LMFDB - Embedded newform 8100.2.a.be.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8100,2,Mod(1,8100)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8100, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8100.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8100 = 2^{2} \cdot 3^{4} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8100.a (trivial)

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:	yes
Analytic conductor:	$64.6788256372$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.8.28356903014400.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 37x^{6} + 399x^{4} - 1195x^{2} + 400 $ x^8 - 37x^6 + 399x^4 - 1195*x^2 + 400
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 1620)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$3.52696$ of defining polynomial
Character	$\chi$	$=$	8100.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.93536 q^{7} +O(q^{10})$	$q-4.93536 q^{7} +2.41269 q^{11} -2.90917 q^{13} +6.86869 q^{17} -4.17891 q^{19} +3.35922 q^{23} -5.19615 q^{29} -6.17891 q^{31} -7.84453 q^{37} +5.87680 q^{41} -4.93536 q^{43} -11.9075 q^{47} +17.3578 q^{49} +8.54830 q^{53} +1.05141 q^{59} +9.17891 q^{61} -4.05239 q^{67} +14.1663 q^{71} -2.02619 q^{73} -11.9075 q^{77} +6.00000 q^{79} -5.18908 q^{83} -3.09334 q^{89} +14.3578 q^{91} -0.882978 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q + 12 q^{19} - 4 q^{31} + 48 q^{49} + 28 q^{61} + 48 q^{79} + 24 q^{91}+O(q^{100}) $ 8 * q + 12 * q^19 - 4 * q^31 + 48 * q^49 + 28 * q^61 + 48 * q^79 + 24 * q^91

Display 100 coefficients 10 coefficients

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
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−4.93536	−1.86539	−0.932696	−	0.360663i	$-0.882550\pi$
$7$	−4.93536	−1.86539	−0.932696	+	0.360663i	$0.882550\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	2.41269	0.727455	0.363727	−	0.931505i	$-0.381504\pi$
$11$	2.41269	0.727455	0.363727	+	0.931505i	$0.381504\pi$
$12$	0	0
$13$	−2.90917	−0.806859	−0.403429	−	0.915011i	$-0.632182\pi$
$13$	−2.90917	−0.806859	−0.403429	+	0.915011i	$0.632182\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	6.86869	1.66590	0.832951	−	0.553347i	$-0.186650\pi$
$17$	6.86869	1.66590	0.832951	+	0.553347i	$0.186650\pi$
$18$	0	0
$19$	−4.17891	−0.958707	−0.479354	−	0.877622i	$-0.659129\pi$
$19$	−4.17891	−0.958707	−0.479354	+	0.877622i	$0.659129\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	3.35922	0.700446	0.350223	−	0.936666i	$-0.386106\pi$
$23$	3.35922	0.700446	0.350223	+	0.936666i	$0.386106\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−5.19615	−0.964901	−0.482451	−	0.875923i	$-0.660253\pi$
$29$	−5.19615	−0.964901	−0.482451	+	0.875923i	$0.660253\pi$
$30$	0	0
$31$	−6.17891	−1.10976	−0.554882	−	0.831929i	$-0.687237\pi$
$31$	−6.17891	−1.10976	−0.554882	+	0.831929i	$0.687237\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−7.84453	−1.28963	−0.644817	−	0.764337i	$-0.723066\pi$
$37$	−7.84453	−1.28963	−0.644817	+	0.764337i	$0.723066\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	5.87680	0.917801	0.458901	−	0.888488i	$-0.348243\pi$
$41$	5.87680	0.917801	0.458901	+	0.888488i	$0.348243\pi$
$42$	0	0
$43$	−4.93536	−0.752636	−0.376318	−	0.926491i	$-0.622810\pi$
$43$	−4.93536	−0.752636	−0.376318	+	0.926491i	$0.622810\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−11.9075	−1.73689	−0.868445	−	0.495785i	$-0.834880\pi$
$47$	−11.9075	−1.73689	−0.868445	+	0.495785i	$0.834880\pi$
$48$	0	0
$49$	17.3578	2.47969
$50$	0	0
$51$	0	0
$52$	0	0
$53$	8.54830	1.17420	0.587100	−	0.809515i	$-0.300270\pi$
$53$	8.54830	1.17420	0.587100	+	0.809515i	$0.300270\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	1.05141	0.136882	0.0684408	−	0.997655i	$-0.478198\pi$
$59$	1.05141	0.136882	0.0684408	+	0.997655i	$0.478198\pi$
$60$	0	0
$61$	9.17891	1.17524	0.587619	−	0.809137i	$-0.300065\pi$
$61$	9.17891	1.17524	0.587619	+	0.809137i	$0.300065\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−4.05239	−0.495078	−0.247539	−	0.968878i	$-0.579622\pi$
$67$	−4.05239	−0.495078	−0.247539	+	0.968878i	$0.579622\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	14.1663	1.68123	0.840614	−	0.541634i	$-0.182194\pi$
$71$	14.1663	1.68123	0.840614	+	0.541634i	$0.182194\pi$
$72$	0	0
$73$	−2.02619	−0.237148	−0.118574	−	0.992945i	$-0.537832\pi$
$73$	−2.02619	−0.237148	−0.118574	+	0.992945i	$0.537832\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−11.9075	−1.35699
$78$	0	0
$79$	6.00000	0.675053	0.337526	−	0.941316i	$-0.390410\pi$
$79$	6.00000	0.675053	0.337526	+	0.941316i	$0.390410\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−5.18908	−0.569576	−0.284788	−	0.958591i	$-0.591923\pi$
$83$	−5.18908	−0.569576	−0.284788	+	0.958591i	$0.591923\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−3.09334	−0.327893	−0.163947	−	0.986469i	$-0.552422\pi$
$89$	−3.09334	−0.327893	−0.163947	+	0.986469i	$0.552422\pi$
$90$	0	0
$91$	14.3578	1.50511
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−0.882978	−0.0896528	−0.0448264	−	0.998995i	$-0.514273\pi$
$97$	−0.882978	−0.0896528	−0.0448264	+	0.998995i	$0.514273\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−5.87680	−0.584763	−0.292382	−	0.956302i	$-0.594448\pi$
$101$	−5.87680	−0.584763	−0.292382	+	0.956302i	$0.594448\pi$
$102$	0	0
$103$	−9.87073	−0.972592	−0.486296	−	0.873794i	$-0.661652\pi$
$103$	−9.87073	−0.972592	−0.486296	+	0.873794i	$0.661652\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	0	0
$109$	7.00000	0.670478	0.335239	−	0.942133i	$-0.391183\pi$
$109$	7.00000	0.670478	0.335239	+	0.942133i	$0.391183\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−13.5871	−1.27817	−0.639085	−	0.769136i	$-0.720687\pi$
$113$	−13.5871	−1.27817	−0.639085	+	0.769136i	$0.720687\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−33.8995	−3.10756
$120$	0	0
$121$	−5.17891	−0.470810
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−4.93536	−0.437943	−0.218971	−	0.975731i	$-0.570270\pi$
$127$	−4.93536	−0.437943	−0.218971	+	0.975731i	$0.570270\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	14.1663	1.23771	0.618857	−	0.785504i	$-0.287596\pi$
$131$	14.1663	1.23771	0.618857	+	0.785504i	$0.287596\pi$
$132$	0	0
$133$	20.6244	1.78837
$134$	0	0
$135$	0	0
$136$	0	0
$137$	5.03883	0.430496	0.215248	−	0.976559i	$-0.430944\pi$
$137$	5.03883	0.430496	0.215248	+	0.976559i	$0.430944\pi$
$138$	0	0
$139$	5.82109	0.493739	0.246869	−	0.969049i	$-0.420598\pi$
$139$	5.82109	0.493739	0.246869	+	0.969049i	$0.420598\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−7.01894	−0.586953
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	13.1758	1.07940	0.539700	−	0.841857i	$-0.318538\pi$
$149$	13.1758	1.07940	0.539700	+	0.841857i	$0.318538\pi$
$150$	0	0
$151$	−6.17891	−0.502832	−0.251416	−	0.967879i	$-0.580896\pi$
$151$	−6.17891	−0.502832	−0.251416	+	0.967879i	$0.580896\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	7.84453	0.626062	0.313031	−	0.949743i	$-0.398656\pi$
$157$	7.84453	0.626062	0.313031	+	0.949743i	$0.398656\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−16.5790	−1.30661
$162$	0	0
$163$	10.7537	0.842295	0.421148	−	0.906992i	$-0.361627\pi$
$163$	10.7537	0.842295	0.421148	+	0.906992i	$0.361627\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	8.54830	0.661487	0.330744	−	0.943721i	$-0.392700\pi$
$167$	8.54830	0.661487	0.330744	+	0.943721i	$0.392700\pi$
$168$	0	0
$169$	−4.53673	−0.348979
$170$	0	0
$171$	0	0
$172$	0	0
$173$	18.7762	1.42753	0.713765	−	0.700386i	$-0.246989\pi$
$173$	18.7762	1.42753	0.713765	+	0.700386i	$0.246989\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	7.97961	0.596424	0.298212	−	0.954500i	$-0.403610\pi$
$179$	7.97961	0.596424	0.298212	+	0.954500i	$0.403610\pi$
$180$	0	0
$181$	3.82109	0.284020	0.142010	−	0.989865i	$-0.454644\pi$
$181$	3.82109	0.284020	0.142010	+	0.989865i	$0.454644\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	16.5720	1.21187
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−6.61832	−0.478885	−0.239443	−	0.970911i	$-0.576965\pi$
$191$	−6.61832	−0.478885	−0.239443	+	0.970911i	$0.576965\pi$
$192$	0	0
$193$	21.7676	1.56687	0.783435	−	0.621474i	$-0.213466\pi$
$193$	21.7676	1.56687	0.783435	+	0.621474i	$0.213466\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−15.4170	−1.09842	−0.549208	−	0.835686i	$-0.685070\pi$
$197$	−15.4170	−1.09842	−0.549208	+	0.835686i	$0.685070\pi$
$198$	0	0
$199$	20.3578	1.44313	0.721564	−	0.692348i	$-0.243424\pi$
$199$	20.3578	1.44313	0.721564	+	0.692348i	$0.243424\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	25.6449	1.79992
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−10.0824	−0.697416
$210$	0	0
$211$	16.1789	1.11380	0.556901	−	0.830579i	$-0.311990\pi$
$211$	16.1789	1.11380	0.556901	+	0.830579i	$0.311990\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	30.4952	2.07015
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−19.9822	−1.34415
$222$	0	0
$223$	−25.5598	−1.71161	−0.855805	−	0.517298i	$-0.826938\pi$
$223$	−25.5598	−1.71161	−0.855805	+	0.517298i	$0.826938\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	18.9265	1.25619	0.628097	−	0.778135i	$-0.283834\pi$
$227$	18.9265	1.25619	0.628097	+	0.778135i	$0.283834\pi$
$228$	0	0
$229$	6.82109	0.450750	0.225375	−	0.974272i	$-0.427639\pi$
$229$	6.82109	0.450750	0.225375	+	0.974272i	$0.427639\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	0.150248	0.00984309	0.00492154	−	0.999988i	$-0.498433\pi$
$233$	0.150248	0.00984309	0.00492154	+	0.999988i	$0.498433\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	9.03102	0.584168	0.292084	−	0.956393i	$-0.405651\pi$
$239$	9.03102	0.584168	0.292084	+	0.956393i	$0.405651\pi$
$240$	0	0
$241$	19.0000	1.22390	0.611949	−	0.790897i	$-0.290386\pi$
$241$	19.0000	1.22390	0.611949	+	0.790897i	$0.290386\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	12.1572	0.773541
$248$	0	0
$249$	0	0
$250$	0	0
$251$	17.3205	1.09326	0.546630	−	0.837374i	$-0.315910\pi$
$251$	17.3205	1.09326	0.546630	+	0.837374i	$0.315910\pi$
$252$	0	0
$253$	8.10477	0.509543
$254$	0	0
$255$	0	0
$256$	0	0
$257$	22.1354	1.38077	0.690385	−	0.723442i	$-0.257441\pi$
$257$	22.1354	1.38077	0.690385	+	0.723442i	$0.257441\pi$
$258$	0	0
$259$	38.7156	2.40567
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−6.71844	−0.414277	−0.207138	−	0.978312i	$-0.566415\pi$
$263$	−6.71844	−0.414277	−0.207138	+	0.978312i	$0.566415\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	30.0646	1.83307	0.916536	−	0.399952i	$-0.130973\pi$
$269$	30.0646	1.83307	0.916536	+	0.399952i	$0.130973\pi$
$270$	0	0
$271$	−22.3578	−1.35814	−0.679070	−	0.734073i	$-0.737617\pi$
$271$	−22.3578	−1.35814	−0.679070	+	0.734073i	$0.737617\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−10.7537	−0.646128	−0.323064	−	0.946377i	$-0.604713\pi$
$277$	−10.7537	−0.646128	−0.323064	+	0.946377i	$0.604713\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−12.4342	−0.741764	−0.370882	−	0.928680i	$-0.620945\pi$
$281$	−12.4342	−0.741764	−0.370882	+	0.928680i	$0.620945\pi$
$282$	0	0
$283$	1.76596	0.104975	0.0524876	−	0.998622i	$-0.483285\pi$
$283$	1.76596	0.104975	0.0524876	+	0.998622i	$0.483285\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−29.0041	−1.71206
$288$	0	0
$289$	30.1789	1.77523
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−20.6061	−1.20382	−0.601910	−	0.798564i	$-0.705593\pi$
$293$	−20.6061	−1.20382	−0.601910	+	0.798564i	$0.705593\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−9.77255	−0.565161
$300$	0	0
$301$	24.3578	1.40396
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	8.98775	0.512958	0.256479	−	0.966550i	$-0.417438\pi$
$307$	8.98775	0.512958	0.256479	+	0.966550i	$0.417438\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	10.7022	0.606865	0.303433	−	0.952853i	$-0.401867\pi$
$311$	10.7022	0.606865	0.303433	+	0.952853i	$0.401867\pi$
$312$	0	0
$313$	6.96156	0.393490	0.196745	−	0.980455i	$-0.436963\pi$
$313$	6.96156	0.393490	0.196745	+	0.980455i	$0.436963\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	34.0430	1.91204	0.956021	−	0.293297i	$-0.0947524\pi$
$317$	34.0430	1.91204	0.956021	+	0.293297i	$0.0947524\pi$
$318$	0	0
$319$	−12.5367	−0.701922
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−28.7036	−1.59711
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	58.7680	3.23998
$330$	0	0
$331$	−32.5367	−1.78838	−0.894190	−	0.447688i	$-0.852248\pi$
$331$	−32.5367	−1.78838	−0.894190	+	0.447688i	$0.852248\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	9.87073	0.537693	0.268846	−	0.963183i	$-0.413358\pi$
$337$	9.87073	0.537693	0.268846	+	0.963183i	$0.413358\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−14.9078	−0.807303
$342$	0	0
$343$	−51.1196	−2.76020
$344$	0	0
$345$	0	0
$346$	0	0
$347$	8.54830	0.458897	0.229448	−	0.973321i	$-0.426308\pi$
$347$	8.54830	0.458897	0.229448	+	0.973321i	$0.426308\pi$
$348$	0	0
$349$	20.1789	1.08015	0.540076	−	0.841616i	$-0.318395\pi$
$349$	20.1789	1.08015	0.540076	+	0.841616i	$0.318395\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−21.9852	−1.17015	−0.585077	−	0.810978i	$-0.698936\pi$
$353$	−21.9852	−1.17015	−0.585077	+	0.810978i	$0.698936\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−0.309878	−0.0163548	−0.00817738	−	0.999967i	$-0.502603\pi$
$359$	−0.309878	−0.0163548	−0.00817738	+	0.999967i	$0.502603\pi$
$360$	0	0
$361$	−1.53673	−0.0808803
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	9.87073	0.515248	0.257624	−	0.966245i	$-0.417060\pi$
$367$	9.87073	0.515248	0.257624	+	0.966245i	$0.417060\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−42.1890	−2.19034
$372$	0	0
$373$	−8.98775	−0.465368	−0.232684	−	0.972552i	$-0.574751\pi$
$373$	−8.98775	−0.465368	−0.232684	+	0.972552i	$0.574751\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	15.1165	0.778539
$378$	0	0
$379$	−0.357817	−0.0183798	−0.00918990	−	0.999958i	$-0.502925\pi$
$379$	−0.357817	−0.0183798	−0.00918990	+	0.999958i	$0.502925\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	13.7374	0.701947	0.350974	−	0.936385i	$-0.385851\pi$
$383$	13.7374	0.701947	0.350974	+	0.936385i	$0.385851\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−25.6100	−1.29848	−0.649239	−	0.760584i	$-0.724913\pi$
$389$	−25.6100	−1.29848	−0.649239	+	0.760584i	$0.724913\pi$
$390$	0	0
$391$	23.0735	1.16687
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−11.0139	−0.552774	−0.276387	−	0.961046i	$-0.589137\pi$
$397$	−11.0139	−0.552774	−0.276387	+	0.961046i	$0.589137\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	20.7237	1.03489	0.517447	−	0.855715i	$-0.326883\pi$
$401$	20.7237	1.03489	0.517447	+	0.855715i	$0.326883\pi$
$402$	0	0
$403$	17.9755	0.895423
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−18.9265	−0.938150
$408$	0	0
$409$	6.82109	0.337281	0.168641	−	0.985678i	$-0.446062\pi$
$409$	6.82109	0.337281	0.168641	+	0.985678i	$0.446062\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−5.18908	−0.255338
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	16.5790	0.809936	0.404968	−	0.914331i	$-0.367283\pi$
$419$	16.5790	0.809936	0.404968	+	0.914331i	$0.367283\pi$
$420$	0	0
$421$	29.7156	1.44825	0.724126	−	0.689668i	$-0.242244\pi$
$421$	29.7156	1.44825	0.724126	+	0.689668i	$0.242244\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−45.3013	−2.19228
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−1.67116	−0.0804972	−0.0402486	−	0.999190i	$-0.512815\pi$
$431$	−1.67116	−0.0804972	−0.0402486	+	0.999190i	$0.512815\pi$
$432$	0	0
$433$	36.5737	1.75762	0.878811	−	0.477170i	$-0.158337\pi$
$433$	36.5737	1.75762	0.878811	+	0.477170i	$0.158337\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−14.0379	−0.671523
$438$	0	0
$439$	18.5367	0.884710	0.442355	−	0.896840i	$-0.354143\pi$
$439$	18.5367	0.884710	0.442355	+	0.896840i	$0.354143\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	18.6260	0.884946	0.442473	−	0.896782i	$-0.354101\pi$
$443$	18.6260	0.884946	0.442473	+	0.896782i	$0.354101\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	25.1783	1.18824	0.594120	−	0.804377i	$-0.297500\pi$
$449$	25.1783	1.18824	0.594120	+	0.804377i	$0.297500\pi$
$450$	0	0
$451$	14.1789	0.667659
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	27.5860	1.29042	0.645209	−	0.764006i	$-0.276770\pi$
$457$	27.5860	1.29042	0.645209	+	0.764006i	$0.276770\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−10.7022	−0.498450	−0.249225	−	0.968446i	$-0.580176\pi$
$461$	−10.7022	−0.498450	−0.249225	+	0.968446i	$0.580176\pi$
$462$	0	0
$463$	29.6122	1.37619	0.688097	−	0.725618i	$-0.258446\pi$
$463$	29.6122	1.37619	0.688097	+	0.725618i	$0.258446\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	15.5672	0.720366	0.360183	−	0.932882i	$-0.382714\pi$
$467$	15.5672	0.720366	0.360183	+	0.932882i	$0.382714\pi$
$468$	0	0
$469$	20.0000	0.923514
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−11.9075	−0.547508
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−25.3001	−1.15599	−0.577996	−	0.816040i	$-0.696165\pi$
$479$	−25.3001	−1.15599	−0.577996	+	0.816040i	$0.696165\pi$
$480$	0	0
$481$	22.8211	1.04055
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−17.9755	−0.814548	−0.407274	−	0.913306i	$-0.633521\pi$
$487$	−17.9755	−0.814548	−0.407274	+	0.913306i	$0.633521\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−35.5707	−1.60528	−0.802641	−	0.596463i	$-0.796572\pi$
$491$	−35.5707	−1.60528	−0.802641	+	0.596463i	$0.796572\pi$
$492$	0	0
$493$	−35.6908	−1.60743
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−69.9158	−3.13615
$498$	0	0
$499$	4.17891	0.187074	0.0935368	−	0.995616i	$-0.470183\pi$
$499$	4.17891	0.187074	0.0935368	+	0.995616i	$0.470183\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	37.2519	1.66098	0.830491	−	0.557032i	$-0.188060\pi$
$503$	37.2519	1.66098	0.830491	+	0.557032i	$0.188060\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−7.54796	−0.334557	−0.167279	−	0.985910i	$-0.553498\pi$
$509$	−7.54796	−0.334557	−0.167279	+	0.985910i	$0.553498\pi$
$510$	0	0
$511$	10.0000	0.442374
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−28.7292	−1.26351
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−9.03102	−0.395656	−0.197828	−	0.980237i	$-0.563389\pi$
$521$	−9.03102	−0.395656	−0.197828	+	0.980237i	$0.563389\pi$
$522$	0	0
$523$	7.58430	0.331638	0.165819	−	0.986156i	$-0.446973\pi$
$523$	7.58430	0.331638	0.165819	+	0.986156i	$0.446973\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−42.4410	−1.84876
$528$	0	0
$529$	−11.7156	−0.509375
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−17.0966	−0.740536
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	41.8791	1.80386
$540$	0	0
$541$	3.35782	0.144364	0.0721819	−	0.997391i	$-0.477004\pi$
$541$	3.35782	0.144364	0.0721819	+	0.997391i	$0.477004\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−19.7415	−0.844084	−0.422042	−	0.906576i	$-0.638686\pi$
$547$	−19.7415	−0.844084	−0.422042	+	0.906576i	$0.638686\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	21.7142	0.925058
$552$	0	0
$553$	−29.6122	−1.25924
$554$	0	0
$555$	0	0
$556$	0	0
$557$	39.2320	1.66231	0.831157	−	0.556037i	$-0.187679\pi$
$557$	39.2320	1.66231	0.831157	+	0.556037i	$0.187679\pi$
$558$	0	0
$559$	14.3578	0.607271
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−11.9075	−0.501842	−0.250921	−	0.968008i	$-0.580733\pi$
$563$	−11.9075	−0.501842	−0.250921	+	0.968008i	$0.580733\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	4.45462	0.186748	0.0933738	−	0.995631i	$-0.470235\pi$
$569$	4.45462	0.186748	0.0933738	+	0.995631i	$0.470235\pi$
$570$	0	0
$571$	−18.1789	−0.760764	−0.380382	−	0.924830i	$-0.624207\pi$
$571$	−18.1789	−0.760764	−0.380382	+	0.924830i	$0.624207\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−7.84453	−0.326572	−0.163286	−	0.986579i	$-0.552209\pi$
$577$	−7.84453	−0.326572	−0.163286	+	0.986579i	$0.552209\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	25.6100	1.06248
$582$	0	0
$583$	20.6244	0.854177
$584$	0	0
$585$	0	0
$586$	0	0
$587$	11.6070	0.479073	0.239537	−	0.970887i	$-0.423004\pi$
$587$	11.6070	0.479073	0.239537	+	0.970887i	$0.423004\pi$
$588$	0	0
$589$	25.8211	1.06394
$590$	0	0
$591$	0	0
$592$	0	0
$593$	5.03883	0.206920	0.103460	−	0.994634i	$-0.467009\pi$
$593$	5.03883	0.206920	0.103460	+	0.994634i	$0.467009\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	17.0106	0.695035	0.347518	−	0.937673i	$-0.387025\pi$
$599$	17.0106	0.695035	0.347518	+	0.937673i	$0.387025\pi$
$600$	0	0
$601$	−3.35782	−0.136968	−0.0684841	−	0.997652i	$-0.521816\pi$
$601$	−3.35782	−0.136968	−0.0684841	+	0.997652i	$0.521816\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	4.93536	0.200320	0.100160	−	0.994971i	$-0.468064\pi$
$607$	4.93536	0.200320	0.100160	+	0.994971i	$0.468064\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	34.6410	1.40143
$612$	0	0
$613$	−37.7170	−1.52337	−0.761687	−	0.647945i	$-0.775628\pi$
$613$	−37.7170	−1.52337	−0.761687	+	0.647945i	$0.775628\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	5.33933	0.214953	0.107477	−	0.994208i	$-0.465723\pi$
$617$	5.33933	0.214953	0.107477	+	0.994208i	$0.465723\pi$
$618$	0	0
$619$	11.6422	0.467939	0.233969	−	0.972244i	$-0.424828\pi$
$619$	11.6422	0.467939	0.233969	+	0.972244i	$0.424828\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	15.2667	0.611649
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−53.8817	−2.14840
$630$	0	0
$631$	−22.5367	−0.897173	−0.448586	−	0.893739i	$-0.648072\pi$
$631$	−22.5367	−0.897173	−0.448586	+	0.893739i	$0.648072\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−50.4969	−2.00076
$638$	0	0
$639$	0	0
$640$	0	0
$641$	18.3110	0.723242	0.361621	−	0.932325i	$-0.382223\pi$
$641$	18.3110	0.723242	0.361621	+	0.932325i	$0.382223\pi$
$642$	0	0
$643$	−20.6244	−0.813348	−0.406674	−	0.913573i	$-0.633312\pi$
$643$	−20.6244	−0.813348	−0.406674	+	0.913573i	$0.633312\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	44.2709	1.74047	0.870234	−	0.492639i	$-0.163968\pi$
$647$	44.2709	1.74047	0.870234	+	0.492639i	$0.163968\pi$
$648$	0	0
$649$	2.53673	0.0995752
$650$	0	0
$651$	0	0
$652$	0	0
$653$	0.300496	0.0117593	0.00587967	−	0.999983i	$-0.498128\pi$
$653$	0.300496	0.0117593	0.00587967	+	0.999983i	$0.498128\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−41.4474	−1.61456	−0.807282	−	0.590166i	$-0.799062\pi$
$659$	−41.4474	−1.61456	−0.807282	+	0.590166i	$0.799062\pi$
$660$	0	0
$661$	−46.4313	−1.80597	−0.902983	−	0.429675i	$-0.858628\pi$
$661$	−46.4313	−1.80597	−0.902983	+	0.429675i	$0.858628\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−17.4550	−0.675861
$668$	0	0
$669$	0	0
$670$	0	0
$671$	22.1459	0.854933
$672$	0	0
$673$	14.5459	0.560701	0.280351	−	0.959898i	$-0.409549\pi$
$673$	14.5459	0.560701	0.280351	+	0.959898i	$0.409549\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	34.4937	1.32570	0.662850	−	0.748752i	$-0.269347\pi$
$677$	34.4937	1.32570	0.662850	+	0.748752i	$0.269347\pi$
$678$	0	0
$679$	4.35782	0.167238
$680$	0	0
$681$	0	0
$682$	0	0
$683$	44.5714	1.70548	0.852738	−	0.522339i	$-0.174940\pi$
$683$	44.5714	1.70548	0.852738	+	0.522339i	$0.174940\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−24.8685	−0.947413
$690$	0	0
$691$	−30.7156	−1.16848	−0.584239	−	0.811582i	$-0.698607\pi$
$691$	−30.7156	−1.16848	−0.584239	+	0.811582i	$0.698607\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	40.3659	1.52897
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−27.3420	−1.03269	−0.516347	−	0.856379i	$-0.672709\pi$
$701$	−27.3420	−1.03269	−0.516347	+	0.856379i	$0.672709\pi$
$702$	0	0
$703$	32.7816	1.23638
$704$	0	0
$705$	0	0
$706$	0	0
$707$	29.0041	1.09081
$708$	0	0
$709$	15.5367	0.583494	0.291747	−	0.956496i	$-0.405763\pi$
$709$	15.5367	0.583494	0.291747	+	0.956496i	$0.405763\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−20.7563	−0.777330
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−40.3960	−1.50652	−0.753259	−	0.657724i	$-0.771519\pi$
$719$	−40.3960	−1.50652	−0.753259	+	0.657724i	$0.771519\pi$
$720$	0	0
$721$	48.7156	1.81426
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−47.0672	−1.74563	−0.872813	−	0.488055i	$-0.837707\pi$
$727$	−47.0672	−1.74563	−0.872813	+	0.488055i	$0.837707\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−33.8995	−1.25382
$732$	0	0
$733$	19.7415	0.729167	0.364584	−	0.931171i	$-0.381211\pi$
$733$	19.7415	0.729167	0.364584	+	0.931171i	$0.381211\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−9.77717	−0.360147
$738$	0	0
$739$	−22.8945	−0.842189	−0.421095	−	0.907017i	$-0.638354\pi$
$739$	−22.8945	−0.842189	−0.421095	+	0.907017i	$0.638354\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−49.7604	−1.82553	−0.912767	−	0.408481i	$-0.866059\pi$
$743$	−49.7604	−1.82553	−0.912767	+	0.408481i	$0.866059\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−6.35782	−0.232000	−0.116000	−	0.993249i	$-0.537007\pi$
$751$	−6.35782	−0.232000	−0.116000	+	0.993249i	$0.537007\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−1.76596	−0.0641847	−0.0320924	−	0.999485i	$-0.510217\pi$
$757$	−1.76596	−0.0641847	−0.0320924	+	0.999485i	$0.510217\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−1.73205	−0.0627868	−0.0313934	−	0.999507i	$-0.509994\pi$
$761$	−1.73205	−0.0627868	−0.0313934	+	0.999507i	$0.509994\pi$
$762$	0	0
$763$	−34.5475	−1.25071
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−3.05872	−0.110444
$768$	0	0
$769$	25.7156	0.927329	0.463665	−	0.886011i	$-0.346534\pi$
$769$	25.7156	0.927329	0.463665	+	0.886011i	$0.346534\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−30.6837	−1.10362	−0.551809	−	0.833971i	$-0.686062\pi$
$773$	−30.6837	−1.10362	−0.551809	+	0.833971i	$0.686062\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−24.5586	−0.879903
$780$	0	0
$781$	34.1789	1.22302
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	47.9502	1.70924	0.854620	−	0.519254i	$-0.173790\pi$
$787$	47.9502	1.70924	0.854620	+	0.519254i	$0.173790\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	67.0574	2.38429
$792$	0	0
$793$	−26.7030	−0.948252
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−11.7573	−0.416464	−0.208232	−	0.978079i	$-0.566771\pi$
$797$	−11.7573	−0.416464	−0.208232	+	0.978079i	$0.566771\pi$
$798$	0	0
$799$	−81.7891	−2.89349
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−4.88858	−0.172514
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−5.19615	−0.182687	−0.0913435	−	0.995819i	$-0.529116\pi$
$809$	−5.19615	−0.182687	−0.0913435	+	0.995819i	$0.529116\pi$
$810$	0	0
$811$	6.89454	0.242100	0.121050	−	0.992646i	$-0.461374\pi$
$811$	6.89454	0.242100	0.121050	+	0.992646i	$0.461374\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	20.6244	0.721558
$818$	0	0
$819$	0	0
$820$	0	0
$821$	52.2105	1.82216	0.911080	−	0.412230i	$-0.135250\pi$
$821$	52.2105	1.82216	0.911080	+	0.412230i	$0.135250\pi$
$822$	0	0
$823$	−11.6367	−0.405629	−0.202815	−	0.979217i	$-0.565009\pi$
$823$	−11.6367	−0.405629	−0.202815	+	0.979217i	$0.565009\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−13.7374	−0.477696	−0.238848	−	0.971057i	$-0.576770\pi$
$827$	−13.7374	−0.477696	−0.238848	+	0.971057i	$0.576770\pi$
$828$	0	0
$829$	−4.17891	−0.145139	−0.0725697	−	0.997363i	$-0.523120\pi$
$829$	−4.17891	−0.145139	−0.0725697	+	0.997363i	$0.523120\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	119.225	4.13092
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−23.8171	−0.822256	−0.411128	−	0.911578i	$-0.634865\pi$
$839$	−23.8171	−0.822256	−0.411128	+	0.911578i	$0.634865\pi$
$840$	0	0
$841$	−2.00000	−0.0689655
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	25.5598	0.878245
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−26.3515	−0.903319
$852$	0	0
$853$	−12.5197	−0.428665	−0.214333	−	0.976761i	$-0.568758\pi$
$853$	−12.5197	−0.428665	−0.214333	+	0.976761i	$0.568758\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−56.3286	−1.92415	−0.962075	−	0.272786i	$-0.912055\pi$
$857$	−56.3286	−1.92415	−0.962075	+	0.272786i	$0.912055\pi$
$858$	0	0
$859$	38.5367	1.31486	0.657428	−	0.753517i	$-0.271644\pi$
$859$	38.5367	1.31486	0.657428	+	0.753517i	$0.271644\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−8.54830	−0.290988	−0.145494	−	0.989359i	$-0.546477\pi$
$863$	−8.54830	−0.290988	−0.145494	+	0.989359i	$0.546477\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	14.4762	0.491070
$870$	0	0
$871$	11.7891	0.399458
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−48.2104	−1.62795	−0.813975	−	0.580900i	$-0.802701\pi$
$877$	−48.2104	−1.62795	−0.813975	+	0.580900i	$0.802701\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−23.9388	−0.806520	−0.403260	−	0.915085i	$-0.632123\pi$
$881$	−23.9388	−0.806520	−0.403260	+	0.915085i	$0.632123\pi$
$882$	0	0
$883$	50.2366	1.69060	0.845298	−	0.534295i	$-0.179423\pi$
$883$	50.2366	1.69060	0.845298	+	0.534295i	$0.179423\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−25.6449	−0.861072	−0.430536	−	0.902574i	$-0.641675\pi$
$887$	−25.6449	−0.861072	−0.430536	+	0.902574i	$0.641675\pi$
$888$	0	0
$889$	24.3578	0.816935
$890$	0	0
$891$	0	0
$892$	0	0
$893$	49.7604	1.66517
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	32.1065	1.07081
$900$	0	0
$901$	58.7156	1.95610
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	59.2244	1.96651	0.983256	−	0.182227i	$-0.0583307\pi$
$907$	59.2244	1.96651	0.983256	+	0.182227i	$0.0583307\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−35.5707	−1.17851	−0.589254	−	0.807948i	$-0.700578\pi$
$911$	−35.5707	−1.17851	−0.589254	+	0.807948i	$0.700578\pi$
$912$	0	0
$913$	−12.5197	−0.414340
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−69.9158	−2.30882
$918$	0	0
$919$	−19.8211	−0.653837	−0.326919	−	0.945052i	$-0.606010\pi$
$919$	−19.8211	−0.653837	−0.326919	+	0.945052i	$0.606010\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−41.2121	−1.35651
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−12.7441	−0.418121	−0.209060	−	0.977903i	$-0.567041\pi$
$929$	−12.7441	−0.418121	−0.209060	+	0.977903i	$0.567041\pi$
$930$	0	0
$931$	−72.5367	−2.37730
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−12.7799	−0.417501	−0.208751	−	0.977969i	$-0.566940\pi$
$937$	−12.7799	−0.417501	−0.208751	+	0.977969i	$0.566940\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	53.8817	1.75649	0.878246	−	0.478209i	$-0.158714\pi$
$941$	53.8817	1.75649	0.878246	+	0.478209i	$0.158714\pi$
$942$	0	0
$943$	19.7415	0.642870
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−7.01894	−0.228085	−0.114042	−	0.993476i	$-0.536380\pi$
$947$	−7.01894	−0.228085	−0.114042	+	0.993476i	$0.536380\pi$
$948$	0	0
$949$	5.89454	0.191345
$950$	0	0
$951$	0	0
$952$	0	0
$953$	11.7573	0.380855	0.190428	−	0.981701i	$-0.439013\pi$
$953$	11.7573	0.380855	0.190428	+	0.981701i	$0.439013\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−24.8685	−0.803045
$960$	0	0
$961$	7.17891	0.231578
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−13.0401	−0.419343	−0.209671	−	0.977772i	$-0.567239\pi$
$967$	−13.0401	−0.419343	−0.209671	+	0.977772i	$0.567239\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−21.7142	−0.696843	−0.348422	−	0.937338i	$-0.613282\pi$
$971$	−21.7142	−0.696843	−0.348422	+	0.937338i	$0.613282\pi$
$972$	0	0
$973$	−28.7292	−0.921016
$974$	0	0
$975$	0	0
$976$	0	0
$977$	27.1743	0.869382	0.434691	−	0.900580i	$-0.356858\pi$
$977$	27.1743	0.869382	0.434691	+	0.900580i	$0.356858\pi$
$978$	0	0
$979$	−7.46327	−0.238527
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−25.9454	−0.827530	−0.413765	−	0.910384i	$-0.635786\pi$
$983$	−25.9454	−0.827530	−0.413765	+	0.910384i	$0.635786\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−16.5790	−0.527181
$990$	0	0
$991$	−32.5367	−1.03356	−0.516782	−	0.856117i	$-0.672870\pi$
$991$	−32.5367	−1.03356	−0.516782	+	0.856117i	$0.672870\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	29.8724	0.946069	0.473035	−	0.881044i	$-0.343159\pi$
$997$	29.8724	0.946069	0.473035	+	0.881044i	$0.343159\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8100.2.a.be.1.2		8
3.2	odd	2	inner	8100.2.a.be.1.1		8
5.2	odd	4		1620.2.d.e.649.8	yes	8
5.3	odd	4		1620.2.d.e.649.7	yes	8
5.4	even	2	inner	8100.2.a.be.1.8		8
15.2	even	4		1620.2.d.e.649.1	✓	8
15.8	even	4		1620.2.d.e.649.2	yes	8
15.14	odd	2	inner	8100.2.a.be.1.7		8
45.2	even	12		1620.2.r.h.109.7		16
45.7	odd	12		1620.2.r.h.109.2		16
45.13	odd	12		1620.2.r.h.1189.2		16
45.22	odd	12		1620.2.r.h.1189.4		16
45.23	even	12		1620.2.r.h.1189.7		16
45.32	even	12		1620.2.r.h.1189.5		16
45.38	even	12		1620.2.r.h.109.5		16
45.43	odd	12		1620.2.r.h.109.4		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1620.2.d.e.649.1	✓	8	15.2	even	4
1620.2.d.e.649.2	yes	8	15.8	even	4
1620.2.d.e.649.7	yes	8	5.3	odd	4
1620.2.d.e.649.8	yes	8	5.2	odd	4
1620.2.r.h.109.2		16	45.7	odd	12
1620.2.r.h.109.4		16	45.43	odd	12
1620.2.r.h.109.5		16	45.38	even	12
1620.2.r.h.109.7		16	45.2	even	12
1620.2.r.h.1189.2		16	45.13	odd	12
1620.2.r.h.1189.4		16	45.22	odd	12
1620.2.r.h.1189.5		16	45.32	even	12
1620.2.r.h.1189.7		16	45.23	even	12
8100.2.a.be.1.1		8	3.2	odd	2	inner
8100.2.a.be.1.2		8	1.1	even	1	trivial
8100.2.a.be.1.7		8	15.14	odd	2	inner
8100.2.a.be.1.8		8	5.4	even	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:	\( N \)	\(=\)	\( 8100 = 2^{2} \cdot 3^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8100.a (trivial)

\(f(q)\)	\(=\)	\(q-4.93536 q^{7} +O(q^{10})\)	\(q-4.93536 q^{7} +2.41269 q^{11} -2.90917 q^{13} +6.86869 q^{17} -4.17891 q^{19} +3.35922 q^{23} -5.19615 q^{29} -6.17891 q^{31} -7.84453 q^{37} +5.87680 q^{41} -4.93536 q^{43} -11.9075 q^{47} +17.3578 q^{49} +8.54830 q^{53} +1.05141 q^{59} +9.17891 q^{61} -4.05239 q^{67} +14.1663 q^{71} -2.02619 q^{73} -11.9075 q^{77} +6.00000 q^{79} -5.18908 q^{83} -3.09334 q^{89} +14.3578 q^{91} -0.882978 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q + 12 q^{19} - 4 q^{31} + 48 q^{49} + 28 q^{61} + 48 q^{79} + 24 q^{91}+O(q^{100}) \) 8 * q + 12 * q^19 - 4 * q^31 + 48 * q^49 + 28 * q^61 + 48 * q^79 + 24 * q^91

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