677i	$-0.249357\pi$
$402$	0	0
$403$	− 10.9452i	− 0.545220i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	19.9089	0.986847
$408$	0	0
$409$	−3.96033	−0.195826	−0.0979129	−	0.995195i	$-0.531217\pi$
$409$	−3.96033	−0.195826	−0.0979129	+	0.995195i	$0.531217\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 21.5405i	− 1.05994i
$414$	0	0
$415$	− 13.3725i	− 0.656429i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−15.1752	−0.741355	−0.370678	−	0.928762i	$-0.620875\pi$
$419$	−15.1752	−0.741355	−0.370678	+	0.928762i	$0.620875\pi$
$420$	0	0
$421$	−24.5641	−1.19718	−0.598592	−	0.801054i	$-0.704273\pi$
$421$	−24.5641	−1.19718	−0.598592	+	0.801054i	$0.704273\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	2.34577i	0.113787i
$426$	0	0
$427$	38.9996i	1.88732i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	27.6482	1.33177	0.665884	−	0.746056i	$-0.268055\pi$
$431$	27.6482	1.33177	0.665884	+	0.746056i	$0.268055\pi$
$432$	0	0
$433$	−33.6176	−1.61556	−0.807779	−	0.589486i	$-0.799330\pi$
$433$	−33.6176	−1.61556	−0.807779	+	0.589486i	$0.799330\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	1.06706i	0.0510445i
$438$	0	0
$439$	− 34.4454i	− 1.64399i	−0.569497	−	0.821994i	$-0.692862\pi$
$439$	− 34.4454i	− 1.64399i	0.569497	−	0.821994i	$-0.307138\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−13.8632	−0.658662	−0.329331	−	0.944214i	$-0.606823\pi$
$443$	−13.8632	−0.658662	−0.329331	+	0.944214i	$0.606823\pi$
$444$	0	0
$445$	7.10776	0.336940
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 15.7089i	− 0.741351i	−0.928762	−	0.370675i	$-0.879126\pi$
$449$	− 15.7089i	− 0.741351i	0.928762	−	0.370675i	$-0.120874\pi$
$450$	0	0
$451$	− 11.0504i	− 0.520344i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−29.3248	−1.37477
$456$	0	0
$457$	−7.23978	−0.338663	−0.169331	−	0.985559i	$-0.554161\pi$
$457$	−7.23978	−0.338663	−0.169331	+	0.985559i	$0.554161\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 19.4060i	− 0.903826i	−0.892062	−	0.451913i	$-0.850742\pi$
$461$	− 19.4060i	− 0.903826i	0.892062	−	0.451913i	$-0.149258\pi$
$462$	0	0
$463$	8.30741i	0.386078i	0.981191	+	0.193039i	$0.0618344\pi$
$463$	8.30741i	0.386078i	−0.981191	+	0.193039i	$0.938166\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	33.2788	1.53996	0.769980	−	0.638068i	$-0.220266\pi$
$467$	33.2788	1.53996	0.769980	+	0.638068i	$0.220266\pi$
$468$	0	0
$469$	5.26751	0.243231
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 17.5469i	− 0.806809i
$474$	0	0
$475$	0.866325i	0.0397497i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−16.7480	−0.765237	−0.382618	−	0.923906i	$-0.624978\pi$
$479$	−16.7480	−0.765237	−0.382618	+	0.923906i	$0.624978\pi$
$480$	0	0
$481$	21.6957	0.989239
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 17.6695i	− 0.802332i
$486$	0	0
$487$	− 15.0304i	− 0.681090i	−0.940228	−	0.340545i	$-0.889388\pi$
$487$	− 15.0304i	− 0.681090i	0.940228	−	0.340545i	$-0.110612\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	30.7982	1.38990	0.694951	−	0.719057i	$-0.255426\pi$
$491$	30.7982	1.38990	0.694951	+	0.719057i	$0.255426\pi$
$492$	0	0
$493$	−6.94366	−0.312727
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 14.1675i	− 0.635500i
$498$	0	0
$499$	− 23.5272i	− 1.05322i	−0.850106	−	0.526612i	$-0.823462\pi$
$499$	− 23.5272i	− 1.05322i	0.850106	−	0.526612i	$-0.176538\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	23.2951	1.03868	0.519338	−	0.854569i	$-0.326178\pi$
$503$	23.2951	1.03868	0.519338	+	0.854569i	$0.326178\pi$
$504$	0	0
$505$	−31.4277	−1.39852
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 1.08852i	− 0.0482478i	−0.999709	−	0.0241239i	$-0.992320\pi$
$509$	− 1.08852i	− 0.0482478i	0.999709	−	0.0241239i	$-0.00767962\pi$
$510$	0	0
$511$	− 0.969841i	− 0.0429032i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−4.97520	−0.219234
$516$	0	0
$517$	−4.93848	−0.217194
$518$	0	0
$519$	0	0
$520$	0	0
$521$	32.0616i	1.40464i	0.711860	+	0.702322i	$0.247853\pi$
$521$	32.0616i	1.40464i	−0.711860	+	0.702322i	$0.752147\pi$
$522$	0	0
$523$	− 36.5989i	− 1.60036i	−0.599761	−	0.800179i	$-0.704738\pi$
$523$	− 36.5989i	− 1.60036i	0.599761	−	0.800179i	$-0.295262\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	6.18273	0.269324
$528$	0	0
$529$	−21.8614	−0.950495
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 12.0422i	− 0.521605i
$534$	0	0
$535$	− 32.0845i	− 1.38713i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−9.03457	−0.389146
$540$	0	0
$541$	−4.53331	−0.194902	−0.0974511	−	0.995240i	$-0.531069\pi$
$541$	−4.53331	−0.194902	−0.0974511	+	0.995240i	$0.531069\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	30.5116i	1.30697i
$546$	0	0
$547$	− 1.16144i	− 0.0496598i	−0.999692	−	0.0248299i	$-0.992096\pi$
$547$	− 1.16144i	− 0.0496598i	0.999692	−	0.0248299i	$-0.00790441\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−2.56438	−0.109246
$552$	0	0
$553$	8.16755	0.347320
$554$	0	0
$555$	0	0
$556$	0	0
$557$	28.2062i	1.19514i	0.801818	+	0.597568i	$0.203866\pi$
$557$	28.2062i	1.19514i	−0.801818	+	0.597568i	$0.796134\pi$
$558$	0	0
$559$	− 19.1218i	− 0.808765i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	22.8312	0.962221	0.481111	−	0.876660i	$-0.340234\pi$
$563$	22.8312	0.962221	0.481111	+	0.876660i	$0.340234\pi$
$564$	0	0
$565$	−18.5292	−0.779529
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 27.5857i	− 1.15645i	−0.815877	−	0.578226i	$-0.803745\pi$
$569$	− 27.5857i	− 1.15645i	0.815877	−	0.578226i	$-0.196255\pi$
$570$	0	0
$571$	0.702859i	0.0294137i	0.999892	+	0.0147069i	$0.00468151\pi$
$571$	0.702859i	0.0294137i	−0.999892	+	0.0147069i	$0.995318\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0.924424	0.0385511
$576$	0	0
$577$	18.7000	0.778490	0.389245	−	0.921134i	$-0.372736\pi$
$577$	18.7000	0.778490	0.389245	+	0.921134i	$0.372736\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 19.7907i	− 0.821058i
$582$	0	0
$583$	− 29.5891i	− 1.22545i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−35.8797	−1.48091	−0.740457	−	0.672104i	$-0.765391\pi$
$587$	−35.8797	−1.48091	−0.740457	+	0.672104i	$0.765391\pi$
$588$	0	0
$589$	2.28336	0.0940844
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 37.3570i	− 1.53407i	−0.641607	−	0.767034i	$-0.721732\pi$
$593$	− 37.3570i	− 1.53407i	0.641607	−	0.767034i	$-0.278268\pi$
$594$	0	0
$595$	− 16.5650i	− 0.679100i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	30.4528	1.24427	0.622134	−	0.782911i	$-0.286266\pi$
$599$	30.4528	1.24427	0.622134	+	0.782911i	$0.286266\pi$
$600$	0	0
$601$	−3.96592	−0.161773	−0.0808866	−	0.996723i	$-0.525775\pi$
$601$	−3.96592	−0.161773	−0.0808866	+	0.996723i	$0.525775\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	16.9735i	0.690069i
$606$	0	0
$607$	− 2.01627i	− 0.0818380i	−0.999162	−	0.0409190i	$-0.986971\pi$
$607$	− 2.01627i	− 0.0818380i	0.999162	−	0.0409190i	$-0.0130286\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−5.38170	−0.217720
$612$	0	0
$613$	39.3109	1.58775	0.793876	−	0.608080i	$-0.208060\pi$
$613$	39.3109	1.58775	0.793876	+	0.608080i	$0.208060\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 5.59842i	− 0.225384i	−0.993630	−	0.112692i	$-0.964053\pi$
$617$	− 5.59842i	− 0.225384i	0.993630	−	0.112692i	$-0.0359473\pi$
$618$	0	0
$619$	43.5734i	1.75136i	0.482888	+	0.875682i	$0.339588\pi$
$619$	43.5734i	1.75136i	−0.482888	+	0.875682i	$0.660412\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	10.5192	0.421443
$624$	0	0
$625$	−19.9179	−0.796714
$626$	0	0
$627$	0	0
$628$	0	0
$629$	12.2555i	0.488658i
$630$	0	0
$631$	12.3862i	0.493085i	0.969132	+	0.246543i	$0.0792945\pi$
$631$	12.3862i	0.493085i	−0.969132	+	0.246543i	$0.920705\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−36.0086	−1.42896
$636$	0	0
$637$	−9.84542	−0.390090
$638$	0	0
$639$	0	0
$640$	0	0
$641$	36.1951i	1.42962i	0.699319	+	0.714810i	$0.253487\pi$
$641$	36.1951i	1.42962i	−0.699319	+	0.714810i	$0.746513\pi$
$642$	0	0
$643$	20.8723i	0.823121i	0.911382	+	0.411561i	$0.135016\pi$
$643$	20.8723i	0.823121i	−0.911382	+	0.411561i	$0.864984\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	4.59359	0.180593	0.0902964	−	0.995915i	$-0.471219\pi$
$647$	4.59359	0.180593	0.0902964	+	0.995915i	$0.471219\pi$
$648$	0	0
$649$	−31.4891	−1.23605
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 24.4420i	− 0.956490i	−0.878226	−	0.478245i	$-0.841273\pi$
$653$	− 24.4420i	− 0.956490i	0.878226	−	0.478245i	$-0.158727\pi$
$654$	0	0
$655$	− 23.6868i	− 0.925521i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	50.0946	1.95141	0.975704	−	0.219091i	$-0.0703093\pi$
$659$	50.0946	1.95141	0.975704	+	0.219091i	$0.0703093\pi$
$660$	0	0
$661$	4.80962	0.187072	0.0935362	−	0.995616i	$-0.470183\pi$
$661$	4.80962	0.187072	0.0935362	+	0.995616i	$0.470183\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 6.11768i	− 0.237233i
$666$	0	0
$667$	2.73636i	0.105952i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	57.0117	2.20091
$672$	0	0
$673$	−14.5173	−0.559599	−0.279800	−	0.960058i	$-0.590268\pi$
$673$	−14.5173	−0.559599	−0.279800	+	0.960058i	$0.590268\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	3.87263i	0.148837i	0.997227	+	0.0744186i	$0.0237101\pi$
$677$	3.87263i	0.148837i	−0.997227	+	0.0744186i	$0.976290\pi$
$678$	0	0
$679$	− 26.1502i	− 1.00355i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−28.1199	−1.07598	−0.537989	−	0.842952i	$-0.680816\pi$
$683$	−28.1199	−1.07598	−0.537989	+	0.842952i	$0.680816\pi$
$684$	0	0
$685$	−6.81383	−0.260343
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 32.2447i	− 1.22842i
$690$	0	0
$691$	− 29.9629i	− 1.13984i	−0.821699	−	0.569922i	$-0.806973\pi$
$691$	− 29.9629i	− 1.13984i	0.821699	−	0.569922i	$-0.193027\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	3.69961	0.140334
$696$	0	0
$697$	6.80240	0.257659
$698$	0	0
$699$	0	0
$700$	0	0
$701$	13.5634i	0.512281i	0.966640	+	0.256141i	$0.0824510\pi$
$701$	13.5634i	0.512281i	−0.966640	+	0.256141i	$0.917549\pi$
$702$	0	0
$703$	4.52611i	0.170705i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−46.5118	−1.74926
$708$	0	0
$709$	−10.5749	−0.397148	−0.198574	−	0.980086i	$-0.563631\pi$
$709$	−10.5749	−0.397148	−0.198574	+	0.980086i	$0.563631\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 2.43649i	− 0.0912474i
$714$	0	0
$715$	42.8686i	1.60319i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	50.6671	1.88956	0.944781	−	0.327701i	$-0.106274\pi$
$719$	50.6671	1.88956	0.944781	+	0.327701i	$0.106274\pi$
$720$	0	0
$721$	−7.36310	−0.274216
$722$	0	0
$723$	0	0
$724$	0	0
$725$	2.22159i	0.0825078i
$726$	0	0
$727$	17.7169i	0.657083i	0.944490	+	0.328541i	$0.106557\pi$
$727$	17.7169i	0.657083i	−0.944490	+	0.328541i	$0.893443\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	10.8015	0.399508
$732$	0	0
$733$	7.76075	0.286650	0.143325	−	0.989676i	$-0.454221\pi$
$733$	7.76075	0.286650	0.143325	+	0.989676i	$0.454221\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 7.70033i	− 0.283645i
$738$	0	0
$739$	− 1.60803i	− 0.0591525i	−0.999563	−	0.0295762i	$-0.990584\pi$
$739$	− 1.60803i	− 0.0591525i	0.999563	−	0.0295762i	$-0.00941578\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	44.1992	1.62151	0.810756	−	0.585385i	$-0.199057\pi$
$743$	44.1992	1.62151	0.810756	+	0.585385i	$0.199057\pi$
$744$	0	0
$745$	−48.5064	−1.77714
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 47.4838i	− 1.73502i
$750$	0	0
$751$	38.2523i	1.39585i	0.716172	+	0.697924i	$0.245893\pi$
$751$	38.2523i	1.39585i	−0.716172	+	0.697924i	$0.754107\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	30.0880	1.09501
$756$	0	0
$757$	41.1231	1.49464	0.747322	−	0.664462i	$-0.231339\pi$
$757$	41.1231	1.49464	0.747322	+	0.664462i	$0.231339\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 22.5026i	− 0.815718i	−0.913045	−	0.407859i	$-0.866275\pi$
$761$	− 22.5026i	− 0.815718i	0.913045	−	0.407859i	$-0.133725\pi$
$762$	0	0
$763$	45.1560i	1.63476i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−34.3152	−1.23905
$768$	0	0
$769$	−3.33370	−0.120216	−0.0601082	−	0.998192i	$-0.519145\pi$
$769$	−3.33370	−0.120216	−0.0601082	+	0.998192i	$0.519145\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	3.88156i	0.139610i	0.997561	+	0.0698050i	$0.0222377\pi$
$773$	3.88156i	0.139610i	−0.997561	+	0.0698050i	$0.977762\pi$
$774$	0	0
$775$	− 1.97814i	− 0.0710568i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	2.51222	0.0900094
$780$	0	0
$781$	−20.7108	−0.741092
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 27.5631i	− 0.983770i
$786$	0	0
$787$	6.07170i	0.216433i	0.994127	+	0.108216i	$0.0345139\pi$
$787$	6.07170i	0.216433i	−0.994127	+	0.108216i	$0.965486\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−27.4225	−0.975032
$792$	0	0
$793$	62.1285	2.20625
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 19.4601i	− 0.689313i	−0.938729	−	0.344657i	$-0.887995\pi$
$797$	− 19.4601i	− 0.689313i	0.938729	−	0.344657i	$-0.112005\pi$
$798$	0	0
$799$	− 3.04002i	− 0.107548i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−1.41777	−0.0500318
$804$	0	0
$805$	−6.52795	−0.230080
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 46.5669i	− 1.63721i	−0.574360	−	0.818603i	$-0.694749\pi$
$809$	− 46.5669i	− 1.63721i	0.574360	−	0.818603i	$-0.305251\pi$
$810$	0	0
$811$	− 55.6852i	− 1.95537i	−0.210068	−	0.977687i	$-0.567369\pi$
$811$	− 55.6852i	− 1.95537i	0.210068	−	0.977687i	$-0.432631\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	9.44090	0.330700
$816$	0	0
$817$	3.98914	0.139562
$818$	0	0
$819$	0	0
$820$	0	0
$821$	23.5583i	0.822190i	0.911593	+	0.411095i	$0.134854\pi$
$821$	23.5583i	0.822190i	−0.911593	+	0.411095i	$0.865146\pi$
$822$	0	0
$823$	− 17.8311i	− 0.621554i	−0.950483	−	0.310777i	$-0.899411\pi$
$823$	− 17.8311i	− 0.621554i	0.950483	−	0.310777i	$-0.100589\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−16.1324	−0.560979	−0.280489	−	0.959857i	$-0.590497\pi$
$827$	−16.1324	−0.560979	−0.280489	+	0.959857i	$0.590497\pi$
$828$	0	0
$829$	−1.73814	−0.0603681	−0.0301841	−	0.999544i	$-0.509609\pi$
$829$	−1.73814	−0.0603681	−0.0301841	+	0.999544i	$0.509609\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 5.56148i	− 0.192694i
$834$	0	0
$835$	17.3696i	0.601100i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−6.53654	−0.225666	−0.112833	−	0.993614i	$-0.535993\pi$
$839$	−6.53654	−0.225666	−0.112833	+	0.993614i	$0.535993\pi$
$840$	0	0
$841$	22.4239	0.773239
$842$	0	0
$843$	0	0
$844$	0	0
$845$	20.2852i	0.697831i
$846$	0	0
$847$	25.1201i	0.863135i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	4.82964	0.165558
$852$	0	0
$853$	−47.0685	−1.61160	−0.805798	−	0.592190i	$-0.798263\pi$
$853$	−47.0685	−1.61160	−0.805798	+	0.592190i	$0.798263\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	2.67316i	0.0913134i	0.998957	+	0.0456567i	$0.0145380\pi$
$857$	2.67316i	0.0913134i	−0.998957	+	0.0456567i	$0.985462\pi$
$858$	0	0
$859$	48.6192i	1.65887i	0.558607	+	0.829433i	$0.311336\pi$
$859$	48.6192i	1.65887i	−0.558607	+	0.829433i	$0.688664\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	11.2319	0.382339	0.191169	−	0.981557i	$-0.438772\pi$
$863$	11.2319	0.382339	0.191169	+	0.981557i	$0.438772\pi$
$864$	0	0
$865$	28.2774	0.961460
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 11.9398i	− 0.405029i
$870$	0	0
$871$	− 8.39143i	− 0.284333i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−35.8883	−1.21325
$876$	0	0
$877$	−15.0409	−0.507894	−0.253947	−	0.967218i	$-0.581729\pi$
$877$	−15.0409	−0.507894	−0.253947	+	0.967218i	$0.581729\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 47.9839i	− 1.61662i	−0.588757	−	0.808310i	$-0.700383\pi$
$881$	− 47.9839i	− 1.61662i	0.588757	−	0.808310i	$-0.299617\pi$
$882$	0	0
$883$	− 10.5172i	− 0.353930i	−0.984217	−	0.176965i	$-0.943372\pi$
$883$	− 10.5172i	− 0.353930i	0.984217	−	0.176965i	$-0.0566280\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	28.0309	0.941184	0.470592	−	0.882351i	$-0.344040\pi$
$887$	28.0309	0.941184	0.470592	+	0.882351i	$0.344040\pi$
$888$	0	0
$889$	−53.2913	−1.78733
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 1.12272i	− 0.0375703i
$894$	0	0
$895$	− 46.6909i	− 1.56070i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	5.85542	0.195289
$900$	0	0
$901$	18.2144	0.606809
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 29.4853i	− 0.980124i
$906$	0	0
$907$	− 22.3296i	− 0.741442i	−0.928744	−	0.370721i	$-0.879111\pi$
$907$	− 22.3296i	− 0.741442i	0.928744	−	0.370721i	$-0.120889\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−32.0293	−1.06118	−0.530589	−	0.847629i	$-0.678029\pi$
$911$	−32.0293	−1.06118	−0.530589	+	0.847629i	$0.678029\pi$
$912$	0	0
$913$	−28.9312	−0.957481
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 35.0556i	− 1.15764i
$918$	0	0
$919$	28.1786i	0.929526i	0.885435	+	0.464763i	$0.153860\pi$
$919$	28.1786i	0.929526i	−0.885435	+	0.464763i	$0.846140\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−22.5696	−0.742888
$924$	0	0
$925$	3.92108	0.128924
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 17.4154i	− 0.571380i	−0.958322	−	0.285690i	$-0.907777\pi$
$929$	− 17.4154i	− 0.571380i	0.958322	−	0.285690i	$-0.0922228\pi$
$930$	0	0
$931$	− 2.05393i	− 0.0673148i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−24.2156	−0.791936
$936$	0	0
$937$	18.9692	0.619695	0.309848	−	0.950786i	$-0.399722\pi$
$937$	18.9692	0.619695	0.309848	+	0.950786i	$0.399722\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	27.2246i	0.887497i	0.896151	+	0.443749i	$0.146352\pi$
$941$	27.2246i	0.887497i	−0.896151	+	0.443749i	$0.853648\pi$
$942$	0	0
$943$	− 2.68069i	− 0.0872953i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	42.6756	1.38677	0.693385	−	0.720568i	$-0.256119\pi$
$947$	42.6756	1.38677	0.693385	+	0.720568i	$0.256119\pi$
$948$	0	0
$949$	−1.54501	−0.0501531
$950$	0	0
$951$	0	0
$952$	0	0
$953$	53.4103i	1.73013i	0.501660	+	0.865065i	$0.332723\pi$
$953$	53.4103i	1.73013i	−0.501660	+	0.865065i	$0.667277\pi$
$954$	0	0
$955$	54.2570i	1.75571i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−10.0842	−0.325636
$960$	0	0
$961$	25.7862	0.831814
$962$	0	0
$963$	0	0
$964$	0	0
$965$	48.9701i	1.57640i
$966$	0	0
$967$	37.1872i	1.19586i	0.801549	+	0.597929i	$0.204010\pi$
$967$	37.1872i	1.19586i	−0.801549	+	0.597929i	$0.795990\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−47.3408	−1.51924	−0.759619	−	0.650368i	$-0.774615\pi$
$971$	−47.3408	−1.51924	−0.759619	+	0.650368i	$0.774615\pi$
$972$	0	0
$973$	5.47527	0.175529
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 7.49046i	− 0.239641i	−0.992796	−	0.119820i	$-0.961768\pi$
$977$	− 7.49046i	− 0.239641i	0.992796	−	0.119820i	$-0.0382319\pi$
$978$	0	0
$979$	− 15.3775i	− 0.491468i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	5.68273	0.181251	0.0906255	−	0.995885i	$-0.471113\pi$
$983$	5.68273	0.181251	0.0906255	+	0.995885i	$0.471113\pi$
$984$	0	0
$985$	29.5826	0.942581
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 4.25666i	− 0.135354i
$990$	0	0
$991$	− 25.7187i	− 0.816981i	−0.912762	−	0.408491i	$-0.866055\pi$
$991$	− 25.7187i	− 0.816981i	0.912762	−	0.408491i	$-0.133945\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	6.19239	0.196312
$996$	0	0
$997$	46.0257	1.45765	0.728825	−	0.684700i	$-0.240067\pi$
$997$	46.0257	1.45765	0.728825	+	0.684700i	$0.240067\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2736.2.d.b.2015.18	yes	24
3.2	odd	2	inner	2736.2.d.b.2015.8	yes	24
4.3	odd	2	inner	2736.2.d.b.2015.17	yes	24
12.11	even	2	inner	2736.2.d.b.2015.7	✓	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2736.2.d.b.2015.7	✓	24	12.11	even	2	inner
2736.2.d.b.2015.8	yes	24	3.2	odd	2	inner
2736.2.d.b.2015.17	yes	24	4.3	odd	2	inner
2736.2.d.b.2015.18	yes	24	1.1	even	1	trivial

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.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.03314i q^{5} +3.00897i q^{7} +O(q^{10})\)	\(q+2.03314i q^{5} +3.00897i q^{7} +4.39868 q^{11} +4.79346 q^{13} +2.70773i q^{17} +1.00000i q^{19} +1.06706 q^{23} +0.866325 q^{25} +2.56438i q^{29} -2.28336i q^{31} -6.11768 q^{35} +4.52611 q^{37} -2.51222i q^{41} -3.98914i q^{43} -1.12272 q^{47} -2.05393 q^{49} -6.72681i q^{53} +8.94315i q^{55} -7.15876 q^{59} +12.9611 q^{61} +9.74579i q^{65} -1.75060i q^{67} -4.70842 q^{71} -0.322316 q^{73} +13.2355i q^{77} -2.71440i q^{79} -6.57724 q^{83} -5.50520 q^{85} -3.49594i q^{89} +14.4234i q^{91} -2.03314 q^{95} -8.69074 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q+O(q^{10}) \) 24 * q	\( 24 q + 8 q^{25} - 32 q^{37} - 32 q^{49} + 8 q^{73} + 40 q^{85} + 16 q^{97}+O(q^{100}) \) 24 * q + 8 * q^25 - 32 * q^37 - 32 * q^49 + 8 * q^73 + 40 * q^85 + 16 * q^97

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