LMFDB - Embedded newform 8820.2.d.b.881.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8820,2,Mod(881,8820)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8820, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8820.881");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8820 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8820.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:	$70.4280545828$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 2 x^{11} - 9 x^{10} + 58 x^{9} - 78 x^{8} - 298 x^{7} + 1341 x^{6} - 2086 x^{5} - 3822 x^{4} + \cdots + 117649 $ x^12 - 2x^11 - 9x^10 + 58x^9 - 78x^8 - 298x^7 + 1341x^6 - 2086x^5 - 3822x^4 + 19894x^3 - 21609x^2 - 33614*x + 117649
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{6}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 1260)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			881.2
Root			$0.260926 + 2.63285i$ of defining polynomial
Character	$\chi$	$=$	8820.881
Dual form			8820.2.d.b.881.11

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{5} -3.53366i q^{11} +0.599777i q^{13} -1.69803 q^{17} +7.78350i q^{19} +3.31319i q^{23} +1.00000 q^{25} -0.456069i q^{29} -1.05585i q^{31} -0.386249 q^{37} -0.478149 q^{41} -4.21237 q^{43} +8.55804 q^{47} -3.98973i q^{53} -3.53366i q^{55} -2.60613 q^{59} +11.0895i q^{61} +0.599777i q^{65} +2.87422 q^{67} +0.336875i q^{71} +3.08479i q^{73} -13.6089 q^{79} -16.2901 q^{83} -1.69803 q^{85} -10.6964 q^{89} +7.78350i q^{95} +2.18092i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 12 q^{5} + 12 q^{25} + 4 q^{37} - 8 q^{41} + 36 q^{43} + 32 q^{47} - 4 q^{67} - 28 q^{79} + 40 q^{83} + 40 q^{89}+O(q^{100}) $ 12 * q + 12 * q^5 + 12 * q^25 + 4 * q^37 - 8 * q^41 + 36 * q^43 + 32 * q^47 - 4 * q^67 - 28 * q^79 + 40 * q^83 + 40 * q^89

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/8820\mathbb{Z}\right)^\times$.

$n$	$1081$	$4411$	$7057$	$7841$
$\chi(n)$	$-1$	$1$	$1$	$-1$

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
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 3.53366i	− 1.06544i	−0.846292	−	0.532719i	$-0.821170\pi$
$11$	− 3.53366i	− 1.06544i	0.846292	−	0.532719i	$-0.178830\pi$
$12$	0	0
$13$	0.599777i	0.166348i	0.996535	+	0.0831742i	$0.0265058\pi$
$13$	0.599777i	0.166348i	−0.996535	+	0.0831742i	$0.973494\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−1.69803	−0.411834	−0.205917	−	0.978569i	$-0.566018\pi$
$17$	−1.69803	−0.411834	−0.205917	+	0.978569i	$0.566018\pi$
$18$	0	0
$19$	7.78350i	1.78566i	0.450397	+	0.892829i	$0.351283\pi$
$19$	7.78350i	1.78566i	−0.450397	+	0.892829i	$0.648717\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	3.31319i	0.690848i	0.938447	+	0.345424i	$0.112265\pi$
$23$	3.31319i	0.690848i	−0.938447	+	0.345424i	$0.887735\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 0.456069i	− 0.0846898i	−0.999103	−	0.0423449i	$-0.986517\pi$
$29$	− 0.456069i	− 0.0846898i	0.999103	−	0.0423449i	$-0.0134828\pi$
$30$	0	0
$31$	− 1.05585i	− 0.189636i	−0.995495	−	0.0948178i	$-0.969773\pi$
$31$	− 1.05585i	− 0.189636i	0.995495	−	0.0948178i	$-0.0302268\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−0.386249	−0.0634989	−0.0317494	−	0.999496i	$-0.510108\pi$
$37$	−0.386249	−0.0634989	−0.0317494	+	0.999496i	$0.510108\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−0.478149	−0.0746743	−0.0373372	−	0.999303i	$-0.511888\pi$
$41$	−0.478149	−0.0746743	−0.0373372	+	0.999303i	$0.511888\pi$
$42$	0	0
$43$	−4.21237	−0.642381	−0.321190	−	0.947015i	$-0.604083\pi$
$43$	−4.21237	−0.642381	−0.321190	+	0.947015i	$0.604083\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	8.55804	1.24832	0.624159	−	0.781297i	$-0.285442\pi$
$47$	8.55804	1.24832	0.624159	+	0.781297i	$0.285442\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 3.98973i	− 0.548031i	−0.961725	−	0.274015i	$-0.911648\pi$
$53$	− 3.98973i	− 0.548031i	0.961725	−	0.274015i	$-0.0883519\pi$
$54$	0	0
$55$	− 3.53366i	− 0.476478i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−2.60613	−0.339290	−0.169645	−	0.985505i	$-0.554262\pi$
$59$	−2.60613	−0.339290	−0.169645	+	0.985505i	$0.554262\pi$
$60$	0	0
$61$	11.0895i	1.41986i	0.704271	+	0.709932i	$0.251274\pi$
$61$	11.0895i	1.41986i	−0.704271	+	0.709932i	$0.748726\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0.599777i	0.0743932i
$66$	0	0
$67$	2.87422	0.351142	0.175571	−	0.984467i	$-0.443823\pi$
$67$	2.87422	0.351142	0.175571	+	0.984467i	$0.443823\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	0.336875i	0.0399797i	0.999800	+	0.0199898i	$0.00636339\pi$
$71$	0.336875i	0.0399797i	−0.999800	+	0.0199898i	$0.993637\pi$
$72$	0	0
$73$	3.08479i	0.361047i	0.983571	+	0.180524i	$0.0577792\pi$
$73$	3.08479i	0.361047i	−0.983571	+	0.180524i	$0.942221\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−13.6089	−1.53112	−0.765562	−	0.643362i	$-0.777539\pi$
$79$	−13.6089	−1.53112	−0.765562	+	0.643362i	$0.777539\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−16.2901	−1.78807	−0.894036	−	0.447996i	$-0.852138\pi$
$83$	−16.2901	−1.78807	−0.894036	+	0.447996i	$0.852138\pi$
$84$	0	0
$85$	−1.69803	−0.184178
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−10.6964	−1.13382	−0.566910	−	0.823779i	$-0.691861\pi$
$89$	−10.6964	−1.13382	−0.566910	+	0.823779i	$0.691861\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	7.78350i	0.798570i
$96$	0	0
$97$	2.18092i	0.221439i	0.993852	+	0.110719i	$0.0353155\pi$
$97$	2.18092i	0.221439i	−0.993852	+	0.110719i	$0.964685\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	7.70835	0.767009	0.383505	−	0.923539i	$-0.374717\pi$
$101$	7.70835	0.767009	0.383505	+	0.923539i	$0.374717\pi$
$102$	0	0
$103$	0.734562i	0.0723785i	0.999345	+	0.0361893i	$0.0115219\pi$
$103$	0.734562i	0.0723785i	−0.999345	+	0.0361893i	$0.988478\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	19.9774i	1.93129i	0.259867	+	0.965645i	$0.416321\pi$
$107$	19.9774i	1.93129i	−0.259867	+	0.965645i	$0.583679\pi$
$108$	0	0
$109$	−0.443025	−0.0424341	−0.0212170	−	0.999775i	$-0.506754\pi$
$109$	−0.443025	−0.0424341	−0.0212170	+	0.999775i	$0.506754\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	14.6235i	1.37566i	0.725871	+	0.687830i	$0.241437\pi$
$113$	14.6235i	1.37566i	−0.725871	+	0.687830i	$0.758563\pi$
$114$	0	0
$115$	3.31319i	0.308957i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−1.48673	−0.135157
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	0.961155	0.0852887	0.0426444	−	0.999090i	$-0.486422\pi$
$127$	0.961155	0.0852887	0.0426444	+	0.999090i	$0.486422\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	15.9947	1.39746	0.698731	−	0.715384i	$-0.253748\pi$
$131$	15.9947	1.39746	0.698731	+	0.715384i	$0.253748\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	9.59969i	0.820157i	0.912050	+	0.410078i	$0.134499\pi$
$137$	9.59969i	0.820157i	−0.912050	+	0.410078i	$0.865501\pi$
$138$	0	0
$139$	− 9.13902i	− 0.775162i	−0.921836	−	0.387581i	$-0.873311\pi$
$139$	− 9.13902i	− 0.775162i	0.921836	−	0.387581i	$-0.126689\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	2.11941	0.177234
$144$	0	0
$145$	− 0.456069i	− 0.0378745i
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 20.2697i	− 1.66056i	−0.557347	−	0.830279i	$-0.688181\pi$
$149$	− 20.2697i	− 1.66056i	0.557347	−	0.830279i	$-0.311819\pi$
$150$	0	0
$151$	−8.01031	−0.651870	−0.325935	−	0.945392i	$-0.605679\pi$
$151$	−8.01031	−0.651870	−0.325935	+	0.945392i	$0.605679\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 1.05585i	− 0.0848076i
$156$	0	0
$157$	− 9.33186i	− 0.744763i	−0.928080	−	0.372382i	$-0.878541\pi$
$157$	− 9.33186i	− 0.744763i	0.928080	−	0.372382i	$-0.121459\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−13.6537	−1.06944	−0.534720	−	0.845029i	$-0.679583\pi$
$163$	−13.6537	−1.06944	−0.534720	+	0.845029i	$0.679583\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	9.94429	0.769512	0.384756	−	0.923018i	$-0.374286\pi$
$167$	9.94429	0.769512	0.384756	+	0.923018i	$0.374286\pi$
$168$	0	0
$169$	12.6403	0.972328
$170$	0	0
$171$	0	0
$172$	0	0
$173$	9.93099	0.755039	0.377520	−	0.926002i	$-0.376777\pi$
$173$	9.93099	0.755039	0.377520	+	0.926002i	$0.376777\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	18.8485i	1.40880i	0.709803	+	0.704400i	$0.248784\pi$
$179$	18.8485i	1.40880i	−0.709803	+	0.704400i	$0.751216\pi$
$180$	0	0
$181$	20.0424i	1.48974i	0.667211	+	0.744869i	$0.267488\pi$
$181$	20.0424i	1.48974i	−0.667211	+	0.744869i	$0.732512\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−0.386249	−0.0283976
$186$	0	0
$187$	6.00027i	0.438783i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	9.22725i	0.667660i	0.942633	+	0.333830i	$0.108341\pi$
$191$	9.22725i	0.667660i	−0.942633	+	0.333830i	$0.891659\pi$
$192$	0	0
$193$	−26.0346	−1.87401	−0.937007	−	0.349310i	$-0.886416\pi$
$193$	−26.0346	−1.87401	−0.937007	+	0.349310i	$0.886416\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 15.6156i	− 1.11256i	−0.830994	−	0.556282i	$-0.812228\pi$
$197$	− 15.6156i	− 1.11256i	0.830994	−	0.556282i	$-0.187772\pi$
$198$	0	0
$199$	6.11946i	0.433797i	0.976194	+	0.216899i	$0.0695941\pi$
$199$	6.11946i	0.433797i	−0.976194	+	0.216899i	$0.930406\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−0.478149	−0.0333954
$206$	0	0
$207$	0	0
$208$	0	0
$209$	27.5042	1.90251
$210$	0	0
$211$	26.7224	1.83964	0.919822	−	0.392337i	$-0.128333\pi$
$211$	26.7224	1.83964	0.919822	+	0.392337i	$0.128333\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−4.21237	−0.287281
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 1.01844i	− 0.0685078i
$222$	0	0
$223$	20.3668i	1.36386i	0.731416	+	0.681932i	$0.238860\pi$
$223$	20.3668i	1.36386i	−0.731416	+	0.681932i	$0.761140\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	19.9390	1.32340	0.661698	−	0.749770i	$-0.269836\pi$
$227$	19.9390	1.32340	0.661698	+	0.749770i	$0.269836\pi$
$228$	0	0
$229$	18.9389i	1.25152i	0.780017	+	0.625758i	$0.215210\pi$
$229$	18.9389i	1.25152i	−0.780017	+	0.625758i	$0.784790\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	26.5531i	1.73955i	0.493448	+	0.869775i	$0.335736\pi$
$233$	26.5531i	1.73955i	−0.493448	+	0.869775i	$0.664264\pi$
$234$	0	0
$235$	8.55804	0.558265
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 6.76076i	− 0.437317i	−0.975801	−	0.218659i	$-0.929832\pi$
$239$	− 6.76076i	− 0.437317i	0.975801	−	0.218659i	$-0.0701681\pi$
$240$	0	0
$241$	− 12.0844i	− 0.778423i	−0.921148	−	0.389211i	$-0.872748\pi$
$241$	− 12.0844i	− 0.778423i	0.921148	−	0.389211i	$-0.127252\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−4.66837	−0.297041
$248$	0	0
$249$	0	0
$250$	0	0
$251$	17.3410	1.09455	0.547276	−	0.836952i	$-0.315665\pi$
$251$	17.3410	1.09455	0.547276	+	0.836952i	$0.315665\pi$
$252$	0	0
$253$	11.7077	0.736055
$254$	0	0
$255$	0	0
$256$	0	0
$257$	8.34255	0.520394	0.260197	−	0.965556i	$-0.416213\pi$
$257$	8.34255	0.520394	0.260197	+	0.965556i	$0.416213\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	13.4187i	0.827433i	0.910406	+	0.413717i	$0.135770\pi$
$263$	13.4187i	0.827433i	−0.910406	+	0.413717i	$0.864230\pi$
$264$	0	0
$265$	− 3.98973i	− 0.245087i
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−7.96289	−0.485506	−0.242753	−	0.970088i	$-0.578050\pi$
$269$	−7.96289	−0.485506	−0.242753	+	0.970088i	$0.578050\pi$
$270$	0	0
$271$	7.46942i	0.453735i	0.973926	+	0.226867i	$0.0728484\pi$
$271$	7.46942i	0.453735i	−0.973926	+	0.226867i	$0.927152\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 3.53366i	− 0.213087i
$276$	0	0
$277$	−31.2617	−1.87833	−0.939166	−	0.343463i	$-0.888400\pi$
$277$	−31.2617	−1.87833	−0.939166	+	0.343463i	$0.888400\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 23.4956i	− 1.40163i	−0.713343	−	0.700815i	$-0.752820\pi$
$281$	− 23.4956i	− 1.40163i	0.713343	−	0.700815i	$-0.247180\pi$
$282$	0	0
$283$	18.5127i	1.10046i	0.835012	+	0.550232i	$0.185461\pi$
$283$	18.5127i	1.10046i	−0.835012	+	0.550232i	$0.814539\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−14.1167	−0.830393
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−14.7388	−0.861048	−0.430524	−	0.902579i	$-0.641671\pi$
$293$	−14.7388	−0.861048	−0.430524	+	0.902579i	$0.641671\pi$
$294$	0	0
$295$	−2.60613	−0.151735
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−1.98718	−0.114921
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	11.0895i	0.634982i
$306$	0	0
$307$	17.6583i	1.00781i	0.863758	+	0.503906i	$0.168104\pi$
$307$	17.6583i	1.00781i	−0.863758	+	0.503906i	$0.831896\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	2.87563	0.163062	0.0815310	−	0.996671i	$-0.474019\pi$
$311$	2.87563	0.163062	0.0815310	+	0.996671i	$0.474019\pi$
$312$	0	0
$313$	− 20.0386i	− 1.13265i	−0.824183	−	0.566324i	$-0.808365\pi$
$313$	− 20.0386i	− 1.13265i	0.824183	−	0.566324i	$-0.191635\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 20.5910i	− 1.15650i	−0.815858	−	0.578252i	$-0.803735\pi$
$317$	− 20.5910i	− 1.15650i	0.815858	−	0.578252i	$-0.196265\pi$
$318$	0	0
$319$	−1.61159	−0.0902317
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 13.2166i	− 0.735394i
$324$	0	0
$325$	0.599777i	0.0332697i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−22.8995	−1.25867	−0.629335	−	0.777134i	$-0.716673\pi$
$331$	−22.8995	−1.25867	−0.629335	+	0.777134i	$0.716673\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	2.87422	0.157035
$336$	0	0
$337$	17.7305	0.965840	0.482920	−	0.875664i	$-0.339576\pi$
$337$	17.7305	0.965840	0.482920	+	0.875664i	$0.339576\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−3.73100	−0.202045
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 18.1899i	− 0.976486i	−0.872708	−	0.488243i	$-0.837638\pi$
$347$	− 18.1899i	− 0.976486i	0.872708	−	0.488243i	$-0.162362\pi$
$348$	0	0
$349$	− 4.44213i	− 0.237782i	−0.992907	−	0.118891i	$-0.962066\pi$
$349$	− 4.44213i	− 0.237782i	0.992907	−	0.118891i	$-0.0379339\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−5.18270	−0.275848	−0.137924	−	0.990443i	$-0.544043\pi$
$353$	−5.18270	−0.275848	−0.137924	+	0.990443i	$0.544043\pi$
$354$	0	0
$355$	0.336875i	0.0178795i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	18.2319i	0.962244i	0.876654	+	0.481122i	$0.159771\pi$
$359$	18.2319i	0.962244i	−0.876654	+	0.481122i	$0.840229\pi$
$360$	0	0
$361$	−41.5829	−2.18857
$362$	0	0
$363$	0	0
$364$	0	0
$365$	3.08479i	0.161465i
$366$	0	0
$367$	20.0597i	1.04711i	0.851993	+	0.523553i	$0.175394\pi$
$367$	20.0597i	1.04711i	−0.851993	+	0.523553i	$0.824606\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	26.5889	1.37672	0.688361	−	0.725368i	$-0.258331\pi$
$373$	26.5889	1.37672	0.688361	+	0.725368i	$0.258331\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0.273540	0.0140880
$378$	0	0
$379$	−18.3623	−0.943210	−0.471605	−	0.881810i	$-0.656325\pi$
$379$	−18.3623	−0.943210	−0.471605	+	0.881810i	$0.656325\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	36.8099	1.88090	0.940449	−	0.339935i	$-0.110405\pi$
$383$	36.8099	1.88090	0.940449	+	0.339935i	$0.110405\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	31.6392i	1.60417i	0.597211	+	0.802084i	$0.296276\pi$
$389$	31.6392i	1.60417i	−0.597211	+	0.802084i	$0.703724\pi$
$390$	0	0
$391$	− 5.62591i	− 0.284515i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−13.6089	−0.684740
$396$	0	0
$397$	26.9858i	1.35438i	0.735808	+	0.677190i	$0.236802\pi$
$397$	26.9858i	1.35438i	−0.735808	+	0.677190i	$0.763198\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	9.80307i	0.489542i	0.969581	+	0.244771i	$0.0787127\pi$
$401$	9.80307i	0.489542i	−0.969581	+	0.244771i	$0.921287\pi$
$402$	0	0
$403$	0.633273	0.0315456
$404$	0	0
$405$	0	0
$406$	0	0
$407$	1.36487i	0.0676541i
$408$	0	0
$409$	8.57996i	0.424252i	0.977242	+	0.212126i	$0.0680387\pi$
$409$	8.57996i	0.424252i	−0.977242	+	0.212126i	$0.931961\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−16.2901	−0.799650
$416$	0	0
$417$	0	0
$418$	0	0
$419$	21.7882	1.06443	0.532213	−	0.846611i	$-0.321361\pi$
$419$	21.7882	1.06443	0.532213	+	0.846611i	$0.321361\pi$
$420$	0	0
$421$	−24.2280	−1.18080	−0.590400	−	0.807111i	$-0.701030\pi$
$421$	−24.2280	−1.18080	−0.590400	+	0.807111i	$0.701030\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−1.69803	−0.0823668
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 23.8330i	− 1.14800i	−0.818857	−	0.573998i	$-0.805392\pi$
$431$	− 23.8330i	− 1.14800i	0.818857	−	0.573998i	$-0.194608\pi$
$432$	0	0
$433$	− 1.39596i	− 0.0670855i	−0.999437	−	0.0335427i	$-0.989321\pi$
$433$	− 1.39596i	− 0.0670855i	0.999437	−	0.0335427i	$-0.0106790\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−25.7882	−1.23362
$438$	0	0
$439$	11.8879i	0.567381i	0.958916	+	0.283690i	$0.0915588\pi$
$439$	11.8879i	0.567381i	−0.958916	+	0.283690i	$0.908441\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	26.7403i	1.27047i	0.772319	+	0.635235i	$0.219097\pi$
$443$	26.7403i	1.27047i	−0.772319	+	0.635235i	$0.780903\pi$
$444$	0	0
$445$	−10.6964	−0.507060
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 32.8394i	− 1.54979i	−0.632091	−	0.774895i	$-0.717803\pi$
$449$	− 32.8394i	− 1.54979i	0.632091	−	0.774895i	$-0.282197\pi$
$450$	0	0
$451$	1.68961i	0.0795608i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−35.0454	−1.63935	−0.819677	−	0.572826i	$-0.805847\pi$
$457$	−35.0454	−1.63935	−0.819677	+	0.572826i	$0.805847\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−8.90991	−0.414976	−0.207488	−	0.978238i	$-0.566529\pi$
$461$	−8.90991	−0.414976	−0.207488	+	0.978238i	$0.566529\pi$
$462$	0	0
$463$	−10.2000	−0.474036	−0.237018	−	0.971505i	$-0.576170\pi$
$463$	−10.2000	−0.474036	−0.237018	+	0.971505i	$0.576170\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−16.8383	−0.779185	−0.389592	−	0.920987i	$-0.627384\pi$
$467$	−16.8383	−0.779185	−0.389592	+	0.920987i	$0.627384\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	14.8851i	0.684417i
$474$	0	0
$475$	7.78350i	0.357131i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	18.3703	0.839360	0.419680	−	0.907672i	$-0.362142\pi$
$479$	18.3703	0.839360	0.419680	+	0.907672i	$0.362142\pi$
$480$	0	0
$481$	− 0.231663i	− 0.0105629i
$482$	0	0
$483$	0	0
$484$	0	0
$485$	2.18092i	0.0990304i
$486$	0	0
$487$	−11.6639	−0.528540	−0.264270	−	0.964449i	$-0.585131\pi$
$487$	−11.6639	−0.528540	−0.264270	+	0.964449i	$0.585131\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	21.4694i	0.968902i	0.874818	+	0.484451i	$0.160981\pi$
$491$	21.4694i	0.968902i	−0.874818	+	0.484451i	$0.839019\pi$
$492$	0	0
$493$	0.774420i	0.0348781i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−14.8218	−0.663515	−0.331757	−	0.943365i	$-0.607642\pi$
$499$	−14.8218	−0.663515	−0.331757	+	0.943365i	$0.607642\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−28.9563	−1.29110	−0.645548	−	0.763719i	$-0.723371\pi$
$503$	−28.9563	−1.29110	−0.645548	+	0.763719i	$0.723371\pi$
$504$	0	0
$505$	7.70835	0.343017
$506$	0	0
$507$	0	0
$508$	0	0
$509$	29.1793	1.29335	0.646675	−	0.762766i	$-0.276159\pi$
$509$	29.1793	1.29335	0.646675	+	0.762766i	$0.276159\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0.734562i	0.0323687i
$516$	0	0
$517$	− 30.2412i	− 1.33001i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	30.6208	1.34152	0.670761	−	0.741674i	$-0.265968\pi$
$521$	30.6208	1.34152	0.670761	+	0.741674i	$0.265968\pi$
$522$	0	0
$523$	12.0824i	0.528324i	0.964478	+	0.264162i	$0.0850954\pi$
$523$	12.0824i	0.528324i	−0.964478	+	0.264162i	$0.914905\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1.79286i	0.0780983i
$528$	0	0
$529$	12.0228	0.522729
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 0.286783i	− 0.0124219i
$534$	0	0
$535$	19.9774i	0.863699i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	5.54879	0.238561	0.119281	−	0.992861i	$-0.461941\pi$
$541$	5.54879	0.238561	0.119281	+	0.992861i	$0.461941\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−0.443025	−0.0189771
$546$	0	0
$547$	4.60440	0.196870	0.0984349	−	0.995143i	$-0.468616\pi$
$547$	4.60440	0.196870	0.0984349	+	0.995143i	$0.468616\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	3.54981	0.151227
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 15.7009i	− 0.665269i	−0.943056	−	0.332634i	$-0.892063\pi$
$557$	− 15.7009i	− 0.665269i	0.943056	−	0.332634i	$-0.107937\pi$
$558$	0	0
$559$	− 2.52649i	− 0.106859i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−32.6851	−1.37751	−0.688756	−	0.724993i	$-0.741843\pi$
$563$	−32.6851	−1.37751	−0.688756	+	0.724993i	$0.741843\pi$
$564$	0	0
$565$	14.6235i	0.615214i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	23.1594i	0.970892i	0.874267	+	0.485446i	$0.161343\pi$
$569$	23.1594i	0.970892i	−0.874267	+	0.485446i	$0.838657\pi$
$570$	0	0
$571$	−3.90494	−0.163417	−0.0817083	−	0.996656i	$-0.526038\pi$
$571$	−3.90494	−0.163417	−0.0817083	+	0.996656i	$0.526038\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	3.31319i	0.138170i
$576$	0	0
$577$	− 7.67922i	− 0.319690i	−0.987142	−	0.159845i	$-0.948901\pi$
$577$	− 7.67922i	− 0.319690i	0.987142	−	0.159845i	$-0.0510995\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−14.0983	−0.583893
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−0.375586	−0.0155021	−0.00775105	−	0.999970i	$-0.502467\pi$
$587$	−0.375586	−0.0155021	−0.00775105	+	0.999970i	$0.502467\pi$
$588$	0	0
$589$	8.21818	0.338624
$590$	0	0
$591$	0	0
$592$	0	0
$593$	23.0404	0.946157	0.473078	−	0.881020i	$-0.343143\pi$
$593$	23.0404	0.946157	0.473078	+	0.881020i	$0.343143\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	8.61675i	0.352071i	0.984384	+	0.176036i	$0.0563274\pi$
$599$	8.61675i	0.352071i	−0.984384	+	0.176036i	$0.943673\pi$
$600$	0	0
$601$	− 27.4954i	− 1.12156i	−0.827965	−	0.560780i	$-0.810501\pi$
$601$	− 27.4954i	− 1.12156i	0.827965	−	0.560780i	$-0.189499\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−1.48673	−0.0604440
$606$	0	0
$607$	35.9277i	1.45826i	0.684375	+	0.729130i	$0.260075\pi$
$607$	35.9277i	1.45826i	−0.684375	+	0.729130i	$0.739925\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	5.13292i	0.207656i
$612$	0	0
$613$	41.8542	1.69048	0.845238	−	0.534389i	$-0.179458\pi$
$613$	41.8542	1.69048	0.845238	+	0.534389i	$0.179458\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	27.6411i	1.11279i	0.830918	+	0.556395i	$0.187816\pi$
$617$	27.6411i	1.11279i	−0.830918	+	0.556395i	$0.812184\pi$
$618$	0	0
$619$	− 13.2580i	− 0.532885i	−0.963851	−	0.266442i	$-0.914152\pi$
$619$	− 13.2580i	− 0.532885i	0.963851	−	0.266442i	$-0.0858482\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0.655863	0.0261510
$630$	0	0
$631$	−12.2455	−0.487485	−0.243742	−	0.969840i	$-0.578375\pi$
$631$	−12.2455	−0.487485	−0.243742	+	0.969840i	$0.578375\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0.961155	0.0381423
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 12.3916i	− 0.489441i	−0.969594	−	0.244720i	$-0.921304\pi$
$641$	− 12.3916i	− 0.489441i	0.969594	−	0.244720i	$-0.0786962\pi$
$642$	0	0
$643$	42.8847i	1.69121i	0.533811	+	0.845604i	$0.320759\pi$
$643$	42.8847i	1.69121i	−0.533811	+	0.845604i	$0.679241\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	26.8700	1.05637	0.528185	−	0.849130i	$-0.322873\pi$
$647$	26.8700	1.05637	0.528185	+	0.849130i	$0.322873\pi$
$648$	0	0
$649$	9.20918i	0.361492i
$650$	0	0
$651$	0	0
$652$	0	0
$653$	0.622092i	0.0243444i	0.999926	+	0.0121722i	$0.00387462\pi$
$653$	0.622092i	0.0243444i	−0.999926	+	0.0121722i	$0.996125\pi$
$654$	0	0
$655$	15.9947	0.624964
$656$	0	0
$657$	0	0
$658$	0	0
$659$	17.9996i	0.701167i	0.936532	+	0.350583i	$0.114017\pi$
$659$	17.9996i	0.701167i	−0.936532	+	0.350583i	$0.885983\pi$
$660$	0	0
$661$	− 30.0737i	− 1.16973i	−0.811130	−	0.584866i	$-0.801147\pi$
$661$	− 30.0737i	− 1.16973i	0.811130	−	0.584866i	$-0.198853\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	1.51104	0.0585078
$668$	0	0
$669$	0	0
$670$	0	0
$671$	39.1864	1.51278
$672$	0	0
$673$	39.4637	1.52121	0.760606	−	0.649213i	$-0.224902\pi$
$673$	39.4637	1.52121	0.760606	+	0.649213i	$0.224902\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	19.6775	0.756267	0.378134	−	0.925751i	$-0.376566\pi$
$677$	19.6775	0.756267	0.378134	+	0.925751i	$0.376566\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	18.0223i	0.689603i	0.938676	+	0.344801i	$0.112054\pi$
$683$	18.0223i	0.689603i	−0.938676	+	0.344801i	$0.887946\pi$
$684$	0	0
$685$	9.59969i	0.366785i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	2.39295	0.0911640
$690$	0	0
$691$	27.2057i	1.03496i	0.855697	+	0.517478i	$0.173129\pi$
$691$	27.2057i	1.03496i	−0.855697	+	0.517478i	$0.826871\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 9.13902i	− 0.346663i
$696$	0	0
$697$	0.811913	0.0307534
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 33.1556i	− 1.25227i	−0.779715	−	0.626134i	$-0.784636\pi$
$701$	− 33.1556i	− 1.25227i	0.779715	−	0.626134i	$-0.215364\pi$
$702$	0	0
$703$	− 3.00637i	− 0.113387i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	1.97919	0.0743301	0.0371650	−	0.999309i	$-0.488167\pi$
$709$	1.97919	0.0743301	0.0371650	+	0.999309i	$0.488167\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	3.49822	0.131009
$714$	0	0
$715$	2.11941	0.0792613
$716$	0	0
$717$	0	0
$718$	0	0
$719$	30.3589	1.13220	0.566098	−	0.824338i	$-0.308452\pi$
$719$	30.3589	1.13220	0.566098	+	0.824338i	$0.308452\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 0.456069i	− 0.0169380i
$726$	0	0
$727$	17.5883i	0.652313i	0.945316	+	0.326156i	$0.105754\pi$
$727$	17.5883i	0.652313i	−0.945316	+	0.326156i	$0.894246\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	7.15275	0.264554
$732$	0	0
$733$	36.9445i	1.36457i	0.731084	+	0.682287i	$0.239015\pi$
$733$	36.9445i	1.36457i	−0.731084	+	0.682287i	$0.760985\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 10.1565i	− 0.374119i
$738$	0	0
$739$	−5.05915	−0.186104	−0.0930520	−	0.995661i	$-0.529662\pi$
$739$	−5.05915	−0.186104	−0.0930520	+	0.995661i	$0.529662\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	6.19725i	0.227355i	0.993518	+	0.113678i	$0.0362631\pi$
$743$	6.19725i	0.227355i	−0.993518	+	0.113678i	$0.963737\pi$
$744$	0	0
$745$	− 20.2697i	− 0.742624i
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−37.8201	−1.38007	−0.690037	−	0.723774i	$-0.742406\pi$
$751$	−37.8201	−1.38007	−0.690037	+	0.723774i	$0.742406\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−8.01031	−0.291525
$756$	0	0
$757$	−13.7436	−0.499519	−0.249760	−	0.968308i	$-0.580352\pi$
$757$	−13.7436	−0.499519	−0.249760	+	0.968308i	$0.580352\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	2.39947	0.0869807	0.0434904	−	0.999054i	$-0.486152\pi$
$761$	2.39947	0.0869807	0.0434904	+	0.999054i	$0.486152\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 1.56310i	− 0.0564403i
$768$	0	0
$769$	− 9.42300i	− 0.339802i	−0.985461	−	0.169901i	$-0.945655\pi$
$769$	− 9.42300i	− 0.339802i	0.985461	−	0.169901i	$-0.0543448\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−22.5676	−0.811700	−0.405850	−	0.913940i	$-0.633025\pi$
$773$	−22.5676	−0.811700	−0.405850	+	0.913940i	$0.633025\pi$
$774$	0	0
$775$	− 1.05585i	− 0.0379271i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 3.72167i	− 0.133343i
$780$	0	0
$781$	1.19040	0.0425959
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 9.33186i	− 0.333068i
$786$	0	0
$787$	− 25.9146i	− 0.923757i	−0.886943	−	0.461878i	$-0.847176\pi$
$787$	− 25.9146i	− 0.923757i	0.886943	−	0.461878i	$-0.152824\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−6.65122	−0.236192
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−6.94772	−0.246101	−0.123050	−	0.992400i	$-0.539268\pi$
$797$	−6.94772	−0.246101	−0.123050	+	0.992400i	$0.539268\pi$
$798$	0	0
$799$	−14.5318	−0.514100
$800$	0	0
$801$	0	0
$802$	0	0
$803$	10.9006	0.384673
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 33.3973i	− 1.17419i	−0.809520	−	0.587093i	$-0.800272\pi$
$809$	− 33.3973i	− 1.17419i	0.809520	−	0.587093i	$-0.199728\pi$
$810$	0	0
$811$	− 28.9701i	− 1.01728i	−0.860979	−	0.508640i	$-0.830149\pi$
$811$	− 28.9701i	− 1.01728i	0.860979	−	0.508640i	$-0.169851\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−13.6537	−0.478268
$816$	0	0
$817$	− 32.7870i	− 1.14707i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	2.60759i	0.0910056i	0.998964	+	0.0455028i	$0.0144890\pi$
$821$	2.60759i	0.0910056i	−0.998964	+	0.0455028i	$0.985511\pi$
$822$	0	0
$823$	8.45695	0.294791	0.147395	−	0.989078i	$-0.452911\pi$
$823$	8.45695	0.294791	0.147395	+	0.989078i	$0.452911\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 13.3362i	− 0.463744i	−0.972746	−	0.231872i	$-0.925515\pi$
$827$	− 13.3362i	− 0.463744i	0.972746	−	0.231872i	$-0.0744851\pi$
$828$	0	0
$829$	11.6374i	0.404183i	0.979367	+	0.202091i	$0.0647738\pi$
$829$	11.6374i	0.404183i	−0.979367	+	0.202091i	$0.935226\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	9.94429	0.344136
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−19.9335	−0.688181	−0.344090	−	0.938937i	$-0.611813\pi$
$839$	−19.9335	−0.688181	−0.344090	+	0.938937i	$0.611813\pi$
$840$	0	0
$841$	28.7920	0.992828
$842$	0	0
$843$	0	0
$844$	0	0
$845$	12.6403	0.434838
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 1.27972i	− 0.0438681i
$852$	0	0
$853$	2.08233i	0.0712975i	0.999364	+	0.0356487i	$0.0113498\pi$
$853$	2.08233i	0.0712975i	−0.999364	+	0.0356487i	$0.988650\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	46.5482	1.59006	0.795028	−	0.606573i	$-0.207456\pi$
$857$	46.5482	1.59006	0.795028	+	0.606573i	$0.207456\pi$
$858$	0	0
$859$	− 5.34497i	− 0.182368i	−0.995834	−	0.0911840i	$-0.970935\pi$
$859$	− 5.34497i	− 0.182368i	0.995834	−	0.0911840i	$-0.0290651\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 13.3070i	− 0.452976i	−0.974014	−	0.226488i	$-0.927276\pi$
$863$	− 13.3070i	− 0.452976i	0.974014	−	0.226488i	$-0.0727243\pi$
$864$	0	0
$865$	9.93099	0.337664
$866$	0	0
$867$	0	0
$868$	0	0
$869$	48.0893i	1.63132i
$870$	0	0
$871$	1.72389i	0.0584118i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−19.7845	−0.668075	−0.334037	−	0.942560i	$-0.608411\pi$
$877$	−19.7845	−0.668075	−0.334037	+	0.942560i	$0.608411\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−34.3203	−1.15628	−0.578141	−	0.815937i	$-0.696222\pi$
$881$	−34.3203	−1.15628	−0.578141	+	0.815937i	$0.696222\pi$
$882$	0	0
$883$	28.1109	0.946006	0.473003	−	0.881061i	$-0.343170\pi$
$883$	28.1109	0.946006	0.473003	+	0.881061i	$0.343170\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−41.7652	−1.40234	−0.701169	−	0.712995i	$-0.747338\pi$
$887$	−41.7652	−1.40234	−0.701169	+	0.712995i	$0.747338\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	66.6115i	2.22907i
$894$	0	0
$895$	18.8485i	0.630035i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−0.481538	−0.0160602
$900$	0	0
$901$	6.77469i	0.225698i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	20.0424i	0.666231i
$906$	0	0
$907$	−7.92101	−0.263013	−0.131506	−	0.991315i	$-0.541981\pi$
$907$	−7.92101	−0.263013	−0.131506	+	0.991315i	$0.541981\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	49.9301i	1.65426i	0.562012	+	0.827129i	$0.310027\pi$
$911$	49.9301i	1.65426i	−0.562012	+	0.827129i	$0.689973\pi$
$912$	0	0
$913$	57.5636i	1.90508i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−0.264145	−0.00871335	−0.00435668	−	0.999991i	$-0.501387\pi$
$919$	−0.264145	−0.00871335	−0.00435668	+	0.999991i	$0.501387\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−0.202050	−0.00665055
$924$	0	0
$925$	−0.386249	−0.0126998
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−56.8692	−1.86582	−0.932910	−	0.360110i	$-0.882739\pi$
$929$	−56.8692	−1.86582	−0.932910	+	0.360110i	$0.882739\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	6.00027i	0.196230i
$936$	0	0
$937$	− 37.3673i	− 1.22074i	−0.792118	−	0.610369i	$-0.791021\pi$
$937$	− 37.3673i	− 1.22074i	0.792118	−	0.610369i	$-0.208979\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	47.7334	1.55607	0.778033	−	0.628224i	$-0.216218\pi$
$941$	47.7334	1.55607	0.778033	+	0.628224i	$0.216218\pi$
$942$	0	0
$943$	− 1.58420i	− 0.0515886i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	33.1741i	1.07801i	0.842302	+	0.539006i	$0.181200\pi$
$947$	33.1741i	1.07801i	−0.842302	+	0.539006i	$0.818800\pi$
$948$	0	0
$949$	−1.85019	−0.0600596
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 28.1591i	− 0.912164i	−0.889938	−	0.456082i	$-0.849252\pi$
$953$	− 28.1591i	− 0.912164i	0.889938	−	0.456082i	$-0.150748\pi$
$954$	0	0
$955$	9.22725i	0.298587i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	29.8852	0.964038
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−26.0346	−0.838085
$966$	0	0
$967$	−53.8983	−1.73325	−0.866626	−	0.498959i	$-0.833716\pi$
$967$	−53.8983	−1.73325	−0.866626	+	0.498959i	$0.833716\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−41.0531	−1.31746	−0.658729	−	0.752380i	$-0.728906\pi$
$971$	−41.0531	−1.31746	−0.658729	+	0.752380i	$0.728906\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	28.9189i	0.925197i	0.886568	+	0.462598i	$0.153083\pi$
$977$	28.9189i	0.925197i	−0.886568	+	0.462598i	$0.846917\pi$
$978$	0	0
$979$	37.7976i	1.20802i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−36.2752	−1.15700	−0.578499	−	0.815683i	$-0.696361\pi$
$983$	−36.2752	−1.15700	−0.578499	+	0.815683i	$0.696361\pi$
$984$	0	0
$985$	− 15.6156i	− 0.497554i
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 13.9564i	− 0.443787i
$990$	0	0
$991$	−39.2201	−1.24587	−0.622934	−	0.782274i	$-0.714060\pi$
$991$	−39.2201	−1.24587	−0.622934	+	0.782274i	$0.714060\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	6.11946i	0.194000i
$996$	0	0
$997$	− 12.4136i	− 0.393142i	−0.980490	−	0.196571i	$-0.937019\pi$
$997$	− 12.4136i	− 0.393142i	0.980490	−	0.196571i	$-0.0629806\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8820.2.d.b.881.2		12
3.2	odd	2		8820.2.d.a.881.11		12
7.4	even	3		1260.2.cg.a.341.3	✓	12
7.5	odd	6		1260.2.cg.b.521.3	yes	12
7.6	odd	2		8820.2.d.a.881.2		12
21.5	even	6		1260.2.cg.a.521.3	yes	12
21.11	odd	6		1260.2.cg.b.341.3	yes	12
21.20	even	2	inner	8820.2.d.b.881.11		12
35.4	even	6		6300.2.ch.c.1601.4		12
35.12	even	12		6300.2.dd.b.4049.1		24
35.18	odd	12		6300.2.dd.c.1349.1		24
35.19	odd	6		6300.2.ch.b.4301.4		12
35.32	odd	12		6300.2.dd.c.1349.12		24
35.33	even	12		6300.2.dd.b.4049.12		24
105.32	even	12		6300.2.dd.b.1349.12		24
105.47	odd	12		6300.2.dd.c.4049.1		24
105.53	even	12		6300.2.dd.b.1349.1		24
105.68	odd	12		6300.2.dd.c.4049.12		24
105.74	odd	6		6300.2.ch.b.1601.4		12
105.89	even	6		6300.2.ch.c.4301.4		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1260.2.cg.a.341.3	✓	12	7.4	even	3
1260.2.cg.a.521.3	yes	12	21.5	even	6
1260.2.cg.b.341.3	yes	12	21.11	odd	6
1260.2.cg.b.521.3	yes	12	7.5	odd	6
6300.2.ch.b.1601.4		12	105.74	odd	6
6300.2.ch.b.4301.4		12	35.19	odd	6
6300.2.ch.c.1601.4		12	35.4	even	6
6300.2.ch.c.4301.4		12	105.89	even	6
6300.2.dd.b.1349.1		24	105.53	even	12
6300.2.dd.b.1349.12		24	105.32	even	12
6300.2.dd.b.4049.1		24	35.12	even	12
6300.2.dd.b.4049.12		24	35.33	even	12
6300.2.dd.c.1349.1		24	35.18	odd	12
6300.2.dd.c.1349.12		24	35.32	odd	12
6300.2.dd.c.4049.1		24	105.47	odd	12
6300.2.dd.c.4049.12		24	105.68	odd	12
8820.2.d.a.881.2		12	7.6	odd	2
8820.2.d.a.881.11		12	3.2	odd	2
8820.2.d.b.881.2		12	1.1	even	1	trivial
8820.2.d.b.881.11		12	21.20	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:	\( N \)	\(=\)	\( 8820 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8820.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000 q^{5} -3.53366i q^{11} +0.599777i q^{13} -1.69803 q^{17} +7.78350i q^{19} +3.31319i q^{23} +1.00000 q^{25} -0.456069i q^{29} -1.05585i q^{31} -0.386249 q^{37} -0.478149 q^{41} -4.21237 q^{43} +8.55804 q^{47} -3.98973i q^{53} -3.53366i q^{55} -2.60613 q^{59} +11.0895i q^{61} +0.599777i q^{65} +2.87422 q^{67} +0.336875i q^{71} +3.08479i q^{73} -13.6089 q^{79} -16.2901 q^{83} -1.69803 q^{85} -10.6964 q^{89} +7.78350i q^{95} +2.18092i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 12 q^{5} + 12 q^{25} + 4 q^{37} - 8 q^{41} + 36 q^{43} + 32 q^{47} - 4 q^{67} - 28 q^{79} + 40 q^{83} + 40 q^{89}+O(q^{100}) \) 12 * q + 12 * q^5 + 12 * q^25 + 4 * q^37 - 8 * q^41 + 36 * q^43 + 32 * q^47 - 4 * q^67 - 28 * q^79 + 40 * q^83 + 40 * q^89

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