LMFDB - Embedded newform 2376.2.a.p.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2376,2,Mod(1,2376)] 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("2376.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2376, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2376 = 2^{3} \cdot 3^{3} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2376.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,0,0,2,0,-3,0,0,0,-3,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	yes
Analytic conductor:	$18.9724555203$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.564.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 5x + 3 $ x^3 - x^2 - 5*x + 3
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$0.571993$ of defining polynomial
Character	$\chi$	$=$	2376.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-2.67282 q^{5} -4.10083 q^{7} -1.00000 q^{11} -3.81681 q^{13} -5.91764 q^{17} -5.81681 q^{19} +3.67282 q^{23} +2.14399 q^{25} +8.38880 q^{29} +6.96080 q^{31} +10.9608 q^{35} +6.52884 q^{37} +7.24482 q^{41} -10.7737 q^{43} -0.143987 q^{47} +9.81681 q^{49} -4.48963 q^{53} +2.67282 q^{55} +11.8745 q^{59} -1.61515 q^{61} +10.2017 q^{65} -6.96080 q^{67} -0.489634 q^{71} -8.48963 q^{73} +4.10083 q^{77} -15.9176 q^{79} -5.16246 q^{83} +15.8168 q^{85} +6.00000 q^{89} +15.6521 q^{91} +15.5473 q^{95} -7.38485 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 2 q^{5} - 3 q^{7} - 3 q^{11} + 3 q^{17} - 6 q^{19} + q^{23} + 5 q^{25} + 13 q^{29} + 8 q^{31} + 20 q^{35} + 11 q^{37} + 11 q^{41} - 13 q^{43} + q^{47} + 18 q^{49} + 8 q^{53} - 2 q^{55} + 7 q^{59}+ \cdots - 15 q^{97}+O(q^{100}) $ 3 * q + 2 * q^5 - 3 * q^7 - 3 * q^11 + 3 * q^17 - 6 * q^19 + q^23 + 5 * q^25 + 13 * q^29 + 8 * q^31 + 20 * q^35 + 11 * q^37 + 11 * q^41 - 13 * q^43 + q^47 + 18 * q^49 + 8 * q^53 - 2 * q^55 + 7 * q^59 - 12 * q^61 + 12 * q^65 - 8 * q^67 + 20 * q^71 - 4 * q^73 + 3 * q^77 - 27 * q^79 + 16 * q^83 + 36 * q^85 + 18 * q^89 - 6 * q^91 + 8 * q^95 - 15 * q^97

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−2.67282	−1.19532	−0.597662	−	0.801749i	$-0.703903\pi$
$5$	−2.67282	−1.19532	−0.597662	+	0.801749i	$0.703903\pi$
$6$	0	0
$7$	−4.10083	−1.54997	−0.774984	−	0.631981i	$-0.782242\pi$
$7$	−4.10083	−1.54997	−0.774984	+	0.631981i	$0.782242\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−3.81681	−1.05859	−0.529296	−	0.848437i	$-0.677544\pi$
$13$	−3.81681	−1.05859	−0.529296	+	0.848437i	$0.677544\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−5.91764	−1.43524	−0.717619	−	0.696436i	$-0.754768\pi$
$17$	−5.91764	−1.43524	−0.717619	+	0.696436i	$0.754768\pi$
$18$	0	0
$19$	−5.81681	−1.33447	−0.667234	−	0.744848i	$-0.732522\pi$
$19$	−5.81681	−1.33447	−0.667234	+	0.744848i	$0.732522\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	3.67282	0.765837	0.382918	−	0.923782i	$-0.374919\pi$
$23$	3.67282	0.765837	0.382918	+	0.923782i	$0.374919\pi$
$24$	0	0
$25$	2.14399	0.428797
$26$	0	0
$27$	0	0
$28$	0	0
$29$	8.38880	1.55776	0.778881	−	0.627172i	$-0.215788\pi$
$29$	8.38880	1.55776	0.778881	+	0.627172i	$0.215788\pi$
$30$	0	0
$31$	6.96080	1.25020	0.625098	−	0.780546i	$-0.285059\pi$
$31$	6.96080	1.25020	0.625098	+	0.780546i	$0.285059\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	10.9608	1.85271
$36$	0	0
$37$	6.52884	1.07333	0.536667	−	0.843794i	$-0.319683\pi$
$37$	6.52884	1.07333	0.536667	+	0.843794i	$0.319683\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	7.24482	1.13145	0.565725	−	0.824594i	$-0.308596\pi$
$41$	7.24482	1.13145	0.565725	+	0.824594i	$0.308596\pi$
$42$	0	0
$43$	−10.7737	−1.64297	−0.821483	−	0.570232i	$-0.806853\pi$
$43$	−10.7737	−1.64297	−0.821483	+	0.570232i	$0.806853\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−0.143987	−0.0210026	−0.0105013	−	0.999945i	$-0.503343\pi$
$47$	−0.143987	−0.0210026	−0.0105013	+	0.999945i	$0.503343\pi$
$48$	0	0
$49$	9.81681	1.40240
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−4.48963	−0.616699	−0.308349	−	0.951273i	$-0.599777\pi$
$53$	−4.48963	−0.616699	−0.308349	+	0.951273i	$0.599777\pi$
$54$	0	0
$55$	2.67282	0.360403
$56$	0	0
$57$	0	0
$58$	0	0
$59$	11.8745	1.54593	0.772963	−	0.634451i	$-0.218774\pi$
$59$	11.8745	1.54593	0.772963	+	0.634451i	$0.218774\pi$
$60$	0	0
$61$	−1.61515	−0.206799	−0.103399	−	0.994640i	$-0.532972\pi$
$61$	−1.61515	−0.206799	−0.103399	+	0.994640i	$0.532972\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	10.2017	1.26536
$66$	0	0
$67$	−6.96080	−0.850397	−0.425198	−	0.905100i	$-0.639796\pi$
$67$	−6.96080	−0.850397	−0.425198	+	0.905100i	$0.639796\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−0.489634	−0.0581089	−0.0290544	−	0.999578i	$-0.509250\pi$
$71$	−0.489634	−0.0581089	−0.0290544	+	0.999578i	$0.509250\pi$
$72$	0	0
$73$	−8.48963	−0.993636	−0.496818	−	0.867855i	$-0.665498\pi$
$73$	−8.48963	−0.993636	−0.496818	+	0.867855i	$0.665498\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	4.10083	0.467333
$78$	0	0
$79$	−15.9176	−1.79087	−0.895437	−	0.445188i	$-0.853137\pi$
$79$	−15.9176	−1.79087	−0.895437	+	0.445188i	$0.853137\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−5.16246	−0.566653	−0.283327	−	0.959023i	$-0.591438\pi$
$83$	−5.16246	−0.566653	−0.283327	+	0.959023i	$0.591438\pi$
$84$	0	0
$85$	15.8168	1.71557
$86$	0	0
$87$	0	0
$88$	0	0
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	0	0
$91$	15.6521	1.64079
$92$	0	0
$93$	0	0
$94$	0	0
$95$	15.5473	1.59512
$96$	0	0
$97$	−7.38485	−0.749818	−0.374909	−	0.927062i	$-0.622326\pi$
$97$	−7.38485	−0.749818	−0.374909	+	0.927062i	$0.622326\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−7.34960	−0.731313	−0.365656	−	0.930750i	$-0.619155\pi$
$101$	−7.34960	−0.731313	−0.365656	+	0.930750i	$0.619155\pi$
$102$	0	0
$103$	13.7305	1.35291	0.676453	−	0.736486i	$-0.263516\pi$
$103$	13.7305	1.35291	0.676453	+	0.736486i	$0.263516\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−8.30644	−0.803014	−0.401507	−	0.915856i	$-0.631514\pi$
$107$	−8.30644	−0.803014	−0.401507	+	0.915856i	$0.631514\pi$
$108$	0	0
$109$	18.1233	1.73589	0.867946	−	0.496658i	$-0.165440\pi$
$109$	18.1233	1.73589	0.867946	+	0.496658i	$0.165440\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	14.1048	1.32687	0.663433	−	0.748236i	$-0.269099\pi$
$113$	14.1048	1.32687	0.663433	+	0.748236i	$0.269099\pi$
$114$	0	0
$115$	−9.81681	−0.915422
$116$	0	0
$117$	0	0
$118$	0	0
$119$	24.2672	2.22457
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	0	0
$124$	0	0
$125$	7.63362	0.682772
$126$	0	0
$127$	4.87448	0.432541	0.216270	−	0.976334i	$-0.430611\pi$
$127$	4.87448	0.432541	0.216270	+	0.976334i	$0.430611\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	10.4896	0.916484	0.458242	−	0.888828i	$-0.348479\pi$
$131$	10.4896	0.916484	0.458242	+	0.888828i	$0.348479\pi$
$132$	0	0
$133$	23.8538	2.06838
$134$	0	0
$135$	0	0
$136$	0	0
$137$	1.79834	0.153642	0.0768212	−	0.997045i	$-0.475523\pi$
$137$	1.79834	0.153642	0.0768212	+	0.997045i	$0.475523\pi$
$138$	0	0
$139$	−5.71598	−0.484823	−0.242412	−	0.970174i	$-0.577938\pi$
$139$	−5.71598	−0.484823	−0.242412	+	0.970174i	$0.577938\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	3.81681	0.319178
$144$	0	0
$145$	−22.4218	−1.86203
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−12.2201	−1.00111	−0.500556	−	0.865704i	$-0.666871\pi$
$149$	−12.2201	−1.00111	−0.500556	+	0.865704i	$0.666871\pi$
$150$	0	0
$151$	13.3641	1.08756	0.543778	−	0.839229i	$-0.316993\pi$
$151$	13.3641	1.08756	0.543778	+	0.839229i	$0.316993\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−18.6050	−1.49439
$156$	0	0
$157$	1.16472	0.0929547	0.0464773	−	0.998919i	$-0.485200\pi$
$157$	1.16472	0.0929547	0.0464773	+	0.998919i	$0.485200\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−15.0616	−1.18702
$162$	0	0
$163$	−22.0000	−1.72317	−0.861586	−	0.507611i	$-0.830529\pi$
$163$	−22.0000	−1.72317	−0.861586	+	0.507611i	$0.830529\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−17.1625	−1.32807	−0.664035	−	0.747701i	$-0.731158\pi$
$167$	−17.1625	−1.32807	−0.664035	+	0.747701i	$0.731158\pi$
$168$	0	0
$169$	1.56804	0.120618
$170$	0	0
$171$	0	0
$172$	0	0
$173$	24.7098	1.87865	0.939324	−	0.343030i	$-0.111453\pi$
$173$	24.7098	1.87865	0.939324	+	0.343030i	$0.111453\pi$
$174$	0	0
$175$	−8.79213	−0.664622
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−4.83528	−0.361406	−0.180703	−	0.983538i	$-0.557837\pi$
$179$	−4.83528	−0.361406	−0.180703	+	0.983538i	$0.557837\pi$
$180$	0	0
$181$	4.71203	0.350242	0.175121	−	0.984547i	$-0.443968\pi$
$181$	4.71203	0.350242	0.175121	+	0.984547i	$0.443968\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−17.4504	−1.28298
$186$	0	0
$187$	5.91764	0.432741
$188$	0	0
$189$	0	0
$190$	0	0
$191$	12.5473	0.907891	0.453946	−	0.891029i	$-0.350016\pi$
$191$	12.5473	0.907891	0.453946	+	0.891029i	$0.350016\pi$
$192$	0	0
$193$	23.3641	1.68179	0.840893	−	0.541201i	$-0.182030\pi$
$193$	23.3641	1.68179	0.840893	+	0.541201i	$0.182030\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	15.5697	1.10930	0.554649	−	0.832085i	$-0.312853\pi$
$197$	15.5697	1.10930	0.554649	+	0.832085i	$0.312853\pi$
$198$	0	0
$199$	−3.73050	−0.264448	−0.132224	−	0.991220i	$-0.542212\pi$
$199$	−3.73050	−0.264448	−0.132224	+	0.991220i	$0.542212\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−34.4011	−2.41448
$204$	0	0
$205$	−19.3641	−1.35245
$206$	0	0
$207$	0	0
$208$	0	0
$209$	5.81681	0.402357
$210$	0	0
$211$	−0.380898	−0.0262221	−0.0131110	−	0.999914i	$-0.504173\pi$
$211$	−0.380898	−0.0262221	−0.0131110	+	0.999914i	$0.504173\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	28.7961	1.96388
$216$	0	0
$217$	−28.5450	−1.93776
$218$	0	0
$219$	0	0
$220$	0	0
$221$	22.5865	1.51933
$222$	0	0
$223$	−6.01847	−0.403027	−0.201513	−	0.979486i	$-0.564586\pi$
$223$	−6.01847	−0.403027	−0.201513	+	0.979486i	$0.564586\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−26.5081	−1.75940	−0.879702	−	0.475525i	$-0.842258\pi$
$227$	−26.5081	−1.75940	−0.879702	+	0.475525i	$0.842258\pi$
$228$	0	0
$229$	−29.3434	−1.93907	−0.969533	−	0.244962i	$-0.921225\pi$
$229$	−29.3434	−1.93907	−0.969533	+	0.244962i	$0.921225\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	12.3064	0.806222	0.403111	−	0.915151i	$-0.367929\pi$
$233$	12.3064	0.806222	0.403111	+	0.915151i	$0.367929\pi$
$234$	0	0
$235$	0.384851	0.0251049
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−6.09688	−0.394374	−0.197187	−	0.980366i	$-0.563181\pi$
$239$	−6.09688	−0.394374	−0.197187	+	0.980366i	$0.563181\pi$
$240$	0	0
$241$	−2.94233	−0.189532	−0.0947659	−	0.995500i	$-0.530210\pi$
$241$	−2.94233	−0.189532	−0.0947659	+	0.995500i	$0.530210\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−26.2386	−1.67632
$246$	0	0
$247$	22.2017	1.41266
$248$	0	0
$249$	0	0
$250$	0	0
$251$	11.7697	0.742897	0.371448	−	0.928454i	$-0.378861\pi$
$251$	11.7697	0.742897	0.371448	+	0.928454i	$0.378861\pi$
$252$	0	0
$253$	−3.67282	−0.230908
$254$	0	0
$255$	0	0
$256$	0	0
$257$	0.960797	0.0599329	0.0299664	−	0.999551i	$-0.490460\pi$
$257$	0.960797	0.0599329	0.0299664	+	0.999551i	$0.490460\pi$
$258$	0	0
$259$	−26.7737	−1.66363
$260$	0	0
$261$	0	0
$262$	0	0
$263$	8.11535	0.500414	0.250207	−	0.968192i	$-0.419501\pi$
$263$	8.11535	0.500414	0.250207	+	0.968192i	$0.419501\pi$
$264$	0	0
$265$	12.0000	0.737154
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−5.03920	−0.307246	−0.153623	−	0.988130i	$-0.549094\pi$
$269$	−5.03920	−0.307246	−0.153623	+	0.988130i	$0.549094\pi$
$270$	0	0
$271$	−15.6297	−0.949435	−0.474717	−	0.880138i	$-0.657450\pi$
$271$	−15.6297	−0.949435	−0.474717	+	0.880138i	$0.657450\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−2.14399	−0.129287
$276$	0	0
$277$	29.9585	1.80003	0.900017	−	0.435855i	$-0.143554\pi$
$277$	29.9585	1.80003	0.900017	+	0.435855i	$0.143554\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	19.1770	1.14400	0.572001	−	0.820253i	$-0.306167\pi$
$281$	19.1770	1.14400	0.572001	+	0.820253i	$0.306167\pi$
$282$	0	0
$283$	14.3064	0.850430	0.425215	−	0.905092i	$-0.360199\pi$
$283$	14.3064	0.850430	0.425215	+	0.905092i	$0.360199\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−29.7098	−1.75371
$288$	0	0
$289$	18.0185	1.05991
$290$	0	0
$291$	0	0
$292$	0	0
$293$	20.1832	1.17911	0.589557	−	0.807727i	$-0.299302\pi$
$293$	20.1832	1.17911	0.589557	+	0.807727i	$0.299302\pi$
$294$	0	0
$295$	−31.7384	−1.84788
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−14.0185	−0.810709
$300$	0	0
$301$	44.1809	2.54655
$302$	0	0
$303$	0	0
$304$	0	0
$305$	4.31701	0.247191
$306$	0	0
$307$	−29.9361	−1.70854	−0.854272	−	0.519826i	$-0.825997\pi$
$307$	−29.9361	−1.70854	−0.854272	+	0.519826i	$0.825997\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−11.3928	−0.646024	−0.323012	−	0.946395i	$-0.604695\pi$
$311$	−11.3928	−0.646024	−0.323012	+	0.946395i	$0.604695\pi$
$312$	0	0
$313$	0.654353	0.0369862	0.0184931	−	0.999829i	$-0.494113\pi$
$313$	0.654353	0.0369862	0.0184931	+	0.999829i	$0.494113\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−3.22239	−0.180988	−0.0904938	−	0.995897i	$-0.528845\pi$
$317$	−3.22239	−0.180988	−0.0904938	+	0.995897i	$0.528845\pi$
$318$	0	0
$319$	−8.38880	−0.469683
$320$	0	0
$321$	0	0
$322$	0	0
$323$	34.4218	1.91528
$324$	0	0
$325$	−8.18319	−0.453922
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0.590464	0.0325534
$330$	0	0
$331$	−13.6151	−0.748356	−0.374178	−	0.927357i	$-0.622075\pi$
$331$	−13.6151	−0.748356	−0.374178	+	0.927357i	$0.622075\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	18.6050	1.01650
$336$	0	0
$337$	22.2280	1.21084	0.605419	−	0.795907i	$-0.293005\pi$
$337$	22.2280	1.21084	0.605419	+	0.795907i	$0.293005\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−6.96080	−0.376948
$342$	0	0
$343$	−11.5513	−0.623709
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−13.6257	−0.731467	−0.365733	−	0.930720i	$-0.619182\pi$
$347$	−13.6257	−0.731467	−0.365733	+	0.930720i	$0.619182\pi$
$348$	0	0
$349$	9.07615	0.485835	0.242917	−	0.970047i	$-0.421896\pi$
$349$	9.07615	0.485835	0.242917	+	0.970047i	$0.421896\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−30.1602	−1.60527	−0.802633	−	0.596474i	$-0.796568\pi$
$353$	−30.1602	−1.60527	−0.802633	+	0.596474i	$0.796568\pi$
$354$	0	0
$355$	1.30871	0.0694589
$356$	0	0
$357$	0	0
$358$	0	0
$359$	4.30644	0.227285	0.113643	−	0.993522i	$-0.463748\pi$
$359$	4.30644	0.227285	0.113643	+	0.993522i	$0.463748\pi$
$360$	0	0
$361$	14.8353	0.780804
$362$	0	0
$363$	0	0
$364$	0	0
$365$	22.6913	1.18772
$366$	0	0
$367$	4.08631	0.213304	0.106652	−	0.994296i	$-0.465987\pi$
$367$	4.08631	0.213304	0.106652	+	0.994296i	$0.465987\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	18.4112	0.955863
$372$	0	0
$373$	12.4712	0.645732	0.322866	−	0.946445i	$-0.395354\pi$
$373$	12.4712	0.645732	0.322866	+	0.946445i	$0.395354\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−32.0185	−1.64904
$378$	0	0
$379$	9.45043	0.485436	0.242718	−	0.970097i	$-0.421961\pi$
$379$	9.45043	0.485436	0.242718	+	0.970097i	$0.421961\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−7.51037	−0.383762	−0.191881	−	0.981418i	$-0.561459\pi$
$383$	−7.51037	−0.383762	−0.191881	+	0.981418i	$0.561459\pi$
$384$	0	0
$385$	−10.9608	−0.558614
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−24.6913	−1.25190	−0.625949	−	0.779864i	$-0.715288\pi$
$389$	−24.6913	−1.25190	−0.625949	+	0.779864i	$0.715288\pi$
$390$	0	0
$391$	−21.7345	−1.09916
$392$	0	0
$393$	0	0
$394$	0	0
$395$	42.5450	2.14067
$396$	0	0
$397$	−3.51827	−0.176577	−0.0882885	−	0.996095i	$-0.528140\pi$
$397$	−3.51827	−0.176577	−0.0882885	+	0.996095i	$0.528140\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	1.14399	0.0571280	0.0285640	−	0.999592i	$-0.490907\pi$
$401$	1.14399	0.0571280	0.0285640	+	0.999592i	$0.490907\pi$
$402$	0	0
$403$	−26.5680	−1.32345
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−6.52884	−0.323622
$408$	0	0
$409$	27.4610	1.35786	0.678929	−	0.734204i	$-0.262444\pi$
$409$	27.4610	1.35786	0.678929	+	0.734204i	$0.262444\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−48.6952	−2.39614
$414$	0	0
$415$	13.7983	0.677334
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−28.4896	−1.39181	−0.695905	−	0.718134i	$-0.744996\pi$
$419$	−28.4896	−1.39181	−0.695905	+	0.718134i	$0.744996\pi$
$420$	0	0
$421$	−27.1602	−1.32371	−0.661853	−	0.749633i	$-0.730230\pi$
$421$	−27.1602	−1.32371	−0.661853	+	0.749633i	$0.730230\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−12.6873	−0.615426
$426$	0	0
$427$	6.62345	0.320531
$428$	0	0
$429$	0	0
$430$	0	0
$431$	7.81681	0.376523	0.188261	−	0.982119i	$-0.439715\pi$
$431$	7.81681	0.376523	0.188261	+	0.982119i	$0.439715\pi$
$432$	0	0
$433$	35.6314	1.71233	0.856167	−	0.516699i	$-0.172839\pi$
$433$	35.6314	1.71233	0.856167	+	0.516699i	$0.172839\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−21.3641	−1.02198
$438$	0	0
$439$	25.3496	1.20987	0.604935	−	0.796275i	$-0.293199\pi$
$439$	25.3496	1.20987	0.604935	+	0.796275i	$0.293199\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−2.07841	−0.0987481	−0.0493740	−	0.998780i	$-0.515723\pi$
$443$	−2.07841	−0.0987481	−0.0493740	+	0.998780i	$0.515723\pi$
$444$	0	0
$445$	−16.0369	−0.760224
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−9.93216	−0.468728	−0.234364	−	0.972149i	$-0.575301\pi$
$449$	−9.93216	−0.468728	−0.234364	+	0.972149i	$0.575301\pi$
$450$	0	0
$451$	−7.24482	−0.341145
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−41.8353	−1.96127
$456$	0	0
$457$	−5.40558	−0.252862	−0.126431	−	0.991975i	$-0.540352\pi$
$457$	−5.40558	−0.252862	−0.126431	+	0.991975i	$0.540352\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−14.9608	−0.696794	−0.348397	−	0.937347i	$-0.613274\pi$
$461$	−14.9608	−0.696794	−0.348397	+	0.937347i	$0.613274\pi$
$462$	0	0
$463$	31.6336	1.47014	0.735070	−	0.677992i	$-0.237150\pi$
$463$	31.6336	1.47014	0.735070	+	0.677992i	$0.237150\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−9.38485	−0.434279	−0.217140	−	0.976141i	$-0.569673\pi$
$467$	−9.38485	−0.434279	−0.217140	+	0.976141i	$0.569673\pi$
$468$	0	0
$469$	28.5450	1.31809
$470$	0	0
$471$	0	0
$472$	0	0
$473$	10.7737	0.495373
$474$	0	0
$475$	−12.4712	−0.572216
$476$	0	0
$477$	0	0
$478$	0	0
$479$	20.7697	0.948992	0.474496	−	0.880258i	$-0.342630\pi$
$479$	20.7697	0.948992	0.474496	+	0.880258i	$0.342630\pi$
$480$	0	0
$481$	−24.9193	−1.13622
$482$	0	0
$483$	0	0
$484$	0	0
$485$	19.7384	0.896275
$486$	0	0
$487$	33.1730	1.50321	0.751607	−	0.659612i	$-0.229279\pi$
$487$	33.1730	1.50321	0.751607	+	0.659612i	$0.229279\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	38.9299	1.75688	0.878441	−	0.477851i	$-0.158584\pi$
$491$	38.9299	1.75688	0.878441	+	0.477851i	$0.158584\pi$
$492$	0	0
$493$	−49.6419	−2.23576
$494$	0	0
$495$	0	0
$496$	0	0
$497$	2.00791	0.0900669
$498$	0	0
$499$	1.04711	0.0468750	0.0234375	−	0.999725i	$-0.492539\pi$
$499$	1.04711	0.0468750	0.0234375	+	0.999725i	$0.492539\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−14.8930	−0.664044	−0.332022	−	0.943272i	$-0.607731\pi$
$503$	−14.8930	−0.664044	−0.332022	+	0.943272i	$0.607731\pi$
$504$	0	0
$505$	19.6442	0.874155
$506$	0	0
$507$	0	0
$508$	0	0
$509$	5.60724	0.248537	0.124268	−	0.992249i	$-0.460342\pi$
$509$	5.60724	0.248537	0.124268	+	0.992249i	$0.460342\pi$
$510$	0	0
$511$	34.8145	1.54010
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−36.6992	−1.61716
$516$	0	0
$517$	0.143987	0.00633252
$518$	0	0
$519$	0	0
$520$	0	0
$521$	1.81681	0.0795959	0.0397980	−	0.999208i	$-0.487329\pi$
$521$	1.81681	0.0795959	0.0397980	+	0.999208i	$0.487329\pi$
$522$	0	0
$523$	−29.0616	−1.27078	−0.635388	−	0.772193i	$-0.719160\pi$
$523$	−29.0616	−1.27078	−0.635388	+	0.772193i	$0.719160\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−41.1915	−1.79433
$528$	0	0
$529$	−9.51037	−0.413494
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−27.6521	−1.19775
$534$	0	0
$535$	22.2017	0.959862
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−9.81681	−0.422840
$540$	0	0
$541$	−14.4527	−0.621370	−0.310685	−	0.950513i	$-0.600558\pi$
$541$	−14.4527	−0.621370	−0.310685	+	0.950513i	$0.600558\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−48.4403	−2.07495
$546$	0	0
$547$	39.1355	1.67331	0.836657	−	0.547727i	$-0.184507\pi$
$547$	39.1355	1.67331	0.836657	+	0.547727i	$0.184507\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−48.7961	−2.07878
$552$	0	0
$553$	65.2755	2.77580
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−44.3210	−1.87794	−0.938970	−	0.344000i	$-0.888218\pi$
$557$	−44.3210	−1.87794	−0.938970	+	0.344000i	$0.888218\pi$
$558$	0	0
$559$	41.1210	1.73923
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−35.8353	−1.51028	−0.755139	−	0.655565i	$-0.772430\pi$
$563$	−35.8353	−1.51028	−0.755139	+	0.655565i	$0.772430\pi$
$564$	0	0
$565$	−37.6996	−1.58603
$566$	0	0
$567$	0	0
$568$	0	0
$569$	22.4218	0.939970	0.469985	−	0.882674i	$-0.344259\pi$
$569$	22.4218	0.939970	0.469985	+	0.882674i	$0.344259\pi$
$570$	0	0
$571$	9.23691	0.386553	0.193276	−	0.981144i	$-0.438089\pi$
$571$	9.23691	0.386553	0.193276	+	0.981144i	$0.438089\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	7.87448	0.328389
$576$	0	0
$577$	−3.03694	−0.126430	−0.0632148	−	0.998000i	$-0.520135\pi$
$577$	−3.03694	−0.126430	−0.0632148	+	0.998000i	$0.520135\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	21.1704	0.878295
$582$	0	0
$583$	4.48963	0.185942
$584$	0	0
$585$	0	0
$586$	0	0
$587$	9.70977	0.400765	0.200382	−	0.979718i	$-0.435782\pi$
$587$	9.70977	0.400765	0.200382	+	0.979718i	$0.435782\pi$
$588$	0	0
$589$	−40.4896	−1.66835
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−35.5513	−1.45992	−0.729958	−	0.683492i	$-0.760460\pi$
$593$	−35.5513	−1.45992	−0.729958	+	0.683492i	$0.760460\pi$
$594$	0	0
$595$	−64.8621	−2.65909
$596$	0	0
$597$	0	0
$598$	0	0
$599$	19.9608	0.815576	0.407788	−	0.913077i	$-0.366300\pi$
$599$	19.9608	0.815576	0.407788	+	0.913077i	$0.366300\pi$
$600$	0	0
$601$	−14.7961	−0.603545	−0.301772	−	0.953380i	$-0.597578\pi$
$601$	−14.7961	−0.603545	−0.301772	+	0.953380i	$0.597578\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−2.67282	−0.108666
$606$	0	0
$607$	−11.5658	−0.469441	−0.234720	−	0.972063i	$-0.575417\pi$
$607$	−11.5658	−0.469441	−0.234720	+	0.972063i	$0.575417\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0.549569	0.0222332
$612$	0	0
$613$	43.6600	1.76341	0.881705	−	0.471800i	$-0.156396\pi$
$613$	43.6600	1.76341	0.881705	+	0.471800i	$0.156396\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	35.1915	1.41676	0.708378	−	0.705833i	$-0.249427\pi$
$617$	35.1915	1.41676	0.708378	+	0.705833i	$0.249427\pi$
$618$	0	0
$619$	−17.7753	−0.714451	−0.357226	−	0.934018i	$-0.616277\pi$
$619$	−17.7753	−0.714451	−0.357226	+	0.934018i	$0.616277\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−24.6050	−0.985778
$624$	0	0
$625$	−31.1233	−1.24493
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−38.6353	−1.54049
$630$	0	0
$631$	24.8745	0.990238	0.495119	−	0.868825i	$-0.335125\pi$
$631$	24.8745	0.990238	0.495119	+	0.868825i	$0.335125\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−13.0286	−0.517026
$636$	0	0
$637$	−37.4689	−1.48457
$638$	0	0
$639$	0	0
$640$	0	0
$641$	24.5944	0.971421	0.485711	−	0.874120i	$-0.338561\pi$
$641$	24.5944	0.971421	0.485711	+	0.874120i	$0.338561\pi$
$642$	0	0
$643$	−1.71993	−0.0678275	−0.0339138	−	0.999425i	$-0.510797\pi$
$643$	−1.71993	−0.0678275	−0.0339138	+	0.999425i	$0.510797\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	31.6336	1.24365	0.621823	−	0.783158i	$-0.286392\pi$
$647$	31.6336	1.24365	0.621823	+	0.783158i	$0.286392\pi$
$648$	0	0
$649$	−11.8745	−0.466114
$650$	0	0
$651$	0	0
$652$	0	0
$653$	2.11535	0.0827800	0.0413900	−	0.999143i	$-0.486821\pi$
$653$	2.11535	0.0827800	0.0413900	+	0.999143i	$0.486821\pi$
$654$	0	0
$655$	−28.0369	−1.09549
$656$	0	0
$657$	0	0
$658$	0	0
$659$	28.8745	1.12479	0.562395	−	0.826869i	$-0.309880\pi$
$659$	28.8745	1.12479	0.562395	+	0.826869i	$0.309880\pi$
$660$	0	0
$661$	−37.7203	−1.46715	−0.733575	−	0.679608i	$-0.762150\pi$
$661$	−37.7203	−1.46715	−0.733575	+	0.679608i	$0.762150\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−63.7569	−2.47239
$666$	0	0
$667$	30.8106	1.19299
$668$	0	0
$669$	0	0
$670$	0	0
$671$	1.61515	0.0623521
$672$	0	0
$673$	−6.55748	−0.252772	−0.126386	−	0.991981i	$-0.540338\pi$
$673$	−6.55748	−0.252772	−0.126386	+	0.991981i	$0.540338\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	1.28176	0.0492620	0.0246310	−	0.999697i	$-0.492159\pi$
$677$	1.28176	0.0492620	0.0246310	+	0.999697i	$0.492159\pi$
$678$	0	0
$679$	30.2840	1.16219
$680$	0	0
$681$	0	0
$682$	0	0
$683$	29.0448	1.11137	0.555685	−	0.831393i	$-0.312456\pi$
$683$	29.0448	1.11137	0.555685	+	0.831393i	$0.312456\pi$
$684$	0	0
$685$	−4.80664	−0.183652
$686$	0	0
$687$	0	0
$688$	0	0
$689$	17.1361	0.652833
$690$	0	0
$691$	19.7569	0.751587	0.375793	−	0.926703i	$-0.377370\pi$
$691$	19.7569	0.751587	0.375793	+	0.926703i	$0.377370\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	15.2778	0.579520
$696$	0	0
$697$	−42.8722	−1.62390
$698$	0	0
$699$	0	0
$700$	0	0
$701$	16.6952	0.630571	0.315285	−	0.948997i	$-0.397900\pi$
$701$	16.6952	0.630571	0.315285	+	0.948997i	$0.397900\pi$
$702$	0	0
$703$	−37.9770	−1.43233
$704$	0	0
$705$	0	0
$706$	0	0
$707$	30.1395	1.13351
$708$	0	0
$709$	17.4712	0.656143	0.328072	−	0.944653i	$-0.393601\pi$
$709$	17.4712	0.656143	0.328072	+	0.944653i	$0.393601\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	25.5658	0.957446
$714$	0	0
$715$	−10.2017	−0.381520
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−23.2179	−0.865880	−0.432940	−	0.901423i	$-0.642524\pi$
$719$	−23.2179	−0.865880	−0.432940	+	0.901423i	$0.642524\pi$
$720$	0	0
$721$	−56.3064	−2.09696
$722$	0	0
$723$	0	0
$724$	0	0
$725$	17.9855	0.667964
$726$	0	0
$727$	24.1338	0.895074	0.447537	−	0.894265i	$-0.352301\pi$
$727$	24.1338	0.895074	0.447537	+	0.894265i	$0.352301\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	63.7546	2.35805
$732$	0	0
$733$	−2.26950	−0.0838260	−0.0419130	−	0.999121i	$-0.513345\pi$
$733$	−2.26950	−0.0838260	−0.0419130	+	0.999121i	$0.513345\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	6.96080	0.256404
$738$	0	0
$739$	26.3994	0.971116	0.485558	−	0.874204i	$-0.338616\pi$
$739$	26.3994	0.971116	0.485558	+	0.874204i	$0.338616\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	4.36638	0.160187	0.0800935	−	0.996787i	$-0.474478\pi$
$743$	4.36638	0.160187	0.0800935	+	0.996787i	$0.474478\pi$
$744$	0	0
$745$	32.6623	1.19665
$746$	0	0
$747$	0	0
$748$	0	0
$749$	34.0633	1.24465
$750$	0	0
$751$	4.06784	0.148438	0.0742188	−	0.997242i	$-0.476354\pi$
$751$	4.06784	0.148438	0.0742188	+	0.997242i	$0.476354\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−35.7199	−1.29998
$756$	0	0
$757$	−35.2096	−1.27971	−0.639857	−	0.768494i	$-0.721006\pi$
$757$	−35.2096	−1.27971	−0.639857	+	0.768494i	$0.721006\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	19.0347	0.690007	0.345003	−	0.938601i	$-0.387878\pi$
$761$	19.0347	0.690007	0.345003	+	0.938601i	$0.387878\pi$
$762$	0	0
$763$	−74.3204	−2.69058
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−45.3227	−1.63651
$768$	0	0
$769$	17.3042	0.624005	0.312002	−	0.950081i	$-0.399000\pi$
$769$	17.3042	0.624005	0.312002	+	0.950081i	$0.399000\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	33.6521	1.21038	0.605191	−	0.796080i	$-0.293097\pi$
$773$	33.6521	1.21038	0.605191	+	0.796080i	$0.293097\pi$
$774$	0	0
$775$	14.9239	0.536081
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−42.1417	−1.50988
$780$	0	0
$781$	0.489634	0.0175205
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−3.11309	−0.111111
$786$	0	0
$787$	−53.6811	−1.91353	−0.956763	−	0.290869i	$-0.906055\pi$
$787$	−53.6811	−1.91353	−0.956763	+	0.290869i	$0.906055\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−57.8413	−2.05660
$792$	0	0
$793$	6.16472	0.218916
$794$	0	0
$795$	0	0
$796$	0	0
$797$	26.9114	0.953252	0.476626	−	0.879106i	$-0.341860\pi$
$797$	26.9114	0.953252	0.476626	+	0.879106i	$0.341860\pi$
$798$	0	0
$799$	0.852061	0.0301437
$800$	0	0
$801$	0	0
$802$	0	0
$803$	8.48963	0.299593
$804$	0	0
$805$	40.2571	1.41888
$806$	0	0
$807$	0	0
$808$	0	0
$809$	34.4337	1.21062	0.605311	−	0.795989i	$-0.293049\pi$
$809$	34.4337	1.21062	0.605311	+	0.795989i	$0.293049\pi$
$810$	0	0
$811$	−12.2241	−0.429246	−0.214623	−	0.976697i	$-0.568852\pi$
$811$	−12.2241	−0.429246	−0.214623	+	0.976697i	$0.568852\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	58.8021	2.05975
$816$	0	0
$817$	62.6683	2.19249
$818$	0	0
$819$	0	0
$820$	0	0
$821$	26.5226	0.925646	0.462823	−	0.886451i	$-0.346837\pi$
$821$	26.5226	0.925646	0.462823	+	0.886451i	$0.346837\pi$
$822$	0	0
$823$	45.1395	1.57346	0.786731	−	0.617295i	$-0.211772\pi$
$823$	45.1395	1.57346	0.786731	+	0.617295i	$0.211772\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	0.261596	0.00909659	0.00454830	−	0.999990i	$-0.498552\pi$
$827$	0.261596	0.00909659	0.00454830	+	0.999990i	$0.498552\pi$
$828$	0	0
$829$	−21.4712	−0.745724	−0.372862	−	0.927887i	$-0.621624\pi$
$829$	−21.4712	−0.745724	−0.372862	+	0.927887i	$0.621624\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−58.0924	−2.01278
$834$	0	0
$835$	45.8722	1.58747
$836$	0	0
$837$	0	0
$838$	0	0
$839$	3.10478	0.107189	0.0535945	−	0.998563i	$-0.482932\pi$
$839$	3.10478	0.107189	0.0535945	+	0.998563i	$0.482932\pi$
$840$	0	0
$841$	41.3720	1.42662
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−4.19110	−0.144178
$846$	0	0
$847$	−4.10083	−0.140906
$848$	0	0
$849$	0	0
$850$	0	0
$851$	23.9793	0.821999
$852$	0	0
$853$	−43.1809	−1.47849	−0.739243	−	0.673438i	$-0.764817\pi$
$853$	−43.1809	−1.47849	−0.739243	+	0.673438i	$0.764817\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−20.3104	−0.693790	−0.346895	−	0.937904i	$-0.612764\pi$
$857$	−20.3104	−0.693790	−0.346895	+	0.937904i	$0.612764\pi$
$858$	0	0
$859$	35.9770	1.22752	0.613760	−	0.789493i	$-0.289656\pi$
$859$	35.9770	1.22752	0.613760	+	0.789493i	$0.289656\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	7.29854	0.248445	0.124223	−	0.992254i	$-0.460356\pi$
$863$	7.29854	0.248445	0.124223	+	0.992254i	$0.460356\pi$
$864$	0	0
$865$	−66.0448	−2.24559
$866$	0	0
$867$	0	0
$868$	0	0
$869$	15.9176	0.539969
$870$	0	0
$871$	26.5680	0.900224
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−31.3042	−1.05827
$876$	0	0
$877$	−15.6257	−0.527643	−0.263821	−	0.964572i	$-0.584983\pi$
$877$	−15.6257	−0.527643	−0.263821	+	0.964572i	$0.584983\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	22.9529	0.773303	0.386651	−	0.922226i	$-0.373632\pi$
$881$	22.9529	0.773303	0.386651	+	0.922226i	$0.373632\pi$
$882$	0	0
$883$	14.4896	0.487615	0.243808	−	0.969824i	$-0.421603\pi$
$883$	14.4896	0.487615	0.243808	+	0.969824i	$0.421603\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−16.1048	−0.540746	−0.270373	−	0.962756i	$-0.587147\pi$
$887$	−16.1048	−0.540746	−0.270373	+	0.962756i	$0.587147\pi$
$888$	0	0
$889$	−19.9894	−0.670424
$890$	0	0
$891$	0	0
$892$	0	0
$893$	0.837542	0.0280273
$894$	0	0
$895$	12.9239	0.431997
$896$	0	0
$897$	0	0
$898$	0	0
$899$	58.3928	1.94751
$900$	0	0
$901$	26.5680	0.885110
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−12.5944	−0.418653
$906$	0	0
$907$	41.8538	1.38973	0.694866	−	0.719140i	$-0.255464\pi$
$907$	41.8538	1.38973	0.694866	+	0.719140i	$0.255464\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−35.9378	−1.19067	−0.595336	−	0.803477i	$-0.702981\pi$
$911$	−35.9378	−1.19067	−0.595336	+	0.803477i	$0.702981\pi$
$912$	0	0
$913$	5.16246	0.170852
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−43.0162	−1.42052
$918$	0	0
$919$	−11.6402	−0.383976	−0.191988	−	0.981397i	$-0.561493\pi$
$919$	−11.6402	−0.383976	−0.191988	+	0.981397i	$0.561493\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	1.86884	0.0615136
$924$	0	0
$925$	13.9977	0.460243
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−6.98717	−0.229242	−0.114621	−	0.993409i	$-0.536565\pi$
$929$	−6.98717	−0.229242	−0.114621	+	0.993409i	$0.536565\pi$
$930$	0	0
$931$	−57.1025	−1.87146
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−15.8168	−0.517265
$936$	0	0
$937$	22.3143	0.728978	0.364489	−	0.931208i	$-0.381244\pi$
$937$	22.3143	0.728978	0.364489	+	0.931208i	$0.381244\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	25.4544	0.829789	0.414895	−	0.909869i	$-0.363818\pi$
$941$	25.4544	0.829789	0.414895	+	0.909869i	$0.363818\pi$
$942$	0	0
$943$	26.6089	0.866506
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−53.2593	−1.73070	−0.865348	−	0.501172i	$-0.832903\pi$
$947$	−53.2593	−1.73070	−0.865348	+	0.501172i	$0.832903\pi$
$948$	0	0
$949$	32.4033	1.05186
$950$	0	0
$951$	0	0
$952$	0	0
$953$	50.9418	1.65017	0.825083	−	0.565012i	$-0.191128\pi$
$953$	50.9418	1.65017	0.825083	+	0.565012i	$0.191128\pi$
$954$	0	0
$955$	−33.5367	−1.08522
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−7.37468	−0.238141
$960$	0	0
$961$	17.4527	0.562990
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−62.4482	−2.01028
$966$	0	0
$967$	25.9546	0.834643	0.417322	−	0.908759i	$-0.362969\pi$
$967$	25.9546	0.834643	0.417322	+	0.908759i	$0.362969\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−35.0946	−1.12624	−0.563120	−	0.826375i	$-0.690399\pi$
$971$	−35.0946	−1.12624	−0.563120	+	0.826375i	$0.690399\pi$
$972$	0	0
$973$	23.4403	0.751460
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−55.3456	−1.77066	−0.885332	−	0.464959i	$-0.846069\pi$
$977$	−55.3456	−1.77066	−0.885332	+	0.464959i	$0.846069\pi$
$978$	0	0
$979$	−6.00000	−0.191761
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−7.25143	−0.231285	−0.115642	−	0.993291i	$-0.536893\pi$
$983$	−7.25143	−0.231285	−0.115642	+	0.993291i	$0.536893\pi$
$984$	0	0
$985$	−41.6151	−1.32597
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−39.5697	−1.25824
$990$	0	0
$991$	−52.0739	−1.65418	−0.827091	−	0.562068i	$-0.810006\pi$
$991$	−52.0739	−1.65418	−0.827091	+	0.562068i	$0.810006\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	9.97096	0.316101
$996$	0	0
$997$	−13.6415	−0.432031	−0.216016	−	0.976390i	$-0.569306\pi$
$997$	−13.6415	−0.432031	−0.216016	+	0.976390i	$0.569306\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2376.2.a.p.1.1	yes	3
3.2	odd	2		2376.2.a.k.1.3	✓	3
4.3	odd	2		4752.2.a.br.1.1		3
12.11	even	2		4752.2.a.bj.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2376.2.a.k.1.3	✓	3	3.2	odd	2
2376.2.a.p.1.1	yes	3	1.1	even	1	trivial
4752.2.a.bj.1.3		3	12.11	even	2
4752.2.a.br.1.1		3	4.3	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2376 = 2^{3} \cdot 3^{3} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2376.a (trivial)

\(f(q)\)	\(=\)	\(q-2.67282 q^{5} -4.10083 q^{7} -1.00000 q^{11} -3.81681 q^{13} -5.91764 q^{17} -5.81681 q^{19} +3.67282 q^{23} +2.14399 q^{25} +8.38880 q^{29} +6.96080 q^{31} +10.9608 q^{35} +6.52884 q^{37} +7.24482 q^{41} -10.7737 q^{43} -0.143987 q^{47} +9.81681 q^{49} -4.48963 q^{53} +2.67282 q^{55} +11.8745 q^{59} -1.61515 q^{61} +10.2017 q^{65} -6.96080 q^{67} -0.489634 q^{71} -8.48963 q^{73} +4.10083 q^{77} -15.9176 q^{79} -5.16246 q^{83} +15.8168 q^{85} +6.00000 q^{89} +15.6521 q^{91} +15.5473 q^{95} -7.38485 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 2 q^{5} - 3 q^{7} - 3 q^{11} + 3 q^{17} - 6 q^{19} + q^{23} + 5 q^{25} + 13 q^{29} + 8 q^{31} + 20 q^{35} + 11 q^{37} + 11 q^{41} - 13 q^{43} + q^{47} + 18 q^{49} + 8 q^{53} - 2 q^{55} + 7 q^{59}+ \cdots - 15 q^{97}+O(q^{100}) \) 3 * q + 2 * q^5 - 3 * q^7 - 3 * q^11 + 3 * q^17 - 6 * q^19 + q^23 + 5 * q^25 + 13 * q^29 + 8 * q^31 + 20 * q^35 + 11 * q^37 + 11 * q^41 - 13 * q^43 + q^47 + 18 * q^49 + 8 * q^53 - 2 * q^55 + 7 * q^59 - 12 * q^61 + 12 * q^65 - 8 * q^67 + 20 * q^71 - 4 * q^73 + 3 * q^77 - 27 * q^79 + 16 * q^83 + 36 * q^85 + 18 * q^89 - 6 * q^91 + 8 * q^95 - 15 * q^97

Introduction

L-functions

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