LMFDB - Embedded newform 8649.2.a.bs.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8649,2,Mod(1,8649)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8649, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("8649.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8649 = 3^{2} \cdot 31^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8649.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:	$69.0626127082$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 24x^{14} + 220x^{12} - 992x^{10} + 2366x^{8} - 2944x^{6} + 1688x^{4} - 288x^{2} + 4 $ x^16 - 24x^14 + 220x^12 - 992x^10 + 2366x^8 - 2944x^6 + 1688x^4 - 288*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 961)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.23447$ of defining polynomial
Character	$\chi$	$=$	8649.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.16383 q^{2} +2.68217 q^{4} +2.65612 q^{5} -3.05284 q^{7} -1.47609 q^{8} +O(q^{10})$	$q-2.16383 q^{2} +2.68217 q^{4} +2.65612 q^{5} -3.05284 q^{7} -1.47609 q^{8} -5.74741 q^{10} +4.20622 q^{11} +4.94542 q^{13} +6.60583 q^{14} -2.17031 q^{16} -0.493982 q^{17} -0.561806 q^{19} +7.12417 q^{20} -9.10154 q^{22} +3.83353 q^{23} +2.05500 q^{25} -10.7011 q^{26} -8.18823 q^{28} -6.08656 q^{29} +7.64838 q^{32} +1.06889 q^{34} -8.10873 q^{35} +5.57017 q^{37} +1.21565 q^{38} -3.92069 q^{40} -2.52547 q^{41} -10.3098 q^{43} +11.2818 q^{44} -8.29512 q^{46} +1.55127 q^{47} +2.31984 q^{49} -4.44667 q^{50} +13.2645 q^{52} +4.37202 q^{53} +11.1722 q^{55} +4.50628 q^{56} +13.1703 q^{58} +7.67905 q^{59} +1.37193 q^{61} -12.2092 q^{64} +13.1357 q^{65} +7.60375 q^{67} -1.32494 q^{68} +17.5459 q^{70} -4.38206 q^{71} +13.9386 q^{73} -12.0529 q^{74} -1.50686 q^{76} -12.8409 q^{77} +2.61080 q^{79} -5.76463 q^{80} +5.46469 q^{82} +13.3751 q^{83} -1.31208 q^{85} +22.3087 q^{86} -6.20877 q^{88} +7.92836 q^{89} -15.0976 q^{91} +10.2822 q^{92} -3.35670 q^{94} -1.49223 q^{95} -14.6782 q^{97} -5.01975 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 8 q^{2} + 8 q^{4} + 16 q^{5} - 16 q^{7}+O(q^{10}) $ 16 * q + 8 * q^2 + 8 * q^4 + 16 * q^5 - 16 * q^7	$ 16 q + 8 q^{2} + 8 q^{4} + 16 q^{5} - 16 q^{7} + 8 q^{10} + 8 q^{14} - 8 q^{16} - 32 q^{19} + 24 q^{20} - 8 q^{28} + 8 q^{32} + 16 q^{35} + 24 q^{38} + 32 q^{41} + 32 q^{47} - 16 q^{49} + 32 q^{50} + 48 q^{56} + 64 q^{59} - 16 q^{64} + 16 q^{67} + 88 q^{70} + 48 q^{71} + 40 q^{76} + 40 q^{80} + 88 q^{82} - 32 q^{94} + 48 q^{95} - 16 q^{98}+O(q^{100}) $ 16 * q + 8 * q^2 + 8 * q^4 + 16 * q^5 - 16 * q^7 + 8 * q^10 + 8 * q^14 - 8 * q^16 - 32 * q^19 + 24 * q^20 - 8 * q^28 + 8 * q^32 + 16 * q^35 + 24 * q^38 + 32 * q^41 + 32 * q^47 - 16 * q^49 + 32 * q^50 + 48 * q^56 + 64 * q^59 - 16 * q^64 + 16 * q^67 + 88 * q^70 + 48 * q^71 + 40 * q^76 + 40 * q^80 + 88 * q^82 - 32 * q^94 + 48 * q^95 - 16 * q^98

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 $

$2$	−2.16383	−1.53006	−0.765030	−	0.643995i	$-0.777276\pi$
$2$	−2.16383	−1.53006	−0.765030	+	0.643995i	$0.777276\pi$
$3$	0	0
$3$	0	0
$4$	2.68217	1.34108
$5$	2.65612	1.18786	0.593928	−	0.804518i	$-0.297577\pi$
$5$	2.65612	1.18786	0.593928	+	0.804518i	$0.297577\pi$
$6$	0	0
$7$	−3.05284	−1.15387	−0.576933	−	0.816792i	$-0.695751\pi$
$7$	−3.05284	−1.15387	−0.576933	+	0.816792i	$0.695751\pi$
$8$	−1.47609	−0.521878
$9$	0	0
$10$	−5.74741	−1.81749
$11$	4.20622	1.26822	0.634111	−	0.773242i	$-0.281366\pi$
$11$	4.20622	1.26822	0.634111	+	0.773242i	$0.281366\pi$
$12$	0	0
$13$	4.94542	1.37161	0.685807	−	0.727784i	$-0.259449\pi$
$13$	4.94542	1.37161	0.685807	+	0.727784i	$0.259449\pi$
$14$	6.60583	1.76548
$15$	0	0
$16$	−2.17031	−0.542579
$17$	−0.493982	−0.119808	−0.0599041	−	0.998204i	$-0.519079\pi$
$17$	−0.493982	−0.119808	−0.0599041	+	0.998204i	$0.519079\pi$
$18$	0	0
$19$	−0.561806	−0.128887	−0.0644436	−	0.997921i	$-0.520527\pi$
$19$	−0.561806	−0.128887	−0.0644436	+	0.997921i	$0.520527\pi$
$20$	7.12417	1.59301
$21$	0	0
$22$	−9.10154	−1.94046
$23$	3.83353	0.799347	0.399674	−	0.916658i	$-0.369123\pi$
$23$	3.83353	0.799347	0.399674	+	0.916658i	$0.369123\pi$
$24$	0	0
$25$	2.05500	0.411000
$26$	−10.7011	−2.09865
$27$	0	0
$28$	−8.18823	−1.54743
$29$	−6.08656	−1.13025	−0.565123	−	0.825007i	$-0.691171\pi$
$29$	−6.08656	−1.13025	−0.565123	+	0.825007i	$0.691171\pi$
$30$	0	0
$31$	0	0
$31$	0	0
$32$	7.64838	1.35206
$33$	0	0
$34$	1.06889	0.183314
$35$	−8.10873	−1.37063
$36$	0	0
$37$	5.57017	0.915730	0.457865	−	0.889022i	$-0.348614\pi$
$37$	5.57017	0.915730	0.457865	+	0.889022i	$0.348614\pi$
$38$	1.21565	0.197205
$39$	0	0
$40$	−3.92069	−0.619916
$41$	−2.52547	−0.394412	−0.197206	−	0.980362i	$-0.563187\pi$
$41$	−2.52547	−0.394412	−0.197206	+	0.980362i	$0.563187\pi$
$42$	0	0
$43$	−10.3098	−1.57223	−0.786116	−	0.618079i	$-0.787911\pi$
$43$	−10.3098	−1.57223	−0.786116	+	0.618079i	$0.787911\pi$
$44$	11.2818	1.70079
$45$	0	0
$46$	−8.29512	−1.22305
$47$	1.55127	0.226277	0.113138	−	0.993579i	$-0.463910\pi$
$47$	1.55127	0.226277	0.113138	+	0.993579i	$0.463910\pi$
$48$	0	0
$49$	2.31984	0.331406
$50$	−4.44667	−0.628855
$51$	0	0
$52$	13.2645	1.83945
$53$	4.37202	0.600543	0.300272	−	0.953854i	$-0.402923\pi$
$53$	4.37202	0.600543	0.300272	+	0.953854i	$0.402923\pi$
$54$	0	0
$55$	11.1722	1.50646
$56$	4.50628	0.602177
$57$	0	0
$58$	13.1703	1.72934
$59$	7.67905	0.999728	0.499864	−	0.866104i	$-0.333383\pi$
$59$	7.67905	0.999728	0.499864	+	0.866104i	$0.333383\pi$
$60$	0	0
$61$	1.37193	0.175658	0.0878288	−	0.996136i	$-0.472007\pi$
$61$	1.37193	0.175658	0.0878288	+	0.996136i	$0.472007\pi$
$62$	0	0
$63$	0	0
$64$	−12.2092	−1.52615
$65$	13.1357	1.62928
$66$	0	0
$67$	7.60375	0.928946	0.464473	−	0.885587i	$-0.346244\pi$
$67$	7.60375	0.928946	0.464473	+	0.885587i	$0.346244\pi$
$68$	−1.32494	−0.160673
$69$	0	0
$70$	17.5459	2.09714
$71$	−4.38206	−0.520055	−0.260028	−	0.965601i	$-0.583732\pi$
$71$	−4.38206	−0.520055	−0.260028	+	0.965601i	$0.583732\pi$
$72$	0	0
$73$	13.9386	1.63139	0.815695	−	0.578483i	$-0.196355\pi$
$73$	13.9386	1.63139	0.815695	+	0.578483i	$0.196355\pi$
$74$	−12.0529	−1.40112
$75$	0	0
$76$	−1.50686	−0.172849
$77$	−12.8409	−1.46336
$78$	0	0
$79$	2.61080	0.293738	0.146869	−	0.989156i	$-0.453080\pi$
$79$	2.61080	0.293738	0.146869	+	0.989156i	$0.453080\pi$
$80$	−5.76463	−0.644505
$81$	0	0
$82$	5.46469	0.603474
$83$	13.3751	1.46811	0.734057	−	0.679088i	$-0.237625\pi$
$83$	13.3751	1.46811	0.734057	+	0.679088i	$0.237625\pi$
$84$	0	0
$85$	−1.31208	−0.142315
$86$	22.3087	2.40561
$87$	0	0
$88$	−6.20877	−0.661857
$89$	7.92836	0.840405	0.420202	−	0.907430i	$-0.361959\pi$
$89$	7.92836	0.840405	0.420202	+	0.907430i	$0.361959\pi$
$90$	0	0
$91$	−15.0976	−1.58266
$92$	10.2822	1.07199
$93$	0	0
$94$	−3.35670	−0.346217
$95$	−1.49223	−0.153099
$96$	0	0
$97$	−14.6782	−1.49034	−0.745172	−	0.666873i	$-0.767633\pi$
$97$	−14.6782	−1.49034	−0.745172	+	0.666873i	$0.767633\pi$
$98$	−5.01975	−0.507071
$99$	0	0
$100$	5.51185	0.551185
$101$	9.19096	0.914535	0.457268	−	0.889329i	$-0.348828\pi$
$101$	9.19096	0.914535	0.457268	+	0.889329i	$0.348828\pi$
$102$	0	0
$103$	−0.325776	−0.0320996	−0.0160498	−	0.999871i	$-0.505109\pi$
$103$	−0.325776	−0.0320996	−0.0160498	+	0.999871i	$0.505109\pi$
$104$	−7.29991	−0.715815
$105$	0	0
$106$	−9.46032	−0.918867
$107$	−2.18451	−0.211184	−0.105592	−	0.994410i	$-0.533674\pi$
$107$	−2.18451	−0.211184	−0.105592	+	0.994410i	$0.533674\pi$
$108$	0	0
$109$	−9.63259	−0.922635	−0.461317	−	0.887235i	$-0.652623\pi$
$109$	−9.63259	−0.922635	−0.461317	+	0.887235i	$0.652623\pi$
$110$	−24.1748	−2.30498
$111$	0	0
$112$	6.62563	0.626063
$113$	−4.83577	−0.454911	−0.227456	−	0.973788i	$-0.573041\pi$
$113$	−4.83577	−0.454911	−0.227456	+	0.973788i	$0.573041\pi$
$114$	0	0
$115$	10.1823	0.949509
$116$	−16.3252	−1.51575
$117$	0	0
$118$	−16.6162	−1.52964
$119$	1.50805	0.138243
$120$	0	0
$121$	6.69225	0.608386
$122$	−2.96862	−0.268767
$123$	0	0
$124$	0	0
$125$	−7.82229	−0.699647
$126$	0	0
$127$	−0.830425	−0.0736883	−0.0368442	−	0.999321i	$-0.511731\pi$
$127$	−0.830425	−0.0736883	−0.0368442	+	0.999321i	$0.511731\pi$
$128$	11.1219	0.983042
$129$	0	0
$130$	−28.4234	−2.49289
$131$	18.6343	1.62808	0.814041	−	0.580807i	$-0.197263\pi$
$131$	18.6343	1.62808	0.814041	+	0.580807i	$0.197263\pi$
$132$	0	0
$133$	1.71511	0.148719
$134$	−16.4532	−1.42134
$135$	0	0
$136$	0.729164	0.0625253
$137$	−2.77208	−0.236835	−0.118418	−	0.992964i	$-0.537782\pi$
$137$	−2.77208	−0.236835	−0.118418	+	0.992964i	$0.537782\pi$
$138$	0	0
$139$	−0.0110333	−0.000935831 0	−0.000467915	−	1.00000i	$-0.500149\pi$
$139$	−0.0110333	−0.000935831 0	−0.000467915		1.00000i	$0.500149\pi$
$140$	−21.7490	−1.83812
$141$	0	0
$142$	9.48205	0.795716
$143$	20.8015	1.73951
$144$	0	0
$145$	−16.1667	−1.34257
$146$	−30.1608	−2.49612
$147$	0	0
$148$	14.9401	1.22807
$149$	5.46420	0.447645	0.223823	−	0.974630i	$-0.428146\pi$
$149$	5.46420	0.447645	0.223823	+	0.974630i	$0.428146\pi$
$150$	0	0
$151$	13.2823	1.08090	0.540451	−	0.841376i	$-0.318254\pi$
$151$	13.2823	1.08090	0.540451	+	0.841376i	$0.318254\pi$
$152$	0.829279	0.0672634
$153$	0	0
$154$	27.7856	2.23902
$155$	0	0
$156$	0	0
$157$	−14.0172	−1.11869	−0.559347	−	0.828933i	$-0.688948\pi$
$157$	−14.0172	−1.11869	−0.559347	+	0.828933i	$0.688948\pi$
$158$	−5.64933	−0.449436
$159$	0	0
$160$	20.3151	1.60605
$161$	−11.7032	−0.922339
$162$	0	0
$163$	−1.51602	−0.118744	−0.0593719	−	0.998236i	$-0.518910\pi$
$163$	−1.51602	−0.118744	−0.0593719	+	0.998236i	$0.518910\pi$
$164$	−6.77372	−0.528939
$165$	0	0
$166$	−28.9416	−2.24630
$167$	9.56374	0.740064	0.370032	−	0.929019i	$-0.379347\pi$
$167$	9.56374	0.740064	0.370032	+	0.929019i	$0.379347\pi$
$168$	0	0
$169$	11.4572	0.881324
$170$	2.83911	0.217750
$171$	0	0
$172$	−27.6527	−2.10849
$173$	12.9171	0.982065	0.491033	−	0.871141i	$-0.336620\pi$
$173$	12.9171	0.982065	0.491033	+	0.871141i	$0.336620\pi$
$174$	0	0
$175$	−6.27359	−0.474239
$176$	−9.12881	−0.688110
$177$	0	0
$178$	−17.1556	−1.28587
$179$	−13.7233	−1.02573	−0.512865	−	0.858469i	$-0.671416\pi$
$179$	−13.7233	−1.02573	−0.512865	+	0.858469i	$0.671416\pi$
$180$	0	0
$181$	−16.8766	−1.25442	−0.627212	−	0.778848i	$-0.715804\pi$
$181$	−16.8766	−1.25442	−0.627212	+	0.778848i	$0.715804\pi$
$182$	32.6687	2.42156
$183$	0	0
$184$	−5.65866	−0.417162
$185$	14.7951	1.08775
$186$	0	0
$187$	−2.07779	−0.151943
$188$	4.16078	0.303456
$189$	0	0
$190$	3.22893	0.234251
$191$	24.1218	1.74539	0.872695	−	0.488266i	$-0.162371\pi$
$191$	24.1218	1.74539	0.872695	+	0.488266i	$0.162371\pi$
$192$	0	0
$193$	10.4950	0.755443	0.377722	−	0.925919i	$-0.376708\pi$
$193$	10.4950	0.755443	0.377722	+	0.925919i	$0.376708\pi$
$194$	31.7611	2.28032
$195$	0	0
$196$	6.22220	0.444443
$197$	−13.8623	−0.987650	−0.493825	−	0.869561i	$-0.664402\pi$
$197$	−13.8623	−0.987650	−0.493825	+	0.869561i	$0.664402\pi$
$198$	0	0
$199$	22.3674	1.58559	0.792793	−	0.609491i	$-0.208626\pi$
$199$	22.3674	1.58559	0.792793	+	0.609491i	$0.208626\pi$
$200$	−3.03337	−0.214492
$201$	0	0
$202$	−19.8877	−1.39929
$203$	18.5813	1.30415
$204$	0	0
$205$	−6.70796	−0.468504
$206$	0.704924	0.0491144
$207$	0	0
$208$	−10.7331	−0.744208
$209$	−2.36308	−0.163458
$210$	0	0
$211$	22.9982	1.58326	0.791631	−	0.611000i	$-0.209232\pi$
$211$	22.9982	1.58326	0.791631	+	0.611000i	$0.209232\pi$
$212$	11.7265	0.805379
$213$	0	0
$214$	4.72691	0.323125
$215$	−27.3842	−1.86758
$216$	0	0
$217$	0	0
$218$	20.8433	1.41169
$219$	0	0
$220$	29.9658	2.02029
$221$	−2.44295	−0.164331
$222$	0	0
$223$	−28.9945	−1.94162	−0.970808	−	0.239860i	$-0.922899\pi$
$223$	−28.9945	−1.94162	−0.970808	+	0.239860i	$0.922899\pi$
$224$	−23.3493	−1.56009
$225$	0	0
$226$	10.4638	0.696042
$227$	5.08516	0.337514	0.168757	−	0.985658i	$-0.446025\pi$
$227$	5.08516	0.337514	0.168757	+	0.985658i	$0.446025\pi$
$228$	0	0
$229$	−8.74533	−0.577908	−0.288954	−	0.957343i	$-0.593307\pi$
$229$	−8.74533	−0.577908	−0.288954	+	0.957343i	$0.593307\pi$
$230$	−22.0329	−1.45281
$231$	0	0
$232$	8.98433	0.589850
$233$	−14.5624	−0.954013	−0.477007	−	0.878900i	$-0.658278\pi$
$233$	−14.5624	−0.954013	−0.477007	+	0.878900i	$0.658278\pi$
$234$	0	0
$235$	4.12038	0.268784
$236$	20.5965	1.34072
$237$	0	0
$238$	−3.26316	−0.211519
$239$	16.0131	1.03580	0.517902	−	0.855440i	$-0.326713\pi$
$239$	16.0131	1.03580	0.517902	+	0.855440i	$0.326713\pi$
$240$	0	0
$241$	27.5794	1.77654	0.888272	−	0.459318i	$-0.151906\pi$
$241$	27.5794	1.77654	0.888272	+	0.459318i	$0.151906\pi$
$242$	−14.4809	−0.930867
$243$	0	0
$244$	3.67974	0.235572
$245$	6.16179	0.393662
$246$	0	0
$247$	−2.77837	−0.176783
$248$	0	0
$249$	0	0
$250$	16.9261	1.07050
$251$	−23.7908	−1.50166	−0.750830	−	0.660495i	$-0.770346\pi$
$251$	−23.7908	−1.50166	−0.750830	+	0.660495i	$0.770346\pi$
$252$	0	0
$253$	16.1247	1.01375
$254$	1.79690	0.112748
$255$	0	0
$256$	0.352558	0.0220349
$257$	7.42887	0.463400	0.231700	−	0.972787i	$-0.425571\pi$
$257$	7.42887	0.463400	0.231700	+	0.972787i	$0.425571\pi$
$258$	0	0
$259$	−17.0048	−1.05663
$260$	35.2320	2.18500
$261$	0	0
$262$	−40.3214	−2.49106
$263$	12.3819	0.763499	0.381749	−	0.924266i	$-0.375322\pi$
$263$	12.3819	0.763499	0.381749	+	0.924266i	$0.375322\pi$
$264$	0	0
$265$	11.6126	0.713359
$266$	−3.71120	−0.227548
$267$	0	0
$268$	20.3945	1.24579
$269$	−8.21200	−0.500694	−0.250347	−	0.968156i	$-0.580545\pi$
$269$	−8.21200	−0.500694	−0.250347	+	0.968156i	$0.580545\pi$
$270$	0	0
$271$	18.3683	1.11579	0.557897	−	0.829910i	$-0.311608\pi$
$271$	18.3683	1.11579	0.557897	+	0.829910i	$0.311608\pi$
$272$	1.07210	0.0650054
$273$	0	0
$274$	5.99832	0.362372
$275$	8.64377	0.521239
$276$	0	0
$277$	−19.1853	−1.15273	−0.576365	−	0.817192i	$-0.695530\pi$
$277$	−19.1853	−1.15273	−0.576365	+	0.817192i	$0.695530\pi$
$278$	0.0238742	0.00143188
$279$	0	0
$280$	11.9692	0.715299
$281$	−6.73207	−0.401601	−0.200801	−	0.979632i	$-0.564354\pi$
$281$	−6.73207	−0.401601	−0.200801	+	0.979632i	$0.564354\pi$
$282$	0	0
$283$	20.0935	1.19443	0.597216	−	0.802080i	$-0.296273\pi$
$283$	20.0935	1.19443	0.597216	+	0.802080i	$0.296273\pi$
$284$	−11.7534	−0.697437
$285$	0	0
$286$	−45.0110	−2.66155
$287$	7.70985	0.455098
$288$	0	0
$289$	−16.7560	−0.985646
$290$	34.9819	2.05421
$291$	0	0
$292$	37.3856	2.18783
$293$	11.3699	0.664237	0.332118	−	0.943238i	$-0.392237\pi$
$293$	11.3699	0.664237	0.332118	+	0.943238i	$0.392237\pi$
$294$	0	0
$295$	20.3965	1.18753
$296$	−8.22209	−0.477899
$297$	0	0
$298$	−11.8236	−0.684924
$299$	18.9584	1.09640
$300$	0	0
$301$	31.4742	1.81415
$302$	−28.7407	−1.65384
$303$	0	0
$304$	1.21930	0.0699315
$305$	3.64402	0.208656
$306$	0	0
$307$	−5.23341	−0.298686	−0.149343	−	0.988785i	$-0.547716\pi$
$307$	−5.23341	−0.298686	−0.149343	+	0.988785i	$0.547716\pi$
$308$	−34.4415	−1.96248
$309$	0	0
$310$	0	0
$311$	−7.85158	−0.445222	−0.222611	−	0.974907i	$-0.571458\pi$
$311$	−7.85158	−0.445222	−0.222611	+	0.974907i	$0.571458\pi$
$312$	0	0
$313$	−7.33101	−0.414373	−0.207187	−	0.978301i	$-0.566431\pi$
$313$	−7.33101	−0.414373	−0.207187	+	0.978301i	$0.566431\pi$
$314$	30.3309	1.71167
$315$	0	0
$316$	7.00260	0.393927
$317$	−7.35696	−0.413208	−0.206604	−	0.978425i	$-0.566241\pi$
$317$	−7.35696	−0.413208	−0.206604	+	0.978425i	$0.566241\pi$
$318$	0	0
$319$	−25.6014	−1.43340
$320$	−32.4291	−1.81284
$321$	0	0
$322$	25.3237	1.41123
$323$	0.277522	0.0154417
$324$	0	0
$325$	10.1628	0.563733
$326$	3.28041	0.181685
$327$	0	0
$328$	3.72783	0.205835
$329$	−4.73579	−0.261093
$330$	0	0
$331$	0.807899	0.0444062	0.0222031	−	0.999753i	$-0.492932\pi$
$331$	0.807899	0.0444062	0.0222031	+	0.999753i	$0.492932\pi$
$332$	35.8744	1.96886
$333$	0	0
$334$	−20.6943	−1.13234
$335$	20.1965	1.10345
$336$	0	0
$337$	−17.1921	−0.936511	−0.468256	−	0.883593i	$-0.655117\pi$
$337$	−17.1921	−0.936511	−0.468256	+	0.883593i	$0.655117\pi$
$338$	−24.7915	−1.34848
$339$	0	0
$340$	−3.51921	−0.190856
$341$	0	0
$342$	0	0
$343$	14.2878	0.771468
$344$	15.2183	0.820514
$345$	0	0
$346$	−27.9503	−1.50262
$347$	−14.9198	−0.800935	−0.400467	−	0.916311i	$-0.631152\pi$
$347$	−14.9198	−0.800935	−0.400467	+	0.916311i	$0.631152\pi$
$348$	0	0
$349$	23.8452	1.27640	0.638202	−	0.769869i	$-0.279679\pi$
$349$	23.8452	1.27640	0.638202	+	0.769869i	$0.279679\pi$
$350$	13.5750	0.725614
$351$	0	0
$352$	32.1707	1.71471
$353$	−13.3682	−0.711515	−0.355757	−	0.934578i	$-0.615777\pi$
$353$	−13.3682	−0.711515	−0.355757	+	0.934578i	$0.615777\pi$
$354$	0	0
$355$	−11.6393	−0.617750
$356$	21.2652	1.12705
$357$	0	0
$358$	29.6950	1.56943
$359$	36.6920	1.93653	0.968265	−	0.249924i	$-0.0804057\pi$
$359$	36.6920	1.93653	0.968265	+	0.249924i	$0.0804057\pi$
$360$	0	0
$361$	−18.6844	−0.983388
$362$	36.5180	1.91934
$363$	0	0
$364$	−40.4943	−2.12248
$365$	37.0226	1.93785
$366$	0	0
$367$	−23.2597	−1.21415	−0.607074	−	0.794645i	$-0.707657\pi$
$367$	−23.2597	−1.21415	−0.607074	+	0.794645i	$0.707657\pi$
$368$	−8.31998	−0.433709
$369$	0	0
$370$	−32.0140	−1.66433
$371$	−13.3471	−0.692946
$372$	0	0
$373$	9.15853	0.474211	0.237105	−	0.971484i	$-0.423801\pi$
$373$	9.15853	0.474211	0.237105	+	0.971484i	$0.423801\pi$
$374$	4.49600	0.232482
$375$	0	0
$376$	−2.28983	−0.118089
$377$	−30.1006	−1.55026
$378$	0	0
$379$	15.7023	0.806571	0.403286	−	0.915074i	$-0.367868\pi$
$379$	15.7023	0.806571	0.403286	+	0.915074i	$0.367868\pi$
$380$	−4.00240	−0.205319
$381$	0	0
$382$	−52.1954	−2.67055
$383$	−3.53055	−0.180403	−0.0902013	−	0.995924i	$-0.528751\pi$
$383$	−3.53055	−0.180403	−0.0902013	+	0.995924i	$0.528751\pi$
$384$	0	0
$385$	−34.1071	−1.73826
$386$	−22.7093	−1.15587
$387$	0	0
$388$	−39.3693	−1.99868
$389$	−7.24656	−0.367415	−0.183708	−	0.982981i	$-0.558810\pi$
$389$	−7.24656	−0.367415	−0.183708	+	0.982981i	$0.558810\pi$
$390$	0	0
$391$	−1.89370	−0.0957683
$392$	−3.42431	−0.172954
$393$	0	0
$394$	29.9957	1.51116
$395$	6.93461	0.348918
$396$	0	0
$397$	−35.1212	−1.76268	−0.881341	−	0.472481i	$-0.843359\pi$
$397$	−35.1212	−1.76268	−0.881341	+	0.472481i	$0.843359\pi$
$398$	−48.3994	−2.42604
$399$	0	0
$400$	−4.46000	−0.223000
$401$	31.5507	1.57557	0.787783	−	0.615953i	$-0.211229\pi$
$401$	31.5507	1.57557	0.787783	+	0.615953i	$0.211229\pi$
$402$	0	0
$403$	0	0
$404$	24.6517	1.22647
$405$	0	0
$406$	−40.2068	−1.99543
$407$	23.4293	1.16135
$408$	0	0
$409$	−16.8986	−0.835580	−0.417790	−	0.908544i	$-0.637195\pi$
$409$	−16.8986	−0.835580	−0.417790	+	0.908544i	$0.637195\pi$
$410$	14.5149	0.716839
$411$	0	0
$412$	−0.873785	−0.0430483
$413$	−23.4429	−1.15355
$414$	0	0
$415$	35.5261	1.74391
$416$	37.8245	1.85450
$417$	0	0
$418$	5.11330	0.250100
$419$	4.37021	0.213499	0.106749	−	0.994286i	$-0.465956\pi$
$419$	4.37021	0.213499	0.106749	+	0.994286i	$0.465956\pi$
$420$	0	0
$421$	22.8400	1.11315	0.556576	−	0.830796i	$-0.312115\pi$
$421$	22.8400	1.11315	0.556576	+	0.830796i	$0.312115\pi$
$422$	−49.7642	−2.42249
$423$	0	0
$424$	−6.45352	−0.313410
$425$	−1.01513	−0.0492412
$426$	0	0
$427$	−4.18828	−0.202685
$428$	−5.85922	−0.283216
$429$	0	0
$430$	59.2547	2.85752
$431$	6.85039	0.329972	0.164986	−	0.986296i	$-0.447242\pi$
$431$	6.85039	0.329972	0.164986	+	0.986296i	$0.447242\pi$
$432$	0	0
$433$	21.6484	1.04036	0.520179	−	0.854058i	$-0.325865\pi$
$433$	21.6484	1.04036	0.520179	+	0.854058i	$0.325865\pi$
$434$	0	0
$435$	0	0
$436$	−25.8362	−1.23733
$437$	−2.15370	−0.103026
$438$	0	0
$439$	−12.4140	−0.592487	−0.296243	−	0.955112i	$-0.595734\pi$
$439$	−12.4140	−0.592487	−0.296243	+	0.955112i	$0.595734\pi$
$440$	−16.4913	−0.786190
$441$	0	0
$442$	5.28613	0.251436
$443$	−2.25969	−0.107361	−0.0536805	−	0.998558i	$-0.517095\pi$
$443$	−2.25969	−0.107361	−0.0536805	+	0.998558i	$0.517095\pi$
$444$	0	0
$445$	21.0587	0.998279
$446$	62.7392	2.97079
$447$	0	0
$448$	37.2727	1.76097
$449$	27.0295	1.27560	0.637801	−	0.770201i	$-0.279844\pi$
$449$	27.0295	1.27560	0.637801	+	0.770201i	$0.279844\pi$
$450$	0	0
$451$	−10.6227	−0.500202
$452$	−12.9704	−0.610074
$453$	0	0
$454$	−11.0034	−0.516417
$455$	−40.1011	−1.87997
$456$	0	0
$457$	15.9888	0.747926	0.373963	−	0.927444i	$-0.377999\pi$
$457$	15.9888	0.747926	0.373963	+	0.927444i	$0.377999\pi$
$458$	18.9234	0.884233
$459$	0	0
$460$	27.3107	1.27337
$461$	−41.4690	−1.93140	−0.965701	−	0.259656i	$-0.916391\pi$
$461$	−41.4690	−1.93140	−0.965701	+	0.259656i	$0.916391\pi$
$462$	0	0
$463$	32.5036	1.51057	0.755286	−	0.655396i	$-0.227498\pi$
$463$	32.5036	1.51057	0.755286	+	0.655396i	$0.227498\pi$
$464$	13.2097	0.613247
$465$	0	0
$466$	31.5105	1.45970
$467$	1.73994	0.0805149	0.0402575	−	0.999189i	$-0.487182\pi$
$467$	1.73994	0.0805149	0.0402575	+	0.999189i	$0.487182\pi$
$468$	0	0
$469$	−23.2130	−1.07188
$470$	−8.91580	−0.411255
$471$	0	0
$472$	−11.3350	−0.521736
$473$	−43.3653	−1.99394
$474$	0	0
$475$	−1.15451	−0.0529726
$476$	4.04484	0.185395
$477$	0	0
$478$	−34.6497	−1.58484
$479$	6.74338	0.308113	0.154056	−	0.988062i	$-0.450766\pi$
$479$	6.74338	0.308113	0.154056	+	0.988062i	$0.450766\pi$
$480$	0	0
$481$	27.5468	1.25603
$482$	−59.6771	−2.71822
$483$	0	0
$484$	17.9497	0.815897
$485$	−38.9871	−1.77031
$486$	0	0
$487$	−26.3634	−1.19464	−0.597321	−	0.802003i	$-0.703768\pi$
$487$	−26.3634	−1.19464	−0.597321	+	0.802003i	$0.703768\pi$
$488$	−2.02510	−0.0916719
$489$	0	0
$490$	−13.3331	−0.602327
$491$	−21.6768	−0.978259	−0.489130	−	0.872211i	$-0.662686\pi$
$491$	−21.6768	−0.978259	−0.489130	+	0.872211i	$0.662686\pi$
$492$	0	0
$493$	3.00665	0.135413
$494$	6.01193	0.270489
$495$	0	0
$496$	0	0
$497$	13.3777	0.600074
$498$	0	0
$499$	−19.3724	−0.867227	−0.433614	−	0.901099i	$-0.642762\pi$
$499$	−19.3724	−0.867227	−0.433614	+	0.901099i	$0.642762\pi$
$500$	−20.9807	−0.938285
$501$	0	0
$502$	51.4792	2.29763
$503$	−20.0681	−0.894792	−0.447396	−	0.894336i	$-0.647649\pi$
$503$	−20.0681	−0.894792	−0.447396	+	0.894336i	$0.647649\pi$
$504$	0	0
$505$	24.4123	1.08634
$506$	−34.8911	−1.55110
$507$	0	0
$508$	−2.22734	−0.0988222
$509$	24.4969	1.08581	0.542904	−	0.839795i	$-0.317325\pi$
$509$	24.4969	1.08581	0.542904	+	0.839795i	$0.317325\pi$
$510$	0	0
$511$	−42.5523	−1.88240
$512$	−23.0066	−1.01676
$513$	0	0
$514$	−16.0748	−0.709030
$515$	−0.865301	−0.0381297
$516$	0	0
$517$	6.52499	0.286969
$518$	36.7956	1.61671
$519$	0	0
$520$	−19.3895	−0.850285
$521$	15.3593	0.672905	0.336452	−	0.941700i	$-0.390773\pi$
$521$	15.3593	0.672905	0.336452	+	0.941700i	$0.390773\pi$
$522$	0	0
$523$	−44.9756	−1.96665	−0.983324	−	0.181863i	$-0.941787\pi$
$523$	−44.9756	−1.96665	−0.983324	+	0.181863i	$0.941787\pi$
$524$	49.9802	2.18339
$525$	0	0
$526$	−26.7923	−1.16820
$527$	0	0
$528$	0	0
$529$	−8.30402	−0.361044
$530$	−25.1278	−1.09148
$531$	0	0
$532$	4.60020	0.199444
$533$	−12.4895	−0.540981
$534$	0	0
$535$	−5.80233	−0.250856
$536$	−11.2239	−0.484796
$537$	0	0
$538$	17.7694	0.766092
$539$	9.75776	0.420296
$540$	0	0
$541$	20.5151	0.882014	0.441007	−	0.897504i	$-0.354621\pi$
$541$	20.5151	0.882014	0.441007	+	0.897504i	$0.354621\pi$
$542$	−39.7459	−1.70723
$543$	0	0
$544$	−3.77816	−0.161987
$545$	−25.5854	−1.09596
$546$	0	0
$547$	−12.2304	−0.522933	−0.261467	−	0.965213i	$-0.584206\pi$
$547$	−12.2304	−0.522933	−0.261467	+	0.965213i	$0.584206\pi$
$548$	−7.43519	−0.317616
$549$	0	0
$550$	−18.7037	−0.797527
$551$	3.41947	0.145674
$552$	0	0
$553$	−7.97035	−0.338934
$554$	41.5137	1.76375
$555$	0	0
$556$	−0.0295931	−0.00125503
$557$	−3.37736	−0.143103	−0.0715516	−	0.997437i	$-0.522795\pi$
$557$	−3.37736	−0.143103	−0.0715516	+	0.997437i	$0.522795\pi$
$558$	0	0
$559$	−50.9864	−2.15650
$560$	17.5985	0.743672
$561$	0	0
$562$	14.5671	0.614474
$563$	−14.3120	−0.603180	−0.301590	−	0.953438i	$-0.597517\pi$
$563$	−14.3120	−0.603180	−0.301590	+	0.953438i	$0.597517\pi$
$564$	0	0
$565$	−12.8444	−0.540369
$566$	−43.4789	−1.82755
$567$	0	0
$568$	6.46834	0.271405
$569$	−21.0164	−0.881051	−0.440526	−	0.897740i	$-0.645208\pi$
$569$	−21.0164	−0.881051	−0.440526	+	0.897740i	$0.645208\pi$
$570$	0	0
$571$	11.5696	0.484173	0.242086	−	0.970255i	$-0.422168\pi$
$571$	11.5696	0.484173	0.242086	+	0.970255i	$0.422168\pi$
$572$	55.7931	2.33283
$573$	0	0
$574$	−16.6828	−0.696327
$575$	7.87791	0.328532
$576$	0	0
$577$	24.5720	1.02295	0.511473	−	0.859299i	$-0.329100\pi$
$577$	24.5720	1.02295	0.511473	+	0.859299i	$0.329100\pi$
$578$	36.2571	1.50810
$579$	0	0
$580$	−43.3617	−1.80050
$581$	−40.8322	−1.69401
$582$	0	0
$583$	18.3897	0.761622
$584$	−20.5747	−0.851386
$585$	0	0
$586$	−24.6025	−1.01632
$587$	10.2999	0.425121	0.212560	−	0.977148i	$-0.431820\pi$
$587$	10.2999	0.425121	0.212560	+	0.977148i	$0.431820\pi$
$588$	0	0
$589$	0	0
$590$	−44.1347	−1.81699
$591$	0	0
$592$	−12.0890	−0.496855
$593$	28.9149	1.18739	0.593697	−	0.804689i	$-0.297668\pi$
$593$	28.9149	1.18739	0.593697	+	0.804689i	$0.297668\pi$
$594$	0	0
$595$	4.00557	0.164212
$596$	14.6559	0.600329
$597$	0	0
$598$	−41.0229	−1.67755
$599$	48.5687	1.98446	0.992231	−	0.124408i	$-0.0397031\pi$
$599$	48.5687	1.98446	0.992231	+	0.124408i	$0.0397031\pi$
$600$	0	0
$601$	−20.0124	−0.816323	−0.408162	−	0.912910i	$-0.633830\pi$
$601$	−20.0124	−0.816323	−0.408162	+	0.912910i	$0.633830\pi$
$602$	−68.1050	−2.77575
$603$	0	0
$604$	35.6255	1.44958
$605$	17.7754	0.722675
$606$	0	0
$607$	−25.9879	−1.05482	−0.527409	−	0.849611i	$-0.676836\pi$
$607$	−25.9879	−1.05482	−0.527409	+	0.849611i	$0.676836\pi$
$608$	−4.29691	−0.174263
$609$	0	0
$610$	−7.88504	−0.319256
$611$	7.67171	0.310364
$612$	0	0
$613$	15.9135	0.642741	0.321370	−	0.946954i	$-0.395857\pi$
$613$	15.9135	0.642741	0.321370	+	0.946954i	$0.395857\pi$
$614$	11.3242	0.457008
$615$	0	0
$616$	18.9544	0.763694
$617$	−16.8884	−0.679901	−0.339950	−	0.940443i	$-0.610410\pi$
$617$	−16.8884	−0.679901	−0.339950	+	0.940443i	$0.610410\pi$
$618$	0	0
$619$	5.26387	0.211573	0.105786	−	0.994389i	$-0.466264\pi$
$619$	5.26387	0.211573	0.105786	+	0.994389i	$0.466264\pi$
$620$	0	0
$621$	0	0
$622$	16.9895	0.681216
$623$	−24.2040	−0.969714
$624$	0	0
$625$	−31.0520	−1.24208
$626$	15.8631	0.634016
$627$	0	0
$628$	−37.5965	−1.50026
$629$	−2.75156	−0.109712
$630$	0	0
$631$	−40.3121	−1.60480	−0.802400	−	0.596787i	$-0.796444\pi$
$631$	−40.3121	−1.60480	−0.802400	+	0.596787i	$0.796444\pi$
$632$	−3.85378	−0.153295
$633$	0	0
$634$	15.9192	0.632233
$635$	−2.20571	−0.0875311
$636$	0	0
$637$	11.4726	0.454561
$638$	55.3971	2.19319
$639$	0	0
$640$	29.5410	1.16771
$641$	0.0796366	0.00314545	0.00157273	−	0.999999i	$-0.499499\pi$
$641$	0.0796366	0.00314545	0.00157273	+	0.999999i	$0.499499\pi$
$642$	0	0
$643$	−10.9289	−0.430994	−0.215497	−	0.976504i	$-0.569137\pi$
$643$	−10.9289	−0.430994	−0.215497	+	0.976504i	$0.569137\pi$
$644$	−31.3899	−1.23693
$645$	0	0
$646$	−0.600511	−0.0236268
$647$	6.88095	0.270518	0.135259	−	0.990810i	$-0.456813\pi$
$647$	6.88095	0.270518	0.135259	+	0.990810i	$0.456813\pi$
$648$	0	0
$649$	32.2998	1.26788
$650$	−21.9907	−0.862546
$651$	0	0
$652$	−4.06622	−0.159245
$653$	−20.7830	−0.813303	−0.406652	−	0.913583i	$-0.633304\pi$
$653$	−20.7830	−0.813303	−0.406652	+	0.913583i	$0.633304\pi$
$654$	0	0
$655$	49.4949	1.93393
$656$	5.48106	0.213999
$657$	0	0
$658$	10.2475	0.399488
$659$	40.6540	1.58366	0.791828	−	0.610744i	$-0.209130\pi$
$659$	40.6540	1.58366	0.791828	+	0.610744i	$0.209130\pi$
$660$	0	0
$661$	10.8366	0.421494	0.210747	−	0.977541i	$-0.432410\pi$
$661$	10.8366	0.421494	0.210747	+	0.977541i	$0.432410\pi$
$662$	−1.74816	−0.0679441
$663$	0	0
$664$	−19.7430	−0.766176
$665$	4.55554	0.176656
$666$	0	0
$667$	−23.3330	−0.903459
$668$	25.6515	0.992488
$669$	0	0
$670$	−43.7018	−1.68835
$671$	5.77063	0.222773
$672$	0	0
$673$	30.1241	1.16120	0.580599	−	0.814190i	$-0.302818\pi$
$673$	30.1241	1.16120	0.580599	+	0.814190i	$0.302818\pi$
$674$	37.2007	1.43292
$675$	0	0
$676$	30.7302	1.18193
$677$	−25.9818	−0.998563	−0.499281	−	0.866440i	$-0.666403\pi$
$677$	−25.9818	−0.998563	−0.499281	+	0.866440i	$0.666403\pi$
$678$	0	0
$679$	44.8102	1.71966
$680$	1.93675	0.0742710
$681$	0	0
$682$	0	0
$683$	−1.20345	−0.0460487	−0.0230243	−	0.999735i	$-0.507330\pi$
$683$	−1.20345	−0.0460487	−0.0230243	+	0.999735i	$0.507330\pi$
$684$	0	0
$685$	−7.36300	−0.281326
$686$	−30.9164	−1.18039
$687$	0	0
$688$	22.3755	0.853060
$689$	21.6215	0.823714
$690$	0	0
$691$	35.0689	1.33409	0.667043	−	0.745019i	$-0.267560\pi$
$691$	35.0689	1.33409	0.667043	+	0.745019i	$0.267560\pi$
$692$	34.6457	1.31703
$693$	0	0
$694$	32.2839	1.22548
$695$	−0.0293058	−0.00111163
$696$	0	0
$697$	1.24754	0.0472538
$698$	−51.5970	−1.95297
$699$	0	0
$700$	−16.8268	−0.635994
$701$	23.8668	0.901435	0.450717	−	0.892667i	$-0.351168\pi$
$701$	23.8668	0.901435	0.450717	+	0.892667i	$0.351168\pi$
$702$	0	0
$703$	−3.12935	−0.118026
$704$	−51.3545	−1.93549
$705$	0	0
$706$	28.9264	1.08866
$707$	−28.0586	−1.05525
$708$	0	0
$709$	13.6597	0.513002	0.256501	−	0.966544i	$-0.417430\pi$
$709$	13.6597	0.513002	0.256501	+	0.966544i	$0.417430\pi$
$710$	25.1855	0.945195
$711$	0	0
$712$	−11.7030	−0.438589
$713$	0	0
$714$	0	0
$715$	55.2514	2.06629
$716$	−36.8083	−1.37559
$717$	0	0
$718$	−79.3954	−2.96301
$719$	12.5553	0.468232	0.234116	−	0.972209i	$-0.424780\pi$
$719$	12.5553	0.468232	0.234116	+	0.972209i	$0.424780\pi$
$720$	0	0
$721$	0.994542	0.0370387
$722$	40.4298	1.50464
$723$	0	0
$724$	−45.2657	−1.68229
$725$	−12.5079	−0.464531
$726$	0	0
$727$	−30.9105	−1.14641	−0.573203	−	0.819414i	$-0.694299\pi$
$727$	−30.9105	−1.14641	−0.573203	+	0.819414i	$0.694299\pi$
$728$	22.2855	0.825955
$729$	0	0
$730$	−80.1108	−2.96503
$731$	5.09286	0.188366
$732$	0	0
$733$	−16.0196	−0.591696	−0.295848	−	0.955235i	$-0.595602\pi$
$733$	−16.0196	−0.591696	−0.295848	+	0.955235i	$0.595602\pi$
$734$	50.3302	1.85772
$735$	0	0
$736$	29.3203	1.08076
$737$	31.9830	1.17811
$738$	0	0
$739$	51.9208	1.90994	0.954969	−	0.296706i	$-0.0958882\pi$
$739$	51.9208	1.90994	0.954969	+	0.296706i	$0.0958882\pi$
$740$	39.6828	1.45877
$741$	0	0
$742$	28.8809	1.06025
$743$	−27.4004	−1.00522	−0.502612	−	0.864512i	$-0.667628\pi$
$743$	−27.4004	−1.00522	−0.502612	+	0.864512i	$0.667628\pi$
$744$	0	0
$745$	14.5136	0.531738
$746$	−19.8175	−0.725571
$747$	0	0
$748$	−5.57299	−0.203769
$749$	6.66896	0.243678
$750$	0	0
$751$	26.2800	0.958971	0.479485	−	0.877550i	$-0.340823\pi$
$751$	26.2800	0.958971	0.479485	+	0.877550i	$0.340823\pi$
$752$	−3.36675	−0.122773
$753$	0	0
$754$	65.1327	2.37199
$755$	35.2796	1.28395
$756$	0	0
$757$	14.7447	0.535907	0.267953	−	0.963432i	$-0.413653\pi$
$757$	14.7447	0.535907	0.267953	+	0.963432i	$0.413653\pi$
$758$	−33.9771	−1.23410
$759$	0	0
$760$	2.20267	0.0798992
$761$	−24.5165	−0.888723	−0.444361	−	0.895848i	$-0.646569\pi$
$761$	−24.5165	−0.888723	−0.444361	+	0.895848i	$0.646569\pi$
$762$	0	0
$763$	29.4068	1.06460
$764$	64.6986	2.34071
$765$	0	0
$766$	7.63951	0.276027
$767$	37.9762	1.37124
$768$	0	0
$769$	31.3760	1.13145	0.565723	−	0.824595i	$-0.308597\pi$
$769$	31.3760	1.13145	0.565723	+	0.824595i	$0.308597\pi$
$770$	73.8019	2.65964
$771$	0	0
$772$	28.1492	1.01311
$773$	36.3296	1.30669	0.653343	−	0.757062i	$-0.273366\pi$
$773$	36.3296	1.30669	0.653343	+	0.757062i	$0.273366\pi$
$774$	0	0
$775$	0	0
$776$	21.6664	0.777778
$777$	0	0
$778$	15.6803	0.562167
$779$	1.41882	0.0508346
$780$	0	0
$781$	−18.4319	−0.659545
$782$	4.09764	0.146531
$783$	0	0
$784$	−5.03479	−0.179814
$785$	−37.2314	−1.32885
$786$	0	0
$787$	11.5370	0.411248	0.205624	−	0.978631i	$-0.434078\pi$
$787$	11.5370	0.411248	0.205624	+	0.978631i	$0.434078\pi$
$788$	−37.1811	−1.32452
$789$	0	0
$790$	−15.0053	−0.533865
$791$	14.7629	0.524907
$792$	0	0
$793$	6.78477	0.240934
$794$	75.9963	2.69701
$795$	0	0
$796$	59.9932	2.12640
$797$	11.8984	0.421464	0.210732	−	0.977544i	$-0.432415\pi$
$797$	11.8984	0.421464	0.210732	+	0.977544i	$0.432415\pi$
$798$	0	0
$799$	−0.766301	−0.0271098
$800$	15.7174	0.555695
$801$	0	0
$802$	−68.2703	−2.41071
$803$	58.6287	2.06896
$804$	0	0
$805$	−31.0851	−1.09561
$806$	0	0
$807$	0	0
$808$	−13.5667	−0.477276
$809$	26.1271	0.918578	0.459289	−	0.888287i	$-0.348104\pi$
$809$	26.1271	0.918578	0.459289	+	0.888287i	$0.348104\pi$
$810$	0	0
$811$	43.0944	1.51325	0.756624	−	0.653850i	$-0.226847\pi$
$811$	43.0944	1.51325	0.756624	+	0.653850i	$0.226847\pi$
$812$	49.8381	1.74898
$813$	0	0
$814$	−50.6971	−1.77693
$815$	−4.02674	−0.141051
$816$	0	0
$817$	5.79212	0.202641
$818$	36.5656	1.27849
$819$	0	0
$820$	−17.9919	−0.628303
$821$	−2.95168	−0.103014	−0.0515071	−	0.998673i	$-0.516402\pi$
$821$	−2.95168	−0.103014	−0.0515071	+	0.998673i	$0.516402\pi$
$822$	0	0
$823$	−16.2427	−0.566186	−0.283093	−	0.959093i	$-0.591360\pi$
$823$	−16.2427	−0.566186	−0.283093	+	0.959093i	$0.591360\pi$
$824$	0.480876	0.0167521
$825$	0	0
$826$	50.7266	1.76500
$827$	51.5167	1.79141	0.895706	−	0.444646i	$-0.146671\pi$
$827$	51.5167	1.79141	0.895706	+	0.444646i	$0.146671\pi$
$828$	0	0
$829$	−29.8166	−1.03557	−0.517786	−	0.855510i	$-0.673244\pi$
$829$	−29.8166	−1.03557	−0.517786	+	0.855510i	$0.673244\pi$
$830$	−76.8724	−2.66828
$831$	0	0
$832$	−60.3796	−2.09329
$833$	−1.14596	−0.0397052
$834$	0	0
$835$	25.4025	0.879089
$836$	−6.33817	−0.219210
$837$	0	0
$838$	−9.45640	−0.326666
$839$	54.6723	1.88750	0.943749	−	0.330664i	$-0.107273\pi$
$839$	54.6723	1.88750	0.943749	+	0.330664i	$0.107273\pi$
$840$	0	0
$841$	8.04620	0.277455
$842$	−49.4219	−1.70319
$843$	0	0
$844$	61.6850	2.12329
$845$	30.4318	1.04689
$846$	0	0
$847$	−20.4304	−0.701996
$848$	−9.48866	−0.325842
$849$	0	0
$850$	2.19658	0.0753419
$851$	21.3534	0.731986
$852$	0	0
$853$	34.2003	1.17100	0.585499	−	0.810673i	$-0.300899\pi$
$853$	34.2003	1.17100	0.585499	+	0.810673i	$0.300899\pi$
$854$	9.06274	0.310121
$855$	0	0
$856$	3.22454	0.110213
$857$	−50.5260	−1.72594	−0.862968	−	0.505258i	$-0.831397\pi$
$857$	−50.5260	−1.72594	−0.862968	+	0.505258i	$0.831397\pi$
$858$	0	0
$859$	−45.8276	−1.56362	−0.781809	−	0.623518i	$-0.785703\pi$
$859$	−45.8276	−1.56362	−0.781809	+	0.623518i	$0.785703\pi$
$860$	−73.4489	−2.50459
$861$	0	0
$862$	−14.8231	−0.504877
$863$	14.6706	0.499394	0.249697	−	0.968324i	$-0.419669\pi$
$863$	14.6706	0.499394	0.249697	+	0.968324i	$0.419669\pi$
$864$	0	0
$865$	34.3093	1.16655
$866$	−46.8436	−1.59181
$867$	0	0
$868$	0	0
$869$	10.9816	0.372525
$870$	0	0
$871$	37.6038	1.27415
$872$	14.2186	0.481503
$873$	0	0
$874$	4.66025	0.157635
$875$	23.8802	0.807298
$876$	0	0
$877$	22.5383	0.761063	0.380532	−	0.924768i	$-0.375741\pi$
$877$	22.5383	0.761063	0.380532	+	0.924768i	$0.375741\pi$
$878$	26.8618	0.906540
$879$	0	0
$880$	−24.2473	−0.817375
$881$	16.3168	0.549726	0.274863	−	0.961483i	$-0.411367\pi$
$881$	16.3168	0.549726	0.274863	+	0.961483i	$0.411367\pi$
$882$	0	0
$883$	46.5638	1.56700	0.783499	−	0.621393i	$-0.213433\pi$
$883$	46.5638	1.56700	0.783499	+	0.621393i	$0.213433\pi$
$884$	−6.55240	−0.220381
$885$	0	0
$886$	4.88959	0.164269
$887$	12.0099	0.403252	0.201626	−	0.979463i	$-0.435378\pi$
$887$	12.0099	0.403252	0.201626	+	0.979463i	$0.435378\pi$
$888$	0	0
$889$	2.53516	0.0850264
$890$	−45.5675	−1.52743
$891$	0	0
$892$	−77.7681	−2.60387
$893$	−0.871516	−0.0291642
$894$	0	0
$895$	−36.4509	−1.21842
$896$	−33.9532	−1.13430
$897$	0	0
$898$	−58.4873	−1.95175
$899$	0	0
$900$	0	0
$901$	−2.15970	−0.0719500
$902$	22.9856	0.765338
$903$	0	0
$904$	7.13806	0.237408
$905$	−44.8262	−1.49007
$906$	0	0
$907$	23.5173	0.780878	0.390439	−	0.920629i	$-0.372323\pi$
$907$	23.5173	0.780878	0.390439	+	0.920629i	$0.372323\pi$
$908$	13.6393	0.452635
$909$	0	0
$910$	86.7720	2.87646
$911$	4.34478	0.143949	0.0719745	−	0.997406i	$-0.477070\pi$
$911$	4.34478	0.143949	0.0719745	+	0.997406i	$0.477070\pi$
$912$	0	0
$913$	56.2587	1.86189
$914$	−34.5972	−1.14437
$915$	0	0
$916$	−23.4564	−0.775022
$917$	−56.8874	−1.87859
$918$	0	0
$919$	7.34792	0.242386	0.121193	−	0.992629i	$-0.461328\pi$
$919$	7.34792	0.242386	0.121193	+	0.992629i	$0.461328\pi$
$920$	−15.0301	−0.495528
$921$	0	0
$922$	89.7319	2.95516
$923$	−21.6712	−0.713315
$924$	0	0
$925$	11.4467	0.376365
$926$	−70.3324	−2.31126
$927$	0	0
$928$	−46.5523	−1.52816
$929$	1.37326	0.0450552	0.0225276	−	0.999746i	$-0.492829\pi$
$929$	1.37326	0.0450552	0.0225276	+	0.999746i	$0.492829\pi$
$930$	0	0
$931$	−1.30330	−0.0427140
$932$	−39.0587	−1.27941
$933$	0	0
$934$	−3.76494	−0.123193
$935$	−5.51888	−0.180487
$936$	0	0
$937$	44.1355	1.44184	0.720922	−	0.693017i	$-0.243719\pi$
$937$	44.1355	1.44184	0.720922	+	0.693017i	$0.243719\pi$
$938$	50.2291	1.64004
$939$	0	0
$940$	11.0515	0.360461
$941$	−15.0592	−0.490916	−0.245458	−	0.969407i	$-0.578938\pi$
$941$	−15.0592	−0.490916	−0.245458	+	0.969407i	$0.578938\pi$
$942$	0	0
$943$	−9.68147	−0.315272
$944$	−16.6660	−0.542431
$945$	0	0
$946$	93.8352	3.05085
$947$	38.0052	1.23500	0.617502	−	0.786569i	$-0.288145\pi$
$947$	38.0052	1.23500	0.617502	+	0.786569i	$0.288145\pi$
$948$	0	0
$949$	68.9323	2.23764
$950$	2.49817	0.0810513
$951$	0	0
$952$	−2.22602	−0.0721458
$953$	−23.3938	−0.757800	−0.378900	−	0.925438i	$-0.623698\pi$
$953$	−23.3938	−0.757800	−0.378900	+	0.925438i	$0.623698\pi$
$954$	0	0
$955$	64.0704	2.07327
$956$	42.9499	1.38910
$957$	0	0
$958$	−14.5915	−0.471431
$959$	8.46273	0.273276
$960$	0	0
$961$	0	0
$962$	−59.6067	−1.92180
$963$	0	0
$964$	73.9725	2.38249
$965$	27.8759	0.897357
$966$	0	0
$967$	4.59862	0.147882	0.0739408	−	0.997263i	$-0.476442\pi$
$967$	4.59862	0.147882	0.0739408	+	0.997263i	$0.476442\pi$
$968$	−9.87839	−0.317503
$969$	0	0
$970$	84.3615	2.70868
$971$	−54.7700	−1.75765	−0.878826	−	0.477142i	$-0.841673\pi$
$971$	−54.7700	−1.75765	−0.878826	+	0.477142i	$0.841673\pi$
$972$	0	0
$973$	0.0336829	0.00107982
$974$	57.0460	1.82787
$975$	0	0
$976$	−2.97752	−0.0953081
$977$	46.1885	1.47770	0.738850	−	0.673870i	$-0.235369\pi$
$977$	46.1885	1.47770	0.738850	+	0.673870i	$0.235369\pi$
$978$	0	0
$979$	33.3484	1.06582
$980$	16.5269	0.527934
$981$	0	0
$982$	46.9049	1.49680
$983$	−41.9239	−1.33716	−0.668582	−	0.743638i	$-0.733099\pi$
$983$	−41.9239	−1.33716	−0.668582	+	0.743638i	$0.733099\pi$
$984$	0	0
$985$	−36.8201	−1.17318
$986$	−6.50588	−0.207190
$987$	0	0
$988$	−7.45205	−0.237081
$989$	−39.5230	−1.25676
$990$	0	0
$991$	−8.11884	−0.257903	−0.128952	−	0.991651i	$-0.541161\pi$
$991$	−8.11884	−0.257903	−0.128952	+	0.991651i	$0.541161\pi$
$992$	0	0
$993$	0	0
$994$	−28.9472	−0.918149
$995$	59.4107	1.88345
$996$	0	0
$997$	12.4657	0.394792	0.197396	−	0.980324i	$-0.436752\pi$
$997$	12.4657	0.394792	0.197396	+	0.980324i	$0.436752\pi$
$998$	41.9186	1.32691
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8649.2.a.bs.1.2		16
3.2	odd	2		961.2.a.l.1.16	yes	16
31.30	odd	2	inner	8649.2.a.bs.1.1		16
93.2	odd	10		961.2.d.s.531.16		64
93.5	odd	6		961.2.c.l.521.15		32
93.8	odd	10		961.2.d.s.374.1		64
93.11	even	30		961.2.g.w.338.2		128
93.14	odd	30		961.2.g.w.816.1		128
93.17	even	30		961.2.g.w.816.2		128
93.20	odd	30		961.2.g.w.338.1		128
93.23	even	10		961.2.d.s.374.2		64
93.26	even	6		961.2.c.l.521.16		32
93.29	even	10		961.2.d.s.531.15		64
93.35	odd	10		961.2.d.s.388.1		64
93.38	odd	30		961.2.g.w.235.2		128
93.41	odd	30		961.2.g.w.844.15		128
93.44	even	30		961.2.g.w.448.15		128
93.47	odd	10		961.2.d.s.628.16		64
93.50	odd	30		961.2.g.w.547.16		128
93.53	even	30		961.2.g.w.732.1		128
93.56	odd	6		961.2.c.l.439.15		32
93.59	odd	30		961.2.g.w.846.15		128
93.65	even	30		961.2.g.w.846.16		128
93.68	even	6		961.2.c.l.439.16		32
93.71	odd	30		961.2.g.w.732.2		128
93.74	even	30		961.2.g.w.547.15		128
93.77	even	10		961.2.d.s.628.15		64
93.80	odd	30		961.2.g.w.448.16		128
93.83	even	30		961.2.g.w.844.16		128
93.86	even	30		961.2.g.w.235.1		128
93.89	even	10		961.2.d.s.388.2		64
93.92	even	2		961.2.a.l.1.15	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
961.2.a.l.1.15	✓	16	93.92	even	2
961.2.a.l.1.16	yes	16	3.2	odd	2
961.2.c.l.439.15		32	93.56	odd	6
961.2.c.l.439.16		32	93.68	even	6
961.2.c.l.521.15		32	93.5	odd	6
961.2.c.l.521.16		32	93.26	even	6
961.2.d.s.374.1		64	93.8	odd	10
961.2.d.s.374.2		64	93.23	even	10
961.2.d.s.388.1		64	93.35	odd	10
961.2.d.s.388.2		64	93.89	even	10
961.2.d.s.531.15		64	93.29	even	10
961.2.d.s.531.16		64	93.2	odd	10
961.2.d.s.628.15		64	93.77	even	10
961.2.d.s.628.16		64	93.47	odd	10
961.2.g.w.235.1		128	93.86	even	30
961.2.g.w.235.2		128	93.38	odd	30
961.2.g.w.338.1		128	93.20	odd	30
961.2.g.w.338.2		128	93.11	even	30
961.2.g.w.448.15		128	93.44	even	30
961.2.g.w.448.16		128	93.80	odd	30
961.2.g.w.547.15		128	93.74	even	30
961.2.g.w.547.16		128	93.50	odd	30
961.2.g.w.732.1		128	93.53	even	30
961.2.g.w.732.2		128	93.71	odd	30
961.2.g.w.816.1		128	93.14	odd	30
961.2.g.w.816.2		128	93.17	even	30
961.2.g.w.844.15		128	93.41	odd	30
961.2.g.w.844.16		128	93.83	even	30
961.2.g.w.846.15		128	93.59	odd	30
961.2.g.w.846.16		128	93.65	even	30
8649.2.a.bs.1.1		16	31.30	odd	2	inner
8649.2.a.bs.1.2		16	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 8649 = 3^{2} \cdot 31^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8649.a (trivial)

\(f(q)\)	\(=\)	\(q-2.16383 q^{2} +2.68217 q^{4} +2.65612 q^{5} -3.05284 q^{7} -1.47609 q^{8} +O(q^{10})\)	\(q-2.16383 q^{2} +2.68217 q^{4} +2.65612 q^{5} -3.05284 q^{7} -1.47609 q^{8} -5.74741 q^{10} +4.20622 q^{11} +4.94542 q^{13} +6.60583 q^{14} -2.17031 q^{16} -0.493982 q^{17} -0.561806 q^{19} +7.12417 q^{20} -9.10154 q^{22} +3.83353 q^{23} +2.05500 q^{25} -10.7011 q^{26} -8.18823 q^{28} -6.08656 q^{29} +7.64838 q^{32} +1.06889 q^{34} -8.10873 q^{35} +5.57017 q^{37} +1.21565 q^{38} -3.92069 q^{40} -2.52547 q^{41} -10.3098 q^{43} +11.2818 q^{44} -8.29512 q^{46} +1.55127 q^{47} +2.31984 q^{49} -4.44667 q^{50} +13.2645 q^{52} +4.37202 q^{53} +11.1722 q^{55} +4.50628 q^{56} +13.1703 q^{58} +7.67905 q^{59} +1.37193 q^{61} -12.2092 q^{64} +13.1357 q^{65} +7.60375 q^{67} -1.32494 q^{68} +17.5459 q^{70} -4.38206 q^{71} +13.9386 q^{73} -12.0529 q^{74} -1.50686 q^{76} -12.8409 q^{77} +2.61080 q^{79} -5.76463 q^{80} +5.46469 q^{82} +13.3751 q^{83} -1.31208 q^{85} +22.3087 q^{86} -6.20877 q^{88} +7.92836 q^{89} -15.0976 q^{91} +10.2822 q^{92} -3.35670 q^{94} -1.49223 q^{95} -14.6782 q^{97} -5.01975 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 8 q^{2} + 8 q^{4} + 16 q^{5} - 16 q^{7}+O(q^{10}) \) 16 * q + 8 * q^2 + 8 * q^4 + 16 * q^5 - 16 * q^7	\( 16 q + 8 q^{2} + 8 q^{4} + 16 q^{5} - 16 q^{7} + 8 q^{10} + 8 q^{14} - 8 q^{16} - 32 q^{19} + 24 q^{20} - 8 q^{28} + 8 q^{32} + 16 q^{35} + 24 q^{38} + 32 q^{41} + 32 q^{47} - 16 q^{49} + 32 q^{50} + 48 q^{56} + 64 q^{59} - 16 q^{64} + 16 q^{67} + 88 q^{70} + 48 q^{71} + 40 q^{76} + 40 q^{80} + 88 q^{82} - 32 q^{94} + 48 q^{95} - 16 q^{98}+O(q^{100}) \) 16 * q + 8 * q^2 + 8 * q^4 + 16 * q^5 - 16 * q^7 + 8 * q^10 + 8 * q^14 - 8 * q^16 - 32 * q^19 + 24 * q^20 - 8 * q^28 + 8 * q^32 + 16 * q^35 + 24 * q^38 + 32 * q^41 + 32 * q^47 - 16 * q^49 + 32 * q^50 + 48 * q^56 + 64 * q^59 - 16 * q^64 + 16 * q^67 + 88 * q^70 + 48 * q^71 + 40 * q^76 + 40 * q^80 + 88 * q^82 - 32 * q^94 + 48 * q^95 - 16 * q^98

Introduction

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