LMFDB - Embedded newform 48.16.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [48,16,Mod(1,48)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(48, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("48.1");

S:= CuspForms(chi, 16);

N := Newforms(S);

Level:	$ N $	$=$	$ 48 = 2^{4} \cdot 3 $
Weight:	$ k $	$=$	$ 16 $
Character orbit:	$[\chi]$	$=$	48.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:	$68.4928824480$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 3)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	48.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-2187.00 q^{3} -221490. q^{5} +2.14900e6 q^{7} +4.78297e6 q^{9} +O(q^{10})$

$q-2187.00 q^{3} -221490. q^{5} +2.14900e6 q^{7} +4.78297e6 q^{9} -3.71693e7 q^{11} -2.79974e8 q^{13} +4.84399e8 q^{15} +2.49291e9 q^{17} +4.66978e9 q^{19} -4.69986e9 q^{21} +1.84679e10 q^{23} +1.85402e10 q^{25} -1.04604e10 q^{27} -1.15953e11 q^{29} +5.61870e10 q^{31} +8.12893e10 q^{33} -4.75982e11 q^{35} +6.14765e11 q^{37} +6.12304e11 q^{39} +5.49860e11 q^{41} +9.82884e11 q^{43} -1.05938e12 q^{45} -2.07614e12 q^{47} -1.29361e11 q^{49} -5.45200e12 q^{51} -1.20484e13 q^{53} +8.23263e12 q^{55} -1.02128e13 q^{57} -2.30879e13 q^{59} -8.50581e12 q^{61} +1.02786e13 q^{63} +6.20115e13 q^{65} +1.23310e13 q^{67} -4.03894e13 q^{69} -5.89892e13 q^{71} -5.60983e12 q^{73} -4.05475e13 q^{75} -7.98769e13 q^{77} -1.59919e14 q^{79} +2.28768e13 q^{81} -5.76759e13 q^{83} -5.52155e14 q^{85} +2.53590e14 q^{87} -3.62288e14 q^{89} -6.01665e14 q^{91} -1.22881e14 q^{93} -1.03431e15 q^{95} -5.39787e14 q^{97} -1.77780e14 q^{99} +O(q^{100})$

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
$3$	−2187.00	−0.577350
$3$	−2187.00	−0.577350
$4$	0	0
$5$	−221490.	−1.26788	−0.633941	−	0.773381i	$-0.718564\pi$
$5$	−221490.	−1.26788	−0.633941	+	0.773381i	$0.718564\pi$
$6$	0	0
$7$	2.14900e6	0.986282	0.493141	−	0.869949i	$-0.335849\pi$
$7$	2.14900e6	0.986282	0.493141	+	0.869949i	$0.335849\pi$
$8$	0	0
$9$	4.78297e6	0.333333
$10$	0	0
$11$	−3.71693e7	−0.575095	−0.287547	−	0.957766i	$-0.592840\pi$
$11$	−3.71693e7	−0.575095	−0.287547	+	0.957766i	$0.592840\pi$
$12$	0	0
$13$	−2.79974e8	−1.23749	−0.618747	−	0.785590i	$-0.712359\pi$
$13$	−2.79974e8	−1.23749	−0.618747	+	0.785590i	$0.712359\pi$
$14$	0	0
$15$	4.84399e8	0.732012
$16$	0	0
$17$	2.49291e9	1.47347	0.736733	−	0.676184i	$-0.236367\pi$
$17$	2.49291e9	1.47347	0.736733	+	0.676184i	$0.236367\pi$
$18$	0	0
$19$	4.66978e9	1.19852	0.599259	−	0.800555i	$-0.295462\pi$
$19$	4.66978e9	1.19852	0.599259	+	0.800555i	$0.295462\pi$
$20$	0	0
$21$	−4.69986e9	−0.569430
$22$	0	0
$23$	1.84679e10	1.13099	0.565496	−	0.824751i	$-0.308685\pi$
$23$	1.84679e10	1.13099	0.565496	+	0.824751i	$0.308685\pi$
$24$	0	0
$25$	1.85402e10	0.607527
$26$	0	0
$27$	−1.04604e10	−0.192450
$28$	0	0
$29$	−1.15953e11	−1.24824	−0.624121	−	0.781328i	$-0.714543\pi$
$29$	−1.15953e11	−1.24824	−0.624121	+	0.781328i	$0.714543\pi$
$30$	0	0
$31$	5.61870e10	0.366795	0.183397	−	0.983039i	$-0.441290\pi$
$31$	5.61870e10	0.366795	0.183397	+	0.983039i	$0.441290\pi$
$32$	0	0
$33$	8.12893e10	0.332031
$34$	0	0
$35$	−4.75982e11	−1.25049
$36$	0	0
$37$	6.14765e11	1.06462	0.532312	−	0.846548i	$-0.321323\pi$
$37$	6.14765e11	1.06462	0.532312	+	0.846548i	$0.321323\pi$
$38$	0	0
$39$	6.12304e11	0.714467
$40$	0	0
$41$	5.49860e11	0.440933	0.220467	−	0.975394i	$-0.429242\pi$
$41$	5.49860e11	0.440933	0.220467	+	0.975394i	$0.429242\pi$
$42$	0	0
$43$	9.82884e11	0.551428	0.275714	−	0.961240i	$-0.411086\pi$
$43$	9.82884e11	0.551428	0.275714	+	0.961240i	$0.411086\pi$
$44$	0	0
$45$	−1.05938e12	−0.422628
$46$	0	0
$47$	−2.07614e12	−0.597756	−0.298878	−	0.954291i	$-0.596612\pi$
$47$	−2.07614e12	−0.597756	−0.298878	+	0.954291i	$0.596612\pi$
$48$	0	0
$49$	−1.29361e11	−0.0272478
$50$	0	0
$51$	−5.45200e12	−0.850706
$52$	0	0
$53$	−1.20484e13	−1.40883	−0.704417	−	0.709787i	$-0.748791\pi$
$53$	−1.20484e13	−1.40883	−0.704417	+	0.709787i	$0.748791\pi$
$54$	0	0
$55$	8.23263e12	0.729153
$56$	0	0
$57$	−1.02128e13	−0.691965
$58$	0	0
$59$	−2.30879e13	−1.20780	−0.603899	−	0.797061i	$-0.706387\pi$
$59$	−2.30879e13	−1.20780	−0.603899	+	0.797061i	$0.706387\pi$
$60$	0	0
$61$	−8.50581e12	−0.346531	−0.173265	−	0.984875i	$-0.555432\pi$
$61$	−8.50581e12	−0.346531	−0.173265	+	0.984875i	$0.555432\pi$
$62$	0	0
$63$	1.02786e13	0.328761
$64$	0	0
$65$	6.20115e13	1.56900
$66$	0	0
$67$	1.23310e13	0.248564	0.124282	−	0.992247i	$-0.460337\pi$
$67$	1.23310e13	0.248564	0.124282	+	0.992247i	$0.460337\pi$
$68$	0	0
$69$	−4.03894e13	−0.652979
$70$	0	0
$71$	−5.89892e13	−0.769724	−0.384862	−	0.922974i	$-0.625751\pi$
$71$	−5.89892e13	−0.769724	−0.384862	+	0.922974i	$0.625751\pi$
$72$	0	0
$73$	−5.60983e12	−0.0594331	−0.0297165	−	0.999558i	$-0.509460\pi$
$73$	−5.60983e12	−0.0594331	−0.0297165	+	0.999558i	$0.509460\pi$
$74$	0	0
$75$	−4.05475e13	−0.350756
$76$	0	0
$77$	−7.98769e13	−0.567206
$78$	0	0
$79$	−1.59919e14	−0.936906	−0.468453	−	0.883488i	$-0.655188\pi$
$79$	−1.59919e14	−0.936906	−0.468453	+	0.883488i	$0.655188\pi$
$80$	0	0
$81$	2.28768e13	0.111111
$82$	0	0
$83$	−5.76759e13	−0.233297	−0.116648	−	0.993173i	$-0.537215\pi$
$83$	−5.76759e13	−0.233297	−0.116648	+	0.993173i	$0.537215\pi$
$84$	0	0
$85$	−5.52155e14	−1.86818
$86$	0	0
$87$	2.53590e14	0.720673
$88$	0	0
$89$	−3.62288e14	−0.868217	−0.434109	−	0.900861i	$-0.642937\pi$
$89$	−3.62288e14	−0.868217	−0.434109	+	0.900861i	$0.642937\pi$
$90$	0	0
$91$	−6.01665e14	−1.22052
$92$	0	0
$93$	−1.22881e14	−0.211769
$94$	0	0
$95$	−1.03431e15	−1.51958
$96$	0	0
$97$	−5.39787e14	−0.678319	−0.339160	−	0.940729i	$-0.610143\pi$
$97$	−5.39787e14	−0.678319	−0.339160	+	0.940729i	$0.610143\pi$
$98$	0	0
$99$	−1.77780e14	−0.191698
$100$	0	0
$101$	−4.22221e14	−0.391859	−0.195929	−	0.980618i	$-0.562772\pi$
$101$	−4.22221e14	−0.391859	−0.195929	+	0.980618i	$0.562772\pi$
$102$	0	0
$103$	1.45689e14	0.116720	0.0583602	−	0.998296i	$-0.481413\pi$
$103$	1.45689e14	0.116720	0.0583602	+	0.998296i	$0.481413\pi$
$104$	0	0
$105$	1.04097e15	0.721971
$106$	0	0
$107$	4.08696e14	0.246049	0.123024	−	0.992404i	$-0.460741\pi$
$107$	4.08696e14	0.246049	0.123024	+	0.992404i	$0.460741\pi$
$108$	0	0
$109$	1.10134e15	0.577060	0.288530	−	0.957471i	$-0.406833\pi$
$109$	1.10134e15	0.577060	0.288530	+	0.957471i	$0.406833\pi$
$110$	0	0
$111$	−1.34449e15	−0.614661
$112$	0	0
$113$	−2.13023e15	−0.851802	−0.425901	−	0.904770i	$-0.640043\pi$
$113$	−2.13023e15	−0.851802	−0.425901	+	0.904770i	$0.640043\pi$
$114$	0	0
$115$	−4.09046e15	−1.43397
$116$	0	0
$117$	−1.33911e15	−0.412498
$118$	0	0
$119$	5.35727e15	1.45325
$120$	0	0
$121$	−2.79569e15	−0.669266
$122$	0	0
$123$	−1.20254e15	−0.254573
$124$	0	0
$125$	2.65286e15	0.497610
$126$	0	0
$127$	6.25962e15	1.04236	0.521182	−	0.853445i	$-0.325491\pi$
$127$	6.25962e15	1.04236	0.521182	+	0.853445i	$0.325491\pi$
$128$	0	0
$129$	−2.14957e15	−0.318367
$130$	0	0
$131$	−1.01059e16	−1.33365	−0.666825	−	0.745214i	$-0.732347\pi$
$131$	−1.01059e16	−1.33365	−0.666825	+	0.745214i	$0.732347\pi$
$132$	0	0
$133$	1.00354e16	1.18208
$134$	0	0
$135$	2.31686e15	0.244004
$136$	0	0
$137$	−9.93739e15	−0.937277	−0.468638	−	0.883390i	$-0.655255\pi$
$137$	−9.93739e15	−0.937277	−0.468638	+	0.883390i	$0.655255\pi$
$138$	0	0
$139$	1.95078e16	1.65043	0.825216	−	0.564817i	$-0.191053\pi$
$139$	1.95078e16	1.65043	0.825216	+	0.564817i	$0.191053\pi$
$140$	0	0
$141$	4.54053e15	0.345114
$142$	0	0
$143$	1.04065e16	0.711676
$144$	0	0
$145$	2.56825e16	1.58262
$146$	0	0
$147$	2.82911e14	0.0157315
$148$	0	0
$149$	1.58935e16	0.798589	0.399295	−	0.916823i	$-0.369255\pi$
$149$	1.58935e16	0.798589	0.399295	+	0.916823i	$0.369255\pi$
$150$	0	0
$151$	−3.10597e16	−1.41211	−0.706057	−	0.708155i	$-0.749528\pi$
$151$	−3.10597e16	−1.41211	−0.706057	+	0.708155i	$0.749528\pi$
$152$	0	0
$153$	1.19235e16	0.491155
$154$	0	0
$155$	−1.24449e16	−0.465053
$156$	0	0
$157$	−5.41372e16	−1.83759	−0.918794	−	0.394738i	$-0.870835\pi$
$157$	−5.41372e16	−1.83759	−0.918794	+	0.394738i	$0.870835\pi$
$158$	0	0
$159$	2.63498e16	0.813390
$160$	0	0
$161$	3.96876e16	1.11548
$162$	0	0
$163$	5.54612e16	1.42096	0.710481	−	0.703717i	$-0.248478\pi$
$163$	5.54612e16	1.42096	0.710481	+	0.703717i	$0.248478\pi$
$164$	0	0
$165$	−1.80048e16	−0.420977
$166$	0	0
$167$	5.83552e15	0.124654	0.0623270	−	0.998056i	$-0.480148\pi$
$167$	5.83552e15	0.124654	0.0623270	+	0.998056i	$0.480148\pi$
$168$	0	0
$169$	2.71997e16	0.531390
$170$	0	0
$171$	2.23354e16	0.399506
$172$	0	0
$173$	4.40958e16	0.722855	0.361428	−	0.932400i	$-0.382289\pi$
$173$	4.40958e16	0.722855	0.361428	+	0.932400i	$0.382289\pi$
$174$	0	0
$175$	3.98430e16	0.599193
$176$	0	0
$177$	5.04932e16	0.697323
$178$	0	0
$179$	5.10594e16	0.648153	0.324077	−	0.946031i	$-0.394946\pi$
$179$	5.10594e16	0.648153	0.324077	+	0.946031i	$0.394946\pi$
$180$	0	0
$181$	3.16143e16	0.369228	0.184614	−	0.982811i	$-0.440897\pi$
$181$	3.16143e16	0.369228	0.184614	+	0.982811i	$0.440897\pi$
$182$	0	0
$183$	1.86022e16	0.200070
$184$	0	0
$185$	−1.36164e17	−1.34982
$186$	0	0
$187$	−9.26599e16	−0.847383
$188$	0	0
$189$	−2.24793e16	−0.189810
$190$	0	0
$191$	−9.44291e16	−0.736810	−0.368405	−	0.929665i	$-0.620096\pi$
$191$	−9.44291e16	−0.736810	−0.368405	+	0.929665i	$0.620096\pi$
$192$	0	0
$193$	4.17744e16	0.301460	0.150730	−	0.988575i	$-0.451838\pi$
$193$	4.17744e16	0.301460	0.150730	+	0.988575i	$0.451838\pi$
$194$	0	0
$195$	−1.35619e17	−0.905861
$196$	0	0
$197$	−1.48383e17	−0.918094	−0.459047	−	0.888412i	$-0.651809\pi$
$197$	−1.48383e17	−0.918094	−0.459047	+	0.888412i	$0.651809\pi$
$198$	0	0
$199$	−1.97555e17	−1.13316	−0.566579	−	0.824007i	$-0.691733\pi$
$199$	−1.97555e17	−1.13316	−0.566579	+	0.824007i	$0.691733\pi$
$200$	0	0
$201$	−2.69679e16	−0.143508
$202$	0	0
$203$	−2.49184e17	−1.23112
$204$	0	0
$205$	−1.21788e17	−0.559052
$206$	0	0
$207$	8.83316e16	0.376997
$208$	0	0
$209$	−1.73573e17	−0.689262
$210$	0	0
$211$	4.48570e17	1.65849	0.829244	−	0.558887i	$-0.188772\pi$
$211$	4.48570e17	1.65849	0.829244	+	0.558887i	$0.188772\pi$
$212$	0	0
$213$	1.29009e17	0.444400
$214$	0	0
$215$	−2.17699e17	−0.699146
$216$	0	0
$217$	1.20746e17	0.361763
$218$	0	0
$219$	1.22687e16	0.0343137
$220$	0	0
$221$	−6.97951e17	−1.82341
$222$	0	0
$223$	−1.59977e17	−0.390634	−0.195317	−	0.980740i	$-0.562574\pi$
$223$	−1.59977e17	−0.390634	−0.195317	+	0.980740i	$0.562574\pi$
$224$	0	0
$225$	8.86774e16	0.202509
$226$	0	0
$227$	−1.45803e17	−0.311582	−0.155791	−	0.987790i	$-0.549793\pi$
$227$	−1.45803e17	−0.311582	−0.155791	+	0.987790i	$0.549793\pi$
$228$	0	0
$229$	6.81194e17	1.36303	0.681514	−	0.731805i	$-0.261322\pi$
$229$	6.81194e17	1.36303	0.681514	+	0.731805i	$0.261322\pi$
$230$	0	0
$231$	1.74691e17	0.327476
$232$	0	0
$233$	−3.89035e17	−0.683628	−0.341814	−	0.939768i	$-0.611041\pi$
$233$	−3.89035e17	−0.683628	−0.341814	+	0.939768i	$0.611041\pi$
$234$	0	0
$235$	4.59845e17	0.757884
$236$	0	0
$237$	3.49742e17	0.540923
$238$	0	0
$239$	−6.76227e16	−0.0981993	−0.0490996	−	0.998794i	$-0.515635\pi$
$239$	−6.76227e16	−0.0981993	−0.0490996	+	0.998794i	$0.515635\pi$
$240$	0	0
$241$	−9.00528e17	−1.22848	−0.614242	−	0.789118i	$-0.710538\pi$
$241$	−9.00528e17	−1.22848	−0.614242	+	0.789118i	$0.710538\pi$
$242$	0	0
$243$	−5.00315e16	−0.0641500
$244$	0	0
$245$	2.86521e16	0.0345470
$246$	0	0
$247$	−1.30742e18	−1.48316
$248$	0	0
$249$	1.26137e17	0.134694
$250$	0	0
$251$	−1.58929e18	−1.59827	−0.799134	−	0.601153i	$-0.794708\pi$
$251$	−1.58929e18	−1.59827	−0.799134	+	0.601153i	$0.794708\pi$
$252$	0	0
$253$	−6.86440e17	−0.650428
$254$	0	0
$255$	1.20756e18	1.07860
$256$	0	0
$257$	2.29738e18	1.93524	0.967618	−	0.252418i	$-0.0812258\pi$
$257$	2.29738e18	1.93524	0.967618	+	0.252418i	$0.0812258\pi$
$258$	0	0
$259$	1.32113e18	1.05002
$260$	0	0
$261$	−5.54602e17	−0.416081
$262$	0	0
$263$	5.05657e17	0.358252	0.179126	−	0.983826i	$-0.442673\pi$
$263$	5.05657e17	0.358252	0.179126	+	0.983826i	$0.442673\pi$
$264$	0	0
$265$	2.66860e18	1.78624
$266$	0	0
$267$	7.92323e17	0.501265
$268$	0	0
$269$	−2.44731e18	−1.46402	−0.732009	−	0.681295i	$-0.761417\pi$
$269$	−2.44731e18	−1.46402	−0.732009	+	0.681295i	$0.761417\pi$
$270$	0	0
$271$	1.53807e18	0.870373	0.435187	−	0.900340i	$-0.356682\pi$
$271$	1.53807e18	0.870373	0.435187	+	0.900340i	$0.356682\pi$
$272$	0	0
$273$	1.31584e18	0.704666
$274$	0	0
$275$	−6.89128e17	−0.349385
$276$	0	0
$277$	−7.89959e17	−0.379321	−0.189660	−	0.981850i	$-0.560739\pi$
$277$	−7.89959e17	−0.379321	−0.189660	+	0.981850i	$0.560739\pi$
$278$	0	0
$279$	2.68741e17	0.122265
$280$	0	0
$281$	−6.15303e17	−0.265333	−0.132667	−	0.991161i	$-0.542354\pi$
$281$	−6.15303e17	−0.265333	−0.132667	+	0.991161i	$0.542354\pi$
$282$	0	0
$283$	1.26279e18	0.516337	0.258169	−	0.966100i	$-0.416881\pi$
$283$	1.26279e18	0.516337	0.258169	+	0.966100i	$0.416881\pi$
$284$	0	0
$285$	2.26204e18	0.877330
$286$	0	0
$287$	1.18165e18	0.434885
$288$	0	0
$289$	3.35219e18	1.17110
$290$	0	0
$291$	1.18051e18	0.391628
$292$	0	0
$293$	−3.49864e18	−1.10253	−0.551267	−	0.834329i	$-0.685855\pi$
$293$	−3.49864e18	−1.10253	−0.551267	+	0.834329i	$0.685855\pi$
$294$	0	0
$295$	5.11374e18	1.53135
$296$	0	0
$297$	3.88804e17	0.110677
$298$	0	0
$299$	−5.17055e18	−1.39960
$300$	0	0
$301$	2.11222e18	0.543864
$302$	0	0
$303$	9.23398e17	0.226240
$304$	0	0
$305$	1.88395e18	0.439360
$306$	0	0
$307$	−1.28861e18	−0.286142	−0.143071	−	0.989712i	$-0.545698\pi$
$307$	−1.28861e18	−0.286142	−0.143071	+	0.989712i	$0.545698\pi$
$308$	0	0
$309$	−3.18621e17	−0.0673886
$310$	0	0
$311$	−8.43030e18	−1.69879	−0.849396	−	0.527756i	$-0.823033\pi$
$311$	−8.43030e18	−1.69879	−0.849396	+	0.527756i	$0.823033\pi$
$312$	0	0
$313$	7.88359e18	1.51405	0.757027	−	0.653383i	$-0.226651\pi$
$313$	7.88359e18	1.51405	0.757027	+	0.653383i	$0.226651\pi$
$314$	0	0
$315$	−2.27661e18	−0.416830
$316$	0	0
$317$	−8.71539e18	−1.52175	−0.760873	−	0.648900i	$-0.775229\pi$
$317$	−8.71539e18	−1.52175	−0.760873	+	0.648900i	$0.775229\pi$
$318$	0	0
$319$	4.30991e18	0.717857
$320$	0	0
$321$	−8.93817e17	−0.142056
$322$	0	0
$323$	1.16414e19	1.76598
$324$	0	0
$325$	−5.19079e18	−0.751810
$326$	0	0
$327$	−2.40862e18	−0.333166
$328$	0	0
$329$	−4.46163e18	−0.589556
$330$	0	0
$331$	−1.26995e19	−1.60353	−0.801763	−	0.597643i	$-0.796104\pi$
$331$	−1.26995e19	−1.60353	−0.801763	+	0.597643i	$0.796104\pi$
$332$	0	0
$333$	2.94040e18	0.354875
$334$	0	0
$335$	−2.73120e18	−0.315150
$336$	0	0
$337$	−6.64250e18	−0.733005	−0.366503	−	0.930417i	$-0.619445\pi$
$337$	−6.64250e18	−0.733005	−0.366503	+	0.930417i	$0.619445\pi$
$338$	0	0
$339$	4.65882e18	0.491788
$340$	0	0
$341$	−2.08843e18	−0.210942
$342$	0	0
$343$	−1.04805e19	−1.01316
$344$	0	0
$345$	8.94584e18	0.827900
$346$	0	0
$347$	−1.87835e18	−0.166458	−0.0832289	−	0.996530i	$-0.526523\pi$
$347$	−1.87835e18	−0.166458	−0.0832289	+	0.996530i	$0.526523\pi$
$348$	0	0
$349$	−1.48545e19	−1.26086	−0.630432	−	0.776245i	$-0.717122\pi$
$349$	−1.48545e19	−1.26086	−0.630432	+	0.776245i	$0.717122\pi$
$350$	0	0
$351$	2.92863e18	0.238156
$352$	0	0
$353$	−8.23380e18	−0.641638	−0.320819	−	0.947140i	$-0.603958\pi$
$353$	−8.23380e18	−0.641638	−0.320819	+	0.947140i	$0.603958\pi$
$354$	0	0
$355$	1.30655e19	0.975919
$356$	0	0
$357$	−1.17163e19	−0.839036
$358$	0	0
$359$	−1.87417e19	−1.28706	−0.643532	−	0.765419i	$-0.722532\pi$
$359$	−1.87417e19	−1.28706	−0.643532	+	0.765419i	$0.722532\pi$
$360$	0	0
$361$	6.62574e18	0.436446
$362$	0	0
$363$	6.11417e18	0.386401
$364$	0	0
$365$	1.24252e18	0.0753542
$366$	0	0
$367$	−8.63117e18	−0.502428	−0.251214	−	0.967932i	$-0.580830\pi$
$367$	−8.63117e18	−0.502428	−0.251214	+	0.967932i	$0.580830\pi$
$368$	0	0
$369$	2.62996e18	0.146978
$370$	0	0
$371$	−2.58920e19	−1.38951
$372$	0	0
$373$	−9.95975e18	−0.513372	−0.256686	−	0.966495i	$-0.582631\pi$
$373$	−9.95975e18	−0.513372	−0.256686	+	0.966495i	$0.582631\pi$
$374$	0	0
$375$	−5.80181e18	−0.287295
$376$	0	0
$377$	3.24640e19	1.54469
$378$	0	0
$379$	−1.00949e19	−0.461645	−0.230823	−	0.972996i	$-0.574142\pi$
$379$	−1.00949e19	−0.461645	−0.230823	+	0.972996i	$0.574142\pi$
$380$	0	0
$381$	−1.36898e19	−0.601810
$382$	0	0
$383$	2.71764e19	1.14869	0.574343	−	0.818614i	$-0.305257\pi$
$383$	2.71764e19	1.14869	0.574343	+	0.818614i	$0.305257\pi$
$384$	0	0
$385$	1.76919e19	0.719150
$386$	0	0
$387$	4.70111e18	0.183809
$388$	0	0
$389$	−9.05285e18	−0.340536	−0.170268	−	0.985398i	$-0.554463\pi$
$389$	−9.05285e18	−0.340536	−0.170268	+	0.985398i	$0.554463\pi$
$390$	0	0
$391$	4.60389e19	1.66648
$392$	0	0
$393$	2.21017e19	0.769984
$394$	0	0
$395$	3.54204e19	1.18789
$396$	0	0
$397$	−1.76495e19	−0.569908	−0.284954	−	0.958541i	$-0.591978\pi$
$397$	−1.76495e19	−0.569908	−0.284954	+	0.958541i	$0.591978\pi$
$398$	0	0
$399$	−2.19473e19	−0.682472
$400$	0	0
$401$	2.81135e19	0.842040	0.421020	−	0.907051i	$-0.361672\pi$
$401$	2.81135e19	0.842040	0.421020	+	0.907051i	$0.361672\pi$
$402$	0	0
$403$	−1.57309e19	−0.453906
$404$	0	0
$405$	−5.06698e18	−0.140876
$406$	0	0
$407$	−2.28504e19	−0.612260
$408$	0	0
$409$	6.42955e19	1.66056	0.830282	−	0.557343i	$-0.188179\pi$
$409$	6.42955e19	1.66056	0.830282	+	0.557343i	$0.188179\pi$
$410$	0	0
$411$	2.17331e19	0.541137
$412$	0	0
$413$	−4.96159e19	−1.19123
$414$	0	0
$415$	1.27746e19	0.295793
$416$	0	0
$417$	−4.26636e19	−0.952878
$418$	0	0
$419$	2.34474e19	0.505230	0.252615	−	0.967567i	$-0.418709\pi$
$419$	2.34474e19	0.505230	0.252615	+	0.967567i	$0.418709\pi$
$420$	0	0
$421$	−7.28227e19	−1.51409	−0.757044	−	0.653364i	$-0.773357\pi$
$421$	−7.28227e19	−1.51409	−0.757044	+	0.653364i	$0.773357\pi$
$422$	0	0
$423$	−9.93013e18	−0.199252
$424$	0	0
$425$	4.62192e19	0.895170
$426$	0	0
$427$	−1.82790e19	−0.341777
$428$	0	0
$429$	−2.27589e19	−0.410886
$430$	0	0
$431$	4.71831e19	0.822635	0.411317	−	0.911492i	$-0.365069\pi$
$431$	4.71831e19	0.822635	0.411317	+	0.911492i	$0.365069\pi$
$432$	0	0
$433$	3.79099e19	0.638400	0.319200	−	0.947687i	$-0.396586\pi$
$433$	3.79099e19	0.638400	0.319200	+	0.947687i	$0.396586\pi$
$434$	0	0
$435$	−5.61677e19	−0.913728
$436$	0	0
$437$	8.62412e19	1.35551
$438$	0	0
$439$	2.26832e19	0.344525	0.172262	−	0.985051i	$-0.444892\pi$
$439$	2.26832e19	0.344525	0.172262	+	0.985051i	$0.444892\pi$
$440$	0	0
$441$	−6.18727e17	−0.00908259
$442$	0	0
$443$	9.94888e19	1.41171	0.705856	−	0.708355i	$-0.250562\pi$
$443$	9.94888e19	1.41171	0.705856	+	0.708355i	$0.250562\pi$
$444$	0	0
$445$	8.02431e19	1.10080
$446$	0	0
$447$	−3.47592e19	−0.461066
$448$	0	0
$449$	1.30159e20	1.66965	0.834825	−	0.550515i	$-0.185568\pi$
$449$	1.30159e20	1.66965	0.834825	+	0.550515i	$0.185568\pi$
$450$	0	0
$451$	−2.04379e19	−0.253579
$452$	0	0
$453$	6.79275e19	0.815285
$454$	0	0
$455$	1.33263e20	1.54747
$456$	0	0
$457$	−1.01910e20	−1.14510	−0.572550	−	0.819870i	$-0.694046\pi$
$457$	−1.01910e20	−1.14510	−0.572550	+	0.819870i	$0.694046\pi$
$458$	0	0
$459$	−2.60767e19	−0.283569
$460$	0	0
$461$	−1.08142e20	−1.13825	−0.569123	−	0.822253i	$-0.692717\pi$
$461$	−1.08142e20	−1.13825	−0.569123	+	0.822253i	$0.692717\pi$
$462$	0	0
$463$	−1.69113e20	−1.72314	−0.861569	−	0.507641i	$-0.830518\pi$
$463$	−1.69113e20	−1.72314	−0.861569	+	0.507641i	$0.830518\pi$
$464$	0	0
$465$	2.72169e19	0.268498
$466$	0	0
$467$	4.58418e19	0.437910	0.218955	−	0.975735i	$-0.429735\pi$
$467$	4.58418e19	0.437910	0.218955	+	0.975735i	$0.429735\pi$
$468$	0	0
$469$	2.64993e19	0.245154
$470$	0	0
$471$	1.18398e20	1.06093
$472$	0	0
$473$	−3.65331e19	−0.317124
$474$	0	0
$475$	8.65789e19	0.728132
$476$	0	0
$477$	−5.76270e19	−0.469611
$478$	0	0
$479$	−1.31513e20	−1.03861	−0.519303	−	0.854590i	$-0.673808\pi$
$479$	−1.31513e20	−1.03861	−0.519303	+	0.854590i	$0.673808\pi$
$480$	0	0
$481$	−1.72118e20	−1.31747
$482$	0	0
$483$	−8.67968e19	−0.644021
$484$	0	0
$485$	1.19557e20	0.860029
$486$	0	0
$487$	−6.13671e19	−0.428024	−0.214012	−	0.976831i	$-0.568653\pi$
$487$	−6.13671e19	−0.428024	−0.214012	+	0.976831i	$0.568653\pi$
$488$	0	0
$489$	−1.21294e20	−0.820392
$490$	0	0
$491$	−2.42057e20	−1.58784	−0.793918	−	0.608024i	$-0.791962\pi$
$491$	−2.42057e20	−1.58784	−0.793918	+	0.608024i	$0.791962\pi$
$492$	0	0
$493$	−2.89062e20	−1.83924
$494$	0	0
$495$	3.93764e19	0.243051
$496$	0	0
$497$	−1.26768e20	−0.759165
$498$	0	0
$499$	−1.46049e20	−0.848680	−0.424340	−	0.905503i	$-0.639494\pi$
$499$	−1.46049e20	−0.848680	−0.424340	+	0.905503i	$0.639494\pi$
$500$	0	0
$501$	−1.27623e19	−0.0719690
$502$	0	0
$503$	1.68041e20	0.919720	0.459860	−	0.887991i	$-0.347900\pi$
$503$	1.68041e20	0.919720	0.459860	+	0.887991i	$0.347900\pi$
$504$	0	0
$505$	9.35178e19	0.496831
$506$	0	0
$507$	−5.94857e19	−0.306798
$508$	0	0
$509$	7.37709e19	0.369404	0.184702	−	0.982795i	$-0.440868\pi$
$509$	7.37709e19	0.369404	0.184702	+	0.982795i	$0.440868\pi$
$510$	0	0
$511$	−1.20555e19	−0.0586178
$512$	0	0
$513$	−4.88476e19	−0.230655
$514$	0	0
$515$	−3.22686e19	−0.147988
$516$	0	0
$517$	7.71689e19	0.343766
$518$	0	0
$519$	−9.64375e19	−0.417341
$520$	0	0
$521$	3.22023e20	1.35396	0.676978	−	0.736003i	$-0.263289\pi$
$521$	3.22023e20	1.35396	0.676978	+	0.736003i	$0.263289\pi$
$522$	0	0
$523$	1.19064e19	0.0486429	0.0243214	−	0.999704i	$-0.492257\pi$
$523$	1.19064e19	0.0486429	0.0243214	+	0.999704i	$0.492257\pi$
$524$	0	0
$525$	−8.71366e19	−0.345944
$526$	0	0
$527$	1.40069e20	0.540460
$528$	0	0
$529$	7.44293e19	0.279143
$530$	0	0
$531$	−1.10429e20	−0.402599
$532$	0	0
$533$	−1.53947e20	−0.545652
$534$	0	0
$535$	−9.05220e19	−0.311961
$536$	0	0
$537$	−1.11667e20	−0.374211
$538$	0	0
$539$	4.80824e18	0.0156701
$540$	0	0
$541$	1.59033e20	0.504089	0.252045	−	0.967716i	$-0.418897\pi$
$541$	1.59033e20	0.504089	0.252045	+	0.967716i	$0.418897\pi$
$542$	0	0
$543$	−6.91405e19	−0.213174
$544$	0	0
$545$	−2.43935e20	−0.731645
$546$	0	0
$547$	4.25405e20	1.24136	0.620680	−	0.784064i	$-0.286857\pi$
$547$	4.25405e20	1.24136	0.620680	+	0.784064i	$0.286857\pi$
$548$	0	0
$549$	−4.06830e19	−0.115510
$550$	0	0
$551$	−5.41477e20	−1.49604
$552$	0	0
$553$	−3.43665e20	−0.924054
$554$	0	0
$555$	2.97791e20	0.779318
$556$	0	0
$557$	2.71579e20	0.691803	0.345901	−	0.938271i	$-0.387573\pi$
$557$	2.71579e20	0.691803	0.345901	+	0.938271i	$0.387573\pi$
$558$	0	0
$559$	−2.75182e20	−0.682389
$560$	0	0
$561$	2.02647e20	0.489237
$562$	0	0
$563$	2.68438e20	0.631002	0.315501	−	0.948925i	$-0.397827\pi$
$563$	2.68438e20	0.631002	0.315501	+	0.948925i	$0.397827\pi$
$564$	0	0
$565$	4.71825e20	1.07999
$566$	0	0
$567$	4.91622e19	0.109587
$568$	0	0
$569$	−1.34893e20	−0.292853	−0.146426	−	0.989222i	$-0.546777\pi$
$569$	−1.34893e20	−0.292853	−0.146426	+	0.989222i	$0.546777\pi$
$570$	0	0
$571$	6.81153e20	1.44037	0.720185	−	0.693782i	$-0.244057\pi$
$571$	6.81153e20	1.44037	0.720185	+	0.693782i	$0.244057\pi$
$572$	0	0
$573$	2.06517e20	0.425398
$574$	0	0
$575$	3.42400e20	0.687108
$576$	0	0
$577$	−2.42927e20	−0.474961	−0.237480	−	0.971392i	$-0.576322\pi$
$577$	−2.42927e20	−0.474961	−0.237480	+	0.971392i	$0.576322\pi$
$578$	0	0
$579$	−9.13605e19	−0.174048
$580$	0	0
$581$	−1.23945e20	−0.230096
$582$	0	0
$583$	4.47830e20	0.810213
$584$	0	0
$585$	2.96599e20	0.522999
$586$	0	0
$587$	1.06241e20	0.182602	0.0913012	−	0.995823i	$-0.470897\pi$
$587$	1.06241e20	0.182602	0.0913012	+	0.995823i	$0.470897\pi$
$588$	0	0
$589$	2.62381e20	0.439610
$590$	0	0
$591$	3.24513e20	0.530062
$592$	0	0
$593$	−7.70415e20	−1.22692	−0.613458	−	0.789727i	$-0.710222\pi$
$593$	−7.70415e20	−1.22692	−0.613458	+	0.789727i	$0.710222\pi$
$594$	0	0
$595$	−1.18658e21	−1.84255
$596$	0	0
$597$	4.32053e20	0.654229
$598$	0	0
$599$	7.59796e19	0.112201	0.0561004	−	0.998425i	$-0.482133\pi$
$599$	7.59796e19	0.112201	0.0561004	+	0.998425i	$0.482133\pi$
$600$	0	0
$601$	7.78003e20	1.12053	0.560264	−	0.828314i	$-0.310700\pi$
$601$	7.78003e20	1.12053	0.560264	+	0.828314i	$0.310700\pi$
$602$	0	0
$603$	5.89788e19	0.0828546
$604$	0	0
$605$	6.19217e20	0.848551
$606$	0	0
$607$	−3.21867e20	−0.430290	−0.215145	−	0.976582i	$-0.569022\pi$
$607$	−3.21867e20	−0.430290	−0.215145	+	0.976582i	$0.569022\pi$
$608$	0	0
$609$	5.44965e20	0.710787
$610$	0	0
$611$	5.81267e20	0.739719
$612$	0	0
$613$	6.68867e20	0.830589	0.415294	−	0.909687i	$-0.363679\pi$
$613$	6.68867e20	0.830589	0.415294	+	0.909687i	$0.363679\pi$
$614$	0	0
$615$	2.66351e20	0.322769
$616$	0	0
$617$	−8.47904e20	−1.00279	−0.501393	−	0.865220i	$-0.667179\pi$
$617$	−8.47904e20	−1.00279	−0.501393	+	0.865220i	$0.667179\pi$
$618$	0	0
$619$	3.51027e20	0.405192	0.202596	−	0.979262i	$-0.435062\pi$
$619$	3.51027e20	0.405192	0.202596	+	0.979262i	$0.435062\pi$
$620$	0	0
$621$	−1.93181e20	−0.217660
$622$	0	0
$623$	−7.78556e20	−0.856307
$624$	0	0
$625$	−1.15339e21	−1.23844
$626$	0	0
$627$	3.79603e20	0.397945
$628$	0	0
$629$	1.53256e21	1.56869
$630$	0	0
$631$	1.30718e21	1.30652	0.653260	−	0.757134i	$-0.273401\pi$
$631$	1.30718e21	1.30652	0.653260	+	0.757134i	$0.273401\pi$
$632$	0	0
$633$	−9.81023e20	−0.957528
$634$	0	0
$635$	−1.38644e21	−1.32160
$636$	0	0
$637$	3.62176e19	0.0337190
$638$	0	0
$639$	−2.82143e20	−0.256575
$640$	0	0
$641$	−1.77975e21	−1.58097	−0.790486	−	0.612480i	$-0.790172\pi$
$641$	−1.77975e21	−1.58097	−0.790486	+	0.612480i	$0.790172\pi$
$642$	0	0
$643$	5.18269e19	0.0449752	0.0224876	−	0.999747i	$-0.492841\pi$
$643$	5.18269e19	0.0449752	0.0224876	+	0.999747i	$0.492841\pi$
$644$	0	0
$645$	4.76108e20	0.403652
$646$	0	0
$647$	−1.80268e21	−1.49327	−0.746634	−	0.665235i	$-0.768331\pi$
$647$	−1.80268e21	−1.49327	−0.746634	+	0.665235i	$0.768331\pi$
$648$	0	0
$649$	8.58162e20	0.694599
$650$	0	0
$651$	−2.64071e20	−0.208864
$652$	0	0
$653$	6.01117e20	0.464633	0.232317	−	0.972640i	$-0.425369\pi$
$653$	6.01117e20	0.464633	0.232317	+	0.972640i	$0.425369\pi$
$654$	0	0
$655$	2.23837e21	1.69091
$656$	0	0
$657$	−2.68316e19	−0.0198110
$658$	0	0
$659$	1.34492e21	0.970635	0.485318	−	0.874338i	$-0.338704\pi$
$659$	1.34492e21	0.970635	0.485318	+	0.874338i	$0.338704\pi$
$660$	0	0
$661$	3.00301e20	0.211858	0.105929	−	0.994374i	$-0.466218\pi$
$661$	3.00301e20	0.211858	0.105929	+	0.994374i	$0.466218\pi$
$662$	0	0
$663$	1.52642e21	1.05274
$664$	0	0
$665$	−2.22273e21	−1.49873
$666$	0	0
$667$	−2.14142e21	−1.41175
$668$	0	0
$669$	3.49869e20	0.225533
$670$	0	0
$671$	3.16155e20	0.199288
$672$	0	0
$673$	−2.48252e21	−1.53031	−0.765156	−	0.643845i	$-0.777338\pi$
$673$	−2.48252e21	−1.53031	−0.765156	+	0.643845i	$0.777338\pi$
$674$	0	0
$675$	−1.93937e20	−0.116919
$676$	0	0
$677$	−1.84000e21	−1.08493	−0.542467	−	0.840077i	$-0.682509\pi$
$677$	−1.84000e21	−1.08493	−0.542467	+	0.840077i	$0.682509\pi$
$678$	0	0
$679$	−1.16000e21	−0.669014
$680$	0	0
$681$	3.18871e20	0.179892
$682$	0	0
$683$	−2.19144e21	−1.20941	−0.604705	−	0.796450i	$-0.706709\pi$
$683$	−2.19144e21	−1.20941	−0.604705	+	0.796450i	$0.706709\pi$
$684$	0	0
$685$	2.20103e21	1.18836
$686$	0	0
$687$	−1.48977e21	−0.786945
$688$	0	0
$689$	3.37324e21	1.74342
$690$	0	0
$691$	5.25387e20	0.265701	0.132851	−	0.991136i	$-0.457587\pi$
$691$	5.25387e20	0.265701	0.132851	+	0.991136i	$0.457587\pi$
$692$	0	0
$693$	−3.82049e20	−0.189069
$694$	0	0
$695$	−4.32079e21	−2.09256
$696$	0	0
$697$	1.37075e21	0.649701
$698$	0	0
$699$	8.50820e20	0.394693
$700$	0	0
$701$	−3.78227e21	−1.71739	−0.858693	−	0.512491i	$-0.828723\pi$
$701$	−3.78227e21	−1.71739	−0.858693	+	0.512491i	$0.828723\pi$
$702$	0	0
$703$	2.87082e21	1.27597
$704$	0	0
$705$	−1.00568e21	−0.437564
$706$	0	0
$707$	−9.07353e20	−0.386483
$708$	0	0
$709$	−4.24647e21	−1.77085	−0.885424	−	0.464784i	$-0.846132\pi$
$709$	−4.24647e21	−1.77085	−0.885424	+	0.464784i	$0.846132\pi$
$710$	0	0
$711$	−7.64886e20	−0.312302
$712$	0	0
$713$	1.03766e21	0.414842
$714$	0	0
$715$	−2.30493e21	−0.902322
$716$	0	0
$717$	1.47891e20	0.0566954
$718$	0	0
$719$	3.17360e20	0.119148	0.0595738	−	0.998224i	$-0.481026\pi$
$719$	3.17360e20	0.119148	0.0595738	+	0.998224i	$0.481026\pi$
$720$	0	0
$721$	3.13085e20	0.115119
$722$	0	0
$723$	1.96946e21	0.709266
$724$	0	0
$725$	−2.14981e21	−0.758340
$726$	0	0
$727$	−1.25112e21	−0.432307	−0.216154	−	0.976359i	$-0.569351\pi$
$727$	−1.25112e21	−0.432307	−0.216154	+	0.976359i	$0.569351\pi$
$728$	0	0
$729$	1.09419e20	0.0370370
$730$	0	0
$731$	2.45025e21	0.812511
$732$	0	0
$733$	4.26993e21	1.38721	0.693603	−	0.720357i	$-0.256022\pi$
$733$	4.26993e21	1.38721	0.693603	+	0.720357i	$0.256022\pi$
$734$	0	0
$735$	−6.26621e19	−0.0199457
$736$	0	0
$737$	−4.58335e20	−0.142948
$738$	0	0
$739$	−1.73302e21	−0.529627	−0.264813	−	0.964300i	$-0.585310\pi$
$739$	−1.73302e21	−0.529627	−0.264813	+	0.964300i	$0.585310\pi$
$740$	0	0
$741$	2.85933e21	0.856302
$742$	0	0
$743$	2.63739e21	0.774032	0.387016	−	0.922073i	$-0.373506\pi$
$743$	2.63739e21	0.774032	0.387016	+	0.922073i	$0.373506\pi$
$744$	0	0
$745$	−3.52026e21	−1.01252
$746$	0	0
$747$	−2.75862e20	−0.0777656
$748$	0	0
$749$	8.78287e20	0.242674
$750$	0	0
$751$	−2.27078e21	−0.615001	−0.307500	−	0.951548i	$-0.599493\pi$
$751$	−2.27078e21	−0.615001	−0.307500	+	0.951548i	$0.599493\pi$
$752$	0	0
$753$	3.47577e21	0.922761
$754$	0	0
$755$	6.87941e21	1.79040
$756$	0	0
$757$	−1.72959e21	−0.441291	−0.220645	−	0.975354i	$-0.570816\pi$
$757$	−1.72959e21	−0.441291	−0.220645	+	0.975354i	$0.570816\pi$
$758$	0	0
$759$	1.50125e21	0.375525
$760$	0	0
$761$	−2.06739e20	−0.0507035	−0.0253518	−	0.999679i	$-0.508071\pi$
$761$	−2.06739e20	−0.0507035	−0.0253518	+	0.999679i	$0.508071\pi$
$762$	0	0
$763$	2.36677e21	0.569144
$764$	0	0
$765$	−2.64094e21	−0.622727
$766$	0	0
$767$	6.46402e21	1.49464
$768$	0	0
$769$	3.01490e21	0.683636	0.341818	−	0.939766i	$-0.388957\pi$
$769$	3.01490e21	0.683636	0.341818	+	0.939766i	$0.388957\pi$
$770$	0	0
$771$	−5.02437e21	−1.11731
$772$	0	0
$773$	8.99357e21	1.96149	0.980745	−	0.195293i	$-0.0625658\pi$
$773$	8.99357e21	1.96149	0.980745	+	0.195293i	$0.0625658\pi$
$774$	0	0
$775$	1.04172e21	0.222838
$776$	0	0
$777$	−2.88931e21	−0.606229
$778$	0	0
$779$	2.56773e21	0.528467
$780$	0	0
$781$	2.19259e21	0.442664
$782$	0	0
$783$	1.21291e21	0.240224
$784$	0	0
$785$	1.19908e22	2.32985
$786$	0	0
$787$	5.92807e21	1.13006	0.565032	−	0.825069i	$-0.308864\pi$
$787$	5.92807e21	1.13006	0.565032	+	0.825069i	$0.308864\pi$
$788$	0	0
$789$	−1.10587e21	−0.206837
$790$	0	0
$791$	−4.57787e21	−0.840117
$792$	0	0
$793$	2.38141e21	0.428830
$794$	0	0
$795$	−5.83622e21	−1.03128
$796$	0	0
$797$	−3.83837e21	−0.665593	−0.332797	−	0.942999i	$-0.607992\pi$
$797$	−3.83837e21	−0.665593	−0.332797	+	0.942999i	$0.607992\pi$
$798$	0	0
$799$	−5.17565e21	−0.880773
$800$	0	0
$801$	−1.73281e21	−0.289406
$802$	0	0
$803$	2.08513e20	0.0341796
$804$	0	0
$805$	−8.79040e21	−1.41429
$806$	0	0
$807$	5.35226e21	0.845251
$808$	0	0
$809$	−6.17948e21	−0.957940	−0.478970	−	0.877831i	$-0.658990\pi$
$809$	−6.17948e21	−0.957940	−0.478970	+	0.877831i	$0.658990\pi$
$810$	0	0
$811$	−1.87862e21	−0.285879	−0.142939	−	0.989731i	$-0.545655\pi$
$811$	−1.87862e21	−0.285879	−0.142939	+	0.989731i	$0.545655\pi$
$812$	0	0
$813$	−3.36375e21	−0.502510
$814$	0	0
$815$	−1.22841e22	−1.80161
$816$	0	0
$817$	4.58986e21	0.660897
$818$	0	0
$819$	−2.87774e21	−0.406839
$820$	0	0
$821$	3.93227e21	0.545845	0.272923	−	0.962036i	$-0.412010\pi$
$821$	3.93227e21	0.545845	0.272923	+	0.962036i	$0.412010\pi$
$822$	0	0
$823$	7.92927e21	1.08077	0.540386	−	0.841417i	$-0.318278\pi$
$823$	7.92927e21	1.08077	0.540386	+	0.841417i	$0.318278\pi$
$824$	0	0
$825$	1.50712e21	0.201718
$826$	0	0
$827$	1.97038e21	0.258976	0.129488	−	0.991581i	$-0.458667\pi$
$827$	1.97038e21	0.258976	0.129488	+	0.991581i	$0.458667\pi$
$828$	0	0
$829$	2.43930e21	0.314851	0.157426	−	0.987531i	$-0.449681\pi$
$829$	2.43930e21	0.314851	0.157426	+	0.987531i	$0.449681\pi$
$830$	0	0
$831$	1.72764e21	0.219001
$832$	0	0
$833$	−3.22484e20	−0.0401487
$834$	0	0
$835$	−1.29251e21	−0.158047
$836$	0	0
$837$	−5.87736e20	−0.0705897
$838$	0	0
$839$	1.35318e22	1.59639	0.798196	−	0.602398i	$-0.205788\pi$
$839$	1.35318e22	1.59639	0.798196	+	0.602398i	$0.205788\pi$
$840$	0	0
$841$	4.81601e21	0.558107
$842$	0	0
$843$	1.34567e21	0.153190
$844$	0	0
$845$	−6.02446e21	−0.673741
$846$	0	0
$847$	−6.00794e21	−0.660085
$848$	0	0
$849$	−2.76173e21	−0.298107
$850$	0	0
$851$	1.13534e22	1.20408
$852$	0	0
$853$	−7.30038e21	−0.760725	−0.380363	−	0.924837i	$-0.624201\pi$
$853$	−7.30038e21	−0.760725	−0.380363	+	0.924837i	$0.624201\pi$
$854$	0	0
$855$	−4.94707e21	−0.506527
$856$	0	0
$857$	1.16475e22	1.17187	0.585934	−	0.810359i	$-0.300728\pi$
$857$	1.16475e22	1.17187	0.585934	+	0.810359i	$0.300728\pi$
$858$	0	0
$859$	1.05473e22	1.04278	0.521388	−	0.853320i	$-0.325414\pi$
$859$	1.05473e22	1.04278	0.521388	+	0.853320i	$0.325414\pi$
$860$	0	0
$861$	−2.58427e21	−0.251081
$862$	0	0
$863$	−1.25514e22	−1.19843	−0.599215	−	0.800588i	$-0.704520\pi$
$863$	−1.25514e22	−1.19843	−0.599215	+	0.800588i	$0.704520\pi$
$864$	0	0
$865$	−9.76678e21	−0.916496
$866$	0	0
$867$	−7.33124e21	−0.676136
$868$	0	0
$869$	5.94407e21	0.538810
$870$	0	0
$871$	−3.45237e21	−0.307596
$872$	0	0
$873$	−2.58178e21	−0.226106
$874$	0	0
$875$	5.70100e21	0.490784
$876$	0	0
$877$	−1.57294e22	−1.33112	−0.665558	−	0.746346i	$-0.731807\pi$
$877$	−1.57294e22	−1.33112	−0.665558	+	0.746346i	$0.731807\pi$
$878$	0	0
$879$	7.65152e21	0.636548
$880$	0	0
$881$	1.80412e22	1.47553	0.737764	−	0.675059i	$-0.235882\pi$
$881$	1.80412e22	1.47553	0.737764	+	0.675059i	$0.235882\pi$
$882$	0	0
$883$	−2.41866e22	−1.94478	−0.972388	−	0.233371i	$-0.925024\pi$
$883$	−2.41866e22	−1.94478	−0.972388	+	0.233371i	$0.925024\pi$
$884$	0	0
$885$	−1.11837e22	−0.884123
$886$	0	0
$887$	−7.05392e21	−0.548282	−0.274141	−	0.961690i	$-0.588393\pi$
$887$	−7.05392e21	−0.548282	−0.274141	+	0.961690i	$0.588393\pi$
$888$	0	0
$889$	1.34519e22	1.02807
$890$	0	0
$891$	−8.50315e20	−0.0638994
$892$	0	0
$893$	−9.69514e21	−0.716421
$894$	0	0
$895$	−1.13091e22	−0.821782
$896$	0	0
$897$	1.13080e22	0.808057
$898$	0	0
$899$	−6.51508e21	−0.457849
$900$	0	0
$901$	−3.00356e22	−2.07587
$902$	0	0
$903$	−4.61942e21	−0.314000
$904$	0	0
$905$	−7.00226e21	−0.468138
$906$	0	0
$907$	−2.48498e22	−1.63406	−0.817030	−	0.576596i	$-0.804381\pi$
$907$	−2.48498e22	−1.63406	−0.817030	+	0.576596i	$0.804381\pi$
$908$	0	0
$909$	−2.01947e21	−0.130620
$910$	0	0
$911$	2.83848e21	0.180592	0.0902958	−	0.995915i	$-0.471219\pi$
$911$	2.83848e21	0.180592	0.0902958	+	0.995915i	$0.471219\pi$
$912$	0	0
$913$	2.14377e21	0.134168
$914$	0	0
$915$	−4.12020e21	−0.253665
$916$	0	0
$917$	−2.17177e22	−1.31536
$918$	0	0
$919$	1.87762e22	1.11877	0.559387	−	0.828907i	$-0.311037\pi$
$919$	1.87762e22	1.11877	0.559387	+	0.828907i	$0.311037\pi$
$920$	0	0
$921$	2.81818e21	0.165204
$922$	0	0
$923$	1.65155e22	0.952528
$924$	0	0
$925$	1.13979e22	0.646787
$926$	0	0
$927$	6.96825e20	0.0389068
$928$	0	0
$929$	−2.61729e21	−0.143792	−0.0718958	−	0.997412i	$-0.522905\pi$
$929$	−2.61729e21	−0.143792	−0.0718958	+	0.997412i	$0.522905\pi$
$930$	0	0
$931$	−6.04085e20	−0.0326570
$932$	0	0
$933$	1.84371e22	0.980798
$934$	0	0
$935$	2.05232e22	1.07438
$936$	0	0
$937$	1.24366e22	0.640701	0.320350	−	0.947299i	$-0.396199\pi$
$937$	1.24366e22	0.640701	0.320350	+	0.947299i	$0.396199\pi$
$938$	0	0
$939$	−1.72414e22	−0.874140
$940$	0	0
$941$	1.71266e22	0.854574	0.427287	−	0.904116i	$-0.359469\pi$
$941$	1.71266e22	0.854574	0.427287	+	0.904116i	$0.359469\pi$
$942$	0	0
$943$	1.01548e22	0.498692
$944$	0	0
$945$	4.97894e21	0.240657
$946$	0	0
$947$	−1.26035e21	−0.0599607	−0.0299804	−	0.999550i	$-0.509544\pi$
$947$	−1.26035e21	−0.0599607	−0.0299804	+	0.999550i	$0.509544\pi$
$948$	0	0
$949$	1.57061e21	0.0735480
$950$	0	0
$951$	1.90606e22	0.878581
$952$	0	0
$953$	−2.19887e22	−0.997706	−0.498853	−	0.866687i	$-0.666245\pi$
$953$	−2.19887e22	−0.997706	−0.498853	+	0.866687i	$0.666245\pi$
$954$	0	0
$955$	2.09151e22	0.934189
$956$	0	0
$957$	−9.42577e21	−0.414455
$958$	0	0
$959$	−2.13554e22	−0.924419
$960$	0	0
$961$	−2.03083e22	−0.865461
$962$	0	0
$963$	1.95478e21	0.0820163
$964$	0	0
$965$	−9.25260e21	−0.382216
$966$	0	0
$967$	−1.21130e21	−0.0492666	−0.0246333	−	0.999697i	$-0.507842\pi$
$967$	−1.21130e21	−0.0492666	−0.0246333	+	0.999697i	$0.507842\pi$
$968$	0	0
$969$	−2.54597e22	−1.01959
$970$	0	0
$971$	−1.26825e22	−0.500103	−0.250051	−	0.968233i	$-0.580448\pi$
$971$	−1.26825e22	−0.500103	−0.250051	+	0.968233i	$0.580448\pi$
$972$	0	0
$973$	4.19223e22	1.62779
$974$	0	0
$975$	1.13523e22	0.434058
$976$	0	0
$977$	−3.29367e21	−0.124014	−0.0620070	−	0.998076i	$-0.519750\pi$
$977$	−3.29367e21	−0.124014	−0.0620070	+	0.998076i	$0.519750\pi$
$978$	0	0
$979$	1.34660e22	0.499307
$980$	0	0
$981$	5.26766e21	0.192353
$982$	0	0
$983$	−3.74166e22	−1.34559	−0.672796	−	0.739828i	$-0.734907\pi$
$983$	−3.74166e22	−1.34559	−0.672796	+	0.739828i	$0.734907\pi$
$984$	0	0
$985$	3.28653e22	1.16404
$986$	0	0
$987$	9.75759e21	0.340380
$988$	0	0
$989$	1.81518e22	0.623661
$990$	0	0
$991$	4.40305e22	1.49005	0.745026	−	0.667035i	$-0.232437\pi$
$991$	4.40305e22	1.49005	0.745026	+	0.667035i	$0.232437\pi$
$992$	0	0
$993$	2.77738e22	0.925796
$994$	0	0
$995$	4.37565e22	1.43671
$996$	0	0
$997$	2.62067e22	0.847614	0.423807	−	0.905753i	$-0.360694\pi$
$997$	2.62067e22	0.847614	0.423807	+	0.905753i	$0.360694\pi$
$998$	0	0
$999$	−6.43066e21	−0.204887

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	48.16.a.a.1.1		1
3.2	odd	2		144.16.a.l.1.1		1
4.3	odd	2		3.16.a.b.1.1	✓	1
12.11	even	2		9.16.a.c.1.1		1
20.3	even	4		75.16.b.b.49.2		2
20.7	even	4		75.16.b.b.49.1		2
20.19	odd	2		75.16.a.a.1.1		1
28.27	even	2		147.16.a.b.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3.16.a.b.1.1	✓	1	4.3	odd	2
9.16.a.c.1.1		1	12.11	even	2
48.16.a.a.1.1		1	1.1	even	1	trivial
75.16.a.a.1.1		1	20.19	odd	2
75.16.b.b.49.1		2	20.7	even	4
75.16.b.b.49.2		2	20.3	even	4
144.16.a.l.1.1		1	3.2	odd	2
147.16.a.b.1.1		1	28.27	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:	\( N \)	\(=\)	\( 48 = 2^{4} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 16 \)
Character orbit:	\([\chi]\)	\(=\)	48.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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