LMFDB - Embedded newform 8784.2.a.bg.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8784,2,Mod(1,8784)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8784, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("8784.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8784 = 2^{4} \cdot 3^{2} \cdot 61 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8784.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:	$70.1405931355$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 366)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.56155$ of defining polynomial
Character	$\chi$	$=$	8784.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.12311 q^{5} +3.56155 q^{7} +3.56155 q^{11} +5.56155 q^{13} -0.561553 q^{17} -4.00000 q^{19} -5.00000 q^{23} +12.0000 q^{25} +5.12311 q^{29} -4.00000 q^{31} +14.6847 q^{35} -3.68466 q^{37} -9.56155 q^{41} +2.56155 q^{43} +7.12311 q^{47} +5.68466 q^{49} -9.12311 q^{53} +14.6847 q^{55} -6.43845 q^{59} +1.00000 q^{61} +22.9309 q^{65} +3.31534 q^{67} +13.9309 q^{71} +16.3693 q^{73} +12.6847 q^{77} +12.9309 q^{79} -10.8078 q^{83} -2.31534 q^{85} -15.6847 q^{89} +19.8078 q^{91} -16.4924 q^{95} -3.43845 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 3 q^{7} + 3 q^{11} + 7 q^{13} + 3 q^{17} - 8 q^{19} - 10 q^{23} + 24 q^{25} + 2 q^{29} - 8 q^{31} + 17 q^{35} + 5 q^{37} - 15 q^{41} + q^{43} + 6 q^{47} - q^{49} - 10 q^{53} + 17 q^{55} - 17 q^{59}+ \cdots - 11 q^{97}+O(q^{100}) $ 2 * q + 3 * q^7 + 3 * q^11 + 7 * q^13 + 3 * q^17 - 8 * q^19 - 10 * q^23 + 24 * q^25 + 2 * q^29 - 8 * q^31 + 17 * q^35 + 5 * q^37 - 15 * q^41 + q^43 + 6 * q^47 - q^49 - 10 * q^53 + 17 * q^55 - 17 * q^59 + 2 * q^61 + 17 * q^65 + 19 * q^67 - q^71 + 8 * q^73 + 13 * q^77 - 3 * q^79 - q^83 - 17 * q^85 - 19 * q^89 + 19 * q^91 - 11 * q^97

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
$5$	4.12311	1.84391	0.921954	−	0.387298i	$-0.126592\pi$
$5$	4.12311	1.84391	0.921954	+	0.387298i	$0.126592\pi$
$6$	0	0
$7$	3.56155	1.34614	0.673070	−	0.739579i	$-0.264975\pi$
$7$	3.56155	1.34614	0.673070	+	0.739579i	$0.264975\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	3.56155	1.07385	0.536924	−	0.843630i	$-0.319586\pi$
$11$	3.56155	1.07385	0.536924	+	0.843630i	$0.319586\pi$
$12$	0	0
$13$	5.56155	1.54250	0.771249	−	0.636534i	$-0.219633\pi$
$13$	5.56155	1.54250	0.771249	+	0.636534i	$0.219633\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−0.561553	−0.136197	−0.0680983	−	0.997679i	$-0.521693\pi$
$17$	−0.561553	−0.136197	−0.0680983	+	0.997679i	$0.521693\pi$
$18$	0	0
$19$	−4.00000	−0.917663	−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000	−0.917663	−0.458831	+	0.888523i	$0.651732\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−5.00000	−1.04257	−0.521286	−	0.853382i	$-0.674548\pi$
$23$	−5.00000	−1.04257	−0.521286	+	0.853382i	$0.674548\pi$
$24$	0	0
$25$	12.0000	2.40000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	5.12311	0.951337	0.475668	−	0.879625i	$-0.342206\pi$
$29$	5.12311	0.951337	0.475668	+	0.879625i	$0.342206\pi$
$30$	0	0
$31$	−4.00000	−0.718421	−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	14.6847	2.48216
$36$	0	0
$37$	−3.68466	−0.605754	−0.302877	−	0.953030i	$-0.597947\pi$
$37$	−3.68466	−0.605754	−0.302877	+	0.953030i	$0.597947\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−9.56155	−1.49326	−0.746632	−	0.665237i	$-0.768330\pi$
$41$	−9.56155	−1.49326	−0.746632	+	0.665237i	$0.768330\pi$
$42$	0	0
$43$	2.56155	0.390633	0.195317	−	0.980740i	$-0.437427\pi$
$43$	2.56155	0.390633	0.195317	+	0.980740i	$0.437427\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	7.12311	1.03901	0.519506	−	0.854467i	$-0.326116\pi$
$47$	7.12311	1.03901	0.519506	+	0.854467i	$0.326116\pi$
$48$	0	0
$49$	5.68466	0.812094
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−9.12311	−1.25315	−0.626577	−	0.779359i	$-0.715545\pi$
$53$	−9.12311	−1.25315	−0.626577	+	0.779359i	$0.715545\pi$
$54$	0	0
$55$	14.6847	1.98008
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−6.43845	−0.838214	−0.419107	−	0.907937i	$-0.637657\pi$
$59$	−6.43845	−0.838214	−0.419107	+	0.907937i	$0.637657\pi$
$60$	0	0
$61$	1.00000	0.128037
$61$	1.00000	0.128037
$62$	0	0
$63$	0	0
$64$	0	0
$65$	22.9309	2.84422
$66$	0	0
$67$	3.31534	0.405033	0.202517	−	0.979279i	$-0.435088\pi$
$67$	3.31534	0.405033	0.202517	+	0.979279i	$0.435088\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	13.9309	1.65329	0.826645	−	0.562724i	$-0.190247\pi$
$71$	13.9309	1.65329	0.826645	+	0.562724i	$0.190247\pi$
$72$	0	0
$73$	16.3693	1.91588	0.957942	−	0.286963i	$-0.0926455\pi$
$73$	16.3693	1.91588	0.957942	+	0.286963i	$0.0926455\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	12.6847	1.44555
$78$	0	0
$79$	12.9309	1.45484	0.727418	−	0.686194i	$-0.240720\pi$
$79$	12.9309	1.45484	0.727418	+	0.686194i	$0.240720\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−10.8078	−1.18631	−0.593153	−	0.805090i	$-0.702117\pi$
$83$	−10.8078	−1.18631	−0.593153	+	0.805090i	$0.702117\pi$
$84$	0	0
$85$	−2.31534	−0.251134
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−15.6847	−1.66257	−0.831285	−	0.555846i	$-0.812394\pi$
$89$	−15.6847	−1.66257	−0.831285	+	0.555846i	$0.812394\pi$
$90$	0	0
$91$	19.8078	2.07642
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−16.4924	−1.69209
$96$	0	0
$97$	−3.43845	−0.349121	−0.174561	−	0.984646i	$-0.555851\pi$
$97$	−3.43845	−0.349121	−0.174561	+	0.984646i	$0.555851\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−6.24621	−0.621521	−0.310761	−	0.950488i	$-0.600584\pi$
$101$	−6.24621	−0.621521	−0.310761	+	0.950488i	$0.600584\pi$
$102$	0	0
$103$	−4.56155	−0.449463	−0.224732	−	0.974421i	$-0.572151\pi$
$103$	−4.56155	−0.449463	−0.224732	+	0.974421i	$0.572151\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	8.56155	0.827677	0.413838	−	0.910350i	$-0.364188\pi$
$107$	8.56155	0.827677	0.413838	+	0.910350i	$0.364188\pi$
$108$	0	0
$109$	6.43845	0.616691	0.308346	−	0.951274i	$-0.400225\pi$
$109$	6.43845	0.616691	0.308346	+	0.951274i	$0.400225\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−1.56155	−0.146899	−0.0734493	−	0.997299i	$-0.523401\pi$
$113$	−1.56155	−0.146899	−0.0734493	+	0.997299i	$0.523401\pi$
$114$	0	0
$115$	−20.6155	−1.92241
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−2.00000	−0.183340
$120$	0	0
$121$	1.68466	0.153151
$122$	0	0
$123$	0	0
$124$	0	0
$125$	28.8617	2.58147
$126$	0	0
$127$	14.8078	1.31398	0.656988	−	0.753901i	$-0.271830\pi$
$127$	14.8078	1.31398	0.656988	+	0.753901i	$0.271830\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−5.68466	−0.496671	−0.248335	−	0.968674i	$-0.579884\pi$
$131$	−5.68466	−0.496671	−0.248335	+	0.968674i	$0.579884\pi$
$132$	0	0
$133$	−14.2462	−1.23530
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−18.0540	−1.54246	−0.771228	−	0.636559i	$-0.780357\pi$
$137$	−18.0540	−1.54246	−0.771228	+	0.636559i	$0.780357\pi$
$138$	0	0
$139$	−13.4924	−1.14441	−0.572206	−	0.820110i	$-0.693912\pi$
$139$	−13.4924	−1.14441	−0.572206	+	0.820110i	$0.693912\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	19.8078	1.65641
$144$	0	0
$145$	21.1231	1.75418
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−8.43845	−0.691305	−0.345652	−	0.938363i	$-0.612342\pi$
$149$	−8.43845	−0.691305	−0.345652	+	0.938363i	$0.612342\pi$
$150$	0	0
$151$	2.68466	0.218474	0.109237	−	0.994016i	$-0.465159\pi$
$151$	2.68466	0.218474	0.109237	+	0.994016i	$0.465159\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−16.4924	−1.32470
$156$	0	0
$157$	−7.75379	−0.618820	−0.309410	−	0.950929i	$-0.600131\pi$
$157$	−7.75379	−0.618820	−0.309410	+	0.950929i	$0.600131\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−17.8078	−1.40345
$162$	0	0
$163$	1.75379	0.137367	0.0686837	−	0.997638i	$-0.478120\pi$
$163$	1.75379	0.137367	0.0686837	+	0.997638i	$0.478120\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	4.24621	0.328582	0.164291	−	0.986412i	$-0.447466\pi$
$167$	4.24621	0.328582	0.164291	+	0.986412i	$0.447466\pi$
$168$	0	0
$169$	17.9309	1.37930
$170$	0	0
$171$	0	0
$172$	0	0
$173$	3.75379	0.285395	0.142698	−	0.989766i	$-0.454422\pi$
$173$	3.75379	0.285395	0.142698	+	0.989766i	$0.454422\pi$
$174$	0	0
$175$	42.7386	3.23074
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−15.6847	−1.17233	−0.586163	−	0.810193i	$-0.699362\pi$
$179$	−15.6847	−1.17233	−0.586163	+	0.810193i	$0.699362\pi$
$180$	0	0
$181$	22.8078	1.69529	0.847644	−	0.530566i	$-0.178020\pi$
$181$	22.8078	1.69529	0.847644	+	0.530566i	$0.178020\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−15.1922	−1.11696
$186$	0	0
$187$	−2.00000	−0.146254
$188$	0	0
$189$	0	0
$190$	0	0
$191$	3.31534	0.239890	0.119945	−	0.992781i	$-0.461728\pi$
$191$	3.31534	0.239890	0.119945	+	0.992781i	$0.461728\pi$
$192$	0	0
$193$	12.8769	0.926899	0.463450	−	0.886123i	$-0.346612\pi$
$193$	12.8769	0.926899	0.463450	+	0.886123i	$0.346612\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	6.19224	0.441179	0.220589	−	0.975367i	$-0.429202\pi$
$197$	6.19224	0.441179	0.220589	+	0.975367i	$0.429202\pi$
$198$	0	0
$199$	−7.43845	−0.527298	−0.263649	−	0.964619i	$-0.584926\pi$
$199$	−7.43845	−0.527298	−0.263649	+	0.964619i	$0.584926\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	18.2462	1.28063
$204$	0	0
$205$	−39.4233	−2.75344
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−14.2462	−0.985431
$210$	0	0
$211$	24.4924	1.68613	0.843064	−	0.537813i	$-0.180749\pi$
$211$	24.4924	1.68613	0.843064	+	0.537813i	$0.180749\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	10.5616	0.720292
$216$	0	0
$217$	−14.2462	−0.967096
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−3.12311	−0.210083
$222$	0	0
$223$	−3.56155	−0.238499	−0.119250	−	0.992864i	$-0.538049\pi$
$223$	−3.56155	−0.238499	−0.119250	+	0.992864i	$0.538049\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−11.5616	−0.767367	−0.383684	−	0.923465i	$-0.625345\pi$
$227$	−11.5616	−0.767367	−0.383684	+	0.923465i	$0.625345\pi$
$228$	0	0
$229$	−21.8078	−1.44110	−0.720549	−	0.693404i	$-0.756110\pi$
$229$	−21.8078	−1.44110	−0.720549	+	0.693404i	$0.756110\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−5.12311	−0.335626	−0.167813	−	0.985819i	$-0.553670\pi$
$233$	−5.12311	−0.335626	−0.167813	+	0.985819i	$0.553670\pi$
$234$	0	0
$235$	29.3693	1.91584
$236$	0	0
$237$	0	0
$238$	0	0
$239$	11.3693	0.735420	0.367710	−	0.929941i	$-0.380142\pi$
$239$	11.3693	0.735420	0.367710	+	0.929941i	$0.380142\pi$
$240$	0	0
$241$	−19.4924	−1.25562	−0.627809	−	0.778368i	$-0.716048\pi$
$241$	−19.4924	−1.25562	−0.627809	+	0.778368i	$0.716048\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	23.4384	1.49743
$246$	0	0
$247$	−22.2462	−1.41549
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−7.36932	−0.465147	−0.232574	−	0.972579i	$-0.574715\pi$
$251$	−7.36932	−0.465147	−0.232574	+	0.972579i	$0.574715\pi$
$252$	0	0
$253$	−17.8078	−1.11956
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−13.3693	−0.833955	−0.416978	−	0.908917i	$-0.636911\pi$
$257$	−13.3693	−0.833955	−0.416978	+	0.908917i	$0.636911\pi$
$258$	0	0
$259$	−13.1231	−0.815430
$260$	0	0
$261$	0	0
$262$	0	0
$263$	28.0000	1.72655	0.863277	−	0.504730i	$-0.168408\pi$
$263$	28.0000	1.72655	0.863277	+	0.504730i	$0.168408\pi$
$264$	0	0
$265$	−37.6155	−2.31070
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−22.0000	−1.34136	−0.670682	−	0.741745i	$-0.733998\pi$
$269$	−22.0000	−1.34136	−0.670682	+	0.741745i	$0.733998\pi$
$270$	0	0
$271$	−16.1771	−0.982688	−0.491344	−	0.870966i	$-0.663494\pi$
$271$	−16.1771	−0.982688	−0.491344	+	0.870966i	$0.663494\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	42.7386	2.57724
$276$	0	0
$277$	6.49242	0.390092	0.195046	−	0.980794i	$-0.437514\pi$
$277$	6.49242	0.390092	0.195046	+	0.980794i	$0.437514\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	2.87689	0.171621	0.0858106	−	0.996311i	$-0.472652\pi$
$281$	2.87689	0.171621	0.0858106	+	0.996311i	$0.472652\pi$
$282$	0	0
$283$	−27.6155	−1.64157	−0.820786	−	0.571235i	$-0.806464\pi$
$283$	−27.6155	−1.64157	−0.820786	+	0.571235i	$0.806464\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−34.0540	−2.01014
$288$	0	0
$289$	−16.6847	−0.981450
$290$	0	0
$291$	0	0
$292$	0	0
$293$	22.1771	1.29560	0.647799	−	0.761811i	$-0.275689\pi$
$293$	22.1771	1.29560	0.647799	+	0.761811i	$0.275689\pi$
$294$	0	0
$295$	−26.5464	−1.54559
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−27.8078	−1.60816
$300$	0	0
$301$	9.12311	0.525847
$302$	0	0
$303$	0	0
$304$	0	0
$305$	4.12311	0.236088
$306$	0	0
$307$	−29.4924	−1.68322	−0.841611	−	0.540085i	$-0.818392\pi$
$307$	−29.4924	−1.68322	−0.841611	+	0.540085i	$0.818392\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−11.8078	−0.669557	−0.334778	−	0.942297i	$-0.608661\pi$
$311$	−11.8078	−0.669557	−0.334778	+	0.942297i	$0.608661\pi$
$312$	0	0
$313$	−13.6155	−0.769595	−0.384798	−	0.923001i	$-0.625729\pi$
$313$	−13.6155	−0.769595	−0.384798	+	0.923001i	$0.625729\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	23.4384	1.31643	0.658217	−	0.752828i	$-0.271311\pi$
$317$	23.4384	1.31643	0.658217	+	0.752828i	$0.271311\pi$
$318$	0	0
$319$	18.2462	1.02159
$320$	0	0
$321$	0	0
$322$	0	0
$323$	2.24621	0.124983
$324$	0	0
$325$	66.7386	3.70199
$326$	0	0
$327$	0	0
$328$	0	0
$329$	25.3693	1.39866
$330$	0	0
$331$	5.00000	0.274825	0.137412	−	0.990514i	$-0.456121\pi$
$331$	5.00000	0.274825	0.137412	+	0.990514i	$0.456121\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	13.6695	0.746845
$336$	0	0
$337$	−29.6155	−1.61326	−0.806630	−	0.591056i	$-0.798711\pi$
$337$	−29.6155	−1.61326	−0.806630	+	0.591056i	$0.798711\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−14.2462	−0.771476
$342$	0	0
$343$	−4.68466	−0.252948
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−29.3002	−1.57292	−0.786458	−	0.617643i	$-0.788087\pi$
$347$	−29.3002	−1.57292	−0.786458	+	0.617643i	$0.788087\pi$
$348$	0	0
$349$	21.0540	1.12699	0.563497	−	0.826118i	$-0.309456\pi$
$349$	21.0540	1.12699	0.563497	+	0.826118i	$0.309456\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−21.3153	−1.13450	−0.567251	−	0.823545i	$-0.691993\pi$
$353$	−21.3153	−1.13450	−0.567251	+	0.823545i	$0.691993\pi$
$354$	0	0
$355$	57.4384	3.04852
$356$	0	0
$357$	0	0
$358$	0	0
$359$	2.56155	0.135194	0.0675968	−	0.997713i	$-0.478467\pi$
$359$	2.56155	0.135194	0.0675968	+	0.997713i	$0.478467\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	0	0
$363$	0	0
$364$	0	0
$365$	67.4924	3.53271
$366$	0	0
$367$	8.00000	0.417597	0.208798	−	0.977959i	$-0.433045\pi$
$367$	8.00000	0.417597	0.208798	+	0.977959i	$0.433045\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−32.4924	−1.68692
$372$	0	0
$373$	10.5616	0.546856	0.273428	−	0.961892i	$-0.411842\pi$
$373$	10.5616	0.546856	0.273428	+	0.961892i	$0.411842\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	28.4924	1.46743
$378$	0	0
$379$	2.24621	0.115380	0.0576901	−	0.998335i	$-0.481626\pi$
$379$	2.24621	0.115380	0.0576901	+	0.998335i	$0.481626\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	2.43845	0.124599	0.0622994	−	0.998058i	$-0.480157\pi$
$383$	2.43845	0.124599	0.0622994	+	0.998058i	$0.480157\pi$
$384$	0	0
$385$	52.3002	2.66546
$386$	0	0
$387$	0	0
$388$	0	0
$389$	8.24621	0.418100	0.209050	−	0.977905i	$-0.432963\pi$
$389$	8.24621	0.418100	0.209050	+	0.977905i	$0.432963\pi$
$390$	0	0
$391$	2.80776	0.141995
$392$	0	0
$393$	0	0
$394$	0	0
$395$	53.3153	2.68259
$396$	0	0
$397$	−10.1771	−0.510773	−0.255387	−	0.966839i	$-0.582203\pi$
$397$	−10.1771	−0.510773	−0.255387	+	0.966839i	$0.582203\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−0.561553	−0.0280426	−0.0140213	−	0.999902i	$-0.504463\pi$
$401$	−0.561553	−0.0280426	−0.0140213	+	0.999902i	$0.504463\pi$
$402$	0	0
$403$	−22.2462	−1.10816
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−13.1231	−0.650488
$408$	0	0
$409$	8.00000	0.395575	0.197787	−	0.980245i	$-0.436624\pi$
$409$	8.00000	0.395575	0.197787	+	0.980245i	$0.436624\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−22.9309	−1.12835
$414$	0	0
$415$	−44.5616	−2.18744
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−4.00000	−0.195413	−0.0977064	−	0.995215i	$-0.531151\pi$
$419$	−4.00000	−0.195413	−0.0977064	+	0.995215i	$0.531151\pi$
$420$	0	0
$421$	−2.31534	−0.112843	−0.0564214	−	0.998407i	$-0.517969\pi$
$421$	−2.31534	−0.112843	−0.0564214	+	0.998407i	$0.517969\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−6.73863	−0.326872
$426$	0	0
$427$	3.56155	0.172356
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−12.7386	−0.613598	−0.306799	−	0.951774i	$-0.599258\pi$
$431$	−12.7386	−0.613598	−0.306799	+	0.951774i	$0.599258\pi$
$432$	0	0
$433$	12.8769	0.618824	0.309412	−	0.950928i	$-0.399868\pi$
$433$	12.8769	0.618824	0.309412	+	0.950928i	$0.399868\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	20.0000	0.956730
$438$	0	0
$439$	22.5616	1.07680	0.538402	−	0.842688i	$-0.319028\pi$
$439$	22.5616	1.07680	0.538402	+	0.842688i	$0.319028\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	4.31534	0.205028	0.102514	−	0.994732i	$-0.467311\pi$
$443$	4.31534	0.205028	0.102514	+	0.994732i	$0.467311\pi$
$444$	0	0
$445$	−64.6695	−3.06563
$446$	0	0
$447$	0	0
$448$	0	0
$449$	23.5616	1.11194	0.555969	−	0.831203i	$-0.312347\pi$
$449$	23.5616	1.11194	0.555969	+	0.831203i	$0.312347\pi$
$450$	0	0
$451$	−34.0540	−1.60354
$452$	0	0
$453$	0	0
$454$	0	0
$455$	81.6695	3.82873
$456$	0	0
$457$	33.1231	1.54943	0.774717	−	0.632308i	$-0.217892\pi$
$457$	33.1231	1.54943	0.774717	+	0.632308i	$0.217892\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−26.1771	−1.21919	−0.609594	−	0.792714i	$-0.708668\pi$
$461$	−26.1771	−1.21919	−0.609594	+	0.792714i	$0.708668\pi$
$462$	0	0
$463$	−25.4384	−1.18222	−0.591112	−	0.806589i	$-0.701311\pi$
$463$	−25.4384	−1.18222	−0.591112	+	0.806589i	$0.701311\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	39.4233	1.82429	0.912146	−	0.409865i	$-0.134424\pi$
$467$	39.4233	1.82429	0.912146	+	0.409865i	$0.134424\pi$
$468$	0	0
$469$	11.8078	0.545232
$470$	0	0
$471$	0	0
$472$	0	0
$473$	9.12311	0.419481
$474$	0	0
$475$	−48.0000	−2.20239
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−7.36932	−0.336713	−0.168356	−	0.985726i	$-0.553846\pi$
$479$	−7.36932	−0.336713	−0.168356	+	0.985726i	$0.553846\pi$
$480$	0	0
$481$	−20.4924	−0.934374
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−14.1771	−0.643748
$486$	0	0
$487$	−14.2462	−0.645557	−0.322779	−	0.946474i	$-0.604617\pi$
$487$	−14.2462	−0.645557	−0.322779	+	0.946474i	$0.604617\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	20.4924	0.924810	0.462405	−	0.886669i	$-0.346987\pi$
$491$	20.4924	0.924810	0.462405	+	0.886669i	$0.346987\pi$
$492$	0	0
$493$	−2.87689	−0.129569
$494$	0	0
$495$	0	0
$496$	0	0
$497$	49.6155	2.22556
$498$	0	0
$499$	−16.3693	−0.732791	−0.366396	−	0.930459i	$-0.619408\pi$
$499$	−16.3693	−0.732791	−0.366396	+	0.930459i	$0.619408\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	6.63068	0.295648	0.147824	−	0.989014i	$-0.452773\pi$
$503$	6.63068	0.295648	0.147824	+	0.989014i	$0.452773\pi$
$504$	0	0
$505$	−25.7538	−1.14603
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−31.6155	−1.40133	−0.700667	−	0.713489i	$-0.747114\pi$
$509$	−31.6155	−1.40133	−0.700667	+	0.713489i	$0.747114\pi$
$510$	0	0
$511$	58.3002	2.57905
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−18.8078	−0.828769
$516$	0	0
$517$	25.3693	1.11574
$518$	0	0
$519$	0	0
$520$	0	0
$521$	18.4924	0.810168	0.405084	−	0.914280i	$-0.367242\pi$
$521$	18.4924	0.810168	0.405084	+	0.914280i	$0.367242\pi$
$522$	0	0
$523$	44.2311	1.93409	0.967045	−	0.254607i	$-0.0819462\pi$
$523$	44.2311	1.93409	0.967045	+	0.254607i	$0.0819462\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	2.24621	0.0978465
$528$	0	0
$529$	2.00000	0.0869565
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−53.1771	−2.30336
$534$	0	0
$535$	35.3002	1.52616
$536$	0	0
$537$	0	0
$538$	0	0
$539$	20.2462	0.872066
$540$	0	0
$541$	8.80776	0.378675	0.189338	−	0.981912i	$-0.439366\pi$
$541$	8.80776	0.378675	0.189338	+	0.981912i	$0.439366\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	26.5464	1.13712
$546$	0	0
$547$	4.68466	0.200302	0.100151	−	0.994972i	$-0.468067\pi$
$547$	4.68466	0.200302	0.100151	+	0.994972i	$0.468067\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−20.4924	−0.873007
$552$	0	0
$553$	46.0540	1.95841
$554$	0	0
$555$	0	0
$556$	0	0
$557$	45.6155	1.93279	0.966396	−	0.257058i	$-0.0827531\pi$
$557$	45.6155	1.93279	0.966396	+	0.257058i	$0.0827531\pi$
$558$	0	0
$559$	14.2462	0.602551
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−17.3693	−0.732029	−0.366015	−	0.930609i	$-0.619278\pi$
$563$	−17.3693	−0.732029	−0.366015	+	0.930609i	$0.619278\pi$
$564$	0	0
$565$	−6.43845	−0.270868
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−23.8078	−0.998073	−0.499037	−	0.866581i	$-0.666313\pi$
$569$	−23.8078	−0.998073	−0.499037	+	0.866581i	$0.666313\pi$
$570$	0	0
$571$	−11.6155	−0.486095	−0.243047	−	0.970014i	$-0.578147\pi$
$571$	−11.6155	−0.486095	−0.243047	+	0.970014i	$0.578147\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−60.0000	−2.50217
$576$	0	0
$577$	−1.36932	−0.0570054	−0.0285027	−	0.999594i	$-0.509074\pi$
$577$	−1.36932	−0.0570054	−0.0285027	+	0.999594i	$0.509074\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−38.4924	−1.59693
$582$	0	0
$583$	−32.4924	−1.34570
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−39.8617	−1.64527	−0.822635	−	0.568570i	$-0.807497\pi$
$587$	−39.8617	−1.64527	−0.822635	+	0.568570i	$0.807497\pi$
$588$	0	0
$589$	16.0000	0.659269
$590$	0	0
$591$	0	0
$592$	0	0
$593$	2.80776	0.115301	0.0576505	−	0.998337i	$-0.481639\pi$
$593$	2.80776	0.115301	0.0576505	+	0.998337i	$0.481639\pi$
$594$	0	0
$595$	−8.24621	−0.338062
$596$	0	0
$597$	0	0
$598$	0	0
$599$	19.2462	0.786379	0.393189	−	0.919457i	$-0.371372\pi$
$599$	19.2462	0.786379	0.393189	+	0.919457i	$0.371372\pi$
$600$	0	0
$601$	25.4924	1.03986	0.519929	−	0.854210i	$-0.325958\pi$
$601$	25.4924	1.03986	0.519929	+	0.854210i	$0.325958\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	6.94602	0.282396
$606$	0	0
$607$	−9.43845	−0.383095	−0.191547	−	0.981483i	$-0.561351\pi$
$607$	−9.43845	−0.383095	−0.191547	+	0.981483i	$0.561351\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	39.6155	1.60267
$612$	0	0
$613$	13.8617	0.559870	0.279935	−	0.960019i	$-0.409687\pi$
$613$	13.8617	0.559870	0.279935	+	0.960019i	$0.409687\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	25.6847	1.03403	0.517013	−	0.855978i	$-0.327044\pi$
$617$	25.6847	1.03403	0.517013	+	0.855978i	$0.327044\pi$
$618$	0	0
$619$	−22.4924	−0.904047	−0.452023	−	0.892006i	$-0.649298\pi$
$619$	−22.4924	−0.904047	−0.452023	+	0.892006i	$0.649298\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−55.8617	−2.23805
$624$	0	0
$625$	59.0000	2.36000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	2.06913	0.0825016
$630$	0	0
$631$	4.05398	0.161386	0.0806931	−	0.996739i	$-0.474287\pi$
$631$	4.05398	0.161386	0.0806931	+	0.996739i	$0.474287\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	61.0540	2.42285
$636$	0	0
$637$	31.6155	1.25265
$638$	0	0
$639$	0	0
$640$	0	0
$641$	33.4384	1.32074	0.660370	−	0.750941i	$-0.270400\pi$
$641$	33.4384	1.32074	0.660370	+	0.750941i	$0.270400\pi$
$642$	0	0
$643$	2.06913	0.0815985	0.0407993	−	0.999167i	$-0.487010\pi$
$643$	2.06913	0.0815985	0.0407993	+	0.999167i	$0.487010\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−10.1231	−0.397980	−0.198990	−	0.980001i	$-0.563766\pi$
$647$	−10.1231	−0.397980	−0.198990	+	0.980001i	$0.563766\pi$
$648$	0	0
$649$	−22.9309	−0.900115
$650$	0	0
$651$	0	0
$652$	0	0
$653$	19.1231	0.748345	0.374172	−	0.927359i	$-0.377927\pi$
$653$	19.1231	0.748345	0.374172	+	0.927359i	$0.377927\pi$
$654$	0	0
$655$	−23.4384	−0.915816
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−15.3002	−0.596011	−0.298005	−	0.954564i	$-0.596321\pi$
$659$	−15.3002	−0.596011	−0.298005	+	0.954564i	$0.596321\pi$
$660$	0	0
$661$	−26.8769	−1.04539	−0.522695	−	0.852520i	$-0.675073\pi$
$661$	−26.8769	−1.04539	−0.522695	+	0.852520i	$0.675073\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−58.7386	−2.27779
$666$	0	0
$667$	−25.6155	−0.991837
$668$	0	0
$669$	0	0
$670$	0	0
$671$	3.56155	0.137492
$672$	0	0
$673$	−4.73863	−0.182661	−0.0913305	−	0.995821i	$-0.529112\pi$
$673$	−4.73863	−0.182661	−0.0913305	+	0.995821i	$0.529112\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	41.8617	1.60888	0.804439	−	0.594036i	$-0.202466\pi$
$677$	41.8617	1.60888	0.804439	+	0.594036i	$0.202466\pi$
$678$	0	0
$679$	−12.2462	−0.469966
$680$	0	0
$681$	0	0
$682$	0	0
$683$	43.6847	1.67155	0.835774	−	0.549074i	$-0.185020\pi$
$683$	43.6847	1.67155	0.835774	+	0.549074i	$0.185020\pi$
$684$	0	0
$685$	−74.4384	−2.84415
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−50.7386	−1.93299
$690$	0	0
$691$	−8.63068	−0.328327	−0.164163	−	0.986433i	$-0.552492\pi$
$691$	−8.63068	−0.328327	−0.164163	+	0.986433i	$0.552492\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−55.6307	−2.11019
$696$	0	0
$697$	5.36932	0.203377
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−22.9848	−0.868126	−0.434063	−	0.900883i	$-0.642920\pi$
$701$	−22.9848	−0.868126	−0.434063	+	0.900883i	$0.642920\pi$
$702$	0	0
$703$	14.7386	0.555878
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−22.2462	−0.836655
$708$	0	0
$709$	−15.9309	−0.598296	−0.299148	−	0.954207i	$-0.596702\pi$
$709$	−15.9309	−0.598296	−0.299148	+	0.954207i	$0.596702\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	20.0000	0.749006
$714$	0	0
$715$	81.6695	3.05427
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−8.49242	−0.316714	−0.158357	−	0.987382i	$-0.550620\pi$
$719$	−8.49242	−0.316714	−0.158357	+	0.987382i	$0.550620\pi$
$720$	0	0
$721$	−16.2462	−0.605041
$722$	0	0
$723$	0	0
$724$	0	0
$725$	61.4773	2.28321
$726$	0	0
$727$	52.1080	1.93258	0.966288	−	0.257462i	$-0.0828862\pi$
$727$	52.1080	1.93258	0.966288	+	0.257462i	$0.0828862\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−1.43845	−0.0532029
$732$	0	0
$733$	−19.5616	−0.722522	−0.361261	−	0.932465i	$-0.617654\pi$
$733$	−19.5616	−0.722522	−0.361261	+	0.932465i	$0.617654\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	11.8078	0.434945
$738$	0	0
$739$	24.6155	0.905497	0.452748	−	0.891638i	$-0.350444\pi$
$739$	24.6155	0.905497	0.452748	+	0.891638i	$0.350444\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−17.6307	−0.646807	−0.323404	−	0.946261i	$-0.604827\pi$
$743$	−17.6307	−0.646807	−0.323404	+	0.946261i	$0.604827\pi$
$744$	0	0
$745$	−34.7926	−1.27470
$746$	0	0
$747$	0	0
$748$	0	0
$749$	30.4924	1.11417
$750$	0	0
$751$	−11.0540	−0.403365	−0.201683	−	0.979451i	$-0.564641\pi$
$751$	−11.0540	−0.403365	−0.201683	+	0.979451i	$0.564641\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	11.0691	0.402847
$756$	0	0
$757$	3.56155	0.129447	0.0647234	−	0.997903i	$-0.479383\pi$
$757$	3.56155	0.129447	0.0647234	+	0.997903i	$0.479383\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−13.5076	−0.489649	−0.244825	−	0.969567i	$-0.578730\pi$
$761$	−13.5076	−0.489649	−0.244825	+	0.969567i	$0.578730\pi$
$762$	0	0
$763$	22.9309	0.830153
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−35.8078	−1.29294
$768$	0	0
$769$	−51.6155	−1.86130	−0.930652	−	0.365906i	$-0.880759\pi$
$769$	−51.6155	−1.86130	−0.930652	+	0.365906i	$0.880759\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	26.6695	0.959235	0.479618	−	0.877478i	$-0.340775\pi$
$773$	26.6695	0.959235	0.479618	+	0.877478i	$0.340775\pi$
$774$	0	0
$775$	−48.0000	−1.72421
$776$	0	0
$777$	0	0
$778$	0	0
$779$	38.2462	1.37031
$780$	0	0
$781$	49.6155	1.77538
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−31.9697	−1.14105
$786$	0	0
$787$	−54.7386	−1.95122	−0.975611	−	0.219508i	$-0.929555\pi$
$787$	−54.7386	−1.95122	−0.975611	+	0.219508i	$0.929555\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−5.56155	−0.197746
$792$	0	0
$793$	5.56155	0.197497
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−24.5464	−0.869478	−0.434739	−	0.900556i	$-0.643159\pi$
$797$	−24.5464	−0.869478	−0.434739	+	0.900556i	$0.643159\pi$
$798$	0	0
$799$	−4.00000	−0.141510
$800$	0	0
$801$	0	0
$802$	0	0
$803$	58.3002	2.05737
$804$	0	0
$805$	−73.4233	−2.58783
$806$	0	0
$807$	0	0
$808$	0	0
$809$	44.0540	1.54886	0.774428	−	0.632662i	$-0.218038\pi$
$809$	44.0540	1.54886	0.774428	+	0.632662i	$0.218038\pi$
$810$	0	0
$811$	−7.49242	−0.263095	−0.131547	−	0.991310i	$-0.541995\pi$
$811$	−7.49242	−0.263095	−0.131547	+	0.991310i	$0.541995\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	7.23106	0.253293
$816$	0	0
$817$	−10.2462	−0.358470
$818$	0	0
$819$	0	0
$820$	0	0
$821$	20.2462	0.706598	0.353299	−	0.935511i	$-0.385060\pi$
$821$	20.2462	0.706598	0.353299	+	0.935511i	$0.385060\pi$
$822$	0	0
$823$	−15.3693	−0.535741	−0.267870	−	0.963455i	$-0.586320\pi$
$823$	−15.3693	−0.535741	−0.267870	+	0.963455i	$0.586320\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	15.0540	0.523478	0.261739	−	0.965139i	$-0.415704\pi$
$827$	15.0540	0.523478	0.261739	+	0.965139i	$0.415704\pi$
$828$	0	0
$829$	−10.1922	−0.353991	−0.176995	−	0.984212i	$-0.556638\pi$
$829$	−10.1922	−0.353991	−0.176995	+	0.984212i	$0.556638\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−3.19224	−0.110604
$834$	0	0
$835$	17.5076	0.605875
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−3.50758	−0.121095	−0.0605475	−	0.998165i	$-0.519285\pi$
$839$	−3.50758	−0.121095	−0.0605475	+	0.998165i	$0.519285\pi$
$840$	0	0
$841$	−2.75379	−0.0949582
$842$	0	0
$843$	0	0
$844$	0	0
$845$	73.9309	2.54330
$846$	0	0
$847$	6.00000	0.206162
$848$	0	0
$849$	0	0
$850$	0	0
$851$	18.4233	0.631542
$852$	0	0
$853$	−11.5616	−0.395860	−0.197930	−	0.980216i	$-0.563422\pi$
$853$	−11.5616	−0.395860	−0.197930	+	0.980216i	$0.563422\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−36.4384	−1.24471	−0.622357	−	0.782734i	$-0.713825\pi$
$857$	−36.4384	−1.24471	−0.622357	+	0.782734i	$0.713825\pi$
$858$	0	0
$859$	54.3542	1.85454	0.927270	−	0.374393i	$-0.122149\pi$
$859$	54.3542	1.85454	0.927270	+	0.374393i	$0.122149\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	24.0000	0.816970	0.408485	−	0.912765i	$-0.366057\pi$
$863$	24.0000	0.816970	0.408485	+	0.912765i	$0.366057\pi$
$864$	0	0
$865$	15.4773	0.526243
$866$	0	0
$867$	0	0
$868$	0	0
$869$	46.0540	1.56227
$870$	0	0
$871$	18.4384	0.624763
$872$	0	0
$873$	0	0
$874$	0	0
$875$	102.793	3.47502
$876$	0	0
$877$	55.5464	1.87567	0.937834	−	0.347083i	$-0.112828\pi$
$877$	55.5464	1.87567	0.937834	+	0.347083i	$0.112828\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	13.4233	0.452242	0.226121	−	0.974099i	$-0.427395\pi$
$881$	13.4233	0.452242	0.226121	+	0.974099i	$0.427395\pi$
$882$	0	0
$883$	−42.4773	−1.42947	−0.714737	−	0.699393i	$-0.753454\pi$
$883$	−42.4773	−1.42947	−0.714737	+	0.699393i	$0.753454\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	6.56155	0.220315	0.110158	−	0.993914i	$-0.464864\pi$
$887$	6.56155	0.220315	0.110158	+	0.993914i	$0.464864\pi$
$888$	0	0
$889$	52.7386	1.76880
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−28.4924	−0.953463
$894$	0	0
$895$	−64.6695	−2.16166
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−20.4924	−0.683461
$900$	0	0
$901$	5.12311	0.170675
$902$	0	0
$903$	0	0
$904$	0	0
$905$	94.0388	3.12596
$906$	0	0
$907$	46.4233	1.54146	0.770730	−	0.637162i	$-0.219892\pi$
$907$	46.4233	1.54146	0.770730	+	0.637162i	$0.219892\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	34.7386	1.15094	0.575471	−	0.817822i	$-0.304819\pi$
$911$	34.7386	1.15094	0.575471	+	0.817822i	$0.304819\pi$
$912$	0	0
$913$	−38.4924	−1.27391
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−20.2462	−0.668589
$918$	0	0
$919$	16.9460	0.558998	0.279499	−	0.960146i	$-0.409832\pi$
$919$	16.9460	0.558998	0.279499	+	0.960146i	$0.409832\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	77.4773	2.55020
$924$	0	0
$925$	−44.2159	−1.45381
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−40.3002	−1.32221	−0.661103	−	0.750295i	$-0.729911\pi$
$929$	−40.3002	−1.32221	−0.661103	+	0.750295i	$0.729911\pi$
$930$	0	0
$931$	−22.7386	−0.745229
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−8.24621	−0.269680
$936$	0	0
$937$	35.4924	1.15949	0.579743	−	0.814799i	$-0.303153\pi$
$937$	35.4924	1.15949	0.579743	+	0.814799i	$0.303153\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	8.00000	0.260793	0.130396	−	0.991462i	$-0.458375\pi$
$941$	8.00000	0.260793	0.130396	+	0.991462i	$0.458375\pi$
$942$	0	0
$943$	47.8078	1.55683
$944$	0	0
$945$	0	0
$946$	0	0
$947$	19.4233	0.631172	0.315586	−	0.948897i	$-0.397799\pi$
$947$	19.4233	0.631172	0.315586	+	0.948897i	$0.397799\pi$
$948$	0	0
$949$	91.0388	2.95525
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−28.5616	−0.925199	−0.462600	−	0.886567i	$-0.653083\pi$
$953$	−28.5616	−0.925199	−0.462600	+	0.886567i	$0.653083\pi$
$954$	0	0
$955$	13.6695	0.442335
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−64.3002	−2.07636
$960$	0	0
$961$	−15.0000	−0.483871
$962$	0	0
$963$	0	0
$964$	0	0
$965$	53.0928	1.70912
$966$	0	0
$967$	−50.1771	−1.61359	−0.806793	−	0.590834i	$-0.798799\pi$
$967$	−50.1771	−1.61359	−0.806793	+	0.590834i	$0.798799\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−20.4233	−0.655415	−0.327707	−	0.944779i	$-0.606276\pi$
$971$	−20.4233	−0.655415	−0.327707	+	0.944779i	$0.606276\pi$
$972$	0	0
$973$	−48.0540	−1.54054
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−1.36932	−0.0438083	−0.0219042	−	0.999760i	$-0.506973\pi$
$977$	−1.36932	−0.0438083	−0.0219042	+	0.999760i	$0.506973\pi$
$978$	0	0
$979$	−55.8617	−1.78535
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−31.2311	−0.996116	−0.498058	−	0.867144i	$-0.665953\pi$
$983$	−31.2311	−0.996116	−0.498058	+	0.867144i	$0.665953\pi$
$984$	0	0
$985$	25.5312	0.813493
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−12.8078	−0.407263
$990$	0	0
$991$	−13.7538	−0.436903	−0.218452	−	0.975848i	$-0.570101\pi$
$991$	−13.7538	−0.436903	−0.218452	+	0.975848i	$0.570101\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−30.6695	−0.972289
$996$	0	0
$997$	39.9309	1.26462	0.632312	−	0.774714i	$-0.282106\pi$
$997$	39.9309	1.26462	0.632312	+	0.774714i	$0.282106\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8784.2.a.bg.1.2		2
3.2	odd	2		2928.2.a.u.1.1		2
4.3	odd	2		1098.2.a.m.1.2		2
12.11	even	2		366.2.a.h.1.1	✓	2
60.59	even	2		9150.2.a.bh.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
366.2.a.h.1.1	✓	2	12.11	even	2
1098.2.a.m.1.2		2	4.3	odd	2
2928.2.a.u.1.1		2	3.2	odd	2
8784.2.a.bg.1.2		2	1.1	even	1	trivial
9150.2.a.bh.1.2		2	60.59	even	2

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:	\( N \)	\(=\)	\( 8784 = 2^{4} \cdot 3^{2} \cdot 61 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8784.a (trivial)

\(f(q)\)	\(=\)	\(q+4.12311 q^{5} +3.56155 q^{7} +3.56155 q^{11} +5.56155 q^{13} -0.561553 q^{17} -4.00000 q^{19} -5.00000 q^{23} +12.0000 q^{25} +5.12311 q^{29} -4.00000 q^{31} +14.6847 q^{35} -3.68466 q^{37} -9.56155 q^{41} +2.56155 q^{43} +7.12311 q^{47} +5.68466 q^{49} -9.12311 q^{53} +14.6847 q^{55} -6.43845 q^{59} +1.00000 q^{61} +22.9309 q^{65} +3.31534 q^{67} +13.9309 q^{71} +16.3693 q^{73} +12.6847 q^{77} +12.9309 q^{79} -10.8078 q^{83} -2.31534 q^{85} -15.6847 q^{89} +19.8078 q^{91} -16.4924 q^{95} -3.43845 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 3 q^{7} + 3 q^{11} + 7 q^{13} + 3 q^{17} - 8 q^{19} - 10 q^{23} + 24 q^{25} + 2 q^{29} - 8 q^{31} + 17 q^{35} + 5 q^{37} - 15 q^{41} + q^{43} + 6 q^{47} - q^{49} - 10 q^{53} + 17 q^{55} - 17 q^{59}+ \cdots - 11 q^{97}+O(q^{100}) \) 2 * q + 3 * q^7 + 3 * q^11 + 7 * q^13 + 3 * q^17 - 8 * q^19 - 10 * q^23 + 24 * q^25 + 2 * q^29 - 8 * q^31 + 17 * q^35 + 5 * q^37 - 15 * q^41 + q^43 + 6 * q^47 - q^49 - 10 * q^53 + 17 * q^55 - 17 * q^59 + 2 * q^61 + 17 * q^65 + 19 * q^67 - q^71 + 8 * q^73 + 13 * q^77 - 3 * q^79 - q^83 - 17 * q^85 - 19 * q^89 + 19 * q^91 - 11 * q^97

Introduction

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