LMFDB - Embedded newform 9680.2.a.br.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [9680,2,Mod(1,9680)] 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("9680.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9680, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 9680 = 2^{4} \cdot 5 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9680.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,2,0,4,0,0,0,0,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	yes
Analytic conductor:	$77.2951891566$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 4840)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.73205$ of defining polynomial
Character	$\chi$	$=$	9680.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.73205 q^{3} +1.00000 q^{5} +0.267949 q^{7} -3.46410 q^{13} +1.73205 q^{15} +2.00000 q^{19} +0.464102 q^{21} -7.46410 q^{23} +1.00000 q^{25} -5.19615 q^{27} +4.92820 q^{29} +1.46410 q^{31} +0.267949 q^{35} -4.00000 q^{37} -6.00000 q^{39} +1.92820 q^{41} +1.73205 q^{43} -6.66025 q^{47} -6.92820 q^{49} -7.46410 q^{53} +3.46410 q^{57} +1.46410 q^{59} -1.53590 q^{61} -3.46410 q^{65} -5.73205 q^{67} -12.9282 q^{69} -2.53590 q^{71} -1.07180 q^{73} +1.73205 q^{75} +14.3923 q^{79} -9.00000 q^{81} -7.46410 q^{83} +8.53590 q^{87} +15.9282 q^{89} -0.928203 q^{91} +2.53590 q^{93} +2.00000 q^{95} -14.3923 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{5} + 4 q^{7} + 4 q^{19} - 6 q^{21} - 8 q^{23} + 2 q^{25} - 4 q^{29} - 4 q^{31} + 4 q^{35} - 8 q^{37} - 12 q^{39} - 10 q^{41} + 4 q^{47} - 8 q^{53} - 4 q^{59} - 10 q^{61} - 8 q^{67} - 12 q^{69}+ \cdots - 8 q^{97}+O(q^{100}) $ 2 * q + 2 * q^5 + 4 * q^7 + 4 * q^19 - 6 * q^21 - 8 * q^23 + 2 * q^25 - 4 * q^29 - 4 * q^31 + 4 * q^35 - 8 * q^37 - 12 * q^39 - 10 * q^41 + 4 * q^47 - 8 * q^53 - 4 * q^59 - 10 * q^61 - 8 * q^67 - 12 * q^69 - 12 * q^71 - 16 * q^73 + 8 * q^79 - 18 * q^81 - 8 * q^83 + 24 * q^87 + 18 * q^89 + 12 * q^91 + 12 * q^93 + 4 * q^95 - 8 * 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
$3$	1.73205	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$3$	1.73205	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	0.267949	0.101275	0.0506376	−	0.998717i	$-0.483875\pi$
$7$	0.267949	0.101275	0.0506376	+	0.998717i	$0.483875\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−3.46410	−0.960769	−0.480384	−	0.877058i	$-0.659503\pi$
$13$	−3.46410	−0.960769	−0.480384	+	0.877058i	$0.659503\pi$
$14$	0	0
$15$	1.73205	0.447214
$16$	0	0
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0	0
$19$	2.00000	0.458831	0.229416	−	0.973329i	$-0.426318\pi$
$19$	2.00000	0.458831	0.229416	+	0.973329i	$0.426318\pi$
$20$	0	0
$21$	0.464102	0.101275
$22$	0	0
$23$	−7.46410	−1.55637	−0.778186	−	0.628033i	$-0.783860\pi$
$23$	−7.46410	−1.55637	−0.778186	+	0.628033i	$0.783860\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−5.19615	−1.00000
$28$	0	0
$29$	4.92820	0.915144	0.457572	−	0.889172i	$-0.348719\pi$
$29$	4.92820	0.915144	0.457572	+	0.889172i	$0.348719\pi$
$30$	0	0
$31$	1.46410	0.262960	0.131480	−	0.991319i	$-0.458027\pi$
$31$	1.46410	0.262960	0.131480	+	0.991319i	$0.458027\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0.267949	0.0452917
$36$	0	0
$37$	−4.00000	−0.657596	−0.328798	−	0.944400i	$-0.606644\pi$
$37$	−4.00000	−0.657596	−0.328798	+	0.944400i	$0.606644\pi$
$38$	0	0
$39$	−6.00000	−0.960769
$40$	0	0
$41$	1.92820	0.301135	0.150567	−	0.988600i	$-0.451890\pi$
$41$	1.92820	0.301135	0.150567	+	0.988600i	$0.451890\pi$
$42$	0	0
$43$	1.73205	0.264135	0.132068	−	0.991241i	$-0.457838\pi$
$43$	1.73205	0.264135	0.132068	+	0.991241i	$0.457838\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−6.66025	−0.971498	−0.485749	−	0.874098i	$-0.661453\pi$
$47$	−6.66025	−0.971498	−0.485749	+	0.874098i	$0.661453\pi$
$48$	0	0
$49$	−6.92820	−0.989743
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−7.46410	−1.02527	−0.512637	−	0.858606i	$-0.671331\pi$
$53$	−7.46410	−1.02527	−0.512637	+	0.858606i	$0.671331\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	3.46410	0.458831
$58$	0	0
$59$	1.46410	0.190610	0.0953049	−	0.995448i	$-0.469617\pi$
$59$	1.46410	0.190610	0.0953049	+	0.995448i	$0.469617\pi$
$60$	0	0
$61$	−1.53590	−0.196652	−0.0983258	−	0.995154i	$-0.531349\pi$
$61$	−1.53590	−0.196652	−0.0983258	+	0.995154i	$0.531349\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−3.46410	−0.429669
$66$	0	0
$67$	−5.73205	−0.700281	−0.350141	−	0.936697i	$-0.613866\pi$
$67$	−5.73205	−0.700281	−0.350141	+	0.936697i	$0.613866\pi$
$68$	0	0
$69$	−12.9282	−1.55637
$70$	0	0
$71$	−2.53590	−0.300956	−0.150478	−	0.988613i	$-0.548081\pi$
$71$	−2.53590	−0.300956	−0.150478	+	0.988613i	$0.548081\pi$
$72$	0	0
$73$	−1.07180	−0.125444	−0.0627222	−	0.998031i	$-0.519978\pi$
$73$	−1.07180	−0.125444	−0.0627222	+	0.998031i	$0.519978\pi$
$74$	0	0
$75$	1.73205	0.200000
$76$	0	0
$77$	0	0
$78$	0	0
$79$	14.3923	1.61926	0.809630	−	0.586940i	$-0.199668\pi$
$79$	14.3923	1.61926	0.809630	+	0.586940i	$0.199668\pi$
$80$	0	0
$81$	−9.00000	−1.00000
$82$	0	0
$83$	−7.46410	−0.819292	−0.409646	−	0.912245i	$-0.634348\pi$
$83$	−7.46410	−0.819292	−0.409646	+	0.912245i	$0.634348\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	8.53590	0.915144
$88$	0	0
$89$	15.9282	1.68839	0.844193	−	0.536039i	$-0.180080\pi$
$89$	15.9282	1.68839	0.844193	+	0.536039i	$0.180080\pi$
$90$	0	0
$91$	−0.928203	−0.0973021
$92$	0	0
$93$	2.53590	0.262960
$94$	0	0
$95$	2.00000	0.205196
$96$	0	0
$97$	−14.3923	−1.46132	−0.730659	−	0.682743i	$-0.760787\pi$
$97$	−14.3923	−1.46132	−0.730659	+	0.682743i	$0.760787\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−11.3923	−1.13358	−0.566788	−	0.823863i	$-0.691814\pi$
$101$	−11.3923	−1.13358	−0.566788	+	0.823863i	$0.691814\pi$
$102$	0	0
$103$	2.39230	0.235721	0.117860	−	0.993030i	$-0.462396\pi$
$103$	2.39230	0.235721	0.117860	+	0.993030i	$0.462396\pi$
$104$	0	0
$105$	0.464102	0.0452917
$106$	0	0
$107$	2.26795	0.219251	0.109625	−	0.993973i	$-0.465035\pi$
$107$	2.26795	0.219251	0.109625	+	0.993973i	$0.465035\pi$
$108$	0	0
$109$	−7.53590	−0.721808	−0.360904	−	0.932603i	$-0.617532\pi$
$109$	−7.53590	−0.721808	−0.360904	+	0.932603i	$0.617532\pi$
$110$	0	0
$111$	−6.92820	−0.657596
$112$	0	0
$113$	−10.0000	−0.940721	−0.470360	−	0.882474i	$-0.655876\pi$
$113$	−10.0000	−0.940721	−0.470360	+	0.882474i	$0.655876\pi$
$114$	0	0
$115$	−7.46410	−0.696031
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	0	0
$122$	0	0
$123$	3.33975	0.301135
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−18.6603	−1.65583	−0.827915	−	0.560854i	$-0.810473\pi$
$127$	−18.6603	−1.65583	−0.827915	+	0.560854i	$0.810473\pi$
$128$	0	0
$129$	3.00000	0.264135
$130$	0	0
$131$	14.0000	1.22319	0.611593	−	0.791173i	$-0.290529\pi$
$131$	14.0000	1.22319	0.611593	+	0.791173i	$0.290529\pi$
$132$	0	0
$133$	0.535898	0.0464683
$134$	0	0
$135$	−5.19615	−0.447214
$136$	0	0
$137$	10.3923	0.887875	0.443937	−	0.896058i	$-0.353581\pi$
$137$	10.3923	0.887875	0.443937	+	0.896058i	$0.353581\pi$
$138$	0	0
$139$	8.39230	0.711826	0.355913	−	0.934519i	$-0.384170\pi$
$139$	8.39230	0.711826	0.355913	+	0.934519i	$0.384170\pi$
$140$	0	0
$141$	−11.5359	−0.971498
$142$	0	0
$143$	0	0
$144$	0	0
$145$	4.92820	0.409265
$146$	0	0
$147$	−12.0000	−0.989743
$148$	0	0
$149$	−6.46410	−0.529560	−0.264780	−	0.964309i	$-0.585299\pi$
$149$	−6.46410	−0.529560	−0.264780	+	0.964309i	$0.585299\pi$
$150$	0	0
$151$	4.39230	0.357441	0.178720	−	0.983900i	$-0.442804\pi$
$151$	4.39230	0.357441	0.178720	+	0.983900i	$0.442804\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	1.46410	0.117599
$156$	0	0
$157$	−20.0000	−1.59617	−0.798087	−	0.602542i	$-0.794154\pi$
$157$	−20.0000	−1.59617	−0.798087	+	0.602542i	$0.794154\pi$
$158$	0	0
$159$	−12.9282	−1.02527
$160$	0	0
$161$	−2.00000	−0.157622
$162$	0	0
$163$	−8.12436	−0.636349	−0.318174	−	0.948032i	$-0.603070\pi$
$163$	−8.12436	−0.636349	−0.318174	+	0.948032i	$0.603070\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−20.5167	−1.58763	−0.793813	−	0.608161i	$-0.791907\pi$
$167$	−20.5167	−1.58763	−0.793813	+	0.608161i	$0.791907\pi$
$168$	0	0
$169$	−1.00000	−0.0769231
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−3.46410	−0.263371	−0.131685	−	0.991292i	$-0.542039\pi$
$173$	−3.46410	−0.263371	−0.131685	+	0.991292i	$0.542039\pi$
$174$	0	0
$175$	0.267949	0.0202551
$176$	0	0
$177$	2.53590	0.190610
$178$	0	0
$179$	−2.39230	−0.178809	−0.0894046	−	0.995995i	$-0.528496\pi$
$179$	−2.39230	−0.178809	−0.0894046	+	0.995995i	$0.528496\pi$
$180$	0	0
$181$	17.3923	1.29276	0.646380	−	0.763016i	$-0.276282\pi$
$181$	17.3923	1.29276	0.646380	+	0.763016i	$0.276282\pi$
$182$	0	0
$183$	−2.66025	−0.196652
$184$	0	0
$185$	−4.00000	−0.294086
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−1.39230	−0.101275
$190$	0	0
$191$	11.0718	0.801127	0.400564	−	0.916269i	$-0.368814\pi$
$191$	11.0718	0.801127	0.400564	+	0.916269i	$0.368814\pi$
$192$	0	0
$193$	−15.8564	−1.14137	−0.570685	−	0.821169i	$-0.693322\pi$
$193$	−15.8564	−1.14137	−0.570685	+	0.821169i	$0.693322\pi$
$194$	0	0
$195$	−6.00000	−0.429669
$196$	0	0
$197$	−0.928203	−0.0661317	−0.0330659	−	0.999453i	$-0.510527\pi$
$197$	−0.928203	−0.0661317	−0.0330659	+	0.999453i	$0.510527\pi$
$198$	0	0
$199$	−6.39230	−0.453138	−0.226569	−	0.973995i	$-0.572751\pi$
$199$	−6.39230	−0.453138	−0.226569	+	0.973995i	$0.572751\pi$
$200$	0	0
$201$	−9.92820	−0.700281
$202$	0	0
$203$	1.32051	0.0926815
$204$	0	0
$205$	1.92820	0.134672
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	6.53590	0.449950	0.224975	−	0.974365i	$-0.427770\pi$
$211$	6.53590	0.449950	0.224975	+	0.974365i	$0.427770\pi$
$212$	0	0
$213$	−4.39230	−0.300956
$214$	0	0
$215$	1.73205	0.118125
$216$	0	0
$217$	0.392305	0.0266314
$218$	0	0
$219$	−1.85641	−0.125444
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−0.267949	−0.0179432	−0.00897160	−	0.999960i	$-0.502856\pi$
$223$	−0.267949	−0.0179432	−0.00897160	+	0.999960i	$0.502856\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−6.80385	−0.451587	−0.225794	−	0.974175i	$-0.572497\pi$
$227$	−6.80385	−0.451587	−0.225794	+	0.974175i	$0.572497\pi$
$228$	0	0
$229$	−4.60770	−0.304485	−0.152243	−	0.988343i	$-0.548649\pi$
$229$	−4.60770	−0.304485	−0.152243	+	0.988343i	$0.548649\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−14.3923	−0.942871	−0.471436	−	0.881900i	$-0.656264\pi$
$233$	−14.3923	−0.942871	−0.471436	+	0.881900i	$0.656264\pi$
$234$	0	0
$235$	−6.66025	−0.434467
$236$	0	0
$237$	24.9282	1.61926
$238$	0	0
$239$	14.0000	0.905585	0.452792	−	0.891616i	$-0.350428\pi$
$239$	14.0000	0.905585	0.452792	+	0.891616i	$0.350428\pi$
$240$	0	0
$241$	−21.0000	−1.35273	−0.676364	−	0.736567i	$-0.736446\pi$
$241$	−21.0000	−1.35273	−0.676364	+	0.736567i	$0.736446\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−6.92820	−0.442627
$246$	0	0
$247$	−6.92820	−0.440831
$248$	0	0
$249$	−12.9282	−0.819292
$250$	0	0
$251$	−10.5359	−0.665020	−0.332510	−	0.943100i	$-0.607896\pi$
$251$	−10.5359	−0.665020	−0.332510	+	0.943100i	$0.607896\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	26.2487	1.63735	0.818675	−	0.574257i	$-0.194709\pi$
$257$	26.2487	1.63735	0.818675	+	0.574257i	$0.194709\pi$
$258$	0	0
$259$	−1.07180	−0.0665982
$260$	0	0
$261$	0	0
$262$	0	0
$263$	26.3923	1.62742	0.813710	−	0.581272i	$-0.197445\pi$
$263$	26.3923	1.62742	0.813710	+	0.581272i	$0.197445\pi$
$264$	0	0
$265$	−7.46410	−0.458516
$266$	0	0
$267$	27.5885	1.68839
$268$	0	0
$269$	−19.2487	−1.17361	−0.586807	−	0.809727i	$-0.699615\pi$
$269$	−19.2487	−1.17361	−0.586807	+	0.809727i	$0.699615\pi$
$270$	0	0
$271$	18.9282	1.14981	0.574903	−	0.818221i	$-0.305040\pi$
$271$	18.9282	1.14981	0.574903	+	0.818221i	$0.305040\pi$
$272$	0	0
$273$	−1.60770	−0.0973021
$274$	0	0
$275$	0	0
$276$	0	0
$277$	22.0000	1.32185	0.660926	−	0.750451i	$-0.270164\pi$
$277$	22.0000	1.32185	0.660926	+	0.750451i	$0.270164\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−31.8564	−1.90039	−0.950197	−	0.311650i	$-0.899118\pi$
$281$	−31.8564	−1.90039	−0.950197	+	0.311650i	$0.899118\pi$
$282$	0	0
$283$	18.5167	1.10070	0.550351	−	0.834934i	$-0.314494\pi$
$283$	18.5167	1.10070	0.550351	+	0.834934i	$0.314494\pi$
$284$	0	0
$285$	3.46410	0.205196
$286$	0	0
$287$	0.516660	0.0304975
$288$	0	0
$289$	−17.0000	−1.00000
$290$	0	0
$291$	−24.9282	−1.46132
$292$	0	0
$293$	−30.2487	−1.76715	−0.883574	−	0.468291i	$-0.844870\pi$
$293$	−30.2487	−1.76715	−0.883574	+	0.468291i	$0.844870\pi$
$294$	0	0
$295$	1.46410	0.0852433
$296$	0	0
$297$	0	0
$298$	0	0
$299$	25.8564	1.49531
$300$	0	0
$301$	0.464102	0.0267504
$302$	0	0
$303$	−19.7321	−1.13358
$304$	0	0
$305$	−1.53590	−0.0879453
$306$	0	0
$307$	26.3923	1.50629	0.753144	−	0.657855i	$-0.228536\pi$
$307$	26.3923	1.50629	0.753144	+	0.657855i	$0.228536\pi$
$308$	0	0
$309$	4.14359	0.235721
$310$	0	0
$311$	−11.0718	−0.627824	−0.313912	−	0.949452i	$-0.601640\pi$
$311$	−11.0718	−0.627824	−0.313912	+	0.949452i	$0.601640\pi$
$312$	0	0
$313$	3.46410	0.195803	0.0979013	−	0.995196i	$-0.468787\pi$
$313$	3.46410	0.195803	0.0979013	+	0.995196i	$0.468787\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−19.8564	−1.11525	−0.557623	−	0.830094i	$-0.688287\pi$
$317$	−19.8564	−1.11525	−0.557623	+	0.830094i	$0.688287\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	3.92820	0.219251
$322$	0	0
$323$	0	0
$324$	0	0
$325$	−3.46410	−0.192154
$326$	0	0
$327$	−13.0526	−0.721808
$328$	0	0
$329$	−1.78461	−0.0983887
$330$	0	0
$331$	20.9282	1.15032	0.575159	−	0.818042i	$-0.304940\pi$
$331$	20.9282	1.15032	0.575159	+	0.818042i	$0.304940\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−5.73205	−0.313175
$336$	0	0
$337$	1.32051	0.0719327	0.0359663	−	0.999353i	$-0.488549\pi$
$337$	1.32051	0.0719327	0.0359663	+	0.999353i	$0.488549\pi$
$338$	0	0
$339$	−17.3205	−0.940721
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−3.73205	−0.201512
$344$	0	0
$345$	−12.9282	−0.696031
$346$	0	0
$347$	−5.73205	−0.307713	−0.153856	−	0.988093i	$-0.549169\pi$
$347$	−5.73205	−0.307713	−0.153856	+	0.988093i	$0.549169\pi$
$348$	0	0
$349$	−29.7128	−1.59049	−0.795245	−	0.606288i	$-0.792658\pi$
$349$	−29.7128	−1.59049	−0.795245	+	0.606288i	$0.792658\pi$
$350$	0	0
$351$	18.0000	0.960769
$352$	0	0
$353$	33.4641	1.78111	0.890557	−	0.454871i	$-0.150315\pi$
$353$	33.4641	1.78111	0.890557	+	0.454871i	$0.150315\pi$
$354$	0	0
$355$	−2.53590	−0.134592
$356$	0	0
$357$	0	0
$358$	0	0
$359$	26.7846	1.41364	0.706819	−	0.707395i	$-0.250130\pi$
$359$	26.7846	1.41364	0.706819	+	0.707395i	$0.250130\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−1.07180	−0.0561004
$366$	0	0
$367$	−33.5885	−1.75330	−0.876652	−	0.481126i	$-0.840228\pi$
$367$	−33.5885	−1.75330	−0.876652	+	0.481126i	$0.840228\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−2.00000	−0.103835
$372$	0	0
$373$	12.3923	0.641649	0.320825	−	0.947139i	$-0.396040\pi$
$373$	12.3923	0.641649	0.320825	+	0.947139i	$0.396040\pi$
$374$	0	0
$375$	1.73205	0.0894427
$376$	0	0
$377$	−17.0718	−0.879242
$378$	0	0
$379$	6.39230	0.328351	0.164175	−	0.986431i	$-0.447504\pi$
$379$	6.39230	0.328351	0.164175	+	0.986431i	$0.447504\pi$
$380$	0	0
$381$	−32.3205	−1.65583
$382$	0	0
$383$	−12.5359	−0.640554	−0.320277	−	0.947324i	$-0.603776\pi$
$383$	−12.5359	−0.640554	−0.320277	+	0.947324i	$0.603776\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−23.2487	−1.17876	−0.589378	−	0.807857i	$-0.700627\pi$
$389$	−23.2487	−1.17876	−0.589378	+	0.807857i	$0.700627\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	24.2487	1.22319
$394$	0	0
$395$	14.3923	0.724155
$396$	0	0
$397$	−30.6410	−1.53783	−0.768914	−	0.639352i	$-0.779203\pi$
$397$	−30.6410	−1.53783	−0.768914	+	0.639352i	$0.779203\pi$
$398$	0	0
$399$	0.928203	0.0464683
$400$	0	0
$401$	−10.8564	−0.542143	−0.271072	−	0.962559i	$-0.587378\pi$
$401$	−10.8564	−0.542143	−0.271072	+	0.962559i	$0.587378\pi$
$402$	0	0
$403$	−5.07180	−0.252644
$404$	0	0
$405$	−9.00000	−0.447214
$406$	0	0
$407$	0	0
$408$	0	0
$409$	26.8564	1.32796	0.663982	−	0.747749i	$-0.268865\pi$
$409$	26.8564	1.32796	0.663982	+	0.747749i	$0.268865\pi$
$410$	0	0
$411$	18.0000	0.887875
$412$	0	0
$413$	0.392305	0.0193041
$414$	0	0
$415$	−7.46410	−0.366398
$416$	0	0
$417$	14.5359	0.711826
$418$	0	0
$419$	−15.8564	−0.774636	−0.387318	−	0.921946i	$-0.626598\pi$
$419$	−15.8564	−0.774636	−0.387318	+	0.921946i	$0.626598\pi$
$420$	0	0
$421$	18.4641	0.899885	0.449943	−	0.893057i	$-0.351444\pi$
$421$	18.4641	0.899885	0.449943	+	0.893057i	$0.351444\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−0.411543	−0.0199159
$428$	0	0
$429$	0	0
$430$	0	0
$431$	30.3923	1.46395	0.731973	−	0.681334i	$-0.238600\pi$
$431$	30.3923	1.46395	0.731973	+	0.681334i	$0.238600\pi$
$432$	0	0
$433$	29.1769	1.40215	0.701077	−	0.713086i	$-0.252703\pi$
$433$	29.1769	1.40215	0.701077	+	0.713086i	$0.252703\pi$
$434$	0	0
$435$	8.53590	0.409265
$436$	0	0
$437$	−14.9282	−0.714113
$438$	0	0
$439$	1.07180	0.0511541	0.0255770	−	0.999673i	$-0.491858\pi$
$439$	1.07180	0.0511541	0.0255770	+	0.999673i	$0.491858\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	20.6603	0.981598	0.490799	−	0.871273i	$-0.336705\pi$
$443$	20.6603	0.981598	0.490799	+	0.871273i	$0.336705\pi$
$444$	0	0
$445$	15.9282	0.755069
$446$	0	0
$447$	−11.1962	−0.529560
$448$	0	0
$449$	22.8564	1.07866	0.539330	−	0.842094i	$-0.318677\pi$
$449$	22.8564	1.07866	0.539330	+	0.842094i	$0.318677\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	7.60770	0.357441
$454$	0	0
$455$	−0.928203	−0.0435148
$456$	0	0
$457$	−6.24871	−0.292302	−0.146151	−	0.989262i	$-0.546689\pi$
$457$	−6.24871	−0.292302	−0.146151	+	0.989262i	$0.546689\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	35.2487	1.64170	0.820848	−	0.571147i	$-0.193501\pi$
$461$	35.2487	1.64170	0.820848	+	0.571147i	$0.193501\pi$
$462$	0	0
$463$	19.7321	0.917026	0.458513	−	0.888688i	$-0.348382\pi$
$463$	19.7321	0.917026	0.458513	+	0.888688i	$0.348382\pi$
$464$	0	0
$465$	2.53590	0.117599
$466$	0	0
$467$	15.3397	0.709839	0.354919	−	0.934897i	$-0.384508\pi$
$467$	15.3397	0.709839	0.354919	+	0.934897i	$0.384508\pi$
$468$	0	0
$469$	−1.53590	−0.0709212
$470$	0	0
$471$	−34.6410	−1.59617
$472$	0	0
$473$	0	0
$474$	0	0
$475$	2.00000	0.0917663
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−28.5359	−1.30384	−0.651919	−	0.758288i	$-0.726036\pi$
$479$	−28.5359	−1.30384	−0.651919	+	0.758288i	$0.726036\pi$
$480$	0	0
$481$	13.8564	0.631798
$482$	0	0
$483$	−3.46410	−0.157622
$484$	0	0
$485$	−14.3923	−0.653521
$486$	0	0
$487$	−6.67949	−0.302677	−0.151338	−	0.988482i	$-0.548358\pi$
$487$	−6.67949	−0.302677	−0.151338	+	0.988482i	$0.548358\pi$
$488$	0	0
$489$	−14.0718	−0.636349
$490$	0	0
$491$	−32.2487	−1.45536	−0.727682	−	0.685915i	$-0.759402\pi$
$491$	−32.2487	−1.45536	−0.727682	+	0.685915i	$0.759402\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−0.679492	−0.0304794
$498$	0	0
$499$	−33.3205	−1.49163	−0.745815	−	0.666153i	$-0.767940\pi$
$499$	−33.3205	−1.49163	−0.745815	+	0.666153i	$0.767940\pi$
$500$	0	0
$501$	−35.5359	−1.58763
$502$	0	0
$503$	38.6603	1.72378	0.861888	−	0.507099i	$-0.169282\pi$
$503$	38.6603	1.72378	0.861888	+	0.507099i	$0.169282\pi$
$504$	0	0
$505$	−11.3923	−0.506951
$506$	0	0
$507$	−1.73205	−0.0769231
$508$	0	0
$509$	−10.4641	−0.463813	−0.231907	−	0.972738i	$-0.574496\pi$
$509$	−10.4641	−0.463813	−0.231907	+	0.972738i	$0.574496\pi$
$510$	0	0
$511$	−0.287187	−0.0127044
$512$	0	0
$513$	−10.3923	−0.458831
$514$	0	0
$515$	2.39230	0.105418
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−6.00000	−0.263371
$520$	0	0
$521$	−17.0000	−0.744784	−0.372392	−	0.928076i	$-0.621462\pi$
$521$	−17.0000	−0.744784	−0.372392	+	0.928076i	$0.621462\pi$
$522$	0	0
$523$	9.32051	0.407557	0.203779	−	0.979017i	$-0.434678\pi$
$523$	9.32051	0.407557	0.203779	+	0.979017i	$0.434678\pi$
$524$	0	0
$525$	0.464102	0.0202551
$526$	0	0
$527$	0	0
$528$	0	0
$529$	32.7128	1.42230
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−6.67949	−0.289321
$534$	0	0
$535$	2.26795	0.0980520
$536$	0	0
$537$	−4.14359	−0.178809
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−12.4641	−0.535874	−0.267937	−	0.963436i	$-0.586342\pi$
$541$	−12.4641	−0.535874	−0.267937	+	0.963436i	$0.586342\pi$
$542$	0	0
$543$	30.1244	1.29276
$544$	0	0
$545$	−7.53590	−0.322802
$546$	0	0
$547$	−22.3923	−0.957426	−0.478713	−	0.877971i	$-0.658897\pi$
$547$	−22.3923	−0.957426	−0.478713	+	0.877971i	$0.658897\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	9.85641	0.419897
$552$	0	0
$553$	3.85641	0.163991
$554$	0	0
$555$	−6.92820	−0.294086
$556$	0	0
$557$	0	0		−	1.00000i	$-0.5\pi$
$557$	0	0			1.00000i	$0.5\pi$
$558$	0	0
$559$	−6.00000	−0.253773
$560$	0	0
$561$	0	0
$562$	0	0
$563$	32.6603	1.37647	0.688233	−	0.725490i	$-0.258387\pi$
$563$	32.6603	1.37647	0.688233	+	0.725490i	$0.258387\pi$
$564$	0	0
$565$	−10.0000	−0.420703
$566$	0	0
$567$	−2.41154	−0.101275
$568$	0	0
$569$	−18.7128	−0.784482	−0.392241	−	0.919863i	$-0.628300\pi$
$569$	−18.7128	−0.784482	−0.392241	+	0.919863i	$0.628300\pi$
$570$	0	0
$571$	16.2487	0.679987	0.339994	−	0.940428i	$-0.389575\pi$
$571$	16.2487	0.679987	0.339994	+	0.940428i	$0.389575\pi$
$572$	0	0
$573$	19.1769	0.801127
$574$	0	0
$575$	−7.46410	−0.311275
$576$	0	0
$577$	8.39230	0.349376	0.174688	−	0.984624i	$-0.444108\pi$
$577$	8.39230	0.349376	0.174688	+	0.984624i	$0.444108\pi$
$578$	0	0
$579$	−27.4641	−1.14137
$580$	0	0
$581$	−2.00000	−0.0829740
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−4.94744	−0.204203	−0.102101	−	0.994774i	$-0.532557\pi$
$587$	−4.94744	−0.204203	−0.102101	+	0.994774i	$0.532557\pi$
$588$	0	0
$589$	2.92820	0.120655
$590$	0	0
$591$	−1.60770	−0.0661317
$592$	0	0
$593$	6.39230	0.262500	0.131250	−	0.991349i	$-0.458101\pi$
$593$	6.39230	0.262500	0.131250	+	0.991349i	$0.458101\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−11.0718	−0.453138
$598$	0	0
$599$	24.9282	1.01854	0.509269	−	0.860607i	$-0.329916\pi$
$599$	24.9282	1.01854	0.509269	+	0.860607i	$0.329916\pi$
$600$	0	0
$601$	−23.8564	−0.973123	−0.486562	−	0.873646i	$-0.661749\pi$
$601$	−23.8564	−0.973123	−0.486562	+	0.873646i	$0.661749\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−27.4641	−1.11473	−0.557367	−	0.830266i	$-0.688188\pi$
$607$	−27.4641	−1.11473	−0.557367	+	0.830266i	$0.688188\pi$
$608$	0	0
$609$	2.28719	0.0926815
$610$	0	0
$611$	23.0718	0.933385
$612$	0	0
$613$	−13.7128	−0.553855	−0.276928	−	0.960891i	$-0.589316\pi$
$613$	−13.7128	−0.553855	−0.276928	+	0.960891i	$0.589316\pi$
$614$	0	0
$615$	3.33975	0.134672
$616$	0	0
$617$	11.3205	0.455746	0.227873	−	0.973691i	$-0.426823\pi$
$617$	11.3205	0.455746	0.227873	+	0.973691i	$0.426823\pi$
$618$	0	0
$619$	17.0718	0.686173	0.343087	−	0.939304i	$-0.388528\pi$
$619$	17.0718	0.686173	0.343087	+	0.939304i	$0.388528\pi$
$620$	0	0
$621$	38.7846	1.55637
$622$	0	0
$623$	4.26795	0.170992
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	27.7128	1.10323	0.551615	−	0.834099i	$-0.314012\pi$
$631$	27.7128	1.10323	0.551615	+	0.834099i	$0.314012\pi$
$632$	0	0
$633$	11.3205	0.449950
$634$	0	0
$635$	−18.6603	−0.740510
$636$	0	0
$637$	24.0000	0.950915
$638$	0	0
$639$	0	0
$640$	0	0
$641$	38.7846	1.53190	0.765950	−	0.642900i	$-0.222269\pi$
$641$	38.7846	1.53190	0.765950	+	0.642900i	$0.222269\pi$
$642$	0	0
$643$	14.8038	0.583807	0.291903	−	0.956448i	$-0.405711\pi$
$643$	14.8038	0.583807	0.291903	+	0.956448i	$0.405711\pi$
$644$	0	0
$645$	3.00000	0.118125
$646$	0	0
$647$	−5.87564	−0.230995	−0.115498	−	0.993308i	$-0.536846\pi$
$647$	−5.87564	−0.230995	−0.115498	+	0.993308i	$0.536846\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0.679492	0.0266314
$652$	0	0
$653$	3.85641	0.150913	0.0754564	−	0.997149i	$-0.475959\pi$
$653$	3.85641	0.150913	0.0754564	+	0.997149i	$0.475959\pi$
$654$	0	0
$655$	14.0000	0.547025
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−47.1769	−1.83775	−0.918876	−	0.394547i	$-0.870902\pi$
$659$	−47.1769	−1.83775	−0.918876	+	0.394547i	$0.870902\pi$
$660$	0	0
$661$	4.32051	0.168048	0.0840241	−	0.996464i	$-0.473223\pi$
$661$	4.32051	0.168048	0.0840241	+	0.996464i	$0.473223\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0.535898	0.0207812
$666$	0	0
$667$	−36.7846	−1.42431
$668$	0	0
$669$	−0.464102	−0.0179432
$670$	0	0
$671$	0	0
$672$	0	0
$673$	29.4641	1.13576	0.567879	−	0.823112i	$-0.307764\pi$
$673$	29.4641	1.13576	0.567879	+	0.823112i	$0.307764\pi$
$674$	0	0
$675$	−5.19615	−0.200000
$676$	0	0
$677$	39.7128	1.52629	0.763144	−	0.646229i	$-0.223655\pi$
$677$	39.7128	1.52629	0.763144	+	0.646229i	$0.223655\pi$
$678$	0	0
$679$	−3.85641	−0.147995
$680$	0	0
$681$	−11.7846	−0.451587
$682$	0	0
$683$	19.5885	0.749531	0.374766	−	0.927120i	$-0.377723\pi$
$683$	19.5885	0.749531	0.374766	+	0.927120i	$0.377723\pi$
$684$	0	0
$685$	10.3923	0.397070
$686$	0	0
$687$	−7.98076	−0.304485
$688$	0	0
$689$	25.8564	0.985051
$690$	0	0
$691$	43.5692	1.65745	0.828726	−	0.559655i	$-0.189066\pi$
$691$	43.5692	1.65745	0.828726	+	0.559655i	$0.189066\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	8.39230	0.318338
$696$	0	0
$697$	0	0
$698$	0	0
$699$	−24.9282	−0.942871
$700$	0	0
$701$	−13.7128	−0.517926	−0.258963	−	0.965887i	$-0.583381\pi$
$701$	−13.7128	−0.517926	−0.258963	+	0.965887i	$0.583381\pi$
$702$	0	0
$703$	−8.00000	−0.301726
$704$	0	0
$705$	−11.5359	−0.434467
$706$	0	0
$707$	−3.05256	−0.114803
$708$	0	0
$709$	−21.3923	−0.803405	−0.401702	−	0.915770i	$-0.631581\pi$
$709$	−21.3923	−0.803405	−0.401702	+	0.915770i	$0.631581\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−10.9282	−0.409264
$714$	0	0
$715$	0	0
$716$	0	0
$717$	24.2487	0.905585
$718$	0	0
$719$	−32.6410	−1.21730	−0.608652	−	0.793437i	$-0.708290\pi$
$719$	−32.6410	−1.21730	−0.608652	+	0.793437i	$0.708290\pi$
$720$	0	0
$721$	0.641016	0.0238727
$722$	0	0
$723$	−36.3731	−1.35273
$724$	0	0
$725$	4.92820	0.183029
$726$	0	0
$727$	−16.8038	−0.623220	−0.311610	−	0.950210i	$-0.600868\pi$
$727$	−16.8038	−0.623220	−0.311610	+	0.950210i	$0.600868\pi$
$728$	0	0
$729$	27.0000	1.00000
$730$	0	0
$731$	0	0
$732$	0	0
$733$	28.0000	1.03420	0.517102	−	0.855924i	$-0.327011\pi$
$733$	28.0000	1.03420	0.517102	+	0.855924i	$0.327011\pi$
$734$	0	0
$735$	−12.0000	−0.442627
$736$	0	0
$737$	0	0
$738$	0	0
$739$	51.1769	1.88257	0.941287	−	0.337609i	$-0.109618\pi$
$739$	51.1769	1.88257	0.941287	+	0.337609i	$0.109618\pi$
$740$	0	0
$741$	−12.0000	−0.440831
$742$	0	0
$743$	−4.26795	−0.156576	−0.0782879	−	0.996931i	$-0.524945\pi$
$743$	−4.26795	−0.156576	−0.0782879	+	0.996931i	$0.524945\pi$
$744$	0	0
$745$	−6.46410	−0.236826
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0.607695	0.0222047
$750$	0	0
$751$	−47.4641	−1.73199	−0.865995	−	0.500053i	$-0.833314\pi$
$751$	−47.4641	−1.73199	−0.865995	+	0.500053i	$0.833314\pi$
$752$	0	0
$753$	−18.2487	−0.665020
$754$	0	0
$755$	4.39230	0.159852
$756$	0	0
$757$	−25.3205	−0.920290	−0.460145	−	0.887844i	$-0.652202\pi$
$757$	−25.3205	−0.920290	−0.460145	+	0.887844i	$0.652202\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−4.14359	−0.150205	−0.0751026	−	0.997176i	$-0.523928\pi$
$761$	−4.14359	−0.150205	−0.0751026	+	0.997176i	$0.523928\pi$
$762$	0	0
$763$	−2.01924	−0.0731013
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−5.07180	−0.183132
$768$	0	0
$769$	−15.8564	−0.571797	−0.285898	−	0.958260i	$-0.592292\pi$
$769$	−15.8564	−0.571797	−0.285898	+	0.958260i	$0.592292\pi$
$770$	0	0
$771$	45.4641	1.63735
$772$	0	0
$773$	9.46410	0.340400	0.170200	−	0.985410i	$-0.445559\pi$
$773$	9.46410	0.340400	0.170200	+	0.985410i	$0.445559\pi$
$774$	0	0
$775$	1.46410	0.0525921
$776$	0	0
$777$	−1.85641	−0.0665982
$778$	0	0
$779$	3.85641	0.138170
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−25.6077	−0.915144
$784$	0	0
$785$	−20.0000	−0.713831
$786$	0	0
$787$	−27.5885	−0.983422	−0.491711	−	0.870758i	$-0.663628\pi$
$787$	−27.5885	−0.983422	−0.491711	+	0.870758i	$0.663628\pi$
$788$	0	0
$789$	45.7128	1.62742
$790$	0	0
$791$	−2.67949	−0.0952718
$792$	0	0
$793$	5.32051	0.188937
$794$	0	0
$795$	−12.9282	−0.458516
$796$	0	0
$797$	−34.5359	−1.22332	−0.611662	−	0.791119i	$-0.709499\pi$
$797$	−34.5359	−1.22332	−0.611662	+	0.791119i	$0.709499\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	0	0
$803$	0	0
$804$	0	0
$805$	−2.00000	−0.0704907
$806$	0	0
$807$	−33.3397	−1.17361
$808$	0	0
$809$	−38.7846	−1.36359	−0.681797	−	0.731541i	$-0.738801\pi$
$809$	−38.7846	−1.36359	−0.681797	+	0.731541i	$0.738801\pi$
$810$	0	0
$811$	−44.3923	−1.55882	−0.779412	−	0.626511i	$-0.784482\pi$
$811$	−44.3923	−1.55882	−0.779412	+	0.626511i	$0.784482\pi$
$812$	0	0
$813$	32.7846	1.14981
$814$	0	0
$815$	−8.12436	−0.284584
$816$	0	0
$817$	3.46410	0.121194
$818$	0	0
$819$	0	0
$820$	0	0
$821$	25.1051	0.876175	0.438087	−	0.898932i	$-0.355656\pi$
$821$	25.1051	0.876175	0.438087	+	0.898932i	$0.355656\pi$
$822$	0	0
$823$	−26.1244	−0.910638	−0.455319	−	0.890328i	$-0.650475\pi$
$823$	−26.1244	−0.910638	−0.455319	+	0.890328i	$0.650475\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−2.80385	−0.0974993	−0.0487497	−	0.998811i	$-0.515524\pi$
$827$	−2.80385	−0.0974993	−0.0487497	+	0.998811i	$0.515524\pi$
$828$	0	0
$829$	−23.2487	−0.807461	−0.403731	−	0.914878i	$-0.632287\pi$
$829$	−23.2487	−0.807461	−0.403731	+	0.914878i	$0.632287\pi$
$830$	0	0
$831$	38.1051	1.32185
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−20.5167	−0.710008
$836$	0	0
$837$	−7.60770	−0.262960
$838$	0	0
$839$	−27.4641	−0.948166	−0.474083	−	0.880480i	$-0.657220\pi$
$839$	−27.4641	−0.948166	−0.474083	+	0.880480i	$0.657220\pi$
$840$	0	0
$841$	−4.71281	−0.162511
$842$	0	0
$843$	−55.1769	−1.90039
$844$	0	0
$845$	−1.00000	−0.0344010
$846$	0	0
$847$	0	0
$848$	0	0
$849$	32.0718	1.10070
$850$	0	0
$851$	29.8564	1.02346
$852$	0	0
$853$	−13.0718	−0.447570	−0.223785	−	0.974639i	$-0.571841\pi$
$853$	−13.0718	−0.447570	−0.223785	+	0.974639i	$0.571841\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	48.9282	1.67136	0.835678	−	0.549220i	$-0.185075\pi$
$857$	48.9282	1.67136	0.835678	+	0.549220i	$0.185075\pi$
$858$	0	0
$859$	39.0718	1.33311	0.666556	−	0.745455i	$-0.267768\pi$
$859$	39.0718	1.33311	0.666556	+	0.745455i	$0.267768\pi$
$860$	0	0
$861$	0.894882	0.0304975
$862$	0	0
$863$	32.5167	1.10688	0.553440	−	0.832889i	$-0.313315\pi$
$863$	32.5167	1.10688	0.553440	+	0.832889i	$0.313315\pi$
$864$	0	0
$865$	−3.46410	−0.117783
$866$	0	0
$867$	−29.4449	−1.00000
$868$	0	0
$869$	0	0
$870$	0	0
$871$	19.8564	0.672809
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0.267949	0.00905834
$876$	0	0
$877$	−35.6077	−1.20239	−0.601193	−	0.799104i	$-0.705308\pi$
$877$	−35.6077	−1.20239	−0.601193	+	0.799104i	$0.705308\pi$
$878$	0	0
$879$	−52.3923	−1.76715
$880$	0	0
$881$	0.856406	0.0288531	0.0144265	−	0.999896i	$-0.495408\pi$
$881$	0.856406	0.0288531	0.0144265	+	0.999896i	$0.495408\pi$
$882$	0	0
$883$	−11.4641	−0.385798	−0.192899	−	0.981219i	$-0.561789\pi$
$883$	−11.4641	−0.385798	−0.192899	+	0.981219i	$0.561789\pi$
$884$	0	0
$885$	2.53590	0.0852433
$886$	0	0
$887$	−12.5167	−0.420268	−0.210134	−	0.977673i	$-0.567390\pi$
$887$	−12.5167	−0.420268	−0.210134	+	0.977673i	$0.567390\pi$
$888$	0	0
$889$	−5.00000	−0.167695
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−13.3205	−0.445754
$894$	0	0
$895$	−2.39230	−0.0799659
$896$	0	0
$897$	44.7846	1.49531
$898$	0	0
$899$	7.21539	0.240647
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0.803848	0.0267504
$904$	0	0
$905$	17.3923	0.578140
$906$	0	0
$907$	52.9090	1.75681	0.878407	−	0.477914i	$-0.158607\pi$
$907$	52.9090	1.75681	0.878407	+	0.477914i	$0.158607\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	12.2487	0.405818	0.202909	−	0.979198i	$-0.434960\pi$
$911$	12.2487	0.405818	0.202909	+	0.979198i	$0.434960\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	−2.66025	−0.0879453
$916$	0	0
$917$	3.75129	0.123878
$918$	0	0
$919$	−42.7846	−1.41133	−0.705667	−	0.708544i	$-0.749353\pi$
$919$	−42.7846	−1.41133	−0.705667	+	0.708544i	$0.749353\pi$
$920$	0	0
$921$	45.7128	1.50629
$922$	0	0
$923$	8.78461	0.289149
$924$	0	0
$925$	−4.00000	−0.131519
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−13.2154	−0.433583	−0.216791	−	0.976218i	$-0.569559\pi$
$929$	−13.2154	−0.433583	−0.216791	+	0.976218i	$0.569559\pi$
$930$	0	0
$931$	−13.8564	−0.454125
$932$	0	0
$933$	−19.1769	−0.627824
$934$	0	0
$935$	0	0
$936$	0	0
$937$	39.0718	1.27642	0.638210	−	0.769862i	$-0.279675\pi$
$937$	39.0718	1.27642	0.638210	+	0.769862i	$0.279675\pi$
$938$	0	0
$939$	6.00000	0.195803
$940$	0	0
$941$	−20.4641	−0.667111	−0.333555	−	0.942731i	$-0.608248\pi$
$941$	−20.4641	−0.667111	−0.333555	+	0.942731i	$0.608248\pi$
$942$	0	0
$943$	−14.3923	−0.468678
$944$	0	0
$945$	−1.39230	−0.0452917
$946$	0	0
$947$	−49.3205	−1.60270	−0.801351	−	0.598195i	$-0.795885\pi$
$947$	−49.3205	−1.60270	−0.801351	+	0.598195i	$0.795885\pi$
$948$	0	0
$949$	3.71281	0.120523
$950$	0	0
$951$	−34.3923	−1.11525
$952$	0	0
$953$	−4.00000	−0.129573	−0.0647864	−	0.997899i	$-0.520637\pi$
$953$	−4.00000	−0.129573	−0.0647864	+	0.997899i	$0.520637\pi$
$954$	0	0
$955$	11.0718	0.358275
$956$	0	0
$957$	0	0
$958$	0	0
$959$	2.78461	0.0899197
$960$	0	0
$961$	−28.8564	−0.930852
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−15.8564	−0.510436
$966$	0	0
$967$	3.17691	0.102163	0.0510813	−	0.998694i	$-0.483733\pi$
$967$	3.17691	0.102163	0.0510813	+	0.998694i	$0.483733\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	0.784610	0.0251793	0.0125897	−	0.999921i	$-0.495992\pi$
$971$	0.784610	0.0251793	0.0125897	+	0.999921i	$0.495992\pi$
$972$	0	0
$973$	2.24871	0.0720904
$974$	0	0
$975$	−6.00000	−0.192154
$976$	0	0
$977$	22.9282	0.733538	0.366769	−	0.930312i	$-0.380464\pi$
$977$	22.9282	0.733538	0.366769	+	0.930312i	$0.380464\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	0	0
$983$	39.7321	1.26726	0.633628	−	0.773638i	$-0.281565\pi$
$983$	39.7321	1.26726	0.633628	+	0.773638i	$0.281565\pi$
$984$	0	0
$985$	−0.928203	−0.0295750
$986$	0	0
$987$	−3.09103	−0.0983887
$988$	0	0
$989$	−12.9282	−0.411093
$990$	0	0
$991$	−6.78461	−0.215520	−0.107760	−	0.994177i	$-0.534368\pi$
$991$	−6.78461	−0.215520	−0.107760	+	0.994177i	$0.534368\pi$
$992$	0	0
$993$	36.2487	1.15032
$994$	0	0
$995$	−6.39230	−0.202650
$996$	0	0
$997$	2.67949	0.0848604	0.0424302	−	0.999099i	$-0.486490\pi$
$997$	2.67949	0.0848604	0.0424302	+	0.999099i	$0.486490\pi$
$998$	0	0
$999$	20.7846	0.657596

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9680.2.a.br.1.2		2
4.3	odd	2		4840.2.a.k.1.1	✓	2
11.10	odd	2		9680.2.a.bq.1.2		2
44.43	even	2		4840.2.a.l.1.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4840.2.a.k.1.1	✓	2	4.3	odd	2
4840.2.a.l.1.1	yes	2	44.43	even	2
9680.2.a.bq.1.2		2	11.10	odd	2
9680.2.a.br.1.2		2	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:	\( N \)	\(=\)	\( 9680 = 2^{4} \cdot 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9680.a (trivial)

\(f(q)\)	\(=\)	\(q+1.73205 q^{3} +1.00000 q^{5} +0.267949 q^{7} -3.46410 q^{13} +1.73205 q^{15} +2.00000 q^{19} +0.464102 q^{21} -7.46410 q^{23} +1.00000 q^{25} -5.19615 q^{27} +4.92820 q^{29} +1.46410 q^{31} +0.267949 q^{35} -4.00000 q^{37} -6.00000 q^{39} +1.92820 q^{41} +1.73205 q^{43} -6.66025 q^{47} -6.92820 q^{49} -7.46410 q^{53} +3.46410 q^{57} +1.46410 q^{59} -1.53590 q^{61} -3.46410 q^{65} -5.73205 q^{67} -12.9282 q^{69} -2.53590 q^{71} -1.07180 q^{73} +1.73205 q^{75} +14.3923 q^{79} -9.00000 q^{81} -7.46410 q^{83} +8.53590 q^{87} +15.9282 q^{89} -0.928203 q^{91} +2.53590 q^{93} +2.00000 q^{95} -14.3923 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{5} + 4 q^{7} + 4 q^{19} - 6 q^{21} - 8 q^{23} + 2 q^{25} - 4 q^{29} - 4 q^{31} + 4 q^{35} - 8 q^{37} - 12 q^{39} - 10 q^{41} + 4 q^{47} - 8 q^{53} - 4 q^{59} - 10 q^{61} - 8 q^{67} - 12 q^{69}+ \cdots - 8 q^{97}+O(q^{100}) \) 2 * q + 2 * q^5 + 4 * q^7 + 4 * q^19 - 6 * q^21 - 8 * q^23 + 2 * q^25 - 4 * q^29 - 4 * q^31 + 4 * q^35 - 8 * q^37 - 12 * q^39 - 10 * q^41 + 4 * q^47 - 8 * q^53 - 4 * q^59 - 10 * q^61 - 8 * q^67 - 12 * q^69 - 12 * q^71 - 16 * q^73 + 8 * q^79 - 18 * q^81 - 8 * q^83 + 24 * q^87 + 18 * q^89 + 12 * q^91 + 12 * q^93 + 4 * q^95 - 8 * q^97

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