LMFDB - Embedded newform 5760.2.a.cf.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5760,2,Mod(1,5760)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5760, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("5760.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5760 = 2^{7} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5760.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:	$45.9938315643$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.56155$ of defining polynomial
Character	$\chi$	$=$	5760.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{5} +3.12311 q^{7} -4.00000 q^{11} +7.12311 q^{13} +1.12311 q^{17} +1.12311 q^{19} -5.12311 q^{23} +1.00000 q^{25} +2.00000 q^{29} +3.12311 q^{31} +3.12311 q^{35} +3.12311 q^{37} -6.24621 q^{41} +4.00000 q^{43} +5.12311 q^{47} +2.75379 q^{49} +12.2462 q^{53} -4.00000 q^{55} -10.2462 q^{59} +6.00000 q^{61} +7.12311 q^{65} +8.00000 q^{67} +10.2462 q^{71} -8.24621 q^{73} -12.4924 q^{77} -15.1231 q^{79} +12.0000 q^{83} +1.12311 q^{85} -2.24621 q^{89} +22.2462 q^{91} +1.12311 q^{95} +8.24621 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{5} - 2 q^{7} - 8 q^{11} + 6 q^{13} - 6 q^{17} - 6 q^{19} - 2 q^{23} + 2 q^{25} + 4 q^{29} - 2 q^{31} - 2 q^{35} - 2 q^{37} + 4 q^{41} + 8 q^{43} + 2 q^{47} + 22 q^{49} + 8 q^{53} - 8 q^{55} - 4 q^{59}+ \cdots - 6 q^{95}+O(q^{100}) $ 2 * q + 2 * q^5 - 2 * q^7 - 8 * q^11 + 6 * q^13 - 6 * q^17 - 6 * q^19 - 2 * q^23 + 2 * q^25 + 4 * q^29 - 2 * q^31 - 2 * q^35 - 2 * q^37 + 4 * q^41 + 8 * q^43 + 2 * q^47 + 22 * q^49 + 8 * q^53 - 8 * q^55 - 4 * q^59 + 12 * q^61 + 6 * q^65 + 16 * q^67 + 4 * q^71 + 8 * q^77 - 22 * q^79 + 24 * q^83 - 6 * q^85 + 12 * q^89 + 28 * q^91 - 6 * q^95

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
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	3.12311	1.18042	0.590211	−	0.807249i	$-0.299044\pi$
$7$	3.12311	1.18042	0.590211	+	0.807249i	$0.299044\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−4.00000	−1.20605	−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−4.00000	−1.20605	−0.603023	+	0.797724i	$0.706037\pi$
$12$	0	0
$13$	7.12311	1.97559	0.987797	−	0.155747i	$-0.0497784\pi$
$13$	7.12311	1.97559	0.987797	+	0.155747i	$0.0497784\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1.12311	0.272393	0.136197	−	0.990682i	$-0.456512\pi$
$17$	1.12311	0.272393	0.136197	+	0.990682i	$0.456512\pi$
$18$	0	0
$19$	1.12311	0.257658	0.128829	−	0.991667i	$-0.458878\pi$
$19$	1.12311	0.257658	0.128829	+	0.991667i	$0.458878\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−5.12311	−1.06824	−0.534121	−	0.845408i	$-0.679357\pi$
$23$	−5.12311	−1.06824	−0.534121	+	0.845408i	$0.679357\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	0	0
$31$	3.12311	0.560926	0.280463	−	0.959865i	$-0.409512\pi$
$31$	3.12311	0.560926	0.280463	+	0.959865i	$0.409512\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	3.12311	0.527901
$36$	0	0
$37$	3.12311	0.513435	0.256718	−	0.966486i	$-0.417359\pi$
$37$	3.12311	0.513435	0.256718	+	0.966486i	$0.417359\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−6.24621	−0.975494	−0.487747	−	0.872985i	$-0.662181\pi$
$41$	−6.24621	−0.975494	−0.487747	+	0.872985i	$0.662181\pi$
$42$	0	0
$43$	4.00000	0.609994	0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000	0.609994	0.304997	+	0.952353i	$0.401344\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	5.12311	0.747282	0.373641	−	0.927573i	$-0.378109\pi$
$47$	5.12311	0.747282	0.373641	+	0.927573i	$0.378109\pi$
$48$	0	0
$49$	2.75379	0.393398
$50$	0	0
$51$	0	0
$52$	0	0
$53$	12.2462	1.68215	0.841073	−	0.540921i	$-0.181924\pi$
$53$	12.2462	1.68215	0.841073	+	0.540921i	$0.181924\pi$
$54$	0	0
$55$	−4.00000	−0.539360
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−10.2462	−1.33394	−0.666972	−	0.745083i	$-0.732410\pi$
$59$	−10.2462	−1.33394	−0.666972	+	0.745083i	$0.732410\pi$
$60$	0	0
$61$	6.00000	0.768221	0.384111	−	0.923287i	$-0.374508\pi$
$61$	6.00000	0.768221	0.384111	+	0.923287i	$0.374508\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	7.12311	0.883513
$66$	0	0
$67$	8.00000	0.977356	0.488678	−	0.872464i	$-0.337479\pi$
$67$	8.00000	0.977356	0.488678	+	0.872464i	$0.337479\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	10.2462	1.21600	0.608001	−	0.793936i	$-0.291972\pi$
$71$	10.2462	1.21600	0.608001	+	0.793936i	$0.291972\pi$
$72$	0	0
$73$	−8.24621	−0.965146	−0.482573	−	0.875856i	$-0.660298\pi$
$73$	−8.24621	−0.965146	−0.482573	+	0.875856i	$0.660298\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−12.4924	−1.42364
$78$	0	0
$79$	−15.1231	−1.70148	−0.850741	−	0.525585i	$-0.823847\pi$
$79$	−15.1231	−1.70148	−0.850741	+	0.525585i	$0.823847\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	12.0000	1.31717	0.658586	−	0.752506i	$-0.271155\pi$
$83$	12.0000	1.31717	0.658586	+	0.752506i	$0.271155\pi$
$84$	0	0
$85$	1.12311	0.121818
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−2.24621	−0.238098	−0.119049	−	0.992888i	$-0.537985\pi$
$89$	−2.24621	−0.238098	−0.119049	+	0.992888i	$0.537985\pi$
$90$	0	0
$91$	22.2462	2.33204
$92$	0	0
$93$	0	0
$94$	0	0
$95$	1.12311	0.115228
$96$	0	0
$97$	8.24621	0.837276	0.418638	−	0.908153i	$-0.362508\pi$
$97$	8.24621	0.837276	0.418638	+	0.908153i	$0.362508\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−14.0000	−1.39305	−0.696526	−	0.717532i	$-0.745272\pi$
$101$	−14.0000	−1.39305	−0.696526	+	0.717532i	$0.745272\pi$
$102$	0	0
$103$	−3.12311	−0.307729	−0.153864	−	0.988092i	$-0.549172\pi$
$103$	−3.12311	−0.307729	−0.153864	+	0.988092i	$0.549172\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	2.24621	0.217149	0.108575	−	0.994088i	$-0.465371\pi$
$107$	2.24621	0.217149	0.108575	+	0.994088i	$0.465371\pi$
$108$	0	0
$109$	−8.24621	−0.789844	−0.394922	−	0.918715i	$-0.629228\pi$
$109$	−8.24621	−0.789844	−0.394922	+	0.918715i	$0.629228\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	13.1231	1.23452	0.617259	−	0.786760i	$-0.288243\pi$
$113$	13.1231	1.23452	0.617259	+	0.786760i	$0.288243\pi$
$114$	0	0
$115$	−5.12311	−0.477732
$116$	0	0
$117$	0	0
$118$	0	0
$119$	3.50758	0.321539
$120$	0	0
$121$	5.00000	0.454545
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−19.1231	−1.69690	−0.848451	−	0.529275i	$-0.822464\pi$
$127$	−19.1231	−1.69690	−0.848451	+	0.529275i	$0.822464\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−2.24621	−0.196252	−0.0981262	−	0.995174i	$-0.531285\pi$
$131$	−2.24621	−0.196252	−0.0981262	+	0.995174i	$0.531285\pi$
$132$	0	0
$133$	3.50758	0.304146
$134$	0	0
$135$	0	0
$136$	0	0
$137$	11.3693	0.971346	0.485673	−	0.874140i	$-0.338575\pi$
$137$	11.3693	0.971346	0.485673	+	0.874140i	$0.338575\pi$
$138$	0	0
$139$	−5.12311	−0.434536	−0.217268	−	0.976112i	$-0.569715\pi$
$139$	−5.12311	−0.434536	−0.217268	+	0.976112i	$0.569715\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−28.4924	−2.38266
$144$	0	0
$145$	2.00000	0.166091
$146$	0	0
$147$	0	0
$148$	0	0
$149$	18.4924	1.51496	0.757479	−	0.652859i	$-0.226431\pi$
$149$	18.4924	1.51496	0.757479	+	0.652859i	$0.226431\pi$
$150$	0	0
$151$	−17.3693	−1.41349	−0.706747	−	0.707466i	$-0.749838\pi$
$151$	−17.3693	−1.41349	−0.706747	+	0.707466i	$0.749838\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	3.12311	0.250854
$156$	0	0
$157$	8.87689	0.708453	0.354227	−	0.935160i	$-0.384744\pi$
$157$	8.87689	0.708453	0.354227	+	0.935160i	$0.384744\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−16.0000	−1.26098
$162$	0	0
$163$	20.0000	1.56652	0.783260	−	0.621694i	$-0.213555\pi$
$163$	20.0000	1.56652	0.783260	+	0.621694i	$0.213555\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	25.6155	1.98219	0.991095	−	0.133160i	$-0.0425125\pi$
$167$	25.6155	1.98219	0.991095	+	0.133160i	$0.0425125\pi$
$168$	0	0
$169$	37.7386	2.90297
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−7.75379	−0.589510	−0.294755	−	0.955573i	$-0.595238\pi$
$173$	−7.75379	−0.589510	−0.294755	+	0.955573i	$0.595238\pi$
$174$	0	0
$175$	3.12311	0.236085
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−10.2462	−0.765838	−0.382919	−	0.923782i	$-0.625081\pi$
$179$	−10.2462	−0.765838	−0.382919	+	0.923782i	$0.625081\pi$
$180$	0	0
$181$	7.75379	0.576335	0.288167	−	0.957580i	$-0.406954\pi$
$181$	7.75379	0.576335	0.288167	+	0.957580i	$0.406954\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	3.12311	0.229615
$186$	0	0
$187$	−4.49242	−0.328518
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−8.00000	−0.578860	−0.289430	−	0.957199i	$-0.593466\pi$
$191$	−8.00000	−0.578860	−0.289430	+	0.957199i	$0.593466\pi$
$192$	0	0
$193$	18.4924	1.33111	0.665557	−	0.746347i	$-0.268194\pi$
$193$	18.4924	1.33111	0.665557	+	0.746347i	$0.268194\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−18.4924	−1.31753	−0.658765	−	0.752349i	$-0.728921\pi$
$197$	−18.4924	−1.31753	−0.658765	+	0.752349i	$0.728921\pi$
$198$	0	0
$199$	−8.87689	−0.629266	−0.314633	−	0.949213i	$-0.601882\pi$
$199$	−8.87689	−0.629266	−0.314633	+	0.949213i	$0.601882\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	6.24621	0.438398
$204$	0	0
$205$	−6.24621	−0.436254
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−4.49242	−0.310747
$210$	0	0
$211$	−25.6155	−1.76345	−0.881723	−	0.471768i	$-0.843616\pi$
$211$	−25.6155	−1.76345	−0.881723	+	0.471768i	$0.843616\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	4.00000	0.272798
$216$	0	0
$217$	9.75379	0.662130
$218$	0	0
$219$	0	0
$220$	0	0
$221$	8.00000	0.538138
$222$	0	0
$223$	−3.12311	−0.209139	−0.104569	−	0.994518i	$-0.533346\pi$
$223$	−3.12311	−0.209139	−0.104569	+	0.994518i	$0.533346\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	24.4924	1.62562	0.812810	−	0.582529i	$-0.197937\pi$
$227$	24.4924	1.62562	0.812810	+	0.582529i	$0.197937\pi$
$228$	0	0
$229$	22.4924	1.48634	0.743171	−	0.669102i	$-0.233321\pi$
$229$	22.4924	1.48634	0.743171	+	0.669102i	$0.233321\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−29.1231	−1.90792	−0.953959	−	0.299937i	$-0.903034\pi$
$233$	−29.1231	−1.90792	−0.953959	+	0.299937i	$0.903034\pi$
$234$	0	0
$235$	5.12311	0.334195
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−18.2462	−1.18025	−0.590125	−	0.807312i	$-0.700921\pi$
$239$	−18.2462	−1.18025	−0.590125	+	0.807312i	$0.700921\pi$
$240$	0	0
$241$	12.2462	0.788848	0.394424	−	0.918929i	$-0.370944\pi$
$241$	12.2462	0.788848	0.394424	+	0.918929i	$0.370944\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	2.75379	0.175933
$246$	0	0
$247$	8.00000	0.509028
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−4.00000	−0.252478	−0.126239	−	0.992000i	$-0.540291\pi$
$251$	−4.00000	−0.252478	−0.126239	+	0.992000i	$0.540291\pi$
$252$	0	0
$253$	20.4924	1.28835
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−10.8769	−0.678482	−0.339241	−	0.940699i	$-0.610170\pi$
$257$	−10.8769	−0.678482	−0.339241	+	0.940699i	$0.610170\pi$
$258$	0	0
$259$	9.75379	0.606071
$260$	0	0
$261$	0	0
$262$	0	0
$263$	7.36932	0.454412	0.227206	−	0.973847i	$-0.427041\pi$
$263$	7.36932	0.454412	0.227206	+	0.973847i	$0.427041\pi$
$264$	0	0
$265$	12.2462	0.752279
$266$	0	0
$267$	0	0
$268$	0	0
$269$	22.4924	1.37139	0.685694	−	0.727890i	$-0.259499\pi$
$269$	22.4924	1.37139	0.685694	+	0.727890i	$0.259499\pi$
$270$	0	0
$271$	−0.876894	−0.0532675	−0.0266338	−	0.999645i	$-0.508479\pi$
$271$	−0.876894	−0.0532675	−0.0266338	+	0.999645i	$0.508479\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−4.00000	−0.241209
$276$	0	0
$277$	12.8769	0.773698	0.386849	−	0.922143i	$-0.373564\pi$
$277$	12.8769	0.773698	0.386849	+	0.922143i	$0.373564\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	10.2462	0.611238	0.305619	−	0.952154i	$-0.401137\pi$
$281$	10.2462	0.611238	0.305619	+	0.952154i	$0.401137\pi$
$282$	0	0
$283$	0	0		−	1.00000i	$-0.5\pi$
$283$	0	0			1.00000i	$0.5\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−19.5076	−1.15150
$288$	0	0
$289$	−15.7386	−0.925802
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−8.24621	−0.481749	−0.240874	−	0.970556i	$-0.577434\pi$
$293$	−8.24621	−0.481749	−0.240874	+	0.970556i	$0.577434\pi$
$294$	0	0
$295$	−10.2462	−0.596557
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−36.4924	−2.11041
$300$	0	0
$301$	12.4924	0.720051
$302$	0	0
$303$	0	0
$304$	0	0
$305$	6.00000	0.343559
$306$	0	0
$307$	22.7386	1.29776	0.648881	−	0.760890i	$-0.275237\pi$
$307$	22.7386	1.29776	0.648881	+	0.760890i	$0.275237\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−14.7386	−0.835751	−0.417876	−	0.908504i	$-0.637225\pi$
$311$	−14.7386	−0.835751	−0.417876	+	0.908504i	$0.637225\pi$
$312$	0	0
$313$	22.0000	1.24351	0.621757	−	0.783210i	$-0.286419\pi$
$313$	22.0000	1.24351	0.621757	+	0.783210i	$0.286419\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	14.4924	0.813976	0.406988	−	0.913434i	$-0.366579\pi$
$317$	14.4924	0.813976	0.406988	+	0.913434i	$0.366579\pi$
$318$	0	0
$319$	−8.00000	−0.447914
$320$	0	0
$321$	0	0
$322$	0	0
$323$	1.26137	0.0701843
$324$	0	0
$325$	7.12311	0.395119
$326$	0	0
$327$	0	0
$328$	0	0
$329$	16.0000	0.882109
$330$	0	0
$331$	18.8769	1.03757	0.518784	−	0.854905i	$-0.326385\pi$
$331$	18.8769	1.03757	0.518784	+	0.854905i	$0.326385\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	8.00000	0.437087
$336$	0	0
$337$	−0.246211	−0.0134120	−0.00670599	−	0.999978i	$-0.502135\pi$
$337$	−0.246211	−0.0134120	−0.00670599	+	0.999978i	$0.502135\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−12.4924	−0.676503
$342$	0	0
$343$	−13.2614	−0.716046
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−9.75379	−0.523611	−0.261805	−	0.965121i	$-0.584318\pi$
$347$	−9.75379	−0.523611	−0.261805	+	0.965121i	$0.584318\pi$
$348$	0	0
$349$	1.50758	0.0806988	0.0403494	−	0.999186i	$-0.487153\pi$
$349$	1.50758	0.0806988	0.0403494	+	0.999186i	$0.487153\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−33.6155	−1.78917	−0.894587	−	0.446894i	$-0.852530\pi$
$353$	−33.6155	−1.78917	−0.894587	+	0.446894i	$0.852530\pi$
$354$	0	0
$355$	10.2462	0.543812
$356$	0	0
$357$	0	0
$358$	0	0
$359$	26.2462	1.38522	0.692611	−	0.721311i	$-0.256460\pi$
$359$	26.2462	1.38522	0.692611	+	0.721311i	$0.256460\pi$
$360$	0	0
$361$	−17.7386	−0.933612
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−8.24621	−0.431626
$366$	0	0
$367$	−25.3693	−1.32427	−0.662134	−	0.749386i	$-0.730349\pi$
$367$	−25.3693	−1.32427	−0.662134	+	0.749386i	$0.730349\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	38.2462	1.98564
$372$	0	0
$373$	−15.1231	−0.783045	−0.391522	−	0.920169i	$-0.628051\pi$
$373$	−15.1231	−0.783045	−0.391522	+	0.920169i	$0.628051\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	14.2462	0.733717
$378$	0	0
$379$	17.1231	0.879555	0.439777	−	0.898107i	$-0.355057\pi$
$379$	17.1231	0.879555	0.439777	+	0.898107i	$0.355057\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−0.630683	−0.0322264	−0.0161132	−	0.999870i	$-0.505129\pi$
$383$	−0.630683	−0.0322264	−0.0161132	+	0.999870i	$0.505129\pi$
$384$	0	0
$385$	−12.4924	−0.636673
$386$	0	0
$387$	0	0
$388$	0	0
$389$	1.50758	0.0764372	0.0382186	−	0.999269i	$-0.487832\pi$
$389$	1.50758	0.0764372	0.0382186	+	0.999269i	$0.487832\pi$
$390$	0	0
$391$	−5.75379	−0.290982
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−15.1231	−0.760926
$396$	0	0
$397$	3.12311	0.156744	0.0783721	−	0.996924i	$-0.475028\pi$
$397$	3.12311	0.156744	0.0783721	+	0.996924i	$0.475028\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	4.00000	0.199750	0.0998752	−	0.995000i	$-0.468156\pi$
$401$	4.00000	0.199750	0.0998752	+	0.995000i	$0.468156\pi$
$402$	0	0
$403$	22.2462	1.10816
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−12.4924	−0.619226
$408$	0	0
$409$	−26.0000	−1.28562	−0.642809	−	0.766027i	$-0.722231\pi$
$409$	−26.0000	−1.28562	−0.642809	+	0.766027i	$0.722231\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−32.0000	−1.57462
$414$	0	0
$415$	12.0000	0.589057
$416$	0	0
$417$	0	0
$418$	0	0
$419$	16.4924	0.805708	0.402854	−	0.915264i	$-0.368018\pi$
$419$	16.4924	0.805708	0.402854	+	0.915264i	$0.368018\pi$
$420$	0	0
$421$	2.00000	0.0974740	0.0487370	−	0.998812i	$-0.484480\pi$
$421$	2.00000	0.0974740	0.0487370	+	0.998812i	$0.484480\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	1.12311	0.0544786
$426$	0	0
$427$	18.7386	0.906826
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−34.2462	−1.64958	−0.824791	−	0.565438i	$-0.808707\pi$
$431$	−34.2462	−1.64958	−0.824791	+	0.565438i	$0.808707\pi$
$432$	0	0
$433$	12.2462	0.588515	0.294258	−	0.955726i	$-0.404928\pi$
$433$	12.2462	0.588515	0.294258	+	0.955726i	$0.404928\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−5.75379	−0.275241
$438$	0	0
$439$	7.12311	0.339967	0.169984	−	0.985447i	$-0.445629\pi$
$439$	7.12311	0.339967	0.169984	+	0.985447i	$0.445629\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−24.4924	−1.16367	−0.581835	−	0.813307i	$-0.697665\pi$
$443$	−24.4924	−1.16367	−0.581835	+	0.813307i	$0.697665\pi$
$444$	0	0
$445$	−2.24621	−0.106481
$446$	0	0
$447$	0	0
$448$	0	0
$449$	12.0000	0.566315	0.283158	−	0.959073i	$-0.408618\pi$
$449$	12.0000	0.566315	0.283158	+	0.959073i	$0.408618\pi$
$450$	0	0
$451$	24.9848	1.17649
$452$	0	0
$453$	0	0
$454$	0	0
$455$	22.2462	1.04292
$456$	0	0
$457$	−4.24621	−0.198629	−0.0993147	−	0.995056i	$-0.531665\pi$
$457$	−4.24621	−0.198629	−0.0993147	+	0.995056i	$0.531665\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	10.4924	0.488681	0.244340	−	0.969690i	$-0.421429\pi$
$461$	10.4924	0.488681	0.244340	+	0.969690i	$0.421429\pi$
$462$	0	0
$463$	17.3693	0.807221	0.403610	−	0.914931i	$-0.367755\pi$
$463$	17.3693	0.807221	0.403610	+	0.914931i	$0.367755\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	30.2462	1.39963	0.699814	−	0.714325i	$-0.253266\pi$
$467$	30.2462	1.39963	0.699814	+	0.714325i	$0.253266\pi$
$468$	0	0
$469$	24.9848	1.15369
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−16.0000	−0.735681
$474$	0	0
$475$	1.12311	0.0515316
$476$	0	0
$477$	0	0
$478$	0	0
$479$	42.2462	1.93028	0.965139	−	0.261737i	$-0.0842952\pi$
$479$	42.2462	1.93028	0.965139	+	0.261737i	$0.0842952\pi$
$480$	0	0
$481$	22.2462	1.01434
$482$	0	0
$483$	0	0
$484$	0	0
$485$	8.24621	0.374441
$486$	0	0
$487$	−4.87689	−0.220993	−0.110497	−	0.993877i	$-0.535244\pi$
$487$	−4.87689	−0.220993	−0.110497	+	0.993877i	$0.535244\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−2.24621	−0.101370	−0.0506850	−	0.998715i	$-0.516140\pi$
$491$	−2.24621	−0.101370	−0.0506850	+	0.998715i	$0.516140\pi$
$492$	0	0
$493$	2.24621	0.101164
$494$	0	0
$495$	0	0
$496$	0	0
$497$	32.0000	1.43540
$498$	0	0
$499$	23.3693	1.04615	0.523077	−	0.852285i	$-0.324784\pi$
$499$	23.3693	1.04615	0.523077	+	0.852285i	$0.324784\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	8.63068	0.384823	0.192412	−	0.981314i	$-0.438369\pi$
$503$	8.63068	0.384823	0.192412	+	0.981314i	$0.438369\pi$
$504$	0	0
$505$	−14.0000	−0.622992
$506$	0	0
$507$	0	0
$508$	0	0
$509$	26.0000	1.15243	0.576215	−	0.817298i	$-0.304529\pi$
$509$	26.0000	1.15243	0.576215	+	0.817298i	$0.304529\pi$
$510$	0	0
$511$	−25.7538	−1.13928
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−3.12311	−0.137620
$516$	0	0
$517$	−20.4924	−0.901256
$518$	0	0
$519$	0	0
$520$	0	0
$521$	28.9848	1.26985	0.634925	−	0.772574i	$-0.281031\pi$
$521$	28.9848	1.26985	0.634925	+	0.772574i	$0.281031\pi$
$522$	0	0
$523$	−38.2462	−1.67239	−0.836195	−	0.548432i	$-0.815225\pi$
$523$	−38.2462	−1.67239	−0.836195	+	0.548432i	$0.815225\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	3.50758	0.152792
$528$	0	0
$529$	3.24621	0.141140
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−44.4924	−1.92718
$534$	0	0
$535$	2.24621	0.0971122
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−11.0152	−0.474456
$540$	0	0
$541$	20.2462	0.870453	0.435226	−	0.900321i	$-0.356668\pi$
$541$	20.2462	0.870453	0.435226	+	0.900321i	$0.356668\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−8.24621	−0.353229
$546$	0	0
$547$	−28.9848	−1.23930	−0.619651	−	0.784877i	$-0.712726\pi$
$547$	−28.9848	−1.23930	−0.619651	+	0.784877i	$0.712726\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	2.24621	0.0956918
$552$	0	0
$553$	−47.2311	−2.00847
$554$	0	0
$555$	0	0
$556$	0	0
$557$	16.2462	0.688374	0.344187	−	0.938901i	$-0.388155\pi$
$557$	16.2462	0.688374	0.344187	+	0.938901i	$0.388155\pi$
$558$	0	0
$559$	28.4924	1.20510
$560$	0	0
$561$	0	0
$562$	0	0
$563$	34.7386	1.46406	0.732029	−	0.681273i	$-0.238573\pi$
$563$	34.7386	1.46406	0.732029	+	0.681273i	$0.238573\pi$
$564$	0	0
$565$	13.1231	0.552093
$566$	0	0
$567$	0	0
$568$	0	0
$569$	40.4924	1.69753	0.848765	−	0.528770i	$-0.177346\pi$
$569$	40.4924	1.69753	0.848765	+	0.528770i	$0.177346\pi$
$570$	0	0
$571$	−0.630683	−0.0263933	−0.0131966	−	0.999913i	$-0.504201\pi$
$571$	−0.630683	−0.0263933	−0.0131966	+	0.999913i	$0.504201\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−5.12311	−0.213648
$576$	0	0
$577$	11.7538	0.489317	0.244658	−	0.969609i	$-0.421324\pi$
$577$	11.7538	0.489317	0.244658	+	0.969609i	$0.421324\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	37.4773	1.55482
$582$	0	0
$583$	−48.9848	−2.02874
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−10.2462	−0.422906	−0.211453	−	0.977388i	$-0.567820\pi$
$587$	−10.2462	−0.422906	−0.211453	+	0.977388i	$0.567820\pi$
$588$	0	0
$589$	3.50758	0.144527
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−23.3693	−0.959663	−0.479831	−	0.877361i	$-0.659302\pi$
$593$	−23.3693	−0.959663	−0.479831	+	0.877361i	$0.659302\pi$
$594$	0	0
$595$	3.50758	0.143797
$596$	0	0
$597$	0	0
$598$	0	0
$599$	34.2462	1.39926	0.699631	−	0.714504i	$-0.253348\pi$
$599$	34.2462	1.39926	0.699631	+	0.714504i	$0.253348\pi$
$600$	0	0
$601$	34.4924	1.40698	0.703488	−	0.710708i	$-0.251625\pi$
$601$	34.4924	1.40698	0.703488	+	0.710708i	$0.251625\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	5.00000	0.203279
$606$	0	0
$607$	33.3693	1.35442	0.677209	−	0.735790i	$-0.263189\pi$
$607$	33.3693	1.35442	0.677209	+	0.735790i	$0.263189\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	36.4924	1.47633
$612$	0	0
$613$	−4.87689	−0.196976	−0.0984880	−	0.995138i	$-0.531401\pi$
$613$	−4.87689	−0.196976	−0.0984880	+	0.995138i	$0.531401\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−46.1080	−1.85624	−0.928118	−	0.372286i	$-0.878574\pi$
$617$	−46.1080	−1.85624	−0.928118	+	0.372286i	$0.878574\pi$
$618$	0	0
$619$	18.8769	0.758726	0.379363	−	0.925248i	$-0.376143\pi$
$619$	18.8769	0.758726	0.379363	+	0.925248i	$0.376143\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−7.01515	−0.281056
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	3.50758	0.139856
$630$	0	0
$631$	40.8769	1.62728	0.813642	−	0.581367i	$-0.197482\pi$
$631$	40.8769	1.62728	0.813642	+	0.581367i	$0.197482\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−19.1231	−0.758877
$636$	0	0
$637$	19.6155	0.777196
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−30.2462	−1.19465	−0.597327	−	0.801998i	$-0.703770\pi$
$641$	−30.2462	−1.19465	−0.597327	+	0.801998i	$0.703770\pi$
$642$	0	0
$643$	−24.4924	−0.965887	−0.482943	−	0.875652i	$-0.660432\pi$
$643$	−24.4924	−0.965887	−0.482943	+	0.875652i	$0.660432\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	6.38447	0.250999	0.125500	−	0.992094i	$-0.459947\pi$
$647$	6.38447	0.250999	0.125500	+	0.992094i	$0.459947\pi$
$648$	0	0
$649$	40.9848	1.60880
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−24.7386	−0.968098	−0.484049	−	0.875041i	$-0.660834\pi$
$653$	−24.7386	−0.968098	−0.484049	+	0.875041i	$0.660834\pi$
$654$	0	0
$655$	−2.24621	−0.0877667
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−38.7386	−1.50904	−0.754521	−	0.656275i	$-0.772131\pi$
$659$	−38.7386	−1.50904	−0.754521	+	0.656275i	$0.772131\pi$
$660$	0	0
$661$	−46.4924	−1.80835	−0.904173	−	0.427167i	$-0.859512\pi$
$661$	−46.4924	−1.80835	−0.904173	+	0.427167i	$0.859512\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	3.50758	0.136018
$666$	0	0
$667$	−10.2462	−0.396735
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−24.0000	−0.926510
$672$	0	0
$673$	28.2462	1.08881	0.544406	−	0.838822i	$-0.316755\pi$
$673$	28.2462	1.08881	0.544406	+	0.838822i	$0.316755\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	40.2462	1.54679	0.773394	−	0.633926i	$-0.218558\pi$
$677$	40.2462	1.54679	0.773394	+	0.633926i	$0.218558\pi$
$678$	0	0
$679$	25.7538	0.988340
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−10.7386	−0.410902	−0.205451	−	0.978667i	$-0.565866\pi$
$683$	−10.7386	−0.410902	−0.205451	+	0.978667i	$0.565866\pi$
$684$	0	0
$685$	11.3693	0.434399
$686$	0	0
$687$	0	0
$688$	0	0
$689$	87.2311	3.32324
$690$	0	0
$691$	−34.1080	−1.29753	−0.648764	−	0.760990i	$-0.724714\pi$
$691$	−34.1080	−1.29753	−0.648764	+	0.760990i	$0.724714\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−5.12311	−0.194330
$696$	0	0
$697$	−7.01515	−0.265718
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−38.9848	−1.47244	−0.736219	−	0.676744i	$-0.763390\pi$
$701$	−38.9848	−1.47244	−0.736219	+	0.676744i	$0.763390\pi$
$702$	0	0
$703$	3.50758	0.132291
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−43.7235	−1.64439
$708$	0	0
$709$	18.0000	0.676004	0.338002	−	0.941145i	$-0.390249\pi$
$709$	18.0000	0.676004	0.338002	+	0.941145i	$0.390249\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−16.0000	−0.599205
$714$	0	0
$715$	−28.4924	−1.06556
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−16.0000	−0.596699	−0.298350	−	0.954457i	$-0.596436\pi$
$719$	−16.0000	−0.596699	−0.298350	+	0.954457i	$0.596436\pi$
$720$	0	0
$721$	−9.75379	−0.363250
$722$	0	0
$723$	0	0
$724$	0	0
$725$	2.00000	0.0742781
$726$	0	0
$727$	−17.3693	−0.644192	−0.322096	−	0.946707i	$-0.604387\pi$
$727$	−17.3693	−0.644192	−0.322096	+	0.946707i	$0.604387\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	4.49242	0.166158
$732$	0	0
$733$	−11.1231	−0.410841	−0.205421	−	0.978674i	$-0.565856\pi$
$733$	−11.1231	−0.410841	−0.205421	+	0.978674i	$0.565856\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−32.0000	−1.17874
$738$	0	0
$739$	−41.6155	−1.53085	−0.765426	−	0.643524i	$-0.777472\pi$
$739$	−41.6155	−1.53085	−0.765426	+	0.643524i	$0.777472\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−10.8769	−0.399035	−0.199517	−	0.979894i	$-0.563937\pi$
$743$	−10.8769	−0.399035	−0.199517	+	0.979894i	$0.563937\pi$
$744$	0	0
$745$	18.4924	0.677510
$746$	0	0
$747$	0	0
$748$	0	0
$749$	7.01515	0.256328
$750$	0	0
$751$	−37.8617	−1.38159	−0.690797	−	0.723049i	$-0.742740\pi$
$751$	−37.8617	−1.38159	−0.690797	+	0.723049i	$0.742740\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−17.3693	−0.632134
$756$	0	0
$757$	−3.12311	−0.113511	−0.0567556	−	0.998388i	$-0.518076\pi$
$757$	−3.12311	−0.113511	−0.0567556	+	0.998388i	$0.518076\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−28.9848	−1.05070	−0.525350	−	0.850886i	$-0.676066\pi$
$761$	−28.9848	−1.05070	−0.525350	+	0.850886i	$0.676066\pi$
$762$	0	0
$763$	−25.7538	−0.932350
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−72.9848	−2.63533
$768$	0	0
$769$	−44.2462	−1.59556	−0.797780	−	0.602949i	$-0.793992\pi$
$769$	−44.2462	−1.59556	−0.797780	+	0.602949i	$0.793992\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−9.50758	−0.341964	−0.170982	−	0.985274i	$-0.554694\pi$
$773$	−9.50758	−0.341964	−0.170982	+	0.985274i	$0.554694\pi$
$774$	0	0
$775$	3.12311	0.112185
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−7.01515	−0.251344
$780$	0	0
$781$	−40.9848	−1.46655
$782$	0	0
$783$	0	0
$784$	0	0
$785$	8.87689	0.316830
$786$	0	0
$787$	−9.75379	−0.347685	−0.173843	−	0.984773i	$-0.555618\pi$
$787$	−9.75379	−0.347685	−0.173843	+	0.984773i	$0.555618\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	40.9848	1.45725
$792$	0	0
$793$	42.7386	1.51769
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−18.0000	−0.637593	−0.318796	−	0.947823i	$-0.603279\pi$
$797$	−18.0000	−0.637593	−0.318796	+	0.947823i	$0.603279\pi$
$798$	0	0
$799$	5.75379	0.203554
$800$	0	0
$801$	0	0
$802$	0	0
$803$	32.9848	1.16401
$804$	0	0
$805$	−16.0000	−0.563926
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−14.2462	−0.500870	−0.250435	−	0.968133i	$-0.580574\pi$
$809$	−14.2462	−0.500870	−0.250435	+	0.968133i	$0.580574\pi$
$810$	0	0
$811$	−26.1080	−0.916774	−0.458387	−	0.888753i	$-0.651573\pi$
$811$	−26.1080	−0.916774	−0.458387	+	0.888753i	$0.651573\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	20.0000	0.700569
$816$	0	0
$817$	4.49242	0.157170
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−10.4924	−0.366188	−0.183094	−	0.983095i	$-0.558611\pi$
$821$	−10.4924	−0.366188	−0.183094	+	0.983095i	$0.558611\pi$
$822$	0	0
$823$	−25.3693	−0.884319	−0.442159	−	0.896936i	$-0.645787\pi$
$823$	−25.3693	−0.884319	−0.442159	+	0.896936i	$0.645787\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−16.9848	−0.590621	−0.295310	−	0.955401i	$-0.595423\pi$
$827$	−16.9848	−0.590621	−0.295310	+	0.955401i	$0.595423\pi$
$828$	0	0
$829$	−28.7386	−0.998134	−0.499067	−	0.866563i	$-0.666324\pi$
$829$	−28.7386	−0.998134	−0.499067	+	0.866563i	$0.666324\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	3.09280	0.107159
$834$	0	0
$835$	25.6155	0.886462
$836$	0	0
$837$	0	0
$838$	0	0
$839$	46.7386	1.61360	0.806798	−	0.590827i	$-0.201198\pi$
$839$	46.7386	1.61360	0.806798	+	0.590827i	$0.201198\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	0	0
$843$	0	0
$844$	0	0
$845$	37.7386	1.29825
$846$	0	0
$847$	15.6155	0.536556
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−16.0000	−0.548473
$852$	0	0
$853$	−13.8617	−0.474617	−0.237308	−	0.971434i	$-0.576265\pi$
$853$	−13.8617	−0.474617	−0.237308	+	0.971434i	$0.576265\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−11.3693	−0.388368	−0.194184	−	0.980965i	$-0.562206\pi$
$857$	−11.3693	−0.388368	−0.194184	+	0.980965i	$0.562206\pi$
$858$	0	0
$859$	−27.3693	−0.933829	−0.466915	−	0.884302i	$-0.654634\pi$
$859$	−27.3693	−0.933829	−0.466915	+	0.884302i	$0.654634\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−47.3693	−1.61247	−0.806235	−	0.591595i	$-0.798498\pi$
$863$	−47.3693	−1.61247	−0.806235	+	0.591595i	$0.798498\pi$
$864$	0	0
$865$	−7.75379	−0.263637
$866$	0	0
$867$	0	0
$868$	0	0
$869$	60.4924	2.05206
$870$	0	0
$871$	56.9848	1.93086
$872$	0	0
$873$	0	0
$874$	0	0
$875$	3.12311	0.105580
$876$	0	0
$877$	−47.1231	−1.59123	−0.795617	−	0.605800i	$-0.792853\pi$
$877$	−47.1231	−1.59123	−0.795617	+	0.605800i	$0.792853\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	42.2462	1.42331	0.711656	−	0.702529i	$-0.247946\pi$
$881$	42.2462	1.42331	0.711656	+	0.702529i	$0.247946\pi$
$882$	0	0
$883$	36.0000	1.21150	0.605748	−	0.795656i	$-0.292874\pi$
$883$	36.0000	1.21150	0.605748	+	0.795656i	$0.292874\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−2.87689	−0.0965967	−0.0482983	−	0.998833i	$-0.515380\pi$
$887$	−2.87689	−0.0965967	−0.0482983	+	0.998833i	$0.515380\pi$
$888$	0	0
$889$	−59.7235	−2.00306
$890$	0	0
$891$	0	0
$892$	0	0
$893$	5.75379	0.192543
$894$	0	0
$895$	−10.2462	−0.342493
$896$	0	0
$897$	0	0
$898$	0	0
$899$	6.24621	0.208323
$900$	0	0
$901$	13.7538	0.458205
$902$	0	0
$903$	0	0
$904$	0	0
$905$	7.75379	0.257745
$906$	0	0
$907$	54.2462	1.80122	0.900608	−	0.434632i	$-0.143122\pi$
$907$	54.2462	1.80122	0.900608	+	0.434632i	$0.143122\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−13.7538	−0.455683	−0.227842	−	0.973698i	$-0.573167\pi$
$911$	−13.7538	−0.455683	−0.227842	+	0.973698i	$0.573167\pi$
$912$	0	0
$913$	−48.0000	−1.58857
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−7.01515	−0.231661
$918$	0	0
$919$	−37.8617	−1.24894	−0.624472	−	0.781047i	$-0.714686\pi$
$919$	−37.8617	−1.24894	−0.624472	+	0.781047i	$0.714686\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	72.9848	2.40233
$924$	0	0
$925$	3.12311	0.102687
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−23.5076	−0.771259	−0.385629	−	0.922654i	$-0.626016\pi$
$929$	−23.5076	−0.771259	−0.385629	+	0.922654i	$0.626016\pi$
$930$	0	0
$931$	3.09280	0.101362
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−4.49242	−0.146918
$936$	0	0
$937$	14.4924	0.473447	0.236723	−	0.971577i	$-0.423926\pi$
$937$	14.4924	0.473447	0.236723	+	0.971577i	$0.423926\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−42.0000	−1.36916	−0.684580	−	0.728937i	$-0.740015\pi$
$941$	−42.0000	−1.36916	−0.684580	+	0.728937i	$0.740015\pi$
$942$	0	0
$943$	32.0000	1.04206
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−2.24621	−0.0729921	−0.0364960	−	0.999334i	$-0.511620\pi$
$947$	−2.24621	−0.0729921	−0.0364960	+	0.999334i	$0.511620\pi$
$948$	0	0
$949$	−58.7386	−1.90674
$950$	0	0
$951$	0	0
$952$	0	0
$953$	14.8769	0.481910	0.240955	−	0.970536i	$-0.422539\pi$
$953$	14.8769	0.481910	0.240955	+	0.970536i	$0.422539\pi$
$954$	0	0
$955$	−8.00000	−0.258874
$956$	0	0
$957$	0	0
$958$	0	0
$959$	35.5076	1.14660
$960$	0	0
$961$	−21.2462	−0.685362
$962$	0	0
$963$	0	0
$964$	0	0
$965$	18.4924	0.595292
$966$	0	0
$967$	20.8769	0.671356	0.335678	−	0.941977i	$-0.391035\pi$
$967$	20.8769	0.671356	0.335678	+	0.941977i	$0.391035\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	20.9848	0.673436	0.336718	−	0.941606i	$-0.390683\pi$
$971$	20.9848	0.673436	0.336718	+	0.941606i	$0.390683\pi$
$972$	0	0
$973$	−16.0000	−0.512936
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−15.8617	−0.507462	−0.253731	−	0.967275i	$-0.581658\pi$
$977$	−15.8617	−0.507462	−0.253731	+	0.967275i	$0.581658\pi$
$978$	0	0
$979$	8.98485	0.287157
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−18.8769	−0.602079	−0.301040	−	0.953612i	$-0.597334\pi$
$983$	−18.8769	−0.602079	−0.301040	+	0.953612i	$0.597334\pi$
$984$	0	0
$985$	−18.4924	−0.589218
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−20.4924	−0.651621
$990$	0	0
$991$	−27.6155	−0.877236	−0.438618	−	0.898674i	$-0.644532\pi$
$991$	−27.6155	−0.877236	−0.438618	+	0.898674i	$0.644532\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−8.87689	−0.281416
$996$	0	0
$997$	−2.63068	−0.0833146	−0.0416573	−	0.999132i	$-0.513264\pi$
$997$	−2.63068	−0.0833146	−0.0416573	+	0.999132i	$0.513264\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5760.2.a.cf.1.2	yes	2
3.2	odd	2		5760.2.a.bz.1.2	yes	2
4.3	odd	2		5760.2.a.cl.1.1	yes	2
8.3	odd	2		5760.2.a.ca.1.1	yes	2
8.5	even	2		5760.2.a.by.1.2	✓	2
12.11	even	2		5760.2.a.cb.1.1	yes	2
24.5	odd	2		5760.2.a.ce.1.2	yes	2
24.11	even	2		5760.2.a.ck.1.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5760.2.a.by.1.2	✓	2	8.5	even	2
5760.2.a.bz.1.2	yes	2	3.2	odd	2
5760.2.a.ca.1.1	yes	2	8.3	odd	2
5760.2.a.cb.1.1	yes	2	12.11	even	2
5760.2.a.ce.1.2	yes	2	24.5	odd	2
5760.2.a.cf.1.2	yes	2	1.1	even	1	trivial
5760.2.a.ck.1.1	yes	2	24.11	even	2
5760.2.a.cl.1.1	yes	2	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 5760 = 2^{7} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5760.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{5} +3.12311 q^{7} -4.00000 q^{11} +7.12311 q^{13} +1.12311 q^{17} +1.12311 q^{19} -5.12311 q^{23} +1.00000 q^{25} +2.00000 q^{29} +3.12311 q^{31} +3.12311 q^{35} +3.12311 q^{37} -6.24621 q^{41} +4.00000 q^{43} +5.12311 q^{47} +2.75379 q^{49} +12.2462 q^{53} -4.00000 q^{55} -10.2462 q^{59} +6.00000 q^{61} +7.12311 q^{65} +8.00000 q^{67} +10.2462 q^{71} -8.24621 q^{73} -12.4924 q^{77} -15.1231 q^{79} +12.0000 q^{83} +1.12311 q^{85} -2.24621 q^{89} +22.2462 q^{91} +1.12311 q^{95} +8.24621 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{5} - 2 q^{7} - 8 q^{11} + 6 q^{13} - 6 q^{17} - 6 q^{19} - 2 q^{23} + 2 q^{25} + 4 q^{29} - 2 q^{31} - 2 q^{35} - 2 q^{37} + 4 q^{41} + 8 q^{43} + 2 q^{47} + 22 q^{49} + 8 q^{53} - 8 q^{55} - 4 q^{59}+ \cdots - 6 q^{95}+O(q^{100}) \) 2 * q + 2 * q^5 - 2 * q^7 - 8 * q^11 + 6 * q^13 - 6 * q^17 - 6 * q^19 - 2 * q^23 + 2 * q^25 + 4 * q^29 - 2 * q^31 - 2 * q^35 - 2 * q^37 + 4 * q^41 + 8 * q^43 + 2 * q^47 + 22 * q^49 + 8 * q^53 - 8 * q^55 - 4 * q^59 + 12 * q^61 + 6 * q^65 + 16 * q^67 + 4 * q^71 + 8 * q^77 - 22 * q^79 + 24 * q^83 - 6 * q^85 + 12 * q^89 + 28 * q^91 - 6 * q^95

Introduction

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