LMFDB - Embedded newform 8280.2.a.bs.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8280,2,Mod(1,8280)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8280, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("8280.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8280 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8280.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:	$66.1161328736$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.13955077.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 14x^{3} - x^{2} + 32x + 16 $ x^5 - 14x^3 - x^2 + 32x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 920)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$3.30649$ of defining polynomial
Character	$\chi$	$=$	8280.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{5} -2.55040 q^{7} +O(q^{10})$	$q+1.00000 q^{5} -2.55040 q^{7} +2.72314 q^{11} +7.12637 q^{13} -0.924010 q^{17} +7.51623 q^{19} +1.00000 q^{23} +1.00000 q^{25} +2.38248 q^{29} +0.866248 q^{31} -2.55040 q^{35} +0.352855 q^{37} -4.34066 q^{41} +13.3239 q^{47} -0.495474 q^{49} -3.99262 q^{53} +2.72314 q^{55} +3.84064 q^{59} -9.14262 q^{61} +7.12637 q^{65} -3.15933 q^{67} +6.07883 q^{71} -11.3239 q^{73} -6.94508 q^{77} -12.0593 q^{79} +6.35285 q^{83} -0.924010 q^{85} +9.71377 q^{89} -18.1751 q^{91} +7.51623 q^{95} +8.76465 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 5 q^{5} - 2 q^{7}+O(q^{10}) $ 5 * q + 5 * q^5 - 2 * q^7	$ 5 q + 5 q^{5} - 2 q^{7} + q^{11} + 4 q^{13} - 4 q^{17} + 7 q^{19} + 5 q^{23} + 5 q^{25} - 4 q^{29} + 19 q^{31} - 2 q^{35} + 15 q^{37} - 25 q^{41} + 11 q^{47} + 25 q^{49} - 3 q^{53} + q^{55} + q^{59} - 5 q^{61} + 4 q^{65} + 9 q^{67} - q^{71} - q^{73} - 18 q^{77} - 2 q^{79} + 45 q^{83} - 4 q^{85} - 6 q^{89} + 11 q^{91} + 7 q^{95} + 25 q^{97}+O(q^{100}) $ 5 * q + 5 * q^5 - 2 * q^7 + q^11 + 4 * q^13 - 4 * q^17 + 7 * q^19 + 5 * q^23 + 5 * q^25 - 4 * q^29 + 19 * q^31 - 2 * q^35 + 15 * q^37 - 25 * q^41 + 11 * q^47 + 25 * q^49 - 3 * q^53 + q^55 + q^59 - 5 * q^61 + 4 * q^65 + 9 * q^67 - q^71 - q^73 - 18 * q^77 - 2 * q^79 + 45 * q^83 - 4 * q^85 - 6 * q^89 + 11 * q^91 + 7 * q^95 + 25 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−2.55040	−0.963960	−0.481980	−	0.876182i	$-0.660082\pi$
$7$	−2.55040	−0.963960	−0.481980	+	0.876182i	$0.660082\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	2.72314	0.821056	0.410528	−	0.911848i	$-0.365344\pi$
$11$	2.72314	0.821056	0.410528	+	0.911848i	$0.365344\pi$
$12$	0	0
$13$	7.12637	1.97650	0.988250	−	0.152845i	$-0.0488435\pi$
$13$	7.12637	1.97650	0.988250	+	0.152845i	$0.0488435\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−0.924010	−0.224105	−0.112053	−	0.993702i	$-0.535743\pi$
$17$	−0.924010	−0.224105	−0.112053	+	0.993702i	$0.535743\pi$
$18$	0	0
$19$	7.51623	1.72434	0.862171	−	0.506617i	$-0.169104\pi$
$19$	7.51623	1.72434	0.862171	+	0.506617i	$0.169104\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	2.38248	0.442415	0.221208	−	0.975227i	$-0.429000\pi$
$29$	2.38248	0.442415	0.221208	+	0.975227i	$0.429000\pi$
$30$	0	0
$31$	0.866248	0.155583	0.0777913	−	0.996970i	$-0.475213\pi$
$31$	0.866248	0.155583	0.0777913	+	0.996970i	$0.475213\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−2.55040	−0.431096
$36$	0	0
$37$	0.352855	0.0580089	0.0290045	−	0.999579i	$-0.490766\pi$
$37$	0.352855	0.0580089	0.0290045	+	0.999579i	$0.490766\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−4.34066	−0.677896	−0.338948	−	0.940805i	$-0.610071\pi$
$41$	−4.34066	−0.677896	−0.338948	+	0.940805i	$0.610071\pi$
$42$	0	0
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	13.3239	1.94349	0.971746	−	0.236027i	$-0.0758454\pi$
$47$	13.3239	1.94349	0.971746	+	0.236027i	$0.0758454\pi$
$48$	0	0
$49$	−0.495474	−0.0707819
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−3.99262	−0.548429	−0.274214	−	0.961669i	$-0.588418\pi$
$53$	−3.99262	−0.548429	−0.274214	+	0.961669i	$0.588418\pi$
$54$	0	0
$55$	2.72314	0.367187
$56$	0	0
$57$	0	0
$58$	0	0
$59$	3.84064	0.500009	0.250004	−	0.968245i	$-0.419568\pi$
$59$	3.84064	0.500009	0.250004	+	0.968245i	$0.419568\pi$
$60$	0	0
$61$	−9.14262	−1.17059	−0.585296	−	0.810820i	$-0.699022\pi$
$61$	−9.14262	−1.17059	−0.585296	+	0.810820i	$0.699022\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	7.12637	0.883918
$66$	0	0
$67$	−3.15933	−0.385974	−0.192987	−	0.981201i	$-0.561817\pi$
$67$	−3.15933	−0.385974	−0.192987	+	0.981201i	$0.561817\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	6.07883	0.721424	0.360712	−	0.932677i	$-0.382534\pi$
$71$	6.07883	0.721424	0.360712	+	0.932677i	$0.382534\pi$
$72$	0	0
$73$	−11.3239	−1.32536	−0.662682	−	0.748901i	$-0.730582\pi$
$73$	−11.3239	−1.32536	−0.662682	+	0.748901i	$0.730582\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−6.94508	−0.791465
$78$	0	0
$79$	−12.0593	−1.35677	−0.678386	−	0.734706i	$-0.737320\pi$
$79$	−12.0593	−1.35677	−0.678386	+	0.734706i	$0.737320\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	6.35285	0.697316	0.348658	−	0.937250i	$-0.386637\pi$
$83$	6.35285	0.697316	0.348658	+	0.937250i	$0.386637\pi$
$84$	0	0
$85$	−0.924010	−0.100223
$86$	0	0
$87$	0	0
$88$	0	0
$89$	9.71377	1.02966	0.514829	−	0.857293i	$-0.327855\pi$
$89$	9.71377	1.02966	0.514829	+	0.857293i	$0.327855\pi$
$90$	0	0
$91$	−18.1751	−1.90527
$92$	0	0
$93$	0	0
$94$	0	0
$95$	7.51623	0.771149
$96$	0	0
$97$	8.76465	0.889916	0.444958	−	0.895552i	$-0.353219\pi$
$97$	8.76465	0.889916	0.444958	+	0.895552i	$0.353219\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−4.74794	−0.472438	−0.236219	−	0.971700i	$-0.575908\pi$
$101$	−4.74794	−0.472438	−0.236219	+	0.971700i	$0.575908\pi$
$102$	0	0
$103$	12.6304	1.24451	0.622255	−	0.782814i	$-0.286217\pi$
$103$	12.6304	1.24451	0.622255	+	0.782814i	$0.286217\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	12.9658	1.25345	0.626727	−	0.779239i	$-0.284394\pi$
$107$	12.9658	1.25345	0.626727	+	0.779239i	$0.284394\pi$
$108$	0	0
$109$	−15.0040	−1.43712	−0.718562	−	0.695463i	$-0.755199\pi$
$109$	−15.0040	−1.43712	−0.718562	+	0.695463i	$0.755199\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−2.13496	−0.200840	−0.100420	−	0.994945i	$-0.532019\pi$
$113$	−2.13496	−0.200840	−0.100420	+	0.994945i	$0.532019\pi$
$114$	0	0
$115$	1.00000	0.0932505
$116$	0	0
$117$	0	0
$118$	0	0
$119$	2.35659	0.216029
$120$	0	0
$121$	−3.58454	−0.325867
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−3.54181	−0.314285	−0.157142	−	0.987576i	$-0.550228\pi$
$127$	−3.54181	−0.314285	−0.157142	+	0.987576i	$0.550228\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−19.5499	−1.70808	−0.854042	−	0.520205i	$-0.825856\pi$
$131$	−19.5499	−1.70808	−0.854042	+	0.520205i	$0.825856\pi$
$132$	0	0
$133$	−19.1694	−1.66220
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−4.41544	−0.377236	−0.188618	−	0.982051i	$-0.560401\pi$
$137$	−4.41544	−0.377236	−0.188618	+	0.982051i	$0.560401\pi$
$138$	0	0
$139$	21.9954	1.86563	0.932814	−	0.360358i	$-0.117345\pi$
$139$	21.9954	1.86563	0.932814	+	0.360358i	$0.117345\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	19.4061	1.62282
$144$	0	0
$145$	2.38248	0.197854
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−19.0780	−1.56293	−0.781466	−	0.623947i	$-0.785528\pi$
$149$	−19.0780	−1.56293	−0.781466	+	0.623947i	$0.785528\pi$
$150$	0	0
$151$	−7.75273	−0.630908	−0.315454	−	0.948941i	$-0.602157\pi$
$151$	−7.75273	−0.630908	−0.315454	+	0.948941i	$0.602157\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0.866248	0.0695787
$156$	0	0
$157$	17.5129	1.39768	0.698841	−	0.715277i	$-0.253700\pi$
$157$	17.5129	1.39768	0.698841	+	0.715277i	$0.253700\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−2.55040	−0.200999
$162$	0	0
$163$	−2.55491	−0.200116	−0.100058	−	0.994982i	$-0.531903\pi$
$163$	−2.55491	−0.200116	−0.100058	+	0.994982i	$0.531903\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	3.61301	0.279583	0.139791	−	0.990181i	$-0.455357\pi$
$167$	3.61301	0.279583	0.139791	+	0.990181i	$0.455357\pi$
$168$	0	0
$169$	37.7852	2.90655
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−12.7824	−0.971827	−0.485913	−	0.874007i	$-0.661513\pi$
$173$	−12.7824	−0.971827	−0.485913	+	0.874007i	$0.661513\pi$
$174$	0	0
$175$	−2.55040	−0.192792
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−10.0149	−0.748546	−0.374273	−	0.927318i	$-0.622108\pi$
$179$	−10.0149	−0.748546	−0.374273	+	0.927318i	$0.622108\pi$
$180$	0	0
$181$	9.96248	0.740505	0.370252	−	0.928931i	$-0.379271\pi$
$181$	9.96248	0.740505	0.370252	+	0.928931i	$0.379271\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0.352855	0.0259424
$186$	0	0
$187$	−2.51621	−0.184003
$188$	0	0
$189$	0	0
$190$	0	0
$191$	6.61298	0.478498	0.239249	−	0.970958i	$-0.423099\pi$
$191$	6.61298	0.478498	0.239249	+	0.970958i	$0.423099\pi$
$192$	0	0
$193$	−19.0540	−1.37154	−0.685768	−	0.727820i	$-0.740534\pi$
$193$	−19.0540	−1.37154	−0.685768	+	0.727820i	$0.740534\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	11.0886	0.790030	0.395015	−	0.918675i	$-0.370739\pi$
$197$	11.0886	0.790030	0.395015	+	0.918675i	$0.370739\pi$
$198$	0	0
$199$	3.71377	0.263263	0.131631	−	0.991299i	$-0.457979\pi$
$199$	3.71377	0.263263	0.131631	+	0.991299i	$0.457979\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−6.07627	−0.426471
$204$	0	0
$205$	−4.34066	−0.303165
$206$	0	0
$207$	0	0
$208$	0	0
$209$	20.4677	1.41578
$210$	0	0
$211$	−21.7656	−1.49841	−0.749204	−	0.662339i	$-0.769564\pi$
$211$	−21.7656	−1.49841	−0.749204	+	0.662339i	$0.769564\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−2.20928	−0.149975
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−6.58484	−0.442944
$222$	0	0
$223$	6.90730	0.462547	0.231273	−	0.972889i	$-0.425711\pi$
$223$	6.90730	0.462547	0.231273	+	0.972889i	$0.425711\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	23.0325	1.52872	0.764359	−	0.644791i	$-0.223055\pi$
$227$	23.0325	1.52872	0.764359	+	0.644791i	$0.223055\pi$
$228$	0	0
$229$	12.9665	0.856854	0.428427	−	0.903576i	$-0.359068\pi$
$229$	12.9665	0.856854	0.428427	+	0.903576i	$0.359068\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	4.80934	0.315071	0.157535	−	0.987513i	$-0.449645\pi$
$233$	4.80934	0.315071	0.157535	+	0.987513i	$0.449645\pi$
$234$	0	0
$235$	13.3239	0.869156
$236$	0	0
$237$	0	0
$238$	0	0
$239$	20.4661	1.32384	0.661922	−	0.749573i	$-0.269741\pi$
$239$	20.4661	1.32384	0.661922	+	0.749573i	$0.269741\pi$
$240$	0	0
$241$	26.8657	1.73057	0.865287	−	0.501277i	$-0.167136\pi$
$241$	26.8657	1.73057	0.865287	+	0.501277i	$0.167136\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−0.495474	−0.0316546
$246$	0	0
$247$	53.5635	3.40816
$248$	0	0
$249$	0	0
$250$	0	0
$251$	15.9625	1.00754	0.503771	−	0.863837i	$-0.331945\pi$
$251$	15.9625	1.00754	0.503771	+	0.863837i	$0.331945\pi$
$252$	0	0
$253$	2.72314	0.171202
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−3.33867	−0.208261	−0.104130	−	0.994564i	$-0.533206\pi$
$257$	−3.33867	−0.208261	−0.104130	+	0.994564i	$0.533206\pi$
$258$	0	0
$259$	−0.899919	−0.0559183
$260$	0	0
$261$	0	0
$262$	0	0
$263$	23.3880	1.44217	0.721083	−	0.692849i	$-0.243645\pi$
$263$	23.3880	1.44217	0.721083	+	0.692849i	$0.243645\pi$
$264$	0	0
$265$	−3.99262	−0.245265
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−14.1378	−0.861995	−0.430998	−	0.902353i	$-0.641838\pi$
$269$	−14.1378	−0.861995	−0.430998	+	0.902353i	$0.641838\pi$
$270$	0	0
$271$	25.9283	1.57503	0.787517	−	0.616293i	$-0.211366\pi$
$271$	25.9283	1.57503	0.787517	+	0.616293i	$0.211366\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	2.72314	0.164211
$276$	0	0
$277$	1.07117	0.0643603	0.0321802	−	0.999482i	$-0.489755\pi$
$277$	1.07117	0.0643603	0.0321802	+	0.999482i	$0.489755\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−12.2260	−0.729341	−0.364671	−	0.931137i	$-0.618818\pi$
$281$	−12.2260	−0.729341	−0.364671	+	0.931137i	$0.618818\pi$
$282$	0	0
$283$	−30.5454	−1.81573	−0.907867	−	0.419259i	$-0.862290\pi$
$283$	−30.5454	−1.81573	−0.907867	+	0.419259i	$0.862290\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	11.0704	0.653465
$288$	0	0
$289$	−16.1462	−0.949777
$290$	0	0
$291$	0	0
$292$	0	0
$293$	8.46838	0.494728	0.247364	−	0.968923i	$-0.420436\pi$
$293$	8.46838	0.494728	0.247364	+	0.968923i	$0.420436\pi$
$294$	0	0
$295$	3.84064	0.223611
$296$	0	0
$297$	0	0
$298$	0	0
$299$	7.12637	0.412129
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−9.14262	−0.523505
$306$	0	0
$307$	−1.32802	−0.0757942	−0.0378971	−	0.999282i	$-0.512066\pi$
$307$	−1.32802	−0.0757942	−0.0378971	+	0.999282i	$0.512066\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−24.2971	−1.37776	−0.688882	−	0.724874i	$-0.741898\pi$
$311$	−24.2971	−1.37776	−0.688882	+	0.724874i	$0.741898\pi$
$312$	0	0
$313$	20.3478	1.15013	0.575063	−	0.818109i	$-0.304977\pi$
$313$	20.3478	1.15013	0.575063	+	0.818109i	$0.304977\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	8.57309	0.481513	0.240756	−	0.970586i	$-0.422604\pi$
$317$	8.57309	0.481513	0.240756	+	0.970586i	$0.422604\pi$
$318$	0	0
$319$	6.48781	0.363248
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−6.94508	−0.386434
$324$	0	0
$325$	7.12637	0.395300
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−33.9813	−1.87345
$330$	0	0
$331$	−20.9767	−1.15299	−0.576493	−	0.817102i	$-0.695579\pi$
$331$	−20.9767	−1.15299	−0.576493	+	0.817102i	$0.695579\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−3.15933	−0.172613
$336$	0	0
$337$	−2.17314	−0.118379	−0.0591894	−	0.998247i	$-0.518852\pi$
$337$	−2.17314	−0.118379	−0.0591894	+	0.998247i	$0.518852\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	2.35891	0.127742
$342$	0	0
$343$	19.1164	1.03219
$344$	0	0
$345$	0	0
$346$	0	0
$347$	32.1335	1.72502	0.862509	−	0.506042i	$-0.168892\pi$
$347$	32.1335	1.72502	0.862509	+	0.506042i	$0.168892\pi$
$348$	0	0
$349$	13.5157	0.723479	0.361740	−	0.932279i	$-0.382183\pi$
$349$	13.5157	0.723479	0.361740	+	0.932279i	$0.382183\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	28.0645	1.49372	0.746861	−	0.664981i	$-0.231560\pi$
$353$	28.0645	1.49372	0.746861	+	0.664981i	$0.231560\pi$
$354$	0	0
$355$	6.07883	0.322631
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−20.2340	−1.06791	−0.533956	−	0.845513i	$-0.679295\pi$
$359$	−20.2340	−1.06791	−0.533956	+	0.845513i	$0.679295\pi$
$360$	0	0
$361$	37.4937	1.97336
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−11.3239	−0.592721
$366$	0	0
$367$	4.76666	0.248818	0.124409	−	0.992231i	$-0.460297\pi$
$367$	4.76666	0.248818	0.124409	+	0.992231i	$0.460297\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	10.1828	0.528663
$372$	0	0
$373$	−29.1510	−1.50938	−0.754690	−	0.656082i	$-0.772213\pi$
$373$	−29.1510	−1.50938	−0.754690	+	0.656082i	$0.772213\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	16.9784	0.874434
$378$	0	0
$379$	4.20856	0.216179	0.108090	−	0.994141i	$-0.465527\pi$
$379$	4.20856	0.216179	0.108090	+	0.994141i	$0.465527\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−3.55206	−0.181502	−0.0907508	−	0.995874i	$-0.528927\pi$
$383$	−3.55206	−0.181502	−0.0907508	+	0.995874i	$0.528927\pi$
$384$	0	0
$385$	−6.94508	−0.353954
$386$	0	0
$387$	0	0
$388$	0	0
$389$	29.2515	1.48311	0.741554	−	0.670893i	$-0.234089\pi$
$389$	29.2515	1.48311	0.741554	+	0.670893i	$0.234089\pi$
$390$	0	0
$391$	−0.924010	−0.0467292
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−12.0593	−0.606767
$396$	0	0
$397$	−17.8000	−0.893356	−0.446678	−	0.894695i	$-0.647393\pi$
$397$	−17.8000	−0.893356	−0.446678	+	0.894695i	$0.647393\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	11.6210	0.580327	0.290164	−	0.956977i	$-0.406290\pi$
$401$	11.6210	0.580327	0.290164	+	0.956977i	$0.406290\pi$
$402$	0	0
$403$	6.17320	0.307509
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0.960871	0.0476286
$408$	0	0
$409$	−13.5710	−0.671041	−0.335521	−	0.942033i	$-0.608912\pi$
$409$	−13.5710	−0.671041	−0.335521	+	0.942033i	$0.608912\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−9.79516	−0.481988
$414$	0	0
$415$	6.35285	0.311849
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−13.4959	−0.659317	−0.329658	−	0.944100i	$-0.606934\pi$
$419$	−13.4959	−0.659317	−0.329658	+	0.944100i	$0.606934\pi$
$420$	0	0
$421$	37.6495	1.83492	0.917462	−	0.397824i	$-0.130234\pi$
$421$	37.6495	1.83492	0.917462	+	0.397824i	$0.130234\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−0.924010	−0.0448211
$426$	0	0
$427$	23.3173	1.12840
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−10.0568	−0.484421	−0.242210	−	0.970224i	$-0.577872\pi$
$431$	−10.0568	−0.484421	−0.242210	+	0.970224i	$0.577872\pi$
$432$	0	0
$433$	2.30001	0.110531	0.0552657	−	0.998472i	$-0.482399\pi$
$433$	2.30001	0.110531	0.0552657	+	0.998472i	$0.482399\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	7.51623	0.359550
$438$	0	0
$439$	12.4273	0.593124	0.296562	−	0.955014i	$-0.404160\pi$
$439$	12.4273	0.593124	0.296562	+	0.955014i	$0.404160\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−2.18268	−0.103702	−0.0518510	−	0.998655i	$-0.516512\pi$
$443$	−2.18268	−0.103702	−0.0518510	+	0.998655i	$0.516512\pi$
$444$	0	0
$445$	9.71377	0.460477
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−3.09603	−0.146111	−0.0730554	−	0.997328i	$-0.523275\pi$
$449$	−3.09603	−0.146111	−0.0730554	+	0.997328i	$0.523275\pi$
$450$	0	0
$451$	−11.8202	−0.556591
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−18.1751	−0.852061
$456$	0	0
$457$	−0.320363	−0.0149860	−0.00749298	−	0.999972i	$-0.502385\pi$
$457$	−0.320363	−0.0149860	−0.00749298	+	0.999972i	$0.502385\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	15.3239	0.713706	0.356853	−	0.934161i	$-0.383850\pi$
$461$	15.3239	0.713706	0.356853	+	0.934161i	$0.383850\pi$
$462$	0	0
$463$	6.81556	0.316746	0.158373	−	0.987379i	$-0.449375\pi$
$463$	6.81556	0.316746	0.158373	+	0.987379i	$0.449375\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−27.2444	−1.26072	−0.630360	−	0.776303i	$-0.717093\pi$
$467$	−27.2444	−1.26072	−0.630360	+	0.776303i	$0.717093\pi$
$468$	0	0
$469$	8.05755	0.372063
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	7.51623	0.344868
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−23.5031	−1.07388	−0.536942	−	0.843619i	$-0.680421\pi$
$479$	−23.5031	−1.07388	−0.536942	+	0.843619i	$0.680421\pi$
$480$	0	0
$481$	2.51457	0.114655
$482$	0	0
$483$	0	0
$484$	0	0
$485$	8.76465	0.397982
$486$	0	0
$487$	7.63058	0.345774	0.172887	−	0.984942i	$-0.444690\pi$
$487$	7.63058	0.345774	0.172887	+	0.984942i	$0.444690\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	19.1206	0.862902	0.431451	−	0.902136i	$-0.358002\pi$
$491$	19.1206	0.862902	0.431451	+	0.902136i	$0.358002\pi$
$492$	0	0
$493$	−2.20144	−0.0991477
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−15.5034	−0.695424
$498$	0	0
$499$	−25.6581	−1.14861	−0.574306	−	0.818641i	$-0.694728\pi$
$499$	−25.6581	−1.14861	−0.574306	+	0.818641i	$0.694728\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	8.42282	0.375555	0.187777	−	0.982212i	$-0.439872\pi$
$503$	8.42282	0.375555	0.187777	+	0.982212i	$0.439872\pi$
$504$	0	0
$505$	−4.74794	−0.211281
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−18.2328	−0.808153	−0.404076	−	0.914725i	$-0.632407\pi$
$509$	−18.2328	−0.808153	−0.404076	+	0.914725i	$0.632407\pi$
$510$	0	0
$511$	28.8805	1.27760
$512$	0	0
$513$	0	0
$514$	0	0
$515$	12.6304	0.556562
$516$	0	0
$517$	36.2828	1.59572
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−19.8738	−0.870689	−0.435345	−	0.900264i	$-0.643373\pi$
$521$	−19.8738	−0.870689	−0.435345	+	0.900264i	$0.643373\pi$
$522$	0	0
$523$	−19.7228	−0.862418	−0.431209	−	0.902252i	$-0.641913\pi$
$523$	−19.7228	−0.862418	−0.431209	+	0.902252i	$0.641913\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−0.800422	−0.0348669
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−30.9331	−1.33986
$534$	0	0
$535$	12.9658	0.560562
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−1.34924	−0.0581160
$540$	0	0
$541$	−16.4662	−0.707938	−0.353969	−	0.935257i	$-0.615168\pi$
$541$	−16.4662	−0.707938	−0.353969	+	0.935257i	$0.615168\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−15.0040	−0.642702
$546$	0	0
$547$	−38.3286	−1.63881	−0.819405	−	0.573215i	$-0.805696\pi$
$547$	−38.3286	−1.63881	−0.819405	+	0.573215i	$0.805696\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	17.9073	0.762875
$552$	0	0
$553$	30.7559	1.30787
$554$	0	0
$555$	0	0
$556$	0	0
$557$	20.3973	0.864263	0.432132	−	0.901811i	$-0.357762\pi$
$557$	20.3973	0.864263	0.432132	+	0.901811i	$0.357762\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	1.72270	0.0726032	0.0363016	−	0.999341i	$-0.488442\pi$
$563$	1.72270	0.0726032	0.0363016	+	0.999341i	$0.488442\pi$
$564$	0	0
$565$	−2.13496	−0.0898184
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−25.3283	−1.06182	−0.530909	−	0.847429i	$-0.678150\pi$
$569$	−25.3283	−1.06182	−0.530909	+	0.847429i	$0.678150\pi$
$570$	0	0
$571$	−5.68968	−0.238106	−0.119053	−	0.992888i	$-0.537986\pi$
$571$	−5.68968	−0.238106	−0.119053	+	0.992888i	$0.537986\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	1.00000	0.0417029
$576$	0	0
$577$	−18.0377	−0.750918	−0.375459	−	0.926839i	$-0.622515\pi$
$577$	−18.0377	−0.750918	−0.375459	+	0.926839i	$0.622515\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−16.2023	−0.672185
$582$	0	0
$583$	−10.8724	−0.450291
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−13.1894	−0.544383	−0.272192	−	0.962243i	$-0.587748\pi$
$587$	−13.1894	−0.544383	−0.272192	+	0.962243i	$0.587748\pi$
$588$	0	0
$589$	6.51092	0.268278
$590$	0	0
$591$	0	0
$592$	0	0
$593$	0.326755	0.0134182	0.00670911	−	0.999977i	$-0.497864\pi$
$593$	0.326755	0.0134182	0.00670911	+	0.999977i	$0.497864\pi$
$594$	0	0
$595$	2.35659	0.0966109
$596$	0	0
$597$	0	0
$598$	0	0
$599$	7.49619	0.306286	0.153143	−	0.988204i	$-0.451061\pi$
$599$	7.49619	0.306286	0.153143	+	0.988204i	$0.451061\pi$
$600$	0	0
$601$	0.121724	0.00496523	0.00248262	−	0.999997i	$-0.499210\pi$
$601$	0.121724	0.00496523	0.00248262	+	0.999997i	$0.499210\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−3.58454	−0.145732
$606$	0	0
$607$	1.52992	0.0620975	0.0310488	−	0.999518i	$-0.490115\pi$
$607$	1.52992	0.0620975	0.0310488	+	0.999518i	$0.490115\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	94.9512	3.84131
$612$	0	0
$613$	24.3927	0.985211	0.492605	−	0.870253i	$-0.336045\pi$
$613$	24.3927	0.985211	0.492605	+	0.870253i	$0.336045\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−27.8906	−1.12283	−0.561416	−	0.827534i	$-0.689743\pi$
$617$	−27.8906	−1.12283	−0.561416	+	0.827534i	$0.689743\pi$
$618$	0	0
$619$	12.8617	0.516957	0.258478	−	0.966017i	$-0.416779\pi$
$619$	12.8617	0.516957	0.258478	+	0.966017i	$0.416779\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−24.7740	−0.992549
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−0.326041	−0.0130001
$630$	0	0
$631$	−11.9518	−0.475794	−0.237897	−	0.971290i	$-0.576458\pi$
$631$	−11.9518	−0.475794	−0.237897	+	0.971290i	$0.576458\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−3.54181	−0.140552
$636$	0	0
$637$	−3.53093	−0.139901
$638$	0	0
$639$	0	0
$640$	0	0
$641$	23.9139	0.944544	0.472272	−	0.881453i	$-0.343434\pi$
$641$	23.9139	0.944544	0.472272	+	0.881453i	$0.343434\pi$
$642$	0	0
$643$	3.87313	0.152741	0.0763707	−	0.997079i	$-0.475667\pi$
$643$	3.87313	0.152741	0.0763707	+	0.997079i	$0.475667\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	10.3330	0.406231	0.203115	−	0.979155i	$-0.434893\pi$
$647$	10.3330	0.406231	0.203115	+	0.979155i	$0.434893\pi$
$648$	0	0
$649$	10.4586	0.410535
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−21.6319	−0.846522	−0.423261	−	0.906008i	$-0.639115\pi$
$653$	−21.6319	−0.846522	−0.423261	+	0.906008i	$0.639115\pi$
$654$	0	0
$655$	−19.5499	−0.763878
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−15.7168	−0.612238	−0.306119	−	0.951993i	$-0.599030\pi$
$659$	−15.7168	−0.612238	−0.306119	+	0.951993i	$0.599030\pi$
$660$	0	0
$661$	−40.3002	−1.56750	−0.783749	−	0.621078i	$-0.786695\pi$
$661$	−40.3002	−1.56750	−0.783749	+	0.621078i	$0.786695\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−19.1694	−0.743357
$666$	0	0
$667$	2.38248	0.0922500
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−24.8966	−0.961122
$672$	0	0
$673$	−6.69888	−0.258223	−0.129111	−	0.991630i	$-0.541212\pi$
$673$	−6.69888	−0.258223	−0.129111	+	0.991630i	$0.541212\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	22.5812	0.867866	0.433933	−	0.900945i	$-0.357125\pi$
$677$	22.5812	0.867866	0.433933	+	0.900945i	$0.357125\pi$
$678$	0	0
$679$	−22.3533	−0.857843
$680$	0	0
$681$	0	0
$682$	0	0
$683$	1.42797	0.0546398	0.0273199	−	0.999627i	$-0.491303\pi$
$683$	1.42797	0.0546398	0.0273199	+	0.999627i	$0.491303\pi$
$684$	0	0
$685$	−4.41544	−0.168705
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−28.4529	−1.08397
$690$	0	0
$691$	−1.50255	−0.0571595	−0.0285798	−	0.999592i	$-0.509098\pi$
$691$	−1.50255	−0.0571595	−0.0285798	+	0.999592i	$0.509098\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	21.9954	0.834334
$696$	0	0
$697$	4.01081	0.151920
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−18.7341	−0.707577	−0.353789	−	0.935325i	$-0.615107\pi$
$701$	−18.7341	−0.707577	−0.353789	+	0.935325i	$0.615107\pi$
$702$	0	0
$703$	2.65214	0.100027
$704$	0	0
$705$	0	0
$706$	0	0
$707$	12.1091	0.455411
$708$	0	0
$709$	−13.5052	−0.507199	−0.253599	−	0.967309i	$-0.581615\pi$
$709$	−13.5052	−0.507199	−0.253599	+	0.967309i	$0.581615\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	0.866248	0.0324412
$714$	0	0
$715$	19.4061	0.725746
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−31.7496	−1.18406	−0.592030	−	0.805916i	$-0.701673\pi$
$719$	−31.7496	−1.18406	−0.592030	+	0.805916i	$0.701673\pi$
$720$	0	0
$721$	−32.2126	−1.19966
$722$	0	0
$723$	0	0
$724$	0	0
$725$	2.38248	0.0884831
$726$	0	0
$727$	21.6531	0.803070	0.401535	−	0.915844i	$-0.368477\pi$
$727$	21.6531	0.803070	0.401535	+	0.915844i	$0.368477\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	0	0
$732$	0	0
$733$	0.640507	0.0236577	0.0118288	−	0.999930i	$-0.496235\pi$
$733$	0.640507	0.0236577	0.0118288	+	0.999930i	$0.496235\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−8.60329	−0.316906
$738$	0	0
$739$	16.8191	0.618700	0.309350	−	0.950948i	$-0.399889\pi$
$739$	16.8191	0.618700	0.309350	+	0.950948i	$0.399889\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	15.9691	0.585851	0.292925	−	0.956135i	$-0.405371\pi$
$743$	15.9691	0.585851	0.292925	+	0.956135i	$0.405371\pi$
$744$	0	0
$745$	−19.0780	−0.698965
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−33.0680	−1.20828
$750$	0	0
$751$	−8.09841	−0.295515	−0.147758	−	0.989024i	$-0.547206\pi$
$751$	−8.09841	−0.295515	−0.147758	+	0.989024i	$0.547206\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−7.75273	−0.282151
$756$	0	0
$757$	20.5691	0.747598	0.373799	−	0.927510i	$-0.378055\pi$
$757$	20.5691	0.747598	0.373799	+	0.927510i	$0.378055\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	17.1840	0.622919	0.311459	−	0.950259i	$-0.399182\pi$
$761$	17.1840	0.622919	0.311459	+	0.950259i	$0.399182\pi$
$762$	0	0
$763$	38.2662	1.38533
$764$	0	0
$765$	0	0
$766$	0	0
$767$	27.3698	0.988268
$768$	0	0
$769$	41.8648	1.50968	0.754841	−	0.655908i	$-0.227714\pi$
$769$	41.8648	1.50968	0.754841	+	0.655908i	$0.227714\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	35.7639	1.28634	0.643170	−	0.765724i	$-0.277619\pi$
$773$	35.7639	1.28634	0.643170	+	0.765724i	$0.277619\pi$
$774$	0	0
$775$	0.866248	0.0311165
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−32.6254	−1.16893
$780$	0	0
$781$	16.5535	0.592330
$782$	0	0
$783$	0	0
$784$	0	0
$785$	17.5129	0.625062
$786$	0	0
$787$	26.9835	0.961858	0.480929	−	0.876759i	$-0.340299\pi$
$787$	26.9835	0.961858	0.480929	+	0.876759i	$0.340299\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	5.44500	0.193602
$792$	0	0
$793$	−65.1537	−2.31368
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−26.7226	−0.946561	−0.473281	−	0.880912i	$-0.656930\pi$
$797$	−26.7226	−0.946561	−0.473281	+	0.880912i	$0.656930\pi$
$798$	0	0
$799$	−12.3114	−0.435547
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−30.8366	−1.08820
$804$	0	0
$805$	−2.55040	−0.0898897
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−50.0319	−1.75903	−0.879513	−	0.475874i	$-0.842132\pi$
$809$	−50.0319	−1.75903	−0.879513	+	0.475874i	$0.842132\pi$
$810$	0	0
$811$	33.5774	1.17906	0.589531	−	0.807746i	$-0.299313\pi$
$811$	33.5774	1.17906	0.589531	+	0.807746i	$0.299313\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−2.55491	−0.0894946
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−27.1429	−0.947295	−0.473647	−	0.880715i	$-0.657063\pi$
$821$	−27.1429	−0.947295	−0.473647	+	0.880715i	$0.657063\pi$
$822$	0	0
$823$	32.6392	1.13773	0.568866	−	0.822431i	$-0.307382\pi$
$823$	32.6392	1.13773	0.568866	+	0.822431i	$0.307382\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	37.0828	1.28949	0.644747	−	0.764396i	$-0.276963\pi$
$827$	37.0828	1.28949	0.644747	+	0.764396i	$0.276963\pi$
$828$	0	0
$829$	33.9405	1.17880	0.589401	−	0.807841i	$-0.299364\pi$
$829$	33.9405	1.17880	0.589401	+	0.807841i	$0.299364\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0.457823	0.0158626
$834$	0	0
$835$	3.61301	0.125033
$836$	0	0
$837$	0	0
$838$	0	0
$839$	1.36929	0.0472730	0.0236365	−	0.999721i	$-0.492476\pi$
$839$	1.36929	0.0472730	0.0236365	+	0.999721i	$0.492476\pi$
$840$	0	0
$841$	−23.3238	−0.804269
$842$	0	0
$843$	0	0
$844$	0	0
$845$	37.7852	1.29985
$846$	0	0
$847$	9.14199	0.314122
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0.352855	0.0120957
$852$	0	0
$853$	9.21382	0.315475	0.157738	−	0.987481i	$-0.449580\pi$
$853$	9.21382	0.315475	0.157738	+	0.987481i	$0.449580\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−14.3564	−0.490405	−0.245202	−	0.969472i	$-0.578854\pi$
$857$	−14.3564	−0.490405	−0.245202	+	0.969472i	$0.578854\pi$
$858$	0	0
$859$	31.3758	1.07053	0.535264	−	0.844685i	$-0.320212\pi$
$859$	31.3758	1.07053	0.535264	+	0.844685i	$0.320212\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−26.1214	−0.889182	−0.444591	−	0.895734i	$-0.646651\pi$
$863$	−26.1214	−0.889182	−0.444591	+	0.895734i	$0.646651\pi$
$864$	0	0
$865$	−12.7824	−0.434614
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−32.8390	−1.11399
$870$	0	0
$871$	−22.5146	−0.762877
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−2.55040	−0.0862192
$876$	0	0
$877$	−11.5530	−0.390118	−0.195059	−	0.980792i	$-0.562490\pi$
$877$	−11.5530	−0.390118	−0.195059	+	0.980792i	$0.562490\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	34.3559	1.15748	0.578740	−	0.815512i	$-0.303544\pi$
$881$	34.3559	1.15748	0.578740	+	0.815512i	$0.303544\pi$
$882$	0	0
$883$	12.8316	0.431818	0.215909	−	0.976414i	$-0.430729\pi$
$883$	12.8316	0.431818	0.215909	+	0.976414i	$0.430729\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	1.46724	0.0492652	0.0246326	−	0.999697i	$-0.492158\pi$
$887$	1.46724	0.0492652	0.0246326	+	0.999697i	$0.492158\pi$
$888$	0	0
$889$	9.03303	0.302958
$890$	0	0
$891$	0	0
$892$	0	0
$893$	100.146	3.35125
$894$	0	0
$895$	−10.0149	−0.334760
$896$	0	0
$897$	0	0
$898$	0	0
$899$	2.06382	0.0688322
$900$	0	0
$901$	3.68922	0.122906
$902$	0	0
$903$	0	0
$904$	0	0
$905$	9.96248	0.331164
$906$	0	0
$907$	4.44558	0.147613	0.0738066	−	0.997273i	$-0.476485\pi$
$907$	4.44558	0.147613	0.0738066	+	0.997273i	$0.476485\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−40.1697	−1.33088	−0.665440	−	0.746451i	$-0.731756\pi$
$911$	−40.1697	−1.33088	−0.665440	+	0.746451i	$0.731756\pi$
$912$	0	0
$913$	17.2997	0.572536
$914$	0	0
$915$	0	0
$916$	0	0
$917$	49.8600	1.64652
$918$	0	0
$919$	46.8063	1.54400	0.771999	−	0.635624i	$-0.219257\pi$
$919$	46.8063	1.54400	0.771999	+	0.635624i	$0.219257\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	43.3200	1.42590
$924$	0	0
$925$	0.352855	0.0116018
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−8.92145	−0.292703	−0.146352	−	0.989233i	$-0.546753\pi$
$929$	−8.92145	−0.292703	−0.146352	+	0.989233i	$0.546753\pi$
$930$	0	0
$931$	−3.72409	−0.122052
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−2.51621	−0.0822887
$936$	0	0
$937$	7.70070	0.251571	0.125785	−	0.992057i	$-0.459855\pi$
$937$	7.70070	0.251571	0.125785	+	0.992057i	$0.459855\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	16.8979	0.550856	0.275428	−	0.961322i	$-0.411180\pi$
$941$	16.8979	0.550856	0.275428	+	0.961322i	$0.411180\pi$
$942$	0	0
$943$	−4.34066	−0.141351
$944$	0	0
$945$	0	0
$946$	0	0
$947$	43.7150	1.42055	0.710273	−	0.703927i	$-0.248572\pi$
$947$	43.7150	1.42055	0.710273	+	0.703927i	$0.248572\pi$
$948$	0	0
$949$	−80.6985	−2.61958
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−30.2353	−0.979418	−0.489709	−	0.871886i	$-0.662897\pi$
$953$	−30.2353	−0.979418	−0.489709	+	0.871886i	$0.662897\pi$
$954$	0	0
$955$	6.61298	0.213991
$956$	0	0
$957$	0	0
$958$	0	0
$959$	11.2611	0.363641
$960$	0	0
$961$	−30.2496	−0.975794
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−19.0540	−0.613370
$966$	0	0
$967$	16.2727	0.523296	0.261648	−	0.965163i	$-0.415734\pi$
$967$	16.2727	0.523296	0.261648	+	0.965163i	$0.415734\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−6.21424	−0.199424	−0.0997122	−	0.995016i	$-0.531792\pi$
$971$	−6.21424	−0.199424	−0.0997122	+	0.995016i	$0.531792\pi$
$972$	0	0
$973$	−56.0971	−1.79839
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−25.6705	−0.821271	−0.410636	−	0.911800i	$-0.634693\pi$
$977$	−25.6705	−0.821271	−0.410636	+	0.911800i	$0.634693\pi$
$978$	0	0
$979$	26.4519	0.845407
$980$	0	0
$981$	0	0
$982$	0	0
$983$	30.7124	0.979574	0.489787	−	0.871842i	$-0.337075\pi$
$983$	30.7124	0.979574	0.489787	+	0.871842i	$0.337075\pi$
$984$	0	0
$985$	11.0886	0.353312
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	29.4052	0.934086	0.467043	−	0.884235i	$-0.345319\pi$
$991$	29.4052	0.934086	0.467043	+	0.884235i	$0.345319\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	3.71377	0.117735
$996$	0	0
$997$	5.59825	0.177298	0.0886492	−	0.996063i	$-0.471745\pi$
$997$	5.59825	0.177298	0.0886492	+	0.996063i	$0.471745\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8280.2.a.bs.1.2		5
3.2	odd	2		920.2.a.j.1.5	✓	5
12.11	even	2		1840.2.a.v.1.1		5
15.2	even	4		4600.2.e.u.4049.2		10
15.8	even	4		4600.2.e.u.4049.9		10
15.14	odd	2		4600.2.a.be.1.1		5
24.5	odd	2		7360.2.a.co.1.1		5
24.11	even	2		7360.2.a.cp.1.5		5
60.59	even	2		9200.2.a.cu.1.5		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
920.2.a.j.1.5	✓	5	3.2	odd	2
1840.2.a.v.1.1		5	12.11	even	2
4600.2.a.be.1.1		5	15.14	odd	2
4600.2.e.u.4049.2		10	15.2	even	4
4600.2.e.u.4049.9		10	15.8	even	4
7360.2.a.co.1.1		5	24.5	odd	2
7360.2.a.cp.1.5		5	24.11	even	2
8280.2.a.bs.1.2		5	1.1	even	1	trivial
9200.2.a.cu.1.5		5	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:	\( N \)	\(=\)	\( 8280 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8280.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{5} -2.55040 q^{7} +O(q^{10})\)	\(q+1.00000 q^{5} -2.55040 q^{7} +2.72314 q^{11} +7.12637 q^{13} -0.924010 q^{17} +7.51623 q^{19} +1.00000 q^{23} +1.00000 q^{25} +2.38248 q^{29} +0.866248 q^{31} -2.55040 q^{35} +0.352855 q^{37} -4.34066 q^{41} +13.3239 q^{47} -0.495474 q^{49} -3.99262 q^{53} +2.72314 q^{55} +3.84064 q^{59} -9.14262 q^{61} +7.12637 q^{65} -3.15933 q^{67} +6.07883 q^{71} -11.3239 q^{73} -6.94508 q^{77} -12.0593 q^{79} +6.35285 q^{83} -0.924010 q^{85} +9.71377 q^{89} -18.1751 q^{91} +7.51623 q^{95} +8.76465 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 5 q^{5} - 2 q^{7}+O(q^{10}) \) 5 * q + 5 * q^5 - 2 * q^7	\( 5 q + 5 q^{5} - 2 q^{7} + q^{11} + 4 q^{13} - 4 q^{17} + 7 q^{19} + 5 q^{23} + 5 q^{25} - 4 q^{29} + 19 q^{31} - 2 q^{35} + 15 q^{37} - 25 q^{41} + 11 q^{47} + 25 q^{49} - 3 q^{53} + q^{55} + q^{59} - 5 q^{61} + 4 q^{65} + 9 q^{67} - q^{71} - q^{73} - 18 q^{77} - 2 q^{79} + 45 q^{83} - 4 q^{85} - 6 q^{89} + 11 q^{91} + 7 q^{95} + 25 q^{97}+O(q^{100}) \) 5 * q + 5 * q^5 - 2 * q^7 + q^11 + 4 * q^13 - 4 * q^17 + 7 * q^19 + 5 * q^23 + 5 * q^25 - 4 * q^29 + 19 * q^31 - 2 * q^35 + 15 * q^37 - 25 * q^41 + 11 * q^47 + 25 * q^49 - 3 * q^53 + q^55 + q^59 - 5 * q^61 + 4 * q^65 + 9 * q^67 - q^71 - q^73 - 18 * q^77 - 2 * q^79 + 45 * q^83 - 4 * q^85 - 6 * q^89 + 11 * q^91 + 7 * q^95 + 25 * q^97

Introduction

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