LMFDB - Embedded newform 3744.2.j.a.2159.18

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3744,2,Mod(2159,3744)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3744, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 1, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3744.2159"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 3744 = 2^{5} \cdot 3^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3744.j (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

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

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

Self dual:	no
Analytic conductor:	$29.8959905168$
Analytic rank:	$0$
Dimension:	$48$
Twist minimal:	no (minimal twist has level 936)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2159.18
Character	$\chi$	$=$	3744.2159
Dual form			3744.2.j.a.2159.17

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-0.743520 q^{5} -1.72000i q^{7} -4.66933i q^{11} -1.00000i q^{13} +6.58087i q^{17} +3.02593 q^{19} +7.02856 q^{23} -4.44718 q^{25} +10.3540 q^{29} +0.619913i q^{31} +1.27886i q^{35} -0.689961i q^{37} +5.53068i q^{41} -8.98383 q^{43} -0.183800 q^{47} +4.04160 q^{49} -10.7059 q^{53} +3.47174i q^{55} -13.2860i q^{59} -13.1575i q^{61} +0.743520i q^{65} -0.268454 q^{67} +10.4124 q^{71} +2.74597 q^{73} -8.03125 q^{77} -11.0954i q^{79} -6.15784i q^{83} -4.89301i q^{85} -13.1131i q^{89} -1.72000 q^{91} -2.24984 q^{95} -17.2096 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q - 32 q^{19} + 48 q^{25} + 32 q^{43} - 48 q^{49} - 32 q^{67} - 32 q^{97}+O(q^{100}) $ 48 * q - 32 * q^19 + 48 * q^25 + 32 * q^43 - 48 * q^49 - 32 * q^67 - 32 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3744\mathbb{Z}\right)^\times$.

$n$	$703$	$2017$	$2081$	$2341$
$\chi(n)$	$-1$	$1$	$-1$	$-1$

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
$5$	−0.743520	−0.332512	−0.166256	−	0.986083i	$-0.553168\pi$
$5$	−0.743520	−0.332512	−0.166256	+	0.986083i	$0.553168\pi$
$6$	0	0
$7$	− 1.72000i	− 0.650099i	−0.945697	−	0.325050i	$-0.894619\pi$
$7$	− 1.72000i	− 0.650099i	0.945697	−	0.325050i	$-0.105381\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 4.66933i	− 1.40786i	−0.710271	−	0.703928i	$-0.751428\pi$
$11$	− 4.66933i	− 1.40786i	0.710271	−	0.703928i	$-0.248572\pi$
$12$	0	0
$13$	− 1.00000i	− 0.277350i
$13$	− 1.00000i	− 0.277350i
$14$	0	0
$15$	0	0
$16$	0	0
$17$	6.58087i	1.59609i	0.602595	+	0.798047i	$0.294134\pi$
$17$	6.58087i	1.59609i	−0.602595	+	0.798047i	$0.705866\pi$
$18$	0	0
$19$	3.02593	0.694196	0.347098	−	0.937829i	$-0.387167\pi$
$19$	3.02593	0.694196	0.347098	+	0.937829i	$0.387167\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	7.02856	1.46556	0.732778	−	0.680468i	$-0.238223\pi$
$23$	7.02856	1.46556	0.732778	+	0.680468i	$0.238223\pi$
$24$	0	0
$25$	−4.44718	−0.889436
$26$	0	0
$27$	0	0
$28$	0	0
$29$	10.3540	1.92269	0.961346	−	0.275342i	$-0.0887911\pi$
$29$	10.3540	1.92269	0.961346	+	0.275342i	$0.0887911\pi$
$30$	0	0
$31$	0.619913i	0.111340i	0.998449	+	0.0556698i	$0.0177294\pi$
$31$	0.619913i	0.111340i	−0.998449	+	0.0556698i	$0.982271\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	1.27886i	0.216166i
$36$	0	0
$37$	− 0.689961i	− 0.113429i	−0.998390	−	0.0567144i	$-0.981938\pi$
$37$	− 0.689961i	− 0.113429i	0.998390	−	0.0567144i	$-0.0180625\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	5.53068i	0.863747i	0.901934	+	0.431874i	$0.142147\pi$
$41$	5.53068i	0.863747i	−0.901934	+	0.431874i	$0.857853\pi$
$42$	0	0
$43$	−8.98383	−1.37002	−0.685011	−	0.728533i	$-0.740203\pi$
$43$	−8.98383	−1.37002	−0.685011	+	0.728533i	$0.740203\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−0.183800	−0.0268100	−0.0134050	−	0.999910i	$-0.504267\pi$
$47$	−0.183800	−0.0268100	−0.0134050	+	0.999910i	$0.504267\pi$
$48$	0	0
$49$	4.04160	0.577371
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−10.7059	−1.47057	−0.735287	−	0.677755i	$-0.762953\pi$
$53$	−10.7059	−1.47057	−0.735287	+	0.677755i	$0.762953\pi$
$54$	0	0
$55$	3.47174i	0.468130i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 13.2860i	− 1.72969i	−0.502036	−	0.864847i	$-0.667416\pi$
$59$	− 13.2860i	− 1.72969i	0.502036	−	0.864847i	$-0.332584\pi$
$60$	0	0
$61$	− 13.1575i	− 1.68465i	−0.538970	−	0.842325i	$-0.681186\pi$
$61$	− 13.1575i	− 1.68465i	0.538970	−	0.842325i	$-0.318814\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0.743520i	0.0922223i
$66$	0	0
$67$	−0.268454	−0.0327969	−0.0163984	−	0.999866i	$-0.505220\pi$
$67$	−0.268454	−0.0327969	−0.0163984	+	0.999866i	$0.505220\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	10.4124	1.23572	0.617861	−	0.786287i	$-0.287999\pi$
$71$	10.4124	1.23572	0.617861	+	0.786287i	$0.287999\pi$
$72$	0	0
$73$	2.74597	0.321391	0.160696	−	0.987004i	$-0.448626\pi$
$73$	2.74597	0.321391	0.160696	+	0.987004i	$0.448626\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−8.03125	−0.915246
$78$	0	0
$79$	− 11.0954i	− 1.24833i	−0.781291	−	0.624167i	$-0.785439\pi$
$79$	− 11.0954i	− 1.24833i	0.781291	−	0.624167i	$-0.214561\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 6.15784i	− 0.675911i	−0.941162	−	0.337955i	$-0.890265\pi$
$83$	− 6.15784i	− 0.675911i	0.941162	−	0.337955i	$-0.109735\pi$
$84$	0	0
$85$	− 4.89301i	− 0.530721i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 13.1131i	− 1.38998i	−0.719018	−	0.694991i	$-0.755408\pi$
$89$	− 13.1131i	− 1.38998i	0.719018	−	0.694991i	$-0.244592\pi$
$90$	0	0
$91$	−1.72000	−0.180305
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−2.24984	−0.230829
$96$	0	0
$97$	−17.2096	−1.74737	−0.873685	−	0.486491i	$-0.838277\pi$
$97$	−17.2096	−1.74737	−0.873685	+	0.486491i	$0.838277\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−0.508663	−0.0506139	−0.0253069	−	0.999680i	$-0.508056\pi$
$101$	−0.508663	−0.0506139	−0.0253069	+	0.999680i	$0.508056\pi$
$102$	0	0
$103$	− 13.6065i	− 1.34069i	−0.742051	−	0.670344i	$-0.766147\pi$
$103$	− 13.6065i	− 1.34069i	0.742051	−	0.670344i	$-0.233853\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	9.51632i	0.919978i	0.887925	+	0.459989i	$0.152147\pi$
$107$	9.51632i	0.919978i	−0.887925	+	0.459989i	$0.847853\pi$
$108$	0	0
$109$	− 5.42029i	− 0.519169i	−0.965720	−	0.259585i	$-0.916414\pi$
$109$	− 5.42029i	− 0.519169i	0.965720	−	0.259585i	$-0.0835857\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	13.4980i	1.26979i	0.772599	+	0.634894i	$0.218956\pi$
$113$	13.4980i	1.26979i	−0.772599	+	0.634894i	$0.781044\pi$
$114$	0	0
$115$	−5.22588	−0.487316
$116$	0	0
$117$	0	0
$118$	0	0
$119$	11.3191	1.03762
$120$	0	0
$121$	−10.8026	−0.982059
$122$	0	0
$123$	0	0
$124$	0	0
$125$	7.02417	0.628261
$126$	0	0
$127$	9.79334i	0.869019i	0.900667	+	0.434509i	$0.143078\pi$
$127$	9.79334i	0.869019i	−0.900667	+	0.434509i	$0.856922\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 2.14770i	− 0.187645i	−0.995589	−	0.0938225i	$-0.970091\pi$
$131$	− 2.14770i	− 0.187645i	0.995589	−	0.0938225i	$-0.0299086\pi$
$132$	0	0
$133$	− 5.20460i	− 0.451296i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 9.47000i	− 0.809077i	−0.914521	−	0.404538i	$-0.867432\pi$
$137$	− 9.47000i	− 0.809077i	0.914521	−	0.404538i	$-0.132568\pi$
$138$	0	0
$139$	−2.93590	−0.249020	−0.124510	−	0.992218i	$-0.539736\pi$
$139$	−2.93590	−0.249020	−0.124510	+	0.992218i	$0.539736\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−4.66933	−0.390469
$144$	0	0
$145$	−7.69842	−0.639319
$146$	0	0
$147$	0	0
$148$	0	0
$149$	6.81718	0.558486	0.279243	−	0.960221i	$-0.409917\pi$
$149$	6.81718	0.558486	0.279243	+	0.960221i	$0.409917\pi$
$150$	0	0
$151$	− 18.6158i	− 1.51493i	−0.652875	−	0.757466i	$-0.726437\pi$
$151$	− 18.6158i	− 1.51493i	0.652875	−	0.757466i	$-0.273563\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 0.460918i	− 0.0370218i
$156$	0	0
$157$	− 9.01440i	− 0.719427i	−0.933063	−	0.359714i	$-0.882874\pi$
$157$	− 9.01440i	− 0.719427i	0.933063	−	0.359714i	$-0.117126\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 12.0891i	− 0.952757i
$162$	0	0
$163$	8.20829	0.642923	0.321462	−	0.946923i	$-0.395826\pi$
$163$	8.20829	0.642923	0.321462	+	0.946923i	$0.395826\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	6.78711	0.525202	0.262601	−	0.964904i	$-0.415420\pi$
$167$	6.78711	0.525202	0.262601	+	0.964904i	$0.415420\pi$
$168$	0	0
$169$	−1.00000	−0.0769231
$170$	0	0
$171$	0	0
$172$	0	0
$173$	6.74312	0.512670	0.256335	−	0.966588i	$-0.417485\pi$
$173$	6.74312	0.512670	0.256335	+	0.966588i	$0.417485\pi$
$174$	0	0
$175$	7.64915i	0.578221i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 7.33246i	− 0.548054i	−0.961722	−	0.274027i	$-0.911644\pi$
$179$	− 7.33246i	− 0.548054i	0.961722	−	0.274027i	$-0.0883558\pi$
$180$	0	0
$181$	1.57125i	0.116790i	0.998294	+	0.0583949i	$0.0185982\pi$
$181$	1.57125i	0.116790i	−0.998294	+	0.0583949i	$0.981402\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0.513000i	0.0377165i
$186$	0	0
$187$	30.7282	2.24707
$188$	0	0
$189$	0	0
$190$	0	0
$191$	20.3374	1.47156	0.735781	−	0.677220i	$-0.236815\pi$
$191$	20.3374	1.47156	0.735781	+	0.677220i	$0.236815\pi$
$192$	0	0
$193$	−2.56171	−0.184396	−0.0921980	−	0.995741i	$-0.529389\pi$
$193$	−2.56171	−0.184396	−0.0921980	+	0.995741i	$0.529389\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	21.0154	1.49728	0.748641	−	0.662975i	$-0.230707\pi$
$197$	21.0154	1.49728	0.748641	+	0.662975i	$0.230707\pi$
$198$	0	0
$199$	20.1058i	1.42527i	0.701537	+	0.712633i	$0.252497\pi$
$199$	20.1058i	1.42527i	−0.701537	+	0.712633i	$0.747503\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 17.8089i	− 1.24994i
$204$	0	0
$205$	− 4.11217i	− 0.287207i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 14.1291i	− 0.977328i
$210$	0	0
$211$	−14.5749	−1.00337	−0.501687	−	0.865049i	$-0.667287\pi$
$211$	−14.5749	−1.00337	−0.501687	+	0.865049i	$0.667287\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	6.67966	0.455549
$216$	0	0
$217$	1.06625	0.0723818
$218$	0	0
$219$	0	0
$220$	0	0
$221$	6.58087	0.442677
$222$	0	0
$223$	− 14.0882i	− 0.943418i	−0.881754	−	0.471709i	$-0.843637\pi$
$223$	− 14.0882i	− 0.943418i	0.881754	−	0.471709i	$-0.156363\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 10.0126i	− 0.664558i	−0.943181	−	0.332279i	$-0.892182\pi$
$227$	− 10.0126i	− 0.664558i	0.943181	−	0.332279i	$-0.107818\pi$
$228$	0	0
$229$	11.3711i	0.751426i	0.926736	+	0.375713i	$0.122602\pi$
$229$	11.3711i	0.751426i	−0.926736	+	0.375713i	$0.877398\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	5.49317i	0.359869i	0.983679	+	0.179935i	$0.0575887\pi$
$233$	5.49317i	0.359869i	−0.983679	+	0.179935i	$0.942411\pi$
$234$	0	0
$235$	0.136659	0.00891465
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−18.0075	−1.16481	−0.582404	−	0.812899i	$-0.697888\pi$
$239$	−18.0075	−1.16481	−0.582404	+	0.812899i	$0.697888\pi$
$240$	0	0
$241$	7.74211	0.498713	0.249357	−	0.968412i	$-0.419781\pi$
$241$	7.74211	0.498713	0.249357	+	0.968412i	$0.419781\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−3.00501	−0.191983
$246$	0	0
$247$	− 3.02593i	− 0.192535i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	19.3627i	1.22216i	0.791567	+	0.611082i	$0.209266\pi$
$251$	19.3627i	1.22216i	−0.791567	+	0.611082i	$0.790734\pi$
$252$	0	0
$253$	− 32.8187i	− 2.06329i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 4.17899i	− 0.260678i	−0.991469	−	0.130339i	$-0.958393\pi$
$257$	− 4.17899i	− 0.260678i	0.991469	−	0.130339i	$-0.0416066\pi$
$258$	0	0
$259$	−1.18673	−0.0737400
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−24.6071	−1.51734	−0.758670	−	0.651475i	$-0.774150\pi$
$263$	−24.6071	−1.51734	−0.758670	+	0.651475i	$0.774150\pi$
$264$	0	0
$265$	7.96009	0.488984
$266$	0	0
$267$	0	0
$268$	0	0
$269$	8.95889	0.546233	0.273116	−	0.961981i	$-0.411946\pi$
$269$	8.95889	0.546233	0.273116	+	0.961981i	$0.411946\pi$
$270$	0	0
$271$	− 21.8098i	− 1.32485i	−0.749128	−	0.662426i	$-0.769527\pi$
$271$	− 21.8098i	− 1.32485i	0.749128	−	0.662426i	$-0.230473\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	20.7653i	1.25220i
$276$	0	0
$277$	− 7.26899i	− 0.436751i	−0.975865	−	0.218376i	$-0.929924\pi$
$277$	− 7.26899i	− 0.436751i	0.975865	−	0.218376i	$-0.0700758\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 0.361353i	− 0.0215565i	−0.999942	−	0.0107783i	$-0.996569\pi$
$281$	− 0.361353i	− 0.0215565i	0.999942	−	0.0107783i	$-0.00343089\pi$
$282$	0	0
$283$	−11.2756	−0.670262	−0.335131	−	0.942172i	$-0.608781\pi$
$283$	−11.2756	−0.670262	−0.335131	+	0.942172i	$0.608781\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	9.51278	0.561521
$288$	0	0
$289$	−26.3078	−1.54752
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−19.6966	−1.15069	−0.575343	−	0.817913i	$-0.695131\pi$
$293$	−19.6966	−1.15069	−0.575343	+	0.817913i	$0.695131\pi$
$294$	0	0
$295$	9.87843i	0.575144i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 7.02856i	− 0.406472i
$300$	0	0
$301$	15.4522i	0.890650i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	9.78290i	0.560167i
$306$	0	0
$307$	7.93961	0.453138	0.226569	−	0.973995i	$-0.427249\pi$
$307$	7.93961	0.453138	0.226569	+	0.973995i	$0.427249\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	23.4955	1.33231	0.666153	−	0.745815i	$-0.267940\pi$
$311$	23.4955	1.33231	0.666153	+	0.745815i	$0.267940\pi$
$312$	0	0
$313$	8.59126	0.485607	0.242803	−	0.970076i	$-0.421933\pi$
$313$	8.59126	0.485607	0.242803	+	0.970076i	$0.421933\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	3.57136	0.200588	0.100294	−	0.994958i	$-0.468022\pi$
$317$	3.57136	0.200588	0.100294	+	0.994958i	$0.468022\pi$
$318$	0	0
$319$	− 48.3463i	− 2.70687i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	19.9132i	1.10800i
$324$	0	0
$325$	4.44718i	0.246685i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0.316136i	0.0174292i
$330$	0	0
$331$	−22.6506	−1.24499	−0.622494	−	0.782625i	$-0.713880\pi$
$331$	−22.6506	−1.24499	−0.622494	+	0.782625i	$0.713880\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0.199601	0.0109054
$336$	0	0
$337$	−13.1565	−0.716679	−0.358340	−	0.933591i	$-0.616657\pi$
$337$	−13.1565	−0.716679	−0.358340	+	0.933591i	$0.616657\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	2.89458	0.156750
$342$	0	0
$343$	− 18.9916i	− 1.02545i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	7.91878i	0.425102i	0.977150	+	0.212551i	$0.0681772\pi$
$347$	7.91878i	0.425102i	−0.977150	+	0.212551i	$0.931823\pi$
$348$	0	0
$349$	7.78876i	0.416923i	0.978031	+	0.208461i	$0.0668456\pi$
$349$	7.78876i	0.416923i	−0.978031	+	0.208461i	$0.933154\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	9.64415i	0.513306i	0.966504	+	0.256653i	$0.0826198\pi$
$353$	9.64415i	0.513306i	−0.966504	+	0.256653i	$0.917380\pi$
$354$	0	0
$355$	−7.74181	−0.410893
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−13.0246	−0.687410	−0.343705	−	0.939078i	$-0.611682\pi$
$359$	−13.0246	−0.687410	−0.343705	+	0.939078i	$0.611682\pi$
$360$	0	0
$361$	−9.84375	−0.518092
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−2.04168	−0.106867
$366$	0	0
$367$	− 9.32427i	− 0.486723i	−0.969936	−	0.243362i	$-0.921750\pi$
$367$	− 9.32427i	− 0.486723i	0.969936	−	0.243362i	$-0.0782502\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	18.4142i	0.956020i
$372$	0	0
$373$	− 33.2225i	− 1.72020i	−0.510129	−	0.860098i	$-0.670402\pi$
$373$	− 33.2225i	− 1.72020i	0.510129	−	0.860098i	$-0.329598\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 10.3540i	− 0.533259i
$378$	0	0
$379$	−4.21971	−0.216752	−0.108376	−	0.994110i	$-0.534565\pi$
$379$	−4.21971	−0.216752	−0.108376	+	0.994110i	$0.534565\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−31.8136	−1.62560	−0.812800	−	0.582542i	$-0.802058\pi$
$383$	−31.8136	−1.62560	−0.812800	+	0.582542i	$0.802058\pi$
$384$	0	0
$385$	5.97140	0.304331
$386$	0	0
$387$	0	0
$388$	0	0
$389$	18.6833	0.947282	0.473641	−	0.880718i	$-0.342939\pi$
$389$	18.6833	0.947282	0.473641	+	0.880718i	$0.342939\pi$
$390$	0	0
$391$	46.2540i	2.33917i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	8.24968i	0.415086i
$396$	0	0
$397$	11.2388i	0.564059i	0.959406	+	0.282029i	$0.0910076\pi$
$397$	11.2388i	0.564059i	−0.959406	+	0.282029i	$0.908992\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 29.1439i	− 1.45537i	−0.685909	−	0.727687i	$-0.740595\pi$
$401$	− 29.1439i	− 1.45537i	0.685909	−	0.727687i	$-0.259405\pi$
$402$	0	0
$403$	0.619913	0.0308801
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−3.22165	−0.159691
$408$	0	0
$409$	32.8717	1.62540	0.812700	−	0.582682i	$-0.197997\pi$
$409$	32.8717	1.62540	0.812700	+	0.582682i	$0.197997\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−22.8520	−1.12447
$414$	0	0
$415$	4.57848i	0.224749i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 9.13194i	− 0.446125i	−0.974804	−	0.223062i	$-0.928395\pi$
$419$	− 9.13194i	− 0.446125i	0.974804	−	0.223062i	$-0.0716053\pi$
$420$	0	0
$421$	31.0989i	1.51567i	0.652449	+	0.757833i	$0.273742\pi$
$421$	31.0989i	1.51567i	−0.652449	+	0.757833i	$0.726258\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 29.2663i	− 1.41962i
$426$	0	0
$427$	−22.6310	−1.09519
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−19.3891	−0.933941	−0.466970	−	0.884273i	$-0.654655\pi$
$431$	−19.3891	−0.933941	−0.466970	+	0.884273i	$0.654655\pi$
$432$	0	0
$433$	34.0276	1.63526	0.817632	−	0.575742i	$-0.195287\pi$
$433$	34.0276	1.63526	0.817632	+	0.575742i	$0.195287\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	21.2679	1.01738
$438$	0	0
$439$	16.8226i	0.802899i	0.915881	+	0.401449i	$0.131493\pi$
$439$	16.8226i	0.802899i	−0.915881	+	0.401449i	$0.868507\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	8.46566i	0.402216i	0.979569	+	0.201108i	$0.0644542\pi$
$443$	8.46566i	0.402216i	−0.979569	+	0.201108i	$0.935546\pi$
$444$	0	0
$445$	9.74983i	0.462186i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	11.6157i	0.548177i	0.961705	+	0.274088i	$0.0883761\pi$
$449$	11.6157i	0.548177i	−0.961705	+	0.274088i	$0.911624\pi$
$450$	0	0
$451$	25.8246	1.21603
$452$	0	0
$453$	0	0
$454$	0	0
$455$	1.27886	0.0599537
$456$	0	0
$457$	25.4077	1.18852	0.594262	−	0.804272i	$-0.297444\pi$
$457$	25.4077	1.18852	0.594262	+	0.804272i	$0.297444\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	10.4445	0.486449	0.243224	−	0.969970i	$-0.421795\pi$
$461$	10.4445	0.486449	0.243224	+	0.969970i	$0.421795\pi$
$462$	0	0
$463$	4.65934i	0.216538i	0.994122	+	0.108269i	$0.0345308\pi$
$463$	4.65934i	0.216538i	−0.994122	+	0.108269i	$0.965469\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 23.2887i	− 1.07767i	−0.842410	−	0.538837i	$-0.818864\pi$
$467$	− 23.2887i	− 1.07767i	0.842410	−	0.538837i	$-0.181136\pi$
$468$	0	0
$469$	0.461741i	0.0213212i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	41.9485i	1.92879i
$474$	0	0
$475$	−13.4568	−0.617442
$476$	0	0
$477$	0	0
$478$	0	0
$479$	35.1469	1.60590	0.802952	−	0.596044i	$-0.203262\pi$
$479$	35.1469	1.60590	0.802952	+	0.596044i	$0.203262\pi$
$480$	0	0
$481$	−0.689961	−0.0314595
$482$	0	0
$483$	0	0
$484$	0	0
$485$	12.7957	0.581022
$486$	0	0
$487$	3.43122i	0.155483i	0.996974	+	0.0777417i	$0.0247710\pi$
$487$	3.43122i	0.155483i	−0.996974	+	0.0777417i	$0.975229\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	6.49646i	0.293181i	0.989197	+	0.146591i	$0.0468300\pi$
$491$	6.49646i	0.293181i	−0.989197	+	0.146591i	$0.953170\pi$
$492$	0	0
$493$	68.1384i	3.06880i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 17.9093i	− 0.803342i
$498$	0	0
$499$	−22.7906	−1.02025	−0.510123	−	0.860101i	$-0.670400\pi$
$499$	−22.7906	−1.02025	−0.510123	+	0.860101i	$0.670400\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	6.22278	0.277460	0.138730	−	0.990330i	$-0.455698\pi$
$503$	6.22278	0.277460	0.138730	+	0.990330i	$0.455698\pi$
$504$	0	0
$505$	0.378201	0.0168297
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−0.0362228	−0.00160555	−0.000802774	−	1.00000i	$-0.500256\pi$
$509$	−0.0362228	−0.00160555	−0.000802774		1.00000i	$0.500256\pi$
$510$	0	0
$511$	− 4.72307i	− 0.208936i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	10.1167i	0.445795i
$516$	0	0
$517$	0.858223i	0.0377446i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 18.2499i	− 0.799542i	−0.916615	−	0.399771i	$-0.869090\pi$
$521$	− 18.2499i	− 0.799542i	0.916615	−	0.399771i	$-0.130910\pi$
$522$	0	0
$523$	−19.9310	−0.871521	−0.435761	−	0.900063i	$-0.643521\pi$
$523$	−19.9310	−0.871521	−0.435761	+	0.900063i	$0.643521\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−4.07957	−0.177709
$528$	0	0
$529$	26.4007	1.14786
$530$	0	0
$531$	0	0
$532$	0	0
$533$	5.53068	0.239560
$534$	0	0
$535$	− 7.07558i	− 0.305904i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 18.8716i	− 0.812855i
$540$	0	0
$541$	− 7.30595i	− 0.314107i	−0.987590	−	0.157054i	$-0.949800\pi$
$541$	− 7.30595i	− 0.314107i	0.987590	−	0.157054i	$-0.0501995\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	4.03009i	0.172630i
$546$	0	0
$547$	−21.5397	−0.920970	−0.460485	−	0.887667i	$-0.652324\pi$
$547$	−21.5397	−0.920970	−0.460485	+	0.887667i	$0.652324\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	31.3305	1.33473
$552$	0	0
$553$	−19.0842	−0.811541
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−31.4672	−1.33331	−0.666653	−	0.745368i	$-0.732274\pi$
$557$	−31.4672	−1.33331	−0.666653	+	0.745368i	$0.732274\pi$
$558$	0	0
$559$	8.98383i	0.379976i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 23.3682i	− 0.984853i	−0.870354	−	0.492427i	$-0.836110\pi$
$563$	− 23.3682i	− 0.984853i	0.870354	−	0.492427i	$-0.163890\pi$
$564$	0	0
$565$	− 10.0361i	− 0.422220i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 9.05265i	− 0.379507i	−0.981832	−	0.189753i	$-0.939231\pi$
$569$	− 9.05265i	− 0.379507i	0.981832	−	0.189753i	$-0.0607688\pi$
$570$	0	0
$571$	−11.1599	−0.467025	−0.233513	−	0.972354i	$-0.575022\pi$
$571$	−11.1599	−0.467025	−0.233513	+	0.972354i	$0.575022\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−31.2573	−1.30352
$576$	0	0
$577$	16.9468	0.705506	0.352753	−	0.935716i	$-0.385246\pi$
$577$	16.9468	0.705506	0.352753	+	0.935716i	$0.385246\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−10.5915	−0.439409
$582$	0	0
$583$	49.9896i	2.07036i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 1.11604i	− 0.0460640i	−0.999735	−	0.0230320i	$-0.992668\pi$
$587$	− 1.11604i	− 0.0460640i	0.999735	−	0.0230320i	$-0.00733196\pi$
$588$	0	0
$589$	1.87581i	0.0772915i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	26.3257i	1.08107i	0.841323	+	0.540533i	$0.181777\pi$
$593$	26.3257i	1.08107i	−0.841323	+	0.540533i	$0.818223\pi$
$594$	0	0
$595$	−8.41598	−0.345021
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−6.69893	−0.273711	−0.136855	−	0.990591i	$-0.543700\pi$
$599$	−6.69893	−0.273711	−0.136855	+	0.990591i	$0.543700\pi$
$600$	0	0
$601$	24.9360	1.01716	0.508580	−	0.861015i	$-0.330170\pi$
$601$	24.9360	1.01716	0.508580	+	0.861015i	$0.330170\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	8.03199	0.326547
$606$	0	0
$607$	− 12.1079i	− 0.491443i	−0.969341	−	0.245721i	$-0.920975\pi$
$607$	− 12.1079i	− 0.491443i	0.969341	−	0.245721i	$-0.0790248\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0.183800i	0.00743575i
$612$	0	0
$613$	− 35.3445i	− 1.42755i	−0.700375	−	0.713775i	$-0.746984\pi$
$613$	− 35.3445i	− 1.42755i	0.700375	−	0.713775i	$-0.253016\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	4.27008i	0.171907i	0.996299	+	0.0859534i	$0.0273936\pi$
$617$	4.27008i	0.171907i	−0.996299	+	0.0859534i	$0.972606\pi$
$618$	0	0
$619$	−0.994984	−0.0399918	−0.0199959	−	0.999800i	$-0.506365\pi$
$619$	−0.994984	−0.0399918	−0.0199959	+	0.999800i	$0.506365\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−22.5545	−0.903626
$624$	0	0
$625$	17.0133	0.680531
$626$	0	0
$627$	0	0
$628$	0	0
$629$	4.54054	0.181043
$630$	0	0
$631$	− 7.71126i	− 0.306981i	−0.988150	−	0.153490i	$-0.950949\pi$
$631$	− 7.71126i	− 0.306981i	0.988150	−	0.153490i	$-0.0490514\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 7.28155i	− 0.288959i
$636$	0	0
$637$	− 4.04160i	− 0.160134i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 1.31766i	− 0.0520443i	−0.999661	−	0.0260221i	$-0.991716\pi$
$641$	− 1.31766i	− 0.0520443i	0.999661	−	0.0260221i	$-0.00828404\pi$
$642$	0	0
$643$	46.5222	1.83466	0.917328	−	0.398133i	$-0.130342\pi$
$643$	46.5222	1.83466	0.917328	+	0.398133i	$0.130342\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	9.96272	0.391675	0.195837	−	0.980636i	$-0.437257\pi$
$647$	9.96272	0.391675	0.195837	+	0.980636i	$0.437257\pi$
$648$	0	0
$649$	−62.0368	−2.43516
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−28.7123	−1.12360	−0.561800	−	0.827273i	$-0.689891\pi$
$653$	−28.7123	−1.12360	−0.561800	+	0.827273i	$0.689891\pi$
$654$	0	0
$655$	1.59685i	0.0623943i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	40.3362i	1.57128i	0.618686	+	0.785638i	$0.287665\pi$
$659$	40.3362i	1.57128i	−0.618686	+	0.785638i	$0.712335\pi$
$660$	0	0
$661$	14.9302i	0.580717i	0.956918	+	0.290358i	$0.0937745\pi$
$661$	14.9302i	0.580717i	−0.956918	+	0.290358i	$0.906225\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	3.86973i	0.150062i
$666$	0	0
$667$	72.7738	2.81781
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−61.4369	−2.37175
$672$	0	0
$673$	−1.18247	−0.0455809	−0.0227905	−	0.999740i	$-0.507255\pi$
$673$	−1.18247	−0.0455809	−0.0227905	+	0.999740i	$0.507255\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−14.2359	−0.547129	−0.273564	−	0.961854i	$-0.588203\pi$
$677$	−14.2359	−0.547129	−0.273564	+	0.961854i	$0.588203\pi$
$678$	0	0
$679$	29.6005i	1.13596i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	45.4893i	1.74060i	0.492521	+	0.870300i	$0.336075\pi$
$683$	45.4893i	1.74060i	−0.492521	+	0.870300i	$0.663925\pi$
$684$	0	0
$685$	7.04114i	0.269028i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	10.7059i	0.407864i
$690$	0	0
$691$	30.8877	1.17502	0.587511	−	0.809216i	$-0.300108\pi$
$691$	30.8877	1.17502	0.587511	+	0.809216i	$0.300108\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	2.18290	0.0828022
$696$	0	0
$697$	−36.3967	−1.37862
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−24.6794	−0.932128	−0.466064	−	0.884751i	$-0.654328\pi$
$701$	−24.6794	−0.932128	−0.466064	+	0.884751i	$0.654328\pi$
$702$	0	0
$703$	− 2.08777i	− 0.0787418i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0.874901i	0.0329040i
$708$	0	0
$709$	41.0911i	1.54321i	0.636103	+	0.771605i	$0.280546\pi$
$709$	41.0911i	1.54321i	−0.636103	+	0.771605i	$0.719454\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	4.35710i	0.163175i
$714$	0	0
$715$	3.47174	0.129836
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−48.3260	−1.80225	−0.901127	−	0.433554i	$-0.857259\pi$
$719$	−48.3260	−1.80225	−0.901127	+	0.433554i	$0.857259\pi$
$720$	0	0
$721$	−23.4032	−0.871580
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−46.0462	−1.71011
$726$	0	0
$727$	− 26.1871i	− 0.971225i	−0.874174	−	0.485612i	$-0.838597\pi$
$727$	− 26.1871i	− 0.971225i	0.874174	−	0.485612i	$-0.161403\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 59.1214i	− 2.18668i
$732$	0	0
$733$	25.1869i	0.930300i	0.885232	+	0.465150i	$0.154000\pi$
$733$	25.1869i	0.930300i	−0.885232	+	0.465150i	$0.846000\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1.25350i	0.0461733i
$738$	0	0
$739$	−21.7993	−0.801900	−0.400950	−	0.916100i	$-0.631320\pi$
$739$	−21.7993	−0.801900	−0.400950	+	0.916100i	$0.631320\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	12.0393	0.441680	0.220840	−	0.975310i	$-0.429120\pi$
$743$	12.0393	0.441680	0.220840	+	0.975310i	$0.429120\pi$
$744$	0	0
$745$	−5.06871	−0.185703
$746$	0	0
$747$	0	0
$748$	0	0
$749$	16.3681	0.598077
$750$	0	0
$751$	7.16475i	0.261446i	0.991419	+	0.130723i	$0.0417298\pi$
$751$	7.16475i	0.261446i	−0.991419	+	0.130723i	$0.958270\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	13.8412i	0.503734i
$756$	0	0
$757$	2.57105i	0.0934464i	0.998908	+	0.0467232i	$0.0148779\pi$
$757$	2.57105i	0.0934464i	−0.998908	+	0.0467232i	$0.985122\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	47.1405i	1.70884i	0.519581	+	0.854421i	$0.326088\pi$
$761$	47.1405i	1.70884i	−0.519581	+	0.854421i	$0.673912\pi$
$762$	0	0
$763$	−9.32290	−0.337512
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−13.2860	−0.479731
$768$	0	0
$769$	54.3586	1.96022	0.980110	−	0.198456i	$-0.0635927\pi$
$769$	54.3586	1.96022	0.980110	+	0.198456i	$0.0635927\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	22.6180	0.813512	0.406756	−	0.913537i	$-0.366660\pi$
$773$	22.6180	0.813512	0.406756	+	0.913537i	$0.366660\pi$
$774$	0	0
$775$	− 2.75686i	− 0.0990295i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	16.7355i	0.599610i
$780$	0	0
$781$	− 48.6188i	− 1.73972i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	6.70238i	0.239218i
$786$	0	0
$787$	38.6165	1.37653	0.688264	−	0.725460i	$-0.258373\pi$
$787$	38.6165	1.37653	0.688264	+	0.725460i	$0.258373\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	23.2166	0.825488
$792$	0	0
$793$	−13.1575	−0.467238
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−32.2392	−1.14197	−0.570986	−	0.820960i	$-0.693439\pi$
$797$	−32.2392	−1.14197	−0.570986	+	0.820960i	$0.693439\pi$
$798$	0	0
$799$	− 1.20956i	− 0.0427913i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 12.8218i	− 0.452473i
$804$	0	0
$805$	8.98851i	0.316803i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	8.34642i	0.293445i	0.989178	+	0.146722i	$0.0468724\pi$
$809$	8.34642i	0.293445i	−0.989178	+	0.146722i	$0.953128\pi$
$810$	0	0
$811$	53.8372	1.89048	0.945241	−	0.326374i	$-0.105827\pi$
$811$	53.8372	1.89048	0.945241	+	0.326374i	$0.105827\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−6.10303	−0.213780
$816$	0	0
$817$	−27.1845	−0.951064
$818$	0	0
$819$	0	0
$820$	0	0
$821$	14.3765	0.501742	0.250871	−	0.968021i	$-0.419283\pi$
$821$	14.3765	0.501742	0.250871	+	0.968021i	$0.419283\pi$
$822$	0	0
$823$	36.1630i	1.26056i	0.776366	+	0.630282i	$0.217061\pi$
$823$	36.1630i	1.26056i	−0.776366	+	0.630282i	$0.782939\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 6.07633i	− 0.211295i	−0.994404	−	0.105647i	$-0.966309\pi$
$827$	− 6.07633i	− 0.211295i	0.994404	−	0.105647i	$-0.0336915\pi$
$828$	0	0
$829$	21.2242i	0.737146i	0.929599	+	0.368573i	$0.120154\pi$
$829$	21.2242i	0.737146i	−0.929599	+	0.368573i	$0.879846\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	26.5972i	0.921539i
$834$	0	0
$835$	−5.04635	−0.174636
$836$	0	0
$837$	0	0
$838$	0	0
$839$	34.8560	1.20336	0.601681	−	0.798736i	$-0.294498\pi$
$839$	34.8560	1.20336	0.601681	+	0.798736i	$0.294498\pi$
$840$	0	0
$841$	78.2057	2.69675
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0.743520	0.0255779
$846$	0	0
$847$	18.5806i	0.638436i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 4.84943i	− 0.166236i
$852$	0	0
$853$	29.9766i	1.02638i	0.858275	+	0.513190i	$0.171536\pi$
$853$	29.9766i	1.02638i	−0.858275	+	0.513190i	$0.828464\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 1.09615i	− 0.0374439i	−0.999825	−	0.0187220i	$-0.994040\pi$
$857$	− 1.09615i	− 0.0374439i	0.999825	−	0.0187220i	$-0.00595974\pi$
$858$	0	0
$859$	1.19251	0.0406879	0.0203440	−	0.999793i	$-0.493524\pi$
$859$	1.19251	0.0406879	0.0203440	+	0.999793i	$0.493524\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−8.53819	−0.290643	−0.145322	−	0.989384i	$-0.546422\pi$
$863$	−8.53819	−0.290643	−0.145322	+	0.989384i	$0.546422\pi$
$864$	0	0
$865$	−5.01365	−0.170469
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−51.8082	−1.75747
$870$	0	0
$871$	0.268454i	0.00909622i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 12.0816i	− 0.408432i
$876$	0	0
$877$	54.5232i	1.84112i	0.390606	+	0.920558i	$0.372266\pi$
$877$	54.5232i	1.84112i	−0.390606	+	0.920558i	$0.627734\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 24.2338i	− 0.816458i	−0.912879	−	0.408229i	$-0.866146\pi$
$881$	− 24.2338i	− 0.816458i	0.912879	−	0.408229i	$-0.133854\pi$
$882$	0	0
$883$	27.7907	0.935232	0.467616	−	0.883932i	$-0.345113\pi$
$883$	27.7907	0.935232	0.467616	+	0.883932i	$0.345113\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	37.5606	1.26116	0.630581	−	0.776123i	$-0.282817\pi$
$887$	37.5606	1.26116	0.630581	+	0.776123i	$0.282817\pi$
$888$	0	0
$889$	16.8446	0.564948
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−0.556166	−0.0186114
$894$	0	0
$895$	5.45183i	0.182235i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	6.41859i	0.214072i
$900$	0	0
$901$	− 70.4544i	− 2.34718i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 1.16825i	− 0.0388340i
$906$	0	0
$907$	26.8748	0.892362	0.446181	−	0.894943i	$-0.352784\pi$
$907$	26.8748	0.892362	0.446181	+	0.894943i	$0.352784\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	22.3775	0.741400	0.370700	−	0.928753i	$-0.379118\pi$
$911$	22.3775	0.741400	0.370700	+	0.928753i	$0.379118\pi$
$912$	0	0
$913$	−28.7530	−0.951585
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−3.69404	−0.121988
$918$	0	0
$919$	38.2831i	1.26284i	0.775439	+	0.631422i	$0.217528\pi$
$919$	38.2831i	1.26284i	−0.775439	+	0.631422i	$0.782472\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 10.4124i	− 0.342728i
$924$	0	0
$925$	3.06838i	0.100888i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 0.192447i	− 0.00631398i	−0.999995	−	0.00315699i	$-0.998995\pi$
$929$	− 0.192447i	− 0.00631398i	0.999995	−	0.00315699i	$-0.00100490\pi$
$930$	0	0
$931$	12.2296	0.400809
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−22.8471	−0.747179
$936$	0	0
$937$	−54.7439	−1.78841	−0.894203	−	0.447662i	$-0.852257\pi$
$937$	−54.7439	−1.78841	−0.894203	+	0.447662i	$0.852257\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	54.6373	1.78113	0.890563	−	0.454859i	$-0.150310\pi$
$941$	54.6373	1.78113	0.890563	+	0.454859i	$0.150310\pi$
$942$	0	0
$943$	38.8727i	1.26587i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	6.29630i	0.204602i	0.994753	+	0.102301i	$0.0326205\pi$
$947$	6.29630i	0.204602i	−0.994753	+	0.102301i	$0.967379\pi$
$948$	0	0
$949$	− 2.74597i	− 0.0891379i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 37.1157i	− 1.20229i	−0.799138	−	0.601147i	$-0.794710\pi$
$953$	− 37.1157i	− 1.20229i	0.799138	−	0.601147i	$-0.205290\pi$
$954$	0	0
$955$	−15.1213	−0.489312
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−16.2884	−0.525980
$960$	0	0
$961$	30.6157	0.987603
$962$	0	0
$963$	0	0
$964$	0	0
$965$	1.90468	0.0613139
$966$	0	0
$967$	− 10.8250i	− 0.348109i	−0.984736	−	0.174054i	$-0.944313\pi$
$967$	− 10.8250i	− 0.348109i	0.984736	−	0.174054i	$-0.0556869\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	16.8746i	0.541531i	0.962645	+	0.270765i	$0.0872768\pi$
$971$	16.8746i	0.541531i	−0.962645	+	0.270765i	$0.912723\pi$
$972$	0	0
$973$	5.04975i	0.161888i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 4.05555i	− 0.129749i	−0.997893	−	0.0648743i	$-0.979335\pi$
$977$	− 4.05555i	− 0.129749i	0.997893	−	0.0648743i	$-0.0206646\pi$
$978$	0	0
$979$	−61.2292	−1.95689
$980$	0	0
$981$	0	0
$982$	0	0
$983$	39.7911	1.26914	0.634569	−	0.772866i	$-0.281178\pi$
$983$	39.7911	1.26914	0.634569	+	0.772866i	$0.281178\pi$
$984$	0	0
$985$	−15.6253	−0.497865
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−63.1434	−2.00784
$990$	0	0
$991$	48.1323i	1.52897i	0.644640	+	0.764486i	$0.277007\pi$
$991$	48.1323i	1.52897i	−0.644640	+	0.764486i	$0.722993\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 14.9491i	− 0.473918i
$996$	0	0
$997$	− 30.0514i	− 0.951738i	−0.879516	−	0.475869i	$-0.842134\pi$
$997$	− 30.0514i	− 0.951738i	0.879516	−	0.475869i	$-0.157866\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3744.2.j.a.2159.18		48
3.2	odd	2	inner	3744.2.j.a.2159.32		48
4.3	odd	2		936.2.j.a.755.33	yes	48
8.3	odd	2	inner	3744.2.j.a.2159.31		48
8.5	even	2		936.2.j.a.755.15	✓	48
12.11	even	2		936.2.j.a.755.16	yes	48
24.5	odd	2		936.2.j.a.755.34	yes	48
24.11	even	2	inner	3744.2.j.a.2159.17		48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
936.2.j.a.755.15	✓	48	8.5	even	2
936.2.j.a.755.16	yes	48	12.11	even	2
936.2.j.a.755.33	yes	48	4.3	odd	2
936.2.j.a.755.34	yes	48	24.5	odd	2
3744.2.j.a.2159.17		48	24.11	even	2	inner
3744.2.j.a.2159.18		48	1.1	even	1	trivial
3744.2.j.a.2159.31		48	8.3	odd	2	inner
3744.2.j.a.2159.32		48	3.2	odd	2	inner

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:	\( N \)	\(=\)	\( 3744 = 2^{5} \cdot 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3744.j (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-0.743520 q^{5} -1.72000i q^{7} -4.66933i q^{11} -1.00000i q^{13} +6.58087i q^{17} +3.02593 q^{19} +7.02856 q^{23} -4.44718 q^{25} +10.3540 q^{29} +0.619913i q^{31} +1.27886i q^{35} -0.689961i q^{37} +5.53068i q^{41} -8.98383 q^{43} -0.183800 q^{47} +4.04160 q^{49} -10.7059 q^{53} +3.47174i q^{55} -13.2860i q^{59} -13.1575i q^{61} +0.743520i q^{65} -0.268454 q^{67} +10.4124 q^{71} +2.74597 q^{73} -8.03125 q^{77} -11.0954i q^{79} -6.15784i q^{83} -4.89301i q^{85} -13.1131i q^{89} -1.72000 q^{91} -2.24984 q^{95} -17.2096 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 32 q^{19} + 48 q^{25} + 32 q^{43} - 48 q^{49} - 32 q^{67} - 32 q^{97}+O(q^{100}) \) 48 * q - 32 * q^19 + 48 * q^25 + 32 * q^43 - 48 * q^49 - 32 * q^67 - 32 * q^97

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