LMFDB - Embedded newform 40.15.e.a.19.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [40,15,Mod(19,40)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(40, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 1])) N = Newforms(chi, 15, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("40.19"); S:= CuspForms(chi, 15); N := Newforms(S);

Level:	$N$	$=$	$40 = 2^{3} \cdot 5$
Weight:	$k$	$=$	$15$
Character orbit:	$[\chi]$	$=$	40.e (of order $2$ , degree $1$ , minimal)

Newform invariants

comment:select newform

sage:traces = [1,-128] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$49.7315872608$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			19.1
Character	$\chi$	$=$	40.19

$q$ -expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

f(q)

=

q-128.000 q^{2} +16384.0 q^{4} +78125.0 q^{5} -67866.0 q^{7} -2.09715e6 q^{8} +4.78297e6 q^{9} -1.00000e7 q^{10} +1.60524e7 q^{11} +1.25277e8 q^{13} +8.68685e6 q^{14} +2.68435e8 q^{16} -6.12220e8 q^{18} +6.44127e8 q^{19} +1.28000e9 q^{20} -2.05470e9 q^{22} -5.95134e9 q^{23} +6.10352e9 q^{25} -1.60354e10 q^{26} -1.11192e9 q^{28} -3.43597e10 q^{32} -5.30203e9 q^{35} +7.83642e10 q^{36} -1.00483e11 q^{37} -8.24483e10 q^{38} -1.63840e11 q^{40} +2.27781e11 q^{41} +2.63002e11 q^{44} +3.73669e11 q^{45} +7.61771e11 q^{46} -9.85930e11 q^{47} -6.73617e11 q^{49} -7.81250e11 q^{50} +2.05253e12 q^{52} +1.80159e12 q^{53} +1.25409e12 q^{55} +1.42325e11 q^{56} +4.68767e12 q^{59} -3.24601e11 q^{63} +4.39805e12 q^{64} +9.78723e12 q^{65} +6.78660e11 q^{70} -1.00306e13 q^{72} +1.28618e13 q^{74} +1.05534e13 q^{76} -1.08941e12 q^{77} +2.09715e13 q^{80} +2.28768e13 q^{81} -2.91560e13 q^{82} -3.36643e13 q^{88} -5.76010e13 q^{89} -4.78297e13 q^{90} -8.50202e12 q^{91} -9.75067e13 q^{92} +1.26199e14 q^{94} +5.03224e13 q^{95} +8.62230e13 q^{98} +7.67780e13 q^{99} +O(q^{100})

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/40\mathbb{Z}\right)^\times$ .

$n$	$17$	$21$	$31$
$\chi(n)$	$-1$	$-1$	$-1$

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

)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$\alpha_n$			$\theta_n$
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$\alpha_p$			$\theta_p$

$2$	−128.000	−1.00000
$2$	−128.000	−1.00000
$3$	0	0	1.00000			$0$
$3$	0	0	−1.00000			$\pi$
$4$	16384.0	1.00000
$5$	78125.0	1.00000
$5$	78125.0	1.00000
$6$	0	0
$7$	−67866.0	−0.0824074	−0.0412037	−	0.999151i	$-0.513119\pi$
$7$	−67866.0	−0.0824074	−0.0412037	+	0.999151i	$0.513119\pi$
$8$	−2.09715e6	−1.00000
$9$	4.78297e6	1.00000
$10$	−1.00000e7	−1.00000
$11$	1.60524e7	0.823741	0.411871	−	0.911242i	$-0.364876\pi$
$11$	1.60524e7	0.823741	0.411871	+	0.911242i	$0.364876\pi$
$12$	0	0
$13$	1.25277e8	1.99649	0.998243	−	0.0592462i	$-0.0188697\pi$
$13$	1.25277e8	1.99649	0.998243	+	0.0592462i	$0.0188697\pi$
$14$	8.68685e6	0.0824074
$15$	0	0
$16$	2.68435e8	1.00000
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	−6.12220e8	−1.00000
$19$	6.44127e8	0.720604	0.360302	−	0.932836i	$-0.382674\pi$
$19$	6.44127e8	0.720604	0.360302	+	0.932836i	$0.382674\pi$
$20$	1.28000e9	1.00000
$21$	0	0
$22$	−2.05470e9	−0.823741
$23$	−5.95134e9	−1.74791	−0.873957	−	0.486004i	$-0.838454\pi$
$23$	−5.95134e9	−1.74791	−0.873957	+	0.486004i	$0.838454\pi$
$24$	0	0
$25$	6.10352e9	1.00000
$26$	−1.60354e10	−1.99649
$27$	0	0
$28$	−1.11192e9	−0.0824074
$29$	0	0	1.00000			$0$
$29$	0	0	−1.00000			$\pi$
$30$	0	0
$31$	0	0	1.00000			$0$
$31$	0	0	−1.00000			$\pi$
$32$	−3.43597e10	−1.00000
$33$	0	0
$34$	0	0
$35$	−5.30203e9	−0.0824074
$36$	7.83642e10	1.00000
$37$	−1.00483e11	−1.05847	−0.529235	−	0.848475i	$-0.677521\pi$
$37$	−1.00483e11	−1.05847	−0.529235	+	0.848475i	$0.677521\pi$
$38$	−8.24483e10	−0.720604
$39$	0	0
$40$	−1.63840e11	−1.00000
$41$	2.27781e11	1.16958	0.584792	−	0.811183i	$-0.301176\pi$
$41$	2.27781e11	1.16958	0.584792	+	0.811183i	$0.301176\pi$
$42$	0	0
$43$	0	0	1.00000			$0$
$43$	0	0	−1.00000			$\pi$
$44$	2.63002e11	0.823741
$45$	3.73669e11	1.00000
$46$	7.61771e11	1.74791
$47$	−9.85930e11	−1.94608	−0.973041	−	0.230633i	$-0.925920\pi$
$47$	−9.85930e11	−1.94608	−0.973041	+	0.230633i	$0.925920\pi$
$48$	0	0
$49$	−6.73617e11	−0.993209
$50$	−7.81250e11	−1.00000
$51$	0	0
$52$	2.05253e12	1.99649
$53$	1.80159e12	1.53364	0.766821	−	0.641861i	$-0.221838\pi$
$53$	1.80159e12	1.53364	0.766821	+	0.641861i	$0.221838\pi$
$54$	0	0
$55$	1.25409e12	0.823741
$56$	1.42325e11	0.0824074
$57$	0	0
$58$	0	0
$59$	4.68767e12	1.88362	0.941809	−	0.336148i	$-0.109124\pi$
$59$	4.68767e12	1.88362	0.941809	+	0.336148i	$0.109124\pi$
$60$	0	0
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	0	0
$63$	−3.24601e11	−0.0824074
$64$	4.39805e12	1.00000
$65$	9.78723e12	1.99649
$66$	0	0
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	0	0
$69$	0	0
$70$	6.78660e11	0.0824074
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	−1.00306e13	−1.00000
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	1.28618e13	1.05847
$75$	0	0
$76$	1.05534e13	0.720604
$77$	−1.08941e12	−0.0678823
$78$	0	0
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	2.09715e13	1.00000
$81$	2.28768e13	1.00000
$82$	−2.91560e13	−1.16958
$83$	0	0	1.00000			$0$
$83$	0	0	−1.00000			$\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	−3.36643e13	−0.823741
$89$	−5.76010e13	−1.30227	−0.651134	−	0.758963i	$-0.725706\pi$
$89$	−5.76010e13	−1.30227	−0.651134	+	0.758963i	$0.725706\pi$
$90$	−4.78297e13	−1.00000
$91$	−8.50202e12	−0.164525
$92$	−9.75067e13	−1.74791
$93$	0	0
$94$	1.26199e14	1.94608
$95$	5.03224e13	0.720604
$96$	0	0
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	8.62230e13	0.993209
$99$	7.67780e13	0.823741
$100$	1.00000e14	1.00000
$101$	0	0	1.00000			$0$
$101$	0	0	−1.00000			$\pi$
$102$	0	0
$103$	2.45853e14	1.99901	0.999504	−	0.0315014i	$-0.0100289\pi$
$103$	2.45853e14	1.99901	0.999504	+	0.0315014i	$0.0100289\pi$
$104$	−2.62724e14	−1.99649
$105$	0	0
$106$	−2.30603e14	−1.53364
$107$	0	0	1.00000			$0$
$107$	0	0	−1.00000			$\pi$
$108$	0	0
$109$	0	0	1.00000			$0$
$109$	0	0	−1.00000			$\pi$
$110$	−1.60524e14	−0.823741
$111$	0	0
$112$	−1.82176e13	−0.0824074
$113$	0	0	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	0	0
$115$	−4.64948e14	−1.74791
$116$	0	0
$117$	5.99194e14	1.99649
$118$	−6.00022e14	−1.88362
$119$	0	0
$120$	0	0
$121$	−1.22071e14	−0.321451
$122$	0	0
$123$	0	0
$124$	0	0
$125$	4.76837e14	1.00000
$126$	4.15489e13	0.0824074
$127$	−2.02035e14	−0.379140	−0.189570	−	0.981867i	$-0.560709\pi$
$127$	−2.02035e14	−0.379140	−0.189570	+	0.981867i	$0.560709\pi$
$128$	−5.62950e14	−1.00000
$129$	0	0
$130$	−1.25277e15	−1.99649
$131$	−9.43716e14	−1.42542	−0.712709	−	0.701460i	$-0.752532\pi$
$131$	−9.43716e14	−1.42542	−0.712709	+	0.701460i	$0.752532\pi$
$132$	0	0
$133$	−4.37143e13	−0.0593830
$134$	0	0
$135$	0	0
$136$	0	0
$137$	0	0	1.00000			$0$
$137$	0	0	−1.00000			$\pi$
$138$	0	0
$139$	1.73492e15	1.73052	0.865259	−	0.501324i	$-0.167154\pi$
$139$	1.73492e15	1.73052	0.865259	+	0.501324i	$0.167154\pi$
$140$	−8.68685e13	−0.0824074
$141$	0	0
$142$	0	0
$143$	2.01099e15	1.64459
$144$	1.28392e15	1.00000
$145$	0	0
$146$	0	0
$147$	0	0
$148$	−1.64631e15	−1.05847
$149$	0	0	1.00000			$0$
$149$	0	0	−1.00000			$\pi$
$150$	0	0
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	−1.35083e15	−0.720604
$153$	0	0
$154$	1.39445e14	0.0678823
$155$	0	0
$156$	0	0
$157$	4.52640e15	1.92511	0.962555	−	0.271088i	$-0.0873835\pi$
$157$	4.52640e15	1.92511	0.962555	+	0.271088i	$0.0873835\pi$
$158$	0	0
$159$	0	0
$160$	−2.68435e15	−1.00000
$161$	4.03894e14	0.144041
$162$	−2.92823e15	−1.00000
$163$	0	0	1.00000			$0$
$163$	0	0	−1.00000			$\pi$
$164$	3.73197e15	1.16958
$165$	0	0
$166$	0	0
$167$	5.50909e15	1.52077	0.760386	−	0.649471i	$-0.225010\pi$
$167$	5.50909e15	1.52077	0.760386	+	0.649471i	$0.225010\pi$
$168$	0	0
$169$	1.17568e16	2.98596
$170$	0	0
$171$	3.08084e15	0.720604
$172$	0	0
$173$	−3.27758e15	−0.706692	−0.353346	−	0.935493i	$-0.614956\pi$
$173$	−3.27758e15	−0.706692	−0.353346	+	0.935493i	$0.614956\pi$
$174$	0	0
$175$	−4.14221e14	−0.0824074
$176$	4.30903e15	0.823741
$177$	0	0
$178$	7.37293e15	1.30227
$179$	−1.15436e16	−1.96051	−0.980254	−	0.197743i	$-0.936639\pi$
$179$	−1.15436e16	−1.96051	−0.980254	+	0.197743i	$0.936639\pi$
$180$	6.12220e15	1.00000
$181$	0	0	1.00000			$0$
$181$	0	0	−1.00000			$\pi$
$182$	1.08826e15	0.164525
$183$	0	0
$184$	1.24809e16	1.74791
$185$	−7.85021e15	−1.05847
$186$	0	0
$187$	0	0
$188$	−1.61535e16	−1.94608
$189$	0	0
$190$	−6.44127e15	−0.720604
$191$	0	0	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	0	0
$193$	0	0	1.00000			$0$
$193$	0	0	−1.00000			$\pi$
$194$	0	0
$195$	0	0
$196$	−1.10365e16	−0.993209
$197$	−2.17426e16	−1.88820	−0.944099	−	0.329661i	$-0.893066\pi$
$197$	−2.17426e16	−1.88820	−0.944099	+	0.329661i	$0.893066\pi$
$198$	−9.82759e15	−0.823741
$199$	0	0	1.00000			$0$
$199$	0	0	−1.00000			$\pi$
$200$	−1.28000e16	−1.00000
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	1.77954e16	1.16958
$206$	−3.14691e16	−1.99901
$207$	−2.84651e16	−1.74791
$208$	3.36287e16	1.99649
$209$	1.03398e16	0.593591
$210$	0	0
$211$	−3.19672e16	−1.71683	−0.858415	−	0.512955i	$-0.828551\pi$
$211$	−3.19672e16	−1.71683	−0.858415	+	0.512955i	$0.828551\pi$
$212$	2.95172e16	1.53364
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	2.05470e16	0.823741
$221$	0	0
$222$	0	0
$223$	−4.11927e16	−1.50206	−0.751028	−	0.660270i	$-0.770442\pi$
$223$	−4.11927e16	−1.50206	−0.751028	+	0.660270i	$0.770442\pi$
$224$	2.33186e15	0.0824074
$225$	2.91929e16	1.00000
$226$	0	0
$227$	0	0	1.00000			$0$
$227$	0	0	−1.00000			$\pi$
$228$	0	0
$229$	0	0	1.00000			$0$
$229$	0	0	−1.00000			$\pi$
$230$	5.95134e16	1.74791
$231$	0	0
$232$	0	0
$233$	0	0	1.00000			$0$
$233$	0	0	−1.00000			$\pi$
$234$	−7.66968e16	−1.99649
$235$	−7.70258e16	−1.94608
$236$	7.68028e16	1.88362
$237$	0	0
$238$	0	0
$239$	0	0	1.00000			$0$
$239$	0	0	−1.00000			$\pi$
$240$	0	0
$241$	6.79936e16	1.43995	0.719977	−	0.693998i	$-0.244152\pi$
$241$	6.79936e16	1.43995	0.719977	+	0.693998i	$0.244152\pi$
$242$	1.56251e16	0.321451
$243$	0	0
$244$	0	0
$245$	−5.26263e16	−0.993209
$246$	0	0
$247$	8.06940e16	1.43868
$248$	0	0
$249$	0	0
$250$	−6.10352e16	−1.00000
$251$	−7.95684e16	−1.26772	−0.633862	−	0.773446i	$-0.718531\pi$
$251$	−7.95684e16	−1.26772	−0.633862	+	0.773446i	$0.718531\pi$
$252$	−5.31826e15	−0.0824074
$253$	−9.55332e16	−1.43983
$254$	2.58604e16	0.379140
$255$	0	0
$256$	7.20576e16	1.00000
$257$	0	0	1.00000			$0$
$257$	0	0	−1.00000			$\pi$
$258$	0	0
$259$	6.81935e15	0.0872258
$260$	1.60354e17	1.99649
$261$	0	0
$262$	1.20796e17	1.42542
$263$	−1.47100e17	−1.69014	−0.845069	−	0.534657i	$-0.820441\pi$
$263$	−1.47100e17	−1.69014	−0.845069	+	0.534657i	$0.820441\pi$
$264$	0	0
$265$	1.40749e17	1.53364
$266$	5.59543e15	0.0593830
$267$	0	0
$268$	0	0
$269$	0	0	1.00000			$0$
$269$	0	0	−1.00000			$\pi$
$270$	0	0
$271$	0	0	1.00000			$0$
$271$	0	0	−1.00000			$\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	9.79760e16	0.823741
$276$	0	0
$277$	3.56086e16	0.284575	0.142287	−	0.989825i	$-0.454554\pi$
$277$	3.56086e16	0.284575	0.142287	+	0.989825i	$0.454554\pi$
$278$	−2.22070e17	−1.73052
$279$	0	0
$280$	1.11192e16	0.0824074
$281$	2.28956e17	1.65504	0.827520	−	0.561436i	$-0.189751\pi$
$281$	2.28956e17	1.65504	0.827520	+	0.561436i	$0.189751\pi$
$282$	0	0
$283$	0	0	1.00000			$0$
$283$	0	0	−1.00000			$\pi$
$284$	0	0
$285$	0	0
$286$	−2.57406e17	−1.64459
$287$	−1.54586e16	−0.0963823
$288$	−1.64342e17	−1.00000
$289$	1.68378e17	1.00000
$290$	0	0
$291$	0	0
$292$	0	0
$293$	9.64995e16	0.520537	0.260269	−	0.965536i	$-0.416189\pi$
$293$	9.64995e16	0.520537	0.260269	+	0.965536i	$0.416189\pi$
$294$	0	0
$295$	3.66224e17	1.88362
$296$	2.10727e17	1.05847
$297$	0	0
$298$	0	0
$299$	−7.45563e17	−3.48969
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	1.72907e17	0.720604
$305$	0	0
$306$	0	0
$307$	0	0	1.00000			$0$
$307$	0	0	−1.00000			$\pi$
$308$	−1.78489e16	−0.0678823
$309$	0	0
$310$	0	0
$311$	0	0	1.00000			$0$
$311$	0	0	−1.00000			$\pi$
$312$	0	0
$313$	0	0	1.00000			$0$
$313$	0	0	−1.00000			$\pi$
$314$	−5.79379e17	−1.92511
$315$	−2.53595e16	−0.0824074
$316$	0	0
$317$	5.03679e17	1.56581	0.782905	−	0.622141i	$-0.213737\pi$
$317$	5.03679e17	1.56581	0.782905	+	0.622141i	$0.213737\pi$
$318$	0	0
$319$	0	0
$320$	3.43597e17	1.00000
$321$	0	0
$322$	−5.16984e16	−0.144041
$323$	0	0
$324$	3.74813e17	1.00000
$325$	7.64628e17	1.99649
$326$	0	0
$327$	0	0
$328$	−4.77692e17	−1.16958
$329$	6.69111e16	0.160371
$330$	0	0
$331$	−4.98299e17	−1.14471	−0.572354	−	0.820007i	$-0.693969\pi$
$331$	−4.98299e17	−1.14471	−0.572354	+	0.820007i	$0.693969\pi$
$332$	0	0
$333$	−4.80605e17	−1.05847
$334$	−7.05163e17	−1.52077
$335$	0	0
$336$	0	0
$337$	0	0	1.00000			$0$
$337$	0	0	−1.00000			$\pi$
$338$	−1.50488e18	−2.98596
$339$	0	0
$340$	0	0
$341$	0	0
$342$	−3.94348e17	−0.720604
$343$	9.17440e16	0.164255
$344$	0	0
$345$	0	0
$346$	4.19530e17	0.706692
$347$	0	0	1.00000			$0$
$347$	0	0	−1.00000			$\pi$
$348$	0	0
$349$	0	0	1.00000			$0$
$349$	0	0	−1.00000			$\pi$
$350$	5.30203e16	0.0824074
$351$	0	0
$352$	−5.51556e17	−0.823741
$353$	0	0	1.00000			$0$
$353$	0	0	−1.00000			$\pi$
$354$	0	0
$355$	0	0
$356$	−9.43735e17	−1.30227
$357$	0	0
$358$	1.47758e18	1.96051
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	−7.83642e17	−1.00000
$361$	−3.84107e17	−0.480731
$362$	0	0
$363$	0	0
$364$	−1.39297e17	−0.164525
$365$	0	0
$366$	0	0
$367$	1.37119e18	1.52910	0.764550	−	0.644565i	$-0.222961\pi$
$367$	1.37119e18	1.52910	0.764550	+	0.644565i	$0.222961\pi$
$368$	−1.59755e18	−1.74791
$369$	1.08947e18	1.16958
$370$	1.00483e18	1.05847
$371$	−1.22266e17	−0.126383
$372$	0	0
$373$	−1.58391e18	−1.57677	−0.788386	−	0.615180i	$-0.789083\pi$
$373$	−1.58391e18	−1.57677	−0.788386	+	0.615180i	$0.789083\pi$
$374$	0	0
$375$	0	0
$376$	2.06764e18	1.94608
$377$	0	0
$378$	0	0
$379$	1.49469e18	1.33069	0.665345	−	0.746536i	$-0.268285\pi$
$379$	1.49469e18	1.33069	0.665345	+	0.746536i	$0.268285\pi$
$380$	8.24483e17	0.720604
$381$	0	0
$382$	0	0
$383$	−4.46539e17	−0.369375	−0.184688	−	0.982797i	$-0.559127\pi$
$383$	−4.46539e17	−0.369375	−0.184688	+	0.982797i	$0.559127\pi$
$384$	0	0
$385$	−8.51102e16	−0.0678823
$386$	0	0
$387$	0	0
$388$	0	0
$389$	0	0	1.00000			$0$
$389$	0	0	−1.00000			$\pi$
$390$	0	0
$391$	0	0
$392$	1.41268e18	0.993209
$393$	0	0
$394$	2.78305e18	1.88820
$395$	0	0
$396$	1.25793e18	0.823741
$397$	1.14115e16	0.00734190	0.00367095	−	0.999993i	$-0.498831\pi$
$397$	1.14115e16	0.00734190	0.00367095	+	0.999993i	$0.498831\pi$
$398$	0	0
$399$	0	0
$400$	1.63840e18	1.00000
$401$	−7.07198e17	−0.424161	−0.212080	−	0.977252i	$-0.568024\pi$
$401$	−7.07198e17	−0.424161	−0.212080	+	0.977252i	$0.568024\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	1.78725e18	1.00000
$406$	0	0
$407$	−1.61299e18	−0.871906
$408$	0	0
$409$	−8.94471e17	−0.467200	−0.233600	−	0.972333i	$-0.575051\pi$
$409$	−8.94471e17	−0.467200	−0.233600	+	0.972333i	$0.575051\pi$
$410$	−2.27781e18	−1.16958
$411$	0	0
$412$	4.02805e18	1.99901
$413$	−3.18133e17	−0.155224
$414$	3.64353e18	1.74791
$415$	0	0
$416$	−4.30447e18	−1.99649
$417$	0	0
$418$	−1.32349e18	−0.593591
$419$	4.43581e18	1.95648	0.978239	−	0.207482i	$-0.0665270\pi$
$419$	4.43581e18	1.95648	0.978239	+	0.207482i	$0.0665270\pi$
$420$	0	0
$421$	0	0	1.00000			$0$
$421$	0	0	−1.00000			$\pi$
$422$	4.09180e18	1.71683
$423$	−4.71567e18	−1.94608
$424$	−3.77820e18	−1.53364
$425$	0	0
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	0	0
$433$	0	0	1.00000			$0$
$433$	0	0	−1.00000			$\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−3.83342e18	−1.25955
$438$	0	0
$439$	0	0	1.00000			$0$
$439$	0	0	−1.00000			$\pi$
$440$	−2.63002e18	−0.823741
$441$	−3.22189e18	−0.993209
$442$	0	0
$443$	0	0	1.00000			$0$
$443$	0	0	−1.00000			$\pi$
$444$	0	0
$445$	−4.50008e18	−1.30227
$446$	5.27267e18	1.50206
$447$	0	0
$448$	−2.98478e17	−0.0824074
$449$	−6.56311e18	−1.78396	−0.891981	−	0.452074i	$-0.850684\pi$
$449$	−6.56311e18	−1.78396	−0.891981	+	0.452074i	$0.850684\pi$
$450$	−3.73669e18	−1.00000
$451$	3.65643e18	0.963434
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−6.64220e17	−0.164525
$456$	0	0
$457$	0	0	1.00000			$0$
$457$	0	0	−1.00000			$\pi$
$458$	0	0
$459$	0	0
$460$	−7.61771e18	−1.74791
$461$	0	0	1.00000			$0$
$461$	0	0	−1.00000			$\pi$
$462$	0	0
$463$	−1.85442e18	−0.406575	−0.203287	−	0.979119i	$-0.565163\pi$
$463$	−1.85442e18	−0.406575	−0.203287	+	0.979119i	$0.565163\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	0	0	1.00000			$0$
$467$	0	0	−1.00000			$\pi$
$468$	9.81719e18	1.99649
$469$	0	0
$470$	9.85930e18	1.94608
$471$	0	0
$472$	−9.83076e18	−1.88362
$473$	0	0
$474$	0	0
$475$	3.93144e18	0.720604
$476$	0	0
$477$	8.61693e18	1.53364
$478$	0	0
$479$	0	0	1.00000			$0$
$479$	0	0	−1.00000			$\pi$
$480$	0	0
$481$	−1.25881e19	−2.11322
$482$	−8.70317e18	−1.43995
$483$	0	0
$484$	−2.00001e18	−0.321451
$485$	0	0
$486$	0	0
$487$	−5.60133e17	−0.0862161	−0.0431081	−	0.999070i	$-0.513726\pi$
$487$	−5.60133e17	−0.0862161	−0.0431081	+	0.999070i	$0.513726\pi$
$488$	0	0
$489$	0	0
$490$	6.73617e18	0.993209
$491$	−1.11016e19	−1.61366	−0.806832	−	0.590780i	$-0.798820\pi$
$491$	−1.11016e19	−1.61366	−0.806832	+	0.590780i	$0.798820\pi$
$492$	0	0
$493$	0	0
$494$	−1.03288e19	−1.43868
$495$	5.99828e18	0.823741
$496$	0	0
$497$	0	0
$498$	0	0
$499$	4.77469e18	0.619785	0.309893	−	0.950772i	$-0.399707\pi$
$499$	4.77469e18	0.619785	0.309893	+	0.950772i	$0.399707\pi$
$500$	7.81250e18	1.00000
$501$	0	0
$502$	1.01848e19	1.26772
$503$	1.61818e19	1.98633	0.993166	−	0.116712i	$-0.0372355\pi$
$503$	1.61818e19	1.98633	0.993166	+	0.116712i	$0.0372355\pi$
$504$	6.80738e17	0.0824074
$505$	0	0
$506$	1.22282e19	1.43983
$507$	0	0
$508$	−3.31013e18	−0.379140
$509$	0	0	1.00000			$0$
$509$	0	0	−1.00000			$\pi$
$510$	0	0
$511$	0	0
$512$	−9.22337e18	−1.00000
$513$	0	0
$514$	0	0
$515$	1.92072e19	1.99901
$516$	0	0
$517$	−1.58265e19	−1.60307
$518$	−8.72877e17	−0.0872258
$519$	0	0
$520$	−2.05253e19	−1.99649
$521$	−7.51842e18	−0.721544	−0.360772	−	0.932654i	$-0.617487\pi$
$521$	−7.51842e18	−0.721544	−0.360772	+	0.932654i	$0.617487\pi$
$522$	0	0
$523$	0	0	1.00000			$0$
$523$	0	0	−1.00000			$\pi$
$524$	−1.54618e19	−1.42542
$525$	0	0
$526$	1.88288e19	1.69014
$527$	0	0
$528$	0	0
$529$	2.38256e19	2.05520
$530$	−1.80159e19	−1.53364
$531$	2.24210e19	1.88362
$532$	−7.16216e17	−0.0593830
$533$	2.85357e19	2.33506
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−1.08132e19	−0.818147
$540$	0	0
$541$	0	0	1.00000			$0$
$541$	0	0	−1.00000			$\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	0	0	1.00000			$0$
$547$	0	0	−1.00000			$\pi$
$548$	0	0
$549$	0	0
$550$	−1.25409e19	−0.823741
$551$	0	0
$552$	0	0
$553$	0	0
$554$	−4.55790e18	−0.284575
$555$	0	0
$556$	2.84250e19	1.73052
$557$	2.08524e19	1.25363	0.626816	−	0.779167i	$-0.284358\pi$
$557$	2.08524e19	1.25363	0.626816	+	0.779167i	$0.284358\pi$
$558$	0	0
$559$	0	0
$560$	−1.42325e18	−0.0824074
$561$	0	0
$562$	−2.93064e19	−1.65504
$563$	0	0	1.00000			$0$
$563$	0	0	−1.00000			$\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−1.55256e18	−0.0824074
$568$	0	0
$569$	−3.15378e19	−1.63322	−0.816610	−	0.577189i	$-0.804149\pi$
$569$	−3.15378e19	−1.63322	−0.816610	+	0.577189i	$0.804149\pi$
$570$	0	0
$571$	−2.53139e19	−1.27911	−0.639553	−	0.768747i	$-0.720880\pi$
$571$	−2.53139e19	−1.27911	−0.639553	+	0.768747i	$0.720880\pi$
$572$	3.29480e19	1.64459
$573$	0	0
$574$	1.97870e18	0.0963823
$575$	−3.63241e19	−1.74791
$576$	2.10357e19	1.00000
$577$	0	0	1.00000			$0$
$577$	0	0	−1.00000			$\pi$
$578$	−2.15524e19	−1.00000
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	2.89197e19	1.26332
$584$	0	0
$585$	4.68120e19	1.99649
$586$	−1.23519e19	−0.520537
$587$	0	0	1.00000			$0$
$587$	0	0	−1.00000			$\pi$
$588$	0	0
$589$	0	0
$590$	−4.68767e19	−1.88362
$591$	0	0
$592$	−2.69731e19	−1.05847
$593$	0	0	1.00000			$0$
$593$	0	0	−1.00000			$\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	9.54321e19	3.48969
$599$	0	0	1.00000			$0$
$599$	0	0	−1.00000			$\pi$
$600$	0	0
$601$	1.47841e19	0.522003	0.261002	−	0.965338i	$-0.415947\pi$
$601$	1.47841e19	0.522003	0.261002	+	0.965338i	$0.415947\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−9.53679e18	−0.321451
$606$	0	0
$607$	4.21288e19	1.38758	0.693790	−	0.720178i	$-0.255940\pi$
$607$	4.21288e19	1.38758	0.693790	+	0.720178i	$0.255940\pi$
$608$	−2.21320e19	−0.720604
$609$	0	0
$610$	0	0
$611$	−1.23514e20	−3.88533
$612$	0	0
$613$	−2.08455e18	−0.0640897	−0.0320448	−	0.999486i	$-0.510202\pi$
$613$	−2.08455e18	−0.0640897	−0.0320448	+	0.999486i	$0.510202\pi$
$614$	0	0
$615$	0	0
$616$	2.28466e18	0.0678823
$617$	0	0	1.00000			$0$
$617$	0	0	−1.00000			$\pi$
$618$	0	0
$619$	4.82069e19	1.38444	0.692221	−	0.721686i	$-0.256632\pi$
$619$	4.82069e19	1.38444	0.692221	+	0.721686i	$0.256632\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	3.90915e18	0.107316
$624$	0	0
$625$	3.72529e19	1.00000
$626$	0	0
$627$	0	0
$628$	7.41605e19	1.92511
$629$	0	0
$630$	3.24601e18	0.0824074
$631$	0	0	1.00000			$0$
$631$	0	0	−1.00000			$\pi$
$632$	0	0
$633$	0	0
$634$	−6.44710e19	−1.56581
$635$	−1.57839e19	−0.379140
$636$	0	0
$637$	−8.43885e19	−1.98293
$638$	0	0
$639$	0	0
$640$	−4.39805e19	−1.00000
$641$	−7.56679e19	−1.70179	−0.850895	−	0.525337i	$-0.823939\pi$
$641$	−7.56679e19	−1.70179	−0.850895	+	0.525337i	$0.823939\pi$
$642$	0	0
$643$	0	0	1.00000			$0$
$643$	0	0	−1.00000			$\pi$
$644$	6.61739e18	0.144041
$645$	0	0
$646$	0	0
$647$	−9.27967e19	−1.95525	−0.977625	−	0.210354i	$-0.932539\pi$
$647$	−9.27967e19	−1.95525	−0.977625	+	0.210354i	$0.932539\pi$
$648$	−4.79761e19	−1.00000
$649$	7.52483e19	1.55161
$650$	−9.78723e19	−1.99649
$651$	0	0
$652$	0	0
$653$	−3.41655e19	−0.674831	−0.337416	−	0.941356i	$-0.609553\pi$
$653$	−3.41655e19	−0.674831	−0.337416	+	0.941356i	$0.609553\pi$
$654$	0	0
$655$	−7.37278e19	−1.42542
$656$	6.11446e19	1.16958
$657$	0	0
$658$	−8.56462e18	−0.160371
$659$	−4.39969e19	−0.815125	−0.407563	−	0.913177i	$-0.633621\pi$
$659$	−4.39969e19	−0.815125	−0.407563	+	0.913177i	$0.633621\pi$
$660$	0	0
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	6.37823e19	1.14471
$663$	0	0
$664$	0	0
$665$	−3.41518e18	−0.0593830
$666$	6.15175e19	1.05847
$667$	0	0
$668$	9.02609e19	1.52077
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	0	0	1.00000			$0$
$673$	0	0	−1.00000			$\pi$
$674$	0	0
$675$	0	0
$676$	1.92624e20	2.98596
$677$	−8.03858e18	−0.123327	−0.0616636	−	0.998097i	$-0.519641\pi$
$677$	−8.03858e18	−0.123327	−0.0616636	+	0.998097i	$0.519641\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	0	0	1.00000			$0$
$683$	0	0	−1.00000			$\pi$
$684$	5.04765e19	0.720604
$685$	0	0
$686$	−1.17432e19	−0.164255
$687$	0	0
$688$	0	0
$689$	2.25697e20	3.06190
$690$	0	0
$691$	−1.39170e20	−1.85012	−0.925058	−	0.379825i	$-0.875984\pi$
$691$	−1.39170e20	−1.85012	−0.925058	+	0.379825i	$0.875984\pi$
$692$	−5.36998e19	−0.706692
$693$	−5.21062e18	−0.0678823
$694$	0	0
$695$	1.35541e20	1.73052
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	−6.78660e18	−0.0824074
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	0	0
$703$	−6.47236e19	−0.762738
$704$	7.05991e19	0.823741
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	0	0	1.00000			$0$
$709$	0	0	−1.00000			$\pi$
$710$	0	0
$711$	0	0
$712$	1.20798e20	1.30227
$713$	0	0
$714$	0	0
$715$	1.57108e20	1.64459
$716$	−1.89130e20	−1.96051
$717$	0	0
$718$	0	0
$719$	0	0	1.00000			$0$
$719$	0	0	−1.00000			$\pi$
$720$	1.00306e20	1.00000
$721$	−1.66850e19	−0.164733
$722$	4.91657e19	0.480731
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−8.52592e19	−0.794329	−0.397164	−	0.917748i	$-0.630006\pi$
$727$	−8.52592e19	−0.794329	−0.397164	+	0.917748i	$0.630006\pi$
$728$	1.78300e19	0.164525
$729$	1.09419e20	1.00000
$730$	0	0
$731$	0	0
$732$	0	0
$733$	2.20448e20	1.93900	0.969499	−	0.245094i	$-0.0788188\pi$
$733$	2.20448e20	1.93900	0.969499	+	0.245094i	$0.0788188\pi$
$734$	−1.75513e20	−1.52910
$735$	0	0
$736$	2.04486e20	1.74791
$737$	0	0
$738$	−1.39452e20	−1.16958
$739$	−1.47880e19	−0.122856	−0.0614281	−	0.998112i	$-0.519565\pi$
$739$	−1.47880e19	−0.122856	−0.0614281	+	0.998112i	$0.519565\pi$
$740$	−1.28618e20	−1.05847
$741$	0	0
$742$	1.56501e19	0.126383
$743$	−1.47711e20	−1.18165	−0.590827	−	0.806798i	$-0.701198\pi$
$743$	−1.47711e20	−1.18165	−0.590827	+	0.806798i	$0.701198\pi$
$744$	0	0
$745$	0	0
$746$	2.02741e20	1.57677
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	−2.64659e20	−1.94608
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	1.65157e20	1.15938	0.579690	−	0.814837i	$-0.303174\pi$
$757$	1.65157e20	1.15938	0.579690	+	0.814837i	$0.303174\pi$
$758$	−1.91320e20	−1.33069
$759$	0	0
$760$	−1.05534e20	−0.720604
$761$	−2.53391e20	−1.71434	−0.857172	−	0.515030i	$-0.827781\pi$
$761$	−2.53391e20	−1.71434	−0.857172	+	0.515030i	$0.827781\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	5.71570e19	0.369375
$767$	5.87255e20	3.76062
$768$	0	0
$769$	−3.17054e20	−1.99365	−0.996823	−	0.0796425i	$-0.974622\pi$
$769$	−3.17054e20	−1.99365	−0.996823	+	0.0796425i	$0.974622\pi$
$770$	1.08941e19	0.0678823
$771$	0	0
$772$	0	0
$773$	−2.91108e20	−1.76522	−0.882609	−	0.470109i	$-0.844215\pi$
$773$	−2.91108e20	−1.76522	−0.882609	+	0.470109i	$0.844215\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	1.46720e20	0.842806
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	−1.80823e20	−0.993209
$785$	3.53625e20	1.92511
$786$	0	0
$787$	0	0	1.00000			$0$
$787$	0	0	−1.00000			$\pi$
$788$	−3.56231e20	−1.88820
$789$	0	0
$790$	0	0
$791$	0	0
$792$	−1.61015e20	−0.823741
$793$	0	0
$794$	−1.46067e18	−0.00734190
$795$	0	0
$796$	0	0
$797$	−4.65475e19	−0.227871	−0.113935	−	0.993488i	$-0.536346\pi$
$797$	−4.65475e19	−0.227871	−0.113935	+	0.993488i	$0.536346\pi$
$798$	0	0
$799$	0	0
$800$	−2.09715e20	−1.00000
$801$	−2.75504e20	−1.30227
$802$	9.05213e19	0.424161
$803$	0	0
$804$	0	0
$805$	3.15542e19	0.144041
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−4.24141e20	−1.87012	−0.935062	−	0.354485i	$-0.884656\pi$
$809$	−4.24141e20	−1.87012	−0.935062	+	0.354485i	$0.884656\pi$
$810$	−2.28768e20	−1.00000
$811$	−4.06829e20	−1.76306	−0.881528	−	0.472132i	$-0.843485\pi$
$811$	−4.06829e20	−1.76306	−0.881528	+	0.472132i	$0.843485\pi$
$812$	0	0
$813$	0	0
$814$	2.06462e20	0.871906
$815$	0	0
$816$	0	0
$817$	0	0
$818$	1.14492e20	0.467200
$819$	−4.06649e19	−0.164525
$820$	2.91560e20	1.16958
$821$	0	0	1.00000			$0$
$821$	0	0	−1.00000			$\pi$
$822$	0	0
$823$	1.14449e20	0.447521	0.223760	−	0.974644i	$-0.428167\pi$
$823$	1.14449e20	0.447521	0.223760	+	0.974644i	$0.428167\pi$
$824$	−5.15590e20	−1.99901
$825$	0	0
$826$	4.07211e19	0.155224
$827$	0	0	1.00000			$0$
$827$	0	0	−1.00000			$\pi$
$828$	−4.66372e20	−1.74791
$829$	0	0	1.00000			$0$
$829$	0	0	−1.00000			$\pi$
$830$	0	0
$831$	0	0
$832$	5.50972e20	1.99649
$833$	0	0
$834$	0	0
$835$	4.30397e20	1.52077
$836$	1.69407e20	0.593591
$837$	0	0
$838$	−5.67784e20	−1.95648
$839$	0	0	1.00000			$0$
$839$	0	0	−1.00000			$\pi$
$840$	0	0
$841$	2.97558e20	1.00000
$842$	0	0
$843$	0	0
$844$	−5.23751e20	−1.71683
$845$	9.18504e20	2.98596
$846$	6.03606e20	1.94608
$847$	8.28446e18	0.0264899
$848$	4.83610e20	1.53364
$849$	0	0
$850$	0	0
$851$	5.98006e20	1.85012
$852$	0	0
$853$	1.38829e20	0.422511	0.211255	−	0.977431i	$-0.432245\pi$
$853$	1.38829e20	0.422511	0.211255	+	0.977431i	$0.432245\pi$
$854$	0	0
$855$	2.40691e20	0.720604
$856$	0	0
$857$	0	0	1.00000			$0$
$857$	0	0	−1.00000			$\pi$
$858$	0	0
$859$	6.71279e20	1.94514	0.972571	−	0.232608i	$-0.0747259\pi$
$859$	6.71279e20	1.94514	0.972571	+	0.232608i	$0.0747259\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−3.15393e20	−0.884661	−0.442330	−	0.896852i	$-0.645848\pi$
$863$	−3.15393e20	−0.884661	−0.442330	+	0.896852i	$0.645848\pi$
$864$	0	0
$865$	−2.56061e20	−0.706692
$866$	0	0
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	4.90678e20	1.25955
$875$	−3.23610e19	−0.0824074
$876$	0	0
$877$	−7.18319e20	−1.80020	−0.900099	−	0.435685i	$-0.856506\pi$
$877$	−7.18319e20	−1.80020	−0.900099	+	0.435685i	$0.856506\pi$
$878$	0	0
$879$	0	0
$880$	3.36643e20	0.823741
$881$	−8.16331e20	−1.98169	−0.990843	−	0.135016i	$-0.956892\pi$
$881$	−8.16331e20	−1.98169	−0.990843	+	0.135016i	$0.956892\pi$
$882$	4.12402e20	0.993209
$883$	0	0	1.00000			$0$
$883$	0	0	−1.00000			$\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−4.28303e20	−0.991484	−0.495742	−	0.868470i	$-0.665104\pi$
$887$	−4.28303e20	−0.991484	−0.495742	+	0.868470i	$0.665104\pi$
$888$	0	0
$889$	1.37113e19	0.0312439
$890$	5.76010e20	1.30227
$891$	3.67227e20	0.823741
$892$	−6.74901e20	−1.50206
$893$	−6.35064e20	−1.40235
$894$	0	0
$895$	−9.01841e20	−1.96051
$896$	3.82052e19	0.0824074
$897$	0	0
$898$	8.40078e20	1.78396
$899$	0	0
$900$	4.78297e20	1.00000
$901$	0	0
$902$	−4.68024e20	−0.963434
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	0	0	1.00000			$0$
$907$	0	0	−1.00000			$\pi$
$908$	0	0
$909$	0	0
$910$	8.50202e19	0.164525
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	6.40462e19	0.117465
$918$	0	0
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	9.75067e20	1.74791
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−6.13297e20	−1.05847
$926$	2.37365e20	0.406575
$927$	1.17591e21	1.99901
$928$	0	0
$929$	4.97319e20	0.832770	0.416385	−	0.909188i	$-0.363297\pi$
$929$	4.97319e20	0.832770	0.416385	+	0.909188i	$0.363297\pi$
$930$	0	0
$931$	−4.33895e20	−0.715710
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	−1.25660e21	−1.99649
$937$	0	0	1.00000			$0$
$937$	0	0	−1.00000			$\pi$
$938$	0	0
$939$	0	0
$940$	−1.26199e21	−1.94608
$941$	0	0	1.00000			$0$
$941$	0	0	−1.00000			$\pi$
$942$	0	0
$943$	−1.35560e21	−2.04433
$944$	1.25834e21	1.88362
$945$	0	0
$946$	0	0
$947$	0	0	1.00000			$0$
$947$	0	0	−1.00000			$\pi$
$948$	0	0
$949$	0	0
$950$	−5.03224e20	−0.720604
$951$	0	0
$952$	0	0
$953$	0	0	1.00000			$0$
$953$	0	0	−1.00000			$\pi$
$954$	−1.10297e21	−1.53364
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	7.56944e20	1.00000
$962$	1.61128e21	2.11322
$963$	0	0
$964$	1.11401e21	1.43995
$965$	0	0
$966$	0	0
$967$	−7.15287e20	−0.904680	−0.452340	−	0.891845i	$-0.649411\pi$
$967$	−7.15287e20	−0.904680	−0.452340	+	0.891845i	$0.649411\pi$
$968$	2.56001e20	0.321451
$969$	0	0
$970$	0	0
$971$	2.39362e20	0.294117	0.147059	−	0.989128i	$-0.453019\pi$
$971$	2.39362e20	0.294117	0.147059	+	0.989128i	$0.453019\pi$
$972$	0	0
$973$	−1.17742e20	−0.142607
$974$	7.16970e19	0.0862161
$975$	0	0
$976$	0	0
$977$	0	0	1.00000			$0$
$977$	0	0	−1.00000			$\pi$
$978$	0	0
$979$	−9.24634e20	−1.07273
$980$	−8.62230e20	−0.993209
$981$	0	0
$982$	1.42100e21	1.61366
$983$	1.67670e21	1.89052	0.945261	−	0.326316i	$-0.105807\pi$
$983$	1.67670e21	1.89052	0.945261	+	0.326316i	$0.105807\pi$
$984$	0	0
$985$	−1.69864e21	−1.88820
$986$	0	0
$987$	0	0
$988$	1.32209e21	1.43868
$989$	0	0
$990$	−7.67780e20	−0.823741
$991$	0	0	1.00000			$0$
$991$	0	0	−1.00000			$\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−1.95441e21	−1.99594	−0.997972	−	0.0636503i	$-0.979726\pi$
$997$	−1.95441e21	−1.99594	−0.997972	+	0.0636503i	$0.979726\pi$
$998$	−6.11160e20	−0.619785
$999$	0	0

Display

a_p

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p

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a_n

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a_n

instead) (See

a_n

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a_n

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n

up to: 50 250 1000 (See only

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a_p

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a_p

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	40.15.e.a.19.1	✓	1
4.3	odd	2		160.15.e.b.79.1		1
5.4	even	2		40.15.e.b.19.1	yes	1
8.3	odd	2		40.15.e.b.19.1	yes	1
8.5	even	2		160.15.e.a.79.1		1
20.19	odd	2		160.15.e.a.79.1		1
40.19	odd	2	CM	40.15.e.a.19.1	✓	1
40.29	even	2		160.15.e.b.79.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
40.15.e.a.19.1	✓	1	1.1	even	1	trivial
40.15.e.a.19.1	✓	1	40.19	odd	2	CM
40.15.e.b.19.1	yes	1	5.4	even	2
40.15.e.b.19.1	yes	1	8.3	odd	2
160.15.e.a.79.1		1	8.5	even	2
160.15.e.a.79.1		1	20.19	odd	2
160.15.e.b.79.1		1	4.3	odd	2
160.15.e.b.79.1		1	40.29	even	2

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Classical	Maass
Hilbert	Bianchi

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

Introduction

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Modular forms

Varieties

Fields

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Groups

Database

Properties

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Character values

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	40.15.e.a.19.1

Level	$40$
Weight	$15$
Character	40.19
Self dual	yes
Analytic conductor	$49.732$
Analytic rank	$0$
Dimension	$1$
CM discriminant	-40
Inner twists	$2$