LMFDB - Embedded newform 9576.2.a.bz.1.3

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:	$3$
Coefficient field:	3.3.316.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 2 $ x^3 - x^2 - 4*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.81361$ 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)

$f(q)$	$=$	$q+1.10278 q^{5} -1.00000 q^{7} +O(q^{10})$	$q+1.10278 q^{5} -1.00000 q^{7} +4.20555 q^{11} +2.52444 q^{17} -1.00000 q^{19} -5.62721 q^{23} -3.78389 q^{25} -0.524438 q^{29} -4.20555 q^{31} -1.10278 q^{35} -5.62721 q^{37} -3.04888 q^{41} -3.83276 q^{43} +12.1517 q^{47} +1.00000 q^{49} -5.68111 q^{53} +4.63778 q^{55} -12.4111 q^{59} -7.04888 q^{61} +12.4111 q^{67} -5.47556 q^{71} -8.20555 q^{73} -4.20555 q^{77} -12.0383 q^{79} -1.27001 q^{83} +2.78389 q^{85} -7.45998 q^{89} -1.10278 q^{95} -16.0978 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 4 q^{5} - 3 q^{7}+O(q^{10}) $ 3 * q - 4 * q^5 - 3 * q^7	$ 3 q - 4 q^{5} - 3 q^{7} - 2 q^{11} + 2 q^{17} - 3 q^{19} - 4 q^{23} + 5 q^{25} + 4 q^{29} + 2 q^{31} + 4 q^{35} - 4 q^{37} + 2 q^{41} + 16 q^{43} + 18 q^{47} + 3 q^{49} - 8 q^{53} + 32 q^{55} - 8 q^{59} - 10 q^{61} + 8 q^{67} - 22 q^{71} - 10 q^{73} + 2 q^{77} + 6 q^{79} - 24 q^{83} - 8 q^{85} + 18 q^{89} + 4 q^{95} - 26 q^{97}+O(q^{100}) $ 3 * q - 4 * q^5 - 3 * q^7 - 2 * q^11 + 2 * q^17 - 3 * q^19 - 4 * q^23 + 5 * q^25 + 4 * q^29 + 2 * q^31 + 4 * q^35 - 4 * q^37 + 2 * q^41 + 16 * q^43 + 18 * q^47 + 3 * q^49 - 8 * q^53 + 32 * q^55 - 8 * q^59 - 10 * q^61 + 8 * q^67 - 22 * q^71 - 10 * q^73 + 2 * q^77 + 6 * q^79 - 24 * q^83 - 8 * q^85 + 18 * q^89 + 4 * q^95 - 26 * 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
$5$	1.10278	0.493176	0.246588	−	0.969120i	$-0.420691\pi$
$5$	1.10278	0.493176	0.246588	+	0.969120i	$0.420691\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	4.20555	1.26802	0.634011	−	0.773324i	$-0.281408\pi$
$11$	4.20555	1.26802	0.634011	+	0.773324i	$0.281408\pi$
$12$	0	0
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	2.52444	0.612266	0.306133	−	0.951989i	$-0.400965\pi$
$17$	2.52444	0.612266	0.306133	+	0.951989i	$0.400965\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−5.62721	−1.17336	−0.586678	−	0.809821i	$-0.699564\pi$
$23$	−5.62721	−1.17336	−0.586678	+	0.809821i	$0.699564\pi$
$24$	0	0
$25$	−3.78389	−0.756777
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−0.524438	−0.0973857	−0.0486928	−	0.998814i	$-0.515506\pi$
$29$	−0.524438	−0.0973857	−0.0486928	+	0.998814i	$0.515506\pi$
$30$	0	0
$31$	−4.20555	−0.755339	−0.377670	−	0.925940i	$-0.623274\pi$
$31$	−4.20555	−0.755339	−0.377670	+	0.925940i	$0.623274\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−1.10278	−0.186403
$36$	0	0
$37$	−5.62721	−0.925108	−0.462554	−	0.886591i	$-0.653067\pi$
$37$	−5.62721	−0.925108	−0.462554	+	0.886591i	$0.653067\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−3.04888	−0.476154	−0.238077	−	0.971246i	$-0.576517\pi$
$41$	−3.04888	−0.476154	−0.238077	+	0.971246i	$0.576517\pi$
$42$	0	0
$43$	−3.83276	−0.584491	−0.292245	−	0.956343i	$-0.594402\pi$
$43$	−3.83276	−0.584491	−0.292245	+	0.956343i	$0.594402\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	12.1517	1.77250	0.886250	−	0.463207i	$-0.153301\pi$
$47$	12.1517	1.77250	0.886250	+	0.463207i	$0.153301\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−5.68111	−0.780361	−0.390180	−	0.920738i	$-0.627587\pi$
$53$	−5.68111	−0.780361	−0.390180	+	0.920738i	$0.627587\pi$
$54$	0	0
$55$	4.63778	0.625358
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−12.4111	−1.61579	−0.807894	−	0.589328i	$-0.799393\pi$
$59$	−12.4111	−1.61579	−0.807894	+	0.589328i	$0.799393\pi$
$60$	0	0
$61$	−7.04888	−0.902516	−0.451258	−	0.892394i	$-0.649025\pi$
$61$	−7.04888	−0.902516	−0.451258	+	0.892394i	$0.649025\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	12.4111	1.51626	0.758129	−	0.652105i	$-0.226114\pi$
$67$	12.4111	1.51626	0.758129	+	0.652105i	$0.226114\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−5.47556	−0.649830	−0.324915	−	0.945743i	$-0.605336\pi$
$71$	−5.47556	−0.649830	−0.324915	+	0.945743i	$0.605336\pi$
$72$	0	0
$73$	−8.20555	−0.960387	−0.480193	−	0.877163i	$-0.659434\pi$
$73$	−8.20555	−0.960387	−0.480193	+	0.877163i	$0.659434\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−4.20555	−0.479267
$78$	0	0
$79$	−12.0383	−1.35442	−0.677208	−	0.735792i	$-0.736810\pi$
$79$	−12.0383	−1.35442	−0.677208	+	0.735792i	$0.736810\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−1.27001	−0.139402	−0.0697010	−	0.997568i	$-0.522205\pi$
$83$	−1.27001	−0.139402	−0.0697010	+	0.997568i	$0.522205\pi$
$84$	0	0
$85$	2.78389	0.301955
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−7.45998	−0.790756	−0.395378	−	0.918519i	$-0.629386\pi$
$89$	−7.45998	−0.790756	−0.395378	+	0.918519i	$0.629386\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−1.10278	−0.113142
$96$	0	0
$97$	−16.0978	−1.63448	−0.817240	−	0.576298i	$-0.804497\pi$
$97$	−16.0978	−1.63448	−0.817240	+	0.576298i	$0.804497\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	3.30833	0.329191	0.164595	−	0.986361i	$-0.447368\pi$
$101$	3.30833	0.329191	0.164595	+	0.986361i	$0.447368\pi$
$102$	0	0
$103$	14.5783	1.43645	0.718223	−	0.695813i	$-0.244956\pi$
$103$	14.5783	1.43645	0.718223	+	0.695813i	$0.244956\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	8.72999	0.843960	0.421980	−	0.906605i	$-0.361335\pi$
$107$	8.72999	0.843960	0.421980	+	0.906605i	$0.361335\pi$
$108$	0	0
$109$	9.51941	0.911795	0.455897	−	0.890032i	$-0.349318\pi$
$109$	9.51941	0.911795	0.455897	+	0.890032i	$0.349318\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	5.98441	0.562966	0.281483	−	0.959566i	$-0.409174\pi$
$113$	5.98441	0.562966	0.281483	+	0.959566i	$0.409174\pi$
$114$	0	0
$115$	−6.20555	−0.578671
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−2.52444	−0.231415
$120$	0	0
$121$	6.68665	0.607877
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−9.68665	−0.866400
$126$	0	0
$127$	0.676089	0.0599932	0.0299966	−	0.999550i	$-0.490450\pi$
$127$	0.676089	0.0599932	0.0299966	+	0.999550i	$0.490450\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−14.7300	−1.28697	−0.643483	−	0.765461i	$-0.722511\pi$
$131$	−14.7300	−1.28697	−0.643483	+	0.765461i	$0.722511\pi$
$132$	0	0
$133$	1.00000	0.0867110
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−20.4111	−1.74384	−0.871919	−	0.489650i	$-0.837124\pi$
$137$	−20.4111	−1.74384	−0.871919	+	0.489650i	$0.837124\pi$
$138$	0	0
$139$	−17.0489	−1.44607	−0.723033	−	0.690813i	$-0.757253\pi$
$139$	−17.0489	−1.44607	−0.723033	+	0.690813i	$0.757253\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−0.578337	−0.0480283
$146$	0	0
$147$	0	0
$148$	0	0
$149$	22.8816	1.87454	0.937268	−	0.348608i	$-0.113346\pi$
$149$	22.8816	1.87454	0.937268	+	0.348608i	$0.113346\pi$
$150$	0	0
$151$	19.2927	1.57002	0.785010	−	0.619483i	$-0.212658\pi$
$151$	19.2927	1.57002	0.785010	+	0.619483i	$0.212658\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−4.63778	−0.372515
$156$	0	0
$157$	−3.04888	−0.243327	−0.121663	−	0.992571i	$-0.538823\pi$
$157$	−3.04888	−0.243327	−0.121663	+	0.992571i	$0.538823\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	5.62721	0.443487
$162$	0	0
$163$	−7.08719	−0.555111	−0.277556	−	0.960710i	$-0.589524\pi$
$163$	−7.08719	−0.555111	−0.277556	+	0.960710i	$0.589524\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	0.637776	0.0493526	0.0246763	−	0.999695i	$-0.492144\pi$
$167$	0.637776	0.0493526	0.0246763	+	0.999695i	$0.492144\pi$
$168$	0	0
$169$	−13.0000	−1.00000
$170$	0	0
$171$	0	0
$172$	0	0
$173$	9.89220	0.752090	0.376045	−	0.926601i	$-0.377284\pi$
$173$	9.89220	0.752090	0.376045	+	0.926601i	$0.377284\pi$
$174$	0	0
$175$	3.78389	0.286035
$176$	0	0
$177$	0	0
$178$	0	0
$179$	10.9355	0.817360	0.408680	−	0.912678i	$-0.365989\pi$
$179$	10.9355	0.817360	0.408680	+	0.912678i	$0.365989\pi$
$180$	0	0
$181$	21.9305	1.63008	0.815041	−	0.579403i	$-0.196714\pi$
$181$	21.9305	1.63008	0.815041	+	0.579403i	$0.196714\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−6.20555	−0.456241
$186$	0	0
$187$	10.6167	0.776366
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−5.21611	−0.377425	−0.188712	−	0.982032i	$-0.560431\pi$
$191$	−5.21611	−0.377425	−0.188712	+	0.982032i	$0.560431\pi$
$192$	0	0
$193$	8.61665	0.620240	0.310120	−	0.950697i	$-0.399631\pi$
$193$	8.61665	0.620240	0.310120	+	0.950697i	$0.399631\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−8.67609	−0.618146	−0.309073	−	0.951038i	$-0.600019\pi$
$197$	−8.67609	−0.618146	−0.309073	+	0.951038i	$0.600019\pi$
$198$	0	0
$199$	23.2544	1.64846	0.824231	−	0.566253i	$-0.191608\pi$
$199$	23.2544	1.64846	0.824231	+	0.566253i	$0.191608\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0.524438	0.0368083
$204$	0	0
$205$	−3.36222	−0.234828
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−4.20555	−0.290904
$210$	0	0
$211$	23.2544	1.60090	0.800450	−	0.599399i	$-0.204594\pi$
$211$	23.2544	1.60090	0.800450	+	0.599399i	$0.204594\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−4.22668	−0.288257
$216$	0	0
$217$	4.20555	0.285491
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−3.45998	−0.231697	−0.115849	−	0.993267i	$-0.536959\pi$
$223$	−3.45998	−0.231697	−0.115849	+	0.993267i	$0.536959\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−26.4011	−1.75230	−0.876150	−	0.482039i	$-0.839896\pi$
$227$	−26.4011	−1.75230	−0.876150	+	0.482039i	$0.839896\pi$
$228$	0	0
$229$	−12.0978	−0.799442	−0.399721	−	0.916637i	$-0.630893\pi$
$229$	−12.0978	−0.799442	−0.399721	+	0.916637i	$0.630893\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−19.5577	−1.28127	−0.640635	−	0.767846i	$-0.721329\pi$
$233$	−19.5577	−1.28127	−0.640635	+	0.767846i	$0.721329\pi$
$234$	0	0
$235$	13.4005	0.874155
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−16.5783	−1.07236	−0.536182	−	0.844103i	$-0.680134\pi$
$239$	−16.5783	−1.07236	−0.536182	+	0.844103i	$0.680134\pi$
$240$	0	0
$241$	25.6655	1.65326	0.826631	−	0.562744i	$-0.190254\pi$
$241$	25.6655	1.65326	0.826631	+	0.562744i	$0.190254\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	1.10278	0.0704537
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	14.7300	0.929749	0.464874	−	0.885377i	$-0.346100\pi$
$251$	14.7300	0.929749	0.464874	+	0.885377i	$0.346100\pi$
$252$	0	0
$253$	−23.6655	−1.48784
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−23.5678	−1.47012	−0.735059	−	0.678003i	$-0.762845\pi$
$257$	−23.5678	−1.47012	−0.735059	+	0.678003i	$0.762845\pi$
$258$	0	0
$259$	5.62721	0.349658
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−31.0872	−1.91692	−0.958459	−	0.285230i	$-0.907930\pi$
$263$	−31.0872	−1.91692	−0.958459	+	0.285230i	$0.907930\pi$
$264$	0	0
$265$	−6.26499	−0.384855
$266$	0	0
$267$	0	0
$268$	0	0
$269$	11.3522	0.692154	0.346077	−	0.938206i	$-0.387514\pi$
$269$	11.3522	0.692154	0.346077	+	0.938206i	$0.387514\pi$
$270$	0	0
$271$	−12.3033	−0.747372	−0.373686	−	0.927555i	$-0.621906\pi$
$271$	−12.3033	−0.747372	−0.373686	+	0.927555i	$0.621906\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−15.9133	−0.959610
$276$	0	0
$277$	−24.9794	−1.50087	−0.750433	−	0.660946i	$-0.770155\pi$
$277$	−24.9794	−1.50087	−0.750433	+	0.660946i	$0.770155\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−3.88666	−0.231859	−0.115929	−	0.993257i	$-0.536985\pi$
$281$	−3.88666	−0.231859	−0.115929	+	0.993257i	$0.536985\pi$
$282$	0	0
$283$	−5.15667	−0.306532	−0.153266	−	0.988185i	$-0.548979\pi$
$283$	−5.15667	−0.306532	−0.153266	+	0.988185i	$0.548979\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	3.04888	0.179969
$288$	0	0
$289$	−10.6272	−0.625130
$290$	0	0
$291$	0	0
$292$	0	0
$293$	17.5577	1.02573	0.512867	−	0.858468i	$-0.328584\pi$
$293$	17.5577	1.02573	0.512867	+	0.858468i	$0.328584\pi$
$294$	0	0
$295$	−13.6867	−0.796868
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	3.83276	0.220917
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−7.77332	−0.445099
$306$	0	0
$307$	−1.21611	−0.0694072	−0.0347036	−	0.999398i	$-0.511049\pi$
$307$	−1.21611	−0.0694072	−0.0347036	+	0.999398i	$0.511049\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−7.62167	−0.432185	−0.216093	−	0.976373i	$-0.569331\pi$
$311$	−7.62167	−0.432185	−0.216093	+	0.976373i	$0.569331\pi$
$312$	0	0
$313$	14.7144	0.831707	0.415854	−	0.909432i	$-0.363483\pi$
$313$	14.7144	0.831707	0.415854	+	0.909432i	$0.363483\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−20.5244	−1.15277	−0.576384	−	0.817179i	$-0.695537\pi$
$317$	−20.5244	−1.15277	−0.576384	+	0.817179i	$0.695537\pi$
$318$	0	0
$319$	−2.20555	−0.123487
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−2.52444	−0.140463
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−12.1517	−0.669942
$330$	0	0
$331$	22.4011	1.23127	0.615637	−	0.788030i	$-0.288899\pi$
$331$	22.4011	1.23127	0.615637	+	0.788030i	$0.288899\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	13.6867	0.747782
$336$	0	0
$337$	−20.0978	−1.09479	−0.547397	−	0.836873i	$-0.684381\pi$
$337$	−20.0978	−1.09479	−0.547397	+	0.836873i	$0.684381\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−17.6867	−0.957786
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	21.6655	1.16307	0.581533	−	0.813523i	$-0.302453\pi$
$347$	21.6655	1.16307	0.581533	+	0.813523i	$0.302453\pi$
$348$	0	0
$349$	−14.8222	−0.793414	−0.396707	−	0.917945i	$-0.629847\pi$
$349$	−14.8222	−0.793414	−0.396707	+	0.917945i	$0.629847\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−16.8378	−0.896185	−0.448092	−	0.893987i	$-0.647896\pi$
$353$	−16.8378	−0.896185	−0.448092	+	0.893987i	$0.647896\pi$
$354$	0	0
$355$	−6.03831	−0.320480
$356$	0	0
$357$	0	0
$358$	0	0
$359$	18.5683	0.979997	0.489998	−	0.871723i	$-0.336997\pi$
$359$	18.5683	0.979997	0.489998	+	0.871723i	$0.336997\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−9.04888	−0.473640
$366$	0	0
$367$	28.0000	1.46159	0.730794	−	0.682598i	$-0.239150\pi$
$367$	28.0000	1.46159	0.730794	+	0.682598i	$0.239150\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	5.68111	0.294949
$372$	0	0
$373$	−5.32391	−0.275662	−0.137831	−	0.990456i	$-0.544013\pi$
$373$	−5.32391	−0.275662	−0.137831	+	0.990456i	$0.544013\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−15.5577	−0.799147	−0.399573	−	0.916701i	$-0.630842\pi$
$379$	−15.5577	−0.799147	−0.399573	+	0.916701i	$0.630842\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	23.3622	1.19375	0.596877	−	0.802333i	$-0.296408\pi$
$383$	23.3622	1.19375	0.596877	+	0.802333i	$0.296408\pi$
$384$	0	0
$385$	−4.63778	−0.236363
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−7.62721	−0.386715	−0.193358	−	0.981128i	$-0.561938\pi$
$389$	−7.62721	−0.386715	−0.193358	+	0.981128i	$0.561938\pi$
$390$	0	0
$391$	−14.2056	−0.718406
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−13.2756	−0.667965
$396$	0	0
$397$	15.3522	0.770504	0.385252	−	0.922811i	$-0.374115\pi$
$397$	15.3522	0.770504	0.385252	+	0.922811i	$0.374115\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−33.3466	−1.66525	−0.832626	−	0.553836i	$-0.813163\pi$
$401$	−33.3466	−1.66525	−0.832626	+	0.553836i	$0.813163\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−23.6655	−1.17306
$408$	0	0
$409$	−11.5194	−0.569598	−0.284799	−	0.958587i	$-0.591927\pi$
$409$	−11.5194	−0.569598	−0.284799	+	0.958587i	$0.591927\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	12.4111	0.610710
$414$	0	0
$415$	−1.40054	−0.0687497
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−32.2978	−1.57785	−0.788924	−	0.614490i	$-0.789362\pi$
$419$	−32.2978	−1.57785	−0.788924	+	0.614490i	$0.789362\pi$
$420$	0	0
$421$	32.9583	1.60629	0.803144	−	0.595785i	$-0.203159\pi$
$421$	32.9583	1.60629	0.803144	+	0.595785i	$0.203159\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−9.55219	−0.463349
$426$	0	0
$427$	7.04888	0.341119
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−7.27001	−0.350184	−0.175092	−	0.984552i	$-0.556022\pi$
$431$	−7.27001	−0.350184	−0.175092	+	0.984552i	$0.556022\pi$
$432$	0	0
$433$	−17.6655	−0.848951	−0.424476	−	0.905439i	$-0.639541\pi$
$433$	−17.6655	−0.848951	−0.424476	+	0.905439i	$0.639541\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	5.62721	0.269186
$438$	0	0
$439$	−17.3622	−0.828654	−0.414327	−	0.910128i	$-0.635983\pi$
$439$	−17.3622	−0.828654	−0.414327	+	0.910128i	$0.635983\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	40.9200	1.94417	0.972083	−	0.234638i	$-0.0753905\pi$
$443$	40.9200	1.94417	0.972083	+	0.234638i	$0.0753905\pi$
$444$	0	0
$445$	−8.22668	−0.389982
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−40.9355	−1.93187	−0.965934	−	0.258789i	$-0.916676\pi$
$449$	−40.9355	−1.93187	−0.965934	+	0.258789i	$0.916676\pi$
$450$	0	0
$451$	−12.8222	−0.603774
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−31.2927	−1.46381	−0.731906	−	0.681405i	$-0.761369\pi$
$457$	−31.2927	−1.46381	−0.731906	+	0.681405i	$0.761369\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	10.5628	0.491956	0.245978	−	0.969275i	$-0.420891\pi$
$461$	10.5628	0.491956	0.245978	+	0.969275i	$0.420891\pi$
$462$	0	0
$463$	27.6655	1.28573	0.642863	−	0.765981i	$-0.277746\pi$
$463$	27.6655	1.28573	0.642863	+	0.765981i	$0.277746\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	30.6988	1.42057	0.710286	−	0.703913i	$-0.248566\pi$
$467$	30.6988	1.42057	0.710286	+	0.703913i	$0.248566\pi$
$468$	0	0
$469$	−12.4111	−0.573091
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−16.1189	−0.741147
$474$	0	0
$475$	3.78389	0.173617
$476$	0	0
$477$	0	0
$478$	0	0
$479$	14.7683	0.674781	0.337390	−	0.941365i	$-0.390456\pi$
$479$	14.7683	0.674781	0.337390	+	0.941365i	$0.390456\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−17.7522	−0.806086
$486$	0	0
$487$	20.8917	0.946693	0.473346	−	0.880876i	$-0.343046\pi$
$487$	20.8917	0.946693	0.473346	+	0.880876i	$0.343046\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−33.1355	−1.49539	−0.747693	−	0.664044i	$-0.768839\pi$
$491$	−33.1355	−1.49539	−0.747693	+	0.664044i	$0.768839\pi$
$492$	0	0
$493$	−1.32391	−0.0596260
$494$	0	0
$495$	0	0
$496$	0	0
$497$	5.47556	0.245613
$498$	0	0
$499$	−31.1355	−1.39382	−0.696909	−	0.717159i	$-0.745442\pi$
$499$	−31.1355	−1.39382	−0.696909	+	0.717159i	$0.745442\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−33.9149	−1.51219	−0.756096	−	0.654461i	$-0.772896\pi$
$503$	−33.9149	−1.51219	−0.756096	+	0.654461i	$0.772896\pi$
$504$	0	0
$505$	3.64834	0.162349
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−13.7844	−0.610983	−0.305491	−	0.952195i	$-0.598821\pi$
$509$	−13.7844	−0.610983	−0.305491	+	0.952195i	$0.598821\pi$
$510$	0	0
$511$	8.20555	0.362992
$512$	0	0
$513$	0	0
$514$	0	0
$515$	16.0766	0.708421
$516$	0	0
$517$	51.1044	2.24757
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−0.0977518	−0.00428259	−0.00214129	−	0.999998i	$-0.500682\pi$
$521$	−0.0977518	−0.00428259	−0.00214129	+	0.999998i	$0.500682\pi$
$522$	0	0
$523$	44.2127	1.93329	0.966643	−	0.256128i	$-0.0824467\pi$
$523$	44.2127	1.93329	0.966643	+	0.256128i	$0.0824467\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−10.6167	−0.462469
$528$	0	0
$529$	8.66553	0.376762
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	9.62721	0.416221
$536$	0	0
$537$	0	0
$538$	0	0
$539$	4.20555	0.181146
$540$	0	0
$541$	−10.1955	−0.438339	−0.219169	−	0.975687i	$-0.570335\pi$
$541$	−10.1955	−0.438339	−0.219169	+	0.975687i	$0.570335\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	10.4978	0.449675
$546$	0	0
$547$	19.9688	0.853805	0.426903	−	0.904298i	$-0.359605\pi$
$547$	19.9688	0.853805	0.426903	+	0.904298i	$0.359605\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0.524438	0.0223418
$552$	0	0
$553$	12.0383	0.511921
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−2.88164	−0.122099	−0.0610495	−	0.998135i	$-0.519445\pi$
$557$	−2.88164	−0.122099	−0.0610495	+	0.998135i	$0.519445\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−18.2056	−0.767272	−0.383636	−	0.923484i	$-0.625328\pi$
$563$	−18.2056	−0.767272	−0.383636	+	0.923484i	$0.625328\pi$
$564$	0	0
$565$	6.59946	0.277641
$566$	0	0
$567$	0	0
$568$	0	0
$569$	14.2111	0.595760	0.297880	−	0.954603i	$-0.403720\pi$
$569$	14.2111	0.595760	0.297880	+	0.954603i	$0.403720\pi$
$570$	0	0
$571$	−30.9200	−1.29396	−0.646980	−	0.762507i	$-0.723968\pi$
$571$	−30.9200	−1.29396	−0.646980	+	0.762507i	$0.723968\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	21.2927	0.887969
$576$	0	0
$577$	−29.7733	−1.23948	−0.619740	−	0.784807i	$-0.712762\pi$
$577$	−29.7733	−1.23948	−0.619740	+	0.784807i	$0.712762\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	1.27001	0.0526890
$582$	0	0
$583$	−23.8922	−0.989514
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−31.0333	−1.28088	−0.640440	−	0.768008i	$-0.721248\pi$
$587$	−31.0333	−1.28088	−0.640440	+	0.768008i	$0.721248\pi$
$588$	0	0
$589$	4.20555	0.173287
$590$	0	0
$591$	0	0
$592$	0	0
$593$	20.6222	0.846852	0.423426	−	0.905931i	$-0.360827\pi$
$593$	20.6222	0.846852	0.423426	+	0.905931i	$0.360827\pi$
$594$	0	0
$595$	−2.78389	−0.114128
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−21.2600	−0.868659	−0.434329	−	0.900754i	$-0.643015\pi$
$599$	−21.2600	−0.868659	−0.434329	+	0.900754i	$0.643015\pi$
$600$	0	0
$601$	−17.4700	−0.712617	−0.356309	−	0.934368i	$-0.615965\pi$
$601$	−17.4700	−0.712617	−0.356309	+	0.934368i	$0.615965\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	7.37387	0.299791
$606$	0	0
$607$	20.8917	0.847967	0.423984	−	0.905670i	$-0.360631\pi$
$607$	20.8917	0.847967	0.423984	+	0.905670i	$0.360631\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−5.52946	−0.223333	−0.111666	−	0.993746i	$-0.535619\pi$
$613$	−5.52946	−0.223333	−0.111666	+	0.993746i	$0.535619\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	18.1744	0.731673	0.365836	−	0.930679i	$-0.380783\pi$
$617$	18.1744	0.731673	0.365836	+	0.930679i	$0.380783\pi$
$618$	0	0
$619$	−16.3799	−0.658365	−0.329182	−	0.944266i	$-0.606773\pi$
$619$	−16.3799	−0.658365	−0.329182	+	0.944266i	$0.606773\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	7.45998	0.298878
$624$	0	0
$625$	8.23724	0.329490
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−14.2056	−0.566412
$630$	0	0
$631$	20.6550	0.822261	0.411131	−	0.911576i	$-0.365134\pi$
$631$	20.6550	0.822261	0.411131	+	0.911576i	$0.365134\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0.745574	0.0295872
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	13.3466	0.527161	0.263580	−	0.964637i	$-0.415097\pi$
$641$	13.3466	0.527161	0.263580	+	0.964637i	$0.415097\pi$
$642$	0	0
$643$	44.4877	1.75442	0.877212	−	0.480103i	$-0.159401\pi$
$643$	44.4877	1.75442	0.877212	+	0.480103i	$0.159401\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	6.38837	0.251153	0.125576	−	0.992084i	$-0.459922\pi$
$647$	6.38837	0.251153	0.125576	+	0.992084i	$0.459922\pi$
$648$	0	0
$649$	−52.1955	−2.04885
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−22.1672	−0.867471	−0.433736	−	0.901040i	$-0.642805\pi$
$653$	−22.1672	−0.867471	−0.433736	+	0.901040i	$0.642805\pi$
$654$	0	0
$655$	−16.2439	−0.634700
$656$	0	0
$657$	0	0
$658$	0	0
$659$	0.729988	0.0284363	0.0142181	−	0.999899i	$-0.495474\pi$
$659$	0.729988	0.0284363	0.0142181	+	0.999899i	$0.495474\pi$
$660$	0	0
$661$	−34.3900	−1.33761	−0.668807	−	0.743436i	$-0.733195\pi$
$661$	−34.3900	−1.33761	−0.668807	+	0.743436i	$0.733195\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	1.10278	0.0427638
$666$	0	0
$667$	2.95112	0.114268
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−29.6444	−1.14441
$672$	0	0
$673$	24.6167	0.948902	0.474451	−	0.880282i	$-0.342647\pi$
$673$	24.6167	0.948902	0.474451	+	0.880282i	$0.342647\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−1.66553	−0.0640114	−0.0320057	−	0.999488i	$-0.510189\pi$
$677$	−1.66553	−0.0640114	−0.0320057	+	0.999488i	$0.510189\pi$
$678$	0	0
$679$	16.0978	0.617775
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−28.6111	−1.09477	−0.547387	−	0.836880i	$-0.684377\pi$
$683$	−28.6111	−1.09477	−0.547387	+	0.836880i	$0.684377\pi$
$684$	0	0
$685$	−22.5089	−0.860019
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	−13.4600	−0.512042	−0.256021	−	0.966671i	$-0.582412\pi$
$691$	−13.4600	−0.512042	−0.256021	+	0.966671i	$0.582412\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−18.8011	−0.713165
$696$	0	0
$697$	−7.69670	−0.291533
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−23.4005	−0.883826	−0.441913	−	0.897058i	$-0.645700\pi$
$701$	−23.4005	−0.883826	−0.441913	+	0.897058i	$0.645700\pi$
$702$	0	0
$703$	5.62721	0.212234
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−3.30833	−0.124422
$708$	0	0
$709$	15.8428	0.594989	0.297495	−	0.954724i	$-0.403849\pi$
$709$	15.8428	0.594989	0.297495	+	0.954724i	$0.403849\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	23.6655	0.886281
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	5.64280	0.210441	0.105220	−	0.994449i	$-0.466445\pi$
$719$	5.64280	0.210441	0.105220	+	0.994449i	$0.466445\pi$
$720$	0	0
$721$	−14.5783	−0.542926
$722$	0	0
$723$	0	0
$724$	0	0
$725$	1.98441	0.0736993
$726$	0	0
$727$	51.2233	1.89977	0.949883	−	0.312607i	$-0.101202\pi$
$727$	51.2233	1.89977	0.949883	+	0.312607i	$0.101202\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−9.67557	−0.357864
$732$	0	0
$733$	−26.1078	−0.964314	−0.482157	−	0.876085i	$-0.660146\pi$
$733$	−26.1078	−0.964314	−0.482157	+	0.876085i	$0.660146\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	52.1955	1.92265
$738$	0	0
$739$	11.8328	0.435275	0.217638	−	0.976030i	$-0.430165\pi$
$739$	11.8328	0.435275	0.217638	+	0.976030i	$0.430165\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−39.8766	−1.46293	−0.731466	−	0.681878i	$-0.761163\pi$
$743$	−39.8766	−1.46293	−0.731466	+	0.681878i	$0.761163\pi$
$744$	0	0
$745$	25.2333	0.924477
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−8.72999	−0.318987
$750$	0	0
$751$	42.0172	1.53323	0.766614	−	0.642108i	$-0.221940\pi$
$751$	42.0172	1.53323	0.766614	+	0.642108i	$0.221940\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	21.2756	0.774297
$756$	0	0
$757$	19.9789	0.726145	0.363072	−	0.931761i	$-0.381728\pi$
$757$	19.9789	0.726145	0.363072	+	0.931761i	$0.381728\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	42.7966	1.55137	0.775687	−	0.631118i	$-0.217403\pi$
$761$	42.7966	1.55137	0.775687	+	0.631118i	$0.217403\pi$
$762$	0	0
$763$	−9.51941	−0.344626
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	45.2233	1.63079	0.815396	−	0.578903i	$-0.196519\pi$
$769$	45.2233	1.63079	0.815396	+	0.578903i	$0.196519\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	25.9688	0.934034	0.467017	−	0.884248i	$-0.345329\pi$
$773$	25.9688	0.934034	0.467017	+	0.884248i	$0.345329\pi$
$774$	0	0
$775$	15.9133	0.571624
$776$	0	0
$777$	0	0
$778$	0	0
$779$	3.04888	0.109237
$780$	0	0
$781$	−23.0278	−0.823998
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−3.36222	−0.120003
$786$	0	0
$787$	−30.2338	−1.07772	−0.538860	−	0.842396i	$-0.681145\pi$
$787$	−30.2338	−1.07772	−0.538860	+	0.842396i	$0.681145\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−5.98441	−0.212781
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	29.5577	1.04699	0.523494	−	0.852029i	$-0.324628\pi$
$797$	29.5577	1.04699	0.523494	+	0.852029i	$0.324628\pi$
$798$	0	0
$799$	30.6761	1.08524
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−34.5089	−1.21779
$804$	0	0
$805$	6.20555	0.218717
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−25.0177	−0.879576	−0.439788	−	0.898102i	$-0.644946\pi$
$809$	−25.0177	−0.879576	−0.439788	+	0.898102i	$0.644946\pi$
$810$	0	0
$811$	15.0388	0.528085	0.264042	−	0.964511i	$-0.414944\pi$
$811$	15.0388	0.528085	0.264042	+	0.964511i	$0.414944\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−7.81558	−0.273768
$816$	0	0
$817$	3.83276	0.134091
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−19.4116	−0.677470	−0.338735	−	0.940882i	$-0.609999\pi$
$821$	−19.4116	−0.677470	−0.338735	+	0.940882i	$0.609999\pi$
$822$	0	0
$823$	23.4217	0.816428	0.408214	−	0.912886i	$-0.366152\pi$
$823$	23.4217	0.816428	0.408214	+	0.912886i	$0.366152\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	39.0122	1.35659	0.678293	−	0.734792i	$-0.262720\pi$
$827$	39.0122	1.35659	0.678293	+	0.734792i	$0.262720\pi$
$828$	0	0
$829$	−35.1638	−1.22129	−0.610645	−	0.791905i	$-0.709090\pi$
$829$	−35.1638	−1.22129	−0.610645	+	0.791905i	$0.709090\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	2.52444	0.0874666
$834$	0	0
$835$	0.703323	0.0243395
$836$	0	0
$837$	0	0
$838$	0	0
$839$	20.4111	0.704669	0.352335	−	0.935874i	$-0.385388\pi$
$839$	20.4111	0.704669	0.352335	+	0.935874i	$0.385388\pi$
$840$	0	0
$841$	−28.7250	−0.990516
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−14.3361	−0.493176
$846$	0	0
$847$	−6.68665	−0.229756
$848$	0	0
$849$	0	0
$850$	0	0
$851$	31.6655	1.08548
$852$	0	0
$853$	−10.9411	−0.374615	−0.187308	−	0.982301i	$-0.559976\pi$
$853$	−10.9411	−0.374615	−0.187308	+	0.982301i	$0.559976\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−29.3622	−1.00299	−0.501497	−	0.865159i	$-0.667217\pi$
$857$	−29.3622	−1.00299	−0.501497	+	0.865159i	$0.667217\pi$
$858$	0	0
$859$	6.43223	0.219465	0.109732	−	0.993961i	$-0.465001\pi$
$859$	6.43223	0.219465	0.109732	+	0.993961i	$0.465001\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−37.7477	−1.28495	−0.642473	−	0.766308i	$-0.722092\pi$
$863$	−37.7477	−1.28495	−0.642473	+	0.766308i	$0.722092\pi$
$864$	0	0
$865$	10.9089	0.370913
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−50.6277	−1.71743
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0	0
$875$	9.68665	0.327469
$876$	0	0
$877$	10.8917	0.367786	0.183893	−	0.982946i	$-0.441130\pi$
$877$	10.8917	0.367786	0.183893	+	0.982946i	$0.441130\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	20.6111	0.694406	0.347203	−	0.937790i	$-0.387131\pi$
$881$	20.6111	0.694406	0.347203	+	0.937790i	$0.387131\pi$
$882$	0	0
$883$	−51.3311	−1.72743	−0.863714	−	0.503983i	$-0.831868\pi$
$883$	−51.3311	−1.72743	−0.863714	+	0.503983i	$0.831868\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−17.1255	−0.575018	−0.287509	−	0.957778i	$-0.592827\pi$
$887$	−17.1255	−0.575018	−0.287509	+	0.957778i	$0.592827\pi$
$888$	0	0
$889$	−0.676089	−0.0226753
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−12.1517	−0.406639
$894$	0	0
$895$	12.0594	0.403103
$896$	0	0
$897$	0	0
$898$	0	0
$899$	2.20555	0.0735592
$900$	0	0
$901$	−14.3416	−0.477788
$902$	0	0
$903$	0	0
$904$	0	0
$905$	24.1844	0.803917
$906$	0	0
$907$	−53.3522	−1.77153	−0.885765	−	0.464134i	$-0.846366\pi$
$907$	−53.3522	−1.77153	−0.885765	+	0.464134i	$0.846366\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−29.9945	−0.993761	−0.496880	−	0.867819i	$-0.665521\pi$
$911$	−29.9945	−0.993761	−0.496880	+	0.867819i	$0.665521\pi$
$912$	0	0
$913$	−5.34110	−0.176765
$914$	0	0
$915$	0	0
$916$	0	0
$917$	14.7300	0.486427
$918$	0	0
$919$	−40.6832	−1.34202	−0.671008	−	0.741450i	$-0.734138\pi$
$919$	−40.6832	−1.34202	−0.671008	+	0.741450i	$0.734138\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	21.2927	0.700101
$926$	0	0
$927$	0	0
$928$	0	0
$929$	10.9355	0.358783	0.179392	−	0.983778i	$-0.442587\pi$
$929$	10.9355	0.358783	0.179392	+	0.983778i	$0.442587\pi$
$930$	0	0
$931$	−1.00000	−0.0327737
$932$	0	0
$933$	0	0
$934$	0	0
$935$	11.7078	0.382885
$936$	0	0
$937$	10.5300	0.343999	0.172000	−	0.985097i	$-0.444977\pi$
$937$	10.5300	0.343999	0.172000	+	0.985097i	$0.444977\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−48.9200	−1.59474	−0.797372	−	0.603488i	$-0.793777\pi$
$941$	−48.9200	−1.59474	−0.797372	+	0.603488i	$0.793777\pi$
$942$	0	0
$943$	17.1567	0.558698
$944$	0	0
$945$	0	0
$946$	0	0
$947$	26.1844	0.850879	0.425440	−	0.904987i	$-0.360119\pi$
$947$	26.1844	0.850879	0.425440	+	0.904987i	$0.360119\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−0.113338	−0.00367137	−0.00183569	−	0.999998i	$-0.500584\pi$
$953$	−0.113338	−0.00367137	−0.00183569	+	0.999998i	$0.500584\pi$
$954$	0	0
$955$	−5.75220	−0.186137
$956$	0	0
$957$	0	0
$958$	0	0
$959$	20.4111	0.659109
$960$	0	0
$961$	−13.3133	−0.429463
$962$	0	0
$963$	0	0
$964$	0	0
$965$	9.50223	0.305888
$966$	0	0
$967$	−26.8716	−0.864132	−0.432066	−	0.901842i	$-0.642215\pi$
$967$	−26.8716	−0.864132	−0.432066	+	0.901842i	$0.642215\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−21.3522	−0.685224	−0.342612	−	0.939477i	$-0.611312\pi$
$971$	−21.3522	−0.685224	−0.342612	+	0.939477i	$0.611312\pi$
$972$	0	0
$973$	17.0489	0.546562
$974$	0	0
$975$	0	0
$976$	0	0
$977$	35.2177	1.12671	0.563357	−	0.826213i	$-0.309509\pi$
$977$	35.2177	1.12671	0.563357	+	0.826213i	$0.309509\pi$
$978$	0	0
$979$	−31.3733	−1.00270
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−28.2922	−0.902382	−0.451191	−	0.892427i	$-0.649001\pi$
$983$	−28.2922	−0.902382	−0.451191	+	0.892427i	$0.649001\pi$
$984$	0	0
$985$	−9.56777	−0.304855
$986$	0	0
$987$	0	0
$988$	0	0
$989$	21.5678	0.685815
$990$	0	0
$991$	20.1461	0.639962	0.319981	−	0.947424i	$-0.396323\pi$
$991$	20.1461	0.639962	0.319981	+	0.947424i	$0.396323\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	25.6444	0.812982
$996$	0	0
$997$	−19.2645	−0.610112	−0.305056	−	0.952334i	$-0.598675\pi$
$997$	−19.2645	−0.610112	−0.305056	+	0.952334i	$0.598675\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9576.2.a.bz.1.3	✓	3
3.2	odd	2		9576.2.a.cf.1.1	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9576.2.a.bz.1.3	✓	3	1.1	even	1	trivial
9576.2.a.cf.1.1	yes	3	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)

\(f(q)\)	\(=\)	\(q+1.10278 q^{5} -1.00000 q^{7} +O(q^{10})\)	\(q+1.10278 q^{5} -1.00000 q^{7} +4.20555 q^{11} +2.52444 q^{17} -1.00000 q^{19} -5.62721 q^{23} -3.78389 q^{25} -0.524438 q^{29} -4.20555 q^{31} -1.10278 q^{35} -5.62721 q^{37} -3.04888 q^{41} -3.83276 q^{43} +12.1517 q^{47} +1.00000 q^{49} -5.68111 q^{53} +4.63778 q^{55} -12.4111 q^{59} -7.04888 q^{61} +12.4111 q^{67} -5.47556 q^{71} -8.20555 q^{73} -4.20555 q^{77} -12.0383 q^{79} -1.27001 q^{83} +2.78389 q^{85} -7.45998 q^{89} -1.10278 q^{95} -16.0978 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 4 q^{5} - 3 q^{7}+O(q^{10}) \) 3 * q - 4 * q^5 - 3 * q^7	\( 3 q - 4 q^{5} - 3 q^{7} - 2 q^{11} + 2 q^{17} - 3 q^{19} - 4 q^{23} + 5 q^{25} + 4 q^{29} + 2 q^{31} + 4 q^{35} - 4 q^{37} + 2 q^{41} + 16 q^{43} + 18 q^{47} + 3 q^{49} - 8 q^{53} + 32 q^{55} - 8 q^{59} - 10 q^{61} + 8 q^{67} - 22 q^{71} - 10 q^{73} + 2 q^{77} + 6 q^{79} - 24 q^{83} - 8 q^{85} + 18 q^{89} + 4 q^{95} - 26 q^{97}+O(q^{100}) \) 3 * q - 4 * q^5 - 3 * q^7 - 2 * q^11 + 2 * q^17 - 3 * q^19 - 4 * q^23 + 5 * q^25 + 4 * q^29 + 2 * q^31 + 4 * q^35 - 4 * q^37 + 2 * q^41 + 16 * q^43 + 18 * q^47 + 3 * q^49 - 8 * q^53 + 32 * q^55 - 8 * q^59 - 10 * q^61 + 8 * q^67 - 22 * q^71 - 10 * q^73 + 2 * q^77 + 6 * q^79 - 24 * q^83 - 8 * q^85 + 18 * q^89 + 4 * q^95 - 26 * q^97

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