LMFDB - Embedded newform 9576.2.a.bf.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9576,2,Mod(1,9576)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9576, 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("9576.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9576 = 2^{3} \cdot 3^{2} \cdot 7 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9576.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:	$76.4647449756$
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:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.73205$ of defining polynomial
Character	$\chi$	$=$	9576.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.73205 q^{5} -1.00000 q^{7} +O(q^{10})$	$q-2.73205 q^{5} -1.00000 q^{7} -3.46410 q^{11} +4.00000 q^{13} -6.73205 q^{17} +1.00000 q^{19} +7.46410 q^{23} +2.46410 q^{25} -0.732051 q^{29} +3.46410 q^{31} +2.73205 q^{35} -0.535898 q^{37} -4.53590 q^{41} +4.92820 q^{43} +11.6603 q^{47} +1.00000 q^{49} +2.19615 q^{53} +9.46410 q^{55} -8.00000 q^{59} +11.4641 q^{61} -10.9282 q^{65} -4.00000 q^{67} -4.19615 q^{71} +7.46410 q^{73} +3.46410 q^{77} +1.46410 q^{79} -10.1962 q^{83} +18.3923 q^{85} -10.3923 q^{89} -4.00000 q^{91} -2.73205 q^{95} +6.00000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{5} - 2 q^{7}+O(q^{10}) $ 2 * q - 2 * q^5 - 2 * q^7	$ 2 q - 2 q^{5} - 2 q^{7} + 8 q^{13} - 10 q^{17} + 2 q^{19} + 8 q^{23} - 2 q^{25} + 2 q^{29} + 2 q^{35} - 8 q^{37} - 16 q^{41} - 4 q^{43} + 6 q^{47} + 2 q^{49} - 6 q^{53} + 12 q^{55} - 16 q^{59} + 16 q^{61} - 8 q^{65} - 8 q^{67} + 2 q^{71} + 8 q^{73} - 4 q^{79} - 10 q^{83} + 16 q^{85} - 8 q^{91} - 2 q^{95} + 12 q^{97}+O(q^{100}) $ 2 * q - 2 * q^5 - 2 * q^7 + 8 * q^13 - 10 * q^17 + 2 * q^19 + 8 * q^23 - 2 * q^25 + 2 * q^29 + 2 * q^35 - 8 * q^37 - 16 * q^41 - 4 * q^43 + 6 * q^47 + 2 * q^49 - 6 * q^53 + 12 * q^55 - 16 * q^59 + 16 * q^61 - 8 * q^65 - 8 * q^67 + 2 * q^71 + 8 * q^73 - 4 * q^79 - 10 * q^83 + 16 * q^85 - 8 * q^91 - 2 * q^95 + 12 * 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
$5$	−2.73205	−1.22181	−0.610905	−	0.791704i	$-0.709194\pi$
$5$	−2.73205	−1.22181	−0.610905	+	0.791704i	$0.709194\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−3.46410	−1.04447	−0.522233	−	0.852803i	$-0.674901\pi$
$11$	−3.46410	−1.04447	−0.522233	+	0.852803i	$0.674901\pi$
$12$	0	0
$13$	4.00000	1.10940	0.554700	−	0.832050i	$-0.312833\pi$
$13$	4.00000	1.10940	0.554700	+	0.832050i	$0.312833\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−6.73205	−1.63276	−0.816381	−	0.577514i	$-0.804023\pi$
$17$	−6.73205	−1.63276	−0.816381	+	0.577514i	$0.804023\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	7.46410	1.55637	0.778186	−	0.628033i	$-0.216140\pi$
$23$	7.46410	1.55637	0.778186	+	0.628033i	$0.216140\pi$
$24$	0	0
$25$	2.46410	0.492820
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−0.732051	−0.135938	−0.0679692	−	0.997687i	$-0.521652\pi$
$29$	−0.732051	−0.135938	−0.0679692	+	0.997687i	$0.521652\pi$
$30$	0	0
$31$	3.46410	0.622171	0.311086	−	0.950382i	$-0.399307\pi$
$31$	3.46410	0.622171	0.311086	+	0.950382i	$0.399307\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	2.73205	0.461801
$36$	0	0
$37$	−0.535898	−0.0881012	−0.0440506	−	0.999029i	$-0.514026\pi$
$37$	−0.535898	−0.0881012	−0.0440506	+	0.999029i	$0.514026\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−4.53590	−0.708388	−0.354194	−	0.935172i	$-0.615245\pi$
$41$	−4.53590	−0.708388	−0.354194	+	0.935172i	$0.615245\pi$
$42$	0	0
$43$	4.92820	0.751544	0.375772	−	0.926712i	$-0.377378\pi$
$43$	4.92820	0.751544	0.375772	+	0.926712i	$0.377378\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	11.6603	1.70082	0.850411	−	0.526118i	$-0.176353\pi$
$47$	11.6603	1.70082	0.850411	+	0.526118i	$0.176353\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	2.19615	0.301665	0.150832	−	0.988559i	$-0.451805\pi$
$53$	2.19615	0.301665	0.150832	+	0.988559i	$0.451805\pi$
$54$	0	0
$55$	9.46410	1.27614
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−8.00000	−1.04151	−0.520756	−	0.853706i	$-0.674350\pi$
$59$	−8.00000	−1.04151	−0.520756	+	0.853706i	$0.674350\pi$
$60$	0	0
$61$	11.4641	1.46783	0.733914	−	0.679243i	$-0.237692\pi$
$61$	11.4641	1.46783	0.733914	+	0.679243i	$0.237692\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−10.9282	−1.35548
$66$	0	0
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−4.19615	−0.497992	−0.248996	−	0.968505i	$-0.580101\pi$
$71$	−4.19615	−0.497992	−0.248996	+	0.968505i	$0.580101\pi$
$72$	0	0
$73$	7.46410	0.873607	0.436804	−	0.899557i	$-0.356111\pi$
$73$	7.46410	0.873607	0.436804	+	0.899557i	$0.356111\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	3.46410	0.394771
$78$	0	0
$79$	1.46410	0.164724	0.0823622	−	0.996602i	$-0.473754\pi$
$79$	1.46410	0.164724	0.0823622	+	0.996602i	$0.473754\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−10.1962	−1.11917	−0.559587	−	0.828772i	$-0.689040\pi$
$83$	−10.1962	−1.11917	−0.559587	+	0.828772i	$0.689040\pi$
$84$	0	0
$85$	18.3923	1.99493
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−10.3923	−1.10158	−0.550791	−	0.834643i	$-0.685674\pi$
$89$	−10.3923	−1.10158	−0.550791	+	0.834643i	$0.685674\pi$
$90$	0	0
$91$	−4.00000	−0.419314
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−2.73205	−0.280302
$96$	0	0
$97$	6.00000	0.609208	0.304604	−	0.952479i	$-0.401476\pi$
$97$	6.00000	0.609208	0.304604	+	0.952479i	$0.401476\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	16.5885	1.65061	0.825307	−	0.564685i	$-0.191002\pi$
$101$	16.5885	1.65061	0.825307	+	0.564685i	$0.191002\pi$
$102$	0	0
$103$	−6.92820	−0.682656	−0.341328	−	0.939944i	$-0.610877\pi$
$103$	−6.92820	−0.682656	−0.341328	+	0.939944i	$0.610877\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	8.19615	0.792352	0.396176	−	0.918175i	$-0.370337\pi$
$107$	8.19615	0.792352	0.396176	+	0.918175i	$0.370337\pi$
$108$	0	0
$109$	−11.8564	−1.13564	−0.567819	−	0.823154i	$-0.692213\pi$
$109$	−11.8564	−1.13564	−0.567819	+	0.823154i	$0.692213\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−0.732051	−0.0688655	−0.0344328	−	0.999407i	$-0.510962\pi$
$113$	−0.732051	−0.0688655	−0.0344328	+	0.999407i	$0.510962\pi$
$114$	0	0
$115$	−20.3923	−1.90159
$116$	0	0
$117$	0	0
$118$	0	0
$119$	6.73205	0.617126
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	0	0
$124$	0	0
$125$	6.92820	0.619677
$126$	0	0
$127$	−10.9282	−0.969721	−0.484861	−	0.874591i	$-0.661130\pi$
$127$	−10.9282	−0.969721	−0.484861	+	0.874591i	$0.661130\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	2.19615	0.191879	0.0959394	−	0.995387i	$-0.469415\pi$
$131$	2.19615	0.191879	0.0959394	+	0.995387i	$0.469415\pi$
$132$	0	0
$133$	−1.00000	−0.0867110
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−2.92820	−0.250173	−0.125087	−	0.992146i	$-0.539921\pi$
$137$	−2.92820	−0.250173	−0.125087	+	0.992146i	$0.539921\pi$
$138$	0	0
$139$	9.46410	0.802735	0.401367	−	0.915917i	$-0.368535\pi$
$139$	9.46410	0.802735	0.401367	+	0.915917i	$0.368535\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−13.8564	−1.15873
$144$	0	0
$145$	2.00000	0.166091
$146$	0	0
$147$	0	0
$148$	0	0
$149$	2.53590	0.207749	0.103874	−	0.994590i	$-0.466876\pi$
$149$	2.53590	0.207749	0.103874	+	0.994590i	$0.466876\pi$
$150$	0	0
$151$	−14.5359	−1.18291	−0.591457	−	0.806336i	$-0.701447\pi$
$151$	−14.5359	−1.18291	−0.591457	+	0.806336i	$0.701447\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−9.46410	−0.760175
$156$	0	0
$157$	−6.39230	−0.510161	−0.255081	−	0.966920i	$-0.582102\pi$
$157$	−6.39230	−0.510161	−0.255081	+	0.966920i	$0.582102\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−7.46410	−0.588254
$162$	0	0
$163$	−10.0000	−0.783260	−0.391630	−	0.920123i	$-0.628089\pi$
$163$	−10.0000	−0.783260	−0.391630	+	0.920123i	$0.628089\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−16.3923	−1.26847	−0.634237	−	0.773138i	$-0.718686\pi$
$167$	−16.3923	−1.26847	−0.634237	+	0.773138i	$0.718686\pi$
$168$	0	0
$169$	3.00000	0.230769
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−10.3923	−0.790112	−0.395056	−	0.918657i	$-0.629275\pi$
$173$	−10.3923	−0.790112	−0.395056	+	0.918657i	$0.629275\pi$
$174$	0	0
$175$	−2.46410	−0.186269
$176$	0	0
$177$	0	0
$178$	0	0
$179$	19.5167	1.45874	0.729372	−	0.684117i	$-0.239812\pi$
$179$	19.5167	1.45874	0.729372	+	0.684117i	$0.239812\pi$
$180$	0	0
$181$	3.85641	0.286644	0.143322	−	0.989676i	$-0.454221\pi$
$181$	3.85641	0.286644	0.143322	+	0.989676i	$0.454221\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	1.46410	0.107643
$186$	0	0
$187$	23.3205	1.70536
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−2.39230	−0.173101	−0.0865506	−	0.996247i	$-0.527584\pi$
$191$	−2.39230	−0.173101	−0.0865506	+	0.996247i	$0.527584\pi$
$192$	0	0
$193$	−17.3205	−1.24676	−0.623379	−	0.781920i	$-0.714240\pi$
$193$	−17.3205	−1.24676	−0.623379	+	0.781920i	$0.714240\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	6.92820	0.493614	0.246807	−	0.969065i	$-0.420619\pi$
$197$	6.92820	0.493614	0.246807	+	0.969065i	$0.420619\pi$
$198$	0	0
$199$	24.7846	1.75693	0.878467	−	0.477803i	$-0.158567\pi$
$199$	24.7846	1.75693	0.878467	+	0.477803i	$0.158567\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0.732051	0.0513799
$204$	0	0
$205$	12.3923	0.865516
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−3.46410	−0.239617
$210$	0	0
$211$	4.00000	0.275371	0.137686	−	0.990476i	$-0.456034\pi$
$211$	4.00000	0.275371	0.137686	+	0.990476i	$0.456034\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−13.4641	−0.918244
$216$	0	0
$217$	−3.46410	−0.235159
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−26.9282	−1.81139
$222$	0	0
$223$	−21.3205	−1.42773	−0.713863	−	0.700285i	$-0.753056\pi$
$223$	−21.3205	−1.42773	−0.713863	+	0.700285i	$0.753056\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−6.53590	−0.433803	−0.216901	−	0.976194i	$-0.569595\pi$
$227$	−6.53590	−0.433803	−0.216901	+	0.976194i	$0.569595\pi$
$228$	0	0
$229$	−12.9282	−0.854320	−0.427160	−	0.904176i	$-0.640486\pi$
$229$	−12.9282	−0.854320	−0.427160	+	0.904176i	$0.640486\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−21.4641	−1.40616	−0.703080	−	0.711111i	$-0.748192\pi$
$233$	−21.4641	−1.40616	−0.703080	+	0.711111i	$0.748192\pi$
$234$	0	0
$235$	−31.8564	−2.07808
$236$	0	0
$237$	0	0
$238$	0	0
$239$	7.85641	0.508189	0.254094	−	0.967179i	$-0.418223\pi$
$239$	7.85641	0.508189	0.254094	+	0.967179i	$0.418223\pi$
$240$	0	0
$241$	28.9282	1.86343	0.931715	−	0.363191i	$-0.118313\pi$
$241$	28.9282	1.86343	0.931715	+	0.363191i	$0.118313\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−2.73205	−0.174544
$246$	0	0
$247$	4.00000	0.254514
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−7.26795	−0.458749	−0.229374	−	0.973338i	$-0.573668\pi$
$251$	−7.26795	−0.458749	−0.229374	+	0.973338i	$0.573668\pi$
$252$	0	0
$253$	−25.8564	−1.62558
$254$	0	0
$255$	0	0
$256$	0	0
$257$	27.8564	1.73763	0.868817	−	0.495133i	$-0.164881\pi$
$257$	27.8564	1.73763	0.868817	+	0.495133i	$0.164881\pi$
$258$	0	0
$259$	0.535898	0.0332991
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−3.07180	−0.189415	−0.0947076	−	0.995505i	$-0.530192\pi$
$263$	−3.07180	−0.189415	−0.0947076	+	0.995505i	$0.530192\pi$
$264$	0	0
$265$	−6.00000	−0.368577
$266$	0	0
$267$	0	0
$268$	0	0
$269$	0.928203	0.0565935	0.0282968	−	0.999600i	$-0.490992\pi$
$269$	0.928203	0.0565935	0.0282968	+	0.999600i	$0.490992\pi$
$270$	0	0
$271$	−11.3205	−0.687672	−0.343836	−	0.939030i	$-0.611726\pi$
$271$	−11.3205	−0.687672	−0.343836	+	0.939030i	$0.611726\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−8.53590	−0.514734
$276$	0	0
$277$	−26.2487	−1.57713	−0.788566	−	0.614950i	$-0.789176\pi$
$277$	−26.2487	−1.57713	−0.788566	+	0.614950i	$0.789176\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−15.2679	−0.910809	−0.455405	−	0.890285i	$-0.650505\pi$
$281$	−15.2679	−0.910809	−0.455405	+	0.890285i	$0.650505\pi$
$282$	0	0
$283$	−5.85641	−0.348127	−0.174064	−	0.984734i	$-0.555690\pi$
$283$	−5.85641	−0.348127	−0.174064	+	0.984734i	$0.555690\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	4.53590	0.267746
$288$	0	0
$289$	28.3205	1.66591
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−27.4641	−1.60447	−0.802235	−	0.597008i	$-0.796356\pi$
$293$	−27.4641	−1.60447	−0.802235	+	0.597008i	$0.796356\pi$
$294$	0	0
$295$	21.8564	1.27253
$296$	0	0
$297$	0	0
$298$	0	0
$299$	29.8564	1.72664
$300$	0	0
$301$	−4.92820	−0.284057
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−31.3205	−1.79341
$306$	0	0
$307$	13.3205	0.760242	0.380121	−	0.924937i	$-0.375882\pi$
$307$	13.3205	0.760242	0.380121	+	0.924937i	$0.375882\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−28.7321	−1.62925	−0.814623	−	0.579991i	$-0.803056\pi$
$311$	−28.7321	−1.62925	−0.814623	+	0.579991i	$0.803056\pi$
$312$	0	0
$313$	−13.3205	−0.752920	−0.376460	−	0.926433i	$-0.622859\pi$
$313$	−13.3205	−0.752920	−0.376460	+	0.926433i	$0.622859\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	11.2679	0.632871	0.316436	−	0.948614i	$-0.397514\pi$
$317$	11.2679	0.632871	0.316436	+	0.948614i	$0.397514\pi$
$318$	0	0
$319$	2.53590	0.141983
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−6.73205	−0.374581
$324$	0	0
$325$	9.85641	0.546735
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−11.6603	−0.642851
$330$	0	0
$331$	4.39230	0.241423	0.120711	−	0.992688i	$-0.461482\pi$
$331$	4.39230	0.241423	0.120711	+	0.992688i	$0.461482\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	10.9282	0.597072
$336$	0	0
$337$	3.07180	0.167331	0.0836657	−	0.996494i	$-0.473337\pi$
$337$	3.07180	0.167331	0.0836657	+	0.996494i	$0.473337\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−12.0000	−0.649836
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−35.8564	−1.92487	−0.962436	−	0.271507i	$-0.912478\pi$
$347$	−35.8564	−1.92487	−0.962436	+	0.271507i	$0.912478\pi$
$348$	0	0
$349$	23.8564	1.27700	0.638502	−	0.769620i	$-0.279554\pi$
$349$	23.8564	1.27700	0.638502	+	0.769620i	$0.279554\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	7.80385	0.415357	0.207678	−	0.978197i	$-0.433409\pi$
$353$	7.80385	0.415357	0.207678	+	0.978197i	$0.433409\pi$
$354$	0	0
$355$	11.4641	0.608451
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−4.53590	−0.239396	−0.119698	−	0.992810i	$-0.538193\pi$
$359$	−4.53590	−0.239396	−0.119698	+	0.992810i	$0.538193\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−20.3923	−1.06738
$366$	0	0
$367$	−25.8564	−1.34969	−0.674847	−	0.737958i	$-0.735790\pi$
$367$	−25.8564	−1.34969	−0.674847	+	0.737958i	$0.735790\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−2.19615	−0.114019
$372$	0	0
$373$	25.7128	1.33136	0.665679	−	0.746238i	$-0.268142\pi$
$373$	25.7128	1.33136	0.665679	+	0.746238i	$0.268142\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−2.92820	−0.150810
$378$	0	0
$379$	−30.2487	−1.55377	−0.776886	−	0.629641i	$-0.783202\pi$
$379$	−30.2487	−1.55377	−0.776886	+	0.629641i	$0.783202\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−21.4641	−1.09676	−0.548382	−	0.836228i	$-0.684756\pi$
$383$	−21.4641	−1.09676	−0.548382	+	0.836228i	$0.684756\pi$
$384$	0	0
$385$	−9.46410	−0.482335
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−2.53590	−0.128575	−0.0642876	−	0.997931i	$-0.520477\pi$
$389$	−2.53590	−0.128575	−0.0642876	+	0.997931i	$0.520477\pi$
$390$	0	0
$391$	−50.2487	−2.54119
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−4.00000	−0.201262
$396$	0	0
$397$	6.00000	0.301131	0.150566	−	0.988600i	$-0.451890\pi$
$397$	6.00000	0.301131	0.150566	+	0.988600i	$0.451890\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	16.0526	0.801627	0.400813	−	0.916160i	$-0.368728\pi$
$401$	16.0526	0.801627	0.400813	+	0.916160i	$0.368728\pi$
$402$	0	0
$403$	13.8564	0.690237
$404$	0	0
$405$	0	0
$406$	0	0
$407$	1.85641	0.0920187
$408$	0	0
$409$	35.7128	1.76588	0.882942	−	0.469481i	$-0.155559\pi$
$409$	35.7128	1.76588	0.882942	+	0.469481i	$0.155559\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	8.00000	0.393654
$414$	0	0
$415$	27.8564	1.36742
$416$	0	0
$417$	0	0
$418$	0	0
$419$	10.1962	0.498115	0.249057	−	0.968489i	$-0.419879\pi$
$419$	10.1962	0.498115	0.249057	+	0.968489i	$0.419879\pi$
$420$	0	0
$421$	−25.3205	−1.23405	−0.617023	−	0.786945i	$-0.711661\pi$
$421$	−25.3205	−1.23405	−0.617023	+	0.786945i	$0.711661\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−16.5885	−0.804658
$426$	0	0
$427$	−11.4641	−0.554787
$428$	0	0
$429$	0	0
$430$	0	0
$431$	22.0526	1.06223	0.531117	−	0.847298i	$-0.321772\pi$
$431$	22.0526	1.06223	0.531117	+	0.847298i	$0.321772\pi$
$432$	0	0
$433$	−4.92820	−0.236834	−0.118417	−	0.992964i	$-0.537782\pi$
$433$	−4.92820	−0.236834	−0.118417	+	0.992964i	$0.537782\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	7.46410	0.357056
$438$	0	0
$439$	11.4641	0.547152	0.273576	−	0.961850i	$-0.411794\pi$
$439$	11.4641	0.547152	0.273576	+	0.961850i	$0.411794\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	19.8564	0.943406	0.471703	−	0.881757i	$-0.343639\pi$
$443$	19.8564	0.943406	0.471703	+	0.881757i	$0.343639\pi$
$444$	0	0
$445$	28.3923	1.34592
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−34.5885	−1.63233	−0.816165	−	0.577819i	$-0.803904\pi$
$449$	−34.5885	−1.63233	−0.816165	+	0.577819i	$0.803904\pi$
$450$	0	0
$451$	15.7128	0.739887
$452$	0	0
$453$	0	0
$454$	0	0
$455$	10.9282	0.512322
$456$	0	0
$457$	−8.39230	−0.392575	−0.196288	−	0.980546i	$-0.562889\pi$
$457$	−8.39230	−0.392575	−0.196288	+	0.980546i	$0.562889\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	3.51666	0.163787	0.0818936	−	0.996641i	$-0.473903\pi$
$461$	3.51666	0.163787	0.0818936	+	0.996641i	$0.473903\pi$
$462$	0	0
$463$	13.8564	0.643962	0.321981	−	0.946746i	$-0.395651\pi$
$463$	13.8564	0.643962	0.321981	+	0.946746i	$0.395651\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	3.26795	0.151223	0.0756113	−	0.997137i	$-0.475909\pi$
$467$	3.26795	0.151223	0.0756113	+	0.997137i	$0.475909\pi$
$468$	0	0
$469$	4.00000	0.184703
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−17.0718	−0.784962
$474$	0	0
$475$	2.46410	0.113061
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−32.7321	−1.49557	−0.747783	−	0.663943i	$-0.768882\pi$
$479$	−32.7321	−1.49557	−0.747783	+	0.663943i	$0.768882\pi$
$480$	0	0
$481$	−2.14359	−0.0977395
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−16.3923	−0.744336
$486$	0	0
$487$	−33.8564	−1.53418	−0.767090	−	0.641539i	$-0.778296\pi$
$487$	−33.8564	−1.53418	−0.767090	+	0.641539i	$0.778296\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−19.8564	−0.896107	−0.448054	−	0.894007i	$-0.647883\pi$
$491$	−19.8564	−0.896107	−0.448054	+	0.894007i	$0.647883\pi$
$492$	0	0
$493$	4.92820	0.221955
$494$	0	0
$495$	0	0
$496$	0	0
$497$	4.19615	0.188223
$498$	0	0
$499$	4.00000	0.179065	0.0895323	−	0.995984i	$-0.471463\pi$
$499$	4.00000	0.179065	0.0895323	+	0.995984i	$0.471463\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−11.6603	−0.519905	−0.259953	−	0.965621i	$-0.583707\pi$
$503$	−11.6603	−0.519905	−0.259953	+	0.965621i	$0.583707\pi$
$504$	0	0
$505$	−45.3205	−2.01674
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−26.7846	−1.18721	−0.593603	−	0.804758i	$-0.702295\pi$
$509$	−26.7846	−1.18721	−0.593603	+	0.804758i	$0.702295\pi$
$510$	0	0
$511$	−7.46410	−0.330192
$512$	0	0
$513$	0	0
$514$	0	0
$515$	18.9282	0.834076
$516$	0	0
$517$	−40.3923	−1.77645
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−37.7128	−1.65223	−0.826114	−	0.563503i	$-0.809453\pi$
$521$	−37.7128	−1.65223	−0.826114	+	0.563503i	$0.809453\pi$
$522$	0	0
$523$	−7.46410	−0.326382	−0.163191	−	0.986594i	$-0.552179\pi$
$523$	−7.46410	−0.326382	−0.163191	+	0.986594i	$0.552179\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−23.3205	−1.01586
$528$	0	0
$529$	32.7128	1.42230
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−18.1436	−0.785886
$534$	0	0
$535$	−22.3923	−0.968104
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−3.46410	−0.149209
$540$	0	0
$541$	4.14359	0.178147	0.0890735	−	0.996025i	$-0.471609\pi$
$541$	4.14359	0.178147	0.0890735	+	0.996025i	$0.471609\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	32.3923	1.38753
$546$	0	0
$547$	−36.1051	−1.54374	−0.771872	−	0.635778i	$-0.780679\pi$
$547$	−36.1051	−1.54374	−0.771872	+	0.635778i	$0.780679\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−0.732051	−0.0311864
$552$	0	0
$553$	−1.46410	−0.0622599
$554$	0	0
$555$	0	0
$556$	0	0
$557$	6.53590	0.276935	0.138467	−	0.990367i	$-0.455782\pi$
$557$	6.53590	0.276935	0.138467	+	0.990367i	$0.455782\pi$
$558$	0	0
$559$	19.7128	0.833763
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−21.4641	−0.904604	−0.452302	−	0.891865i	$-0.649397\pi$
$563$	−21.4641	−0.904604	−0.452302	+	0.891865i	$0.649397\pi$
$564$	0	0
$565$	2.00000	0.0841406
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−16.0526	−0.672958	−0.336479	−	0.941691i	$-0.609236\pi$
$569$	−16.0526	−0.672958	−0.336479	+	0.941691i	$0.609236\pi$
$570$	0	0
$571$	−30.9282	−1.29431	−0.647153	−	0.762361i	$-0.724040\pi$
$571$	−30.9282	−1.29431	−0.647153	+	0.762361i	$0.724040\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	18.3923	0.767012
$576$	0	0
$577$	10.6795	0.444593	0.222297	−	0.974979i	$-0.428645\pi$
$577$	10.6795	0.444593	0.222297	+	0.974979i	$0.428645\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	10.1962	0.423008
$582$	0	0
$583$	−7.60770	−0.315079
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−16.0526	−0.662560	−0.331280	−	0.943532i	$-0.607480\pi$
$587$	−16.0526	−0.662560	−0.331280	+	0.943532i	$0.607480\pi$
$588$	0	0
$589$	3.46410	0.142736
$590$	0	0
$591$	0	0
$592$	0	0
$593$	20.9808	0.861577	0.430788	−	0.902453i	$-0.358236\pi$
$593$	20.9808	0.861577	0.430788	+	0.902453i	$0.358236\pi$
$594$	0	0
$595$	−18.3923	−0.754011
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−18.0526	−0.737608	−0.368804	−	0.929507i	$-0.620233\pi$
$599$	−18.0526	−0.737608	−0.368804	+	0.929507i	$0.620233\pi$
$600$	0	0
$601$	−28.6410	−1.16829	−0.584146	−	0.811649i	$-0.698570\pi$
$601$	−28.6410	−1.16829	−0.584146	+	0.811649i	$0.698570\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−2.73205	−0.111074
$606$	0	0
$607$	16.7846	0.681266	0.340633	−	0.940196i	$-0.389359\pi$
$607$	16.7846	0.681266	0.340633	+	0.940196i	$0.389359\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	46.6410	1.88689
$612$	0	0
$613$	28.3923	1.14675	0.573377	−	0.819292i	$-0.305633\pi$
$613$	28.3923	1.14675	0.573377	+	0.819292i	$0.305633\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	19.7128	0.793608	0.396804	−	0.917903i	$-0.370119\pi$
$617$	19.7128	0.793608	0.396804	+	0.917903i	$0.370119\pi$
$618$	0	0
$619$	−6.53590	−0.262700	−0.131350	−	0.991336i	$-0.541931\pi$
$619$	−6.53590	−0.262700	−0.131350	+	0.991336i	$0.541931\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	10.3923	0.416359
$624$	0	0
$625$	−31.2487	−1.24995
$626$	0	0
$627$	0	0
$628$	0	0
$629$	3.60770	0.143848
$630$	0	0
$631$	−7.07180	−0.281524	−0.140762	−	0.990043i	$-0.544955\pi$
$631$	−7.07180	−0.281524	−0.140762	+	0.990043i	$0.544955\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	29.8564	1.18482
$636$	0	0
$637$	4.00000	0.158486
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−17.1244	−0.676371	−0.338186	−	0.941079i	$-0.609813\pi$
$641$	−17.1244	−0.676371	−0.338186	+	0.941079i	$0.609813\pi$
$642$	0	0
$643$	−41.5692	−1.63933	−0.819665	−	0.572843i	$-0.805840\pi$
$643$	−41.5692	−1.63933	−0.819665	+	0.572843i	$0.805840\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−5.12436	−0.201459	−0.100730	−	0.994914i	$-0.532118\pi$
$647$	−5.12436	−0.201459	−0.100730	+	0.994914i	$0.532118\pi$
$648$	0	0
$649$	27.7128	1.08782
$650$	0	0
$651$	0	0
$652$	0	0
$653$	26.9282	1.05378	0.526891	−	0.849933i	$-0.323358\pi$
$653$	26.9282	1.05378	0.526891	+	0.849933i	$0.323358\pi$
$654$	0	0
$655$	−6.00000	−0.234439
$656$	0	0
$657$	0	0
$658$	0	0
$659$	13.2679	0.516846	0.258423	−	0.966032i	$-0.416797\pi$
$659$	13.2679	0.516846	0.258423	+	0.966032i	$0.416797\pi$
$660$	0	0
$661$	−42.6410	−1.65854	−0.829272	−	0.558846i	$-0.811244\pi$
$661$	−42.6410	−1.65854	−0.829272	+	0.558846i	$0.811244\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	2.73205	0.105944
$666$	0	0
$667$	−5.46410	−0.211571
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−39.7128	−1.53310
$672$	0	0
$673$	16.2487	0.626342	0.313171	−	0.949697i	$-0.398609\pi$
$673$	16.2487	0.626342	0.313171	+	0.949697i	$0.398609\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	8.92820	0.343139	0.171569	−	0.985172i	$-0.445116\pi$
$677$	8.92820	0.343139	0.171569	+	0.985172i	$0.445116\pi$
$678$	0	0
$679$	−6.00000	−0.230259
$680$	0	0
$681$	0	0
$682$	0	0
$683$	18.7321	0.716762	0.358381	−	0.933575i	$-0.383329\pi$
$683$	18.7321	0.716762	0.358381	+	0.933575i	$0.383329\pi$
$684$	0	0
$685$	8.00000	0.305664
$686$	0	0
$687$	0	0
$688$	0	0
$689$	8.78461	0.334667
$690$	0	0
$691$	−39.3205	−1.49582	−0.747911	−	0.663799i	$-0.768943\pi$
$691$	−39.3205	−1.49582	−0.747911	+	0.663799i	$0.768943\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−25.8564	−0.980789
$696$	0	0
$697$	30.5359	1.15663
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−12.0000	−0.453234	−0.226617	−	0.973984i	$-0.572767\pi$
$701$	−12.0000	−0.453234	−0.226617	+	0.973984i	$0.572767\pi$
$702$	0	0
$703$	−0.535898	−0.0202118
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−16.5885	−0.623873
$708$	0	0
$709$	−34.2487	−1.28624	−0.643119	−	0.765767i	$-0.722360\pi$
$709$	−34.2487	−1.28624	−0.643119	+	0.765767i	$0.722360\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	25.8564	0.968330
$714$	0	0
$715$	37.8564	1.41575
$716$	0	0
$717$	0	0
$718$	0	0
$719$	38.5885	1.43911	0.719553	−	0.694437i	$-0.244347\pi$
$719$	38.5885	1.43911	0.719553	+	0.694437i	$0.244347\pi$
$720$	0	0
$721$	6.92820	0.258020
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−1.80385	−0.0669932
$726$	0	0
$727$	12.3923	0.459605	0.229803	−	0.973237i	$-0.426192\pi$
$727$	12.3923	0.459605	0.229803	+	0.973237i	$0.426192\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−33.1769	−1.22709
$732$	0	0
$733$	0.535898	0.0197939	0.00989693	−	0.999951i	$-0.496850\pi$
$733$	0.535898	0.0197939	0.00989693	+	0.999951i	$0.496850\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	13.8564	0.510407
$738$	0	0
$739$	40.9282	1.50557	0.752784	−	0.658267i	$-0.228710\pi$
$739$	40.9282	1.50557	0.752784	+	0.658267i	$0.228710\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−52.3013	−1.91875	−0.959374	−	0.282138i	$-0.908956\pi$
$743$	−52.3013	−1.91875	−0.959374	+	0.282138i	$0.908956\pi$
$744$	0	0
$745$	−6.92820	−0.253830
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−8.19615	−0.299481
$750$	0	0
$751$	4.39230	0.160277	0.0801387	−	0.996784i	$-0.474464\pi$
$751$	4.39230	0.160277	0.0801387	+	0.996784i	$0.474464\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	39.7128	1.44530
$756$	0	0
$757$	49.7128	1.80684	0.903421	−	0.428754i	$-0.141047\pi$
$757$	49.7128	1.80684	0.903421	+	0.428754i	$0.141047\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−10.7321	−0.389037	−0.194518	−	0.980899i	$-0.562314\pi$
$761$	−10.7321	−0.389037	−0.194518	+	0.980899i	$0.562314\pi$
$762$	0	0
$763$	11.8564	0.429231
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−32.0000	−1.15545
$768$	0	0
$769$	32.5359	1.17327	0.586637	−	0.809850i	$-0.300451\pi$
$769$	32.5359	1.17327	0.586637	+	0.809850i	$0.300451\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	30.3923	1.09314	0.546568	−	0.837415i	$-0.315934\pi$
$773$	30.3923	1.09314	0.546568	+	0.837415i	$0.315934\pi$
$774$	0	0
$775$	8.53590	0.306619
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−4.53590	−0.162515
$780$	0	0
$781$	14.5359	0.520135
$782$	0	0
$783$	0	0
$784$	0	0
$785$	17.4641	0.623321
$786$	0	0
$787$	23.4641	0.836405	0.418202	−	0.908354i	$-0.362660\pi$
$787$	23.4641	0.836405	0.418202	+	0.908354i	$0.362660\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0.732051	0.0260287
$792$	0	0
$793$	45.8564	1.62841
$794$	0	0
$795$	0	0
$796$	0	0
$797$	12.2487	0.433872	0.216936	−	0.976186i	$-0.430394\pi$
$797$	12.2487	0.433872	0.216936	+	0.976186i	$0.430394\pi$
$798$	0	0
$799$	−78.4974	−2.77704
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−25.8564	−0.912453
$804$	0	0
$805$	20.3923	0.718734
$806$	0	0
$807$	0	0
$808$	0	0
$809$	25.0718	0.881477	0.440739	−	0.897635i	$-0.354717\pi$
$809$	25.0718	0.881477	0.440739	+	0.897635i	$0.354717\pi$
$810$	0	0
$811$	27.7128	0.973128	0.486564	−	0.873645i	$-0.338250\pi$
$811$	27.7128	0.973128	0.486564	+	0.873645i	$0.338250\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	27.3205	0.956996
$816$	0	0
$817$	4.92820	0.172416
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−15.3205	−0.534689	−0.267345	−	0.963601i	$-0.586146\pi$
$821$	−15.3205	−0.534689	−0.267345	+	0.963601i	$0.586146\pi$
$822$	0	0
$823$	−10.0000	−0.348578	−0.174289	−	0.984695i	$-0.555763\pi$
$823$	−10.0000	−0.348578	−0.174289	+	0.984695i	$0.555763\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−0.483340	−0.0168074	−0.00840368	−	0.999965i	$-0.502675\pi$
$827$	−0.483340	−0.0168074	−0.00840368	+	0.999965i	$0.502675\pi$
$828$	0	0
$829$	52.9282	1.83827	0.919136	−	0.393940i	$-0.128888\pi$
$829$	52.9282	1.83827	0.919136	+	0.393940i	$0.128888\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−6.73205	−0.233252
$834$	0	0
$835$	44.7846	1.54984
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−8.78461	−0.303278	−0.151639	−	0.988436i	$-0.548455\pi$
$839$	−8.78461	−0.303278	−0.151639	+	0.988436i	$0.548455\pi$
$840$	0	0
$841$	−28.4641	−0.981521
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−8.19615	−0.281956
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−4.00000	−0.137118
$852$	0	0
$853$	−32.9282	−1.12744	−0.563720	−	0.825966i	$-0.690630\pi$
$853$	−32.9282	−1.12744	−0.563720	+	0.825966i	$0.690630\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	6.39230	0.218357	0.109178	−	0.994022i	$-0.465178\pi$
$857$	6.39230	0.218357	0.109178	+	0.994022i	$0.465178\pi$
$858$	0	0
$859$	0.784610	0.0267705	0.0133853	−	0.999910i	$-0.495739\pi$
$859$	0.784610	0.0267705	0.0133853	+	0.999910i	$0.495739\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−22.0526	−0.750678	−0.375339	−	0.926888i	$-0.622474\pi$
$863$	−22.0526	−0.750678	−0.375339	+	0.926888i	$0.622474\pi$
$864$	0	0
$865$	28.3923	0.965367
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−5.07180	−0.172049
$870$	0	0
$871$	−16.0000	−0.542139
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−6.92820	−0.234216
$876$	0	0
$877$	54.7846	1.84994	0.924972	−	0.380034i	$-0.124088\pi$
$877$	54.7846	1.84994	0.924972	+	0.380034i	$0.124088\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	27.5167	0.927060	0.463530	−	0.886081i	$-0.346583\pi$
$881$	27.5167	0.927060	0.463530	+	0.886081i	$0.346583\pi$
$882$	0	0
$883$	15.7128	0.528778	0.264389	−	0.964416i	$-0.414830\pi$
$883$	15.7128	0.528778	0.264389	+	0.964416i	$0.414830\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−40.1051	−1.34660	−0.673299	−	0.739370i	$-0.735123\pi$
$887$	−40.1051	−1.34660	−0.673299	+	0.739370i	$0.735123\pi$
$888$	0	0
$889$	10.9282	0.366520
$890$	0	0
$891$	0	0
$892$	0	0
$893$	11.6603	0.390196
$894$	0	0
$895$	−53.3205	−1.78231
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−2.53590	−0.0845769
$900$	0	0
$901$	−14.7846	−0.492547
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−10.5359	−0.350225
$906$	0	0
$907$	−30.6410	−1.01742	−0.508709	−	0.860938i	$-0.669877\pi$
$907$	−30.6410	−1.01742	−0.508709	+	0.860938i	$0.669877\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−9.94744	−0.329573	−0.164787	−	0.986329i	$-0.552694\pi$
$911$	−9.94744	−0.329573	−0.164787	+	0.986329i	$0.552694\pi$
$912$	0	0
$913$	35.3205	1.16894
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−2.19615	−0.0725233
$918$	0	0
$919$	−13.8564	−0.457081	−0.228540	−	0.973534i	$-0.573395\pi$
$919$	−13.8564	−0.457081	−0.228540	+	0.973534i	$0.573395\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−16.7846	−0.552472
$924$	0	0
$925$	−1.32051	−0.0434180
$926$	0	0
$927$	0	0
$928$	0	0
$929$	32.1962	1.05632	0.528161	−	0.849144i	$-0.322882\pi$
$929$	32.1962	1.05632	0.528161	+	0.849144i	$0.322882\pi$
$930$	0	0
$931$	1.00000	0.0327737
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−63.7128	−2.08363
$936$	0	0
$937$	12.1436	0.396714	0.198357	−	0.980130i	$-0.436439\pi$
$937$	12.1436	0.396714	0.198357	+	0.980130i	$0.436439\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	28.6410	0.933670	0.466835	−	0.884344i	$-0.345394\pi$
$941$	28.6410	0.933670	0.466835	+	0.884344i	$0.345394\pi$
$942$	0	0
$943$	−33.8564	−1.10252
$944$	0	0
$945$	0	0
$946$	0	0
$947$	33.0333	1.07344	0.536719	−	0.843761i	$-0.319663\pi$
$947$	33.0333	1.07344	0.536719	+	0.843761i	$0.319663\pi$
$948$	0	0
$949$	29.8564	0.969180
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−34.5885	−1.12043	−0.560215	−	0.828347i	$-0.689281\pi$
$953$	−34.5885	−1.12043	−0.560215	+	0.828347i	$0.689281\pi$
$954$	0	0
$955$	6.53590	0.211497
$956$	0	0
$957$	0	0
$958$	0	0
$959$	2.92820	0.0945566
$960$	0	0
$961$	−19.0000	−0.612903
$962$	0	0
$963$	0	0
$964$	0	0
$965$	47.3205	1.52330
$966$	0	0
$967$	2.78461	0.0895470	0.0447735	−	0.998997i	$-0.485743\pi$
$967$	2.78461	0.0895470	0.0447735	+	0.998997i	$0.485743\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	54.9282	1.76273	0.881365	−	0.472436i	$-0.156625\pi$
$971$	54.9282	1.76273	0.881365	+	0.472436i	$0.156625\pi$
$972$	0	0
$973$	−9.46410	−0.303405
$974$	0	0
$975$	0	0
$976$	0	0
$977$	21.1244	0.675828	0.337914	−	0.941177i	$-0.390279\pi$
$977$	21.1244	0.675828	0.337914	+	0.941177i	$0.390279\pi$
$978$	0	0
$979$	36.0000	1.15056
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−28.7846	−0.918086	−0.459043	−	0.888414i	$-0.651808\pi$
$983$	−28.7846	−0.918086	−0.459043	+	0.888414i	$0.651808\pi$
$984$	0	0
$985$	−18.9282	−0.603103
$986$	0	0
$987$	0	0
$988$	0	0
$989$	36.7846	1.16968
$990$	0	0
$991$	−11.2154	−0.356269	−0.178134	−	0.984006i	$-0.557006\pi$
$991$	−11.2154	−0.356269	−0.178134	+	0.984006i	$0.557006\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−67.7128	−2.14664
$996$	0	0
$997$	30.1051	0.953439	0.476719	−	0.879056i	$-0.341826\pi$
$997$	30.1051	0.953439	0.476719	+	0.879056i	$0.341826\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9576.2.a.bf.1.1	✓	2
3.2	odd	2		9576.2.a.br.1.2	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9576.2.a.bf.1.1	✓	2	1.1	even	1	trivial
9576.2.a.br.1.2	yes	2	3.2	odd	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:	\( N \)	\(=\)	\( 9576 = 2^{3} \cdot 3^{2} \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9576.a (trivial)

\(f(q)\)	\(=\)	\(q-2.73205 q^{5} -1.00000 q^{7} +O(q^{10})\)	\(q-2.73205 q^{5} -1.00000 q^{7} -3.46410 q^{11} +4.00000 q^{13} -6.73205 q^{17} +1.00000 q^{19} +7.46410 q^{23} +2.46410 q^{25} -0.732051 q^{29} +3.46410 q^{31} +2.73205 q^{35} -0.535898 q^{37} -4.53590 q^{41} +4.92820 q^{43} +11.6603 q^{47} +1.00000 q^{49} +2.19615 q^{53} +9.46410 q^{55} -8.00000 q^{59} +11.4641 q^{61} -10.9282 q^{65} -4.00000 q^{67} -4.19615 q^{71} +7.46410 q^{73} +3.46410 q^{77} +1.46410 q^{79} -10.1962 q^{83} +18.3923 q^{85} -10.3923 q^{89} -4.00000 q^{91} -2.73205 q^{95} +6.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{5} - 2 q^{7}+O(q^{10}) \) 2 * q - 2 * q^5 - 2 * q^7	\( 2 q - 2 q^{5} - 2 q^{7} + 8 q^{13} - 10 q^{17} + 2 q^{19} + 8 q^{23} - 2 q^{25} + 2 q^{29} + 2 q^{35} - 8 q^{37} - 16 q^{41} - 4 q^{43} + 6 q^{47} + 2 q^{49} - 6 q^{53} + 12 q^{55} - 16 q^{59} + 16 q^{61} - 8 q^{65} - 8 q^{67} + 2 q^{71} + 8 q^{73} - 4 q^{79} - 10 q^{83} + 16 q^{85} - 8 q^{91} - 2 q^{95} + 12 q^{97}+O(q^{100}) \) 2 * q - 2 * q^5 - 2 * q^7 + 8 * q^13 - 10 * q^17 + 2 * q^19 + 8 * q^23 - 2 * q^25 + 2 * q^29 + 2 * q^35 - 8 * q^37 - 16 * q^41 - 4 * q^43 + 6 * q^47 + 2 * q^49 - 6 * q^53 + 12 * q^55 - 16 * q^59 + 16 * q^61 - 8 * q^65 - 8 * q^67 + 2 * q^71 + 8 * q^73 - 4 * q^79 - 10 * q^83 + 16 * q^85 - 8 * q^91 - 2 * q^95 + 12 * q^97

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