LMFDB - Embedded newform 8.16.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8,16,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, 16, names="a")

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

chi := DirichletCharacter("8.1");

S:= CuspForms(chi, 16);

N := Newforms(S);

Level:	$ N $	$=$	$ 8 = 2^{3} $
Weight:	$ k $	$=$	$ 16 $
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:	$11.4154804080$
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)

$f(q)$

$=$

$q-3444.00 q^{3} +313358. q^{5} -2.32462e6 q^{7} -2.48777e6 q^{9} +O(q^{10})$

$q-3444.00 q^{3} +313358. q^{5} -2.32462e6 q^{7} -2.48777e6 q^{9} -5.52491e7 q^{11} -1.10260e8 q^{13} -1.07920e9 q^{15} -2.60143e9 q^{17} +1.95212e9 q^{19} +8.00598e9 q^{21} -2.54303e10 q^{23} +6.76757e10 q^{25} +5.79855e10 q^{27} -2.27722e9 q^{29} -1.90667e11 q^{31} +1.90278e11 q^{33} -7.28437e11 q^{35} -2.88229e11 q^{37} +3.79734e11 q^{39} +7.56412e11 q^{41} -3.54187e11 q^{43} -7.79563e11 q^{45} +6.03592e12 q^{47} +6.56278e11 q^{49} +8.95932e12 q^{51} -1.21989e13 q^{53} -1.73127e13 q^{55} -6.72312e12 q^{57} -4.09091e12 q^{59} +1.75659e13 q^{61} +5.78311e12 q^{63} -3.45507e13 q^{65} -3.93125e12 q^{67} +8.75821e13 q^{69} +5.88254e13 q^{71} +1.07572e14 q^{73} -2.33075e14 q^{75} +1.28433e14 q^{77} +6.15439e13 q^{79} -1.64005e14 q^{81} +1.34321e13 q^{83} -8.15179e14 q^{85} +7.84276e12 q^{87} +2.69696e14 q^{89} +2.56311e14 q^{91} +6.56658e14 q^{93} +6.11714e14 q^{95} -7.93797e14 q^{97} +1.37447e14 q^{99} +O(q^{100})$

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
$3$	−3444.00	−0.909188	−0.454594	−	0.890699i	$-0.650216\pi$
$3$	−3444.00	−0.909188	−0.454594	+	0.890699i	$0.650216\pi$
$4$	0	0
$5$	313358.	1.79377	0.896883	−	0.442268i	$-0.145826\pi$
$5$	313358.	1.79377	0.896883	+	0.442268i	$0.145826\pi$
$6$	0	0
$7$	−2.32462e6	−1.06688	−0.533440	−	0.845838i	$-0.679101\pi$
$7$	−2.32462e6	−1.06688	−0.533440	+	0.845838i	$0.679101\pi$
$8$	0	0
$9$	−2.48777e6	−0.173377
$10$	0	0
$11$	−5.52491e7	−0.854830	−0.427415	−	0.904055i	$-0.640576\pi$
$11$	−5.52491e7	−0.854830	−0.427415	+	0.904055i	$0.640576\pi$
$12$	0	0
$13$	−1.10260e8	−0.487350	−0.243675	−	0.969857i	$-0.578353\pi$
$13$	−1.10260e8	−0.487350	−0.243675	+	0.969857i	$0.578353\pi$
$14$	0	0
$15$	−1.07920e9	−1.63087
$16$	0	0
$17$	−2.60143e9	−1.53761	−0.768803	−	0.639486i	$-0.779147\pi$
$17$	−2.60143e9	−1.53761	−0.768803	+	0.639486i	$0.779147\pi$
$18$	0	0
$19$	1.95212e9	0.501020	0.250510	−	0.968114i	$-0.419402\pi$
$19$	1.95212e9	0.501020	0.250510	+	0.968114i	$0.419402\pi$
$20$	0	0
$21$	8.00598e9	0.969995
$22$	0	0
$23$	−2.54303e10	−1.55738	−0.778688	−	0.627412i	$-0.784114\pi$
$23$	−2.54303e10	−1.55738	−0.778688	+	0.627412i	$0.784114\pi$
$24$	0	0
$25$	6.76757e10	2.21760
$26$	0	0
$27$	5.79855e10	1.06682
$28$	0	0
$29$	−2.27722e9	−0.0245144	−0.0122572	−	0.999925i	$-0.503902\pi$
$29$	−2.27722e9	−0.0245144	−0.0122572	+	0.999925i	$0.503902\pi$
$30$	0	0
$31$	−1.90667e11	−1.24470	−0.622348	−	0.782741i	$-0.713821\pi$
$31$	−1.90667e11	−1.24470	−0.622348	+	0.782741i	$0.713821\pi$
$32$	0	0
$33$	1.90278e11	0.777202
$34$	0	0
$35$	−7.28437e11	−1.91373
$36$	0	0
$37$	−2.88229e11	−0.499144	−0.249572	−	0.968356i	$-0.580290\pi$
$37$	−2.88229e11	−0.499144	−0.249572	+	0.968356i	$0.580290\pi$
$38$	0	0
$39$	3.79734e11	0.443093
$40$	0	0
$41$	7.56412e11	0.606568	0.303284	−	0.952900i	$-0.401917\pi$
$41$	7.56412e11	0.606568	0.303284	+	0.952900i	$0.401917\pi$
$42$	0	0
$43$	−3.54187e11	−0.198710	−0.0993548	−	0.995052i	$-0.531678\pi$
$43$	−3.54187e11	−0.198710	−0.0993548	+	0.995052i	$0.531678\pi$
$44$	0	0
$45$	−7.79563e11	−0.310998
$46$	0	0
$47$	6.03592e12	1.73784	0.868920	−	0.494953i	$-0.164815\pi$
$47$	6.03592e12	1.73784	0.868920	+	0.494953i	$0.164815\pi$
$48$	0	0
$49$	6.56278e11	0.138235
$50$	0	0
$51$	8.95932e12	1.39797
$52$	0	0
$53$	−1.21989e13	−1.42644	−0.713218	−	0.700942i	$-0.752763\pi$
$53$	−1.21989e13	−1.42644	−0.713218	+	0.700942i	$0.752763\pi$
$54$	0	0
$55$	−1.73127e13	−1.53337
$56$	0	0
$57$	−6.72312e12	−0.455522
$58$	0	0
$59$	−4.09091e12	−0.214008	−0.107004	−	0.994259i	$-0.534126\pi$
$59$	−4.09091e12	−0.214008	−0.107004	+	0.994259i	$0.534126\pi$
$60$	0	0
$61$	1.75659e13	0.715644	0.357822	−	0.933790i	$-0.383520\pi$
$61$	1.75659e13	0.715644	0.357822	+	0.933790i	$0.383520\pi$
$62$	0	0
$63$	5.78311e12	0.184973
$64$	0	0
$65$	−3.45507e13	−0.874192
$66$	0	0
$67$	−3.93125e12	−0.0792445	−0.0396223	−	0.999215i	$-0.512615\pi$
$67$	−3.93125e12	−0.0792445	−0.0396223	+	0.999215i	$0.512615\pi$
$68$	0	0
$69$	8.75821e13	1.41595
$70$	0	0
$71$	5.88254e13	0.767587	0.383793	−	0.923419i	$-0.374617\pi$
$71$	5.88254e13	0.767587	0.383793	+	0.923419i	$0.374617\pi$
$72$	0	0
$73$	1.07572e14	1.13966	0.569831	−	0.821762i	$-0.307009\pi$
$73$	1.07572e14	1.13966	0.569831	+	0.821762i	$0.307009\pi$
$74$	0	0
$75$	−2.33075e14	−2.01621
$76$	0	0
$77$	1.28433e14	0.912002
$78$	0	0
$79$	6.15439e13	0.360563	0.180282	−	0.983615i	$-0.442299\pi$
$79$	6.15439e13	0.360563	0.180282	+	0.983615i	$0.442299\pi$
$80$	0	0
$81$	−1.64005e14	−0.796563
$82$	0	0
$83$	1.34321e13	0.0543322	0.0271661	−	0.999631i	$-0.491352\pi$
$83$	1.34321e13	0.0543322	0.0271661	+	0.999631i	$0.491352\pi$
$84$	0	0
$85$	−8.15179e14	−2.75810
$86$	0	0
$87$	7.84276e12	0.0222882
$88$	0	0
$89$	2.69696e14	0.646324	0.323162	−	0.946344i	$-0.395254\pi$
$89$	2.69696e14	0.646324	0.323162	+	0.946344i	$0.395254\pi$
$90$	0	0
$91$	2.56311e14	0.519945
$92$	0	0
$93$	6.56658e14	1.13166
$94$	0	0
$95$	6.11714e14	0.898713
$96$	0	0
$97$	−7.93797e14	−0.997519	−0.498760	−	0.866740i	$-0.666211\pi$
$97$	−7.93797e14	−0.997519	−0.498760	+	0.866740i	$0.666211\pi$
$98$	0	0
$99$	1.37447e14	0.148208
$100$	0	0
$101$	−5.00218e14	−0.464247	−0.232124	−	0.972686i	$-0.574567\pi$
$101$	−5.00218e14	−0.464247	−0.232124	+	0.972686i	$0.574567\pi$
$102$	0	0
$103$	1.26374e15	1.01246	0.506230	−	0.862398i	$-0.331039\pi$
$103$	1.26374e15	1.01246	0.506230	+	0.862398i	$0.331039\pi$
$104$	0	0
$105$	2.50874e15	1.73994
$106$	0	0
$107$	−2.55213e15	−1.53647	−0.768236	−	0.640167i	$-0.778865\pi$
$107$	−2.55213e15	−1.53647	−0.768236	+	0.640167i	$0.778865\pi$
$108$	0	0
$109$	−5.94488e14	−0.311490	−0.155745	−	0.987797i	$-0.549778\pi$
$109$	−5.94488e14	−0.311490	−0.155745	+	0.987797i	$0.549778\pi$
$110$	0	0
$111$	9.92662e14	0.453815
$112$	0	0
$113$	−1.95194e15	−0.780509	−0.390254	−	0.920707i	$-0.627613\pi$
$113$	−1.95194e15	−0.780509	−0.390254	+	0.920707i	$0.627613\pi$
$114$	0	0
$115$	−7.96880e15	−2.79357
$116$	0	0
$117$	2.74301e14	0.0844953
$118$	0	0
$119$	6.04732e15	1.64044
$120$	0	0
$121$	−1.12479e15	−0.269265
$122$	0	0
$123$	−2.60508e15	−0.551485
$124$	0	0
$125$	1.16438e16	2.18408
$126$	0	0
$127$	−5.47351e15	−0.911460	−0.455730	−	0.890118i	$-0.650622\pi$
$127$	−5.47351e15	−0.911460	−0.455730	+	0.890118i	$0.650622\pi$
$128$	0	0
$129$	1.21982e15	0.180664
$130$	0	0
$131$	9.07856e15	1.19807	0.599035	−	0.800723i	$-0.295551\pi$
$131$	9.07856e15	1.19807	0.599035	+	0.800723i	$0.295551\pi$
$132$	0	0
$133$	−4.53794e15	−0.534529
$134$	0	0
$135$	1.81702e16	1.91363
$136$	0	0
$137$	−8.11213e15	−0.765122	−0.382561	−	0.923930i	$-0.624958\pi$
$137$	−8.11213e15	−0.765122	−0.382561	+	0.923930i	$0.624958\pi$
$138$	0	0
$139$	−7.64263e15	−0.646594	−0.323297	−	0.946298i	$-0.604791\pi$
$139$	−7.64263e15	−0.646594	−0.323297	+	0.946298i	$0.604791\pi$
$140$	0	0
$141$	−2.07877e16	−1.58002
$142$	0	0
$143$	6.09174e15	0.416602
$144$	0	0
$145$	−7.13586e14	−0.0439730
$146$	0	0
$147$	−2.26022e15	−0.125681
$148$	0	0
$149$	9.28584e15	0.466578	0.233289	−	0.972407i	$-0.425051\pi$
$149$	9.28584e15	0.466578	0.233289	+	0.972407i	$0.425051\pi$
$150$	0	0
$151$	−7.69461e15	−0.349832	−0.174916	−	0.984583i	$-0.555965\pi$
$151$	−7.69461e15	−0.349832	−0.174916	+	0.984583i	$0.555965\pi$
$152$	0	0
$153$	6.47176e15	0.266586
$154$	0	0
$155$	−5.97471e16	−2.23269
$156$	0	0
$157$	−4.29390e16	−1.45749	−0.728744	−	0.684786i	$-0.759896\pi$
$157$	−4.29390e16	−1.45749	−0.728744	+	0.684786i	$0.759896\pi$
$158$	0	0
$159$	4.20131e16	1.29690
$160$	0	0
$161$	5.91158e16	1.66153
$162$	0	0
$163$	3.58947e16	0.919651	0.459826	−	0.888009i	$-0.347912\pi$
$163$	3.58947e16	0.919651	0.459826	+	0.888009i	$0.347912\pi$
$164$	0	0
$165$	5.96251e16	1.39412
$166$	0	0
$167$	2.28104e16	0.487259	0.243629	−	0.969868i	$-0.421662\pi$
$167$	2.28104e16	0.487259	0.243629	+	0.969868i	$0.421662\pi$
$168$	0	0
$169$	−3.90287e16	−0.762490
$170$	0	0
$171$	−4.85644e15	−0.0868654
$172$	0	0
$173$	5.52708e16	0.906045	0.453022	−	0.891499i	$-0.350346\pi$
$173$	5.52708e16	0.906045	0.453022	+	0.891499i	$0.350346\pi$
$174$	0	0
$175$	−1.57320e17	−2.36591
$176$	0	0
$177$	1.40891e16	0.194574
$178$	0	0
$179$	3.83039e16	0.486234	0.243117	−	0.969997i	$-0.421830\pi$
$179$	3.83039e16	0.486234	0.243117	+	0.969997i	$0.421830\pi$
$180$	0	0
$181$	1.59603e15	0.0186403	0.00932015	−	0.999957i	$-0.497033\pi$
$181$	1.59603e15	0.0186403	0.00932015	+	0.999957i	$0.497033\pi$
$182$	0	0
$183$	−6.04970e16	−0.650655
$184$	0	0
$185$	−9.03190e16	−0.895347
$186$	0	0
$187$	1.43727e17	1.31439
$188$	0	0
$189$	−1.34794e17	−1.13817
$190$	0	0
$191$	−9.70225e16	−0.757046	−0.378523	−	0.925592i	$-0.623568\pi$
$191$	−9.70225e16	−0.757046	−0.378523	+	0.925592i	$0.623568\pi$
$192$	0	0
$193$	−5.75620e16	−0.415390	−0.207695	−	0.978194i	$-0.566596\pi$
$193$	−5.75620e16	−0.415390	−0.207695	+	0.978194i	$0.566596\pi$
$194$	0	0
$195$	1.18993e17	0.794805
$196$	0	0
$197$	−2.01588e16	−0.124729	−0.0623646	−	0.998053i	$-0.519864\pi$
$197$	−2.01588e16	−0.124729	−0.0623646	+	0.998053i	$0.519864\pi$
$198$	0	0
$199$	1.34502e17	0.771490	0.385745	−	0.922606i	$-0.373945\pi$
$199$	1.34502e17	0.771490	0.385745	+	0.922606i	$0.373945\pi$
$200$	0	0
$201$	1.35392e16	0.0720482
$202$	0	0
$203$	5.29367e15	0.0261539
$204$	0	0
$205$	2.37028e17	1.08804
$206$	0	0
$207$	6.32649e16	0.270013
$208$	0	0
$209$	−1.07853e17	−0.428288
$210$	0	0
$211$	4.14319e17	1.53185	0.765925	−	0.642930i	$-0.222281\pi$
$211$	4.14319e17	1.53185	0.765925	+	0.642930i	$0.222281\pi$
$212$	0	0
$213$	−2.02595e17	−0.697881
$214$	0	0
$215$	−1.10987e17	−0.356438
$216$	0	0
$217$	4.43228e17	1.32794
$218$	0	0
$219$	−3.70476e17	−1.03617
$220$	0	0
$221$	2.86832e17	0.749353
$222$	0	0
$223$	−7.01722e17	−1.71348	−0.856739	−	0.515750i	$-0.827513\pi$
$223$	−7.01722e17	−1.71348	−0.856739	+	0.515750i	$0.827513\pi$
$224$	0	0
$225$	−1.68362e17	−0.384480
$226$	0	0
$227$	−6.58732e17	−1.40772	−0.703858	−	0.710341i	$-0.748541\pi$
$227$	−6.58732e17	−1.40772	−0.703858	+	0.710341i	$0.748541\pi$
$228$	0	0
$229$	−5.96455e17	−1.19347	−0.596735	−	0.802439i	$-0.703536\pi$
$229$	−5.96455e17	−1.19347	−0.596735	+	0.802439i	$0.703536\pi$
$230$	0	0
$231$	−4.42323e17	−0.829181
$232$	0	0
$233$	5.13755e17	0.902789	0.451395	−	0.892324i	$-0.350927\pi$
$233$	5.13755e17	0.902789	0.451395	+	0.892324i	$0.350927\pi$
$234$	0	0
$235$	1.89140e18	3.11728
$236$	0	0
$237$	−2.11957e17	−0.327820
$238$	0	0
$239$	−1.80527e17	−0.262154	−0.131077	−	0.991372i	$-0.541844\pi$
$239$	−1.80527e17	−0.262154	−0.131077	+	0.991372i	$0.541844\pi$
$240$	0	0
$241$	5.46261e16	0.0745199	0.0372600	−	0.999306i	$-0.488137\pi$
$241$	5.46261e16	0.0745199	0.0372600	+	0.999306i	$0.488137\pi$
$242$	0	0
$243$	−2.67194e17	−0.342595
$244$	0	0
$245$	2.05650e17	0.247961
$246$	0	0
$247$	−2.15240e17	−0.244172
$248$	0	0
$249$	−4.62601e16	−0.0493982
$250$	0	0
$251$	−1.15126e18	−1.15777	−0.578884	−	0.815410i	$-0.696512\pi$
$251$	−1.15126e18	−1.15777	−0.578884	+	0.815410i	$0.696512\pi$
$252$	0	0
$253$	1.40500e18	1.33129
$254$	0	0
$255$	2.80747e18	2.50764
$256$	0	0
$257$	−1.29138e17	−0.108782	−0.0543909	−	0.998520i	$-0.517322\pi$
$257$	−1.29138e17	−0.108782	−0.0543909	+	0.998520i	$0.517322\pi$
$258$	0	0
$259$	6.70023e17	0.532527
$260$	0	0
$261$	5.66521e15	0.00425023
$262$	0	0
$263$	1.35782e16	0.00961997	0.00480999	−	0.999988i	$-0.498469\pi$
$263$	1.35782e16	0.00961997	0.00480999	+	0.999988i	$0.498469\pi$
$264$	0	0
$265$	−3.82263e18	−2.55869
$266$	0	0
$267$	−9.28834e17	−0.587630
$268$	0	0
$269$	−9.78092e17	−0.585110	−0.292555	−	0.956249i	$-0.594505\pi$
$269$	−9.78092e17	−0.585110	−0.292555	+	0.956249i	$0.594505\pi$
$270$	0	0
$271$	2.68164e18	1.51751	0.758754	−	0.651377i	$-0.225808\pi$
$271$	2.68164e18	1.51751	0.758754	+	0.651377i	$0.225808\pi$
$272$	0	0
$273$	−8.82736e17	−0.472727
$274$	0	0
$275$	−3.73902e18	−1.89567
$276$	0	0
$277$	−7.47243e17	−0.358809	−0.179405	−	0.983775i	$-0.557417\pi$
$277$	−7.47243e17	−0.358809	−0.179405	+	0.983775i	$0.557417\pi$
$278$	0	0
$279$	4.74336e17	0.215802
$280$	0	0
$281$	4.27286e18	1.84256	0.921279	−	0.388902i	$-0.127145\pi$
$281$	4.27286e18	1.84256	0.921279	+	0.388902i	$0.127145\pi$
$282$	0	0
$283$	−2.36954e18	−0.968870	−0.484435	−	0.874827i	$-0.660975\pi$
$283$	−2.36954e18	−0.968870	−0.484435	+	0.874827i	$0.660975\pi$
$284$	0	0
$285$	−2.10674e18	−0.817099
$286$	0	0
$287$	−1.75837e18	−0.647136
$288$	0	0
$289$	3.90501e18	1.36423
$290$	0	0
$291$	2.73384e18	0.906933
$292$	0	0
$293$	5.73591e18	1.80757	0.903785	−	0.427987i	$-0.140777\pi$
$293$	5.73591e18	1.80757	0.903785	+	0.427987i	$0.140777\pi$
$294$	0	0
$295$	−1.28192e18	−0.383880
$296$	0	0
$297$	−3.20365e18	−0.911950
$298$	0	0
$299$	2.80394e18	0.758987
$300$	0	0
$301$	8.23348e17	0.211999
$302$	0	0
$303$	1.72275e18	0.422088
$304$	0	0
$305$	5.50442e18	1.28370
$306$	0	0
$307$	−7.37962e18	−1.63869	−0.819344	−	0.573302i	$-0.805662\pi$
$307$	−7.37962e18	−1.63869	−0.819344	+	0.573302i	$0.805662\pi$
$308$	0	0
$309$	−4.35231e18	−0.920517
$310$	0	0
$311$	3.24057e18	0.653008	0.326504	−	0.945196i	$-0.394129\pi$
$311$	3.24057e18	0.653008	0.326504	+	0.945196i	$0.394129\pi$
$312$	0	0
$313$	−5.95866e18	−1.14437	−0.572185	−	0.820125i	$-0.693904\pi$
$313$	−5.95866e18	−1.14437	−0.572185	+	0.820125i	$0.693904\pi$
$314$	0	0
$315$	1.81218e18	0.331798
$316$	0	0
$317$	−5.46868e17	−0.0954856	−0.0477428	−	0.998860i	$-0.515203\pi$
$317$	−5.46868e17	−0.0954856	−0.0477428	+	0.998860i	$0.515203\pi$
$318$	0	0
$319$	1.25815e17	0.0209556
$320$	0	0
$321$	8.78954e18	1.39694
$322$	0	0
$323$	−5.07831e18	−0.770372
$324$	0	0
$325$	−7.46189e18	−1.08075
$326$	0	0
$327$	2.04742e18	0.283203
$328$	0	0
$329$	−1.40312e19	−1.85407
$330$	0	0
$331$	6.15884e17	0.0777658	0.0388829	−	0.999244i	$-0.487620\pi$
$331$	6.15884e17	0.0777658	0.0388829	+	0.999244i	$0.487620\pi$
$332$	0	0
$333$	7.17049e17	0.0865400
$334$	0	0
$335$	−1.23189e18	−0.142146
$336$	0	0
$337$	−1.59933e19	−1.76488	−0.882438	−	0.470428i	$-0.844099\pi$
$337$	−1.59933e19	−1.76488	−0.882438	+	0.470428i	$0.844099\pi$
$338$	0	0
$339$	6.72248e18	0.709629
$340$	0	0
$341$	1.05342e19	1.06400
$342$	0	0
$343$	9.51066e18	0.919401
$344$	0	0
$345$	2.74445e19	2.53988
$346$	0	0
$347$	5.10448e18	0.452356	0.226178	−	0.974086i	$-0.427377\pi$
$347$	5.10448e18	0.452356	0.226178	+	0.974086i	$0.427377\pi$
$348$	0	0
$349$	6.56067e18	0.556874	0.278437	−	0.960454i	$-0.410184\pi$
$349$	6.56067e18	0.556874	0.278437	+	0.960454i	$0.410184\pi$
$350$	0	0
$351$	−6.39346e18	−0.519915
$352$	0	0
$353$	−3.92233e18	−0.305657	−0.152828	−	0.988253i	$-0.548838\pi$
$353$	−3.92233e18	−0.305657	−0.152828	+	0.988253i	$0.548838\pi$
$354$	0	0
$355$	1.84334e19	1.37687
$356$	0	0
$357$	−2.08270e19	−1.49147
$358$	0	0
$359$	−2.14658e19	−1.47414	−0.737069	−	0.675817i	$-0.763791\pi$
$359$	−2.14658e19	−1.47414	−0.737069	+	0.675817i	$0.763791\pi$
$360$	0	0
$361$	−1.13703e19	−0.748979
$362$	0	0
$363$	3.87377e18	0.244813
$364$	0	0
$365$	3.37084e19	2.04429
$366$	0	0
$367$	−1.99400e19	−1.16073	−0.580363	−	0.814358i	$-0.697089\pi$
$367$	−1.99400e19	−1.16073	−0.580363	+	0.814358i	$0.697089\pi$
$368$	0	0
$369$	−1.88178e18	−0.105165
$370$	0	0
$371$	2.83578e19	1.52184
$372$	0	0
$373$	−7.57819e18	−0.390615	−0.195308	−	0.980742i	$-0.562571\pi$
$373$	−7.57819e18	−0.390615	−0.195308	+	0.980742i	$0.562571\pi$
$374$	0	0
$375$	−4.01012e19	−1.98574
$376$	0	0
$377$	2.51086e17	0.0119471
$378$	0	0
$379$	−4.16502e19	−1.90468	−0.952342	−	0.305034i	$-0.901332\pi$
$379$	−4.16502e19	−1.90468	−0.952342	+	0.305034i	$0.901332\pi$
$380$	0	0
$381$	1.88508e19	0.828689
$382$	0	0
$383$	−2.54105e19	−1.07405	−0.537023	−	0.843568i	$-0.680451\pi$
$383$	−2.54105e19	−1.07405	−0.537023	+	0.843568i	$0.680451\pi$
$384$	0	0
$385$	4.02455e19	1.63592
$386$	0	0
$387$	8.81135e17	0.0344517
$388$	0	0
$389$	−4.48730e19	−1.68796	−0.843981	−	0.536373i	$-0.819794\pi$
$389$	−4.48730e19	−1.68796	−0.843981	+	0.536373i	$0.819794\pi$
$390$	0	0
$391$	6.61552e19	2.39463
$392$	0	0
$393$	−3.12666e19	−1.08927
$394$	0	0
$395$	1.92853e19	0.646766
$396$	0	0
$397$	−1.82905e18	−0.0590606	−0.0295303	−	0.999564i	$-0.509401\pi$
$397$	−1.82905e18	−0.0590606	−0.0295303	+	0.999564i	$0.509401\pi$
$398$	0	0
$399$	1.56287e19	0.485988
$400$	0	0
$401$	2.95566e18	0.0885261	0.0442630	−	0.999020i	$-0.485906\pi$
$401$	2.95566e18	0.0885261	0.0442630	+	0.999020i	$0.485906\pi$
$402$	0	0
$403$	2.10229e19	0.606603
$404$	0	0
$405$	−5.13924e19	−1.42885
$406$	0	0
$407$	1.59244e19	0.426683
$408$	0	0
$409$	6.46247e19	1.66907	0.834534	−	0.550957i	$-0.185737\pi$
$409$	6.46247e19	1.66907	0.834534	+	0.550957i	$0.185737\pi$
$410$	0	0
$411$	2.79382e19	0.695640
$412$	0	0
$413$	9.50980e18	0.228321
$414$	0	0
$415$	4.20905e18	0.0974592
$416$	0	0
$417$	2.63212e19	0.587876
$418$	0	0
$419$	−3.18045e19	−0.685304	−0.342652	−	0.939462i	$-0.611325\pi$
$419$	−3.18045e19	−0.685304	−0.342652	+	0.939462i	$0.611325\pi$
$420$	0	0
$421$	7.92911e19	1.64858	0.824288	−	0.566171i	$-0.191576\pi$
$421$	7.92911e19	1.64858	0.824288	+	0.566171i	$0.191576\pi$
$422$	0	0
$423$	−1.50160e19	−0.301302
$424$	0	0
$425$	−1.76053e20	−3.40979
$426$	0	0
$427$	−4.08340e19	−0.763507
$428$	0	0
$429$	−2.09800e19	−0.378769
$430$	0	0
$431$	2.55606e19	0.445648	0.222824	−	0.974859i	$-0.428472\pi$
$431$	2.55606e19	0.445648	0.222824	+	0.974859i	$0.428472\pi$
$432$	0	0
$433$	−1.75851e19	−0.296133	−0.148066	−	0.988977i	$-0.547305\pi$
$433$	−1.75851e19	−0.296133	−0.148066	+	0.988977i	$0.547305\pi$
$434$	0	0
$435$	2.45759e18	0.0399798
$436$	0	0
$437$	−4.96432e19	−0.780277
$438$	0	0
$439$	−6.95087e19	−1.05574	−0.527868	−	0.849326i	$-0.677008\pi$
$439$	−6.95087e19	−1.05574	−0.527868	+	0.849326i	$0.677008\pi$
$440$	0	0
$441$	−1.63267e18	−0.0239667
$442$	0	0
$443$	4.99022e19	0.708096	0.354048	−	0.935227i	$-0.384805\pi$
$443$	4.99022e19	0.708096	0.354048	+	0.935227i	$0.384805\pi$
$444$	0	0
$445$	8.45115e19	1.15935
$446$	0	0
$447$	−3.19804e19	−0.424207
$448$	0	0
$449$	1.49870e20	1.92250	0.961250	−	0.275679i	$-0.0889026\pi$
$449$	1.49870e20	1.92250	0.961250	+	0.275679i	$0.0889026\pi$
$450$	0	0
$451$	−4.17911e19	−0.518513
$452$	0	0
$453$	2.65002e19	0.318063
$454$	0	0
$455$	8.03172e19	0.932659
$456$	0	0
$457$	−1.52710e20	−1.71591	−0.857956	−	0.513723i	$-0.828266\pi$
$457$	−1.52710e20	−1.71591	−0.857956	+	0.513723i	$0.828266\pi$
$458$	0	0
$459$	−1.50845e20	−1.64035
$460$	0	0
$461$	4.30759e19	0.453396	0.226698	−	0.973965i	$-0.427207\pi$
$461$	4.30759e19	0.453396	0.226698	+	0.973965i	$0.427207\pi$
$462$	0	0
$463$	7.28785e19	0.742578	0.371289	−	0.928517i	$-0.378916\pi$
$463$	7.28785e19	0.742578	0.371289	+	0.928517i	$0.378916\pi$
$464$	0	0
$465$	2.05769e20	2.02994
$466$	0	0
$467$	−1.55854e20	−1.48882	−0.744409	−	0.667724i	$-0.767269\pi$
$467$	−1.55854e20	−1.48882	−0.744409	+	0.667724i	$0.767269\pi$
$468$	0	0
$469$	9.13864e18	0.0845445
$470$	0	0
$471$	1.47882e20	1.32513
$472$	0	0
$473$	1.95685e19	0.169863
$474$	0	0
$475$	1.32111e20	1.11106
$476$	0	0
$477$	3.03481e19	0.247311
$478$	0	0
$479$	1.49800e20	1.18303	0.591514	−	0.806295i	$-0.298530\pi$
$479$	1.49800e20	1.18303	0.591514	+	0.806295i	$0.298530\pi$
$480$	0	0
$481$	3.17801e19	0.243258
$482$	0	0
$483$	−2.03595e20	−1.51065
$484$	0	0
$485$	−2.48743e20	−1.78932
$486$	0	0
$487$	−6.20773e19	−0.432978	−0.216489	−	0.976285i	$-0.569460\pi$
$487$	−6.20773e19	−0.432978	−0.216489	+	0.976285i	$0.569460\pi$
$488$	0	0
$489$	−1.23621e20	−0.836136
$490$	0	0
$491$	−5.25617e19	−0.344793	−0.172397	−	0.985028i	$-0.555151\pi$
$491$	−5.25617e19	−0.344793	−0.172397	+	0.985028i	$0.555151\pi$
$492$	0	0
$493$	5.92404e18	0.0376934
$494$	0	0
$495$	4.30701e19	0.265850
$496$	0	0
$497$	−1.36747e20	−0.818924
$498$	0	0
$499$	6.41491e19	0.372766	0.186383	−	0.982477i	$-0.440323\pi$
$499$	6.41491e19	0.372766	0.186383	+	0.982477i	$0.440323\pi$
$500$	0	0
$501$	−7.85591e19	−0.443010
$502$	0	0
$503$	5.33679e19	0.292093	0.146046	−	0.989278i	$-0.453345\pi$
$503$	5.33679e19	0.292093	0.146046	+	0.989278i	$0.453345\pi$
$504$	0	0
$505$	−1.56747e20	−0.832751
$506$	0	0
$507$	1.34415e20	0.693247
$508$	0	0
$509$	1.33649e19	0.0669240	0.0334620	−	0.999440i	$-0.489347\pi$
$509$	1.33649e19	0.0669240	0.0334620	+	0.999440i	$0.489347\pi$
$510$	0	0
$511$	−2.50062e20	−1.21588
$512$	0	0
$513$	1.13195e20	0.534499
$514$	0	0
$515$	3.96002e20	1.81612
$516$	0	0
$517$	−3.33479e20	−1.48556
$518$	0	0
$519$	−1.90353e20	−0.823765
$520$	0	0
$521$	2.10261e20	0.884049	0.442025	−	0.897003i	$-0.354260\pi$
$521$	2.10261e20	0.884049	0.442025	+	0.897003i	$0.354260\pi$
$522$	0	0
$523$	−4.33927e19	−0.177278	−0.0886388	−	0.996064i	$-0.528252\pi$
$523$	−4.33927e19	−0.177278	−0.0886388	+	0.996064i	$0.528252\pi$
$524$	0	0
$525$	5.41810e20	2.15106
$526$	0	0
$527$	4.96007e20	1.91385
$528$	0	0
$529$	3.80067e20	1.42542
$530$	0	0
$531$	1.01773e19	0.0371041
$532$	0	0
$533$	−8.34017e19	−0.295611
$534$	0	0
$535$	−7.99731e20	−2.75607
$536$	0	0
$537$	−1.31919e20	−0.442079
$538$	0	0
$539$	−3.62588e19	−0.118167
$540$	0	0
$541$	2.52702e20	0.800993	0.400497	−	0.916298i	$-0.368838\pi$
$541$	2.52702e20	0.800993	0.400497	+	0.916298i	$0.368838\pi$
$542$	0	0
$543$	−5.49674e18	−0.0169475
$544$	0	0
$545$	−1.86288e20	−0.558740
$546$	0	0
$547$	4.66972e18	0.0136266	0.00681328	−	0.999977i	$-0.497831\pi$
$547$	4.66972e18	0.0136266	0.00681328	+	0.999977i	$0.497831\pi$
$548$	0	0
$549$	−4.37000e19	−0.124076
$550$	0	0
$551$	−4.44542e18	−0.0122822
$552$	0	0
$553$	−1.43066e20	−0.384678
$554$	0	0
$555$	3.11059e20	0.814038
$556$	0	0
$557$	5.45097e20	1.38854	0.694272	−	0.719712i	$-0.255726\pi$
$557$	5.45097e20	1.38854	0.694272	+	0.719712i	$0.255726\pi$
$558$	0	0
$559$	3.90525e19	0.0968411
$560$	0	0
$561$	−4.94994e20	−1.19503
$562$	0	0
$563$	−5.43403e20	−1.27735	−0.638674	−	0.769477i	$-0.720517\pi$
$563$	−5.43403e20	−1.27735	−0.638674	+	0.769477i	$0.720517\pi$
$564$	0	0
$565$	−6.11656e20	−1.40005
$566$	0	0
$567$	3.81249e20	0.849838
$568$	0	0
$569$	−1.60405e20	−0.348238	−0.174119	−	0.984725i	$-0.555708\pi$
$569$	−1.60405e20	−0.348238	−0.174119	+	0.984725i	$0.555708\pi$
$570$	0	0
$571$	−2.32798e20	−0.492275	−0.246138	−	0.969235i	$-0.579162\pi$
$571$	−2.32798e20	−0.492275	−0.246138	+	0.969235i	$0.579162\pi$
$572$	0	0
$573$	3.34145e20	0.688297
$574$	0	0
$575$	−1.72102e21	−3.45363
$576$	0	0
$577$	5.64435e20	1.10356	0.551779	−	0.833990i	$-0.313949\pi$
$577$	5.64435e20	1.10356	0.551779	+	0.833990i	$0.313949\pi$
$578$	0	0
$579$	1.98243e20	0.377668
$580$	0	0
$581$	−3.12244e19	−0.0579660
$582$	0	0
$583$	6.73979e20	1.21936
$584$	0	0
$585$	8.59543e19	0.151565
$586$	0	0
$587$	8.53706e20	1.46731	0.733656	−	0.679521i	$-0.237812\pi$
$587$	8.53706e20	1.46731	0.733656	+	0.679521i	$0.237812\pi$
$588$	0	0
$589$	−3.72206e20	−0.623618
$590$	0	0
$591$	6.94269e19	0.113402
$592$	0	0
$593$	7.01535e20	1.11722	0.558610	−	0.829430i	$-0.311335\pi$
$593$	7.01535e20	1.11722	0.558610	+	0.829430i	$0.311335\pi$
$594$	0	0
$595$	1.89498e21	2.94257
$596$	0	0
$597$	−4.63224e20	−0.701429
$598$	0	0
$599$	4.70886e20	0.695367	0.347684	−	0.937612i	$-0.386968\pi$
$599$	4.70886e20	0.695367	0.347684	+	0.937612i	$0.386968\pi$
$600$	0	0
$601$	−1.14703e21	−1.65202	−0.826011	−	0.563655i	$-0.809395\pi$
$601$	−1.14703e21	−1.65202	−0.826011	+	0.563655i	$0.809395\pi$
$602$	0	0
$603$	9.78004e18	0.0137392
$604$	0	0
$605$	−3.52461e20	−0.482998
$606$	0	0
$607$	2.81356e20	0.376132	0.188066	−	0.982156i	$-0.439778\pi$
$607$	2.81356e20	0.376132	0.188066	+	0.982156i	$0.439778\pi$
$608$	0	0
$609$	−1.82314e19	−0.0237788
$610$	0	0
$611$	−6.65518e20	−0.846937
$612$	0	0
$613$	6.03142e20	0.748973	0.374486	−	0.927232i	$-0.377819\pi$
$613$	6.03142e20	0.748973	0.374486	+	0.927232i	$0.377819\pi$
$614$	0	0
$615$	−8.16324e20	−0.989234
$616$	0	0
$617$	7.24780e20	0.857171	0.428585	−	0.903501i	$-0.359012\pi$
$617$	7.24780e20	0.857171	0.428585	+	0.903501i	$0.359012\pi$
$618$	0	0
$619$	−3.24109e20	−0.374120	−0.187060	−	0.982348i	$-0.559896\pi$
$619$	−3.24109e20	−0.374120	−0.187060	+	0.982348i	$0.559896\pi$
$620$	0	0
$621$	−1.47459e21	−1.66144
$622$	0	0
$623$	−6.26940e20	−0.689550
$624$	0	0
$625$	1.58337e21	1.70014
$626$	0	0
$627$	3.71446e20	0.389394
$628$	0	0
$629$	7.49808e20	0.767486
$630$	0	0
$631$	−1.63168e20	−0.163085	−0.0815427	−	0.996670i	$-0.525985\pi$
$631$	−1.63168e20	−0.163085	−0.0815427	+	0.996670i	$0.525985\pi$
$632$	0	0
$633$	−1.42691e21	−1.39274
$634$	0	0
$635$	−1.71517e21	−1.63495
$636$	0	0
$637$	−7.23609e19	−0.0673687
$638$	0	0
$639$	−1.46344e20	−0.133082
$640$	0	0
$641$	−3.93853e20	−0.349864	−0.174932	−	0.984581i	$-0.555971\pi$
$641$	−3.93853e20	−0.349864	−0.174932	+	0.984581i	$0.555971\pi$
$642$	0	0
$643$	−1.41070e21	−1.22420	−0.612098	−	0.790782i	$-0.709674\pi$
$643$	−1.41070e21	−1.22420	−0.612098	+	0.790782i	$0.709674\pi$
$644$	0	0
$645$	3.82240e20	0.324070
$646$	0	0
$647$	2.21676e21	1.83627	0.918135	−	0.396268i	$-0.129695\pi$
$647$	2.21676e21	1.83627	0.918135	+	0.396268i	$0.129695\pi$
$648$	0	0
$649$	2.26019e20	0.182941
$650$	0	0
$651$	−1.52648e21	−1.20735
$652$	0	0
$653$	1.49836e21	1.15815	0.579077	−	0.815273i	$-0.303413\pi$
$653$	1.49836e21	1.15815	0.579077	+	0.815273i	$0.303413\pi$
$654$	0	0
$655$	2.84484e21	2.14906
$656$	0	0
$657$	−2.67613e20	−0.197591
$658$	0	0
$659$	−1.96051e21	−1.41491	−0.707454	−	0.706760i	$-0.750156\pi$
$659$	−1.96051e21	−1.41491	−0.707454	+	0.706760i	$0.750156\pi$
$660$	0	0
$661$	1.41983e21	1.00167	0.500835	−	0.865543i	$-0.333026\pi$
$661$	1.41983e21	1.00167	0.500835	+	0.865543i	$0.333026\pi$
$662$	0	0
$663$	−9.87851e20	−0.681302
$664$	0	0
$665$	−1.42200e21	−0.958820
$666$	0	0
$667$	5.79106e19	0.0381781
$668$	0	0
$669$	2.41673e21	1.55787
$670$	0	0
$671$	−9.70500e20	−0.611754
$672$	0	0
$673$	3.83634e20	0.236485	0.118243	−	0.992985i	$-0.462274\pi$
$673$	3.83634e20	0.236485	0.118243	+	0.992985i	$0.462274\pi$
$674$	0	0
$675$	3.92421e21	2.36578
$676$	0	0
$677$	−2.81127e21	−1.65763	−0.828816	−	0.559521i	$-0.810985\pi$
$677$	−2.81127e21	−1.65763	−0.828816	+	0.559521i	$0.810985\pi$
$678$	0	0
$679$	1.84527e21	1.06423
$680$	0	0
$681$	2.26867e21	1.27988
$682$	0	0
$683$	1.24673e21	0.688045	0.344022	−	0.938961i	$-0.388210\pi$
$683$	1.24673e21	0.688045	0.344022	+	0.938961i	$0.388210\pi$
$684$	0	0
$685$	−2.54200e21	−1.37245
$686$	0	0
$687$	2.05419e21	1.08509
$688$	0	0
$689$	1.34505e21	0.695174
$690$	0	0
$691$	−3.09282e21	−1.56412	−0.782059	−	0.623204i	$-0.785831\pi$
$691$	−3.09282e21	−1.56412	−0.782059	+	0.623204i	$0.785831\pi$
$692$	0	0
$693$	−3.19512e20	−0.158120
$694$	0	0
$695$	−2.39488e21	−1.15984
$696$	0	0
$697$	−1.96775e21	−0.932663
$698$	0	0
$699$	−1.76937e21	−0.820805
$700$	0	0
$701$	−1.71155e21	−0.777149	−0.388574	−	0.921417i	$-0.627032\pi$
$701$	−1.71155e21	−0.777149	−0.388574	+	0.921417i	$0.627032\pi$
$702$	0	0
$703$	−5.62660e20	−0.250081
$704$	0	0
$705$	−6.51400e21	−2.83419
$706$	0	0
$707$	1.16282e21	0.495297
$708$	0	0
$709$	1.85721e21	0.774486	0.387243	−	0.921978i	$-0.373427\pi$
$709$	1.85721e21	0.774486	0.387243	+	0.921978i	$0.373427\pi$
$710$	0	0
$711$	−1.53107e20	−0.0625134
$712$	0	0
$713$	4.84873e21	1.93846
$714$	0	0
$715$	1.90890e21	0.747286
$716$	0	0
$717$	6.21733e20	0.238347
$718$	0	0
$719$	7.53757e20	0.282986	0.141493	−	0.989939i	$-0.454810\pi$
$719$	7.53757e20	0.282986	0.141493	+	0.989939i	$0.454810\pi$
$720$	0	0
$721$	−2.93771e21	−1.08017
$722$	0	0
$723$	−1.88132e20	−0.0677526
$724$	0	0
$725$	−1.54113e20	−0.0543630
$726$	0	0
$727$	1.15397e21	0.398736	0.199368	−	0.979925i	$-0.436111\pi$
$727$	1.15397e21	0.398736	0.199368	+	0.979925i	$0.436111\pi$
$728$	0	0
$729$	3.27351e21	1.10805
$730$	0	0
$731$	9.21391e20	0.305537
$732$	0	0
$733$	−4.12177e21	−1.33907	−0.669536	−	0.742780i	$-0.733507\pi$
$733$	−4.12177e21	−1.33907	−0.669536	+	0.742780i	$0.733507\pi$
$734$	0	0
$735$	−7.08259e20	−0.225443
$736$	0	0
$737$	2.17198e20	0.0677406
$738$	0	0
$739$	−2.97710e21	−0.909831	−0.454915	−	0.890535i	$-0.650331\pi$
$739$	−2.97710e21	−0.909831	−0.454915	+	0.890535i	$0.650331\pi$
$740$	0	0
$741$	7.41288e20	0.221999
$742$	0	0
$743$	3.28788e21	0.964938	0.482469	−	0.875913i	$-0.339740\pi$
$743$	3.28788e21	0.964938	0.482469	+	0.875913i	$0.339740\pi$
$744$	0	0
$745$	2.90979e21	0.836931
$746$	0	0
$747$	−3.34159e19	−0.00941996
$748$	0	0
$749$	5.93273e21	1.63923
$750$	0	0
$751$	1.25297e20	0.0339344	0.0169672	−	0.999856i	$-0.494599\pi$
$751$	1.25297e20	0.0339344	0.0169672	+	0.999856i	$0.494599\pi$
$752$	0	0
$753$	3.96495e21	1.05263
$754$	0	0
$755$	−2.41117e21	−0.627517
$756$	0	0
$757$	−1.29679e21	−0.330866	−0.165433	−	0.986221i	$-0.552902\pi$
$757$	−1.29679e21	−0.330866	−0.165433	+	0.986221i	$0.552902\pi$
$758$	0	0
$759$	−4.83883e21	−1.21039
$760$	0	0
$761$	3.01343e21	0.739054	0.369527	−	0.929220i	$-0.379520\pi$
$761$	3.01343e21	0.739054	0.369527	+	0.929220i	$0.379520\pi$
$762$	0	0
$763$	1.38196e21	0.332323
$764$	0	0
$765$	2.02798e21	0.478192
$766$	0	0
$767$	4.51062e20	0.104297
$768$	0	0
$769$	2.77174e20	0.0628500	0.0314250	−	0.999506i	$-0.489995\pi$
$769$	2.77174e20	0.0628500	0.0314250	+	0.999506i	$0.489995\pi$
$770$	0	0
$771$	4.44752e20	0.0989032
$772$	0	0
$773$	−4.63967e20	−0.101191	−0.0505954	−	0.998719i	$-0.516112\pi$
$773$	−4.63967e20	−0.101191	−0.0505954	+	0.998719i	$0.516112\pi$
$774$	0	0
$775$	−1.29035e22	−2.76023
$776$	0	0
$777$	−2.30756e21	−0.484167
$778$	0	0
$779$	1.47661e21	0.303903
$780$	0	0
$781$	−3.25005e21	−0.656157
$782$	0	0
$783$	−1.32046e20	−0.0261524
$784$	0	0
$785$	−1.34553e22	−2.61439
$786$	0	0
$787$	9.16403e21	1.74693	0.873466	−	0.486885i	$-0.161867\pi$
$787$	9.16403e21	1.74693	0.873466	+	0.486885i	$0.161867\pi$
$788$	0	0
$789$	−4.67633e19	−0.00874636
$790$	0	0
$791$	4.53751e21	0.832710
$792$	0	0
$793$	−1.93681e21	−0.348769
$794$	0	0
$795$	1.31651e22	2.32633
$796$	0	0
$797$	−8.40446e21	−1.45738	−0.728689	−	0.684844i	$-0.759870\pi$
$797$	−8.40446e21	−1.45738	−0.728689	+	0.684844i	$0.759870\pi$
$798$	0	0
$799$	−1.57020e22	−2.67211
$800$	0	0
$801$	−6.70943e20	−0.112058
$802$	0	0
$803$	−5.94323e21	−0.974217
$804$	0	0
$805$	1.85244e22	2.98040
$806$	0	0
$807$	3.36855e21	0.531975
$808$	0	0
$809$	−2.07188e21	−0.321182	−0.160591	−	0.987021i	$-0.551340\pi$
$809$	−2.07188e21	−0.321182	−0.160591	+	0.987021i	$0.551340\pi$
$810$	0	0
$811$	4.27452e21	0.650475	0.325238	−	0.945632i	$-0.394556\pi$
$811$	4.27452e21	0.650475	0.325238	+	0.945632i	$0.394556\pi$
$812$	0	0
$813$	−9.23558e21	−1.37970
$814$	0	0
$815$	1.12479e22	1.64964
$816$	0	0
$817$	−6.91416e20	−0.0995575
$818$	0	0
$819$	−6.37644e20	−0.0901465
$820$	0	0
$821$	1.27979e22	1.77650	0.888250	−	0.459361i	$-0.151921\pi$
$821$	1.27979e22	1.77650	0.888250	+	0.459361i	$0.151921\pi$
$822$	0	0
$823$	−3.06111e21	−0.417235	−0.208617	−	0.977997i	$-0.566896\pi$
$823$	−3.06111e21	−0.417235	−0.208617	+	0.977997i	$0.566896\pi$
$824$	0	0
$825$	1.28772e22	1.72352
$826$	0	0
$827$	1.33452e21	0.175402	0.0877010	−	0.996147i	$-0.472048\pi$
$827$	1.33452e21	0.175402	0.0877010	+	0.996147i	$0.472048\pi$
$828$	0	0
$829$	−2.91788e21	−0.376624	−0.188312	−	0.982109i	$-0.560302\pi$
$829$	−2.91788e21	−0.376624	−0.188312	+	0.982109i	$0.560302\pi$
$830$	0	0
$831$	2.57351e21	0.326225
$832$	0	0
$833$	−1.70726e21	−0.212551
$834$	0	0
$835$	7.14783e21	0.874028
$836$	0	0
$837$	−1.10559e22	−1.32787
$838$	0	0
$839$	2.77423e21	0.327287	0.163643	−	0.986520i	$-0.447675\pi$
$839$	2.77423e21	0.327287	0.163643	+	0.986520i	$0.447675\pi$
$840$	0	0
$841$	−8.62400e21	−0.999399
$842$	0	0
$843$	−1.47157e22	−1.67523
$844$	0	0
$845$	−1.22300e22	−1.36773
$846$	0	0
$847$	2.61470e21	0.287274
$848$	0	0
$849$	8.16069e21	0.880885
$850$	0	0
$851$	7.32977e21	0.777354
$852$	0	0
$853$	7.08371e21	0.738148	0.369074	−	0.929400i	$-0.379675\pi$
$853$	7.08371e21	0.738148	0.369074	+	0.929400i	$0.379675\pi$
$854$	0	0
$855$	−1.52180e21	−0.155816
$856$	0	0
$857$	2.74289e21	0.275964	0.137982	−	0.990435i	$-0.455938\pi$
$857$	2.74289e21	0.275964	0.137982	+	0.990435i	$0.455938\pi$
$858$	0	0
$859$	−1.53540e22	−1.51800	−0.759000	−	0.651090i	$-0.774312\pi$
$859$	−1.53540e22	−1.51800	−0.759000	+	0.651090i	$0.774312\pi$
$860$	0	0
$861$	6.05582e21	0.588368
$862$	0	0
$863$	4.60080e21	0.439292	0.219646	−	0.975580i	$-0.429510\pi$
$863$	4.60080e21	0.439292	0.219646	+	0.975580i	$0.429510\pi$
$864$	0	0
$865$	1.73195e22	1.62523
$866$	0	0
$867$	−1.34488e22	−1.24034
$868$	0	0
$869$	−3.40024e21	−0.308220
$870$	0	0
$871$	4.33458e20	0.0386198
$872$	0	0
$873$	1.97478e21	0.172947
$874$	0	0
$875$	−2.70673e22	−2.33016
$876$	0	0
$877$	−1.60904e22	−1.36166	−0.680831	−	0.732441i	$-0.738381\pi$
$877$	−1.60904e22	−1.36166	−0.680831	+	0.732441i	$0.738381\pi$
$878$	0	0
$879$	−1.97545e22	−1.64342
$880$	0	0
$881$	−2.11475e22	−1.72958	−0.864788	−	0.502138i	$-0.832547\pi$
$881$	−2.11475e22	−1.72958	−0.864788	+	0.502138i	$0.832547\pi$
$882$	0	0
$883$	2.01715e22	1.62193	0.810966	−	0.585093i	$-0.198942\pi$
$883$	2.01715e22	1.62193	0.810966	+	0.585093i	$0.198942\pi$
$884$	0	0
$885$	4.41493e21	0.349019
$886$	0	0
$887$	7.82890e21	0.608519	0.304259	−	0.952589i	$-0.401591\pi$
$887$	7.82890e21	0.608519	0.304259	+	0.952589i	$0.401591\pi$
$888$	0	0
$889$	1.27238e22	0.972420
$890$	0	0
$891$	9.06114e21	0.680927
$892$	0	0
$893$	1.17829e22	0.870693
$894$	0	0
$895$	1.20028e22	0.872191
$896$	0	0
$897$	−9.65676e21	−0.690062
$898$	0	0
$899$	4.34192e20	0.0305130
$900$	0	0
$901$	3.17346e22	2.19330
$902$	0	0
$903$	−2.83561e21	−0.192747
$904$	0	0
$905$	5.00130e20	0.0334363
$906$	0	0
$907$	1.83291e22	1.20527	0.602637	−	0.798016i	$-0.294117\pi$
$907$	1.83291e22	1.20527	0.602637	+	0.798016i	$0.294117\pi$
$908$	0	0
$909$	1.24443e21	0.0804898
$910$	0	0
$911$	2.79476e21	0.177810	0.0889050	−	0.996040i	$-0.471663\pi$
$911$	2.79476e21	0.177810	0.0889050	+	0.996040i	$0.471663\pi$
$912$	0	0
$913$	−7.42110e20	−0.0464448
$914$	0	0
$915$	−1.89572e22	−1.16712
$916$	0	0
$917$	−2.11042e22	−1.27820
$918$	0	0
$919$	2.35019e22	1.40035	0.700174	−	0.713972i	$-0.253106\pi$
$919$	2.35019e22	1.40035	0.700174	+	0.713972i	$0.253106\pi$
$920$	0	0
$921$	2.54154e22	1.48988
$922$	0	0
$923$	−6.48607e21	−0.374084
$924$	0	0
$925$	−1.95061e22	−1.10690
$926$	0	0
$927$	−3.14389e21	−0.175537
$928$	0	0
$929$	−2.31449e22	−1.27156	−0.635780	−	0.771871i	$-0.719321\pi$
$929$	−2.31449e22	−1.27156	−0.635780	+	0.771871i	$0.719321\pi$
$930$	0	0
$931$	1.28114e21	0.0692584
$932$	0	0
$933$	−1.11605e22	−0.593707
$934$	0	0
$935$	4.50379e22	2.35771
$936$	0	0
$937$	−3.19217e22	−1.64452	−0.822260	−	0.569112i	$-0.807287\pi$
$937$	−3.19217e22	−1.64452	−0.822260	+	0.569112i	$0.807287\pi$
$938$	0	0
$939$	2.05216e22	1.04045
$940$	0	0
$941$	3.38688e22	1.68996	0.844982	−	0.534795i	$-0.179611\pi$
$941$	3.38688e22	1.68996	0.844982	+	0.534795i	$0.179611\pi$
$942$	0	0
$943$	−1.92358e22	−0.944655
$944$	0	0
$945$	−4.22388e22	−2.04161
$946$	0	0
$947$	−3.82082e22	−1.81774	−0.908870	−	0.417079i	$-0.863054\pi$
$947$	−3.82082e22	−1.81774	−0.908870	+	0.417079i	$0.863054\pi$
$948$	0	0
$949$	−1.18608e22	−0.555414
$950$	0	0
$951$	1.88341e21	0.0868144
$952$	0	0
$953$	2.83541e22	1.28653	0.643263	−	0.765645i	$-0.277580\pi$
$953$	2.83541e22	1.28653	0.643263	+	0.765645i	$0.277580\pi$
$954$	0	0
$955$	−3.04028e22	−1.35796
$956$	0	0
$957$	−4.33305e20	−0.0190526
$958$	0	0
$959$	1.88576e22	0.816294
$960$	0	0
$961$	1.28887e22	0.549269
$962$	0	0
$963$	6.34912e21	0.266389
$964$	0	0
$965$	−1.80375e22	−0.745112
$966$	0	0
$967$	−2.35320e22	−0.957107	−0.478553	−	0.878058i	$-0.658839\pi$
$967$	−2.35320e22	−0.957107	−0.478553	+	0.878058i	$0.658839\pi$
$968$	0	0
$969$	1.74897e22	0.700413
$970$	0	0
$971$	−2.92750e22	−1.15439	−0.577196	−	0.816606i	$-0.695853\pi$
$971$	−2.92750e22	−1.15439	−0.577196	+	0.816606i	$0.695853\pi$
$972$	0	0
$973$	1.77662e22	0.689839
$974$	0	0
$975$	2.56987e22	0.982601
$976$	0	0
$977$	3.15113e22	1.18647	0.593236	−	0.805029i	$-0.297850\pi$
$977$	3.15113e22	1.18647	0.593236	+	0.805029i	$0.297850\pi$
$978$	0	0
$979$	−1.49005e22	−0.552497
$980$	0	0
$981$	1.47895e21	0.0540052
$982$	0	0
$983$	−3.86669e22	−1.39056	−0.695278	−	0.718741i	$-0.744719\pi$
$983$	−3.86669e22	−1.39056	−0.695278	+	0.718741i	$0.744719\pi$
$984$	0	0
$985$	−6.31692e21	−0.223735
$986$	0	0
$987$	4.83235e22	1.68570
$988$	0	0
$989$	9.00709e21	0.309465
$990$	0	0
$991$	−2.34935e21	−0.0795050	−0.0397525	−	0.999210i	$-0.512657\pi$
$991$	−2.34935e21	−0.0795050	−0.0397525	+	0.999210i	$0.512657\pi$
$992$	0	0
$993$	−2.12110e21	−0.0707038
$994$	0	0
$995$	4.21472e22	1.38387
$996$	0	0
$997$	4.91914e21	0.159102	0.0795509	−	0.996831i	$-0.474651\pi$
$997$	4.91914e21	0.159102	0.0795509	+	0.996831i	$0.474651\pi$
$998$	0	0
$999$	−1.67131e22	−0.532496

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8.16.a.a.1.1	✓	1
3.2	odd	2		72.16.a.a.1.1		1
4.3	odd	2		16.16.a.e.1.1		1
8.3	odd	2		64.16.a.b.1.1		1
8.5	even	2		64.16.a.j.1.1		1
12.11	even	2		144.16.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8.16.a.a.1.1	✓	1	1.1	even	1	trivial
16.16.a.e.1.1		1	4.3	odd	2
64.16.a.b.1.1		1	8.3	odd	2
64.16.a.j.1.1		1	8.5	even	2
72.16.a.a.1.1		1	3.2	odd	2
144.16.a.a.1.1		1	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 8 = 2^{3} \)
Weight:	\( k \)	\(=\)	\( 16 \)
Character orbit:	\([\chi]\)	\(=\)	8.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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