LMFDB - Embedded newform 8281.2.a.cw.1.18

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8281,2,Mod(1,8281)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8281, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("8281.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8281 = 7^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8281.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:	$66.1241179138$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	no (minimal twist has level 1183)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.18
Character	$\chi$	$=$	8281.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.36142 q^{2} +1.31293 q^{3} -0.146524 q^{4} -3.05867 q^{5} +1.78745 q^{6} -2.92233 q^{8} -1.27622 q^{9} -4.16415 q^{10} -1.55093 q^{11} -0.192376 q^{12} -4.01582 q^{15} -3.68548 q^{16} -5.82376 q^{17} -1.73747 q^{18} +1.44521 q^{19} +0.448169 q^{20} -2.11147 q^{22} +6.27773 q^{23} -3.83681 q^{24} +4.35546 q^{25} -5.61437 q^{27} -9.88196 q^{29} -5.46723 q^{30} +1.52667 q^{31} +0.827155 q^{32} -2.03626 q^{33} -7.92861 q^{34} +0.186996 q^{36} -7.75148 q^{37} +1.96754 q^{38} +8.93844 q^{40} +7.17386 q^{41} +5.03082 q^{43} +0.227249 q^{44} +3.90352 q^{45} +8.54665 q^{46} +5.21844 q^{47} -4.83878 q^{48} +5.92963 q^{50} -7.64619 q^{51} +7.77747 q^{53} -7.64354 q^{54} +4.74378 q^{55} +1.89746 q^{57} -13.4535 q^{58} +2.79061 q^{59} +0.588415 q^{60} -13.7572 q^{61} +2.07845 q^{62} +8.49707 q^{64} -2.77222 q^{66} +0.683026 q^{67} +0.853322 q^{68} +8.24222 q^{69} +0.582838 q^{71} +3.72952 q^{72} +4.32488 q^{73} -10.5531 q^{74} +5.71842 q^{75} -0.211758 q^{76} +6.41818 q^{79} +11.2727 q^{80} -3.54263 q^{81} +9.76667 q^{82} +14.7890 q^{83} +17.8130 q^{85} +6.84909 q^{86} -12.9743 q^{87} +4.53233 q^{88} -2.06579 q^{89} +5.31435 q^{90} -0.919839 q^{92} +2.00442 q^{93} +7.10451 q^{94} -4.42042 q^{95} +1.08600 q^{96} +5.78897 q^{97} +1.97932 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + q^{2} + 23 q^{4} + 13 q^{5} + 14 q^{6} + 26 q^{9} - 5 q^{10} + q^{11} - 5 q^{12} - 5 q^{15} + 17 q^{16} + 5 q^{17} + 24 q^{19} + 34 q^{20} - 14 q^{22} + 11 q^{23} + 32 q^{24} + 33 q^{25} + 21 q^{27}+ \cdots + 39 q^{99}+O(q^{100}) $ 24 * q + q^2 + 23 * q^4 + 13 * q^5 + 14 * q^6 + 26 * q^9 - 5 * q^10 + q^11 - 5 * q^12 - 5 * q^15 + 17 * q^16 + 5 * q^17 + 24 * q^19 + 34 * q^20 - 14 * q^22 + 11 * q^23 + 32 * q^24 + 33 * q^25 + 21 * q^27 + 4 * q^29 - 22 * q^30 + 40 * q^31 + 6 * q^32 + 24 * q^33 + 36 * q^34 - 15 * q^36 + 4 * q^37 + 29 * q^38 + 4 * q^40 + 49 * q^41 + 13 * q^43 - 10 * q^44 + 58 * q^45 + 10 * q^46 + 62 * q^47 - 89 * q^48 + 23 * q^50 - 21 * q^51 - 18 * q^53 + 12 * q^54 + 14 * q^55 + 13 * q^57 - 56 * q^58 + 79 * q^59 - 22 * q^60 - 13 * q^61 - 12 * q^62 + 18 * q^64 + 38 * q^66 + 2 * q^67 + 12 * q^68 + 28 * q^69 + 19 * q^71 - 81 * q^72 + 17 * q^73 + 17 * q^74 - 24 * q^75 + 58 * q^76 - 9 * q^79 + 63 * q^80 + 16 * q^81 + 22 * q^82 + 81 * q^83 + 34 * q^85 - 22 * q^86 - 70 * q^87 - 33 * q^88 + 72 * q^89 - q^90 - 4 * q^92 - 19 * q^93 + 30 * q^94 + 13 * q^95 + 11 * q^96 + 45 * q^97 + 39 * q^99

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 $

$2$	1.36142	0.962672	0.481336	−	0.876536i	$-0.340152\pi$
$2$	1.36142	0.962672	0.481336	+	0.876536i	$0.340152\pi$
$3$	1.31293	0.758020	0.379010	−	0.925393i	$-0.376265\pi$
$3$	1.31293	0.758020	0.379010	+	0.925393i	$0.376265\pi$
$4$	−0.146524	−0.0732621
$5$	−3.05867	−1.36788	−0.683939	−	0.729539i	$-0.739735\pi$
$5$	−3.05867	−1.36788	−0.683939	+	0.729539i	$0.739735\pi$
$6$	1.78745	0.729725
$7$	0	0
$7$	0	0
$8$	−2.92233	−1.03320
$9$	−1.27622	−0.425405
$10$	−4.16415	−1.31682
$11$	−1.55093	−0.467623	−0.233812	−	0.972282i	$-0.575120\pi$
$11$	−1.55093	−0.467623	−0.233812	+	0.972282i	$0.575120\pi$
$12$	−0.192376	−0.0555342
$13$	0	0
$13$	0	0
$14$	0	0
$15$	−4.01582	−1.03688
$16$	−3.68548	−0.921371
$17$	−5.82376	−1.41247	−0.706235	−	0.707977i	$-0.749608\pi$
$17$	−5.82376	−1.41247	−0.706235	+	0.707977i	$0.749608\pi$
$18$	−1.73747	−0.409526
$19$	1.44521	0.331554	0.165777	−	0.986163i	$-0.446987\pi$
$19$	1.44521	0.331554	0.165777	+	0.986163i	$0.446987\pi$
$20$	0.448169	0.100214
$21$	0	0
$22$	−2.11147	−0.450168
$23$	6.27773	1.30900	0.654498	−	0.756063i	$-0.272880\pi$
$23$	6.27773	1.30900	0.654498	+	0.756063i	$0.272880\pi$
$24$	−3.83681	−0.783186
$25$	4.35546	0.871092
$26$	0	0
$27$	−5.61437	−1.08049
$28$	0	0
$29$	−9.88196	−1.83503	−0.917517	−	0.397698i	$-0.869809\pi$
$29$	−9.88196	−1.83503	−0.917517	+	0.397698i	$0.869809\pi$
$30$	−5.46723	−0.998176
$31$	1.52667	0.274199	0.137099	−	0.990557i	$-0.456222\pi$
$31$	1.52667	0.274199	0.137099	+	0.990557i	$0.456222\pi$
$32$	0.827155	0.146222
$33$	−2.03626	−0.354468
$34$	−7.92861	−1.35975
$35$	0	0
$36$	0.186996	0.0311661
$37$	−7.75148	−1.27434	−0.637168	−	0.770725i	$-0.719894\pi$
$37$	−7.75148	−1.27434	−0.637168	+	0.770725i	$0.719894\pi$
$38$	1.96754	0.319178
$39$	0	0
$40$	8.93844	1.41329
$41$	7.17386	1.12037	0.560184	−	0.828368i	$-0.310730\pi$
$41$	7.17386	1.12037	0.560184	+	0.828368i	$0.310730\pi$
$42$	0	0
$43$	5.03082	0.767194	0.383597	−	0.923501i	$-0.374685\pi$
$43$	5.03082	0.767194	0.383597	+	0.923501i	$0.374685\pi$
$44$	0.227249	0.0342590
$45$	3.90352	0.581903
$46$	8.54665	1.26013
$47$	5.21844	0.761188	0.380594	−	0.924742i	$-0.375720\pi$
$47$	5.21844	0.761188	0.380594	+	0.924742i	$0.375720\pi$
$48$	−4.83878	−0.698418
$49$	0	0
$50$	5.92963	0.838576
$51$	−7.64619	−1.07068
$52$	0	0
$53$	7.77747	1.06832	0.534159	−	0.845384i	$-0.320628\pi$
$53$	7.77747	1.06832	0.534159	+	0.845384i	$0.320628\pi$
$54$	−7.64354	−1.04015
$55$	4.74378	0.639652
$56$	0	0
$57$	1.89746	0.251325
$58$	−13.4535	−1.76654
$59$	2.79061	0.363306	0.181653	−	0.983363i	$-0.441855\pi$
$59$	2.79061	0.363306	0.181653	+	0.983363i	$0.441855\pi$
$60$	0.588415	0.0759640
$61$	−13.7572	−1.76143	−0.880714	−	0.473649i	$-0.842937\pi$
$61$	−13.7572	−1.76143	−0.880714	+	0.473649i	$0.842937\pi$
$62$	2.07845	0.263964
$63$	0	0
$64$	8.49707	1.06213
$65$	0	0
$66$	−2.77222	−0.341236
$67$	0.683026	0.0834449	0.0417225	−	0.999129i	$-0.486715\pi$
$67$	0.683026	0.0834449	0.0417225	+	0.999129i	$0.486715\pi$
$68$	0.853322	0.103481
$69$	8.24222	0.992246
$70$	0	0
$71$	0.582838	0.0691702	0.0345851	−	0.999402i	$-0.488989\pi$
$71$	0.582838	0.0691702	0.0345851	+	0.999402i	$0.488989\pi$
$72$	3.72952	0.439529
$73$	4.32488	0.506189	0.253095	−	0.967441i	$-0.418552\pi$
$73$	4.32488	0.506189	0.253095	+	0.967441i	$0.418552\pi$
$74$	−10.5531	−1.22677
$75$	5.71842	0.660306
$76$	−0.211758	−0.0242903
$77$	0	0
$78$	0	0
$79$	6.41818	0.722102	0.361051	−	0.932546i	$-0.382418\pi$
$79$	6.41818	0.722102	0.361051	+	0.932546i	$0.382418\pi$
$80$	11.2727	1.26032
$81$	−3.54263	−0.393625
$82$	9.76667	1.07855
$83$	14.7890	1.62331	0.811653	−	0.584140i	$-0.198568\pi$
$83$	14.7890	1.62331	0.811653	+	0.584140i	$0.198568\pi$
$84$	0	0
$85$	17.8130	1.93209
$86$	6.84909	0.738556
$87$	−12.9743	−1.39099
$88$	4.53233	0.483148
$89$	−2.06579	−0.218973	−0.109487	−	0.993988i	$-0.534921\pi$
$89$	−2.06579	−0.218973	−0.109487	+	0.993988i	$0.534921\pi$
$90$	5.31435	0.560182
$91$	0	0
$92$	−0.919839	−0.0958999
$93$	2.00442	0.207848
$94$	7.10451	0.732774
$95$	−4.42042	−0.453526
$96$	1.08600	0.110839
$97$	5.78897	0.587781	0.293890	−	0.955839i	$-0.405050\pi$
$97$	5.78897	0.587781	0.293890	+	0.955839i	$0.405050\pi$
$98$	0	0
$99$	1.97932	0.198929
$100$	−0.638181	−0.0638181
$101$	16.8223	1.67388	0.836942	−	0.547291i	$-0.184341\pi$
$101$	16.8223	1.67388	0.836942	+	0.547291i	$0.184341\pi$
$102$	−10.4097	−1.03071
$103$	0.532797	0.0524980	0.0262490	−	0.999655i	$-0.491644\pi$
$103$	0.532797	0.0524980	0.0262490	+	0.999655i	$0.491644\pi$
$104$	0	0
$105$	0	0
$106$	10.5884	1.02844
$107$	−15.6549	−1.51341	−0.756707	−	0.653754i	$-0.773193\pi$
$107$	−15.6549	−1.51341	−0.756707	+	0.653754i	$0.773193\pi$
$108$	0.822641	0.0791587
$109$	5.85539	0.560844	0.280422	−	0.959877i	$-0.409526\pi$
$109$	5.85539	0.560844	0.280422	+	0.959877i	$0.409526\pi$
$110$	6.45830	0.615775
$111$	−10.1771	−0.965972
$112$	0	0
$113$	−6.38603	−0.600747	−0.300373	−	0.953822i	$-0.597111\pi$
$113$	−6.38603	−0.600747	−0.300373	+	0.953822i	$0.597111\pi$
$114$	2.58325	0.241943
$115$	−19.2015	−1.79055
$116$	1.44795	0.134438
$117$	0	0
$118$	3.79920	0.349745
$119$	0	0
$120$	11.7355	1.07130
$121$	−8.59462	−0.781329
$122$	−18.7294	−1.69568
$123$	9.41877	0.849262
$124$	−0.223695	−0.0200884
$125$	1.97143	0.176330
$126$	0	0
$127$	−15.2760	−1.35552	−0.677762	−	0.735281i	$-0.737050\pi$
$127$	−15.2760	−1.35552	−0.677762	+	0.735281i	$0.737050\pi$
$128$	9.91381	0.876265
$129$	6.60512	0.581548
$130$	0	0
$131$	8.60251	0.751605	0.375802	−	0.926700i	$-0.377367\pi$
$131$	8.60251	0.751605	0.375802	+	0.926700i	$0.377367\pi$
$132$	0.298362	0.0259691
$133$	0	0
$134$	0.929889	0.0803301
$135$	17.1725	1.47797
$136$	17.0190	1.45936
$137$	−8.80304	−0.752094	−0.376047	−	0.926601i	$-0.622717\pi$
$137$	−8.80304	−0.752094	−0.376047	+	0.926601i	$0.622717\pi$
$138$	11.2212	0.955208
$139$	9.68708	0.821647	0.410824	−	0.911715i	$-0.365241\pi$
$139$	9.68708	0.821647	0.410824	+	0.911715i	$0.365241\pi$
$140$	0	0
$141$	6.85145	0.576996
$142$	0.793490	0.0665882
$143$	0	0
$144$	4.70347	0.391956
$145$	30.2256	2.51010
$146$	5.88800	0.487295
$147$	0	0
$148$	1.13578	0.0933605
$149$	−0.447444	−0.0366561	−0.0183280	−	0.999832i	$-0.505834\pi$
$149$	−0.447444	−0.0366561	−0.0183280	+	0.999832i	$0.505834\pi$
$150$	7.78519	0.635658
$151$	14.2769	1.16184	0.580918	−	0.813962i	$-0.302694\pi$
$151$	14.2769	1.16184	0.580918	+	0.813962i	$0.302694\pi$
$152$	−4.22338	−0.342561
$153$	7.43238	0.600872
$154$	0	0
$155$	−4.66959	−0.375071
$156$	0	0
$157$	−6.97329	−0.556529	−0.278265	−	0.960504i	$-0.589759\pi$
$157$	−6.97329	−0.556529	−0.278265	+	0.960504i	$0.589759\pi$
$158$	8.73787	0.695147
$159$	10.2113	0.809807
$160$	−2.52999	−0.200014
$161$	0	0
$162$	−4.82302	−0.378932
$163$	−22.6114	−1.77106	−0.885530	−	0.464582i	$-0.846204\pi$
$163$	−22.6114	−1.77106	−0.885530	+	0.464582i	$0.846204\pi$
$164$	−1.05114	−0.0820806
$165$	6.22825	0.484869
$166$	20.1341	1.56271
$167$	5.44039	0.420990	0.210495	−	0.977595i	$-0.432492\pi$
$167$	5.44039	0.420990	0.210495	+	0.977595i	$0.432492\pi$
$168$	0	0
$169$	0	0
$170$	24.2510	1.85997
$171$	−1.84440	−0.141045
$172$	−0.737138	−0.0562062
$173$	−18.6522	−1.41810	−0.709049	−	0.705159i	$-0.750875\pi$
$173$	−18.6522	−1.41810	−0.709049	+	0.705159i	$0.750875\pi$
$174$	−17.6635	−1.33907
$175$	0	0
$176$	5.71593	0.430854
$177$	3.66387	0.275393
$178$	−2.81241	−0.210799
$179$	1.43984	0.107618	0.0538092	−	0.998551i	$-0.482864\pi$
$179$	1.43984	0.107618	0.0538092	+	0.998551i	$0.482864\pi$
$180$	−0.571960	−0.0426314
$181$	−1.93813	−0.144060	−0.0720300	−	0.997402i	$-0.522948\pi$
$181$	−1.93813	−0.144060	−0.0720300	+	0.997402i	$0.522948\pi$
$182$	0	0
$183$	−18.0622	−1.33520
$184$	−18.3456	−1.35246
$185$	23.7092	1.74314
$186$	2.72886	0.200090
$187$	9.03225	0.660504
$188$	−0.764628	−0.0557662
$189$	0	0
$190$	−6.01807	−0.436596
$191$	5.13781	0.371759	0.185879	−	0.982573i	$-0.440487\pi$
$191$	5.13781	0.371759	0.185879	+	0.982573i	$0.440487\pi$
$192$	11.1561	0.805119
$193$	8.66938	0.624035	0.312018	−	0.950076i	$-0.398995\pi$
$193$	8.66938	0.624035	0.312018	+	0.950076i	$0.398995\pi$
$194$	7.88124	0.565840
$195$	0	0
$196$	0	0
$197$	17.2707	1.23049	0.615243	−	0.788337i	$-0.289058\pi$
$197$	17.2707	1.23049	0.615243	+	0.788337i	$0.289058\pi$
$198$	2.69470	0.191504
$199$	10.2717	0.728141	0.364071	−	0.931371i	$-0.381387\pi$
$199$	10.2717	0.728141	0.364071	+	0.931371i	$0.381387\pi$
$200$	−12.7281	−0.900012
$201$	0.896766	0.0632530
$202$	22.9023	1.61140
$203$	0	0
$204$	1.12035	0.0784403
$205$	−21.9425	−1.53253
$206$	0.725362	0.0505384
$207$	−8.01174	−0.556854
$208$	0	0
$209$	−2.24142	−0.155042
$210$	0	0
$211$	−27.7889	−1.91307	−0.956535	−	0.291618i	$-0.905806\pi$
$211$	−27.7889	−1.91307	−0.956535	+	0.291618i	$0.905806\pi$
$212$	−1.13959	−0.0782672
$213$	0.765226	0.0524324
$214$	−21.3129	−1.45692
$215$	−15.3876	−1.04943
$216$	16.4070	1.11636
$217$	0	0
$218$	7.97167	0.539909
$219$	5.67827	0.383702
$220$	−0.695079	−0.0468622
$221$	0	0
$222$	−13.8554	−0.929915
$223$	20.2750	1.35771	0.678856	−	0.734271i	$-0.262476\pi$
$223$	20.2750	1.35771	0.678856	+	0.734271i	$0.262476\pi$
$224$	0	0
$225$	−5.55851	−0.370567
$226$	−8.69409	−0.578322
$227$	1.79509	0.119144	0.0595720	−	0.998224i	$-0.481026\pi$
$227$	1.79509	0.119144	0.0595720	+	0.998224i	$0.481026\pi$
$228$	−0.278024	−0.0184126
$229$	4.81218	0.317998	0.158999	−	0.987279i	$-0.449173\pi$
$229$	4.81218	0.317998	0.158999	+	0.987279i	$0.449173\pi$
$230$	−26.1414	−1.72371
$231$	0	0
$232$	28.8783	1.89596
$233$	12.8778	0.843651	0.421826	−	0.906677i	$-0.361389\pi$
$233$	12.8778	0.843651	0.421826	+	0.906677i	$0.361389\pi$
$234$	0	0
$235$	−15.9615	−1.04121
$236$	−0.408891	−0.0266166
$237$	8.42662	0.547368
$238$	0	0
$239$	−9.97163	−0.645011	−0.322505	−	0.946568i	$-0.604525\pi$
$239$	−9.97163	−0.645011	−0.322505	+	0.946568i	$0.604525\pi$
$240$	14.8002	0.955351
$241$	23.5714	1.51836	0.759182	−	0.650878i	$-0.225599\pi$
$241$	23.5714	1.51836	0.759182	+	0.650878i	$0.225599\pi$
$242$	−11.7009	−0.752163
$243$	12.1919	0.782110
$244$	2.01576	0.129046
$245$	0	0
$246$	12.8229	0.817561
$247$	0	0
$248$	−4.46145	−0.283302
$249$	19.4169	1.23050
$250$	2.68395	0.169748
$251$	12.5979	0.795175	0.397587	−	0.917564i	$-0.369848\pi$
$251$	12.5979	0.795175	0.397587	+	0.917564i	$0.369848\pi$
$252$	0	0
$253$	−9.73632	−0.612117
$254$	−20.7971	−1.30493
$255$	23.3872	1.46456
$256$	−3.49724	−0.218578
$257$	5.97339	0.372610	0.186305	−	0.982492i	$-0.440349\pi$
$257$	5.97339	0.372610	0.186305	+	0.982492i	$0.440349\pi$
$258$	8.99237	0.559840
$259$	0	0
$260$	0	0
$261$	12.6115	0.780633
$262$	11.7117	0.723549
$263$	29.2598	1.80424	0.902118	−	0.431489i	$-0.142012\pi$
$263$	29.2598	1.80424	0.902118	+	0.431489i	$0.142012\pi$
$264$	5.95063	0.366236
$265$	−23.7887	−1.46133
$266$	0	0
$267$	−2.71224	−0.165986
$268$	−0.100080	−0.00611335
$269$	−3.44421	−0.209997	−0.104999	−	0.994472i	$-0.533484\pi$
$269$	−3.44421	−0.209997	−0.104999	+	0.994472i	$0.533484\pi$
$270$	23.3791	1.42280
$271$	−31.2275	−1.89694	−0.948468	−	0.316873i	$-0.897367\pi$
$271$	−31.2275	−1.89694	−0.948468	+	0.316873i	$0.897367\pi$
$272$	21.4634	1.30141
$273$	0	0
$274$	−11.9847	−0.724020
$275$	−6.75502	−0.407343
$276$	−1.20768	−0.0726940
$277$	23.2363	1.39613	0.698067	−	0.716032i	$-0.254044\pi$
$277$	23.2363	1.39613	0.698067	+	0.716032i	$0.254044\pi$
$278$	13.1882	0.790977
$279$	−1.94837	−0.116646
$280$	0	0
$281$	−5.30362	−0.316387	−0.158194	−	0.987408i	$-0.550567\pi$
$281$	−5.30362	−0.316387	−0.158194	+	0.987408i	$0.550567\pi$
$282$	9.32772	0.555458
$283$	−9.79277	−0.582120	−0.291060	−	0.956705i	$-0.594008\pi$
$283$	−9.79277	−0.582120	−0.291060	+	0.956705i	$0.594008\pi$
$284$	−0.0853999	−0.00506755
$285$	−5.80370	−0.343782
$286$	0	0
$287$	0	0
$288$	−1.05563	−0.0622035
$289$	16.9162	0.995072
$290$	41.1499	2.41641
$291$	7.60051	0.445550
$292$	−0.633700	−0.0370845
$293$	24.2189	1.41488	0.707441	−	0.706772i	$-0.249849\pi$
$293$	24.2189	1.41488	0.707441	+	0.706772i	$0.249849\pi$
$294$	0	0
$295$	−8.53555	−0.496959
$296$	22.6524	1.31664
$297$	8.70750	0.505260
$298$	−0.609161	−0.0352878
$299$	0	0
$300$	−0.837886	−0.0483754
$301$	0	0
$302$	19.4369	1.11847
$303$	22.0865	1.26884
$304$	−5.32630	−0.305484
$305$	42.0787	2.40942
$306$	10.1186	0.578443
$307$	6.52654	0.372490	0.186245	−	0.982503i	$-0.440368\pi$
$307$	6.52654	0.372490	0.186245	+	0.982503i	$0.440368\pi$
$308$	0	0
$309$	0.699524	0.0397946
$310$	−6.35730	−0.361070
$311$	1.71314	0.0971430	0.0485715	−	0.998820i	$-0.484533\pi$
$311$	1.71314	0.0971430	0.0485715	+	0.998820i	$0.484533\pi$
$312$	0	0
$313$	4.07898	0.230558	0.115279	−	0.993333i	$-0.463224\pi$
$313$	4.07898	0.230558	0.115279	+	0.993333i	$0.463224\pi$
$314$	−9.49360	−0.535755
$315$	0	0
$316$	−0.940419	−0.0529027
$317$	0.497659	0.0279513	0.0139756	−	0.999902i	$-0.495551\pi$
$317$	0.497659	0.0279513	0.0139756	+	0.999902i	$0.495551\pi$
$318$	13.9019	0.779579
$319$	15.3262	0.858104
$320$	−25.9897	−1.45287
$321$	−20.5538	−1.14720
$322$	0	0
$323$	−8.41656	−0.468310
$324$	0.519081	0.0288378
$325$	0	0
$326$	−30.7837	−1.70495
$327$	7.68771	0.425131
$328$	−20.9644	−1.15756
$329$	0	0
$330$	8.47930	0.466770
$331$	10.0544	0.552638	0.276319	−	0.961066i	$-0.410885\pi$
$331$	10.0544	0.552638	0.276319	+	0.961066i	$0.410885\pi$
$332$	−2.16695	−0.118927
$333$	9.89256	0.542109
$334$	7.40668	0.405275
$335$	−2.08915	−0.114143
$336$	0	0
$337$	−20.4206	−1.11238	−0.556189	−	0.831056i	$-0.687737\pi$
$337$	−20.4206	−1.11238	−0.556189	+	0.831056i	$0.687737\pi$
$338$	0	0
$339$	−8.38440	−0.455378
$340$	−2.61003	−0.141549
$341$	−2.36777	−0.128222
$342$	−2.51101	−0.135780
$343$	0	0
$344$	−14.7017	−0.792664
$345$	−25.2102	−1.35727
$346$	−25.3935	−1.36516
$347$	−27.3668	−1.46912	−0.734562	−	0.678541i	$-0.762613\pi$
$347$	−27.3668	−1.46912	−0.734562	+	0.678541i	$0.762613\pi$
$348$	1.90105	0.101907
$349$	−22.2070	−1.18871	−0.594356	−	0.804202i	$-0.702593\pi$
$349$	−22.2070	−1.18871	−0.594356	+	0.804202i	$0.702593\pi$
$350$	0	0
$351$	0	0
$352$	−1.28286	−0.0683767
$353$	27.8506	1.48234	0.741168	−	0.671319i	$-0.234272\pi$
$353$	27.8506	1.48234	0.741168	+	0.671319i	$0.234272\pi$
$354$	4.98808	0.265114
$355$	−1.78271	−0.0946164
$356$	0.302688	0.0160424
$357$	0	0
$358$	1.96023	0.103601
$359$	−20.8005	−1.09781	−0.548904	−	0.835885i	$-0.684955\pi$
$359$	−20.8005	−1.09781	−0.548904	+	0.835885i	$0.684955\pi$
$360$	−11.4074	−0.601222
$361$	−16.9114	−0.890072
$362$	−2.63862	−0.138683
$363$	−11.2841	−0.592263
$364$	0	0
$365$	−13.2284	−0.692406
$366$	−24.5903	−1.28536
$367$	15.7858	0.824012	0.412006	−	0.911181i	$-0.364828\pi$
$367$	15.7858	0.824012	0.412006	+	0.911181i	$0.364828\pi$
$368$	−23.1365	−1.20607
$369$	−9.15539	−0.476611
$370$	32.2783	1.67807
$371$	0	0
$372$	−0.293695	−0.0152274
$373$	9.01000	0.466520	0.233260	−	0.972414i	$-0.425061\pi$
$373$	9.01000	0.466520	0.233260	+	0.972414i	$0.425061\pi$
$374$	12.2967	0.635849
$375$	2.58835	0.133662
$376$	−15.2500	−0.786459
$377$	0	0
$378$	0	0
$379$	−11.6146	−0.596603	−0.298301	−	0.954472i	$-0.596420\pi$
$379$	−11.6146	−0.596603	−0.298301	+	0.954472i	$0.596420\pi$
$380$	0.647698	0.0332262
$381$	−20.0563	−1.02752
$382$	6.99474	0.357882
$383$	4.33633	0.221576	0.110788	−	0.993844i	$-0.464663\pi$
$383$	4.33633	0.221576	0.110788	+	0.993844i	$0.464663\pi$
$384$	13.0161	0.664227
$385$	0	0
$386$	11.8027	0.600742
$387$	−6.42042	−0.326368
$388$	−0.848224	−0.0430621
$389$	6.81733	0.345653	0.172826	−	0.984952i	$-0.444710\pi$
$389$	6.81733	0.345653	0.172826	+	0.984952i	$0.444710\pi$
$390$	0	0
$391$	−36.5600	−1.84892
$392$	0	0
$393$	11.2945	0.569732
$394$	23.5128	1.18456
$395$	−19.6311	−0.987748
$396$	−0.290018	−0.0145740
$397$	−24.8275	−1.24605	−0.623027	−	0.782200i	$-0.714097\pi$
$397$	−24.8275	−1.24605	−0.623027	+	0.782200i	$0.714097\pi$
$398$	13.9841	0.700961
$399$	0	0
$400$	−16.0520	−0.802599
$401$	10.6211	0.530393	0.265197	−	0.964194i	$-0.414563\pi$
$401$	10.6211	0.530393	0.265197	+	0.964194i	$0.414563\pi$
$402$	1.22088	0.0608919
$403$	0	0
$404$	−2.46488	−0.122632
$405$	10.8357	0.538432
$406$	0	0
$407$	12.0220	0.595909
$408$	22.3447	1.10623
$409$	10.6699	0.527592	0.263796	−	0.964579i	$-0.415025\pi$
$409$	10.6699	0.527592	0.263796	+	0.964579i	$0.415025\pi$
$410$	−29.8730	−1.47532
$411$	−11.5578	−0.570103
$412$	−0.0780676	−0.00384611
$413$	0	0
$414$	−10.9074	−0.536068
$415$	−45.2347	−2.22049
$416$	0	0
$417$	12.7185	0.622825
$418$	−3.05152	−0.149255
$419$	8.54940	0.417666	0.208833	−	0.977951i	$-0.433034\pi$
$419$	8.54940	0.417666	0.208833	+	0.977951i	$0.433034\pi$
$420$	0	0
$421$	−0.523234	−0.0255009	−0.0127504	−	0.999919i	$-0.504059\pi$
$421$	−0.523234	−0.0255009	−0.0127504	+	0.999919i	$0.504059\pi$
$422$	−37.8325	−1.84166
$423$	−6.65985	−0.323813
$424$	−22.7283	−1.10379
$425$	−25.3652	−1.23039
$426$	1.04180	0.0504752
$427$	0	0
$428$	2.29382	0.110876
$429$	0	0
$430$	−20.9491	−1.01026
$431$	−26.6501	−1.28369	−0.641846	−	0.766834i	$-0.721831\pi$
$431$	−26.6501	−1.28369	−0.641846	+	0.766834i	$0.721831\pi$
$432$	20.6917	0.995528
$433$	−13.2489	−0.636700	−0.318350	−	0.947973i	$-0.603129\pi$
$433$	−13.2489	−0.636700	−0.318350	+	0.947973i	$0.603129\pi$
$434$	0	0
$435$	39.6841	1.90271
$436$	−0.857956	−0.0410886
$437$	9.07264	0.434003
$438$	7.73053	0.369379
$439$	7.90856	0.377455	0.188728	−	0.982029i	$-0.439564\pi$
$439$	7.90856	0.377455	0.188728	+	0.982029i	$0.439564\pi$
$440$	−13.8629	−0.660888
$441$	0	0
$442$	0	0
$443$	0.741911	0.0352493	0.0176246	−	0.999845i	$-0.494390\pi$
$443$	0.741911	0.0352493	0.0176246	+	0.999845i	$0.494390\pi$
$444$	1.49120	0.0707692
$445$	6.31857	0.299529
$446$	27.6028	1.30703
$447$	−0.587463	−0.0277860
$448$	0	0
$449$	26.1281	1.23306	0.616530	−	0.787332i	$-0.288538\pi$
$449$	26.1281	1.23306	0.616530	+	0.787332i	$0.288538\pi$
$450$	−7.56749	−0.356735
$451$	−11.1262	−0.523910
$452$	0.935707	0.0440120
$453$	18.7445	0.880695
$454$	2.44387	0.114697
$455$	0	0
$456$	−5.54500	−0.259668
$457$	2.95826	0.138381	0.0691907	−	0.997603i	$-0.477958\pi$
$457$	2.95826	0.138381	0.0691907	+	0.997603i	$0.477958\pi$
$458$	6.55142	0.306128
$459$	32.6968	1.52615
$460$	2.81348	0.131179
$461$	−0.916982	−0.0427081	−0.0213541	−	0.999772i	$-0.506798\pi$
$461$	−0.916982	−0.0427081	−0.0213541	+	0.999772i	$0.506798\pi$
$462$	0	0
$463$	−10.1773	−0.472981	−0.236490	−	0.971634i	$-0.575997\pi$
$463$	−10.1773	−0.472981	−0.236490	+	0.971634i	$0.575997\pi$
$464$	36.4198	1.69075
$465$	−6.13085	−0.284311
$466$	17.5321	0.812160
$467$	27.2677	1.26180	0.630900	−	0.775864i	$-0.282686\pi$
$467$	27.2677	1.26180	0.630900	+	0.775864i	$0.282686\pi$
$468$	0	0
$469$	0	0
$470$	−21.7304	−1.00235
$471$	−9.15544	−0.421860
$472$	−8.15507	−0.375368
$473$	−7.80246	−0.358757
$474$	11.4722	0.526936
$475$	6.29456	0.288814
$476$	0	0
$477$	−9.92573	−0.454468
$478$	−13.5756	−0.620934
$479$	33.4296	1.52744	0.763720	−	0.645548i	$-0.223371\pi$
$479$	33.4296	1.52744	0.763720	+	0.645548i	$0.223371\pi$
$480$	−3.32171	−0.151614
$481$	0	0
$482$	32.0906	1.46169
$483$	0	0
$484$	1.25932	0.0572418
$485$	−17.7065	−0.804013
$486$	16.5983	0.752916
$487$	−36.6328	−1.65999	−0.829996	−	0.557769i	$-0.811658\pi$
$487$	−36.6328	−1.65999	−0.829996	+	0.557769i	$0.811658\pi$
$488$	40.2030	1.81991
$489$	−29.6871	−1.34250
$490$	0	0
$491$	13.1049	0.591415	0.295708	−	0.955278i	$-0.404445\pi$
$491$	13.1049	0.591415	0.295708	+	0.955278i	$0.404445\pi$
$492$	−1.38008	−0.0622187
$493$	57.5502	2.59193
$494$	0	0
$495$	−6.05409	−0.272111
$496$	−5.62653	−0.252639
$497$	0	0
$498$	26.4347	1.18457
$499$	4.17391	0.186850	0.0934250	−	0.995626i	$-0.470218\pi$
$499$	4.17391	0.186850	0.0934250	+	0.995626i	$0.470218\pi$
$500$	−0.288862	−0.0129183
$501$	7.14285	0.319119
$502$	17.1511	0.765492
$503$	−18.9926	−0.846838	−0.423419	−	0.905934i	$-0.639170\pi$
$503$	−18.9926	−0.846838	−0.423419	+	0.905934i	$0.639170\pi$
$504$	0	0
$505$	−51.4540	−2.28967
$506$	−13.2553	−0.589268
$507$	0	0
$508$	2.23830	0.0993086
$509$	−4.25731	−0.188702	−0.0943509	−	0.995539i	$-0.530078\pi$
$509$	−4.25731	−0.188702	−0.0943509	+	0.995539i	$0.530078\pi$
$510$	31.8399	1.40989
$511$	0	0
$512$	−24.5889	−1.08668
$513$	−8.11394	−0.358239
$514$	8.13232	0.358701
$515$	−1.62965	−0.0718109
$516$	−0.967810	−0.0426054
$517$	−8.09344	−0.355949
$518$	0	0
$519$	−24.4890	−1.07495
$520$	0	0
$521$	2.35828	0.103318	0.0516590	−	0.998665i	$-0.483549\pi$
$521$	2.35828	0.103318	0.0516590	+	0.998665i	$0.483549\pi$
$522$	17.1696	0.751493
$523$	−5.57416	−0.243741	−0.121870	−	0.992546i	$-0.538889\pi$
$523$	−5.57416	−0.243741	−0.121870	+	0.992546i	$0.538889\pi$
$524$	−1.26048	−0.0550641
$525$	0	0
$526$	39.8350	1.73689
$527$	−8.89099	−0.387298
$528$	7.50461	0.326596
$529$	16.4099	0.713473
$530$	−32.3865	−1.40678
$531$	−3.56142	−0.154552
$532$	0	0
$533$	0	0
$534$	−3.69250	−0.159790
$535$	47.8831	2.07017
$536$	−1.99603	−0.0862153
$537$	1.89040	0.0815769
$538$	−4.68904	−0.202159
$539$	0	0
$540$	−2.51619	−0.108279
$541$	4.64124	0.199542	0.0997712	−	0.995010i	$-0.468189\pi$
$541$	4.64124	0.199542	0.0997712	+	0.995010i	$0.468189\pi$
$542$	−42.5139	−1.82613
$543$	−2.54463	−0.109200
$544$	−4.81716	−0.206534
$545$	−17.9097	−0.767167
$546$	0	0
$547$	−7.82685	−0.334652	−0.167326	−	0.985902i	$-0.553513\pi$
$547$	−7.82685	−0.334652	−0.167326	+	0.985902i	$0.553513\pi$
$548$	1.28986	0.0551000
$549$	17.5571	0.749321
$550$	−9.19644	−0.392138
$551$	−14.2815	−0.608412
$552$	−24.0865	−1.02519
$553$	0	0
$554$	31.6345	1.34402
$555$	31.1285	1.32133
$556$	−1.41939	−0.0601956
$557$	−34.0460	−1.44258	−0.721288	−	0.692635i	$-0.756450\pi$
$557$	−34.0460	−1.44258	−0.721288	+	0.692635i	$0.756450\pi$
$558$	−2.65255	−0.112291
$559$	0	0
$560$	0	0
$561$	11.8587	0.500675
$562$	−7.22048	−0.304577
$563$	4.13227	0.174154	0.0870772	−	0.996202i	$-0.472247\pi$
$563$	4.13227	0.174154	0.0870772	+	0.996202i	$0.472247\pi$
$564$	−1.00390	−0.0422719
$565$	19.5327	0.821749
$566$	−13.3321	−0.560390
$567$	0	0
$568$	−1.70325	−0.0714666
$569$	10.0458	0.421142	0.210571	−	0.977579i	$-0.432468\pi$
$569$	10.0458	0.421142	0.210571	+	0.977579i	$0.432468\pi$
$570$	−7.90130	−0.330949
$571$	10.7360	0.449289	0.224645	−	0.974441i	$-0.427878\pi$
$571$	10.7360	0.449289	0.224645	+	0.974441i	$0.427878\pi$
$572$	0	0
$573$	6.74558	0.281801
$574$	0	0
$575$	27.3424	1.14026
$576$	−10.8441	−0.451837
$577$	42.5070	1.76959	0.884795	−	0.465980i	$-0.154298\pi$
$577$	42.5070	1.76959	0.884795	+	0.465980i	$0.154298\pi$
$578$	23.0302	0.957928
$579$	11.3823	0.473032
$580$	−4.42879	−0.183895
$581$	0	0
$582$	10.3475	0.428918
$583$	−12.0623	−0.499570
$584$	−12.6387	−0.522995
$585$	0	0
$586$	32.9722	1.36207
$587$	18.8276	0.777098	0.388549	−	0.921428i	$-0.372976\pi$
$587$	18.8276	0.777098	0.388549	+	0.921428i	$0.372976\pi$
$588$	0	0
$589$	2.20637	0.0909117
$590$	−11.6205	−0.478408
$591$	22.6752	0.932734
$592$	28.5679	1.17414
$593$	8.45175	0.347072	0.173536	−	0.984828i	$-0.444481\pi$
$593$	8.45175	0.347072	0.173536	+	0.984828i	$0.444481\pi$
$594$	11.8546	0.486400
$595$	0	0
$596$	0.0655614	0.00268550
$597$	13.4860	0.551946
$598$	0	0
$599$	34.7524	1.41994	0.709972	−	0.704230i	$-0.248707\pi$
$599$	34.7524	1.41994	0.709972	+	0.704230i	$0.248707\pi$
$600$	−16.7111	−0.682228
$601$	−33.2606	−1.35673	−0.678364	−	0.734726i	$-0.737311\pi$
$601$	−33.2606	−1.35673	−0.678364	+	0.734726i	$0.737311\pi$
$602$	0	0
$603$	−0.871689	−0.0354979
$604$	−2.09191	−0.0851185
$605$	26.2881	1.06876
$606$	30.0692	1.22148
$607$	26.6989	1.08367	0.541837	−	0.840484i	$-0.317729\pi$
$607$	26.6989	1.08367	0.541837	+	0.840484i	$0.317729\pi$
$608$	1.19541	0.0484804
$609$	0	0
$610$	57.2870	2.31948
$611$	0	0
$612$	−1.08902	−0.0440212
$613$	21.0806	0.851438	0.425719	−	0.904855i	$-0.360021\pi$
$613$	21.0806	0.851438	0.425719	+	0.904855i	$0.360021\pi$
$614$	8.88539	0.358585
$615$	−28.8089	−1.16169
$616$	0	0
$617$	6.37659	0.256712	0.128356	−	0.991728i	$-0.459030\pi$
$617$	6.37659	0.256712	0.128356	+	0.991728i	$0.459030\pi$
$618$	0.952349	0.0383091
$619$	−39.0440	−1.56931	−0.784656	−	0.619931i	$-0.787160\pi$
$619$	−39.0440	−1.56931	−0.784656	+	0.619931i	$0.787160\pi$
$620$	0.684208	0.0274785
$621$	−35.2455	−1.41435
$622$	2.33230	0.0935169
$623$	0	0
$624$	0	0
$625$	−27.8073	−1.11229
$626$	5.55323	0.221952
$627$	−2.94283	−0.117525
$628$	1.02176	0.0407725
$629$	45.1428	1.79996
$630$	0	0
$631$	11.7857	0.469181	0.234590	−	0.972094i	$-0.424625\pi$
$631$	11.7857	0.469181	0.234590	+	0.972094i	$0.424625\pi$
$632$	−18.7560	−0.746075
$633$	−36.4849	−1.45015
$634$	0.677524	0.0269079
$635$	46.7242	1.85419
$636$	−1.49620	−0.0593281
$637$	0	0
$638$	20.8655	0.826073
$639$	−0.743827	−0.0294254
$640$	−30.3231	−1.19862
$641$	0.620887	0.0245236	0.0122618	−	0.999925i	$-0.496097\pi$
$641$	0.620887	0.0245236	0.0122618	+	0.999925i	$0.496097\pi$
$642$	−27.9824	−1.10438
$643$	6.61337	0.260806	0.130403	−	0.991461i	$-0.458373\pi$
$643$	6.61337	0.260806	0.130403	+	0.991461i	$0.458373\pi$
$644$	0	0
$645$	−20.2029	−0.795488
$646$	−11.4585	−0.450829
$647$	−6.45338	−0.253709	−0.126854	−	0.991921i	$-0.540488\pi$
$647$	−6.45338	−0.253709	−0.126854	+	0.991921i	$0.540488\pi$
$648$	10.3527	0.406693
$649$	−4.32804	−0.169890
$650$	0	0
$651$	0	0
$652$	3.31311	0.129752
$653$	−4.52898	−0.177233	−0.0886164	−	0.996066i	$-0.528245\pi$
$653$	−4.52898	−0.177233	−0.0886164	+	0.996066i	$0.528245\pi$
$654$	10.4662	0.409262
$655$	−26.3122	−1.02810
$656$	−26.4391	−1.03227
$657$	−5.51949	−0.215336
$658$	0	0
$659$	−27.2666	−1.06216	−0.531078	−	0.847323i	$-0.678213\pi$
$659$	−27.2666	−1.06216	−0.531078	+	0.847323i	$0.678213\pi$
$660$	−0.912590	−0.0355225
$661$	3.89031	0.151315	0.0756577	−	0.997134i	$-0.475894\pi$
$661$	3.89031	0.151315	0.0756577	+	0.997134i	$0.475894\pi$
$662$	13.6883	0.532009
$663$	0	0
$664$	−43.2184	−1.67720
$665$	0	0
$666$	13.4680	0.521873
$667$	−62.0362	−2.40205
$668$	−0.797148	−0.0308426
$669$	26.6196	1.02917
$670$	−2.84422	−0.109882
$671$	21.3364	0.823684
$672$	0	0
$673$	−4.18325	−0.161253	−0.0806263	−	0.996744i	$-0.525692\pi$
$673$	−4.18325	−0.161253	−0.0806263	+	0.996744i	$0.525692\pi$
$674$	−27.8010	−1.07086
$675$	−24.4532	−0.941203
$676$	0	0
$677$	−43.5653	−1.67435	−0.837175	−	0.546935i	$-0.815795\pi$
$677$	−43.5653	−1.67435	−0.837175	+	0.546935i	$0.815795\pi$
$678$	−11.4147	−0.438380
$679$	0	0
$680$	−52.0554	−1.99623
$681$	2.35682	0.0903136
$682$	−3.22353	−0.123435
$683$	14.7994	0.566283	0.283142	−	0.959078i	$-0.408623\pi$
$683$	14.7994	0.566283	0.283142	+	0.959078i	$0.408623\pi$
$684$	0.270249	0.0103332
$685$	26.9256	1.02877
$686$	0	0
$687$	6.31806	0.241049
$688$	−18.5410	−0.706870
$689$	0	0
$690$	−34.3218	−1.30661
$691$	−4.73188	−0.180009	−0.0900047	−	0.995941i	$-0.528688\pi$
$691$	−4.73188	−0.180009	−0.0900047	+	0.995941i	$0.528688\pi$
$692$	2.73299	0.103893
$693$	0	0
$694$	−37.2578	−1.41429
$695$	−29.6296	−1.12391
$696$	37.9152	1.43717
$697$	−41.7789	−1.58249
$698$	−30.2331	−1.14434
$699$	16.9076	0.639505
$700$	0	0
$701$	36.8398	1.39142	0.695709	−	0.718324i	$-0.255090\pi$
$701$	36.8398	1.39142	0.695709	+	0.718324i	$0.255090\pi$
$702$	0	0
$703$	−11.2025	−0.422511
$704$	−13.1784	−0.496678
$705$	−20.9563	−0.789260
$706$	37.9164	1.42700
$707$	0	0
$708$	−0.536846	−0.0201759
$709$	17.0716	0.641136	0.320568	−	0.947225i	$-0.396126\pi$
$709$	17.0716	0.641136	0.320568	+	0.947225i	$0.396126\pi$
$710$	−2.42702	−0.0910846
$711$	−8.19098	−0.307186
$712$	6.03692	0.226243
$713$	9.58405	0.358925
$714$	0	0
$715$	0	0
$716$	−0.210971	−0.00788435
$717$	−13.0920	−0.488931
$718$	−28.3183	−1.05683
$719$	−22.2490	−0.829747	−0.414874	−	0.909879i	$-0.636174\pi$
$719$	−22.2490	−0.829747	−0.414874	+	0.909879i	$0.636174\pi$
$720$	−14.3864	−0.536148
$721$	0	0
$722$	−23.0235	−0.856848
$723$	30.9475	1.15095
$724$	0.283983	0.0105541
$725$	−43.0405	−1.59848
$726$	−15.3625	−0.570155
$727$	47.9699	1.77910	0.889552	−	0.456833i	$-0.151016\pi$
$727$	47.9699	1.77910	0.889552	+	0.456833i	$0.151016\pi$
$728$	0	0
$729$	26.6350	0.986481
$730$	−18.0095	−0.666560
$731$	−29.2983	−1.08364
$732$	2.64655	0.0978194
$733$	−37.9957	−1.40340	−0.701701	−	0.712471i	$-0.747576\pi$
$733$	−37.9957	−1.40340	−0.701701	+	0.712471i	$0.747576\pi$
$734$	21.4912	0.793254
$735$	0	0
$736$	5.19266	0.191404
$737$	−1.05933	−0.0390208
$738$	−12.4644	−0.458820
$739$	34.5887	1.27237	0.636184	−	0.771538i	$-0.280512\pi$
$739$	34.5887	1.27237	0.636184	+	0.771538i	$0.280512\pi$
$740$	−3.47397	−0.127706
$741$	0	0
$742$	0	0
$743$	25.1252	0.921752	0.460876	−	0.887464i	$-0.347535\pi$
$743$	25.1252	0.921752	0.460876	+	0.887464i	$0.347535\pi$
$744$	−5.85757	−0.214749
$745$	1.36858	0.0501410
$746$	12.2664	0.449106
$747$	−18.8740	−0.690563
$748$	−1.32344	−0.0483899
$749$	0	0
$750$	3.52384	0.128672
$751$	−10.5969	−0.386685	−0.193343	−	0.981131i	$-0.561933\pi$
$751$	−10.5969	−0.386685	−0.193343	+	0.981131i	$0.561933\pi$
$752$	−19.2325	−0.701336
$753$	16.5402	0.602758
$754$	0	0
$755$	−43.6682	−1.58925
$756$	0	0
$757$	20.4219	0.742248	0.371124	−	0.928583i	$-0.378972\pi$
$757$	20.4219	0.742248	0.371124	+	0.928583i	$0.378972\pi$
$758$	−15.8124	−0.574333
$759$	−12.7831	−0.463997
$760$	12.9179	0.468582
$761$	8.57045	0.310678	0.155339	−	0.987861i	$-0.450353\pi$
$761$	8.57045	0.310678	0.155339	+	0.987861i	$0.450353\pi$
$762$	−27.3051	−0.989160
$763$	0	0
$764$	−0.752813	−0.0272358
$765$	−22.7332	−0.821920
$766$	5.90358	0.213305
$767$	0	0
$768$	−4.59164	−0.165686
$769$	14.8952	0.537133	0.268567	−	0.963261i	$-0.413450\pi$
$769$	14.8952	0.537133	0.268567	+	0.963261i	$0.413450\pi$
$770$	0	0
$771$	7.84264	0.282446
$772$	−1.27027	−0.0457181
$773$	−49.2303	−1.77069	−0.885345	−	0.464935i	$-0.846078\pi$
$773$	−49.2303	−1.77069	−0.885345	+	0.464935i	$0.846078\pi$
$774$	−8.74091	−0.314186
$775$	6.64937	0.238853
$776$	−16.9173	−0.607295
$777$	0	0
$778$	9.28128	0.332750
$779$	10.3677	0.371463
$780$	0	0
$781$	−0.903942	−0.0323456
$782$	−49.7737	−1.77990
$783$	55.4810	1.98273
$784$	0	0
$785$	21.3290	0.761264
$786$	15.3766	0.548465
$787$	18.8851	0.673180	0.336590	−	0.941651i	$-0.390726\pi$
$787$	18.8851	0.673180	0.336590	+	0.941651i	$0.390726\pi$
$788$	−2.53058	−0.0901480
$789$	38.4160	1.36765
$790$	−26.7263	−0.950877
$791$	0	0
$792$	−5.78423	−0.205534
$793$	0	0
$794$	−33.8007	−1.19954
$795$	−31.2329	−1.10772
$796$	−1.50505	−0.0533451
$797$	−11.8478	−0.419672	−0.209836	−	0.977737i	$-0.567293\pi$
$797$	−11.8478	−0.419672	−0.209836	+	0.977737i	$0.567293\pi$
$798$	0	0
$799$	−30.3910	−1.07515
$800$	3.60264	0.127373
$801$	2.63639	0.0931523
$802$	14.4599	0.510595
$803$	−6.70759	−0.236706
$804$	−0.131398	−0.00463404
$805$	0	0
$806$	0	0
$807$	−4.52201	−0.159182
$808$	−49.1604	−1.72946
$809$	−23.3622	−0.821371	−0.410686	−	0.911777i	$-0.634711\pi$
$809$	−23.3622	−0.821371	−0.410686	+	0.911777i	$0.634711\pi$
$810$	14.7520	0.518333
$811$	−17.8384	−0.626390	−0.313195	−	0.949689i	$-0.601399\pi$
$811$	−17.8384	−0.626390	−0.313195	+	0.949689i	$0.601399\pi$
$812$	0	0
$813$	−40.9995	−1.43792
$814$	16.3671	0.573665
$815$	69.1607	2.42259
$816$	28.1799	0.986494
$817$	7.27060	0.254366
$818$	14.5262	0.507898
$819$	0	0
$820$	3.21510	0.112276
$821$	27.0060	0.942516	0.471258	−	0.881996i	$-0.343800\pi$
$821$	27.0060	0.942516	0.471258	+	0.881996i	$0.343800\pi$
$822$	−15.7350	−0.548822
$823$	−49.8512	−1.73770	−0.868852	−	0.495072i	$-0.835142\pi$
$823$	−49.8512	−1.73770	−0.868852	+	0.495072i	$0.835142\pi$
$824$	−1.55701	−0.0542409
$825$	−8.86886	−0.308774
$826$	0	0
$827$	29.2002	1.01539	0.507695	−	0.861537i	$-0.330498\pi$
$827$	29.2002	1.01539	0.507695	+	0.861537i	$0.330498\pi$
$828$	1.17391	0.0407963
$829$	−15.3988	−0.534822	−0.267411	−	0.963583i	$-0.586168\pi$
$829$	−15.3988	−0.534822	−0.267411	+	0.963583i	$0.586168\pi$
$830$	−61.5837	−2.13760
$831$	30.5076	1.05830
$832$	0	0
$833$	0	0
$834$	17.3152	0.599577
$835$	−16.6404	−0.575863
$836$	0.328422	0.0113587
$837$	−8.57132	−0.296268
$838$	11.6394	0.402075
$839$	29.6007	1.02193	0.510966	−	0.859601i	$-0.329288\pi$
$839$	29.6007	1.02193	0.510966	+	0.859601i	$0.329288\pi$
$840$	0	0
$841$	68.6530	2.36735
$842$	−0.712343	−0.0245490
$843$	−6.96328	−0.239828
$844$	4.07175	0.140156
$845$	0	0
$846$	−9.06689	−0.311726
$847$	0	0
$848$	−28.6637	−0.984317
$849$	−12.8572	−0.441258
$850$	−34.5328	−1.18446
$851$	−48.6617	−1.66810
$852$	−0.112124	−0.00384131
$853$	−27.5135	−0.942046	−0.471023	−	0.882121i	$-0.656115\pi$
$853$	−27.5135	−0.942046	−0.471023	+	0.882121i	$0.656115\pi$
$854$	0	0
$855$	5.64141	0.192932
$856$	45.7487	1.56366
$857$	−13.3991	−0.457706	−0.228853	−	0.973461i	$-0.573497\pi$
$857$	−13.3991	−0.457706	−0.228853	+	0.973461i	$0.573497\pi$
$858$	0	0
$859$	−44.4736	−1.51742	−0.758710	−	0.651429i	$-0.774170\pi$
$859$	−44.4736	−1.51742	−0.758710	+	0.651429i	$0.774170\pi$
$860$	2.25466	0.0768833
$861$	0	0
$862$	−36.2821	−1.23577
$863$	−27.5554	−0.937996	−0.468998	−	0.883199i	$-0.655385\pi$
$863$	−27.5554	−0.937996	−0.468998	+	0.883199i	$0.655385\pi$
$864$	−4.64396	−0.157991
$865$	57.0508	1.93979
$866$	−18.0373	−0.612934
$867$	22.2098	0.754285
$868$	0	0
$869$	−9.95415	−0.337672
$870$	54.0269	1.83169
$871$	0	0
$872$	−17.1114	−0.579464
$873$	−7.38797	−0.250045
$874$	12.3517	0.417803
$875$	0	0
$876$	−0.832004	−0.0281108
$877$	−19.4434	−0.656558	−0.328279	−	0.944581i	$-0.606469\pi$
$877$	−19.4434	−0.656558	−0.328279	+	0.944581i	$0.606469\pi$
$878$	10.7669	0.363366
$879$	31.7977	1.07251
$880$	−17.4831	−0.589356
$881$	−5.21331	−0.175641	−0.0878205	−	0.996136i	$-0.527990\pi$
$881$	−5.21331	−0.175641	−0.0878205	+	0.996136i	$0.527990\pi$
$882$	0	0
$883$	−25.9305	−0.872632	−0.436316	−	0.899793i	$-0.643717\pi$
$883$	−25.9305	−0.872632	−0.436316	+	0.899793i	$0.643717\pi$
$884$	0	0
$885$	−11.2066	−0.376705
$886$	1.01006	0.0339335
$887$	−19.5294	−0.655732	−0.327866	−	0.944724i	$-0.606330\pi$
$887$	−19.5294	−0.655732	−0.327866	+	0.944724i	$0.606330\pi$
$888$	29.7410	0.998042
$889$	0	0
$890$	8.60225	0.288348
$891$	5.49437	0.184068
$892$	−2.97077	−0.0994688
$893$	7.54174	0.252375
$894$	−0.799786	−0.0267488
$895$	−4.40398	−0.147209
$896$	0	0
$897$	0	0
$898$	35.5714	1.18703
$899$	−15.0865	−0.503164
$900$	0.814456	0.0271485
$901$	−45.2942	−1.50897
$902$	−15.1474	−0.504354
$903$	0	0
$904$	18.6621	0.620691
$905$	5.92810	0.197057
$906$	25.5192	0.847820
$907$	−27.8808	−0.925766	−0.462883	−	0.886419i	$-0.653185\pi$
$907$	−27.8808	−0.925766	−0.462883	+	0.886419i	$0.653185\pi$
$908$	−0.263023	−0.00872874
$909$	−21.4689	−0.712079
$910$	0	0
$911$	26.1656	0.866906	0.433453	−	0.901176i	$-0.357295\pi$
$911$	26.1656	0.866906	0.433453	+	0.901176i	$0.357295\pi$
$912$	−6.99305	−0.231563
$913$	−22.9367	−0.759095
$914$	4.02744	0.133216
$915$	55.2464	1.82639
$916$	−0.705101	−0.0232972
$917$	0	0
$918$	44.5142	1.46919
$919$	−27.3730	−0.902951	−0.451476	−	0.892283i	$-0.649102\pi$
$919$	−27.3730	−0.902951	−0.451476	+	0.892283i	$0.649102\pi$
$920$	56.1131	1.84999
$921$	8.56889	0.282355
$922$	−1.24840	−0.0411139
$923$	0	0
$924$	0	0
$925$	−33.7613	−1.11006
$926$	−13.8557	−0.455325
$927$	−0.679963	−0.0223329
$928$	−8.17391	−0.268322
$929$	−9.86006	−0.323498	−0.161749	−	0.986832i	$-0.551714\pi$
$929$	−9.86006	−0.323498	−0.161749	+	0.986832i	$0.551714\pi$
$930$	−8.34668	−0.273699
$931$	0	0
$932$	−1.88691	−0.0618077
$933$	2.24923	0.0736364
$934$	37.1230	1.21470
$935$	−27.6267	−0.903489
$936$	0	0
$937$	48.2540	1.57639	0.788194	−	0.615426i	$-0.211016\pi$
$937$	48.2540	1.57639	0.788194	+	0.615426i	$0.211016\pi$
$938$	0	0
$939$	5.35542	0.174767
$940$	2.33874	0.0762814
$941$	2.42957	0.0792018	0.0396009	−	0.999216i	$-0.487391\pi$
$941$	2.42957	0.0792018	0.0396009	+	0.999216i	$0.487391\pi$
$942$	−12.4644	−0.406113
$943$	45.0355	1.46656
$944$	−10.2847	−0.334740
$945$	0	0
$946$	−10.6225	−0.345366
$947$	0.696419	0.0226306	0.0113153	−	0.999936i	$-0.496398\pi$
$947$	0.696419	0.0226306	0.0113153	+	0.999936i	$0.496398\pi$
$948$	−1.23470	−0.0401013
$949$	0	0
$950$	8.56956	0.278033
$951$	0.653391	0.0211876
$952$	0	0
$953$	34.9112	1.13088	0.565442	−	0.824788i	$-0.308706\pi$
$953$	34.9112	1.13088	0.565442	+	0.824788i	$0.308706\pi$
$954$	−13.5131	−0.437504
$955$	−15.7149	−0.508521
$956$	1.46108	0.0472549
$957$	20.1223	0.650460
$958$	45.5119	1.47042
$959$	0	0
$960$	−34.1227	−1.10131
$961$	−28.6693	−0.924815
$962$	0	0
$963$	19.9790	0.643814
$964$	−3.45377	−0.111239
$965$	−26.5168	−0.853605
$966$	0	0
$967$	12.1384	0.390345	0.195172	−	0.980769i	$-0.437473\pi$
$967$	12.1384	0.390345	0.195172	+	0.980769i	$0.437473\pi$
$968$	25.1163	0.807269
$969$	−11.0504	−0.354988
$970$	−24.1061	−0.774001
$971$	33.1210	1.06290	0.531451	−	0.847089i	$-0.321647\pi$
$971$	33.1210	1.06290	0.531451	+	0.847089i	$0.321647\pi$
$972$	−1.78641	−0.0572990
$973$	0	0
$974$	−49.8728	−1.59803
$975$	0	0
$976$	50.7019	1.62293
$977$	43.1916	1.38182	0.690911	−	0.722940i	$-0.257210\pi$
$977$	43.1916	1.38182	0.690911	+	0.722940i	$0.257210\pi$
$978$	−40.4168	−1.29239
$979$	3.20389	0.102397
$980$	0	0
$981$	−7.47274	−0.238586
$982$	17.8413	0.569339
$983$	39.1839	1.24977	0.624887	−	0.780716i	$-0.285145\pi$
$983$	39.1839	1.24977	0.624887	+	0.780716i	$0.285145\pi$
$984$	−27.5248	−0.877457
$985$	−52.8254	−1.68316
$986$	78.3502	2.49518
$987$	0	0
$988$	0	0
$989$	31.5822	1.00425
$990$	−8.24219	−0.261954
$991$	8.45729	0.268655	0.134327	−	0.990937i	$-0.457113\pi$
$991$	8.45729	0.268655	0.134327	+	0.990937i	$0.457113\pi$
$992$	1.26280	0.0400938
$993$	13.2007	0.418911
$994$	0	0
$995$	−31.4177	−0.996009
$996$	−2.84505	−0.0901489
$997$	29.7433	0.941980	0.470990	−	0.882139i	$-0.343897\pi$
$997$	29.7433	0.941980	0.470990	+	0.882139i	$0.343897\pi$
$998$	5.68247	0.179875
$999$	43.5197	1.37690

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8281.2.a.cw.1.18		24
7.3	odd	6		1183.2.e.k.170.7	✓	48
7.5	odd	6		1183.2.e.k.508.7	yes	48
7.6	odd	2		8281.2.a.cv.1.18		24
13.12	even	2		8281.2.a.ct.1.7		24
91.12	odd	6		1183.2.e.l.508.18	yes	48
91.38	odd	6		1183.2.e.l.170.18	yes	48
91.90	odd	2		8281.2.a.cu.1.7		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1183.2.e.k.170.7	✓	48	7.3	odd	6
1183.2.e.k.508.7	yes	48	7.5	odd	6
1183.2.e.l.170.18	yes	48	91.38	odd	6
1183.2.e.l.508.18	yes	48	91.12	odd	6
8281.2.a.ct.1.7		24	13.12	even	2
8281.2.a.cu.1.7		24	91.90	odd	2
8281.2.a.cv.1.18		24	7.6	odd	2
8281.2.a.cw.1.18		24	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:	\( N \)	\(=\)	\( 8281 = 7^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8281.a (trivial)

\(f(q)\)	\(=\)	\(q+1.36142 q^{2} +1.31293 q^{3} -0.146524 q^{4} -3.05867 q^{5} +1.78745 q^{6} -2.92233 q^{8} -1.27622 q^{9} -4.16415 q^{10} -1.55093 q^{11} -0.192376 q^{12} -4.01582 q^{15} -3.68548 q^{16} -5.82376 q^{17} -1.73747 q^{18} +1.44521 q^{19} +0.448169 q^{20} -2.11147 q^{22} +6.27773 q^{23} -3.83681 q^{24} +4.35546 q^{25} -5.61437 q^{27} -9.88196 q^{29} -5.46723 q^{30} +1.52667 q^{31} +0.827155 q^{32} -2.03626 q^{33} -7.92861 q^{34} +0.186996 q^{36} -7.75148 q^{37} +1.96754 q^{38} +8.93844 q^{40} +7.17386 q^{41} +5.03082 q^{43} +0.227249 q^{44} +3.90352 q^{45} +8.54665 q^{46} +5.21844 q^{47} -4.83878 q^{48} +5.92963 q^{50} -7.64619 q^{51} +7.77747 q^{53} -7.64354 q^{54} +4.74378 q^{55} +1.89746 q^{57} -13.4535 q^{58} +2.79061 q^{59} +0.588415 q^{60} -13.7572 q^{61} +2.07845 q^{62} +8.49707 q^{64} -2.77222 q^{66} +0.683026 q^{67} +0.853322 q^{68} +8.24222 q^{69} +0.582838 q^{71} +3.72952 q^{72} +4.32488 q^{73} -10.5531 q^{74} +5.71842 q^{75} -0.211758 q^{76} +6.41818 q^{79} +11.2727 q^{80} -3.54263 q^{81} +9.76667 q^{82} +14.7890 q^{83} +17.8130 q^{85} +6.84909 q^{86} -12.9743 q^{87} +4.53233 q^{88} -2.06579 q^{89} +5.31435 q^{90} -0.919839 q^{92} +2.00442 q^{93} +7.10451 q^{94} -4.42042 q^{95} +1.08600 q^{96} +5.78897 q^{97} +1.97932 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + q^{2} + 23 q^{4} + 13 q^{5} + 14 q^{6} + 26 q^{9} - 5 q^{10} + q^{11} - 5 q^{12} - 5 q^{15} + 17 q^{16} + 5 q^{17} + 24 q^{19} + 34 q^{20} - 14 q^{22} + 11 q^{23} + 32 q^{24} + 33 q^{25} + 21 q^{27}+ \cdots + 39 q^{99}+O(q^{100}) \) 24 * q + q^2 + 23 * q^4 + 13 * q^5 + 14 * q^6 + 26 * q^9 - 5 * q^10 + q^11 - 5 * q^12 - 5 * q^15 + 17 * q^16 + 5 * q^17 + 24 * q^19 + 34 * q^20 - 14 * q^22 + 11 * q^23 + 32 * q^24 + 33 * q^25 + 21 * q^27 + 4 * q^29 - 22 * q^30 + 40 * q^31 + 6 * q^32 + 24 * q^33 + 36 * q^34 - 15 * q^36 + 4 * q^37 + 29 * q^38 + 4 * q^40 + 49 * q^41 + 13 * q^43 - 10 * q^44 + 58 * q^45 + 10 * q^46 + 62 * q^47 - 89 * q^48 + 23 * q^50 - 21 * q^51 - 18 * q^53 + 12 * q^54 + 14 * q^55 + 13 * q^57 - 56 * q^58 + 79 * q^59 - 22 * q^60 - 13 * q^61 - 12 * q^62 + 18 * q^64 + 38 * q^66 + 2 * q^67 + 12 * q^68 + 28 * q^69 + 19 * q^71 - 81 * q^72 + 17 * q^73 + 17 * q^74 - 24 * q^75 + 58 * q^76 - 9 * q^79 + 63 * q^80 + 16 * q^81 + 22 * q^82 + 81 * q^83 + 34 * q^85 - 22 * q^86 - 70 * q^87 - 33 * q^88 + 72 * q^89 - q^90 - 4 * q^92 - 19 * q^93 + 30 * q^94 + 13 * q^95 + 11 * q^96 + 45 * q^97 + 39 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more