LMFDB - Embedded newform 8032.2.a.i.1.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8032,2,Mod(1,8032)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8032, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("8032.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8032 = 2^{5} \cdot 251 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8032.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:	$64.1358429035$
Analytic rank:	$0$
Dimension:	$30$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.14
Character	$\chi$	$=$	8032.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.0957135 q^{3} +3.87548 q^{5} +3.14340 q^{7} -2.99084 q^{9} +O(q^{10})$	$q-0.0957135 q^{3} +3.87548 q^{5} +3.14340 q^{7} -2.99084 q^{9} +1.99009 q^{11} +3.58369 q^{13} -0.370936 q^{15} -2.39463 q^{17} +4.12285 q^{19} -0.300866 q^{21} -2.01919 q^{23} +10.0194 q^{25} +0.573404 q^{27} +8.01240 q^{29} +6.88587 q^{31} -0.190478 q^{33} +12.1822 q^{35} -2.67007 q^{37} -0.343008 q^{39} -2.97990 q^{41} -5.35779 q^{43} -11.5909 q^{45} -11.1484 q^{47} +2.88095 q^{49} +0.229198 q^{51} +4.98417 q^{53} +7.71255 q^{55} -0.394613 q^{57} +11.3719 q^{59} -6.37044 q^{61} -9.40140 q^{63} +13.8885 q^{65} +11.1673 q^{67} +0.193263 q^{69} -6.00682 q^{71} -3.84922 q^{73} -0.958989 q^{75} +6.25564 q^{77} -6.21181 q^{79} +8.91763 q^{81} +6.53463 q^{83} -9.28034 q^{85} -0.766894 q^{87} -12.5172 q^{89} +11.2650 q^{91} -0.659071 q^{93} +15.9781 q^{95} -5.14098 q^{97} -5.95203 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 30 q + 3 q^{3} + 13 q^{7} + 35 q^{9}+O(q^{10}) $ 30 * q + 3 * q^3 + 13 * q^7 + 35 * q^9	$ 30 q + 3 q^{3} + 13 q^{7} + 35 q^{9} + 13 q^{11} - 7 q^{13} + 28 q^{15} - 9 q^{17} - 17 q^{19} - 6 q^{21} + 43 q^{23} + 34 q^{25} + 12 q^{27} - q^{29} + 39 q^{31} + 17 q^{35} - q^{37} + 48 q^{39} - 3 q^{41} - 19 q^{43} + 66 q^{47} + 25 q^{49} - 14 q^{51} + 3 q^{53} + 50 q^{55} - 14 q^{57} + 27 q^{59} + 15 q^{61} + 75 q^{63} - 6 q^{65} + 8 q^{67} + 18 q^{69} + 64 q^{71} - 15 q^{73} + 9 q^{75} + 71 q^{79} + 6 q^{81} + 60 q^{83} + 15 q^{85} + 64 q^{87} - 32 q^{89} - 26 q^{91} - 4 q^{93} + 72 q^{95} - 4 q^{97} + 13 q^{99}+O(q^{100}) $ 30 * q + 3 * q^3 + 13 * q^7 + 35 * q^9 + 13 * q^11 - 7 * q^13 + 28 * q^15 - 9 * q^17 - 17 * q^19 - 6 * q^21 + 43 * q^23 + 34 * q^25 + 12 * q^27 - q^29 + 39 * q^31 + 17 * q^35 - q^37 + 48 * q^39 - 3 * q^41 - 19 * q^43 + 66 * q^47 + 25 * q^49 - 14 * q^51 + 3 * q^53 + 50 * q^55 - 14 * q^57 + 27 * q^59 + 15 * q^61 + 75 * q^63 - 6 * q^65 + 8 * q^67 + 18 * q^69 + 64 * q^71 - 15 * q^73 + 9 * q^75 + 71 * q^79 + 6 * q^81 + 60 * q^83 + 15 * q^85 + 64 * q^87 - 32 * q^89 - 26 * q^91 - 4 * q^93 + 72 * q^95 - 4 * q^97 + 13 * q^99

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
$3$	−0.0957135	−0.0552602	−0.0276301	−	0.999618i	$-0.508796\pi$
$3$	−0.0957135	−0.0552602	−0.0276301	+	0.999618i	$0.508796\pi$
$4$	0	0
$5$	3.87548	1.73317	0.866584	−	0.499031i	$-0.166310\pi$
$5$	3.87548	1.73317	0.866584	+	0.499031i	$0.166310\pi$
$6$	0	0
$7$	3.14340	1.18809	0.594046	−	0.804431i	$-0.297530\pi$
$7$	3.14340	1.18809	0.594046	+	0.804431i	$0.297530\pi$
$8$	0	0
$9$	−2.99084	−0.996946
$10$	0	0
$11$	1.99009	0.600034	0.300017	−	0.953934i	$-0.403008\pi$
$11$	1.99009	0.600034	0.300017	+	0.953934i	$0.403008\pi$
$12$	0	0
$13$	3.58369	0.993938	0.496969	−	0.867768i	$-0.334446\pi$
$13$	3.58369	0.993938	0.496969	+	0.867768i	$0.334446\pi$
$14$	0	0
$15$	−0.370936	−0.0957753
$16$	0	0
$17$	−2.39463	−0.580783	−0.290391	−	0.956908i	$-0.593786\pi$
$17$	−2.39463	−0.580783	−0.290391	+	0.956908i	$0.593786\pi$
$18$	0	0
$19$	4.12285	0.945848	0.472924	−	0.881103i	$-0.343199\pi$
$19$	4.12285	0.945848	0.472924	+	0.881103i	$0.343199\pi$
$20$	0	0
$21$	−0.300866	−0.0656543
$22$	0	0
$23$	−2.01919	−0.421029	−0.210515	−	0.977591i	$-0.567514\pi$
$23$	−2.01919	−0.421029	−0.210515	+	0.977591i	$0.567514\pi$
$24$	0	0
$25$	10.0194	2.00387
$26$	0	0
$27$	0.573404	0.110352
$28$	0	0
$29$	8.01240	1.48786	0.743932	−	0.668255i	$-0.232958\pi$
$29$	8.01240	1.48786	0.743932	+	0.668255i	$0.232958\pi$
$30$	0	0
$31$	6.88587	1.23674	0.618369	−	0.785888i	$-0.287794\pi$
$31$	6.88587	1.23674	0.618369	+	0.785888i	$0.287794\pi$
$32$	0	0
$33$	−0.190478	−0.0331580
$34$	0	0
$35$	12.1822	2.05917
$36$	0	0
$37$	−2.67007	−0.438957	−0.219478	−	0.975617i	$-0.570436\pi$
$37$	−2.67007	−0.438957	−0.219478	+	0.975617i	$0.570436\pi$
$38$	0	0
$39$	−0.343008	−0.0549252
$40$	0	0
$41$	−2.97990	−0.465383	−0.232691	−	0.972551i	$-0.574753\pi$
$41$	−2.97990	−0.465383	−0.232691	+	0.972551i	$0.574753\pi$
$42$	0	0
$43$	−5.35779	−0.817055	−0.408527	−	0.912746i	$-0.633958\pi$
$43$	−5.35779	−0.817055	−0.408527	+	0.912746i	$0.633958\pi$
$44$	0	0
$45$	−11.5909	−1.72788
$46$	0	0
$47$	−11.1484	−1.62617	−0.813084	−	0.582147i	$-0.802213\pi$
$47$	−11.1484	−1.62617	−0.813084	+	0.582147i	$0.802213\pi$
$48$	0	0
$49$	2.88095	0.411564
$50$	0	0
$51$	0.229198	0.0320942
$52$	0	0
$53$	4.98417	0.684628	0.342314	−	0.939586i	$-0.388789\pi$
$53$	4.98417	0.684628	0.342314	+	0.939586i	$0.388789\pi$
$54$	0	0
$55$	7.71255	1.03996
$56$	0	0
$57$	−0.394613	−0.0522677
$58$	0	0
$59$	11.3719	1.48050	0.740251	−	0.672331i	$-0.234707\pi$
$59$	11.3719	1.48050	0.740251	+	0.672331i	$0.234707\pi$
$60$	0	0
$61$	−6.37044	−0.815651	−0.407826	−	0.913060i	$-0.633713\pi$
$61$	−6.37044	−0.815651	−0.407826	+	0.913060i	$0.633713\pi$
$62$	0	0
$63$	−9.40140	−1.18446
$64$	0	0
$65$	13.8885	1.72266
$66$	0	0
$67$	11.1673	1.36430	0.682150	−	0.731212i	$-0.261045\pi$
$67$	11.1673	1.36430	0.682150	+	0.731212i	$0.261045\pi$
$68$	0	0
$69$	0.193263	0.0232662
$70$	0	0
$71$	−6.00682	−0.712878	−0.356439	−	0.934319i	$-0.616009\pi$
$71$	−6.00682	−0.712878	−0.356439	+	0.934319i	$0.616009\pi$
$72$	0	0
$73$	−3.84922	−0.450517	−0.225258	−	0.974299i	$-0.572323\pi$
$73$	−3.84922	−0.450517	−0.225258	+	0.974299i	$0.572323\pi$
$74$	0	0
$75$	−0.958989	−0.110735
$76$	0	0
$77$	6.25564	0.712896
$78$	0	0
$79$	−6.21181	−0.698883	−0.349442	−	0.936958i	$-0.613629\pi$
$79$	−6.21181	−0.698883	−0.349442	+	0.936958i	$0.613629\pi$
$80$	0	0
$81$	8.91763	0.990848
$82$	0	0
$83$	6.53463	0.717269	0.358634	−	0.933478i	$-0.383243\pi$
$83$	6.53463	0.717269	0.358634	+	0.933478i	$0.383243\pi$
$84$	0	0
$85$	−9.28034	−1.00659
$86$	0	0
$87$	−0.766894	−0.0822197
$88$	0	0
$89$	−12.5172	−1.32682	−0.663408	−	0.748258i	$-0.730891\pi$
$89$	−12.5172	−1.32682	−0.663408	+	0.748258i	$0.730891\pi$
$90$	0	0
$91$	11.2650	1.18089
$92$	0	0
$93$	−0.659071	−0.0683425
$94$	0	0
$95$	15.9781	1.63931
$96$	0	0
$97$	−5.14098	−0.521988	−0.260994	−	0.965340i	$-0.584050\pi$
$97$	−5.14098	−0.521988	−0.260994	+	0.965340i	$0.584050\pi$
$98$	0	0
$99$	−5.95203	−0.598202
$100$	0	0
$101$	−2.81025	−0.279630	−0.139815	−	0.990178i	$-0.544651\pi$
$101$	−2.81025	−0.279630	−0.139815	+	0.990178i	$0.544651\pi$
$102$	0	0
$103$	−4.42639	−0.436145	−0.218073	−	0.975933i	$-0.569977\pi$
$103$	−4.42639	−0.436145	−0.218073	+	0.975933i	$0.569977\pi$
$104$	0	0
$105$	−1.16600	−0.113790
$106$	0	0
$107$	15.0067	1.45076	0.725378	−	0.688351i	$-0.241665\pi$
$107$	15.0067	1.45076	0.725378	+	0.688351i	$0.241665\pi$
$108$	0	0
$109$	2.84413	0.272418	0.136209	−	0.990680i	$-0.456508\pi$
$109$	2.84413	0.272418	0.136209	+	0.990680i	$0.456508\pi$
$110$	0	0
$111$	0.255562	0.0242569
$112$	0	0
$113$	4.69330	0.441509	0.220754	−	0.975329i	$-0.429148\pi$
$113$	4.69330	0.441509	0.220754	+	0.975329i	$0.429148\pi$
$114$	0	0
$115$	−7.82532	−0.729715
$116$	0	0
$117$	−10.7182	−0.990903
$118$	0	0
$119$	−7.52727	−0.690024
$120$	0	0
$121$	−7.03955	−0.639959
$122$	0	0
$123$	0.285217	0.0257171
$124$	0	0
$125$	19.4525	1.73988
$126$	0	0
$127$	8.37015	0.742731	0.371366	−	0.928487i	$-0.378890\pi$
$127$	8.37015	0.742731	0.371366	+	0.928487i	$0.378890\pi$
$128$	0	0
$129$	0.512813	0.0451506
$130$	0	0
$131$	−7.94797	−0.694417	−0.347209	−	0.937788i	$-0.612870\pi$
$131$	−7.94797	−0.694417	−0.347209	+	0.937788i	$0.612870\pi$
$132$	0	0
$133$	12.9598	1.12375
$134$	0	0
$135$	2.22222	0.191258
$136$	0	0
$137$	−3.75316	−0.320654	−0.160327	−	0.987064i	$-0.551255\pi$
$137$	−3.75316	−0.320654	−0.160327	+	0.987064i	$0.551255\pi$
$138$	0	0
$139$	14.4227	1.22332	0.611658	−	0.791122i	$-0.290503\pi$
$139$	14.4227	1.22332	0.611658	+	0.791122i	$0.290503\pi$
$140$	0	0
$141$	1.06706	0.0898624
$142$	0	0
$143$	7.13186	0.596396
$144$	0	0
$145$	31.0519	2.57872
$146$	0	0
$147$	−0.275746	−0.0227431
$148$	0	0
$149$	−10.0027	−0.819450	−0.409725	−	0.912209i	$-0.634375\pi$
$149$	−10.0027	−0.819450	−0.409725	+	0.912209i	$0.634375\pi$
$150$	0	0
$151$	3.07508	0.250246	0.125123	−	0.992141i	$-0.460067\pi$
$151$	3.07508	0.250246	0.125123	+	0.992141i	$0.460067\pi$
$152$	0	0
$153$	7.16195	0.579009
$154$	0	0
$155$	26.6861	2.14348
$156$	0	0
$157$	−19.5094	−1.55702	−0.778512	−	0.627630i	$-0.784025\pi$
$157$	−19.5094	−1.55702	−0.778512	+	0.627630i	$0.784025\pi$
$158$	0	0
$159$	−0.477052	−0.0378327
$160$	0	0
$161$	−6.34711	−0.500222
$162$	0	0
$163$	15.5119	1.21498	0.607492	−	0.794325i	$-0.292176\pi$
$163$	15.5119	1.21498	0.607492	+	0.794325i	$0.292176\pi$
$164$	0	0
$165$	−0.738195	−0.0574684
$166$	0	0
$167$	20.9423	1.62056	0.810282	−	0.586041i	$-0.199314\pi$
$167$	20.9423	1.62056	0.810282	+	0.586041i	$0.199314\pi$
$168$	0	0
$169$	−0.157142	−0.0120879
$170$	0	0
$171$	−12.3308	−0.942959
$172$	0	0
$173$	−6.78477	−0.515836	−0.257918	−	0.966167i	$-0.583036\pi$
$173$	−6.78477	−0.515836	−0.257918	+	0.966167i	$0.583036\pi$
$174$	0	0
$175$	31.4949	2.38079
$176$	0	0
$177$	−1.08845	−0.0818128
$178$	0	0
$179$	2.45807	0.183725	0.0918623	−	0.995772i	$-0.470718\pi$
$179$	2.45807	0.183725	0.0918623	+	0.995772i	$0.470718\pi$
$180$	0	0
$181$	−8.54066	−0.634822	−0.317411	−	0.948288i	$-0.602813\pi$
$181$	−8.54066	−0.634822	−0.317411	+	0.948288i	$0.602813\pi$
$182$	0	0
$183$	0.609737	0.0450731
$184$	0	0
$185$	−10.3478	−0.760786
$186$	0	0
$187$	−4.76552	−0.348489
$188$	0	0
$189$	1.80244	0.131108
$190$	0	0
$191$	13.9644	1.01043	0.505213	−	0.862994i	$-0.331414\pi$
$191$	13.9644	1.01043	0.505213	+	0.862994i	$0.331414\pi$
$192$	0	0
$193$	−21.8516	−1.57291	−0.786456	−	0.617647i	$-0.788086\pi$
$193$	−21.8516	−1.57291	−0.786456	+	0.617647i	$0.788086\pi$
$194$	0	0
$195$	−1.32932	−0.0951947
$196$	0	0
$197$	11.3747	0.810414	0.405207	−	0.914225i	$-0.367199\pi$
$197$	11.3747	0.810414	0.405207	+	0.914225i	$0.367199\pi$
$198$	0	0
$199$	7.83887	0.555683	0.277841	−	0.960627i	$-0.410381\pi$
$199$	7.83887	0.555683	0.277841	+	0.960627i	$0.410381\pi$
$200$	0	0
$201$	−1.06886	−0.0753915
$202$	0	0
$203$	25.1861	1.76772
$204$	0	0
$205$	−11.5486	−0.806587
$206$	0	0
$207$	6.03906	0.419744
$208$	0	0
$209$	8.20484	0.567541
$210$	0	0
$211$	1.91800	0.132041	0.0660203	−	0.997818i	$-0.478970\pi$
$211$	1.91800	0.132041	0.0660203	+	0.997818i	$0.478970\pi$
$212$	0	0
$213$	0.574933	0.0393938
$214$	0	0
$215$	−20.7640	−1.41609
$216$	0	0
$217$	21.6450	1.46936
$218$	0	0
$219$	0.368422	0.0248956
$220$	0	0
$221$	−8.58162	−0.577262
$222$	0	0
$223$	−25.6453	−1.71734	−0.858668	−	0.512533i	$-0.828707\pi$
$223$	−25.6453	−1.71734	−0.858668	+	0.512533i	$0.828707\pi$
$224$	0	0
$225$	−29.9663	−1.99775
$226$	0	0
$227$	5.45459	0.362034	0.181017	−	0.983480i	$-0.442061\pi$
$227$	5.45459	0.362034	0.181017	+	0.983480i	$0.442061\pi$
$228$	0	0
$229$	−19.6809	−1.30055	−0.650275	−	0.759699i	$-0.725347\pi$
$229$	−19.6809	−1.30055	−0.650275	+	0.759699i	$0.725347\pi$
$230$	0	0
$231$	−0.598749	−0.0393948
$232$	0	0
$233$	20.6003	1.34957	0.674787	−	0.738013i	$-0.264236\pi$
$233$	20.6003	1.34957	0.674787	+	0.738013i	$0.264236\pi$
$234$	0	0
$235$	−43.2056	−2.81842
$236$	0	0
$237$	0.594554	0.0386204
$238$	0	0
$239$	−18.0081	−1.16485	−0.582424	−	0.812885i	$-0.697896\pi$
$239$	−18.0081	−1.16485	−0.582424	+	0.812885i	$0.697896\pi$
$240$	0	0
$241$	−8.67183	−0.558602	−0.279301	−	0.960204i	$-0.590103\pi$
$241$	−8.67183	−0.558602	−0.279301	+	0.960204i	$0.590103\pi$
$242$	0	0
$243$	−2.57375	−0.165106
$244$	0	0
$245$	11.1651	0.713310
$246$	0	0
$247$	14.7750	0.940114
$248$	0	0
$249$	−0.625452	−0.0396364
$250$	0	0
$251$	1.00000	0.0631194
$251$	1.00000	0.0631194
$252$	0	0
$253$	−4.01836	−0.252632
$254$	0	0
$255$	0.888254	0.0556246
$256$	0	0
$257$	8.64861	0.539485	0.269743	−	0.962932i	$-0.413061\pi$
$257$	8.64861	0.539485	0.269743	+	0.962932i	$0.413061\pi$
$258$	0	0
$259$	−8.39309	−0.521522
$260$	0	0
$261$	−23.9638	−1.48332
$262$	0	0
$263$	8.91179	0.549524	0.274762	−	0.961512i	$-0.411401\pi$
$263$	8.91179	0.549524	0.274762	+	0.961512i	$0.411401\pi$
$264$	0	0
$265$	19.3161	1.18658
$266$	0	0
$267$	1.19806	0.0733201
$268$	0	0
$269$	−12.4448	−0.758771	−0.379385	−	0.925239i	$-0.623865\pi$
$269$	−12.4448	−0.758771	−0.379385	+	0.925239i	$0.623865\pi$
$270$	0	0
$271$	−8.77758	−0.533200	−0.266600	−	0.963807i	$-0.585900\pi$
$271$	−8.77758	−0.533200	−0.266600	+	0.963807i	$0.585900\pi$
$272$	0	0
$273$	−1.07821	−0.0652562
$274$	0	0
$275$	19.9394	1.20239
$276$	0	0
$277$	−23.2025	−1.39410	−0.697051	−	0.717021i	$-0.745505\pi$
$277$	−23.2025	−1.39410	−0.697051	+	0.717021i	$0.745505\pi$
$278$	0	0
$279$	−20.5945	−1.23296
$280$	0	0
$281$	−14.6135	−0.871765	−0.435883	−	0.900003i	$-0.643564\pi$
$281$	−14.6135	−0.871765	−0.435883	+	0.900003i	$0.643564\pi$
$282$	0	0
$283$	−27.4094	−1.62932	−0.814661	−	0.579938i	$-0.803077\pi$
$283$	−27.4094	−1.62932	−0.814661	+	0.579938i	$0.803077\pi$
$284$	0	0
$285$	−1.52932	−0.0905888
$286$	0	0
$287$	−9.36702	−0.552918
$288$	0	0
$289$	−11.2658	−0.662691
$290$	0	0
$291$	0.492062	0.0288452
$292$	0	0
$293$	32.4637	1.89655	0.948276	−	0.317447i	$-0.102826\pi$
$293$	32.4637	1.89655	0.948276	+	0.317447i	$0.102826\pi$
$294$	0	0
$295$	44.0718	2.56596
$296$	0	0
$297$	1.14112	0.0662147
$298$	0	0
$299$	−7.23614	−0.418477
$300$	0	0
$301$	−16.8417	−0.970737
$302$	0	0
$303$	0.268979	0.0154524
$304$	0	0
$305$	−24.6885	−1.41366
$306$	0	0
$307$	−24.9735	−1.42531	−0.712656	−	0.701514i	$-0.752508\pi$
$307$	−24.9735	−1.42531	−0.712656	+	0.701514i	$0.752508\pi$
$308$	0	0
$309$	0.423665	0.0241015
$310$	0	0
$311$	3.31489	0.187970	0.0939851	−	0.995574i	$-0.470039\pi$
$311$	3.31489	0.187970	0.0939851	+	0.995574i	$0.470039\pi$
$312$	0	0
$313$	16.6983	0.943841	0.471921	−	0.881641i	$-0.343561\pi$
$313$	16.6983	0.943841	0.471921	+	0.881641i	$0.343561\pi$
$314$	0	0
$315$	−36.4350	−2.05288
$316$	0	0
$317$	19.1747	1.07696	0.538481	−	0.842638i	$-0.318999\pi$
$317$	19.1747	1.07696	0.538481	+	0.842638i	$0.318999\pi$
$318$	0	0
$319$	15.9454	0.892769
$320$	0	0
$321$	−1.43635	−0.0801691
$322$	0	0
$323$	−9.87271	−0.549332
$324$	0	0
$325$	35.9063	1.99173
$326$	0	0
$327$	−0.272222	−0.0150539
$328$	0	0
$329$	−35.0440	−1.93204
$330$	0	0
$331$	7.40356	0.406936	0.203468	−	0.979082i	$-0.434779\pi$
$331$	7.40356	0.406936	0.203468	+	0.979082i	$0.434779\pi$
$332$	0	0
$333$	7.98575	0.437616
$334$	0	0
$335$	43.2786	2.36456
$336$	0	0
$337$	3.99403	0.217568	0.108784	−	0.994065i	$-0.465304\pi$
$337$	3.99403	0.217568	0.108784	+	0.994065i	$0.465304\pi$
$338$	0	0
$339$	−0.449212	−0.0243979
$340$	0	0
$341$	13.7035	0.742085
$342$	0	0
$343$	−12.9478	−0.699116
$344$	0	0
$345$	0.748989	0.0403242
$346$	0	0
$347$	−6.96214	−0.373747	−0.186874	−	0.982384i	$-0.559835\pi$
$347$	−6.96214	−0.373747	−0.186874	+	0.982384i	$0.559835\pi$
$348$	0	0
$349$	−13.7833	−0.737806	−0.368903	−	0.929468i	$-0.620266\pi$
$349$	−13.7833	−0.737806	−0.368903	+	0.929468i	$0.620266\pi$
$350$	0	0
$351$	2.05490	0.109683
$352$	0	0
$353$	−14.8705	−0.791478	−0.395739	−	0.918363i	$-0.629512\pi$
$353$	−14.8705	−0.791478	−0.395739	+	0.918363i	$0.629512\pi$
$354$	0	0
$355$	−23.2793	−1.23554
$356$	0	0
$357$	0.720462	0.0381309
$358$	0	0
$359$	32.7299	1.72742	0.863709	−	0.503991i	$-0.168135\pi$
$359$	32.7299	1.72742	0.863709	+	0.503991i	$0.168135\pi$
$360$	0	0
$361$	−2.00207	−0.105372
$362$	0	0
$363$	0.673780	0.0353643
$364$	0	0
$365$	−14.9176	−0.780821
$366$	0	0
$367$	−8.77027	−0.457804	−0.228902	−	0.973449i	$-0.573514\pi$
$367$	−8.77027	−0.457804	−0.228902	+	0.973449i	$0.573514\pi$
$368$	0	0
$369$	8.91241	0.463961
$370$	0	0
$371$	15.6672	0.813402
$372$	0	0
$373$	9.22194	0.477494	0.238747	−	0.971082i	$-0.423263\pi$
$373$	9.22194	0.477494	0.238747	+	0.971082i	$0.423263\pi$
$374$	0	0
$375$	−1.86187	−0.0961463
$376$	0	0
$377$	28.7140	1.47884
$378$	0	0
$379$	32.7190	1.68066	0.840330	−	0.542075i	$-0.182361\pi$
$379$	32.7190	1.68066	0.840330	+	0.542075i	$0.182361\pi$
$380$	0	0
$381$	−0.801137	−0.0410435
$382$	0	0
$383$	−35.1828	−1.79776	−0.898880	−	0.438196i	$-0.855618\pi$
$383$	−35.1828	−1.79776	−0.898880	+	0.438196i	$0.855618\pi$
$384$	0	0
$385$	24.2436	1.23557
$386$	0	0
$387$	16.0243	0.814560
$388$	0	0
$389$	0.858487	0.0435270	0.0217635	−	0.999763i	$-0.493072\pi$
$389$	0.858487	0.0435270	0.0217635	+	0.999763i	$0.493072\pi$
$390$	0	0
$391$	4.83520	0.244527
$392$	0	0
$393$	0.760728	0.0383736
$394$	0	0
$395$	−24.0738	−1.21128
$396$	0	0
$397$	19.8190	0.994686	0.497343	−	0.867554i	$-0.334309\pi$
$397$	19.8190	0.994686	0.497343	+	0.867554i	$0.334309\pi$
$398$	0	0
$399$	−1.24043	−0.0620989
$400$	0	0
$401$	16.4763	0.822788	0.411394	−	0.911458i	$-0.365042\pi$
$401$	16.4763	0.822788	0.411394	+	0.911458i	$0.365042\pi$
$402$	0	0
$403$	24.6768	1.22924
$404$	0	0
$405$	34.5601	1.71731
$406$	0	0
$407$	−5.31367	−0.263389
$408$	0	0
$409$	6.85804	0.339108	0.169554	−	0.985521i	$-0.445767\pi$
$409$	6.85804	0.339108	0.169554	+	0.985521i	$0.445767\pi$
$410$	0	0
$411$	0.359228	0.0177194
$412$	0	0
$413$	35.7465	1.75897
$414$	0	0
$415$	25.3248	1.24315
$416$	0	0
$417$	−1.38045	−0.0676007
$418$	0	0
$419$	−9.70102	−0.473926	−0.236963	−	0.971519i	$-0.576152\pi$
$419$	−9.70102	−0.473926	−0.236963	+	0.971519i	$0.576152\pi$
$420$	0	0
$421$	16.4395	0.801210	0.400605	−	0.916251i	$-0.368800\pi$
$421$	16.4395	0.801210	0.400605	+	0.916251i	$0.368800\pi$
$422$	0	0
$423$	33.3432	1.62120
$424$	0	0
$425$	−23.9927	−1.16382
$426$	0	0
$427$	−20.0248	−0.969070
$428$	0	0
$429$	−0.682615	−0.0329570
$430$	0	0
$431$	4.06932	0.196012	0.0980060	−	0.995186i	$-0.468754\pi$
$431$	4.06932	0.196012	0.0980060	+	0.995186i	$0.468754\pi$
$432$	0	0
$433$	3.57352	0.171732	0.0858661	−	0.996307i	$-0.472634\pi$
$433$	3.57352	0.171732	0.0858661	+	0.996307i	$0.472634\pi$
$434$	0	0
$435$	−2.97209	−0.142501
$436$	0	0
$437$	−8.32481	−0.398230
$438$	0	0
$439$	33.0352	1.57668	0.788341	−	0.615238i	$-0.210940\pi$
$439$	33.0352	1.57668	0.788341	+	0.615238i	$0.210940\pi$
$440$	0	0
$441$	−8.61646	−0.410308
$442$	0	0
$443$	−21.1773	−1.00617	−0.503083	−	0.864238i	$-0.667801\pi$
$443$	−21.1773	−1.00617	−0.503083	+	0.864238i	$0.667801\pi$
$444$	0	0
$445$	−48.5100	−2.29960
$446$	0	0
$447$	0.957390	0.0452830
$448$	0	0
$449$	32.4019	1.52914	0.764570	−	0.644541i	$-0.222951\pi$
$449$	32.4019	1.52914	0.764570	+	0.644541i	$0.222951\pi$
$450$	0	0
$451$	−5.93027	−0.279245
$452$	0	0
$453$	−0.294326	−0.0138287
$454$	0	0
$455$	43.6572	2.04668
$456$	0	0
$457$	−22.3713	−1.04649	−0.523243	−	0.852183i	$-0.675278\pi$
$457$	−22.3713	−1.04649	−0.523243	+	0.852183i	$0.675278\pi$
$458$	0	0
$459$	−1.37309	−0.0640904
$460$	0	0
$461$	14.4915	0.674937	0.337468	−	0.941337i	$-0.390429\pi$
$461$	14.4915	0.674937	0.337468	+	0.941337i	$0.390429\pi$
$462$	0	0
$463$	13.5093	0.627828	0.313914	−	0.949451i	$-0.398360\pi$
$463$	13.5093	0.627828	0.313914	+	0.949451i	$0.398360\pi$
$464$	0	0
$465$	−2.55422	−0.118449
$466$	0	0
$467$	−22.8617	−1.05791	−0.528956	−	0.848649i	$-0.677416\pi$
$467$	−22.8617	−1.05791	−0.528956	+	0.848649i	$0.677416\pi$
$468$	0	0
$469$	35.1032	1.62091
$470$	0	0
$471$	1.86732	0.0860414
$472$	0	0
$473$	−10.6625	−0.490261
$474$	0	0
$475$	41.3084	1.89536
$476$	0	0
$477$	−14.9069	−0.682538
$478$	0	0
$479$	−32.1517	−1.46905	−0.734524	−	0.678582i	$-0.762595\pi$
$479$	−32.1517	−1.46905	−0.734524	+	0.678582i	$0.762595\pi$
$480$	0	0
$481$	−9.56871	−0.436296
$482$	0	0
$483$	0.607504	0.0276424
$484$	0	0
$485$	−19.9238	−0.904693
$486$	0	0
$487$	41.1445	1.86443	0.932217	−	0.361900i	$-0.117872\pi$
$487$	41.1445	1.86443	0.932217	+	0.361900i	$0.117872\pi$
$488$	0	0
$489$	−1.48470	−0.0671403
$490$	0	0
$491$	−9.47742	−0.427710	−0.213855	−	0.976865i	$-0.568602\pi$
$491$	−9.47742	−0.427710	−0.213855	+	0.976865i	$0.568602\pi$
$492$	0	0
$493$	−19.1867	−0.864126
$494$	0	0
$495$	−23.0670	−1.03678
$496$	0	0
$497$	−18.8818	−0.846965
$498$	0	0
$499$	−21.9513	−0.982673	−0.491337	−	0.870970i	$-0.663492\pi$
$499$	−21.9513	−0.982673	−0.491337	+	0.870970i	$0.663492\pi$
$500$	0	0
$501$	−2.00446	−0.0895527
$502$	0	0
$503$	−17.7185	−0.790027	−0.395014	−	0.918675i	$-0.629260\pi$
$503$	−17.7185	−0.790027	−0.395014	+	0.918675i	$0.629260\pi$
$504$	0	0
$505$	−10.8911	−0.484646
$506$	0	0
$507$	0.0150406	0.000667977 0
$508$	0	0
$509$	−38.1419	−1.69061	−0.845304	−	0.534286i	$-0.820581\pi$
$509$	−38.1419	−1.69061	−0.845304	+	0.534286i	$0.820581\pi$
$510$	0	0
$511$	−12.0996	−0.535256
$512$	0	0
$513$	2.36406	0.104376
$514$	0	0
$515$	−17.1544	−0.755913
$516$	0	0
$517$	−22.1864	−0.975756
$518$	0	0
$519$	0.649394	0.0285052
$520$	0	0
$521$	5.66708	0.248279	0.124140	−	0.992265i	$-0.460383\pi$
$521$	5.66708	0.248279	0.124140	+	0.992265i	$0.460383\pi$
$522$	0	0
$523$	−20.3115	−0.888162	−0.444081	−	0.895987i	$-0.646470\pi$
$523$	−20.3115	−0.888162	−0.444081	+	0.895987i	$0.646470\pi$
$524$	0	0
$525$	−3.01448	−0.131563
$526$	0	0
$527$	−16.4891	−0.718277
$528$	0	0
$529$	−18.9229	−0.822734
$530$	0	0
$531$	−34.0117	−1.47598
$532$	0	0
$533$	−10.6791	−0.462561
$534$	0	0
$535$	58.1584	2.51441
$536$	0	0
$537$	−0.235270	−0.0101527
$538$	0	0
$539$	5.73334	0.246953
$540$	0	0
$541$	34.2169	1.47110	0.735551	−	0.677470i	$-0.236924\pi$
$541$	34.2169	1.47110	0.735551	+	0.677470i	$0.236924\pi$
$542$	0	0
$543$	0.817456	0.0350804
$544$	0	0
$545$	11.0224	0.472147
$546$	0	0
$547$	19.0671	0.815249	0.407625	−	0.913150i	$-0.366357\pi$
$547$	19.0671	0.815249	0.407625	+	0.913150i	$0.366357\pi$
$548$	0	0
$549$	19.0530	0.813161
$550$	0	0
$551$	33.0339	1.40729
$552$	0	0
$553$	−19.5262	−0.830338
$554$	0	0
$555$	0.990426	0.0420412
$556$	0	0
$557$	10.9294	0.463091	0.231546	−	0.972824i	$-0.425622\pi$
$557$	10.9294	0.463091	0.231546	+	0.972824i	$0.425622\pi$
$558$	0	0
$559$	−19.2007	−0.812102
$560$	0	0
$561$	0.456125	0.0192576
$562$	0	0
$563$	29.1517	1.22860	0.614300	−	0.789073i	$-0.289439\pi$
$563$	29.1517	1.22860	0.614300	+	0.789073i	$0.289439\pi$
$564$	0	0
$565$	18.1888	0.765209
$566$	0	0
$567$	28.0317	1.17722
$568$	0	0
$569$	21.1829	0.888032	0.444016	−	0.896019i	$-0.353553\pi$
$569$	21.1829	0.888032	0.444016	+	0.896019i	$0.353553\pi$
$570$	0	0
$571$	−11.1621	−0.467120	−0.233560	−	0.972342i	$-0.575038\pi$
$571$	−11.1621	−0.467120	−0.233560	+	0.972342i	$0.575038\pi$
$572$	0	0
$573$	−1.33658	−0.0558364
$574$	0	0
$575$	−20.2310	−0.843690
$576$	0	0
$577$	−28.3160	−1.17881	−0.589405	−	0.807838i	$-0.700638\pi$
$577$	−28.3160	−1.17881	−0.589405	+	0.807838i	$0.700638\pi$
$578$	0	0
$579$	2.09149	0.0869194
$580$	0	0
$581$	20.5409	0.852182
$582$	0	0
$583$	9.91893	0.410800
$584$	0	0
$585$	−41.5384	−1.71740
$586$	0	0
$587$	24.8001	1.02361	0.511805	−	0.859102i	$-0.328977\pi$
$587$	24.8001	1.02361	0.511805	+	0.859102i	$0.328977\pi$
$588$	0	0
$589$	28.3894	1.16977
$590$	0	0
$591$	−1.08871	−0.0447837
$592$	0	0
$593$	17.9490	0.737079	0.368539	−	0.929612i	$-0.379858\pi$
$593$	17.9490	0.737079	0.368539	+	0.929612i	$0.379858\pi$
$594$	0	0
$595$	−29.1718	−1.19593
$596$	0	0
$597$	−0.750286	−0.0307071
$598$	0	0
$599$	−34.2687	−1.40018	−0.700090	−	0.714054i	$-0.746857\pi$
$599$	−34.2687	−1.40018	−0.700090	+	0.714054i	$0.746857\pi$
$600$	0	0
$601$	−38.3769	−1.56542	−0.782712	−	0.622384i	$-0.786164\pi$
$601$	−38.3769	−1.56542	−0.782712	+	0.622384i	$0.786164\pi$
$602$	0	0
$603$	−33.3995	−1.36013
$604$	0	0
$605$	−27.2817	−1.10916
$606$	0	0
$607$	−17.9709	−0.729417	−0.364709	−	0.931122i	$-0.618831\pi$
$607$	−17.9709	−0.729417	−0.364709	+	0.931122i	$0.618831\pi$
$608$	0	0
$609$	−2.41065	−0.0976846
$610$	0	0
$611$	−39.9526	−1.61631
$612$	0	0
$613$	28.6496	1.15715	0.578573	−	0.815631i	$-0.303610\pi$
$613$	28.6496	1.15715	0.578573	+	0.815631i	$0.303610\pi$
$614$	0	0
$615$	1.10535	0.0445721
$616$	0	0
$617$	15.7509	0.634106	0.317053	−	0.948408i	$-0.397307\pi$
$617$	15.7509	0.634106	0.317053	+	0.948408i	$0.397307\pi$
$618$	0	0
$619$	−11.9616	−0.480777	−0.240389	−	0.970677i	$-0.577275\pi$
$619$	−11.9616	−0.480777	−0.240389	+	0.970677i	$0.577275\pi$
$620$	0	0
$621$	−1.15781	−0.0464613
$622$	0	0
$623$	−39.3464	−1.57638
$624$	0	0
$625$	25.2909	1.01164
$626$	0	0
$627$	−0.785314	−0.0313624
$628$	0	0
$629$	6.39383	0.254939
$630$	0	0
$631$	29.9798	1.19348	0.596738	−	0.802436i	$-0.296463\pi$
$631$	29.9798	1.19348	0.596738	+	0.802436i	$0.296463\pi$
$632$	0	0
$633$	−0.183579	−0.00729660
$634$	0	0
$635$	32.4384	1.28728
$636$	0	0
$637$	10.3244	0.409069
$638$	0	0
$639$	17.9654	0.710701
$640$	0	0
$641$	14.5539	0.574845	0.287423	−	0.957804i	$-0.407202\pi$
$641$	14.5539	0.574845	0.287423	+	0.957804i	$0.407202\pi$
$642$	0	0
$643$	41.4931	1.63633	0.818163	−	0.574986i	$-0.194992\pi$
$643$	41.4931	1.63633	0.818163	+	0.574986i	$0.194992\pi$
$644$	0	0
$645$	1.98740	0.0782537
$646$	0	0
$647$	24.1946	0.951188	0.475594	−	0.879665i	$-0.342233\pi$
$647$	24.1946	0.951188	0.475594	+	0.879665i	$0.342233\pi$
$648$	0	0
$649$	22.6312	0.888351
$650$	0	0
$651$	−2.07172	−0.0811972
$652$	0	0
$653$	−35.2239	−1.37842	−0.689209	−	0.724563i	$-0.742042\pi$
$653$	−35.2239	−1.37842	−0.689209	+	0.724563i	$0.742042\pi$
$654$	0	0
$655$	−30.8022	−1.20354
$656$	0	0
$657$	11.5124	0.449141
$658$	0	0
$659$	−12.4264	−0.484065	−0.242032	−	0.970268i	$-0.577814\pi$
$659$	−12.4264	−0.484065	−0.242032	+	0.970268i	$0.577814\pi$
$660$	0	0
$661$	36.5315	1.42091	0.710456	−	0.703742i	$-0.248489\pi$
$661$	36.5315	1.42091	0.710456	+	0.703742i	$0.248489\pi$
$662$	0	0
$663$	0.821376	0.0318996
$664$	0	0
$665$	50.2254	1.94766
$666$	0	0
$667$	−16.1785	−0.626435
$668$	0	0
$669$	2.45460	0.0949003
$670$	0	0
$671$	−12.6777	−0.489418
$672$	0	0
$673$	50.8065	1.95845	0.979224	−	0.202782i	$-0.0649981\pi$
$673$	50.8065	1.95845	0.979224	+	0.202782i	$0.0649981\pi$
$674$	0	0
$675$	5.74515	0.221131
$676$	0	0
$677$	42.5876	1.63678	0.818388	−	0.574666i	$-0.194868\pi$
$677$	42.5876	1.63678	0.818388	+	0.574666i	$0.194868\pi$
$678$	0	0
$679$	−16.1602	−0.620170
$680$	0	0
$681$	−0.522078	−0.0200061
$682$	0	0
$683$	−3.15713	−0.120804	−0.0604020	−	0.998174i	$-0.519238\pi$
$683$	−3.15713	−0.120804	−0.0604020	+	0.998174i	$0.519238\pi$
$684$	0	0
$685$	−14.5453	−0.555748
$686$	0	0
$687$	1.88373	0.0718687
$688$	0	0
$689$	17.8617	0.680478
$690$	0	0
$691$	11.7384	0.446551	0.223276	−	0.974755i	$-0.428325\pi$
$691$	11.7384	0.446551	0.223276	+	0.974755i	$0.428325\pi$
$692$	0	0
$693$	−18.7096	−0.710719
$694$	0	0
$695$	55.8948	2.12021
$696$	0	0
$697$	7.13576	0.270286
$698$	0	0
$699$	−1.97173	−0.0745777
$700$	0	0
$701$	−12.0340	−0.454516	−0.227258	−	0.973835i	$-0.572976\pi$
$701$	−12.0340	−0.454516	−0.227258	+	0.973835i	$0.572976\pi$
$702$	0	0
$703$	−11.0083	−0.415186
$704$	0	0
$705$	4.13536	0.155747
$706$	0	0
$707$	−8.83373	−0.332227
$708$	0	0
$709$	−10.7693	−0.404449	−0.202225	−	0.979339i	$-0.564817\pi$
$709$	−10.7693	−0.404449	−0.202225	+	0.979339i	$0.564817\pi$
$710$	0	0
$711$	18.5785	0.696749
$712$	0	0
$713$	−13.9039	−0.520703
$714$	0	0
$715$	27.6394	1.03366
$716$	0	0
$717$	1.72362	0.0643698
$718$	0	0
$719$	−36.4939	−1.36099	−0.680496	−	0.732752i	$-0.738236\pi$
$719$	−36.4939	−1.36099	−0.680496	+	0.732752i	$0.738236\pi$
$720$	0	0
$721$	−13.9139	−0.518181
$722$	0	0
$723$	0.830011	0.0308685
$724$	0	0
$725$	80.2792	2.98149
$726$	0	0
$727$	18.9440	0.702594	0.351297	−	0.936264i	$-0.385741\pi$
$727$	18.9440	0.702594	0.351297	+	0.936264i	$0.385741\pi$
$728$	0	0
$729$	−26.5066	−0.981724
$730$	0	0
$731$	12.8299	0.474531
$732$	0	0
$733$	24.3144	0.898073	0.449037	−	0.893513i	$-0.351767\pi$
$733$	24.3144	0.898073	0.449037	+	0.893513i	$0.351767\pi$
$734$	0	0
$735$	−1.06865	−0.0394177
$736$	0	0
$737$	22.2238	0.818626
$738$	0	0
$739$	−15.5649	−0.572566	−0.286283	−	0.958145i	$-0.592420\pi$
$739$	−15.5649	−0.572566	−0.286283	+	0.958145i	$0.592420\pi$
$740$	0	0
$741$	−1.41417	−0.0519509
$742$	0	0
$743$	16.9359	0.621317	0.310658	−	0.950522i	$-0.399451\pi$
$743$	16.9359	0.621317	0.310658	+	0.950522i	$0.399451\pi$
$744$	0	0
$745$	−38.7651	−1.42024
$746$	0	0
$747$	−19.5440	−0.715078
$748$	0	0
$749$	47.1722	1.72363
$750$	0	0
$751$	37.2144	1.35797	0.678987	−	0.734150i	$-0.262419\pi$
$751$	37.2144	1.35797	0.678987	+	0.734150i	$0.262419\pi$
$752$	0	0
$753$	−0.0957135	−0.00348799
$754$	0	0
$755$	11.9174	0.433719
$756$	0	0
$757$	−32.8787	−1.19499	−0.597497	−	0.801871i	$-0.703838\pi$
$757$	−32.8787	−1.19499	−0.597497	+	0.801871i	$0.703838\pi$
$758$	0	0
$759$	0.384611	0.0139605
$760$	0	0
$761$	−1.06413	−0.0385747	−0.0192873	−	0.999814i	$-0.506140\pi$
$761$	−1.06413	−0.0385747	−0.0192873	+	0.999814i	$0.506140\pi$
$762$	0	0
$763$	8.94023	0.323658
$764$	0	0
$765$	27.7560	1.00352
$766$	0	0
$767$	40.7536	1.47153
$768$	0	0
$769$	7.73677	0.278995	0.139497	−	0.990222i	$-0.455451\pi$
$769$	7.73677	0.278995	0.139497	+	0.990222i	$0.455451\pi$
$770$	0	0
$771$	−0.827788	−0.0298121
$772$	0	0
$773$	−7.68491	−0.276407	−0.138203	−	0.990404i	$-0.544133\pi$
$773$	−7.68491	−0.276407	−0.138203	+	0.990404i	$0.544133\pi$
$774$	0	0
$775$	68.9921	2.47827
$776$	0	0
$777$	0.803332	0.0288194
$778$	0	0
$779$	−12.2857	−0.440181
$780$	0	0
$781$	−11.9541	−0.427751
$782$	0	0
$783$	4.59434	0.164188
$784$	0	0
$785$	−75.6085	−2.69858
$786$	0	0
$787$	6.78127	0.241726	0.120863	−	0.992669i	$-0.461434\pi$
$787$	6.78127	0.241726	0.120863	+	0.992669i	$0.461434\pi$
$788$	0	0
$789$	−0.852978	−0.0303668
$790$	0	0
$791$	14.7529	0.524553
$792$	0	0
$793$	−22.8297	−0.810707
$794$	0	0
$795$	−1.84881	−0.0655705
$796$	0	0
$797$	−30.4234	−1.07765	−0.538827	−	0.842417i	$-0.681132\pi$
$797$	−30.4234	−1.07765	−0.538827	+	0.842417i	$0.681132\pi$
$798$	0	0
$799$	26.6964	0.944450
$800$	0	0
$801$	37.4368	1.32276
$802$	0	0
$803$	−7.66027	−0.270325
$804$	0	0
$805$	−24.5981	−0.866969
$806$	0	0
$807$	1.19113	0.0419298
$808$	0	0
$809$	−23.0238	−0.809473	−0.404737	−	0.914433i	$-0.632637\pi$
$809$	−23.0238	−0.809473	−0.404737	+	0.914433i	$0.632637\pi$
$810$	0	0
$811$	33.3504	1.17109	0.585546	−	0.810639i	$-0.300880\pi$
$811$	33.3504	1.17109	0.585546	+	0.810639i	$0.300880\pi$
$812$	0	0
$813$	0.840133	0.0294647
$814$	0	0
$815$	60.1161	2.10577
$816$	0	0
$817$	−22.0894	−0.772809
$818$	0	0
$819$	−33.6917	−1.17728
$820$	0	0
$821$	−40.3229	−1.40728	−0.703639	−	0.710558i	$-0.748443\pi$
$821$	−40.3229	−1.40728	−0.703639	+	0.710558i	$0.748443\pi$
$822$	0	0
$823$	1.25986	0.0439161	0.0219581	−	0.999759i	$-0.493010\pi$
$823$	1.25986	0.0439161	0.0219581	+	0.999759i	$0.493010\pi$
$824$	0	0
$825$	−1.90847	−0.0664445
$826$	0	0
$827$	−10.4882	−0.364709	−0.182355	−	0.983233i	$-0.558372\pi$
$827$	−10.4882	−0.364709	−0.182355	+	0.983233i	$0.558372\pi$
$828$	0	0
$829$	−4.32631	−0.150259	−0.0751295	−	0.997174i	$-0.523937\pi$
$829$	−4.32631	−0.150259	−0.0751295	+	0.997174i	$0.523937\pi$
$830$	0	0
$831$	2.22079	0.0770384
$832$	0	0
$833$	−6.89881	−0.239029
$834$	0	0
$835$	81.1615	2.80871
$836$	0	0
$837$	3.94839	0.136476
$838$	0	0
$839$	17.6873	0.610635	0.305318	−	0.952251i	$-0.401237\pi$
$839$	17.6873	0.610635	0.305318	+	0.952251i	$0.401237\pi$
$840$	0	0
$841$	35.1985	1.21374
$842$	0	0
$843$	1.39870	0.0481739
$844$	0	0
$845$	−0.609002	−0.0209503
$846$	0	0
$847$	−22.1281	−0.760331
$848$	0	0
$849$	2.62345	0.0900367
$850$	0	0
$851$	5.39137	0.184814
$852$	0	0
$853$	−12.7078	−0.435108	−0.217554	−	0.976048i	$-0.569808\pi$
$853$	−12.7078	−0.435108	−0.217554	+	0.976048i	$0.569808\pi$
$854$	0	0
$855$	−47.7878	−1.63431
$856$	0	0
$857$	46.7992	1.59863	0.799314	−	0.600913i	$-0.205196\pi$
$857$	46.7992	1.59863	0.799314	+	0.600913i	$0.205196\pi$
$858$	0	0
$859$	−12.5437	−0.427985	−0.213993	−	0.976835i	$-0.568647\pi$
$859$	−12.5437	−0.427985	−0.213993	+	0.976835i	$0.568647\pi$
$860$	0	0
$861$	0.896550	0.0305543
$862$	0	0
$863$	43.6146	1.48466	0.742329	−	0.670035i	$-0.233721\pi$
$863$	43.6146	1.48466	0.742329	+	0.670035i	$0.233721\pi$
$864$	0	0
$865$	−26.2942	−0.894031
$866$	0	0
$867$	1.07828	0.0366205
$868$	0	0
$869$	−12.3620	−0.419354
$870$	0	0
$871$	40.0201	1.35603
$872$	0	0
$873$	15.3759	0.520394
$874$	0	0
$875$	61.1469	2.06714
$876$	0	0
$877$	−30.1565	−1.01831	−0.509156	−	0.860674i	$-0.670042\pi$
$877$	−30.1565	−1.01831	−0.509156	+	0.860674i	$0.670042\pi$
$878$	0	0
$879$	−3.10722	−0.104804
$880$	0	0
$881$	4.59359	0.154762	0.0773811	−	0.997002i	$-0.475344\pi$
$881$	4.59359	0.154762	0.0773811	+	0.997002i	$0.475344\pi$
$882$	0	0
$883$	−27.9404	−0.940270	−0.470135	−	0.882595i	$-0.655795\pi$
$883$	−27.9404	−0.940270	−0.470135	+	0.882595i	$0.655795\pi$
$884$	0	0
$885$	−4.21826	−0.141795
$886$	0	0
$887$	−27.5636	−0.925496	−0.462748	−	0.886490i	$-0.653136\pi$
$887$	−27.5636	−0.925496	−0.462748	+	0.886490i	$0.653136\pi$
$888$	0	0
$889$	26.3107	0.882433
$890$	0	0
$891$	17.7469	0.594542
$892$	0	0
$893$	−45.9634	−1.53811
$894$	0	0
$895$	9.52620	0.318426
$896$	0	0
$897$	0.692597	0.0231251
$898$	0	0
$899$	55.1723	1.84010
$900$	0	0
$901$	−11.9352	−0.397620
$902$	0	0
$903$	1.61197	0.0536431
$904$	0	0
$905$	−33.0992	−1.10025
$906$	0	0
$907$	−3.18167	−0.105646	−0.0528228	−	0.998604i	$-0.516822\pi$
$907$	−3.18167	−0.105646	−0.0528228	+	0.998604i	$0.516822\pi$
$908$	0	0
$909$	8.40500	0.278776
$910$	0	0
$911$	−7.81500	−0.258922	−0.129461	−	0.991584i	$-0.541325\pi$
$911$	−7.81500	−0.258922	−0.129461	+	0.991584i	$0.541325\pi$
$912$	0	0
$913$	13.0045	0.430385
$914$	0	0
$915$	2.36303	0.0781192
$916$	0	0
$917$	−24.9836	−0.825032
$918$	0	0
$919$	−9.39761	−0.309999	−0.154999	−	0.987915i	$-0.549538\pi$
$919$	−9.39761	−0.309999	−0.154999	+	0.987915i	$0.549538\pi$
$920$	0	0
$921$	2.39030	0.0787630
$922$	0	0
$923$	−21.5266	−0.708556
$924$	0	0
$925$	−26.7524	−0.879614
$926$	0	0
$927$	13.2386	0.434813
$928$	0	0
$929$	−24.2094	−0.794285	−0.397143	−	0.917757i	$-0.629998\pi$
$929$	−24.2094	−0.794285	−0.397143	+	0.917757i	$0.629998\pi$
$930$	0	0
$931$	11.8777	0.389277
$932$	0	0
$933$	−0.317280	−0.0103873
$934$	0	0
$935$	−18.4687	−0.603991
$936$	0	0
$937$	−43.0839	−1.40749	−0.703746	−	0.710452i	$-0.748491\pi$
$937$	−43.0839	−1.40749	−0.703746	+	0.710452i	$0.748491\pi$
$938$	0	0
$939$	−1.59825	−0.0521569
$940$	0	0
$941$	−30.9110	−1.00767	−0.503835	−	0.863800i	$-0.668078\pi$
$941$	−30.9110	−1.00767	−0.503835	+	0.863800i	$0.668078\pi$
$942$	0	0
$943$	6.01698	0.195940
$944$	0	0
$945$	6.98532	0.227232
$946$	0	0
$947$	−45.2715	−1.47113	−0.735563	−	0.677456i	$-0.763082\pi$
$947$	−45.2715	−1.47113	−0.735563	+	0.677456i	$0.763082\pi$
$948$	0	0
$949$	−13.7944	−0.447785
$950$	0	0
$951$	−1.83528	−0.0595131
$952$	0	0
$953$	51.1301	1.65627	0.828133	−	0.560531i	$-0.189403\pi$
$953$	51.1301	1.65627	0.828133	+	0.560531i	$0.189403\pi$
$954$	0	0
$955$	54.1187	1.75124
$956$	0	0
$957$	−1.52619	−0.0493346
$958$	0	0
$959$	−11.7977	−0.380967
$960$	0	0
$961$	16.4152	0.529523
$962$	0	0
$963$	−44.8827	−1.44633
$964$	0	0
$965$	−84.6854	−2.72612
$966$	0	0
$967$	20.2583	0.651464	0.325732	−	0.945462i	$-0.394389\pi$
$967$	20.2583	0.651464	0.325732	+	0.945462i	$0.394389\pi$
$968$	0	0
$969$	0.944951	0.0303562
$970$	0	0
$971$	−18.7250	−0.600912	−0.300456	−	0.953796i	$-0.597139\pi$
$971$	−18.7250	−0.600912	−0.300456	+	0.953796i	$0.597139\pi$
$972$	0	0
$973$	45.3362	1.45341
$974$	0	0
$975$	−3.43672	−0.110063
$976$	0	0
$977$	−44.0949	−1.41072	−0.705361	−	0.708848i	$-0.749215\pi$
$977$	−44.0949	−1.41072	−0.705361	+	0.708848i	$0.749215\pi$
$978$	0	0
$979$	−24.9102	−0.796134
$980$	0	0
$981$	−8.50633	−0.271586
$982$	0	0
$983$	−7.10138	−0.226499	−0.113249	−	0.993567i	$-0.536126\pi$
$983$	−7.10138	−0.226499	−0.113249	+	0.993567i	$0.536126\pi$
$984$	0	0
$985$	44.0825	1.40458
$986$	0	0
$987$	3.35418	0.106765
$988$	0	0
$989$	10.8184	0.344004
$990$	0	0
$991$	29.8365	0.947787	0.473893	−	0.880582i	$-0.342848\pi$
$991$	29.8365	0.947787	0.473893	+	0.880582i	$0.342848\pi$
$992$	0	0
$993$	−0.708621	−0.0224874
$994$	0	0
$995$	30.3794	0.963092
$996$	0	0
$997$	−35.2893	−1.11762	−0.558811	−	0.829295i	$-0.688742\pi$
$997$	−35.2893	−1.11762	−0.558811	+	0.829295i	$0.688742\pi$
$998$	0	0
$999$	−1.53103	−0.0484396

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8032.2.a.i.1.14	yes	30
4.3	odd	2		8032.2.a.h.1.17	✓	30

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8032.2.a.h.1.17	✓	30	4.3	odd	2
8032.2.a.i.1.14	yes	30	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 8032 = 2^{5} \cdot 251 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8032.a (trivial)

\(f(q)\)	\(=\)	\(q-0.0957135 q^{3} +3.87548 q^{5} +3.14340 q^{7} -2.99084 q^{9} +O(q^{10})\)	\(q-0.0957135 q^{3} +3.87548 q^{5} +3.14340 q^{7} -2.99084 q^{9} +1.99009 q^{11} +3.58369 q^{13} -0.370936 q^{15} -2.39463 q^{17} +4.12285 q^{19} -0.300866 q^{21} -2.01919 q^{23} +10.0194 q^{25} +0.573404 q^{27} +8.01240 q^{29} +6.88587 q^{31} -0.190478 q^{33} +12.1822 q^{35} -2.67007 q^{37} -0.343008 q^{39} -2.97990 q^{41} -5.35779 q^{43} -11.5909 q^{45} -11.1484 q^{47} +2.88095 q^{49} +0.229198 q^{51} +4.98417 q^{53} +7.71255 q^{55} -0.394613 q^{57} +11.3719 q^{59} -6.37044 q^{61} -9.40140 q^{63} +13.8885 q^{65} +11.1673 q^{67} +0.193263 q^{69} -6.00682 q^{71} -3.84922 q^{73} -0.958989 q^{75} +6.25564 q^{77} -6.21181 q^{79} +8.91763 q^{81} +6.53463 q^{83} -9.28034 q^{85} -0.766894 q^{87} -12.5172 q^{89} +11.2650 q^{91} -0.659071 q^{93} +15.9781 q^{95} -5.14098 q^{97} -5.95203 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q + 3 q^{3} + 13 q^{7} + 35 q^{9}+O(q^{10}) \) 30 * q + 3 * q^3 + 13 * q^7 + 35 * q^9	\( 30 q + 3 q^{3} + 13 q^{7} + 35 q^{9} + 13 q^{11} - 7 q^{13} + 28 q^{15} - 9 q^{17} - 17 q^{19} - 6 q^{21} + 43 q^{23} + 34 q^{25} + 12 q^{27} - q^{29} + 39 q^{31} + 17 q^{35} - q^{37} + 48 q^{39} - 3 q^{41} - 19 q^{43} + 66 q^{47} + 25 q^{49} - 14 q^{51} + 3 q^{53} + 50 q^{55} - 14 q^{57} + 27 q^{59} + 15 q^{61} + 75 q^{63} - 6 q^{65} + 8 q^{67} + 18 q^{69} + 64 q^{71} - 15 q^{73} + 9 q^{75} + 71 q^{79} + 6 q^{81} + 60 q^{83} + 15 q^{85} + 64 q^{87} - 32 q^{89} - 26 q^{91} - 4 q^{93} + 72 q^{95} - 4 q^{97} + 13 q^{99}+O(q^{100}) \) 30 * q + 3 * q^3 + 13 * q^7 + 35 * q^9 + 13 * q^11 - 7 * q^13 + 28 * q^15 - 9 * q^17 - 17 * q^19 - 6 * q^21 + 43 * q^23 + 34 * q^25 + 12 * q^27 - q^29 + 39 * q^31 + 17 * q^35 - q^37 + 48 * q^39 - 3 * q^41 - 19 * q^43 + 66 * q^47 + 25 * q^49 - 14 * q^51 + 3 * q^53 + 50 * q^55 - 14 * q^57 + 27 * q^59 + 15 * q^61 + 75 * q^63 - 6 * q^65 + 8 * q^67 + 18 * q^69 + 64 * q^71 - 15 * q^73 + 9 * q^75 + 71 * q^79 + 6 * q^81 + 60 * q^83 + 15 * q^85 + 64 * q^87 - 32 * q^89 - 26 * q^91 - 4 * q^93 + 72 * q^95 - 4 * q^97 + 13 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more