LMFDB - Embedded newform 8.12.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8,12,Mod(1,8)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 12, names="a")

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

chi := DirichletCharacter("8.1");

S:= CuspForms(chi, 12);

N := Newforms(S);

Level:	$ N $	$=$	$ 8 = 2^{3} $
Weight:	$ k $	$=$	$ 12 $
Character orbit:	$[\chi]$	$=$	8.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:	$6.14674544448$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	8.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-36.0000 q^{3} -3490.00 q^{5} -55464.0 q^{7} -175851. q^{9} +O(q^{10})$

$q-36.0000 q^{3} -3490.00 q^{5} -55464.0 q^{7} -175851. q^{9} -597004. q^{11} +1.37388e6 q^{13} +125640. q^{15} +1.01408e7 q^{17} -7.29740e6 q^{19} +1.99670e6 q^{21} -3.20575e7 q^{23} -3.66480e7 q^{25} +1.27079e7 q^{27} -1.36054e7 q^{29} +2.33161e8 q^{31} +2.14921e7 q^{33} +1.93569e8 q^{35} -2.57786e8 q^{37} -4.94596e7 q^{39} -2.21439e8 q^{41} -1.69776e9 q^{43} +6.13720e8 q^{45} +5.27509e8 q^{47} +1.09893e9 q^{49} -3.65071e8 q^{51} +3.27738e9 q^{53} +2.08354e9 q^{55} +2.62706e8 q^{57} -3.00191e9 q^{59} -1.16300e10 q^{61} +9.75340e9 q^{63} -4.79483e9 q^{65} -1.71890e10 q^{67} +1.15407e9 q^{69} +2.61695e10 q^{71} -7.03902e9 q^{73} +1.31933e9 q^{75} +3.31122e10 q^{77} -4.19991e9 q^{79} +3.06940e10 q^{81} -3.97399e10 q^{83} -3.53916e10 q^{85} +4.89794e8 q^{87} +1.05653e10 q^{89} -7.62008e10 q^{91} -8.39379e9 q^{93} +2.54679e10 q^{95} -6.98516e10 q^{97} +1.04984e11 q^{99} +O(q^{100})$

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
$3$	−36.0000	−0.0855334	−0.0427667	−	0.999085i	$-0.513617\pi$
$3$	−36.0000	−0.0855334	−0.0427667	+	0.999085i	$0.513617\pi$
$4$	0	0
$5$	−3490.00	−0.499448	−0.249724	−	0.968317i	$-0.580340\pi$
$5$	−3490.00	−0.499448	−0.249724	+	0.968317i	$0.580340\pi$
$6$	0	0
$7$	−55464.0	−1.24730	−0.623652	−	0.781703i	$-0.714352\pi$
$7$	−55464.0	−1.24730	−0.623652	+	0.781703i	$0.714352\pi$
$8$	0	0
$9$	−175851.	−0.992684
$10$	0	0
$11$	−597004.	−1.11768	−0.558840	−	0.829276i	$-0.688753\pi$
$11$	−597004.	−1.11768	−0.558840	+	0.829276i	$0.688753\pi$
$12$	0	0
$13$	1.37388e6	1.02627	0.513133	−	0.858309i	$-0.328485\pi$
$13$	1.37388e6	1.02627	0.513133	+	0.858309i	$0.328485\pi$
$14$	0	0
$15$	125640.	0.0427195
$16$	0	0
$17$	1.01408e7	1.73223	0.866114	−	0.499846i	$-0.166610\pi$
$17$	1.01408e7	1.73223	0.866114	+	0.499846i	$0.166610\pi$
$18$	0	0
$19$	−7.29740e6	−0.676119	−0.338059	−	0.941125i	$-0.609771\pi$
$19$	−7.29740e6	−0.676119	−0.338059	+	0.941125i	$0.609771\pi$
$20$	0	0
$21$	1.99670e6	0.106686
$22$	0	0
$23$	−3.20575e7	−1.03855	−0.519273	−	0.854608i	$-0.673797\pi$
$23$	−3.20575e7	−1.03855	−0.519273	+	0.854608i	$0.673797\pi$
$24$	0	0
$25$	−3.66480e7	−0.750552
$26$	0	0
$27$	1.27079e7	0.170441
$28$	0	0
$29$	−1.36054e7	−0.123175	−0.0615875	−	0.998102i	$-0.519616\pi$
$29$	−1.36054e7	−0.123175	−0.0615875	+	0.998102i	$0.519616\pi$
$30$	0	0
$31$	2.33161e8	1.46274	0.731368	−	0.681983i	$-0.238882\pi$
$31$	2.33161e8	1.46274	0.731368	+	0.681983i	$0.238882\pi$
$32$	0	0
$33$	2.14921e7	0.0955989
$34$	0	0
$35$	1.93569e8	0.622963
$36$	0	0
$37$	−2.57786e8	−0.611153	−0.305577	−	0.952167i	$-0.598849\pi$
$37$	−2.57786e8	−0.611153	−0.305577	+	0.952167i	$0.598849\pi$
$38$	0	0
$39$	−4.94596e7	−0.0877800
$40$	0	0
$41$	−2.21439e8	−0.298498	−0.149249	−	0.988800i	$-0.547686\pi$
$41$	−2.21439e8	−0.298498	−0.149249	+	0.988800i	$0.547686\pi$
$42$	0	0
$43$	−1.69776e9	−1.76116	−0.880581	−	0.473895i	$-0.842848\pi$
$43$	−1.69776e9	−1.76116	−0.880581	+	0.473895i	$0.842848\pi$
$44$	0	0
$45$	6.13720e8	0.495794
$46$	0	0
$47$	5.27509e8	0.335500	0.167750	−	0.985830i	$-0.446350\pi$
$47$	5.27509e8	0.335500	0.167750	+	0.985830i	$0.446350\pi$
$48$	0	0
$49$	1.09893e9	0.555765
$50$	0	0
$51$	−3.65071e8	−0.148163
$52$	0	0
$53$	3.27738e9	1.07649	0.538244	−	0.842789i	$-0.319088\pi$
$53$	3.27738e9	1.07649	0.538244	+	0.842789i	$0.319088\pi$
$54$	0	0
$55$	2.08354e9	0.558223
$56$	0	0
$57$	2.62706e8	0.0578307
$58$	0	0
$59$	−3.00191e9	−0.546653	−0.273326	−	0.961921i	$-0.588124\pi$
$59$	−3.00191e9	−0.546653	−0.273326	+	0.961921i	$0.588124\pi$
$60$	0	0
$61$	−1.16300e10	−1.76306	−0.881529	−	0.472130i	$-0.843485\pi$
$61$	−1.16300e10	−1.76306	−0.881529	+	0.472130i	$0.843485\pi$
$62$	0	0
$63$	9.75340e9	1.23818
$64$	0	0
$65$	−4.79483e9	−0.512566
$66$	0	0
$67$	−1.71890e10	−1.55539	−0.777695	−	0.628642i	$-0.783611\pi$
$67$	−1.71890e10	−1.55539	−0.777695	+	0.628642i	$0.783611\pi$
$68$	0	0
$69$	1.15407e9	0.0888304
$70$	0	0
$71$	2.61695e10	1.72137	0.860687	−	0.509135i	$-0.170034\pi$
$71$	2.61695e10	1.72137	0.860687	+	0.509135i	$0.170034\pi$
$72$	0	0
$73$	−7.03902e9	−0.397408	−0.198704	−	0.980060i	$-0.563673\pi$
$73$	−7.03902e9	−0.397408	−0.198704	+	0.980060i	$0.563673\pi$
$74$	0	0
$75$	1.31933e9	0.0641972
$76$	0	0
$77$	3.31122e10	1.39409
$78$	0	0
$79$	−4.19991e9	−0.153565	−0.0767823	−	0.997048i	$-0.524465\pi$
$79$	−4.19991e9	−0.153565	−0.0767823	+	0.997048i	$0.524465\pi$
$80$	0	0
$81$	3.06940e10	0.978106
$82$	0	0
$83$	−3.97399e10	−1.10738	−0.553691	−	0.832722i	$-0.686781\pi$
$83$	−3.97399e10	−1.10738	−0.553691	+	0.832722i	$0.686781\pi$
$84$	0	0
$85$	−3.53916e10	−0.865159
$86$	0	0
$87$	4.89794e8	0.0105356
$88$	0	0
$89$	1.05653e10	0.200557	0.100279	−	0.994959i	$-0.468027\pi$
$89$	1.05653e10	0.200557	0.100279	+	0.994959i	$0.468027\pi$
$90$	0	0
$91$	−7.62008e10	−1.28006
$92$	0	0
$93$	−8.39379e9	−0.125113
$94$	0	0
$95$	2.54679e10	0.337686
$96$	0	0
$97$	−6.98516e10	−0.825909	−0.412954	−	0.910752i	$-0.635503\pi$
$97$	−6.98516e10	−0.825909	−0.412954	+	0.910752i	$0.635503\pi$
$98$	0	0
$99$	1.04984e11	1.10950
$100$	0	0
$101$	7.45194e10	0.705508	0.352754	−	0.935716i	$-0.385245\pi$
$101$	7.45194e10	0.705508	0.352754	+	0.935716i	$0.385245\pi$
$102$	0	0
$103$	7.86642e10	0.668609	0.334304	−	0.942465i	$-0.391499\pi$
$103$	7.86642e10	0.668609	0.334304	+	0.942465i	$0.391499\pi$
$104$	0	0
$105$	−6.96850e9	−0.0532841
$106$	0	0
$107$	7.88490e10	0.543482	0.271741	−	0.962370i	$-0.412401\pi$
$107$	7.88490e10	0.543482	0.271741	+	0.962370i	$0.412401\pi$
$108$	0	0
$109$	−7.08680e10	−0.441169	−0.220584	−	0.975368i	$-0.570796\pi$
$109$	−7.08680e10	−0.441169	−0.220584	+	0.975368i	$0.570796\pi$
$110$	0	0
$111$	9.28030e9	0.0522740
$112$	0	0
$113$	1.96816e10	0.100491	0.0502457	−	0.998737i	$-0.484000\pi$
$113$	1.96816e10	0.100491	0.0502457	+	0.998737i	$0.484000\pi$
$114$	0	0
$115$	1.11881e11	0.518700
$116$	0	0
$117$	−2.41598e11	−1.01876
$118$	0	0
$119$	−5.62452e11	−2.16061
$120$	0	0
$121$	7.11021e10	0.249209
$122$	0	0
$123$	7.97179e9	0.0255316
$124$	0	0
$125$	2.98312e11	0.874310
$126$	0	0
$127$	−1.90252e11	−0.510985	−0.255492	−	0.966811i	$-0.582238\pi$
$127$	−1.90252e11	−0.510985	−0.255492	+	0.966811i	$0.582238\pi$
$128$	0	0
$129$	6.11193e10	0.150638
$130$	0	0
$131$	1.14525e11	0.259363	0.129681	−	0.991556i	$-0.458605\pi$
$131$	1.14525e11	0.259363	0.129681	+	0.991556i	$0.458605\pi$
$132$	0	0
$133$	4.04743e11	0.843325
$134$	0	0
$135$	−4.43507e10	−0.0851264
$136$	0	0
$137$	1.17101e11	0.207299	0.103650	−	0.994614i	$-0.466948\pi$
$137$	1.17101e11	0.207299	0.103650	+	0.994614i	$0.466948\pi$
$138$	0	0
$139$	9.89561e11	1.61756	0.808781	−	0.588110i	$-0.200128\pi$
$139$	9.89561e11	1.61756	0.808781	+	0.588110i	$0.200128\pi$
$140$	0	0
$141$	−1.89903e10	−0.0286964
$142$	0	0
$143$	−8.20211e11	−1.14704
$144$	0	0
$145$	4.74829e10	0.0615195
$146$	0	0
$147$	−3.95614e10	−0.0475364
$148$	0	0
$149$	−7.97579e11	−0.889712	−0.444856	−	0.895602i	$-0.646745\pi$
$149$	−7.97579e11	−0.889712	−0.444856	+	0.895602i	$0.646745\pi$
$150$	0	0
$151$	1.18323e12	1.22658	0.613290	−	0.789858i	$-0.289846\pi$
$151$	1.18323e12	1.22658	0.613290	+	0.789858i	$0.289846\pi$
$152$	0	0
$153$	−1.78328e12	−1.71956
$154$	0	0
$155$	−8.13731e11	−0.730561
$156$	0	0
$157$	−7.53512e11	−0.630438	−0.315219	−	0.949019i	$-0.602078\pi$
$157$	−7.53512e11	−0.630438	−0.315219	+	0.949019i	$0.602078\pi$
$158$	0	0
$159$	−1.17986e11	−0.0920757
$160$	0	0
$161$	1.77804e12	1.29538
$162$	0	0
$163$	8.34305e11	0.567928	0.283964	−	0.958835i	$-0.408350\pi$
$163$	8.34305e11	0.567928	0.283964	+	0.958835i	$0.408350\pi$
$164$	0	0
$165$	−7.50076e10	−0.0477467
$166$	0	0
$167$	−1.01514e12	−0.604761	−0.302380	−	0.953187i	$-0.597781\pi$
$167$	−1.01514e12	−0.604761	−0.302380	+	0.953187i	$0.597781\pi$
$168$	0	0
$169$	9.53804e10	0.0532209
$170$	0	0
$171$	1.28325e12	0.671172
$172$	0	0
$173$	2.89496e12	1.42033	0.710165	−	0.704035i	$-0.248620\pi$
$173$	2.89496e12	1.42033	0.710165	+	0.704035i	$0.248620\pi$
$174$	0	0
$175$	2.03265e12	0.936165
$176$	0	0
$177$	1.08069e11	0.0467570
$178$	0	0
$179$	−4.09263e12	−1.66461	−0.832303	−	0.554322i	$-0.812978\pi$
$179$	−4.09263e12	−1.66461	−0.832303	+	0.554322i	$0.812978\pi$
$180$	0	0
$181$	5.08615e11	0.194606	0.0973032	−	0.995255i	$-0.468978\pi$
$181$	5.08615e11	0.194606	0.0973032	+	0.995255i	$0.468978\pi$
$182$	0	0
$183$	4.18681e11	0.150800
$184$	0	0
$185$	8.99674e11	0.305239
$186$	0	0
$187$	−6.05413e12	−1.93608
$188$	0	0
$189$	−7.04833e11	−0.212592
$190$	0	0
$191$	2.99951e12	0.853822	0.426911	−	0.904294i	$-0.359602\pi$
$191$	2.99951e12	0.853822	0.426911	+	0.904294i	$0.359602\pi$
$192$	0	0
$193$	−2.18193e12	−0.586509	−0.293255	−	0.956034i	$-0.594738\pi$
$193$	−2.18193e12	−0.586509	−0.293255	+	0.956034i	$0.594738\pi$
$194$	0	0
$195$	1.72614e11	0.0438415
$196$	0	0
$197$	−4.77475e12	−1.14653	−0.573267	−	0.819369i	$-0.694324\pi$
$197$	−4.77475e12	−1.14653	−0.573267	+	0.819369i	$0.694324\pi$
$198$	0	0
$199$	2.95210e11	0.0670563	0.0335281	−	0.999438i	$-0.489326\pi$
$199$	2.95210e11	0.0670563	0.0335281	+	0.999438i	$0.489326\pi$
$200$	0	0
$201$	6.18804e11	0.133038
$202$	0	0
$203$	7.54610e11	0.153636
$204$	0	0
$205$	7.72821e11	0.149085
$206$	0	0
$207$	5.63734e12	1.03095
$208$	0	0
$209$	4.35657e12	0.755685
$210$	0	0
$211$	4.76886e12	0.784984	0.392492	−	0.919755i	$-0.371613\pi$
$211$	4.76886e12	0.784984	0.392492	+	0.919755i	$0.371613\pi$
$212$	0	0
$213$	−9.42103e11	−0.147235
$214$	0	0
$215$	5.92518e12	0.879609
$216$	0	0
$217$	−1.29320e13	−1.82448
$218$	0	0
$219$	2.53405e11	0.0339916
$220$	0	0
$221$	1.39323e13	1.77773
$222$	0	0
$223$	−9.21455e12	−1.11892	−0.559458	−	0.828859i	$-0.688991\pi$
$223$	−9.21455e12	−1.11892	−0.559458	+	0.828859i	$0.688991\pi$
$224$	0	0
$225$	6.44459e12	0.745061
$226$	0	0
$227$	−5.85444e12	−0.644678	−0.322339	−	0.946624i	$-0.604469\pi$
$227$	−5.85444e12	−0.644678	−0.322339	+	0.946624i	$0.604469\pi$
$228$	0	0
$229$	−1.40273e13	−1.47191	−0.735953	−	0.677033i	$-0.763266\pi$
$229$	−1.40273e13	−1.47191	−0.735953	+	0.677033i	$0.763266\pi$
$230$	0	0
$231$	−1.19204e12	−0.119241
$232$	0	0
$233$	−5.29610e12	−0.505241	−0.252621	−	0.967565i	$-0.581292\pi$
$233$	−5.29610e12	−0.505241	−0.252621	+	0.967565i	$0.581292\pi$
$234$	0	0
$235$	−1.84101e12	−0.167565
$236$	0	0
$237$	1.51197e11	0.0131349
$238$	0	0
$239$	−1.31646e13	−1.09199	−0.545997	−	0.837787i	$-0.683849\pi$
$239$	−1.31646e13	−1.09199	−0.545997	+	0.837787i	$0.683849\pi$
$240$	0	0
$241$	2.25897e13	1.78985	0.894924	−	0.446219i	$-0.147230\pi$
$241$	2.25897e13	1.78985	0.894924	+	0.446219i	$0.147230\pi$
$242$	0	0
$243$	−3.35616e12	−0.254102
$244$	0	0
$245$	−3.83526e12	−0.277576
$246$	0	0
$247$	−1.00257e13	−0.693878
$248$	0	0
$249$	1.43064e12	0.0947182
$250$	0	0
$251$	2.61185e13	1.65479	0.827396	−	0.561619i	$-0.189821\pi$
$251$	2.61185e13	1.65479	0.827396	+	0.561619i	$0.189821\pi$
$252$	0	0
$253$	1.91384e13	1.16076
$254$	0	0
$255$	1.27410e12	0.0739999
$256$	0	0
$257$	−1.26580e13	−0.704259	−0.352129	−	0.935951i	$-0.614542\pi$
$257$	−1.26580e13	−0.704259	−0.352129	+	0.935951i	$0.614542\pi$
$258$	0	0
$259$	1.42979e13	0.762293
$260$	0	0
$261$	2.39252e12	0.122274
$262$	0	0
$263$	5.37023e12	0.263170	0.131585	−	0.991305i	$-0.457993\pi$
$263$	5.37023e12	0.263170	0.131585	+	0.991305i	$0.457993\pi$
$264$	0	0
$265$	−1.14381e13	−0.537650
$266$	0	0
$267$	−3.80352e11	−0.0171543
$268$	0	0
$269$	−8.60140e12	−0.372333	−0.186167	−	0.982518i	$-0.559606\pi$
$269$	−8.60140e12	−0.372333	−0.186167	+	0.982518i	$0.559606\pi$
$270$	0	0
$271$	−6.01371e12	−0.249926	−0.124963	−	0.992161i	$-0.539881\pi$
$271$	−6.01371e12	−0.249926	−0.124963	+	0.992161i	$0.539881\pi$
$272$	0	0
$273$	2.74323e12	0.109488
$274$	0	0
$275$	2.18790e13	0.838876
$276$	0	0
$277$	−2.63312e13	−0.970134	−0.485067	−	0.874477i	$-0.661205\pi$
$277$	−2.63312e13	−0.970134	−0.485067	+	0.874477i	$0.661205\pi$
$278$	0	0
$279$	−4.10016e13	−1.45204
$280$	0	0
$281$	−5.11748e13	−1.74250	−0.871248	−	0.490843i	$-0.836689\pi$
$281$	−5.11748e13	−1.74250	−0.871248	+	0.490843i	$0.836689\pi$
$282$	0	0
$283$	8.94896e12	0.293054	0.146527	−	0.989207i	$-0.453191\pi$
$283$	8.94896e12	0.293054	0.146527	+	0.989207i	$0.453191\pi$
$284$	0	0
$285$	−9.16845e11	−0.0288835
$286$	0	0
$287$	1.22819e13	0.372318
$288$	0	0
$289$	6.85649e13	2.00062
$290$	0	0
$291$	2.51466e12	0.0706428
$292$	0	0
$293$	−2.68480e13	−0.726339	−0.363170	−	0.931723i	$-0.618305\pi$
$293$	−2.68480e13	−0.726339	−0.363170	+	0.931723i	$0.618305\pi$
$294$	0	0
$295$	1.04767e13	0.273025
$296$	0	0
$297$	−7.58668e12	−0.190498
$298$	0	0
$299$	−4.40430e13	−1.06582
$300$	0	0
$301$	9.41645e13	2.19670
$302$	0	0
$303$	−2.68270e12	−0.0603445
$304$	0	0
$305$	4.05888e13	0.880556
$306$	0	0
$307$	6.62433e12	0.138638	0.0693188	−	0.997595i	$-0.477917\pi$
$307$	6.62433e12	0.138638	0.0693188	+	0.997595i	$0.477917\pi$
$308$	0	0
$309$	−2.83191e12	−0.0571884
$310$	0	0
$311$	4.02852e13	0.785170	0.392585	−	0.919716i	$-0.371581\pi$
$311$	4.02852e13	0.785170	0.392585	+	0.919716i	$0.371581\pi$
$312$	0	0
$313$	−2.15423e13	−0.405320	−0.202660	−	0.979249i	$-0.564959\pi$
$313$	−2.15423e13	−0.405320	−0.202660	+	0.979249i	$0.564959\pi$
$314$	0	0
$315$	−3.40394e13	−0.618406
$316$	0	0
$317$	−2.79479e12	−0.0490369	−0.0245184	−	0.999699i	$-0.507805\pi$
$317$	−2.79479e12	−0.0490369	−0.0245184	+	0.999699i	$0.507805\pi$
$318$	0	0
$319$	8.12248e12	0.137670
$320$	0	0
$321$	−2.83856e12	−0.0464859
$322$	0	0
$323$	−7.40018e13	−1.17119
$324$	0	0
$325$	−5.03499e13	−0.770265
$326$	0	0
$327$	2.55125e12	0.0377346
$328$	0	0
$329$	−2.92578e13	−0.418470
$330$	0	0
$331$	−4.48560e13	−0.620536	−0.310268	−	0.950649i	$-0.600419\pi$
$331$	−4.48560e13	−0.620536	−0.310268	+	0.950649i	$0.600419\pi$
$332$	0	0
$333$	4.53320e13	0.606682
$334$	0	0
$335$	5.99896e13	0.776836
$336$	0	0
$337$	−5.23428e13	−0.655982	−0.327991	−	0.944681i	$-0.606372\pi$
$337$	−5.23428e13	−0.655982	−0.327991	+	0.944681i	$0.606372\pi$
$338$	0	0
$339$	−7.08538e11	−0.00859537
$340$	0	0
$341$	−1.39198e14	−1.63487
$342$	0	0
$343$	4.87195e13	0.554096
$344$	0	0
$345$	−4.02770e12	−0.0443662
$346$	0	0
$347$	1.62475e14	1.73370	0.866849	−	0.498571i	$-0.166142\pi$
$347$	1.62475e14	1.73370	0.866849	+	0.498571i	$0.166142\pi$
$348$	0	0
$349$	5.60706e13	0.579689	0.289845	−	0.957074i	$-0.406396\pi$
$349$	5.60706e13	0.579689	0.289845	+	0.957074i	$0.406396\pi$
$350$	0	0
$351$	1.74591e13	0.174918
$352$	0	0
$353$	1.21433e14	1.17917	0.589585	−	0.807707i	$-0.299291\pi$
$353$	1.21433e14	1.17917	0.589585	+	0.807707i	$0.299291\pi$
$354$	0	0
$355$	−9.13317e13	−0.859737
$356$	0	0
$357$	2.02483e13	0.184805
$358$	0	0
$359$	9.93082e13	0.878953	0.439476	−	0.898254i	$-0.355164\pi$
$359$	9.93082e13	0.878953	0.439476	+	0.898254i	$0.355164\pi$
$360$	0	0
$361$	−6.32383e13	−0.542863
$362$	0	0
$363$	−2.55968e12	−0.0213156
$364$	0	0
$365$	2.45662e13	0.198485
$366$	0	0
$367$	5.80965e13	0.455498	0.227749	−	0.973720i	$-0.426863\pi$
$367$	5.80965e13	0.455498	0.227749	+	0.973720i	$0.426863\pi$
$368$	0	0
$369$	3.89402e13	0.296315
$370$	0	0
$371$	−1.81777e14	−1.34271
$372$	0	0
$373$	1.06336e13	0.0762576	0.0381288	−	0.999273i	$-0.487860\pi$
$373$	1.06336e13	0.0762576	0.0381288	+	0.999273i	$0.487860\pi$
$374$	0	0
$375$	−1.07392e13	−0.0747827
$376$	0	0
$377$	−1.86922e13	−0.126410
$378$	0	0
$379$	−1.94564e14	−1.27805	−0.639023	−	0.769188i	$-0.720661\pi$
$379$	−1.94564e14	−1.27805	−0.639023	+	0.769188i	$0.720661\pi$
$380$	0	0
$381$	6.84906e12	0.0437062
$382$	0	0
$383$	−2.19102e14	−1.35848	−0.679240	−	0.733916i	$-0.737691\pi$
$383$	−2.19102e14	−1.35848	−0.679240	+	0.733916i	$0.737691\pi$
$384$	0	0
$385$	−1.15562e14	−0.696273
$386$	0	0
$387$	2.98553e14	1.74828
$388$	0	0
$389$	2.58688e14	1.47250	0.736248	−	0.676712i	$-0.236596\pi$
$389$	2.58688e14	1.47250	0.736248	+	0.676712i	$0.236596\pi$
$390$	0	0
$391$	−3.25090e14	−1.79900
$392$	0	0
$393$	−4.12289e12	−0.0221842
$394$	0	0
$395$	1.46577e13	0.0766975
$396$	0	0
$397$	−2.53970e14	−1.29251	−0.646256	−	0.763121i	$-0.723666\pi$
$397$	−2.53970e14	−1.29251	−0.646256	+	0.763121i	$0.723666\pi$
$398$	0	0
$399$	−1.45707e13	−0.0721324
$400$	0	0
$401$	1.76194e14	0.848590	0.424295	−	0.905524i	$-0.360522\pi$
$401$	1.76194e14	0.848590	0.424295	+	0.905524i	$0.360522\pi$
$402$	0	0
$403$	3.20334e14	1.50116
$404$	0	0
$405$	−1.07122e14	−0.488513
$406$	0	0
$407$	1.53899e14	0.683074
$408$	0	0
$409$	2.54652e14	1.10019	0.550097	−	0.835101i	$-0.314591\pi$
$409$	2.54652e14	1.10019	0.550097	+	0.835101i	$0.314591\pi$
$410$	0	0
$411$	−4.21564e12	−0.0177310
$412$	0	0
$413$	1.66498e14	0.681841
$414$	0	0
$415$	1.38692e14	0.553080
$416$	0	0
$417$	−3.56242e13	−0.138356
$418$	0	0
$419$	−1.09764e13	−0.0415224	−0.0207612	−	0.999784i	$-0.506609\pi$
$419$	−1.09764e13	−0.0415224	−0.0207612	+	0.999784i	$0.506609\pi$
$420$	0	0
$421$	−3.06791e14	−1.13056	−0.565278	−	0.824901i	$-0.691231\pi$
$421$	−3.06791e14	−1.13056	−0.565278	+	0.824901i	$0.691231\pi$
$422$	0	0
$423$	−9.27631e13	−0.333045
$424$	0	0
$425$	−3.71642e14	−1.30013
$426$	0	0
$427$	6.45048e14	2.19907
$428$	0	0
$429$	2.95276e13	0.0981099
$430$	0	0
$431$	−1.94137e14	−0.628756	−0.314378	−	0.949298i	$-0.601796\pi$
$431$	−1.94137e14	−0.628756	−0.314378	+	0.949298i	$0.601796\pi$
$432$	0	0
$433$	−1.97021e14	−0.622055	−0.311027	−	0.950401i	$-0.600673\pi$
$433$	−1.97021e14	−0.622055	−0.311027	+	0.950401i	$0.600673\pi$
$434$	0	0
$435$	−1.70938e12	−0.00526197
$436$	0	0
$437$	2.33936e14	0.702181
$438$	0	0
$439$	−4.81833e14	−1.41040	−0.705199	−	0.709009i	$-0.749143\pi$
$439$	−4.81833e14	−1.41040	−0.705199	+	0.709009i	$0.749143\pi$
$440$	0	0
$441$	−1.93248e14	−0.551699
$442$	0	0
$443$	−4.72499e14	−1.31577	−0.657886	−	0.753118i	$-0.728549\pi$
$443$	−4.72499e14	−1.31577	−0.657886	+	0.753118i	$0.728549\pi$
$444$	0	0
$445$	−3.68730e13	−0.100168
$446$	0	0
$447$	2.87129e13	0.0761001
$448$	0	0
$449$	−3.52414e13	−0.0911377	−0.0455688	−	0.998961i	$-0.514510\pi$
$449$	−3.52414e13	−0.0911377	−0.0455688	+	0.998961i	$0.514510\pi$
$450$	0	0
$451$	1.32200e14	0.333626
$452$	0	0
$453$	−4.25962e13	−0.104913
$454$	0	0
$455$	2.65941e14	0.639326
$456$	0	0
$457$	5.41522e14	1.27080	0.635400	−	0.772183i	$-0.280835\pi$
$457$	5.41522e14	1.27080	0.635400	+	0.772183i	$0.280835\pi$
$458$	0	0
$459$	1.28869e14	0.295243
$460$	0	0
$461$	−2.63903e14	−0.590322	−0.295161	−	0.955447i	$-0.595373\pi$
$461$	−2.63903e14	−0.590322	−0.295161	+	0.955447i	$0.595373\pi$
$462$	0	0
$463$	−3.44968e13	−0.0753499	−0.0376750	−	0.999290i	$-0.511995\pi$
$463$	−3.44968e13	−0.0753499	−0.0376750	+	0.999290i	$0.511995\pi$
$464$	0	0
$465$	2.92943e13	0.0624874
$466$	0	0
$467$	−5.88838e14	−1.22674	−0.613370	−	0.789795i	$-0.710187\pi$
$467$	−5.88838e14	−1.22674	−0.613370	+	0.789795i	$0.710187\pi$
$468$	0	0
$469$	9.53371e14	1.94004
$470$	0	0
$471$	2.71264e13	0.0539235
$472$	0	0
$473$	1.01357e15	1.96842
$474$	0	0
$475$	2.67435e14	0.507462
$476$	0	0
$477$	−5.76331e14	−1.06861
$478$	0	0
$479$	−1.73690e14	−0.314723	−0.157362	−	0.987541i	$-0.550299\pi$
$479$	−1.73690e14	−0.314723	−0.157362	+	0.987541i	$0.550299\pi$
$480$	0	0
$481$	−3.54167e14	−0.627206
$482$	0	0
$483$	−6.40093e13	−0.110798
$484$	0	0
$485$	2.43782e14	0.412499
$486$	0	0
$487$	8.42617e14	1.39387	0.696933	−	0.717136i	$-0.254547\pi$
$487$	8.42617e14	1.39387	0.696933	+	0.717136i	$0.254547\pi$
$488$	0	0
$489$	−3.00350e13	−0.0485768
$490$	0	0
$491$	−5.38408e14	−0.851458	−0.425729	−	0.904851i	$-0.639982\pi$
$491$	−5.38408e14	−0.851458	−0.425729	+	0.904851i	$0.639982\pi$
$492$	0	0
$493$	−1.37970e14	−0.213367
$494$	0	0
$495$	−3.66393e14	−0.554139
$496$	0	0
$497$	−1.45147e15	−2.14707
$498$	0	0
$499$	−1.36234e14	−0.197121	−0.0985603	−	0.995131i	$-0.531424\pi$
$499$	−1.36234e14	−0.197121	−0.0985603	+	0.995131i	$0.531424\pi$
$500$	0	0
$501$	3.65449e13	0.0517272
$502$	0	0
$503$	4.88252e14	0.676115	0.338057	−	0.941125i	$-0.390230\pi$
$503$	4.88252e14	0.676115	0.338057	+	0.941125i	$0.390230\pi$
$504$	0	0
$505$	−2.60073e14	−0.352365
$506$	0	0
$507$	−3.43369e12	−0.00455216
$508$	0	0
$509$	−4.88014e14	−0.633117	−0.316559	−	0.948573i	$-0.602527\pi$
$509$	−4.88014e14	−0.633117	−0.316559	+	0.948573i	$0.602527\pi$
$510$	0	0
$511$	3.90412e14	0.495688
$512$	0	0
$513$	−9.27348e13	−0.115238
$514$	0	0
$515$	−2.74538e14	−0.333935
$516$	0	0
$517$	−3.14925e14	−0.374981
$518$	0	0
$519$	−1.04219e14	−0.121486
$520$	0	0
$521$	1.10790e15	1.26442	0.632211	−	0.774796i	$-0.282148\pi$
$521$	1.10790e15	1.26442	0.632211	+	0.774796i	$0.282148\pi$
$522$	0	0
$523$	−6.18140e14	−0.690761	−0.345380	−	0.938463i	$-0.612250\pi$
$523$	−6.18140e14	−0.690761	−0.345380	+	0.938463i	$0.612250\pi$
$524$	0	0
$525$	−7.31753e13	−0.0800734
$526$	0	0
$527$	2.36445e15	2.53380
$528$	0	0
$529$	7.48712e13	0.0785794
$530$	0	0
$531$	5.27889e14	0.542653
$532$	0	0
$533$	−3.04230e14	−0.306339
$534$	0	0
$535$	−2.75183e14	−0.271441
$536$	0	0
$537$	1.47335e14	0.142379
$538$	0	0
$539$	−6.56065e14	−0.621167
$540$	0	0
$541$	7.61295e14	0.706265	0.353133	−	0.935573i	$-0.385116\pi$
$541$	7.61295e14	0.706265	0.353133	+	0.935573i	$0.385116\pi$
$542$	0	0
$543$	−1.83101e13	−0.0166453
$544$	0	0
$545$	2.47329e14	0.220341
$546$	0	0
$547$	−5.65709e14	−0.493927	−0.246964	−	0.969025i	$-0.579433\pi$
$547$	−5.65709e14	−0.493927	−0.246964	+	0.969025i	$0.579433\pi$
$548$	0	0
$549$	2.04515e15	1.75016
$550$	0	0
$551$	9.92840e13	0.0832809
$552$	0	0
$553$	2.32944e14	0.191542
$554$	0	0
$555$	−3.23883e13	−0.0261082
$556$	0	0
$557$	−1.88089e15	−1.48649	−0.743243	−	0.669021i	$-0.766713\pi$
$557$	−1.88089e15	−1.48649	−0.743243	+	0.669021i	$0.766713\pi$
$558$	0	0
$559$	−2.33251e15	−1.80742
$560$	0	0
$561$	2.17949e14	0.165599
$562$	0	0
$563$	−1.76231e14	−0.131306	−0.0656531	−	0.997843i	$-0.520913\pi$
$563$	−1.76231e14	−0.131306	−0.0656531	+	0.997843i	$0.520913\pi$
$564$	0	0
$565$	−6.86888e13	−0.0501903
$566$	0	0
$567$	−1.70241e15	−1.21999
$568$	0	0
$569$	−4.91855e14	−0.345716	−0.172858	−	0.984947i	$-0.555300\pi$
$569$	−4.91855e14	−0.345716	−0.172858	+	0.984947i	$0.555300\pi$
$570$	0	0
$571$	9.18553e14	0.633295	0.316647	−	0.948543i	$-0.397443\pi$
$571$	9.18553e14	0.633295	0.316647	+	0.948543i	$0.397443\pi$
$572$	0	0
$573$	−1.07982e14	−0.0730303
$574$	0	0
$575$	1.17484e15	0.779483
$576$	0	0
$577$	−1.53842e15	−1.00140	−0.500701	−	0.865620i	$-0.666924\pi$
$577$	−1.53842e15	−1.00140	−0.500701	+	0.865620i	$0.666924\pi$
$578$	0	0
$579$	7.85493e13	0.0501661
$580$	0	0
$581$	2.20414e15	1.38124
$582$	0	0
$583$	−1.95661e15	−1.20317
$584$	0	0
$585$	8.43176e14	0.508816
$586$	0	0
$587$	5.66027e12	0.00335218	0.00167609	−	0.999999i	$-0.499466\pi$
$587$	5.66027e12	0.00335218	0.00167609	+	0.999999i	$0.499466\pi$
$588$	0	0
$589$	−1.70147e15	−0.988984
$590$	0	0
$591$	1.71891e14	0.0980668
$592$	0	0
$593$	−4.11514e14	−0.230454	−0.115227	−	0.993339i	$-0.536760\pi$
$593$	−4.11514e14	−0.230454	−0.115227	+	0.993339i	$0.536760\pi$
$594$	0	0
$595$	1.96296e15	1.07911
$596$	0	0
$597$	−1.06276e13	−0.00573555
$598$	0	0
$599$	3.35375e15	1.77698	0.888492	−	0.458893i	$-0.151754\pi$
$599$	3.35375e15	1.77698	0.888492	+	0.458893i	$0.151754\pi$
$600$	0	0
$601$	1.02502e15	0.533240	0.266620	−	0.963802i	$-0.414093\pi$
$601$	1.02502e15	0.533240	0.266620	+	0.963802i	$0.414093\pi$
$602$	0	0
$603$	3.02270e15	1.54401
$604$	0	0
$605$	−2.48146e14	−0.124467
$606$	0	0
$607$	−1.99079e15	−0.980591	−0.490296	−	0.871556i	$-0.663111\pi$
$607$	−1.99079e15	−0.980591	−0.490296	+	0.871556i	$0.663111\pi$
$608$	0	0
$609$	−2.71660e13	−0.0131410
$610$	0	0
$611$	7.24734e14	0.344312
$612$	0	0
$613$	−5.16975e14	−0.241233	−0.120617	−	0.992699i	$-0.538487\pi$
$613$	−5.16975e14	−0.241233	−0.120617	+	0.992699i	$0.538487\pi$
$614$	0	0
$615$	−2.78215e13	−0.0127517
$616$	0	0
$617$	−6.78133e14	−0.305314	−0.152657	−	0.988279i	$-0.548783\pi$
$617$	−6.78133e14	−0.305314	−0.152657	+	0.988279i	$0.548783\pi$
$618$	0	0
$619$	−6.55793e14	−0.290047	−0.145023	−	0.989428i	$-0.546326\pi$
$619$	−6.55793e14	−0.290047	−0.145023	+	0.989428i	$0.546326\pi$
$620$	0	0
$621$	−4.07384e14	−0.177011
$622$	0	0
$623$	−5.85996e14	−0.250156
$624$	0	0
$625$	7.48346e14	0.313879
$626$	0	0
$627$	−1.56837e14	−0.0646363
$628$	0	0
$629$	−2.61417e15	−1.05866
$630$	0	0
$631$	3.36388e14	0.133869	0.0669343	−	0.997757i	$-0.478678\pi$
$631$	3.36388e14	0.133869	0.0669343	+	0.997757i	$0.478678\pi$
$632$	0	0
$633$	−1.71679e14	−0.0671423
$634$	0	0
$635$	6.63978e14	0.255210
$636$	0	0
$637$	1.50979e15	0.570362
$638$	0	0
$639$	−4.60194e15	−1.70878
$640$	0	0
$641$	−2.68016e15	−0.978233	−0.489116	−	0.872218i	$-0.662681\pi$
$641$	−2.68016e15	−0.978233	−0.489116	+	0.872218i	$0.662681\pi$
$642$	0	0
$643$	1.63908e14	0.0588085	0.0294043	−	0.999568i	$-0.490639\pi$
$643$	1.63908e14	0.0588085	0.0294043	+	0.999568i	$0.490639\pi$
$644$	0	0
$645$	−2.13306e14	−0.0752360
$646$	0	0
$647$	−3.11319e15	−1.07952	−0.539762	−	0.841818i	$-0.681486\pi$
$647$	−3.11319e15	−1.07952	−0.539762	+	0.841818i	$0.681486\pi$
$648$	0	0
$649$	1.79215e15	0.610983
$650$	0	0
$651$	4.65553e14	0.156054
$652$	0	0
$653$	3.15674e15	1.04044	0.520218	−	0.854033i	$-0.325851\pi$
$653$	3.15674e15	1.04044	0.520218	+	0.854033i	$0.325851\pi$
$654$	0	0
$655$	−3.99691e14	−0.129538
$656$	0	0
$657$	1.23782e15	0.394501
$658$	0	0
$659$	−2.21997e15	−0.695787	−0.347894	−	0.937534i	$-0.613103\pi$
$659$	−2.21997e15	−0.695787	−0.347894	+	0.937534i	$0.613103\pi$
$660$	0	0
$661$	2.94533e15	0.907873	0.453937	−	0.891034i	$-0.350019\pi$
$661$	2.94533e15	0.907873	0.453937	+	0.891034i	$0.350019\pi$
$662$	0	0
$663$	−5.01562e14	−0.152055
$664$	0	0
$665$	−1.41255e15	−0.421197
$666$	0	0
$667$	4.36155e14	0.127923
$668$	0	0
$669$	3.31724e14	0.0957046
$670$	0	0
$671$	6.94317e15	1.97053
$672$	0	0
$673$	2.03972e15	0.569492	0.284746	−	0.958603i	$-0.408091\pi$
$673$	2.03972e15	0.569492	0.284746	+	0.958603i	$0.408091\pi$
$674$	0	0
$675$	−4.65720e14	−0.127925
$676$	0	0
$677$	1.47929e14	0.0399775	0.0199887	−	0.999800i	$-0.493637\pi$
$677$	1.47929e14	0.0399775	0.0199887	+	0.999800i	$0.493637\pi$
$678$	0	0
$679$	3.87425e15	1.03016
$680$	0	0
$681$	2.10760e14	0.0551415
$682$	0	0
$683$	−4.94016e15	−1.27183	−0.635913	−	0.771761i	$-0.719376\pi$
$683$	−4.94016e15	−1.27183	−0.635913	+	0.771761i	$0.719376\pi$
$684$	0	0
$685$	−4.08683e14	−0.103535
$686$	0	0
$687$	5.04984e14	0.125897
$688$	0	0
$689$	4.50272e15	1.10476
$690$	0	0
$691$	2.96403e15	0.715737	0.357868	−	0.933772i	$-0.383504\pi$
$691$	2.96403e15	0.715737	0.357868	+	0.933772i	$0.383504\pi$
$692$	0	0
$693$	−5.82282e15	−1.38389
$694$	0	0
$695$	−3.45357e15	−0.807888
$696$	0	0
$697$	−2.24558e15	−0.517068
$698$	0	0
$699$	1.90660e14	0.0432150
$700$	0	0
$701$	8.45018e14	0.188546	0.0942729	−	0.995546i	$-0.469947\pi$
$701$	8.45018e14	0.188546	0.0942729	+	0.995546i	$0.469947\pi$
$702$	0	0
$703$	1.88117e15	0.413212
$704$	0	0
$705$	6.62763e13	0.0143324
$706$	0	0
$707$	−4.13314e15	−0.879982
$708$	0	0
$709$	−6.16194e15	−1.29171	−0.645853	−	0.763462i	$-0.723498\pi$
$709$	−6.16194e15	−1.29171	−0.645853	+	0.763462i	$0.723498\pi$
$710$	0	0
$711$	7.38558e14	0.152441
$712$	0	0
$713$	−7.47454e15	−1.51912
$714$	0	0
$715$	2.86254e15	0.572885
$716$	0	0
$717$	4.73927e14	0.0934019
$718$	0	0
$719$	7.67467e15	1.48954	0.744768	−	0.667324i	$-0.232560\pi$
$719$	7.67467e15	1.48954	0.744768	+	0.667324i	$0.232560\pi$
$720$	0	0
$721$	−4.36303e15	−0.833958
$722$	0	0
$723$	−8.13228e14	−0.153092
$724$	0	0
$725$	4.98611e14	0.0924491
$726$	0	0
$727$	2.55382e15	0.466391	0.233196	−	0.972430i	$-0.425082\pi$
$727$	2.55382e15	0.466391	0.233196	+	0.972430i	$0.425082\pi$
$728$	0	0
$729$	−5.31653e15	−0.956371
$730$	0	0
$731$	−1.72167e16	−3.05074
$732$	0	0
$733$	1.06639e16	1.86142	0.930709	−	0.365760i	$-0.119191\pi$
$733$	1.06639e16	1.86142	0.930709	+	0.365760i	$0.119191\pi$
$734$	0	0
$735$	1.38069e14	0.0237420
$736$	0	0
$737$	1.02619e16	1.73843
$738$	0	0
$739$	4.93967e15	0.824430	0.412215	−	0.911087i	$-0.364755\pi$
$739$	4.93967e15	0.824430	0.412215	+	0.911087i	$0.364755\pi$
$740$	0	0
$741$	3.60926e14	0.0593497
$742$	0	0
$743$	1.04838e16	1.69855	0.849277	−	0.527948i	$-0.177039\pi$
$743$	1.04838e16	1.69855	0.849277	+	0.527948i	$0.177039\pi$
$744$	0	0
$745$	2.78355e15	0.444365
$746$	0	0
$747$	6.98831e15	1.09928
$748$	0	0
$749$	−4.37328e15	−0.677887
$750$	0	0
$751$	−9.78697e15	−1.49496	−0.747478	−	0.664287i	$-0.768735\pi$
$751$	−9.78697e15	−1.49496	−0.747478	+	0.664287i	$0.768735\pi$
$752$	0	0
$753$	−9.40268e14	−0.141540
$754$	0	0
$755$	−4.12947e15	−0.612613
$756$	0	0
$757$	−4.93317e14	−0.0721271	−0.0360636	−	0.999349i	$-0.511482\pi$
$757$	−4.93317e14	−0.0721271	−0.0360636	+	0.999349i	$0.511482\pi$
$758$	0	0
$759$	−6.88984e14	−0.0992840
$760$	0	0
$761$	8.97168e15	1.27426	0.637130	−	0.770756i	$-0.280121\pi$
$761$	8.97168e15	1.27426	0.637130	+	0.770756i	$0.280121\pi$
$762$	0	0
$763$	3.93062e15	0.550271
$764$	0	0
$765$	6.22364e15	0.858829
$766$	0	0
$767$	−4.12426e15	−0.561011
$768$	0	0
$769$	3.72650e15	0.499697	0.249849	−	0.968285i	$-0.419619\pi$
$769$	3.72650e15	0.499697	0.249849	+	0.968285i	$0.419619\pi$
$770$	0	0
$771$	4.55687e14	0.0602376
$772$	0	0
$773$	1.12337e15	0.146399	0.0731993	−	0.997317i	$-0.476679\pi$
$773$	1.12337e15	0.146399	0.0731993	+	0.997317i	$0.476679\pi$
$774$	0	0
$775$	−8.54488e15	−1.09786
$776$	0	0
$777$	−5.14723e14	−0.0652015
$778$	0	0
$779$	1.61593e15	0.201820
$780$	0	0
$781$	−1.56233e16	−1.92394
$782$	0	0
$783$	−1.72896e14	−0.0209941
$784$	0	0
$785$	2.62976e15	0.314871
$786$	0	0
$787$	−1.39683e16	−1.64923	−0.824617	−	0.565692i	$-0.808609\pi$
$787$	−1.39683e16	−1.64923	−0.824617	+	0.565692i	$0.808609\pi$
$788$	0	0
$789$	−1.93328e14	−0.0225098
$790$	0	0
$791$	−1.09162e15	−0.125343
$792$	0	0
$793$	−1.59782e16	−1.80937
$794$	0	0
$795$	4.11770e14	0.0459870
$796$	0	0
$797$	1.77226e16	1.95213	0.976063	−	0.217490i	$-0.0697869\pi$
$797$	1.77226e16	1.95213	0.976063	+	0.217490i	$0.0697869\pi$
$798$	0	0
$799$	5.34939e15	0.581162
$800$	0	0
$801$	−1.85792e15	−0.199090
$802$	0	0
$803$	4.20232e15	0.444175
$804$	0	0
$805$	−6.20534e15	−0.646976
$806$	0	0
$807$	3.09650e14	0.0318469
$808$	0	0
$809$	−1.35271e16	−1.37242	−0.686211	−	0.727402i	$-0.740727\pi$
$809$	−1.35271e16	−1.37242	−0.686211	+	0.727402i	$0.740727\pi$
$810$	0	0
$811$	1.11634e16	1.11733	0.558666	−	0.829393i	$-0.311313\pi$
$811$	1.11634e16	1.11733	0.558666	+	0.829393i	$0.311313\pi$
$812$	0	0
$813$	2.16494e14	0.0213770
$814$	0	0
$815$	−2.91173e15	−0.283651
$816$	0	0
$817$	1.23892e16	1.19076
$818$	0	0
$819$	1.34000e16	1.27070
$820$	0	0
$821$	−1.29639e16	−1.21297	−0.606483	−	0.795097i	$-0.707420\pi$
$821$	−1.29639e16	−1.21297	−0.606483	+	0.795097i	$0.707420\pi$
$822$	0	0
$823$	−3.20238e15	−0.295647	−0.147824	−	0.989014i	$-0.547227\pi$
$823$	−3.20238e15	−0.295647	−0.147824	+	0.989014i	$0.547227\pi$
$824$	0	0
$825$	−7.87645e14	−0.0717519
$826$	0	0
$827$	9.92879e15	0.892516	0.446258	−	0.894904i	$-0.352756\pi$
$827$	9.92879e15	0.892516	0.446258	+	0.894904i	$0.352756\pi$
$828$	0	0
$829$	1.17935e16	1.04615	0.523073	−	0.852288i	$-0.324785\pi$
$829$	1.17935e16	1.04615	0.523073	+	0.852288i	$0.324785\pi$
$830$	0	0
$831$	9.47923e14	0.0829789
$832$	0	0
$833$	1.11441e16	0.962712
$834$	0	0
$835$	3.54282e15	0.302047
$836$	0	0
$837$	2.96299e15	0.249310
$838$	0	0
$839$	−2.90602e14	−0.0241328	−0.0120664	−	0.999927i	$-0.503841\pi$
$839$	−2.90602e14	−0.0241328	−0.0120664	+	0.999927i	$0.503841\pi$
$840$	0	0
$841$	−1.20154e16	−0.984828
$842$	0	0
$843$	1.84229e15	0.149042
$844$	0	0
$845$	−3.32877e14	−0.0265811
$846$	0	0
$847$	−3.94361e15	−0.310839
$848$	0	0
$849$	−3.22162e14	−0.0250659
$850$	0	0
$851$	8.26397e15	0.634711
$852$	0	0
$853$	−1.57981e16	−1.19780	−0.598902	−	0.800823i	$-0.704396\pi$
$853$	−1.57981e16	−1.19780	−0.598902	+	0.800823i	$0.704396\pi$
$854$	0	0
$855$	−4.47856e15	−0.335216
$856$	0	0
$857$	1.64826e16	1.21796	0.608978	−	0.793187i	$-0.291580\pi$
$857$	1.64826e16	1.21796	0.608978	+	0.793187i	$0.291580\pi$
$858$	0	0
$859$	−6.21838e13	−0.00453644	−0.00226822	−	0.999997i	$-0.500722\pi$
$859$	−6.21838e13	−0.00453644	−0.00226822	+	0.999997i	$0.500722\pi$
$860$	0	0
$861$	−4.42147e14	−0.0318456
$862$	0	0
$863$	−1.77134e16	−1.25963	−0.629815	−	0.776745i	$-0.716869\pi$
$863$	−1.77134e16	−1.25963	−0.629815	+	0.776745i	$0.716869\pi$
$864$	0	0
$865$	−1.01034e16	−0.709382
$866$	0	0
$867$	−2.46834e15	−0.171120
$868$	0	0
$869$	2.50736e15	0.171636
$870$	0	0
$871$	−2.36156e16	−1.59624
$872$	0	0
$873$	1.22835e16	0.819867
$874$	0	0
$875$	−1.65456e16	−1.09053
$876$	0	0
$877$	8.21556e15	0.534736	0.267368	−	0.963595i	$-0.413846\pi$
$877$	8.21556e15	0.534736	0.267368	+	0.963595i	$0.413846\pi$
$878$	0	0
$879$	9.66527e14	0.0621263
$880$	0	0
$881$	−1.91858e16	−1.21790	−0.608950	−	0.793209i	$-0.708409\pi$
$881$	−1.91858e16	−1.21790	−0.608950	+	0.793209i	$0.708409\pi$
$882$	0	0
$883$	1.04412e16	0.654584	0.327292	−	0.944923i	$-0.393864\pi$
$883$	1.04412e16	0.654584	0.327292	+	0.944923i	$0.393864\pi$
$884$	0	0
$885$	−3.77160e14	−0.0233527
$886$	0	0
$887$	9.33912e15	0.571118	0.285559	−	0.958361i	$-0.407821\pi$
$887$	9.33912e15	0.571118	0.285559	+	0.958361i	$0.407821\pi$
$888$	0	0
$889$	1.05521e16	0.637353
$890$	0	0
$891$	−1.83244e16	−1.09321
$892$	0	0
$893$	−3.84944e15	−0.226838
$894$	0	0
$895$	1.42833e16	0.831384
$896$	0	0
$897$	1.58555e15	0.0911636
$898$	0	0
$899$	−3.17225e15	−0.180173
$900$	0	0
$901$	3.32354e16	1.86472
$902$	0	0
$903$	−3.38992e15	−0.187891
$904$	0	0
$905$	−1.77507e15	−0.0971958
$906$	0	0
$907$	−1.35976e16	−0.735565	−0.367782	−	0.929912i	$-0.619883\pi$
$907$	−1.35976e16	−0.735565	−0.367782	+	0.929912i	$0.619883\pi$
$908$	0	0
$909$	−1.31043e16	−0.700346
$910$	0	0
$911$	2.44918e16	1.29321	0.646607	−	0.762823i	$-0.276187\pi$
$911$	2.44918e16	1.29321	0.646607	+	0.762823i	$0.276187\pi$
$912$	0	0
$913$	2.37249e16	1.23770
$914$	0	0
$915$	−1.46120e15	−0.0753169
$916$	0	0
$917$	−6.35200e15	−0.323504
$918$	0	0
$919$	1.94715e16	0.979860	0.489930	−	0.871762i	$-0.337022\pi$
$919$	1.94715e16	0.979860	0.489930	+	0.871762i	$0.337022\pi$
$920$	0	0
$921$	−2.38476e14	−0.0118581
$922$	0	0
$923$	3.59538e16	1.76659
$924$	0	0
$925$	9.44735e15	0.458702
$926$	0	0
$927$	−1.38332e16	−0.663717
$928$	0	0
$929$	−3.72465e16	−1.76604	−0.883018	−	0.469340i	$-0.844492\pi$
$929$	−3.72465e16	−1.76604	−0.883018	+	0.469340i	$0.844492\pi$
$930$	0	0
$931$	−8.01932e15	−0.375763
$932$	0	0
$933$	−1.45027e15	−0.0671582
$934$	0	0
$935$	2.11289e16	0.966970
$936$	0	0
$937$	−3.42521e15	−0.154924	−0.0774622	−	0.996995i	$-0.524682\pi$
$937$	−3.42521e15	−0.154924	−0.0774622	+	0.996995i	$0.524682\pi$
$938$	0	0
$939$	7.75523e14	0.0346684
$940$	0	0
$941$	−2.69228e16	−1.18954	−0.594768	−	0.803898i	$-0.702756\pi$
$941$	−2.69228e16	−1.18954	−0.594768	+	0.803898i	$0.702756\pi$
$942$	0	0
$943$	7.09876e15	0.310005
$944$	0	0
$945$	2.45987e15	0.106178
$946$	0	0
$947$	−2.87322e16	−1.22587	−0.612935	−	0.790133i	$-0.710011\pi$
$947$	−2.87322e16	−1.22587	−0.612935	+	0.790133i	$0.710011\pi$
$948$	0	0
$949$	−9.67076e15	−0.407846
$950$	0	0
$951$	1.00612e14	0.00419429
$952$	0	0
$953$	−1.77454e16	−0.731265	−0.365633	−	0.930759i	$-0.619147\pi$
$953$	−1.77454e16	−0.731265	−0.365633	+	0.930759i	$0.619147\pi$
$954$	0	0
$955$	−1.04683e16	−0.426440
$956$	0	0
$957$	−2.92409e14	−0.0117754
$958$	0	0
$959$	−6.49490e15	−0.258565
$960$	0	0
$961$	2.89555e16	1.13960
$962$	0	0
$963$	−1.38657e16	−0.539506
$964$	0	0
$965$	7.61492e15	0.292931
$966$	0	0
$967$	2.45602e16	0.934086	0.467043	−	0.884235i	$-0.345319\pi$
$967$	2.45602e16	0.934086	0.467043	+	0.884235i	$0.345319\pi$
$968$	0	0
$969$	2.66406e15	0.100176
$970$	0	0
$971$	−4.69158e15	−0.174427	−0.0872135	−	0.996190i	$-0.527796\pi$
$971$	−4.69158e15	−0.174427	−0.0872135	+	0.996190i	$0.527796\pi$
$972$	0	0
$973$	−5.48850e16	−2.01759
$974$	0	0
$975$	1.81260e15	0.0658834
$976$	0	0
$977$	−2.77872e16	−0.998676	−0.499338	−	0.866407i	$-0.666423\pi$
$977$	−2.77872e16	−0.998676	−0.499338	+	0.866407i	$0.666423\pi$
$978$	0	0
$979$	−6.30755e15	−0.224159
$980$	0	0
$981$	1.24622e16	0.437941
$982$	0	0
$983$	−1.09519e16	−0.380581	−0.190290	−	0.981728i	$-0.560943\pi$
$983$	−1.09519e16	−0.380581	−0.190290	+	0.981728i	$0.560943\pi$
$984$	0	0
$985$	1.66639e16	0.572634
$986$	0	0
$987$	1.05328e15	0.0357931
$988$	0	0
$989$	5.44258e16	1.82905
$990$	0	0
$991$	3.36280e16	1.11762	0.558811	−	0.829295i	$-0.311258\pi$
$991$	3.36280e16	1.11762	0.558811	+	0.829295i	$0.311258\pi$
$992$	0	0
$993$	1.61482e15	0.0530765
$994$	0	0
$995$	−1.03028e15	−0.0334911
$996$	0	0
$997$	5.27110e16	1.69464	0.847321	−	0.531082i	$-0.178214\pi$
$997$	5.27110e16	1.69464	0.847321	+	0.531082i	$0.178214\pi$
$998$	0	0
$999$	−3.27593e15	−0.104166

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8.12.a.a.1.1	✓	1
3.2	odd	2		72.12.a.c.1.1		1
4.3	odd	2		16.12.a.b.1.1		1
5.2	odd	4		200.12.c.b.49.2		2
5.3	odd	4		200.12.c.b.49.1		2
5.4	even	2		200.12.a.b.1.1		1
8.3	odd	2		64.12.a.c.1.1		1
8.5	even	2		64.12.a.e.1.1		1
12.11	even	2		144.12.a.j.1.1		1
16.3	odd	4		256.12.b.a.129.2		2
16.5	even	4		256.12.b.g.129.2		2
16.11	odd	4		256.12.b.a.129.1		2
16.13	even	4		256.12.b.g.129.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8.12.a.a.1.1	✓	1	1.1	even	1	trivial
16.12.a.b.1.1		1	4.3	odd	2
64.12.a.c.1.1		1	8.3	odd	2
64.12.a.e.1.1		1	8.5	even	2
72.12.a.c.1.1		1	3.2	odd	2
144.12.a.j.1.1		1	12.11	even	2
200.12.a.b.1.1		1	5.4	even	2
200.12.c.b.49.1		2	5.3	odd	4
200.12.c.b.49.2		2	5.2	odd	4
256.12.b.a.129.1		2	16.11	odd	4
256.12.b.a.129.2		2	16.3	odd	4
256.12.b.g.129.1		2	16.13	even	4
256.12.b.g.129.2		2	16.5	even	4

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:	\( N \)	\(=\)	\( 8 = 2^{3} \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	8.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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