LMFDB - Embedded newform 4680.2.a.bh.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4680,2,Mod(1,4680)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4680, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("4680.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4680 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4680.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:	$37.3699881460$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.940.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 7x - 4 $ x^3 - 7*x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1560)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.602705$ of defining polynomial
Character	$\chi$	$=$	4680.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{5} +3.43134 q^{7} +O(q^{10})$	$q-1.00000 q^{5} +3.43134 q^{7} +2.63675 q^{11} -1.00000 q^{13} -7.84216 q^{17} +0.794590 q^{19} +3.43134 q^{23} +1.00000 q^{25} +4.06808 q^{29} +9.27349 q^{31} -3.43134 q^{35} -0.636747 q^{37} +4.63675 q^{41} +2.41082 q^{43} -5.27349 q^{47} +4.77407 q^{49} -11.4994 q^{53} -2.63675 q^{55} +12.1362 q^{59} -11.4994 q^{61} +1.00000 q^{65} -6.41082 q^{67} +10.6367 q^{71} +16.0681 q^{73} +9.04757 q^{77} +10.6367 q^{79} +16.1362 q^{83} +7.84216 q^{85} -8.63675 q^{89} -3.43134 q^{91} -0.794590 q^{95} +12.2940 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{5} + q^{7}+O(q^{10}) $ 3 * q - 3 * q^5 + q^7	$ 3 q - 3 q^{5} + q^{7} - 5 q^{11} - 3 q^{13} - 7 q^{17} + 6 q^{19} + q^{23} + 3 q^{25} - 10 q^{29} + 2 q^{31} - q^{35} + 11 q^{37} + q^{41} + 10 q^{47} + 20 q^{49} - 3 q^{53} + 5 q^{55} - 8 q^{59} - 3 q^{61} + 3 q^{65} - 12 q^{67} + 19 q^{71} + 26 q^{73} + 7 q^{77} + 19 q^{79} + 4 q^{83} + 7 q^{85} - 13 q^{89} - q^{91} - 6 q^{95} + 9 q^{97}+O(q^{100}) $ 3 * q - 3 * q^5 + q^7 - 5 * q^11 - 3 * q^13 - 7 * q^17 + 6 * q^19 + q^23 + 3 * q^25 - 10 * q^29 + 2 * q^31 - q^35 + 11 * q^37 + q^41 + 10 * q^47 + 20 * q^49 - 3 * q^53 + 5 * q^55 - 8 * q^59 - 3 * q^61 + 3 * q^65 - 12 * q^67 + 19 * q^71 + 26 * q^73 + 7 * q^77 + 19 * q^79 + 4 * q^83 + 7 * q^85 - 13 * q^89 - q^91 - 6 * q^95 + 9 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	3.43134	1.29692	0.648462	−	0.761247i	$-0.275413\pi$
$7$	3.43134	1.29692	0.648462	+	0.761247i	$0.275413\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	2.63675	0.795009	0.397505	−	0.917600i	$-0.369876\pi$
$11$	2.63675	0.795009	0.397505	+	0.917600i	$0.369876\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−7.84216	−1.90200	−0.951001	−	0.309187i	$-0.899943\pi$
$17$	−7.84216	−1.90200	−0.951001	+	0.309187i	$0.899943\pi$
$18$	0	0
$19$	0.794590	0.182291	0.0911457	−	0.995838i	$-0.470947\pi$
$19$	0.794590	0.182291	0.0911457	+	0.995838i	$0.470947\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	3.43134	0.715483	0.357742	−	0.933821i	$-0.383547\pi$
$23$	3.43134	0.715483	0.357742	+	0.933821i	$0.383547\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	4.06808	0.755424	0.377712	−	0.925923i	$-0.376711\pi$
$29$	4.06808	0.755424	0.377712	+	0.925923i	$0.376711\pi$
$30$	0	0
$31$	9.27349	1.66557	0.832784	−	0.553598i	$-0.186745\pi$
$31$	9.27349	1.66557	0.832784	+	0.553598i	$0.186745\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−3.43134	−0.580002
$36$	0	0
$37$	−0.636747	−0.104681	−0.0523403	−	0.998629i	$-0.516668\pi$
$37$	−0.636747	−0.104681	−0.0523403	+	0.998629i	$0.516668\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	4.63675	0.724138	0.362069	−	0.932151i	$-0.382070\pi$
$41$	4.63675	0.724138	0.362069	+	0.932151i	$0.382070\pi$
$42$	0	0
$43$	2.41082	0.367647	0.183823	−	0.982959i	$-0.441153\pi$
$43$	2.41082	0.367647	0.183823	+	0.982959i	$0.441153\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−5.27349	−0.769218	−0.384609	−	0.923080i	$-0.625664\pi$
$47$	−5.27349	−0.769218	−0.384609	+	0.923080i	$0.625664\pi$
$48$	0	0
$49$	4.77407	0.682010
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−11.4994	−1.57957	−0.789783	−	0.613386i	$-0.789807\pi$
$53$	−11.4994	−1.57957	−0.789783	+	0.613386i	$0.789807\pi$
$54$	0	0
$55$	−2.63675	−0.355539
$56$	0	0
$57$	0	0
$58$	0	0
$59$	12.1362	1.57999	0.789997	−	0.613110i	$-0.210082\pi$
$59$	12.1362	1.57999	0.789997	+	0.613110i	$0.210082\pi$
$60$	0	0
$61$	−11.4994	−1.47235	−0.736175	−	0.676791i	$-0.763370\pi$
$61$	−11.4994	−1.47235	−0.736175	+	0.676791i	$0.763370\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	1.00000	0.124035
$66$	0	0
$67$	−6.41082	−0.783206	−0.391603	−	0.920134i	$-0.628079\pi$
$67$	−6.41082	−0.783206	−0.391603	+	0.920134i	$0.628079\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	10.6367	1.26235	0.631175	−	0.775641i	$-0.282573\pi$
$71$	10.6367	1.26235	0.631175	+	0.775641i	$0.282573\pi$
$72$	0	0
$73$	16.0681	1.88063	0.940313	−	0.340310i	$-0.110532\pi$
$73$	16.0681	1.88063	0.940313	+	0.340310i	$0.110532\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	9.04757	1.03107
$78$	0	0
$79$	10.6367	1.19673	0.598364	−	0.801225i	$-0.295818\pi$
$79$	10.6367	1.19673	0.598364	+	0.801225i	$0.295818\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	16.1362	1.77117	0.885587	−	0.464473i	$-0.153756\pi$
$83$	16.1362	1.77117	0.885587	+	0.464473i	$0.153756\pi$
$84$	0	0
$85$	7.84216	0.850601
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−8.63675	−0.915493	−0.457747	−	0.889083i	$-0.651343\pi$
$89$	−8.63675	−0.915493	−0.457747	+	0.889083i	$0.651343\pi$
$90$	0	0
$91$	−3.43134	−0.359702
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−0.794590	−0.0815232
$96$	0	0
$97$	12.2940	1.24827	0.624134	−	0.781317i	$-0.285452\pi$
$97$	12.2940	1.24827	0.624134	+	0.781317i	$0.285452\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−5.34158	−0.531507	−0.265753	−	0.964041i	$-0.585621\pi$
$101$	−5.34158	−0.531507	−0.265753	+	0.964041i	$0.585621\pi$
$102$	0	0
$103$	−10.4108	−1.02581	−0.512904	−	0.858446i	$-0.671430\pi$
$103$	−10.4108	−1.02581	−0.512904	+	0.858446i	$0.671430\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−5.04757	−0.487967	−0.243983	−	0.969779i	$-0.578454\pi$
$107$	−5.04757	−0.487967	−0.243983	+	0.969779i	$0.578454\pi$
$108$	0	0
$109$	6.79459	0.650804	0.325402	−	0.945576i	$-0.394500\pi$
$109$	6.79459	0.650804	0.325402	+	0.945576i	$0.394500\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	8.06808	0.758981	0.379491	−	0.925196i	$-0.376099\pi$
$113$	8.06808	0.758981	0.379491	+	0.925196i	$0.376099\pi$
$114$	0	0
$115$	−3.43134	−0.319974
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−26.9091	−2.46675
$120$	0	0
$121$	−4.04757	−0.367961
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−15.6843	−1.39176	−0.695879	−	0.718159i	$-0.744985\pi$
$127$	−15.6843	−1.39176	−0.695879	+	0.718159i	$0.744985\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−12.9308	−1.12977	−0.564883	−	0.825171i	$-0.691079\pi$
$131$	−12.9308	−1.12977	−0.564883	+	0.825171i	$0.691079\pi$
$132$	0	0
$133$	2.72651	0.236418
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−2.00000	−0.170872	−0.0854358	−	0.996344i	$-0.527228\pi$
$137$	−2.00000	−0.170872	−0.0854358	+	0.996344i	$0.527228\pi$
$138$	0	0
$139$	19.9102	1.68876	0.844382	−	0.535741i	$-0.179968\pi$
$139$	19.9102	1.68876	0.844382	+	0.535741i	$0.179968\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−2.63675	−0.220496
$144$	0	0
$145$	−4.06808	−0.337836
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−8.63675	−0.707550	−0.353775	−	0.935331i	$-0.615102\pi$
$149$	−8.63675	−0.707550	−0.353775	+	0.935331i	$0.615102\pi$
$150$	0	0
$151$	−17.7253	−1.44247	−0.721234	−	0.692691i	$-0.756425\pi$
$151$	−17.7253	−1.44247	−0.721234	+	0.692691i	$0.756425\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−9.27349	−0.744865
$156$	0	0
$157$	20.5470	1.63983	0.819914	−	0.572487i	$-0.194021\pi$
$157$	20.5470	1.63983	0.819914	+	0.572487i	$0.194021\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	11.7741	0.927927
$162$	0	0
$163$	−4.22593	−0.331000	−0.165500	−	0.986210i	$-0.552924\pi$
$163$	−4.22593	−0.331000	−0.165500	+	0.986210i	$0.552924\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	14.4108	1.11514	0.557571	−	0.830129i	$-0.311733\pi$
$167$	14.4108	1.11514	0.557571	+	0.830129i	$0.311733\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	0	0
$172$	0	0
$173$	6.00000	0.456172	0.228086	−	0.973641i	$-0.426753\pi$
$173$	6.00000	0.456172	0.228086	+	0.973641i	$0.426753\pi$
$174$	0	0
$175$	3.43134	0.259385
$176$	0	0
$177$	0	0
$178$	0	0
$179$	20.9308	1.56444	0.782219	−	0.623003i	$-0.214088\pi$
$179$	20.9308	1.56444	0.782219	+	0.623003i	$0.214088\pi$
$180$	0	0
$181$	−5.08860	−0.378233	−0.189116	−	0.981955i	$-0.560562\pi$
$181$	−5.08860	−0.378233	−0.189116	+	0.981955i	$0.560562\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0.636747	0.0468146
$186$	0	0
$187$	−20.6778	−1.51211
$188$	0	0
$189$	0	0
$190$	0	0
$191$	13.7253	0.993131	0.496566	−	0.867999i	$-0.334594\pi$
$191$	13.7253	0.993131	0.496566	+	0.867999i	$0.334594\pi$
$192$	0	0
$193$	−5.11565	−0.368233	−0.184116	−	0.982904i	$-0.558942\pi$
$193$	−5.11565	−0.368233	−0.184116	+	0.982904i	$0.558942\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	20.9988	1.49611	0.748053	−	0.663639i	$-0.230989\pi$
$197$	20.9988	1.49611	0.748053	+	0.663639i	$0.230989\pi$
$198$	0	0
$199$	9.58918	0.679759	0.339879	−	0.940469i	$-0.389614\pi$
$199$	9.58918	0.679759	0.339879	+	0.940469i	$0.389614\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	13.9590	0.979727
$204$	0	0
$205$	−4.63675	−0.323844
$206$	0	0
$207$	0	0
$208$	0	0
$209$	2.09513	0.144923
$210$	0	0
$211$	−8.82164	−0.607307	−0.303653	−	0.952783i	$-0.598206\pi$
$211$	−8.82164	−0.607307	−0.303653	+	0.952783i	$0.598206\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−2.41082	−0.164417
$216$	0	0
$217$	31.8205	2.16011
$218$	0	0
$219$	0	0
$220$	0	0
$221$	7.84216	0.527521
$222$	0	0
$223$	−9.93192	−0.665090	−0.332545	−	0.943087i	$-0.607907\pi$
$223$	−9.93192	−0.665090	−0.332545	+	0.943087i	$0.607907\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−9.27349	−0.615503	−0.307752	−	0.951467i	$-0.599577\pi$
$227$	−9.27349	−0.615503	−0.307752	+	0.951467i	$0.599577\pi$
$228$	0	0
$229$	12.0681	0.797481	0.398741	−	0.917064i	$-0.369447\pi$
$229$	12.0681	0.797481	0.398741	+	0.917064i	$0.369447\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−13.1157	−0.859235	−0.429617	−	0.903011i	$-0.641352\pi$
$233$	−13.1157	−0.859235	−0.429617	+	0.903011i	$0.641352\pi$
$234$	0	0
$235$	5.27349	0.344005
$236$	0	0
$237$	0	0
$238$	0	0
$239$	22.7729	1.47306	0.736529	−	0.676406i	$-0.236464\pi$
$239$	22.7729	1.47306	0.736529	+	0.676406i	$0.236464\pi$
$240$	0	0
$241$	6.82164	0.439420	0.219710	−	0.975565i	$-0.429489\pi$
$241$	6.82164	0.439420	0.219710	+	0.975565i	$0.429489\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−4.77407	−0.305004
$246$	0	0
$247$	−0.794590	−0.0505586
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−9.24644	−0.583630	−0.291815	−	0.956475i	$-0.594259\pi$
$251$	−9.24644	−0.583630	−0.291815	+	0.956475i	$0.594259\pi$
$252$	0	0
$253$	9.04757	0.568816
$254$	0	0
$255$	0	0
$256$	0	0
$257$	11.2464	0.701534	0.350767	−	0.936463i	$-0.385921\pi$
$257$	11.2464	0.701534	0.350767	+	0.936463i	$0.385921\pi$
$258$	0	0
$259$	−2.18489	−0.135763
$260$	0	0
$261$	0	0
$262$	0	0
$263$	22.0681	1.36078	0.680388	−	0.732852i	$-0.261811\pi$
$263$	22.0681	1.36078	0.680388	+	0.732852i	$0.261811\pi$
$264$	0	0
$265$	11.4994	0.706404
$266$	0	0
$267$	0	0
$268$	0	0
$269$	13.2054	0.805148	0.402574	−	0.915387i	$-0.368116\pi$
$269$	13.2054	0.805148	0.402574	+	0.915387i	$0.368116\pi$
$270$	0	0
$271$	14.0951	0.856218	0.428109	−	0.903727i	$-0.359180\pi$
$271$	14.0951	0.856218	0.428109	+	0.903727i	$0.359180\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	2.63675	0.159002
$276$	0	0
$277$	−30.2723	−1.81889	−0.909444	−	0.415826i	$-0.863493\pi$
$277$	−30.2723	−1.81889	−0.909444	+	0.415826i	$0.863493\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−2.13617	−0.127433	−0.0637165	−	0.997968i	$-0.520295\pi$
$281$	−2.13617	−0.127433	−0.0637165	+	0.997968i	$0.520295\pi$
$282$	0	0
$283$	0.136167	0.00809431	0.00404715	−	0.999992i	$-0.498712\pi$
$283$	0.136167	0.00809431	0.00404715	+	0.999992i	$0.498712\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	15.9102	0.939152
$288$	0	0
$289$	44.4994	2.61761
$290$	0	0
$291$	0	0
$292$	0	0
$293$	15.2735	0.892287	0.446144	−	0.894961i	$-0.352797\pi$
$293$	15.2735	0.892287	0.446144	+	0.894961i	$0.352797\pi$
$294$	0	0
$295$	−12.1362	−0.706595
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−3.43134	−0.198439
$300$	0	0
$301$	8.27233	0.476809
$302$	0	0
$303$	0	0
$304$	0	0
$305$	11.4994	0.658455
$306$	0	0
$307$	7.91024	0.451461	0.225731	−	0.974190i	$-0.427523\pi$
$307$	7.91024	0.451461	0.225731	+	0.974190i	$0.427523\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−2.54699	−0.144426	−0.0722132	−	0.997389i	$-0.523006\pi$
$311$	−2.54699	−0.144426	−0.0722132	+	0.997389i	$0.523006\pi$
$312$	0	0
$313$	19.5892	1.10725	0.553623	−	0.832767i	$-0.313245\pi$
$313$	19.5892	1.10725	0.553623	+	0.832767i	$0.313245\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	23.2735	1.30717	0.653585	−	0.756853i	$-0.273264\pi$
$317$	23.2735	1.30717	0.653585	+	0.756853i	$0.273264\pi$
$318$	0	0
$319$	10.7265	0.600569
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−6.23130	−0.346719
$324$	0	0
$325$	−1.00000	−0.0554700
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−18.0951	−0.997617
$330$	0	0
$331$	−20.4789	−1.12562	−0.562811	−	0.826586i	$-0.690280\pi$
$331$	−20.4789	−1.12562	−0.562811	+	0.826586i	$0.690280\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	6.41082	0.350260
$336$	0	0
$337$	−25.6843	−1.39911	−0.699557	−	0.714577i	$-0.746619\pi$
$337$	−25.6843	−1.39911	−0.699557	+	0.714577i	$0.746619\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	24.4519	1.32414
$342$	0	0
$343$	−7.63791	−0.412408
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−9.81511	−0.526903	−0.263451	−	0.964673i	$-0.584861\pi$
$347$	−9.81511	−0.526903	−0.263451	+	0.964673i	$0.584861\pi$
$348$	0	0
$349$	12.5199	0.670177	0.335088	−	0.942187i	$-0.391234\pi$
$349$	12.5199	0.670177	0.335088	+	0.942187i	$0.391234\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	14.0000	0.745145	0.372572	−	0.928003i	$-0.378476\pi$
$353$	14.0000	0.745145	0.372572	+	0.928003i	$0.378476\pi$
$354$	0	0
$355$	−10.6367	−0.564540
$356$	0	0
$357$	0	0
$358$	0	0
$359$	28.1362	1.48497	0.742485	−	0.669863i	$-0.233647\pi$
$359$	28.1362	1.48497	0.742485	+	0.669863i	$0.233647\pi$
$360$	0	0
$361$	−18.3686	−0.966770
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−16.0681	−0.841042
$366$	0	0
$367$	2.41082	0.125844	0.0629219	−	0.998018i	$-0.479958\pi$
$367$	2.41082	0.125844	0.0629219	+	0.998018i	$0.479958\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−39.4584	−2.04858
$372$	0	0
$373$	−5.54815	−0.287272	−0.143636	−	0.989631i	$-0.545879\pi$
$373$	−5.54815	−0.287272	−0.143636	+	0.989631i	$0.545879\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−4.06808	−0.209517
$378$	0	0
$379$	−14.5199	−0.745839	−0.372920	−	0.927864i	$-0.621643\pi$
$379$	−14.5199	−0.745839	−0.372920	+	0.927864i	$0.621643\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	22.8627	1.16823	0.584114	−	0.811672i	$-0.301442\pi$
$383$	22.8627	1.16823	0.584114	+	0.811672i	$0.301442\pi$
$384$	0	0
$385$	−9.04757	−0.461107
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−5.34158	−0.270829	−0.135414	−	0.990789i	$-0.543237\pi$
$389$	−5.34158	−0.270829	−0.135414	+	0.990789i	$0.543237\pi$
$390$	0	0
$391$	−26.9091	−1.36085
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−10.6367	−0.535193
$396$	0	0
$397$	−21.4584	−1.07697	−0.538483	−	0.842637i	$-0.681002\pi$
$397$	−21.4584	−1.07697	−0.538483	+	0.842637i	$0.681002\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	17.3145	0.864646	0.432323	−	0.901719i	$-0.357694\pi$
$401$	17.3145	0.864646	0.432323	+	0.901719i	$0.357694\pi$
$402$	0	0
$403$	−9.27349	−0.461946
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−1.67894	−0.0832220
$408$	0	0
$409$	16.4108	0.811463	0.405731	−	0.913992i	$-0.367017\pi$
$409$	16.4108	0.811463	0.405731	+	0.913992i	$0.367017\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	41.6433	2.04913
$414$	0	0
$415$	−16.1362	−0.792093
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−25.7524	−1.25809	−0.629043	−	0.777370i	$-0.716553\pi$
$419$	−25.7524	−1.25809	−0.629043	+	0.777370i	$0.716553\pi$
$420$	0	0
$421$	13.4777	0.656865	0.328433	−	0.944527i	$-0.393480\pi$
$421$	13.4777	0.656865	0.328433	+	0.944527i	$0.393480\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−7.84216	−0.380400
$426$	0	0
$427$	−39.4584	−1.90953
$428$	0	0
$429$	0	0
$430$	0	0
$431$	24.2723	1.16916	0.584579	−	0.811337i	$-0.301260\pi$
$431$	24.2723	1.16916	0.584579	+	0.811337i	$0.301260\pi$
$432$	0	0
$433$	3.13733	0.150770	0.0753851	−	0.997154i	$-0.475981\pi$
$433$	3.13733	0.150770	0.0753851	+	0.997154i	$0.475981\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	2.72651	0.130426
$438$	0	0
$439$	8.59571	0.410251	0.205125	−	0.978736i	$-0.434240\pi$
$439$	8.59571	0.410251	0.205125	+	0.978736i	$0.434240\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−0.911399	−0.0433019	−0.0216509	−	0.999766i	$-0.506892\pi$
$443$	−0.911399	−0.0433019	−0.0216509	+	0.999766i	$0.506892\pi$
$444$	0	0
$445$	8.63675	0.409421
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−38.8139	−1.83174	−0.915872	−	0.401471i	$-0.868499\pi$
$449$	−38.8139	−1.83174	−0.915872	+	0.401471i	$0.868499\pi$
$450$	0	0
$451$	12.2259	0.575696
$452$	0	0
$453$	0	0
$454$	0	0
$455$	3.43134	0.160864
$456$	0	0
$457$	−6.25298	−0.292502	−0.146251	−	0.989248i	$-0.546721\pi$
$457$	−6.25298	−0.292502	−0.146251	+	0.989248i	$0.546721\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	27.9513	1.30182	0.650910	−	0.759155i	$-0.274387\pi$
$461$	27.9513	1.30182	0.650910	+	0.759155i	$0.274387\pi$
$462$	0	0
$463$	−11.8832	−0.552259	−0.276129	−	0.961120i	$-0.589052\pi$
$463$	−11.8832	−0.552259	−0.276129	+	0.961120i	$0.589052\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	28.8680	1.33585	0.667927	−	0.744227i	$-0.267182\pi$
$467$	28.8680	1.33585	0.667927	+	0.744227i	$0.267182\pi$
$468$	0	0
$469$	−21.9977	−1.01576
$470$	0	0
$471$	0	0
$472$	0	0
$473$	6.35672	0.292282
$474$	0	0
$475$	0.794590	0.0364583
$476$	0	0
$477$	0	0
$478$	0	0
$479$	12.0464	0.550414	0.275207	−	0.961385i	$-0.411254\pi$
$479$	12.0464	0.550414	0.275207	+	0.961385i	$0.411254\pi$
$480$	0	0
$481$	0.636747	0.0290332
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−12.2940	−0.558242
$486$	0	0
$487$	11.2518	0.509869	0.254934	−	0.966958i	$-0.417946\pi$
$487$	11.2518	0.509869	0.254934	+	0.966958i	$0.417946\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−31.0259	−1.40018	−0.700089	−	0.714055i	$-0.746857\pi$
$491$	−31.0259	−1.40018	−0.700089	+	0.714055i	$0.746857\pi$
$492$	0	0
$493$	−31.9025	−1.43682
$494$	0	0
$495$	0	0
$496$	0	0
$497$	36.4983	1.63717
$498$	0	0
$499$	−5.61623	−0.251417	−0.125708	−	0.992067i	$-0.540120\pi$
$499$	−5.61623	−0.251417	−0.125708	+	0.992067i	$0.540120\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	3.52110	0.156998	0.0784990	−	0.996914i	$-0.474987\pi$
$503$	3.52110	0.156998	0.0784990	+	0.996914i	$0.474987\pi$
$504$	0	0
$505$	5.34158	0.237697
$506$	0	0
$507$	0	0
$508$	0	0
$509$	13.7741	0.610525	0.305263	−	0.952268i	$-0.401256\pi$
$509$	13.7741	0.610525	0.305263	+	0.952268i	$0.401256\pi$
$510$	0	0
$511$	55.1350	2.43903
$512$	0	0
$513$	0	0
$514$	0	0
$515$	10.4108	0.458756
$516$	0	0
$517$	−13.9049	−0.611535
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−25.3145	−1.10905	−0.554525	−	0.832167i	$-0.687100\pi$
$521$	−25.3145	−1.10905	−0.554525	+	0.832167i	$0.687100\pi$
$522$	0	0
$523$	−4.00000	−0.174908	−0.0874539	−	0.996169i	$-0.527873\pi$
$523$	−4.00000	−0.174908	−0.0874539	+	0.996169i	$0.527873\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−72.7242	−3.16792
$528$	0	0
$529$	−11.2259	−0.488084
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−4.63675	−0.200840
$534$	0	0
$535$	5.04757	0.218225
$536$	0	0
$537$	0	0
$538$	0	0
$539$	12.5880	0.542204
$540$	0	0
$541$	−26.6151	−1.14427	−0.572136	−	0.820159i	$-0.693885\pi$
$541$	−26.6151	−1.14427	−0.572136	+	0.820159i	$0.693885\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−6.79459	−0.291048
$546$	0	0
$547$	−17.7253	−0.757881	−0.378941	−	0.925421i	$-0.623712\pi$
$547$	−17.7253	−0.757881	−0.378941	+	0.925421i	$0.623712\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	3.23246	0.137707
$552$	0	0
$553$	36.4983	1.55206
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−30.2723	−1.28268	−0.641340	−	0.767257i	$-0.721621\pi$
$557$	−30.2723	−1.28268	−0.641340	+	0.767257i	$0.721621\pi$
$558$	0	0
$559$	−2.41082	−0.101967
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−22.6367	−0.954025	−0.477013	−	0.878896i	$-0.658280\pi$
$563$	−22.6367	−0.954025	−0.477013	+	0.878896i	$0.658280\pi$
$564$	0	0
$565$	−8.06808	−0.339427
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−20.0410	−0.840164	−0.420082	−	0.907486i	$-0.637999\pi$
$569$	−20.0410	−0.840164	−0.420082	+	0.907486i	$0.637999\pi$
$570$	0	0
$571$	20.3621	0.852127	0.426064	−	0.904693i	$-0.359900\pi$
$571$	20.3621	0.852127	0.426064	+	0.904693i	$0.359900\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	3.43134	0.143097
$576$	0	0
$577$	2.70483	0.112604	0.0563018	−	0.998414i	$-0.482069\pi$
$577$	2.70483	0.112604	0.0563018	+	0.998414i	$0.482069\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	55.3686	2.29708
$582$	0	0
$583$	−30.3211	−1.25577
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−33.2735	−1.37334	−0.686672	−	0.726967i	$-0.740929\pi$
$587$	−33.2735	−1.37334	−0.686672	+	0.726967i	$0.740929\pi$
$588$	0	0
$589$	7.36863	0.303619
$590$	0	0
$591$	0	0
$592$	0	0
$593$	28.2313	1.15932	0.579660	−	0.814858i	$-0.303185\pi$
$593$	28.2313	1.15932	0.579660	+	0.814858i	$0.303185\pi$
$594$	0	0
$595$	26.9091	1.10316
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−29.0940	−1.18875	−0.594374	−	0.804189i	$-0.702600\pi$
$599$	−29.0940	−1.18875	−0.594374	+	0.804189i	$0.702600\pi$
$600$	0	0
$601$	46.0464	1.87827	0.939136	−	0.343545i	$-0.111628\pi$
$601$	46.0464	1.87827	0.939136	+	0.343545i	$0.111628\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	4.04757	0.164557
$606$	0	0
$607$	24.1362	0.979657	0.489828	−	0.871819i	$-0.337059\pi$
$607$	24.1362	0.979657	0.489828	+	0.871819i	$0.337059\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	5.27349	0.213343
$612$	0	0
$613$	3.73304	0.150776	0.0753880	−	0.997154i	$-0.475980\pi$
$613$	3.73304	0.150776	0.0753880	+	0.997154i	$0.475980\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	13.3686	0.538201	0.269100	−	0.963112i	$-0.413274\pi$
$617$	13.3686	0.538201	0.269100	+	0.963112i	$0.413274\pi$
$618$	0	0
$619$	−21.8886	−0.879776	−0.439888	−	0.898053i	$-0.644982\pi$
$619$	−21.8886	−0.879776	−0.439888	+	0.898053i	$0.644982\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−29.6356	−1.18732
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	4.99347	0.199103
$630$	0	0
$631$	21.5892	0.859452	0.429726	−	0.902959i	$-0.358610\pi$
$631$	21.5892	0.859452	0.429726	+	0.902959i	$0.358610\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	15.6843	0.622413
$636$	0	0
$637$	−4.77407	−0.189156
$638$	0	0
$639$	0	0
$640$	0	0
$641$	5.54815	0.219139	0.109569	−	0.993979i	$-0.465053\pi$
$641$	5.54815	0.219139	0.109569	+	0.993979i	$0.465053\pi$
$642$	0	0
$643$	−34.6367	−1.36594	−0.682970	−	0.730446i	$-0.739312\pi$
$643$	−34.6367	−1.36594	−0.682970	+	0.730446i	$0.739312\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−29.9783	−1.17857	−0.589285	−	0.807925i	$-0.700590\pi$
$647$	−29.9783	−1.17857	−0.589285	+	0.807925i	$0.700590\pi$
$648$	0	0
$649$	32.0000	1.25611
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−32.6832	−1.27899	−0.639495	−	0.768795i	$-0.720857\pi$
$653$	−32.6832	−1.27899	−0.639495	+	0.768795i	$0.720857\pi$
$654$	0	0
$655$	12.9308	0.505247
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−23.9296	−0.932165	−0.466082	−	0.884741i	$-0.654335\pi$
$659$	−23.9296	−0.932165	−0.466082	+	0.884741i	$0.654335\pi$
$660$	0	0
$661$	−34.1632	−1.32880	−0.664398	−	0.747379i	$-0.731312\pi$
$661$	−34.1632	−1.32880	−0.664398	+	0.747379i	$0.731312\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−2.72651	−0.105729
$666$	0	0
$667$	13.9590	0.540493
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−30.3211	−1.17053
$672$	0	0
$673$	−6.90371	−0.266118	−0.133059	−	0.991108i	$-0.542480\pi$
$673$	−6.90371	−0.266118	−0.133059	+	0.991108i	$0.542480\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−49.2789	−1.89394	−0.946970	−	0.321321i	$-0.895873\pi$
$677$	−49.2789	−1.89394	−0.946970	+	0.321321i	$0.895873\pi$
$678$	0	0
$679$	42.1849	1.61891
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−3.04219	−0.116406	−0.0582031	−	0.998305i	$-0.518537\pi$
$683$	−3.04219	−0.116406	−0.0582031	+	0.998305i	$0.518537\pi$
$684$	0	0
$685$	2.00000	0.0764161
$686$	0	0
$687$	0	0
$688$	0	0
$689$	11.4994	0.438093
$690$	0	0
$691$	13.8886	0.528346	0.264173	−	0.964475i	$-0.414901\pi$
$691$	13.8886	0.528346	0.264173	+	0.964475i	$0.414901\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−19.9102	−0.755238
$696$	0	0
$697$	−36.3621	−1.37731
$698$	0	0
$699$	0	0
$700$	0	0
$701$	5.20541	0.196606	0.0983028	−	0.995157i	$-0.468659\pi$
$701$	5.20541	0.196606	0.0983028	+	0.995157i	$0.468659\pi$
$702$	0	0
$703$	−0.505953	−0.0190824
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−18.3288	−0.689324
$708$	0	0
$709$	−19.7524	−0.741817	−0.370908	−	0.928669i	$-0.620954\pi$
$709$	−19.7524	−0.741817	−0.370908	+	0.928669i	$0.620954\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	31.8205	1.19169
$714$	0	0
$715$	2.63675	0.0986087
$716$	0	0
$717$	0	0
$718$	0	0
$719$	4.13617	0.154253	0.0771265	−	0.997021i	$-0.475425\pi$
$719$	4.13617	0.154253	0.0771265	+	0.997021i	$0.475425\pi$
$720$	0	0
$721$	−35.7230	−1.33040
$722$	0	0
$723$	0	0
$724$	0	0
$725$	4.06808	0.151085
$726$	0	0
$727$	6.09513	0.226056	0.113028	−	0.993592i	$-0.463945\pi$
$727$	6.09513	0.226056	0.113028	+	0.993592i	$0.463945\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−18.9060	−0.699265
$732$	0	0
$733$	−40.0054	−1.47763	−0.738816	−	0.673907i	$-0.764615\pi$
$733$	−40.0054	−1.47763	−0.738816	+	0.673907i	$0.764615\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−16.9037	−0.622656
$738$	0	0
$739$	12.0270	0.442422	0.221211	−	0.975226i	$-0.428999\pi$
$739$	12.0270	0.442422	0.221211	+	0.975226i	$0.428999\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−10.0410	−0.368370	−0.184185	−	0.982892i	$-0.558965\pi$
$743$	−10.0410	−0.368370	−0.184185	+	0.982892i	$0.558965\pi$
$744$	0	0
$745$	8.63675	0.316426
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−17.3199	−0.632855
$750$	0	0
$751$	12.6778	0.462619	0.231309	−	0.972880i	$-0.425699\pi$
$751$	12.6778	0.462619	0.231309	+	0.972880i	$0.425699\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	17.7253	0.645091
$756$	0	0
$757$	13.6302	0.495399	0.247699	−	0.968837i	$-0.420326\pi$
$757$	13.6302	0.495399	0.247699	+	0.968837i	$0.420326\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	11.8615	0.429980	0.214990	−	0.976616i	$-0.431028\pi$
$761$	11.8615	0.429980	0.214990	+	0.976616i	$0.431028\pi$
$762$	0	0
$763$	23.3145	0.844043
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−12.1362	−0.438212
$768$	0	0
$769$	20.9988	0.757238	0.378619	−	0.925553i	$-0.376399\pi$
$769$	20.9988	0.757238	0.378619	+	0.925553i	$0.376399\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−14.4519	−0.519797	−0.259899	−	0.965636i	$-0.583689\pi$
$773$	−14.4519	−0.519797	−0.259899	+	0.965636i	$0.583689\pi$
$774$	0	0
$775$	9.27349	0.333114
$776$	0	0
$777$	0	0
$778$	0	0
$779$	3.68431	0.132004
$780$	0	0
$781$	28.0464	1.00358
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−20.5470	−0.733353
$786$	0	0
$787$	−10.5470	−0.375959	−0.187980	−	0.982173i	$-0.560194\pi$
$787$	−10.5470	−0.375959	−0.187980	+	0.982173i	$0.560194\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	27.6843	0.984341
$792$	0	0
$793$	11.4994	0.408356
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−40.7729	−1.44425	−0.722125	−	0.691762i	$-0.756835\pi$
$797$	−40.7729	−1.44425	−0.722125	+	0.691762i	$0.756835\pi$
$798$	0	0
$799$	41.3556	1.46305
$800$	0	0
$801$	0	0
$802$	0	0
$803$	42.3675	1.49512
$804$	0	0
$805$	−11.7741	−0.414982
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−20.2747	−0.712819	−0.356409	−	0.934330i	$-0.615999\pi$
$809$	−20.2747	−0.712819	−0.356409	+	0.934330i	$0.615999\pi$
$810$	0	0
$811$	−15.0259	−0.527630	−0.263815	−	0.964573i	$-0.584981\pi$
$811$	−15.0259	−0.527630	−0.263815	+	0.964573i	$0.584981\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	4.22593	0.148028
$816$	0	0
$817$	1.91561	0.0670188
$818$	0	0
$819$	0	0
$820$	0	0
$821$	2.36209	0.0824377	0.0412188	−	0.999150i	$-0.486876\pi$
$821$	2.36209	0.0824377	0.0412188	+	0.999150i	$0.486876\pi$
$822$	0	0
$823$	−16.1362	−0.562471	−0.281236	−	0.959639i	$-0.590744\pi$
$823$	−16.1362	−0.562471	−0.281236	+	0.959639i	$0.590744\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−24.5880	−0.855009	−0.427505	−	0.904013i	$-0.640607\pi$
$827$	−24.5880	−0.855009	−0.427505	+	0.904013i	$0.640607\pi$
$828$	0	0
$829$	−22.8216	−0.792628	−0.396314	−	0.918115i	$-0.629711\pi$
$829$	−22.8216	−0.792628	−0.396314	+	0.918115i	$0.629711\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−37.4390	−1.29719
$834$	0	0
$835$	−14.4108	−0.498707
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−46.3211	−1.59918	−0.799590	−	0.600546i	$-0.794950\pi$
$839$	−46.3211	−1.59918	−0.799590	+	0.600546i	$0.794950\pi$
$840$	0	0
$841$	−12.4507	−0.429334
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−1.00000	−0.0344010
$846$	0	0
$847$	−13.8886	−0.477217
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−2.18489	−0.0748972
$852$	0	0
$853$	47.1837	1.61554	0.807770	−	0.589498i	$-0.200674\pi$
$853$	47.1837	1.61554	0.807770	+	0.589498i	$0.200674\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	11.8422	0.404520	0.202260	−	0.979332i	$-0.435171\pi$
$857$	11.8422	0.404520	0.202260	+	0.979332i	$0.435171\pi$
$858$	0	0
$859$	10.3211	0.352150	0.176075	−	0.984377i	$-0.443660\pi$
$859$	10.3211	0.352150	0.176075	+	0.984377i	$0.443660\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−1.13733	−0.0387150	−0.0193575	−	0.999813i	$-0.506162\pi$
$863$	−1.13733	−0.0387150	−0.0193575	+	0.999813i	$0.506162\pi$
$864$	0	0
$865$	−6.00000	−0.204006
$866$	0	0
$867$	0	0
$868$	0	0
$869$	28.0464	0.951409
$870$	0	0
$871$	6.41082	0.217222
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−3.43134	−0.116000
$876$	0	0
$877$	−54.5447	−1.84184	−0.920921	−	0.389749i	$-0.872562\pi$
$877$	−54.5447	−1.84184	−0.920921	+	0.389749i	$0.872562\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−48.9555	−1.64935	−0.824676	−	0.565605i	$-0.808643\pi$
$881$	−48.9555	−1.64935	−0.824676	+	0.565605i	$0.808643\pi$
$882$	0	0
$883$	27.3686	0.921028	0.460514	−	0.887653i	$-0.347665\pi$
$883$	27.3686	0.921028	0.460514	+	0.887653i	$0.347665\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	21.5265	0.722788	0.361394	−	0.932413i	$-0.382301\pi$
$887$	21.5265	0.722788	0.361394	+	0.932413i	$0.382301\pi$
$888$	0	0
$889$	−53.8182	−1.80500
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−4.19027	−0.140222
$894$	0	0
$895$	−20.9308	−0.699638
$896$	0	0
$897$	0	0
$898$	0	0
$899$	37.7253	1.25821
$900$	0	0
$901$	90.1803	3.00434
$902$	0	0
$903$	0	0
$904$	0	0
$905$	5.08860	0.169151
$906$	0	0
$907$	−39.6843	−1.31770	−0.658848	−	0.752276i	$-0.728956\pi$
$907$	−39.6843	−1.31770	−0.658848	+	0.752276i	$0.728956\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	4.13617	0.137037	0.0685187	−	0.997650i	$-0.478173\pi$
$911$	4.13617	0.137037	0.0685187	+	0.997650i	$0.478173\pi$
$912$	0	0
$913$	42.5470	1.40810
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−44.3698	−1.46522
$918$	0	0
$919$	17.6789	0.583174	0.291587	−	0.956544i	$-0.405817\pi$
$919$	17.6789	0.583174	0.291587	+	0.956544i	$0.405817\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−10.6367	−0.350113
$924$	0	0
$925$	−0.636747	−0.0209361
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−27.8151	−0.912584	−0.456292	−	0.889830i	$-0.650823\pi$
$929$	−27.8151	−0.912584	−0.456292	+	0.889830i	$0.650823\pi$
$930$	0	0
$931$	3.79343	0.124325
$932$	0	0
$933$	0	0
$934$	0	0
$935$	20.6778	0.676236
$936$	0	0
$937$	−54.5447	−1.78190	−0.890948	−	0.454105i	$-0.849959\pi$
$937$	−54.5447	−1.78190	−0.890948	+	0.454105i	$0.849959\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	12.1849	0.397216	0.198608	−	0.980079i	$-0.436358\pi$
$941$	12.1849	0.397216	0.198608	+	0.980079i	$0.436358\pi$
$942$	0	0
$943$	15.9102	0.518109
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−39.5048	−1.28373	−0.641867	−	0.766816i	$-0.721840\pi$
$947$	−39.5048	−1.28373	−0.641867	+	0.766816i	$0.721840\pi$
$948$	0	0
$949$	−16.0681	−0.521592
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−19.0205	−0.616135	−0.308067	−	0.951365i	$-0.599682\pi$
$953$	−19.0205	−0.616135	−0.308067	+	0.951365i	$0.599682\pi$
$954$	0	0
$955$	−13.7253	−0.444142
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−6.86267	−0.221607
$960$	0	0
$961$	54.9977	1.77412
$962$	0	0
$963$	0	0
$964$	0	0
$965$	5.11565	0.164679
$966$	0	0
$967$	−38.3404	−1.23294	−0.616472	−	0.787377i	$-0.711439\pi$
$967$	−38.3404	−1.23294	−0.616472	+	0.787377i	$0.711439\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−15.7114	−0.504202	−0.252101	−	0.967701i	$-0.581121\pi$
$971$	−15.7114	−0.504202	−0.252101	+	0.967701i	$0.581121\pi$
$972$	0	0
$973$	68.3187	2.19020
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−53.9566	−1.72623	−0.863113	−	0.505011i	$-0.831489\pi$
$977$	−53.9566	−1.72623	−0.863113	+	0.505011i	$0.831489\pi$
$978$	0	0
$979$	−22.7729	−0.727825
$980$	0	0
$981$	0	0
$982$	0	0
$983$	45.0399	1.43655	0.718274	−	0.695760i	$-0.244932\pi$
$983$	45.0399	1.43655	0.718274	+	0.695760i	$0.244932\pi$
$984$	0	0
$985$	−20.9988	−0.669079
$986$	0	0
$987$	0	0
$988$	0	0
$989$	8.27233	0.263045
$990$	0	0
$991$	7.63791	0.242626	0.121313	−	0.992614i	$-0.461290\pi$
$991$	7.63791	0.242626	0.121313	+	0.992614i	$0.461290\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−9.58918	−0.303997
$996$	0	0
$997$	−16.0951	−0.509738	−0.254869	−	0.966976i	$-0.582032\pi$
$997$	−16.0951	−0.509738	−0.254869	+	0.966976i	$0.582032\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4680.2.a.bh.1.3		3
3.2	odd	2		1560.2.a.q.1.3	✓	3
4.3	odd	2		9360.2.a.cy.1.1		3
12.11	even	2		3120.2.a.bi.1.1		3
15.14	odd	2		7800.2.a.bi.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1560.2.a.q.1.3	✓	3	3.2	odd	2
3120.2.a.bi.1.1		3	12.11	even	2
4680.2.a.bh.1.3		3	1.1	even	1	trivial
7800.2.a.bi.1.1		3	15.14	odd	2
9360.2.a.cy.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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 4680 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4680.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{5} +3.43134 q^{7} +O(q^{10})\)	\(q-1.00000 q^{5} +3.43134 q^{7} +2.63675 q^{11} -1.00000 q^{13} -7.84216 q^{17} +0.794590 q^{19} +3.43134 q^{23} +1.00000 q^{25} +4.06808 q^{29} +9.27349 q^{31} -3.43134 q^{35} -0.636747 q^{37} +4.63675 q^{41} +2.41082 q^{43} -5.27349 q^{47} +4.77407 q^{49} -11.4994 q^{53} -2.63675 q^{55} +12.1362 q^{59} -11.4994 q^{61} +1.00000 q^{65} -6.41082 q^{67} +10.6367 q^{71} +16.0681 q^{73} +9.04757 q^{77} +10.6367 q^{79} +16.1362 q^{83} +7.84216 q^{85} -8.63675 q^{89} -3.43134 q^{91} -0.794590 q^{95} +12.2940 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{5} + q^{7}+O(q^{10}) \) 3 * q - 3 * q^5 + q^7	\( 3 q - 3 q^{5} + q^{7} - 5 q^{11} - 3 q^{13} - 7 q^{17} + 6 q^{19} + q^{23} + 3 q^{25} - 10 q^{29} + 2 q^{31} - q^{35} + 11 q^{37} + q^{41} + 10 q^{47} + 20 q^{49} - 3 q^{53} + 5 q^{55} - 8 q^{59} - 3 q^{61} + 3 q^{65} - 12 q^{67} + 19 q^{71} + 26 q^{73} + 7 q^{77} + 19 q^{79} + 4 q^{83} + 7 q^{85} - 13 q^{89} - q^{91} - 6 q^{95} + 9 q^{97}+O(q^{100}) \) 3 * q - 3 * q^5 + q^7 - 5 * q^11 - 3 * q^13 - 7 * q^17 + 6 * q^19 + q^23 + 3 * q^25 - 10 * q^29 + 2 * q^31 - q^35 + 11 * q^37 + q^41 + 10 * q^47 + 20 * q^49 - 3 * q^53 + 5 * q^55 - 8 * q^59 - 3 * q^61 + 3 * q^65 - 12 * q^67 + 19 * q^71 + 26 * q^73 + 7 * q^77 + 19 * q^79 + 4 * q^83 + 7 * q^85 - 13 * q^89 - q^91 - 6 * q^95 + 9 * q^97

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