LMFDB - Embedded newform 9680.2.a.cy.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9680,2,Mod(1,9680)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9680, 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("9680.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9680 = 2^{4} \cdot 5 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9680.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:	$77.2951891566$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.45753625.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 13x^{4} + 11x^{3} + 41x^{2} - 30x - 20 $ x^6 - x^5 - 13x^4 + 11x^3 + 41x^2 - 30x - 20
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 440)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$3.05921$ of defining polynomial
Character	$\chi$	$=$	9680.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.89070 q^{3} -1.00000 q^{5} -2.74075 q^{7} +0.574737 q^{9} -0.196821 q^{13} +1.89070 q^{15} +1.46809 q^{17} -2.68154 q^{19} +5.18193 q^{21} -1.81736 q^{23} +1.00000 q^{25} +4.58544 q^{27} -1.37946 q^{29} -5.90411 q^{31} +2.74075 q^{35} +4.57623 q^{37} +0.372129 q^{39} -1.23428 q^{41} +2.96514 q^{43} -0.574737 q^{45} -9.30107 q^{47} +0.511708 q^{49} -2.77571 q^{51} +2.69388 q^{53} +5.06997 q^{57} -14.2092 q^{59} -7.54027 q^{61} -1.57521 q^{63} +0.196821 q^{65} -10.6500 q^{67} +3.43607 q^{69} +6.87672 q^{71} -10.4314 q^{73} -1.89070 q^{75} +1.87328 q^{79} -10.3939 q^{81} -5.79397 q^{83} -1.46809 q^{85} +2.60815 q^{87} +17.3060 q^{89} +0.539438 q^{91} +11.1629 q^{93} +2.68154 q^{95} -18.3188 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 2 q^{3} - 6 q^{5} + 6 q^{7} + 10 q^{9} + 6 q^{13} + 2 q^{15} - 11 q^{17} - 11 q^{19} - 2 q^{21} - 18 q^{23} + 6 q^{25} + q^{27} + 6 q^{29} - q^{31} - 6 q^{35} + 4 q^{37} + 27 q^{39} + 4 q^{41} + 3 q^{43}+ \cdots + 20 q^{97}+O(q^{100}) $ 6 * q - 2 * q^3 - 6 * q^5 + 6 * q^7 + 10 * q^9 + 6 * q^13 + 2 * q^15 - 11 * q^17 - 11 * q^19 - 2 * q^21 - 18 * q^23 + 6 * q^25 + q^27 + 6 * q^29 - q^31 - 6 * q^35 + 4 * q^37 + 27 * q^39 + 4 * q^41 + 3 * q^43 - 10 * q^45 - 14 * q^47 + 8 * q^49 + 31 * q^51 + 14 * q^53 + 5 * q^57 - 2 * q^59 + 4 * q^61 - 16 * q^63 - 6 * q^65 - 11 * q^67 + 8 * q^69 - 7 * q^71 + 9 * q^73 - 2 * q^75 + 36 * q^79 + 30 * q^81 - 45 * q^83 + 11 * q^85 + 25 * q^87 + q^89 + 8 * q^91 + 55 * q^93 + 11 * q^95 + 20 * q^97

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−1.89070	−1.09159	−0.545797	−	0.837917i	$-0.683773\pi$
$3$	−1.89070	−1.09159	−0.545797	+	0.837917i	$0.683773\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−2.74075	−1.03591	−0.517953	−	0.855409i	$-0.673306\pi$
$7$	−2.74075	−1.03591	−0.517953	+	0.855409i	$0.673306\pi$
$8$	0	0
$9$	0.574737	0.191579
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−0.196821	−0.0545884	−0.0272942	−	0.999627i	$-0.508689\pi$
$13$	−0.196821	−0.0545884	−0.0272942	+	0.999627i	$0.508689\pi$
$14$	0	0
$15$	1.89070	0.488176
$16$	0	0
$17$	1.46809	0.356063	0.178032	−	0.984025i	$-0.443027\pi$
$17$	1.46809	0.356063	0.178032	+	0.984025i	$0.443027\pi$
$18$	0	0
$19$	−2.68154	−0.615187	−0.307593	−	0.951518i	$-0.599524\pi$
$19$	−2.68154	−0.615187	−0.307593	+	0.951518i	$0.599524\pi$
$20$	0	0
$21$	5.18193	1.13079
$22$	0	0
$23$	−1.81736	−0.378945	−0.189473	−	0.981886i	$-0.560678\pi$
$23$	−1.81736	−0.378945	−0.189473	+	0.981886i	$0.560678\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	4.58544	0.882468
$28$	0	0
$29$	−1.37946	−0.256160	−0.128080	−	0.991764i	$-0.540881\pi$
$29$	−1.37946	−0.256160	−0.128080	+	0.991764i	$0.540881\pi$
$30$	0	0
$31$	−5.90411	−1.06041	−0.530205	−	0.847870i	$-0.677885\pi$
$31$	−5.90411	−1.06041	−0.530205	+	0.847870i	$0.677885\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	2.74075	0.463271
$36$	0	0
$37$	4.57623	0.752328	0.376164	−	0.926553i	$-0.377243\pi$
$37$	4.57623	0.752328	0.376164	+	0.926553i	$0.377243\pi$
$38$	0	0
$39$	0.372129	0.0595884
$40$	0	0
$41$	−1.23428	−0.192762	−0.0963812	−	0.995344i	$-0.530727\pi$
$41$	−1.23428	−0.192762	−0.0963812	+	0.995344i	$0.530727\pi$
$42$	0	0
$43$	2.96514	0.452180	0.226090	−	0.974106i	$-0.427406\pi$
$43$	2.96514	0.452180	0.226090	+	0.974106i	$0.427406\pi$
$44$	0	0
$45$	−0.574737	−0.0856768
$46$	0	0
$47$	−9.30107	−1.35670	−0.678350	−	0.734739i	$-0.737305\pi$
$47$	−9.30107	−1.35670	−0.678350	+	0.734739i	$0.737305\pi$
$48$	0	0
$49$	0.511708	0.0731012
$50$	0	0
$51$	−2.77571	−0.388677
$52$	0	0
$53$	2.69388	0.370032	0.185016	−	0.982735i	$-0.440766\pi$
$53$	2.69388	0.370032	0.185016	+	0.982735i	$0.440766\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	5.06997	0.671535
$58$	0	0
$59$	−14.2092	−1.84988	−0.924942	−	0.380108i	$-0.875887\pi$
$59$	−14.2092	−1.84988	−0.924942	+	0.380108i	$0.875887\pi$
$60$	0	0
$61$	−7.54027	−0.965433	−0.482716	−	0.875777i	$-0.660350\pi$
$61$	−7.54027	−0.965433	−0.482716	+	0.875777i	$0.660350\pi$
$62$	0	0
$63$	−1.57521	−0.198458
$64$	0	0
$65$	0.196821	0.0244127
$66$	0	0
$67$	−10.6500	−1.30111	−0.650553	−	0.759461i	$-0.725463\pi$
$67$	−10.6500	−1.30111	−0.650553	+	0.759461i	$0.725463\pi$
$68$	0	0
$69$	3.43607	0.413655
$70$	0	0
$71$	6.87672	0.816116	0.408058	−	0.912956i	$-0.366206\pi$
$71$	6.87672	0.816116	0.408058	+	0.912956i	$0.366206\pi$
$72$	0	0
$73$	−10.4314	−1.22090	−0.610452	−	0.792053i	$-0.709012\pi$
$73$	−10.4314	−1.22090	−0.610452	+	0.792053i	$0.709012\pi$
$74$	0	0
$75$	−1.89070	−0.218319
$76$	0	0
$77$	0	0
$78$	0	0
$79$	1.87328	0.210761	0.105380	−	0.994432i	$-0.466394\pi$
$79$	1.87328	0.210761	0.105380	+	0.994432i	$0.466394\pi$
$80$	0	0
$81$	−10.3939	−1.15488
$82$	0	0
$83$	−5.79397	−0.635971	−0.317986	−	0.948096i	$-0.603006\pi$
$83$	−5.79397	−0.635971	−0.317986	+	0.948096i	$0.603006\pi$
$84$	0	0
$85$	−1.46809	−0.159236
$86$	0	0
$87$	2.60815	0.279623
$88$	0	0
$89$	17.3060	1.83443	0.917214	−	0.398396i	$-0.130433\pi$
$89$	17.3060	1.83443	0.917214	+	0.398396i	$0.130433\pi$
$90$	0	0
$91$	0.539438	0.0565484
$92$	0	0
$93$	11.1629	1.15754
$94$	0	0
$95$	2.68154	0.275120
$96$	0	0
$97$	−18.3188	−1.86000	−0.929998	−	0.367564i	$-0.880192\pi$
$97$	−18.3188	−1.86000	−0.929998	+	0.367564i	$0.880192\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−5.00039	−0.497558	−0.248779	−	0.968560i	$-0.580029\pi$
$101$	−5.00039	−0.497558	−0.248779	+	0.968560i	$0.580029\pi$
$102$	0	0
$103$	−9.17856	−0.904391	−0.452195	−	0.891919i	$-0.649359\pi$
$103$	−9.17856	−0.904391	−0.452195	+	0.891919i	$0.649359\pi$
$104$	0	0
$105$	−5.18193	−0.505704
$106$	0	0
$107$	−16.8116	−1.62524	−0.812621	−	0.582792i	$-0.801960\pi$
$107$	−16.8116	−1.62524	−0.812621	+	0.582792i	$0.801960\pi$
$108$	0	0
$109$	9.91076	0.949279	0.474639	−	0.880180i	$-0.342578\pi$
$109$	9.91076	0.949279	0.474639	+	0.880180i	$0.342578\pi$
$110$	0	0
$111$	−8.65227	−0.821238
$112$	0	0
$113$	12.6713	1.19202	0.596009	−	0.802978i	$-0.296752\pi$
$113$	12.6713	1.19202	0.596009	+	0.802978i	$0.296752\pi$
$114$	0	0
$115$	1.81736	0.169470
$116$	0	0
$117$	−0.113120	−0.0104580
$118$	0	0
$119$	−4.02366	−0.368848
$120$	0	0
$121$	0	0
$122$	0	0
$123$	2.33365	0.210418
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	10.4404	0.926437	0.463218	−	0.886244i	$-0.346695\pi$
$127$	10.4404	0.926437	0.463218	+	0.886244i	$0.346695\pi$
$128$	0	0
$129$	−5.60619	−0.493597
$130$	0	0
$131$	−20.9537	−1.83073	−0.915367	−	0.402621i	$-0.868099\pi$
$131$	−20.9537	−1.83073	−0.915367	+	0.402621i	$0.868099\pi$
$132$	0	0
$133$	7.34942	0.637276
$134$	0	0
$135$	−4.58544	−0.394652
$136$	0	0
$137$	2.96083	0.252961	0.126481	−	0.991969i	$-0.459632\pi$
$137$	2.96083	0.252961	0.126481	+	0.991969i	$0.459632\pi$
$138$	0	0
$139$	16.4253	1.39318	0.696589	−	0.717470i	$-0.254700\pi$
$139$	16.4253	1.39318	0.696589	+	0.717470i	$0.254700\pi$
$140$	0	0
$141$	17.5855	1.48097
$142$	0	0
$143$	0	0
$144$	0	0
$145$	1.37946	0.114558
$146$	0	0
$147$	−0.967485	−0.0797968
$148$	0	0
$149$	19.1836	1.57158	0.785790	−	0.618494i	$-0.212257\pi$
$149$	19.1836	1.57158	0.785790	+	0.618494i	$0.212257\pi$
$150$	0	0
$151$	−9.04992	−0.736472	−0.368236	−	0.929732i	$-0.620038\pi$
$151$	−9.04992	−0.736472	−0.368236	+	0.929732i	$0.620038\pi$
$152$	0	0
$153$	0.843764	0.0682142
$154$	0	0
$155$	5.90411	0.474230
$156$	0	0
$157$	−18.1894	−1.45167	−0.725837	−	0.687867i	$-0.758547\pi$
$157$	−18.1894	−1.45167	−0.725837	+	0.687867i	$0.758547\pi$
$158$	0	0
$159$	−5.09331	−0.403925
$160$	0	0
$161$	4.98092	0.392552
$162$	0	0
$163$	−6.34677	−0.497117	−0.248559	−	0.968617i	$-0.579957\pi$
$163$	−6.34677	−0.497117	−0.248559	+	0.968617i	$0.579957\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	13.9261	1.07763	0.538817	−	0.842423i	$-0.318871\pi$
$167$	13.9261	1.07763	0.538817	+	0.842423i	$0.318871\pi$
$168$	0	0
$169$	−12.9613	−0.997020
$170$	0	0
$171$	−1.54118	−0.117857
$172$	0	0
$173$	−24.2172	−1.84120	−0.920598	−	0.390511i	$-0.872299\pi$
$173$	−24.2172	−1.84120	−0.920598	+	0.390511i	$0.872299\pi$
$174$	0	0
$175$	−2.74075	−0.207181
$176$	0	0
$177$	26.8654	2.01932
$178$	0	0
$179$	−1.41972	−0.106115	−0.0530573	−	0.998591i	$-0.516897\pi$
$179$	−1.41972	−0.106115	−0.0530573	+	0.998591i	$0.516897\pi$
$180$	0	0
$181$	5.62889	0.418392	0.209196	−	0.977874i	$-0.432915\pi$
$181$	5.62889	0.418392	0.209196	+	0.977874i	$0.432915\pi$
$182$	0	0
$183$	14.2564	1.05386
$184$	0	0
$185$	−4.57623	−0.336451
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−12.5675	−0.914154
$190$	0	0
$191$	2.24259	0.162268	0.0811341	−	0.996703i	$-0.474146\pi$
$191$	2.24259	0.162268	0.0811341	+	0.996703i	$0.474146\pi$
$192$	0	0
$193$	9.30699	0.669932	0.334966	−	0.942230i	$-0.391275\pi$
$193$	9.30699	0.669932	0.334966	+	0.942230i	$0.391275\pi$
$194$	0	0
$195$	−0.372129	−0.0266487
$196$	0	0
$197$	−14.3935	−1.02549	−0.512747	−	0.858540i	$-0.671372\pi$
$197$	−14.3935	−1.02549	−0.512747	+	0.858540i	$0.671372\pi$
$198$	0	0
$199$	−3.00915	−0.213313	−0.106657	−	0.994296i	$-0.534015\pi$
$199$	−3.00915	−0.213313	−0.106657	+	0.994296i	$0.534015\pi$
$200$	0	0
$201$	20.1360	1.42028
$202$	0	0
$203$	3.78076	0.265358
$204$	0	0
$205$	1.23428	0.0862059
$206$	0	0
$207$	−1.04450	−0.0725980
$208$	0	0
$209$	0	0
$210$	0	0
$211$	15.6428	1.07689	0.538447	−	0.842659i	$-0.319011\pi$
$211$	15.6428	1.07689	0.538447	+	0.842659i	$0.319011\pi$
$212$	0	0
$213$	−13.0018	−0.890868
$214$	0	0
$215$	−2.96514	−0.202221
$216$	0	0
$217$	16.1817	1.09848
$218$	0	0
$219$	19.7226	1.33273
$220$	0	0
$221$	−0.288950	−0.0194369
$222$	0	0
$223$	−5.21134	−0.348977	−0.174488	−	0.984659i	$-0.555827\pi$
$223$	−5.21134	−0.348977	−0.174488	+	0.984659i	$0.555827\pi$
$224$	0	0
$225$	0.574737	0.0383158
$226$	0	0
$227$	−13.7838	−0.914861	−0.457430	−	0.889245i	$-0.651230\pi$
$227$	−13.7838	−0.914861	−0.457430	+	0.889245i	$0.651230\pi$
$228$	0	0
$229$	24.5800	1.62429	0.812147	−	0.583453i	$-0.198299\pi$
$229$	24.5800	1.62429	0.812147	+	0.583453i	$0.198299\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−8.02886	−0.525988	−0.262994	−	0.964797i	$-0.584710\pi$
$233$	−8.02886	−0.525988	−0.262994	+	0.964797i	$0.584710\pi$
$234$	0	0
$235$	9.30107	0.606735
$236$	0	0
$237$	−3.54181	−0.230066
$238$	0	0
$239$	−1.36215	−0.0881102	−0.0440551	−	0.999029i	$-0.514028\pi$
$239$	−1.36215	−0.0881102	−0.0440551	+	0.999029i	$0.514028\pi$
$240$	0	0
$241$	2.94349	0.189607	0.0948035	−	0.995496i	$-0.469778\pi$
$241$	2.94349	0.189607	0.0948035	+	0.995496i	$0.469778\pi$
$242$	0	0
$243$	5.89538	0.378189
$244$	0	0
$245$	−0.511708	−0.0326918
$246$	0	0
$247$	0.527783	0.0335820
$248$	0	0
$249$	10.9547	0.694223
$250$	0	0
$251$	−15.7107	−0.991651	−0.495826	−	0.868422i	$-0.665134\pi$
$251$	−15.7107	−0.991651	−0.495826	+	0.868422i	$0.665134\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	2.77571	0.173821
$256$	0	0
$257$	−8.90470	−0.555460	−0.277730	−	0.960659i	$-0.589582\pi$
$257$	−8.90470	−0.555460	−0.277730	+	0.960659i	$0.589582\pi$
$258$	0	0
$259$	−12.5423	−0.779341
$260$	0	0
$261$	−0.792829	−0.0490749
$262$	0	0
$263$	0.671733	0.0414208	0.0207104	−	0.999786i	$-0.493407\pi$
$263$	0.671733	0.0414208	0.0207104	+	0.999786i	$0.493407\pi$
$264$	0	0
$265$	−2.69388	−0.165484
$266$	0	0
$267$	−32.7203	−2.00245
$268$	0	0
$269$	26.7105	1.62857	0.814283	−	0.580468i	$-0.197130\pi$
$269$	26.7105	1.62857	0.814283	+	0.580468i	$0.197130\pi$
$270$	0	0
$271$	22.3089	1.35517	0.677584	−	0.735446i	$-0.263027\pi$
$271$	22.3089	1.35517	0.677584	+	0.735446i	$0.263027\pi$
$272$	0	0
$273$	−1.01991	−0.0617280
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−22.9001	−1.37593	−0.687967	−	0.725742i	$-0.741497\pi$
$277$	−22.9001	−1.37593	−0.687967	+	0.725742i	$0.741497\pi$
$278$	0	0
$279$	−3.39331	−0.203152
$280$	0	0
$281$	28.3433	1.69082	0.845411	−	0.534117i	$-0.179356\pi$
$281$	28.3433	1.69082	0.845411	+	0.534117i	$0.179356\pi$
$282$	0	0
$283$	−31.0561	−1.84609	−0.923046	−	0.384690i	$-0.874308\pi$
$283$	−31.0561	−1.84609	−0.923046	+	0.384690i	$0.874308\pi$
$284$	0	0
$285$	−5.06997	−0.300319
$286$	0	0
$287$	3.38286	0.199684
$288$	0	0
$289$	−14.8447	−0.873219
$290$	0	0
$291$	34.6354	2.03036
$292$	0	0
$293$	−7.84981	−0.458591	−0.229295	−	0.973357i	$-0.573642\pi$
$293$	−7.84981	−0.458591	−0.229295	+	0.973357i	$0.573642\pi$
$294$	0	0
$295$	14.2092	0.827293
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0.357695	0.0206860
$300$	0	0
$301$	−8.12671	−0.468416
$302$	0	0
$303$	9.45423	0.543131
$304$	0	0
$305$	7.54027	0.431755
$306$	0	0
$307$	10.0601	0.574160	0.287080	−	0.957907i	$-0.407315\pi$
$307$	10.0601	0.574160	0.287080	+	0.957907i	$0.407315\pi$
$308$	0	0
$309$	17.3539	0.987228
$310$	0	0
$311$	22.3650	1.26820	0.634101	−	0.773250i	$-0.281370\pi$
$311$	22.3650	1.26820	0.634101	+	0.773250i	$0.281370\pi$
$312$	0	0
$313$	18.1486	1.02582	0.512910	−	0.858442i	$-0.328567\pi$
$313$	18.1486	1.02582	0.512910	+	0.858442i	$0.328567\pi$
$314$	0	0
$315$	1.57521	0.0887531
$316$	0	0
$317$	−2.51660	−0.141346	−0.0706732	−	0.997500i	$-0.522515\pi$
$317$	−2.51660	−0.141346	−0.0706732	+	0.997500i	$0.522515\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	31.7857	1.77411
$322$	0	0
$323$	−3.93673	−0.219045
$324$	0	0
$325$	−0.196821	−0.0109177
$326$	0	0
$327$	−18.7383	−1.03623
$328$	0	0
$329$	25.4919	1.40541
$330$	0	0
$331$	7.54546	0.414736	0.207368	−	0.978263i	$-0.433510\pi$
$331$	7.54546	0.414736	0.207368	+	0.978263i	$0.433510\pi$
$332$	0	0
$333$	2.63013	0.144130
$334$	0	0
$335$	10.6500	0.581872
$336$	0	0
$337$	23.0017	1.25298	0.626491	−	0.779429i	$-0.284490\pi$
$337$	23.0017	1.25298	0.626491	+	0.779429i	$0.284490\pi$
$338$	0	0
$339$	−23.9576	−1.30120
$340$	0	0
$341$	0	0
$342$	0	0
$343$	17.7828	0.960180
$344$	0	0
$345$	−3.43607	−0.184992
$346$	0	0
$347$	4.07466	0.218739	0.109370	−	0.994001i	$-0.465117\pi$
$347$	4.07466	0.218739	0.109370	+	0.994001i	$0.465117\pi$
$348$	0	0
$349$	23.8075	1.27439	0.637193	−	0.770704i	$-0.280095\pi$
$349$	23.8075	1.27439	0.637193	+	0.770704i	$0.280095\pi$
$350$	0	0
$351$	−0.902511	−0.0481725
$352$	0	0
$353$	−7.00060	−0.372605	−0.186302	−	0.982492i	$-0.559650\pi$
$353$	−7.00060	−0.372605	−0.186302	+	0.982492i	$0.559650\pi$
$354$	0	0
$355$	−6.87672	−0.364978
$356$	0	0
$357$	7.60752	0.402632
$358$	0	0
$359$	36.4700	1.92482	0.962408	−	0.271609i	$-0.0875557\pi$
$359$	36.4700	1.92482	0.962408	+	0.271609i	$0.0875557\pi$
$360$	0	0
$361$	−11.8094	−0.621545
$362$	0	0
$363$	0	0
$364$	0	0
$365$	10.4314	0.546005
$366$	0	0
$367$	29.1854	1.52346	0.761732	−	0.647892i	$-0.224349\pi$
$367$	29.1854	1.52346	0.761732	+	0.647892i	$0.224349\pi$
$368$	0	0
$369$	−0.709387	−0.0369292
$370$	0	0
$371$	−7.38324	−0.383319
$372$	0	0
$373$	−20.0215	−1.03667	−0.518337	−	0.855176i	$-0.673449\pi$
$373$	−20.0215	−1.03667	−0.518337	+	0.855176i	$0.673449\pi$
$374$	0	0
$375$	1.89070	0.0976352
$376$	0	0
$377$	0.271508	0.0139833
$378$	0	0
$379$	−8.18971	−0.420677	−0.210338	−	0.977629i	$-0.567457\pi$
$379$	−8.18971	−0.420677	−0.210338	+	0.977629i	$0.567457\pi$
$380$	0	0
$381$	−19.7397	−1.01129
$382$	0	0
$383$	−20.2852	−1.03653	−0.518263	−	0.855221i	$-0.673421\pi$
$383$	−20.2852	−1.03653	−0.518263	+	0.855221i	$0.673421\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	1.70418	0.0866282
$388$	0	0
$389$	13.0260	0.660443	0.330221	−	0.943904i	$-0.392877\pi$
$389$	13.0260	0.660443	0.330221	+	0.943904i	$0.392877\pi$
$390$	0	0
$391$	−2.66804	−0.134928
$392$	0	0
$393$	39.6171	1.99842
$394$	0	0
$395$	−1.87328	−0.0942552
$396$	0	0
$397$	−0.593277	−0.0297757	−0.0148878	−	0.999889i	$-0.504739\pi$
$397$	−0.593277	−0.0297757	−0.0148878	+	0.999889i	$0.504739\pi$
$398$	0	0
$399$	−13.8955	−0.695647
$400$	0	0
$401$	−9.04664	−0.451768	−0.225884	−	0.974154i	$-0.572527\pi$
$401$	−9.04664	−0.451768	−0.225884	+	0.974154i	$0.572527\pi$
$402$	0	0
$403$	1.16205	0.0578860
$404$	0	0
$405$	10.3939	0.516476
$406$	0	0
$407$	0	0
$408$	0	0
$409$	17.7566	0.878008	0.439004	−	0.898485i	$-0.355331\pi$
$409$	17.7566	0.878008	0.439004	+	0.898485i	$0.355331\pi$
$410$	0	0
$411$	−5.59804	−0.276131
$412$	0	0
$413$	38.9439	1.91631
$414$	0	0
$415$	5.79397	0.284415
$416$	0	0
$417$	−31.0553	−1.52079
$418$	0	0
$419$	9.43701	0.461028	0.230514	−	0.973069i	$-0.425959\pi$
$419$	9.43701	0.461028	0.230514	+	0.973069i	$0.425959\pi$
$420$	0	0
$421$	−24.2461	−1.18168	−0.590842	−	0.806787i	$-0.701204\pi$
$421$	−24.2461	−1.18168	−0.590842	+	0.806787i	$0.701204\pi$
$422$	0	0
$423$	−5.34567	−0.259915
$424$	0	0
$425$	1.46809	0.0712126
$426$	0	0
$427$	20.6660	1.00010
$428$	0	0
$429$	0	0
$430$	0	0
$431$	9.61485	0.463131	0.231565	−	0.972819i	$-0.425615\pi$
$431$	9.61485	0.463131	0.231565	+	0.972819i	$0.425615\pi$
$432$	0	0
$433$	−9.44277	−0.453791	−0.226895	−	0.973919i	$-0.572858\pi$
$433$	−9.44277	−0.453791	−0.226895	+	0.973919i	$0.572858\pi$
$434$	0	0
$435$	−2.60815	−0.125051
$436$	0	0
$437$	4.87331	0.233122
$438$	0	0
$439$	17.8211	0.850555	0.425278	−	0.905063i	$-0.360176\pi$
$439$	17.8211	0.850555	0.425278	+	0.905063i	$0.360176\pi$
$440$	0	0
$441$	0.294098	0.0140047
$442$	0	0
$443$	17.6167	0.836995	0.418497	−	0.908218i	$-0.362557\pi$
$443$	17.6167	0.836995	0.418497	+	0.908218i	$0.362557\pi$
$444$	0	0
$445$	−17.3060	−0.820381
$446$	0	0
$447$	−36.2703	−1.71553
$448$	0	0
$449$	−8.01122	−0.378073	−0.189036	−	0.981970i	$-0.560536\pi$
$449$	−8.01122	−0.378073	−0.189036	+	0.981970i	$0.560536\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	17.1107	0.803929
$454$	0	0
$455$	−0.539438	−0.0252892
$456$	0	0
$457$	−8.21578	−0.384318	−0.192159	−	0.981364i	$-0.561549\pi$
$457$	−8.21578	−0.384318	−0.192159	+	0.981364i	$0.561549\pi$
$458$	0	0
$459$	6.73182	0.314214
$460$	0	0
$461$	−5.47559	−0.255024	−0.127512	−	0.991837i	$-0.540699\pi$
$461$	−5.47559	−0.255024	−0.127512	+	0.991837i	$0.540699\pi$
$462$	0	0
$463$	−27.1843	−1.26336	−0.631681	−	0.775228i	$-0.717635\pi$
$463$	−27.1843	−1.26336	−0.631681	+	0.775228i	$0.717635\pi$
$464$	0	0
$465$	−11.1629	−0.517667
$466$	0	0
$467$	−7.05376	−0.326409	−0.163205	−	0.986592i	$-0.552183\pi$
$467$	−7.05376	−0.326409	−0.163205	+	0.986592i	$0.552183\pi$
$468$	0	0
$469$	29.1890	1.34782
$470$	0	0
$471$	34.3907	1.58464
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−2.68154	−0.123037
$476$	0	0
$477$	1.54827	0.0708905
$478$	0	0
$479$	−5.31048	−0.242642	−0.121321	−	0.992613i	$-0.538713\pi$
$479$	−5.31048	−0.242642	−0.121321	+	0.992613i	$0.538713\pi$
$480$	0	0
$481$	−0.900700	−0.0410684
$482$	0	0
$483$	−9.41742	−0.428507
$484$	0	0
$485$	18.3188	0.831816
$486$	0	0
$487$	−11.8480	−0.536884	−0.268442	−	0.963296i	$-0.586509\pi$
$487$	−11.8480	−0.536884	−0.268442	+	0.963296i	$0.586509\pi$
$488$	0	0
$489$	11.9998	0.542651
$490$	0	0
$491$	38.3787	1.73201	0.866003	−	0.500038i	$-0.166681\pi$
$491$	38.3787	1.73201	0.866003	+	0.500038i	$0.166681\pi$
$492$	0	0
$493$	−2.02517	−0.0912091
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−18.8474	−0.845420
$498$	0	0
$499$	−10.6869	−0.478411	−0.239205	−	0.970969i	$-0.576887\pi$
$499$	−10.6869	−0.478411	−0.239205	+	0.970969i	$0.576887\pi$
$500$	0	0
$501$	−26.3300	−1.17634
$502$	0	0
$503$	35.5077	1.58321	0.791606	−	0.611032i	$-0.209245\pi$
$503$	35.5077	1.58321	0.791606	+	0.611032i	$0.209245\pi$
$504$	0	0
$505$	5.00039	0.222515
$506$	0	0
$507$	24.5058	1.08834
$508$	0	0
$509$	−21.3976	−0.948433	−0.474216	−	0.880408i	$-0.657269\pi$
$509$	−21.3976	−0.948433	−0.474216	+	0.880408i	$0.657269\pi$
$510$	0	0
$511$	28.5899	1.26474
$512$	0	0
$513$	−12.2960	−0.542883
$514$	0	0
$515$	9.17856	0.404456
$516$	0	0
$517$	0	0
$518$	0	0
$519$	45.7873	2.00984
$520$	0	0
$521$	−19.3712	−0.848669	−0.424334	−	0.905506i	$-0.639492\pi$
$521$	−19.3712	−0.848669	−0.424334	+	0.905506i	$0.639492\pi$
$522$	0	0
$523$	0.894305	0.0391052	0.0195526	−	0.999809i	$-0.493776\pi$
$523$	0.894305	0.0391052	0.0195526	+	0.999809i	$0.493776\pi$
$524$	0	0
$525$	5.18193	0.226158
$526$	0	0
$527$	−8.66774	−0.377573
$528$	0	0
$529$	−19.6972	−0.856400
$530$	0	0
$531$	−8.16657	−0.354399
$532$	0	0
$533$	0.242933	0.0105226
$534$	0	0
$535$	16.8116	0.726831
$536$	0	0
$537$	2.68425	0.115834
$538$	0	0
$539$	0	0
$540$	0	0
$541$	40.1527	1.72630	0.863150	−	0.504948i	$-0.168488\pi$
$541$	40.1527	1.72630	0.863150	+	0.504948i	$0.168488\pi$
$542$	0	0
$543$	−10.6425	−0.456715
$544$	0	0
$545$	−9.91076	−0.424530
$546$	0	0
$547$	−4.49729	−0.192290	−0.0961451	−	0.995367i	$-0.530651\pi$
$547$	−4.49729	−0.192290	−0.0961451	+	0.995367i	$0.530651\pi$
$548$	0	0
$549$	−4.33367	−0.184957
$550$	0	0
$551$	3.69908	0.157586
$552$	0	0
$553$	−5.13420	−0.218329
$554$	0	0
$555$	8.65227	0.367269
$556$	0	0
$557$	23.9736	1.01580	0.507898	−	0.861417i	$-0.330423\pi$
$557$	23.9736	1.01580	0.507898	+	0.861417i	$0.330423\pi$
$558$	0	0
$559$	−0.583603	−0.0246838
$560$	0	0
$561$	0	0
$562$	0	0
$563$	18.3442	0.773116	0.386558	−	0.922265i	$-0.373664\pi$
$563$	18.3442	0.773116	0.386558	+	0.922265i	$0.373664\pi$
$564$	0	0
$565$	−12.6713	−0.533087
$566$	0	0
$567$	28.4870	1.19634
$568$	0	0
$569$	2.70511	0.113404	0.0567020	−	0.998391i	$-0.481942\pi$
$569$	2.70511	0.113404	0.0567020	+	0.998391i	$0.481942\pi$
$570$	0	0
$571$	13.6593	0.571626	0.285813	−	0.958285i	$-0.407736\pi$
$571$	13.6593	0.571626	0.285813	+	0.958285i	$0.407736\pi$
$572$	0	0
$573$	−4.24006	−0.177131
$574$	0	0
$575$	−1.81736	−0.0757891
$576$	0	0
$577$	8.23640	0.342886	0.171443	−	0.985194i	$-0.445157\pi$
$577$	8.23640	0.342886	0.171443	+	0.985194i	$0.445157\pi$
$578$	0	0
$579$	−17.5967	−0.731294
$580$	0	0
$581$	15.8798	0.658806
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0.113120	0.00467696
$586$	0	0
$587$	−22.3766	−0.923581	−0.461791	−	0.886989i	$-0.652793\pi$
$587$	−22.3766	−0.923581	−0.461791	+	0.886989i	$0.652793\pi$
$588$	0	0
$589$	15.8321	0.652350
$590$	0	0
$591$	27.2138	1.11942
$592$	0	0
$593$	44.5253	1.82844	0.914218	−	0.405223i	$-0.132806\pi$
$593$	44.5253	1.82844	0.914218	+	0.405223i	$0.132806\pi$
$594$	0	0
$595$	4.02366	0.164954
$596$	0	0
$597$	5.68940	0.232851
$598$	0	0
$599$	−39.9292	−1.63146	−0.815732	−	0.578430i	$-0.803666\pi$
$599$	−39.9292	−1.63146	−0.815732	+	0.578430i	$0.803666\pi$
$600$	0	0
$601$	−30.9828	−1.26381	−0.631907	−	0.775044i	$-0.717727\pi$
$601$	−30.9828	−1.26381	−0.631907	+	0.775044i	$0.717727\pi$
$602$	0	0
$603$	−6.12096	−0.249265
$604$	0	0
$605$	0	0
$606$	0	0
$607$	26.5634	1.07817	0.539087	−	0.842250i	$-0.318769\pi$
$607$	26.5634	1.07817	0.539087	+	0.842250i	$0.318769\pi$
$608$	0	0
$609$	−7.14828	−0.289663
$610$	0	0
$611$	1.83065	0.0740601
$612$	0	0
$613$	−28.2647	−1.14160	−0.570801	−	0.821089i	$-0.693367\pi$
$613$	−28.2647	−1.14160	−0.570801	+	0.821089i	$0.693367\pi$
$614$	0	0
$615$	−2.33365	−0.0941020
$616$	0	0
$617$	−3.98722	−0.160520	−0.0802598	−	0.996774i	$-0.525575\pi$
$617$	−3.98722	−0.160520	−0.0802598	+	0.996774i	$0.525575\pi$
$618$	0	0
$619$	−29.6186	−1.19047	−0.595236	−	0.803551i	$-0.702941\pi$
$619$	−29.6186	−1.19047	−0.595236	+	0.803551i	$0.702941\pi$
$620$	0	0
$621$	−8.33338	−0.334407
$622$	0	0
$623$	−47.4313	−1.90029
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	6.71831	0.267876
$630$	0	0
$631$	34.7749	1.38437	0.692183	−	0.721722i	$-0.256649\pi$
$631$	34.7749	1.38437	0.692183	+	0.721722i	$0.256649\pi$
$632$	0	0
$633$	−29.5758	−1.17553
$634$	0	0
$635$	−10.4404	−0.414315
$636$	0	0
$637$	−0.100715	−0.00399047
$638$	0	0
$639$	3.95231	0.156351
$640$	0	0
$641$	27.0319	1.06770	0.533848	−	0.845580i	$-0.320745\pi$
$641$	27.0319	1.06770	0.533848	+	0.845580i	$0.320745\pi$
$642$	0	0
$643$	31.3855	1.23772	0.618861	−	0.785500i	$-0.287594\pi$
$643$	31.3855	1.23772	0.618861	+	0.785500i	$0.287594\pi$
$644$	0	0
$645$	5.60619	0.220743
$646$	0	0
$647$	43.1624	1.69689	0.848444	−	0.529284i	$-0.177540\pi$
$647$	43.1624	1.69689	0.848444	+	0.529284i	$0.177540\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	−30.5947	−1.19910
$652$	0	0
$653$	9.89228	0.387115	0.193557	−	0.981089i	$-0.437997\pi$
$653$	9.89228	0.387115	0.193557	+	0.981089i	$0.437997\pi$
$654$	0	0
$655$	20.9537	0.818729
$656$	0	0
$657$	−5.99532	−0.233900
$658$	0	0
$659$	12.3139	0.479683	0.239842	−	0.970812i	$-0.422904\pi$
$659$	12.3139	0.479683	0.239842	+	0.970812i	$0.422904\pi$
$660$	0	0
$661$	−26.4390	−1.02836	−0.514179	−	0.857683i	$-0.671903\pi$
$661$	−26.4390	−1.02836	−0.514179	+	0.857683i	$0.671903\pi$
$662$	0	0
$663$	0.546318	0.0212172
$664$	0	0
$665$	−7.34942	−0.284998
$666$	0	0
$667$	2.50698	0.0970706
$668$	0	0
$669$	9.85306	0.380941
$670$	0	0
$671$	0	0
$672$	0	0
$673$	14.6219	0.563632	0.281816	−	0.959469i	$-0.409063\pi$
$673$	14.6219	0.563632	0.281816	+	0.959469i	$0.409063\pi$
$674$	0	0
$675$	4.58544	0.176494
$676$	0	0
$677$	29.1645	1.12088	0.560441	−	0.828195i	$-0.310632\pi$
$677$	29.1645	1.12088	0.560441	+	0.828195i	$0.310632\pi$
$678$	0	0
$679$	50.2074	1.92678
$680$	0	0
$681$	26.0609	0.998657
$682$	0	0
$683$	−25.7011	−0.983427	−0.491713	−	0.870757i	$-0.663629\pi$
$683$	−25.7011	−0.983427	−0.491713	+	0.870757i	$0.663629\pi$
$684$	0	0
$685$	−2.96083	−0.113128
$686$	0	0
$687$	−46.4734	−1.77307
$688$	0	0
$689$	−0.530212	−0.0201995
$690$	0	0
$691$	−1.87725	−0.0714139	−0.0357070	−	0.999362i	$-0.511368\pi$
$691$	−1.87725	−0.0714139	−0.0357070	+	0.999362i	$0.511368\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−16.4253	−0.623048
$696$	0	0
$697$	−1.81203	−0.0686356
$698$	0	0
$699$	15.1801	0.574166
$700$	0	0
$701$	−39.4735	−1.49089	−0.745447	−	0.666565i	$-0.767764\pi$
$701$	−39.4735	−1.49089	−0.745447	+	0.666565i	$0.767764\pi$
$702$	0	0
$703$	−12.2713	−0.462822
$704$	0	0
$705$	−17.5855	−0.662308
$706$	0	0
$707$	13.7048	0.515423
$708$	0	0
$709$	−34.7390	−1.30465	−0.652325	−	0.757939i	$-0.726206\pi$
$709$	−34.7390	−1.30465	−0.652325	+	0.757939i	$0.726206\pi$
$710$	0	0
$711$	1.07665	0.0403774
$712$	0	0
$713$	10.7299	0.401837
$714$	0	0
$715$	0	0
$716$	0	0
$717$	2.57542	0.0961807
$718$	0	0
$719$	−21.7972	−0.812897	−0.406449	−	0.913674i	$-0.633233\pi$
$719$	−21.7972	−0.812897	−0.406449	+	0.913674i	$0.633233\pi$
$720$	0	0
$721$	25.1561	0.936864
$722$	0	0
$723$	−5.56525	−0.206974
$724$	0	0
$725$	−1.37946	−0.0512320
$726$	0	0
$727$	−33.0916	−1.22730	−0.613650	−	0.789578i	$-0.710299\pi$
$727$	−33.0916	−1.22730	−0.613650	+	0.789578i	$0.710299\pi$
$728$	0	0
$729$	20.0353	0.742047
$730$	0	0
$731$	4.35308	0.161005
$732$	0	0
$733$	37.8016	1.39624	0.698118	−	0.715983i	$-0.254021\pi$
$733$	37.8016	1.39624	0.698118	+	0.715983i	$0.254021\pi$
$734$	0	0
$735$	0.967485	0.0356862
$736$	0	0
$737$	0	0
$738$	0	0
$739$	27.8114	1.02306	0.511529	−	0.859266i	$-0.329079\pi$
$739$	27.8114	1.02306	0.511529	+	0.859266i	$0.329079\pi$
$740$	0	0
$741$	−0.997878	−0.0366580
$742$	0	0
$743$	22.6766	0.831925	0.415963	−	0.909382i	$-0.363445\pi$
$743$	22.6766	0.831925	0.415963	+	0.909382i	$0.363445\pi$
$744$	0	0
$745$	−19.1836	−0.702832
$746$	0	0
$747$	−3.33001	−0.121839
$748$	0	0
$749$	46.0765	1.68360
$750$	0	0
$751$	−1.70546	−0.0622330	−0.0311165	−	0.999516i	$-0.509906\pi$
$751$	−1.70546	−0.0622330	−0.0311165	+	0.999516i	$0.509906\pi$
$752$	0	0
$753$	29.7042	1.08248
$754$	0	0
$755$	9.04992	0.329360
$756$	0	0
$757$	−8.27807	−0.300872	−0.150436	−	0.988620i	$-0.548068\pi$
$757$	−8.27807	−0.300872	−0.150436	+	0.988620i	$0.548068\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	14.2064	0.514982	0.257491	−	0.966281i	$-0.417104\pi$
$761$	14.2064	0.514982	0.257491	+	0.966281i	$0.417104\pi$
$762$	0	0
$763$	−27.1629	−0.983364
$764$	0	0
$765$	−0.843764	−0.0305063
$766$	0	0
$767$	2.79668	0.100982
$768$	0	0
$769$	31.0174	1.11852	0.559258	−	0.828993i	$-0.311086\pi$
$769$	31.0174	1.11852	0.559258	+	0.828993i	$0.311086\pi$
$770$	0	0
$771$	16.8361	0.606337
$772$	0	0
$773$	−12.2223	−0.439607	−0.219803	−	0.975544i	$-0.570542\pi$
$773$	−12.2223	−0.439607	−0.219803	+	0.975544i	$0.570542\pi$
$774$	0	0
$775$	−5.90411	−0.212082
$776$	0	0
$777$	23.7137	0.850725
$778$	0	0
$779$	3.30977	0.118585
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−6.32544	−0.226053
$784$	0	0
$785$	18.1894	0.649208
$786$	0	0
$787$	7.68049	0.273780	0.136890	−	0.990586i	$-0.456289\pi$
$787$	7.68049	0.273780	0.136890	+	0.990586i	$0.456289\pi$
$788$	0	0
$789$	−1.27004	−0.0452147
$790$	0	0
$791$	−34.7289	−1.23482
$792$	0	0
$793$	1.48408	0.0527014
$794$	0	0
$795$	5.09331	0.180641
$796$	0	0
$797$	−1.14757	−0.0406491	−0.0203245	−	0.999793i	$-0.506470\pi$
$797$	−1.14757	−0.0406491	−0.0203245	+	0.999793i	$0.506470\pi$
$798$	0	0
$799$	−13.6548	−0.483071
$800$	0	0
$801$	9.94637	0.351438
$802$	0	0
$803$	0	0
$804$	0	0
$805$	−4.98092	−0.175554
$806$	0	0
$807$	−50.5014	−1.77773
$808$	0	0
$809$	−0.494854	−0.0173982	−0.00869908	−	0.999962i	$-0.502769\pi$
$809$	−0.494854	−0.0173982	−0.00869908	+	0.999962i	$0.502769\pi$
$810$	0	0
$811$	23.7818	0.835091	0.417546	−	0.908656i	$-0.362890\pi$
$811$	23.7818	0.835091	0.417546	+	0.908656i	$0.362890\pi$
$812$	0	0
$813$	−42.1793	−1.47929
$814$	0	0
$815$	6.34677	0.222318
$816$	0	0
$817$	−7.95113	−0.278175
$818$	0	0
$819$	0.310035	0.0108335
$820$	0	0
$821$	−41.7817	−1.45819	−0.729096	−	0.684412i	$-0.760059\pi$
$821$	−41.7817	−1.45819	−0.729096	+	0.684412i	$0.760059\pi$
$822$	0	0
$823$	23.3527	0.814023	0.407011	−	0.913423i	$-0.366571\pi$
$823$	23.3527	0.814023	0.407011	+	0.913423i	$0.366571\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−19.2469	−0.669280	−0.334640	−	0.942346i	$-0.608615\pi$
$827$	−19.2469	−0.669280	−0.334640	+	0.942346i	$0.608615\pi$
$828$	0	0
$829$	17.6334	0.612434	0.306217	−	0.951962i	$-0.400937\pi$
$829$	17.6334	0.612434	0.306217	+	0.951962i	$0.400937\pi$
$830$	0	0
$831$	43.2972	1.50196
$832$	0	0
$833$	0.751232	0.0260286
$834$	0	0
$835$	−13.9261	−0.481932
$836$	0	0
$837$	−27.0729	−0.935778
$838$	0	0
$839$	29.8854	1.03176	0.515879	−	0.856662i	$-0.327465\pi$
$839$	29.8854	1.03176	0.515879	+	0.856662i	$0.327465\pi$
$840$	0	0
$841$	−27.0971	−0.934382
$842$	0	0
$843$	−53.5887	−1.84569
$844$	0	0
$845$	12.9613	0.445881
$846$	0	0
$847$	0	0
$848$	0	0
$849$	58.7176	2.01518
$850$	0	0
$851$	−8.31666	−0.285091
$852$	0	0
$853$	8.78136	0.300668	0.150334	−	0.988635i	$-0.451965\pi$
$853$	8.78136	0.300668	0.150334	+	0.988635i	$0.451965\pi$
$854$	0	0
$855$	1.54118	0.0527072
$856$	0	0
$857$	−25.6711	−0.876909	−0.438454	−	0.898753i	$-0.644474\pi$
$857$	−25.6711	−0.876909	−0.438454	+	0.898753i	$0.644474\pi$
$858$	0	0
$859$	41.5338	1.41711	0.708557	−	0.705654i	$-0.249346\pi$
$859$	41.5338	1.41711	0.708557	+	0.705654i	$0.249346\pi$
$860$	0	0
$861$	−6.39596	−0.217974
$862$	0	0
$863$	2.83297	0.0964353	0.0482177	−	0.998837i	$-0.484646\pi$
$863$	2.83297	0.0964353	0.0482177	+	0.998837i	$0.484646\pi$
$864$	0	0
$865$	24.2172	0.823408
$866$	0	0
$867$	28.0669	0.953201
$868$	0	0
$869$	0	0
$870$	0	0
$871$	2.09615	0.0710253
$872$	0	0
$873$	−10.5285	−0.356336
$874$	0	0
$875$	2.74075	0.0926542
$876$	0	0
$877$	−26.0319	−0.879035	−0.439518	−	0.898234i	$-0.644851\pi$
$877$	−26.0319	−0.879035	−0.439518	+	0.898234i	$0.644851\pi$
$878$	0	0
$879$	14.8416	0.500595
$880$	0	0
$881$	−37.6862	−1.26968	−0.634841	−	0.772643i	$-0.718934\pi$
$881$	−37.6862	−1.26968	−0.634841	+	0.772643i	$0.718934\pi$
$882$	0	0
$883$	47.5341	1.59965	0.799825	−	0.600233i	$-0.204926\pi$
$883$	47.5341	1.59965	0.799825	+	0.600233i	$0.204926\pi$
$884$	0	0
$885$	−26.8654	−0.903069
$886$	0	0
$887$	47.7310	1.60265	0.801326	−	0.598228i	$-0.204128\pi$
$887$	47.7310	1.60265	0.801326	+	0.598228i	$0.204128\pi$
$888$	0	0
$889$	−28.6146	−0.959701
$890$	0	0
$891$	0	0
$892$	0	0
$893$	24.9412	0.834624
$894$	0	0
$895$	1.41972	0.0474559
$896$	0	0
$897$	−0.676292	−0.0225807
$898$	0	0
$899$	8.14450	0.271634
$900$	0	0
$901$	3.95484	0.131755
$902$	0	0
$903$	15.3651	0.511320
$904$	0	0
$905$	−5.62889	−0.187111
$906$	0	0
$907$	−33.7132	−1.11943	−0.559714	−	0.828686i	$-0.689089\pi$
$907$	−33.7132	−1.11943	−0.559714	+	0.828686i	$0.689089\pi$
$908$	0	0
$909$	−2.87391	−0.0953216
$910$	0	0
$911$	−38.9640	−1.29094	−0.645468	−	0.763788i	$-0.723338\pi$
$911$	−38.9640	−1.29094	−0.645468	+	0.763788i	$0.723338\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	−14.2564	−0.471301
$916$	0	0
$917$	57.4289	1.89647
$918$	0	0
$919$	12.5411	0.413692	0.206846	−	0.978374i	$-0.433680\pi$
$919$	12.5411	0.413692	0.206846	+	0.978374i	$0.433680\pi$
$920$	0	0
$921$	−19.0206	−0.626750
$922$	0	0
$923$	−1.35348	−0.0445505
$924$	0	0
$925$	4.57623	0.150466
$926$	0	0
$927$	−5.27526	−0.173262
$928$	0	0
$929$	26.5557	0.871263	0.435631	−	0.900125i	$-0.356525\pi$
$929$	26.5557	0.871263	0.435631	+	0.900125i	$0.356525\pi$
$930$	0	0
$931$	−1.37216	−0.0449709
$932$	0	0
$933$	−42.2854	−1.38436
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−0.566670	−0.0185123	−0.00925615	−	0.999957i	$-0.502946\pi$
$937$	−0.566670	−0.0185123	−0.00925615	+	0.999957i	$0.502946\pi$
$938$	0	0
$939$	−34.3136	−1.11978
$940$	0	0
$941$	−11.4578	−0.373514	−0.186757	−	0.982406i	$-0.559798\pi$
$941$	−11.4578	−0.373514	−0.186757	+	0.982406i	$0.559798\pi$
$942$	0	0
$943$	2.24313	0.0730464
$944$	0	0
$945$	12.5675	0.408822
$946$	0	0
$947$	−30.2412	−0.982707	−0.491354	−	0.870960i	$-0.663498\pi$
$947$	−30.2412	−0.982707	−0.491354	+	0.870960i	$0.663498\pi$
$948$	0	0
$949$	2.05312	0.0666472
$950$	0	0
$951$	4.75813	0.154293
$952$	0	0
$953$	−41.7753	−1.35324	−0.676618	−	0.736334i	$-0.736555\pi$
$953$	−41.7753	−1.35324	−0.676618	+	0.736334i	$0.736555\pi$
$954$	0	0
$955$	−2.24259	−0.0725685
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−8.11490	−0.262044
$960$	0	0
$961$	3.85852	0.124468
$962$	0	0
$963$	−9.66227	−0.311362
$964$	0	0
$965$	−9.30699	−0.299603
$966$	0	0
$967$	13.9026	0.447078	0.223539	−	0.974695i	$-0.428239\pi$
$967$	13.9026	0.447078	0.223539	+	0.974695i	$0.428239\pi$
$968$	0	0
$969$	7.44316	0.239109
$970$	0	0
$971$	5.24997	0.168480	0.0842398	−	0.996446i	$-0.473154\pi$
$971$	5.24997	0.168480	0.0842398	+	0.996446i	$0.473154\pi$
$972$	0	0
$973$	−45.0177	−1.44320
$974$	0	0
$975$	0.372129	0.0119177
$976$	0	0
$977$	53.0523	1.69729	0.848646	−	0.528961i	$-0.177418\pi$
$977$	53.0523	1.69729	0.848646	+	0.528961i	$0.177418\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	5.69608	0.181862
$982$	0	0
$983$	−1.43729	−0.0458425	−0.0229212	−	0.999737i	$-0.507297\pi$
$983$	−1.43729	−0.0458425	−0.0229212	+	0.999737i	$0.507297\pi$
$984$	0	0
$985$	14.3935	0.458615
$986$	0	0
$987$	−48.1975	−1.53414
$988$	0	0
$989$	−5.38872	−0.171351
$990$	0	0
$991$	22.6101	0.718234	0.359117	−	0.933292i	$-0.383078\pi$
$991$	22.6101	0.718234	0.359117	+	0.933292i	$0.383078\pi$
$992$	0	0
$993$	−14.2662	−0.452724
$994$	0	0
$995$	3.00915	0.0953965
$996$	0	0
$997$	59.0567	1.87034	0.935172	−	0.354194i	$-0.115245\pi$
$997$	59.0567	1.87034	0.935172	+	0.354194i	$0.115245\pi$
$998$	0	0
$999$	20.9840	0.663906

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9680.2.a.cy.1.2		6
4.3	odd	2		4840.2.a.be.1.5		6
11.2	odd	10		880.2.bo.j.81.1		12
11.6	odd	10		880.2.bo.j.641.1		12
11.10	odd	2		9680.2.a.cx.1.2		6
44.35	even	10		440.2.y.b.81.3	✓	12
44.39	even	10		440.2.y.b.201.3	yes	12
44.43	even	2		4840.2.a.bf.1.5		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
440.2.y.b.81.3	✓	12	44.35	even	10
440.2.y.b.201.3	yes	12	44.39	even	10
880.2.bo.j.81.1		12	11.2	odd	10
880.2.bo.j.641.1		12	11.6	odd	10
4840.2.a.be.1.5		6	4.3	odd	2
4840.2.a.bf.1.5		6	44.43	even	2
9680.2.a.cx.1.2		6	11.10	odd	2
9680.2.a.cy.1.2		6	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 9680 = 2^{4} \cdot 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9680.a (trivial)

\(f(q)\)	\(=\)	\(q-1.89070 q^{3} -1.00000 q^{5} -2.74075 q^{7} +0.574737 q^{9} -0.196821 q^{13} +1.89070 q^{15} +1.46809 q^{17} -2.68154 q^{19} +5.18193 q^{21} -1.81736 q^{23} +1.00000 q^{25} +4.58544 q^{27} -1.37946 q^{29} -5.90411 q^{31} +2.74075 q^{35} +4.57623 q^{37} +0.372129 q^{39} -1.23428 q^{41} +2.96514 q^{43} -0.574737 q^{45} -9.30107 q^{47} +0.511708 q^{49} -2.77571 q^{51} +2.69388 q^{53} +5.06997 q^{57} -14.2092 q^{59} -7.54027 q^{61} -1.57521 q^{63} +0.196821 q^{65} -10.6500 q^{67} +3.43607 q^{69} +6.87672 q^{71} -10.4314 q^{73} -1.89070 q^{75} +1.87328 q^{79} -10.3939 q^{81} -5.79397 q^{83} -1.46809 q^{85} +2.60815 q^{87} +17.3060 q^{89} +0.539438 q^{91} +11.1629 q^{93} +2.68154 q^{95} -18.3188 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 2 q^{3} - 6 q^{5} + 6 q^{7} + 10 q^{9} + 6 q^{13} + 2 q^{15} - 11 q^{17} - 11 q^{19} - 2 q^{21} - 18 q^{23} + 6 q^{25} + q^{27} + 6 q^{29} - q^{31} - 6 q^{35} + 4 q^{37} + 27 q^{39} + 4 q^{41} + 3 q^{43}+ \cdots + 20 q^{97}+O(q^{100}) \) 6 * q - 2 * q^3 - 6 * q^5 + 6 * q^7 + 10 * q^9 + 6 * q^13 + 2 * q^15 - 11 * q^17 - 11 * q^19 - 2 * q^21 - 18 * q^23 + 6 * q^25 + q^27 + 6 * q^29 - q^31 - 6 * q^35 + 4 * q^37 + 27 * q^39 + 4 * q^41 + 3 * q^43 - 10 * q^45 - 14 * q^47 + 8 * q^49 + 31 * q^51 + 14 * q^53 + 5 * q^57 - 2 * q^59 + 4 * q^61 - 16 * q^63 - 6 * q^65 - 11 * q^67 + 8 * q^69 - 7 * q^71 + 9 * q^73 - 2 * q^75 + 36 * q^79 + 30 * q^81 - 45 * q^83 + 11 * q^85 + 25 * q^87 + q^89 + 8 * q^91 + 55 * q^93 + 11 * q^95 + 20 * q^97

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