LMFDB - Embedded newform 8208.2.a.bl.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8208,2,Mod(1,8208)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("8208.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8208, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 8208 = 2^{4} \cdot 3^{3} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8208.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,0,0,2,0,2,0,0,0,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$65.5412099791$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 4104)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.311108$ of defining polynomial
Character	$\chi$	$=$	8208.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.90321 q^{5} -1.52543 q^{7} +4.73975 q^{11} -1.37778 q^{13} -5.05086 q^{17} +1.00000 q^{19} -2.90321 q^{23} +3.42864 q^{25} -5.49532 q^{29} +5.70964 q^{31} -4.42864 q^{35} -10.7096 q^{37} -12.3985 q^{41} -6.47457 q^{43} -3.02074 q^{47} -4.67307 q^{49} -2.45875 q^{53} +13.7605 q^{55} +4.66815 q^{59} +8.13828 q^{61} -4.00000 q^{65} -1.28100 q^{67} -14.8573 q^{71} -5.90321 q^{73} -7.23014 q^{77} +0.244431 q^{79} +8.07160 q^{83} -14.6637 q^{85} -9.92396 q^{89} +2.10171 q^{91} +2.90321 q^{95} -9.03657 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 2 q^{5} + 2 q^{7} + q^{11} - 4 q^{13} - 2 q^{17} + 3 q^{19} - 2 q^{23} - 3 q^{25} - 3 q^{29} - 3 q^{31} - 12 q^{37} - 17 q^{41} - 26 q^{43} + 11 q^{47} - q^{49} - q^{53} + 8 q^{55} + 34 q^{59}+ \cdots - 20 q^{97}+O(q^{100}) $ 3 * q + 2 * q^5 + 2 * q^7 + q^11 - 4 * q^13 - 2 * q^17 + 3 * q^19 - 2 * q^23 - 3 * q^25 - 3 * q^29 - 3 * q^31 - 12 * q^37 - 17 * q^41 - 26 * q^43 + 11 * q^47 - q^49 - q^53 + 8 * q^55 + 34 * q^59 - 9 * q^61 - 12 * q^65 + 3 * q^67 - 18 * q^71 - 11 * q^73 - 28 * q^77 + q^79 - 9 * q^83 - 4 * q^85 - 3 * q^89 - 20 * q^91 + 2 * q^95 - 20 * 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
$5$	2.90321	1.29836	0.649178	−	0.760637i	$-0.275113\pi$
$5$	2.90321	1.29836	0.649178	+	0.760637i	$0.275113\pi$
$6$	0	0
$7$	−1.52543	−0.576557	−0.288279	−	0.957547i	$-0.593083\pi$
$7$	−1.52543	−0.576557	−0.288279	+	0.957547i	$0.593083\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	4.73975	1.42909	0.714544	−	0.699591i	$-0.246634\pi$
$11$	4.73975	1.42909	0.714544	+	0.699591i	$0.246634\pi$
$12$	0	0
$13$	−1.37778	−0.382129	−0.191064	−	0.981578i	$-0.561194\pi$
$13$	−1.37778	−0.382129	−0.191064	+	0.981578i	$0.561194\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−5.05086	−1.22501	−0.612506	−	0.790466i	$-0.709839\pi$
$17$	−5.05086	−1.22501	−0.612506	+	0.790466i	$0.709839\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−2.90321	−0.605362	−0.302681	−	0.953092i	$-0.597882\pi$
$23$	−2.90321	−0.605362	−0.302681	+	0.953092i	$0.597882\pi$
$24$	0	0
$25$	3.42864	0.685728
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−5.49532	−1.02045	−0.510227	−	0.860040i	$-0.670439\pi$
$29$	−5.49532	−1.02045	−0.510227	+	0.860040i	$0.670439\pi$
$30$	0	0
$31$	5.70964	1.02548	0.512740	−	0.858544i	$-0.328630\pi$
$31$	5.70964	1.02548	0.512740	+	0.858544i	$0.328630\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−4.42864	−0.748577
$36$	0	0
$37$	−10.7096	−1.76065	−0.880327	−	0.474368i	$-0.842677\pi$
$37$	−10.7096	−1.76065	−0.880327	+	0.474368i	$0.842677\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−12.3985	−1.93632	−0.968162	−	0.250323i	$-0.919463\pi$
$41$	−12.3985	−1.93632	−0.968162	+	0.250323i	$0.919463\pi$
$42$	0	0
$43$	−6.47457	−0.987363	−0.493682	−	0.869643i	$-0.664349\pi$
$43$	−6.47457	−0.987363	−0.493682	+	0.869643i	$0.664349\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−3.02074	−0.440621	−0.220310	−	0.975430i	$-0.570707\pi$
$47$	−3.02074	−0.440621	−0.220310	+	0.975430i	$0.570707\pi$
$48$	0	0
$49$	−4.67307	−0.667582
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−2.45875	−0.337735	−0.168868	−	0.985639i	$-0.554011\pi$
$53$	−2.45875	−0.337735	−0.168868	+	0.985639i	$0.554011\pi$
$54$	0	0
$55$	13.7605	1.85546
$56$	0	0
$57$	0	0
$58$	0	0
$59$	4.66815	0.607741	0.303871	−	0.952713i	$-0.401721\pi$
$59$	4.66815	0.607741	0.303871	+	0.952713i	$0.401721\pi$
$60$	0	0
$61$	8.13828	1.04200	0.521000	−	0.853557i	$-0.325559\pi$
$61$	8.13828	1.04200	0.521000	+	0.853557i	$0.325559\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−4.00000	−0.496139
$66$	0	0
$67$	−1.28100	−0.156499	−0.0782493	−	0.996934i	$-0.524933\pi$
$67$	−1.28100	−0.156499	−0.0782493	+	0.996934i	$0.524933\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−14.8573	−1.76323	−0.881617	−	0.471965i	$-0.843545\pi$
$71$	−14.8573	−1.76323	−0.881617	+	0.471965i	$0.843545\pi$
$72$	0	0
$73$	−5.90321	−0.690919	−0.345459	−	0.938434i	$-0.612277\pi$
$73$	−5.90321	−0.690919	−0.345459	+	0.938434i	$0.612277\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−7.23014	−0.823951
$78$	0	0
$79$	0.244431	0.0275007	0.0137503	−	0.999905i	$-0.495623\pi$
$79$	0.244431	0.0275007	0.0137503	+	0.999905i	$0.495623\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	8.07160	0.885973	0.442987	−	0.896528i	$-0.353919\pi$
$83$	8.07160	0.885973	0.442987	+	0.896528i	$0.353919\pi$
$84$	0	0
$85$	−14.6637	−1.59050
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−9.92396	−1.05194	−0.525969	−	0.850504i	$-0.676297\pi$
$89$	−9.92396	−1.05194	−0.525969	+	0.850504i	$0.676297\pi$
$90$	0	0
$91$	2.10171	0.220319
$92$	0	0
$93$	0	0
$94$	0	0
$95$	2.90321	0.297863
$96$	0	0
$97$	−9.03657	−0.917524	−0.458762	−	0.888559i	$-0.651707\pi$
$97$	−9.03657	−0.917524	−0.458762	+	0.888559i	$0.651707\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−0.668149	−0.0664833	−0.0332416	−	0.999447i	$-0.510583\pi$
$101$	−0.668149	−0.0664833	−0.0332416	+	0.999447i	$0.510583\pi$
$102$	0	0
$103$	1.37778	0.135757	0.0678786	−	0.997694i	$-0.478377\pi$
$103$	1.37778	0.135757	0.0678786	+	0.997694i	$0.478377\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	11.3319	1.09549	0.547746	−	0.836645i	$-0.315486\pi$
$107$	11.3319	1.09549	0.547746	+	0.836645i	$0.315486\pi$
$108$	0	0
$109$	20.0558	1.92100	0.960498	−	0.278288i	$-0.0897671\pi$
$109$	20.0558	1.92100	0.960498	+	0.278288i	$0.0897671\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−3.68889	−0.347022	−0.173511	−	0.984832i	$-0.555511\pi$
$113$	−3.68889	−0.347022	−0.173511	+	0.984832i	$0.555511\pi$
$114$	0	0
$115$	−8.42864	−0.785975
$116$	0	0
$117$	0	0
$118$	0	0
$119$	7.70471	0.706290
$120$	0	0
$121$	11.4652	1.04229
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−4.56199	−0.408037
$126$	0	0
$127$	−8.29529	−0.736088	−0.368044	−	0.929808i	$-0.619972\pi$
$127$	−8.29529	−0.736088	−0.368044	+	0.929808i	$0.619972\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−7.76049	−0.678037	−0.339019	−	0.940780i	$-0.610095\pi$
$131$	−7.76049	−0.678037	−0.339019	+	0.940780i	$0.610095\pi$
$132$	0	0
$133$	−1.52543	−0.132271
$134$	0	0
$135$	0	0
$136$	0	0
$137$	11.1240	0.950386	0.475193	−	0.879882i	$-0.342378\pi$
$137$	11.1240	0.950386	0.475193	+	0.879882i	$0.342378\pi$
$138$	0	0
$139$	−8.10171	−0.687178	−0.343589	−	0.939120i	$-0.611643\pi$
$139$	−8.10171	−0.687178	−0.343589	+	0.939120i	$0.611643\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−6.53035	−0.546095
$144$	0	0
$145$	−15.9541	−1.32491
$146$	0	0
$147$	0	0
$148$	0	0
$149$	7.28592	0.596886	0.298443	−	0.954428i	$-0.403533\pi$
$149$	7.28592	0.596886	0.298443	+	0.954428i	$0.403533\pi$
$150$	0	0
$151$	−19.0415	−1.54957	−0.774787	−	0.632222i	$-0.782143\pi$
$151$	−19.0415	−1.54957	−0.774787	+	0.632222i	$0.782143\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	16.5763	1.33144
$156$	0	0
$157$	14.9956	1.19678	0.598388	−	0.801207i	$-0.295808\pi$
$157$	14.9956	1.19678	0.598388	+	0.801207i	$0.295808\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	4.42864	0.349026
$162$	0	0
$163$	−8.08742	−0.633456	−0.316728	−	0.948516i	$-0.602584\pi$
$163$	−8.08742	−0.633456	−0.316728	+	0.948516i	$0.602584\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	13.6731	1.05805	0.529027	−	0.848605i	$-0.322557\pi$
$167$	13.6731	1.05805	0.529027	+	0.848605i	$0.322557\pi$
$168$	0	0
$169$	−11.1017	−0.853978
$170$	0	0
$171$	0	0
$172$	0	0
$173$	2.88739	0.219524	0.109762	−	0.993958i	$-0.464991\pi$
$173$	2.88739	0.219524	0.109762	+	0.993958i	$0.464991\pi$
$174$	0	0
$175$	−5.23014	−0.395362
$176$	0	0
$177$	0	0
$178$	0	0
$179$	1.95407	0.146054	0.0730269	−	0.997330i	$-0.476734\pi$
$179$	1.95407	0.146054	0.0730269	+	0.997330i	$0.476734\pi$
$180$	0	0
$181$	25.6686	1.90793	0.953967	−	0.299912i	$-0.0969573\pi$
$181$	25.6686	1.90793	0.953967	+	0.299912i	$0.0969573\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−31.0923	−2.28595
$186$	0	0
$187$	−23.9398	−1.75065
$188$	0	0
$189$	0	0
$190$	0	0
$191$	9.46367	0.684767	0.342384	−	0.939560i	$-0.388766\pi$
$191$	9.46367	0.684767	0.342384	+	0.939560i	$0.388766\pi$
$192$	0	0
$193$	−16.0415	−1.15469	−0.577346	−	0.816500i	$-0.695912\pi$
$193$	−16.0415	−1.15469	−0.577346	+	0.816500i	$0.695912\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	20.1748	1.43740	0.718699	−	0.695321i	$-0.244738\pi$
$197$	20.1748	1.43740	0.718699	+	0.695321i	$0.244738\pi$
$198$	0	0
$199$	−3.18421	−0.225723	−0.112861	−	0.993611i	$-0.536002\pi$
$199$	−3.18421	−0.225723	−0.112861	+	0.993611i	$0.536002\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	8.38271	0.588351
$204$	0	0
$205$	−35.9956	−2.51404
$206$	0	0
$207$	0	0
$208$	0	0
$209$	4.73975	0.327855
$210$	0	0
$211$	−20.9224	−1.44036	−0.720180	−	0.693788i	$-0.755941\pi$
$211$	−20.9224	−1.44036	−0.720180	+	0.693788i	$0.755941\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−18.7971	−1.28195
$216$	0	0
$217$	−8.70964	−0.591249
$218$	0	0
$219$	0	0
$220$	0	0
$221$	6.95899	0.468112
$222$	0	0
$223$	−23.1432	−1.54978	−0.774892	−	0.632094i	$-0.782196\pi$
$223$	−23.1432	−1.54978	−0.774892	+	0.632094i	$0.782196\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	15.9398	1.05796	0.528980	−	0.848634i	$-0.322575\pi$
$227$	15.9398	1.05796	0.528980	+	0.848634i	$0.322575\pi$
$228$	0	0
$229$	−8.53972	−0.564320	−0.282160	−	0.959367i	$-0.591051\pi$
$229$	−8.53972	−0.564320	−0.282160	+	0.959367i	$0.591051\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−11.0366	−0.723029	−0.361515	−	0.932366i	$-0.617740\pi$
$233$	−11.0366	−0.723029	−0.361515	+	0.932366i	$0.617740\pi$
$234$	0	0
$235$	−8.76986	−0.572083
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−1.58274	−0.102379	−0.0511893	−	0.998689i	$-0.516301\pi$
$239$	−1.58274	−0.102379	−0.0511893	+	0.998689i	$0.516301\pi$
$240$	0	0
$241$	20.5575	1.32423	0.662114	−	0.749403i	$-0.269659\pi$
$241$	20.5575	1.32423	0.662114	+	0.749403i	$0.269659\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−13.5669	−0.866758
$246$	0	0
$247$	−1.37778	−0.0876663
$248$	0	0
$249$	0	0
$250$	0	0
$251$	30.9862	1.95583	0.977915	−	0.209001i	$-0.0670213\pi$
$251$	30.9862	1.95583	0.977915	+	0.209001i	$0.0670213\pi$
$252$	0	0
$253$	−13.7605	−0.865115
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−23.9541	−1.49421	−0.747107	−	0.664704i	$-0.768558\pi$
$257$	−23.9541	−1.49421	−0.747107	+	0.664704i	$0.768558\pi$
$258$	0	0
$259$	16.3368	1.01512
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−3.98418	−0.245675	−0.122837	−	0.992427i	$-0.539199\pi$
$263$	−3.98418	−0.245675	−0.122837	+	0.992427i	$0.539199\pi$
$264$	0	0
$265$	−7.13828	−0.438501
$266$	0	0
$267$	0	0
$268$	0	0
$269$	6.14764	0.374828	0.187414	−	0.982281i	$-0.439989\pi$
$269$	6.14764	0.374828	0.187414	+	0.982281i	$0.439989\pi$
$270$	0	0
$271$	12.9032	0.783814	0.391907	−	0.920005i	$-0.371815\pi$
$271$	12.9032	0.783814	0.391907	+	0.920005i	$0.371815\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	16.2509	0.979965
$276$	0	0
$277$	15.7971	0.949153	0.474577	−	0.880214i	$-0.342601\pi$
$277$	15.7971	0.949153	0.474577	+	0.880214i	$0.342601\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	1.03657	0.0618363	0.0309181	−	0.999522i	$-0.490157\pi$
$281$	1.03657	0.0618363	0.0309181	+	0.999522i	$0.490157\pi$
$282$	0	0
$283$	20.8988	1.24230	0.621151	−	0.783691i	$-0.286665\pi$
$283$	20.8988	1.24230	0.621151	+	0.783691i	$0.286665\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	18.9131	1.11640
$288$	0	0
$289$	8.51114	0.500655
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−15.6987	−0.917130	−0.458565	−	0.888661i	$-0.651636\pi$
$293$	−15.6987	−0.917130	−0.458565	+	0.888661i	$0.651636\pi$
$294$	0	0
$295$	13.5526	0.789064
$296$	0	0
$297$	0	0
$298$	0	0
$299$	4.00000	0.231326
$300$	0	0
$301$	9.87649	0.569271
$302$	0	0
$303$	0	0
$304$	0	0
$305$	23.6271	1.35289
$306$	0	0
$307$	25.3783	1.44841	0.724207	−	0.689583i	$-0.242206\pi$
$307$	25.3783	1.44841	0.724207	+	0.689583i	$0.242206\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	29.3620	1.66496	0.832482	−	0.554052i	$-0.186919\pi$
$311$	29.3620	1.66496	0.832482	+	0.554052i	$0.186919\pi$
$312$	0	0
$313$	−17.2494	−0.974991	−0.487496	−	0.873125i	$-0.662089\pi$
$313$	−17.2494	−0.974991	−0.487496	+	0.873125i	$0.662089\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−13.3319	−0.748791	−0.374396	−	0.927269i	$-0.622150\pi$
$317$	−13.3319	−0.748791	−0.374396	+	0.927269i	$0.622150\pi$
$318$	0	0
$319$	−26.0464	−1.45832
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−5.05086	−0.281037
$324$	0	0
$325$	−4.72393	−0.262036
$326$	0	0
$327$	0	0
$328$	0	0
$329$	4.60793	0.254043
$330$	0	0
$331$	11.6543	0.640580	0.320290	−	0.947320i	$-0.396220\pi$
$331$	11.6543	0.640580	0.320290	+	0.947320i	$0.396220\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−3.71900	−0.203191
$336$	0	0
$337$	−3.23014	−0.175957	−0.0879785	−	0.996122i	$-0.528041\pi$
$337$	−3.23014	−0.175957	−0.0879785	+	0.996122i	$0.528041\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	27.0622	1.46550
$342$	0	0
$343$	17.8064	0.961457
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−6.24935	−0.335483	−0.167741	−	0.985831i	$-0.553647\pi$
$347$	−6.24935	−0.335483	−0.167741	+	0.985831i	$0.553647\pi$
$348$	0	0
$349$	−28.3876	−1.51955	−0.759777	−	0.650183i	$-0.774692\pi$
$349$	−28.3876	−1.51955	−0.759777	+	0.650183i	$0.774692\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	18.7368	0.997261	0.498630	−	0.866815i	$-0.333836\pi$
$353$	18.7368	0.997261	0.498630	+	0.866815i	$0.333836\pi$
$354$	0	0
$355$	−43.1338	−2.28931
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−20.5763	−1.08597	−0.542987	−	0.839741i	$-0.682707\pi$
$359$	−20.5763	−1.08597	−0.542987	+	0.839741i	$0.682707\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−17.1383	−0.897058
$366$	0	0
$367$	−18.3970	−0.960315	−0.480158	−	0.877182i	$-0.659421\pi$
$367$	−18.3970	−0.960315	−0.480158	+	0.877182i	$0.659421\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	3.75065	0.194724
$372$	0	0
$373$	−1.23014	−0.0636943	−0.0318471	−	0.999493i	$-0.510139\pi$
$373$	−1.23014	−0.0636943	−0.0318471	+	0.999493i	$0.510139\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	7.57136	0.389945
$378$	0	0
$379$	−8.18421	−0.420395	−0.210197	−	0.977659i	$-0.567411\pi$
$379$	−8.18421	−0.420395	−0.210197	+	0.977659i	$0.567411\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−1.12843	−0.0576601	−0.0288301	−	0.999584i	$-0.509178\pi$
$383$	−1.12843	−0.0576601	−0.0288301	+	0.999584i	$0.509178\pi$
$384$	0	0
$385$	−20.9906	−1.06978
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−13.7190	−0.695581	−0.347791	−	0.937572i	$-0.613068\pi$
$389$	−13.7190	−0.695581	−0.347791	+	0.937572i	$0.613068\pi$
$390$	0	0
$391$	14.6637	0.741575
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0.709636	0.0357057
$396$	0	0
$397$	−31.2766	−1.56973	−0.784863	−	0.619670i	$-0.787266\pi$
$397$	−31.2766	−1.56973	−0.784863	+	0.619670i	$0.787266\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−14.2810	−0.713159	−0.356579	−	0.934265i	$-0.616057\pi$
$401$	−14.2810	−0.713159	−0.356579	+	0.934265i	$0.616057\pi$
$402$	0	0
$403$	−7.86665	−0.391866
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−50.7610	−2.51613
$408$	0	0
$409$	−7.00492	−0.346371	−0.173186	−	0.984889i	$-0.555406\pi$
$409$	−7.00492	−0.346371	−0.173186	+	0.984889i	$0.555406\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−7.12092	−0.350398
$414$	0	0
$415$	23.4336	1.15031
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−18.3067	−0.894339	−0.447169	−	0.894449i	$-0.647568\pi$
$419$	−18.3067	−0.894339	−0.447169	+	0.894449i	$0.647568\pi$
$420$	0	0
$421$	−16.2667	−0.792791	−0.396395	−	0.918080i	$-0.629739\pi$
$421$	−16.2667	−0.792791	−0.396395	+	0.918080i	$0.629739\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−17.3176	−0.840025
$426$	0	0
$427$	−12.4143	−0.600772
$428$	0	0
$429$	0	0
$430$	0	0
$431$	17.9224	0.863293	0.431646	−	0.902043i	$-0.357933\pi$
$431$	17.9224	0.863293	0.431646	+	0.902043i	$0.357933\pi$
$432$	0	0
$433$	13.4050	0.644202	0.322101	−	0.946705i	$-0.395611\pi$
$433$	13.4050	0.644202	0.322101	+	0.946705i	$0.395611\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−2.90321	−0.138879
$438$	0	0
$439$	38.3689	1.83125	0.915624	−	0.402036i	$-0.131697\pi$
$439$	38.3689	1.83125	0.915624	+	0.402036i	$0.131697\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	19.0622	0.905674	0.452837	−	0.891593i	$-0.350412\pi$
$443$	19.0622	0.905674	0.452837	+	0.891593i	$0.350412\pi$
$444$	0	0
$445$	−28.8113	−1.36579
$446$	0	0
$447$	0	0
$448$	0	0
$449$	37.0306	1.74758	0.873791	−	0.486302i	$-0.161655\pi$
$449$	37.0306	1.74758	0.873791	+	0.486302i	$0.161655\pi$
$450$	0	0
$451$	−58.7659	−2.76718
$452$	0	0
$453$	0	0
$454$	0	0
$455$	6.10171	0.286053
$456$	0	0
$457$	−26.9670	−1.26146	−0.630731	−	0.776002i	$-0.717245\pi$
$457$	−26.9670	−1.26146	−0.630731	+	0.776002i	$0.717245\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−0.847435	−0.0394690	−0.0197345	−	0.999805i	$-0.506282\pi$
$461$	−0.847435	−0.0394690	−0.0197345	+	0.999805i	$0.506282\pi$
$462$	0	0
$463$	0.876015	0.0407119	0.0203559	−	0.999793i	$-0.493520\pi$
$463$	0.876015	0.0407119	0.0203559	+	0.999793i	$0.493520\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−19.1827	−0.887668	−0.443834	−	0.896109i	$-0.646382\pi$
$467$	−19.1827	−0.887668	−0.443834	+	0.896109i	$0.646382\pi$
$468$	0	0
$469$	1.95407	0.0902304
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−30.6878	−1.41103
$474$	0	0
$475$	3.42864	0.157317
$476$	0	0
$477$	0	0
$478$	0	0
$479$	25.4050	1.16078	0.580392	−	0.814338i	$-0.302899\pi$
$479$	25.4050	1.16078	0.580392	+	0.814338i	$0.302899\pi$
$480$	0	0
$481$	14.7556	0.672796
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−26.2351	−1.19127
$486$	0	0
$487$	−44.0178	−1.99464	−0.997319	−	0.0731750i	$-0.976687\pi$
$487$	−44.0178	−1.99464	−0.997319	+	0.0731750i	$0.976687\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	25.0020	1.12832	0.564162	−	0.825664i	$-0.309199\pi$
$491$	25.0020	1.12832	0.564162	+	0.825664i	$0.309199\pi$
$492$	0	0
$493$	27.7560	1.25007
$494$	0	0
$495$	0	0
$496$	0	0
$497$	22.6637	1.01661
$498$	0	0
$499$	9.19405	0.411582	0.205791	−	0.978596i	$-0.434023\pi$
$499$	9.19405	0.411582	0.205791	+	0.978596i	$0.434023\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−20.8415	−0.929275	−0.464637	−	0.885501i	$-0.653815\pi$
$503$	−20.8415	−0.929275	−0.464637	+	0.885501i	$0.653815\pi$
$504$	0	0
$505$	−1.93978	−0.0863189
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−36.1417	−1.60195	−0.800976	−	0.598697i	$-0.795685\pi$
$509$	−36.1417	−1.60195	−0.800976	+	0.598697i	$0.795685\pi$
$510$	0	0
$511$	9.00492	0.398354
$512$	0	0
$513$	0	0
$514$	0	0
$515$	4.00000	0.176261
$516$	0	0
$517$	−14.3176	−0.629686
$518$	0	0
$519$	0	0
$520$	0	0
$521$	26.8716	1.17726	0.588632	−	0.808401i	$-0.299667\pi$
$521$	26.8716	1.17726	0.588632	+	0.808401i	$0.299667\pi$
$522$	0	0
$523$	−15.9032	−0.695399	−0.347699	−	0.937606i	$-0.613037\pi$
$523$	−15.9032	−0.695399	−0.347699	+	0.937606i	$0.613037\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−28.8385	−1.25623
$528$	0	0
$529$	−14.5714	−0.633537
$530$	0	0
$531$	0	0
$532$	0	0
$533$	17.0825	0.739925
$534$	0	0
$535$	32.8988	1.42234
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−22.1492	−0.954032
$540$	0	0
$541$	6.39207	0.274817	0.137408	−	0.990514i	$-0.456123\pi$
$541$	6.39207	0.274817	0.137408	+	0.990514i	$0.456123\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	58.2262	2.49414
$546$	0	0
$547$	−12.0000	−0.513083	−0.256541	−	0.966533i	$-0.582583\pi$
$547$	−12.0000	−0.513083	−0.256541	+	0.966533i	$0.582583\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−5.49532	−0.234108
$552$	0	0
$553$	−0.372862	−0.0158557
$554$	0	0
$555$	0	0
$556$	0	0
$557$	18.2395	0.772833	0.386416	−	0.922324i	$-0.373713\pi$
$557$	18.2395	0.772833	0.386416	+	0.922324i	$0.373713\pi$
$558$	0	0
$559$	8.92056	0.377300
$560$	0	0
$561$	0	0
$562$	0	0
$563$	2.37286	0.100004	0.0500021	−	0.998749i	$-0.484077\pi$
$563$	2.37286	0.100004	0.0500021	+	0.998749i	$0.484077\pi$
$564$	0	0
$565$	−10.7096	−0.450558
$566$	0	0
$567$	0	0
$568$	0	0
$569$	5.45230	0.228572	0.114286	−	0.993448i	$-0.463542\pi$
$569$	5.45230	0.228572	0.114286	+	0.993448i	$0.463542\pi$
$570$	0	0
$571$	36.7926	1.53972	0.769861	−	0.638211i	$-0.220325\pi$
$571$	36.7926	1.53972	0.769861	+	0.638211i	$0.220325\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−9.95407	−0.415113
$576$	0	0
$577$	40.4514	1.68401	0.842007	−	0.539467i	$-0.181374\pi$
$577$	40.4514	1.68401	0.842007	+	0.539467i	$0.181374\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−12.3126	−0.510814
$582$	0	0
$583$	−11.6539	−0.482654
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−18.3827	−0.758735	−0.379368	−	0.925246i	$-0.623858\pi$
$587$	−18.3827	−0.758735	−0.379368	+	0.925246i	$0.623858\pi$
$588$	0	0
$589$	5.70964	0.235261
$590$	0	0
$591$	0	0
$592$	0	0
$593$	21.9857	0.902845	0.451422	−	0.892310i	$-0.350917\pi$
$593$	21.9857	0.902845	0.451422	+	0.892310i	$0.350917\pi$
$594$	0	0
$595$	22.3684	0.917016
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−0.207866	−0.00849318	−0.00424659	−	0.999991i	$-0.501352\pi$
$599$	−0.207866	−0.00849318	−0.00424659	+	0.999991i	$0.501352\pi$
$600$	0	0
$601$	−15.4652	−0.630839	−0.315419	−	0.948952i	$-0.602145\pi$
$601$	−15.4652	−0.630839	−0.315419	+	0.948952i	$0.602145\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	33.2859	1.35327
$606$	0	0
$607$	39.5861	1.60675	0.803376	−	0.595473i	$-0.203035\pi$
$607$	39.5861	1.60675	0.803376	+	0.595473i	$0.203035\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	4.16193	0.168374
$612$	0	0
$613$	10.2252	0.412993	0.206496	−	0.978447i	$-0.433794\pi$
$613$	10.2252	0.412993	0.206496	+	0.978447i	$0.433794\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−21.9857	−0.885111	−0.442556	−	0.896741i	$-0.645928\pi$
$617$	−21.9857	−0.885111	−0.442556	+	0.896741i	$0.645928\pi$
$618$	0	0
$619$	−47.8578	−1.92357	−0.961783	−	0.273814i	$-0.911715\pi$
$619$	−47.8578	−1.92357	−0.961783	+	0.273814i	$0.911715\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	15.1383	0.606502
$624$	0	0
$625$	−30.3876	−1.21551
$626$	0	0
$627$	0	0
$628$	0	0
$629$	54.0928	2.15682
$630$	0	0
$631$	12.8760	0.512586	0.256293	−	0.966599i	$-0.417499\pi$
$631$	12.8760	0.512586	0.256293	+	0.966599i	$0.417499\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−24.0830	−0.955704
$636$	0	0
$637$	6.43848	0.255102
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−9.89538	−0.390844	−0.195422	−	0.980719i	$-0.562608\pi$
$641$	−9.89538	−0.390844	−0.195422	+	0.980719i	$0.562608\pi$
$642$	0	0
$643$	−46.7195	−1.84244	−0.921218	−	0.389046i	$-0.872805\pi$
$643$	−46.7195	−1.84244	−0.921218	+	0.389046i	$0.872805\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	29.3990	1.15579	0.577897	−	0.816110i	$-0.303874\pi$
$647$	29.3990	1.15579	0.577897	+	0.816110i	$0.303874\pi$
$648$	0	0
$649$	22.1258	0.868515
$650$	0	0
$651$	0	0
$652$	0	0
$653$	36.5402	1.42993	0.714964	−	0.699161i	$-0.246443\pi$
$653$	36.5402	1.42993	0.714964	+	0.699161i	$0.246443\pi$
$654$	0	0
$655$	−22.5303	−0.880334
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−9.98571	−0.388988	−0.194494	−	0.980904i	$-0.562306\pi$
$659$	−9.98571	−0.388988	−0.194494	+	0.980904i	$0.562306\pi$
$660$	0	0
$661$	26.0415	1.01290	0.506448	−	0.862270i	$-0.330958\pi$
$661$	26.0415	1.01290	0.506448	+	0.862270i	$0.330958\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−4.42864	−0.171735
$666$	0	0
$667$	15.9541	0.617744
$668$	0	0
$669$	0	0
$670$	0	0
$671$	38.5734	1.48911
$672$	0	0
$673$	−34.7239	−1.33851	−0.669254	−	0.743033i	$-0.733386\pi$
$673$	−34.7239	−1.33851	−0.669254	+	0.743033i	$0.733386\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−50.1802	−1.92858	−0.964292	−	0.264843i	$-0.914680\pi$
$677$	−50.1802	−1.92858	−0.964292	+	0.264843i	$0.914680\pi$
$678$	0	0
$679$	13.7846	0.529005
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−30.8430	−1.18017	−0.590087	−	0.807340i	$-0.700907\pi$
$683$	−30.8430	−1.18017	−0.590087	+	0.807340i	$0.700907\pi$
$684$	0	0
$685$	32.2953	1.23394
$686$	0	0
$687$	0	0
$688$	0	0
$689$	3.38763	0.129058
$690$	0	0
$691$	−40.0874	−1.52500	−0.762499	−	0.646990i	$-0.776028\pi$
$691$	−40.0874	−1.52500	−0.762499	+	0.646990i	$0.776028\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−23.5210	−0.892202
$696$	0	0
$697$	62.6232	2.37202
$698$	0	0
$699$	0	0
$700$	0	0
$701$	2.96343	0.111927	0.0559637	−	0.998433i	$-0.482177\pi$
$701$	2.96343	0.111927	0.0559637	+	0.998433i	$0.482177\pi$
$702$	0	0
$703$	−10.7096	−0.403922
$704$	0	0
$705$	0	0
$706$	0	0
$707$	1.01921	0.0383314
$708$	0	0
$709$	11.1240	0.417770	0.208885	−	0.977940i	$-0.433017\pi$
$709$	11.1240	0.417770	0.208885	+	0.977940i	$0.433017\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−16.5763	−0.620787
$714$	0	0
$715$	−18.9590	−0.709026
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−20.8143	−0.776241	−0.388120	−	0.921609i	$-0.626876\pi$
$719$	−20.8143	−0.776241	−0.388120	+	0.921609i	$0.626876\pi$
$720$	0	0
$721$	−2.10171	−0.0782718
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−18.8415	−0.699754
$726$	0	0
$727$	−43.6356	−1.61835	−0.809177	−	0.587564i	$-0.800087\pi$
$727$	−43.6356	−1.61835	−0.809177	+	0.587564i	$0.800087\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	32.7021	1.20953
$732$	0	0
$733$	−12.5955	−0.465225	−0.232613	−	0.972569i	$-0.574727\pi$
$733$	−12.5955	−0.465225	−0.232613	+	0.972569i	$0.574727\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−6.07160	−0.223650
$738$	0	0
$739$	−23.3002	−0.857112	−0.428556	−	0.903515i	$-0.640978\pi$
$739$	−23.3002	−0.857112	−0.428556	+	0.903515i	$0.640978\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	14.9906	0.549953	0.274976	−	0.961451i	$-0.411330\pi$
$743$	14.9906	0.549953	0.274976	+	0.961451i	$0.411330\pi$
$744$	0	0
$745$	21.1526	0.774970
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−17.2859	−0.631614
$750$	0	0
$751$	4.54909	0.165998	0.0829992	−	0.996550i	$-0.473550\pi$
$751$	4.54909	0.165998	0.0829992	+	0.996550i	$0.473550\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−55.2815	−2.01190
$756$	0	0
$757$	18.4509	0.670610	0.335305	−	0.942110i	$-0.391161\pi$
$757$	18.4509	0.670610	0.335305	+	0.942110i	$0.391161\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	6.35551	0.230387	0.115194	−	0.993343i	$-0.463251\pi$
$761$	6.35551	0.230387	0.115194	+	0.993343i	$0.463251\pi$
$762$	0	0
$763$	−30.5936	−1.10756
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−6.43170	−0.232235
$768$	0	0
$769$	−15.2159	−0.548698	−0.274349	−	0.961630i	$-0.588462\pi$
$769$	−15.2159	−0.548698	−0.274349	+	0.961630i	$0.588462\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	18.7126	0.673044	0.336522	−	0.941676i	$-0.390749\pi$
$773$	18.7126	0.673044	0.336522	+	0.941676i	$0.390749\pi$
$774$	0	0
$775$	19.5763	0.703201
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−12.3985	−0.444223
$780$	0	0
$781$	−70.4197	−2.51982
$782$	0	0
$783$	0	0
$784$	0	0
$785$	43.5353	1.55384
$786$	0	0
$787$	1.38271	0.0492882	0.0246441	−	0.999696i	$-0.492155\pi$
$787$	1.38271	0.0492882	0.0246441	+	0.999696i	$0.492155\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	5.62714	0.200078
$792$	0	0
$793$	−11.2128	−0.398178
$794$	0	0
$795$	0	0
$796$	0	0
$797$	5.30619	0.187955	0.0939774	−	0.995574i	$-0.470042\pi$
$797$	5.30619	0.187955	0.0939774	+	0.995574i	$0.470042\pi$
$798$	0	0
$799$	15.2573	0.539766
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−27.9797	−0.987383
$804$	0	0
$805$	12.8573	0.453160
$806$	0	0
$807$	0	0
$808$	0	0
$809$	8.90321	0.313020	0.156510	−	0.987676i	$-0.449976\pi$
$809$	8.90321	0.313020	0.156510	+	0.987676i	$0.449976\pi$
$810$	0	0
$811$	7.02858	0.246807	0.123403	−	0.992357i	$-0.460619\pi$
$811$	7.02858	0.246807	0.123403	+	0.992357i	$0.460619\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−23.4795	−0.822451
$816$	0	0
$817$	−6.47457	−0.226517
$818$	0	0
$819$	0	0
$820$	0	0
$821$	34.6494	1.20927	0.604636	−	0.796502i	$-0.293318\pi$
$821$	34.6494	1.20927	0.604636	+	0.796502i	$0.293318\pi$
$822$	0	0
$823$	11.8840	0.414250	0.207125	−	0.978314i	$-0.433589\pi$
$823$	11.8840	0.414250	0.207125	+	0.978314i	$0.433589\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−2.18913	−0.0761235	−0.0380618	−	0.999275i	$-0.512118\pi$
$827$	−2.18913	−0.0761235	−0.0380618	+	0.999275i	$0.512118\pi$
$828$	0	0
$829$	0.161933	0.00562418	0.00281209	−	0.999996i	$-0.499105\pi$
$829$	0.161933	0.00562418	0.00281209	+	0.999996i	$0.499105\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	23.6030	0.817796
$834$	0	0
$835$	39.6958	1.37373
$836$	0	0
$837$	0	0
$838$	0	0
$839$	9.87649	0.340974	0.170487	−	0.985360i	$-0.445466\pi$
$839$	9.87649	0.340974	0.170487	+	0.985360i	$0.445466\pi$
$840$	0	0
$841$	1.19850	0.0413275
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−32.2306	−1.10877
$846$	0	0
$847$	−17.4893	−0.600941
$848$	0	0
$849$	0	0
$850$	0	0
$851$	31.0923	1.06583
$852$	0	0
$853$	−11.9906	−0.410551	−0.205276	−	0.978704i	$-0.565809\pi$
$853$	−11.9906	−0.410551	−0.205276	+	0.978704i	$0.565809\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−21.3205	−0.728294	−0.364147	−	0.931342i	$-0.618639\pi$
$857$	−21.3205	−0.728294	−0.364147	+	0.931342i	$0.618639\pi$
$858$	0	0
$859$	5.59994	0.191068	0.0955338	−	0.995426i	$-0.469544\pi$
$859$	5.59994	0.191068	0.0955338	+	0.995426i	$0.469544\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−23.0825	−0.785737	−0.392869	−	0.919595i	$-0.628517\pi$
$863$	−23.0825	−0.785737	−0.392869	+	0.919595i	$0.628517\pi$
$864$	0	0
$865$	8.38271	0.285021
$866$	0	0
$867$	0	0
$868$	0	0
$869$	1.15854	0.0393009
$870$	0	0
$871$	1.76494	0.0598026
$872$	0	0
$873$	0	0
$874$	0	0
$875$	6.95899	0.235257
$876$	0	0
$877$	−39.1481	−1.32194	−0.660969	−	0.750413i	$-0.729855\pi$
$877$	−39.1481	−1.32194	−0.660969	+	0.750413i	$0.729855\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−51.7057	−1.74201	−0.871004	−	0.491276i	$-0.836531\pi$
$881$	−51.7057	−1.74201	−0.871004	+	0.491276i	$0.836531\pi$
$882$	0	0
$883$	47.9166	1.61252	0.806261	−	0.591560i	$-0.201488\pi$
$883$	47.9166	1.61252	0.806261	+	0.591560i	$0.201488\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	31.7418	1.06578	0.532892	−	0.846183i	$-0.321105\pi$
$887$	31.7418	1.06578	0.532892	+	0.846183i	$0.321105\pi$
$888$	0	0
$889$	12.6539	0.424397
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−3.02074	−0.101085
$894$	0	0
$895$	5.67307	0.189630
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−31.3763	−1.04646
$900$	0	0
$901$	12.4188	0.413730
$902$	0	0
$903$	0	0
$904$	0	0
$905$	74.5215	2.47718
$906$	0	0
$907$	36.1530	1.20044	0.600221	−	0.799834i	$-0.295079\pi$
$907$	36.1530	1.20044	0.600221	+	0.799834i	$0.295079\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−27.5669	−0.913333	−0.456666	−	0.889638i	$-0.650957\pi$
$911$	−27.5669	−0.913333	−0.456666	+	0.889638i	$0.650957\pi$
$912$	0	0
$913$	38.2573	1.26613
$914$	0	0
$915$	0	0
$916$	0	0
$917$	11.8381	0.390927
$918$	0	0
$919$	−23.7101	−0.782124	−0.391062	−	0.920364i	$-0.627892\pi$
$919$	−23.7101	−0.782124	−0.391062	+	0.920364i	$0.627892\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	20.4701	0.673782
$924$	0	0
$925$	−36.7195	−1.20733
$926$	0	0
$927$	0	0
$928$	0	0
$929$	6.54770	0.214823	0.107412	−	0.994215i	$-0.465744\pi$
$929$	6.54770	0.214823	0.107412	+	0.994215i	$0.465744\pi$
$930$	0	0
$931$	−4.67307	−0.153154
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−69.5022	−2.27297
$936$	0	0
$937$	−52.4390	−1.71311	−0.856553	−	0.516059i	$-0.827399\pi$
$937$	−52.4390	−1.71311	−0.856553	+	0.516059i	$0.827399\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	9.16992	0.298931	0.149465	−	0.988767i	$-0.452245\pi$
$941$	9.16992	0.298931	0.149465	+	0.988767i	$0.452245\pi$
$942$	0	0
$943$	35.9956	1.17218
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−19.0983	−0.620612	−0.310306	−	0.950637i	$-0.600432\pi$
$947$	−19.0983	−0.620612	−0.310306	+	0.950637i	$0.600432\pi$
$948$	0	0
$949$	8.13335	0.264020
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−54.4815	−1.76483	−0.882414	−	0.470473i	$-0.844083\pi$
$953$	−54.4815	−1.76483	−0.882414	+	0.470473i	$0.844083\pi$
$954$	0	0
$955$	27.4750	0.889072
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−16.9688	−0.547952
$960$	0	0
$961$	1.59994	0.0516110
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−46.5718	−1.49920
$966$	0	0
$967$	−31.9037	−1.02595	−0.512977	−	0.858403i	$-0.671457\pi$
$967$	−31.9037	−1.02595	−0.512977	+	0.858403i	$0.671457\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	41.5526	1.33349	0.666744	−	0.745287i	$-0.267688\pi$
$971$	41.5526	1.33349	0.666744	+	0.745287i	$0.267688\pi$
$972$	0	0
$973$	12.3586	0.396198
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−3.54125	−0.113295	−0.0566473	−	0.998394i	$-0.518041\pi$
$977$	−3.54125	−0.113295	−0.0566473	+	0.998394i	$0.518041\pi$
$978$	0	0
$979$	−47.0370	−1.50331
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−26.3511	−0.840468	−0.420234	−	0.907416i	$-0.638052\pi$
$983$	−26.3511	−0.840468	−0.420234	+	0.907416i	$0.638052\pi$
$984$	0	0
$985$	58.5718	1.86625
$986$	0	0
$987$	0	0
$988$	0	0
$989$	18.7971	0.597712
$990$	0	0
$991$	13.8983	0.441494	0.220747	−	0.975331i	$-0.429151\pi$
$991$	13.8983	0.441494	0.220747	+	0.975331i	$0.429151\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−9.24443	−0.293068
$996$	0	0
$997$	37.4005	1.18449	0.592243	−	0.805759i	$-0.298242\pi$
$997$	37.4005	1.18449	0.592243	+	0.805759i	$0.298242\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8208.2.a.bl.1.3		3
3.2	odd	2		8208.2.a.bi.1.1		3
4.3	odd	2		4104.2.a.h.1.3	yes	3
12.11	even	2		4104.2.a.e.1.1	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4104.2.a.e.1.1	✓	3	12.11	even	2
4104.2.a.h.1.3	yes	3	4.3	odd	2
8208.2.a.bi.1.1		3	3.2	odd	2
8208.2.a.bl.1.3		3	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 8208 = 2^{4} \cdot 3^{3} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8208.a (trivial)

\(f(q)\)	\(=\)	\(q+2.90321 q^{5} -1.52543 q^{7} +4.73975 q^{11} -1.37778 q^{13} -5.05086 q^{17} +1.00000 q^{19} -2.90321 q^{23} +3.42864 q^{25} -5.49532 q^{29} +5.70964 q^{31} -4.42864 q^{35} -10.7096 q^{37} -12.3985 q^{41} -6.47457 q^{43} -3.02074 q^{47} -4.67307 q^{49} -2.45875 q^{53} +13.7605 q^{55} +4.66815 q^{59} +8.13828 q^{61} -4.00000 q^{65} -1.28100 q^{67} -14.8573 q^{71} -5.90321 q^{73} -7.23014 q^{77} +0.244431 q^{79} +8.07160 q^{83} -14.6637 q^{85} -9.92396 q^{89} +2.10171 q^{91} +2.90321 q^{95} -9.03657 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 2 q^{5} + 2 q^{7} + q^{11} - 4 q^{13} - 2 q^{17} + 3 q^{19} - 2 q^{23} - 3 q^{25} - 3 q^{29} - 3 q^{31} - 12 q^{37} - 17 q^{41} - 26 q^{43} + 11 q^{47} - q^{49} - q^{53} + 8 q^{55} + 34 q^{59}+ \cdots - 20 q^{97}+O(q^{100}) \) 3 * q + 2 * q^5 + 2 * q^7 + q^11 - 4 * q^13 - 2 * q^17 + 3 * q^19 - 2 * q^23 - 3 * q^25 - 3 * q^29 - 3 * q^31 - 12 * q^37 - 17 * q^41 - 26 * q^43 + 11 * q^47 - q^49 - q^53 + 8 * q^55 + 34 * q^59 - 9 * q^61 - 12 * q^65 + 3 * q^67 - 18 * q^71 - 11 * q^73 - 28 * q^77 + q^79 - 9 * q^83 - 4 * q^85 - 3 * q^89 - 20 * q^91 + 2 * q^95 - 20 * q^97

Introduction

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