LMFDB - Embedded newform 12.14.a.a.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [12,14,Mod(1,12)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("12.1"); S:= CuspForms(chi, 14); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(12, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 14, names="a")

Level:	$ N $	$=$	$ 12 = 2^{2} \cdot 3 $
Weight:	$ k $	$=$	$ 14 $
Character orbit:	$[\chi]$	$=$	12.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,0,-729] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	yes
Analytic conductor:	$12.8677114742$
Analytic rank:	$0$
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	$\chi$	$=$	12.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

$q-729.000 q^{3} -14850.0 q^{5} -62896.0 q^{7} +531441. q^{9} +5.10484e6 q^{11} +1.14841e7 q^{13} +1.08256e7 q^{15} +1.19965e8 q^{17} +3.32601e8 q^{19} +4.58512e7 q^{21} +3.50924e8 q^{23} -1.00018e9 q^{25} -3.87420e8 q^{27} -1.76110e9 q^{29} -3.93422e9 q^{31} -3.72143e9 q^{33} +9.34006e8 q^{35} -7.80357e9 q^{37} -8.37192e9 q^{39} +5.28826e10 q^{41} +2.60184e10 q^{43} -7.89190e9 q^{45} +1.42371e11 q^{47} -9.29331e10 q^{49} -8.74544e10 q^{51} +1.37700e10 q^{53} -7.58068e10 q^{55} -2.42466e11 q^{57} +3.36465e11 q^{59} -6.77261e11 q^{61} -3.34255e10 q^{63} -1.70539e11 q^{65} +2.62302e11 q^{67} -2.55824e11 q^{69} +1.59496e12 q^{71} +5.78813e11 q^{73} +7.29132e11 q^{75} -3.21074e11 q^{77} +2.49582e12 q^{79} +2.82430e11 q^{81} -2.69324e12 q^{83} -1.78148e12 q^{85} +1.28384e12 q^{87} -7.93554e12 q^{89} -7.22305e11 q^{91} +2.86805e12 q^{93} -4.93913e12 q^{95} -7.85860e9 q^{97} +2.71292e12 q^{99} +O(q^{100})$

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
$3$	−729.000	−0.577350
$3$	−729.000	−0.577350
$4$	0	0
$5$	−14850.0	−0.425032	−0.212516	−	0.977158i	$-0.568166\pi$
$5$	−14850.0	−0.425032	−0.212516	+	0.977158i	$0.568166\pi$
$6$	0	0
$7$	−62896.0	−0.202063	−0.101031	−	0.994883i	$-0.532214\pi$
$7$	−62896.0	−0.202063	−0.101031	+	0.994883i	$0.532214\pi$
$8$	0	0
$9$	531441.	0.333333
$10$	0	0
$11$	5.10484e6	0.868819	0.434410	−	0.900715i	$-0.356957\pi$
$11$	5.10484e6	0.868819	0.434410	+	0.900715i	$0.356957\pi$
$12$	0	0
$13$	1.14841e7	0.659881	0.329940	−	0.944002i	$-0.392971\pi$
$13$	1.14841e7	0.659881	0.329940	+	0.944002i	$0.392971\pi$
$14$	0	0
$15$	1.08256e7	0.245392
$16$	0	0
$17$	1.19965e8	1.20541	0.602707	−	0.797963i	$-0.294089\pi$
$17$	1.19965e8	1.20541	0.602707	+	0.797963i	$0.294089\pi$
$18$	0	0
$19$	3.32601e8	1.62190	0.810952	−	0.585113i	$-0.198950\pi$
$19$	3.32601e8	1.62190	0.810952	+	0.585113i	$0.198950\pi$
$20$	0	0
$21$	4.58512e7	0.116661
$22$	0	0
$23$	3.50924e8	0.494291	0.247145	−	0.968978i	$-0.420507\pi$
$23$	3.50924e8	0.494291	0.247145	+	0.968978i	$0.420507\pi$
$24$	0	0
$25$	−1.00018e9	−0.819348
$26$	0	0
$27$	−3.87420e8	−0.192450
$28$	0	0
$29$	−1.76110e9	−0.549791	−0.274895	−	0.961474i	$-0.588643\pi$
$29$	−1.76110e9	−0.549791	−0.274895	+	0.961474i	$0.588643\pi$
$30$	0	0
$31$	−3.93422e9	−0.796174	−0.398087	−	0.917348i	$-0.630326\pi$
$31$	−3.93422e9	−0.796174	−0.398087	+	0.917348i	$0.630326\pi$
$32$	0	0
$33$	−3.72143e9	−0.501613
$34$	0	0
$35$	9.34006e8	0.0858830
$36$	0	0
$37$	−7.80357e9	−0.500014	−0.250007	−	0.968244i	$-0.580433\pi$
$37$	−7.80357e9	−0.500014	−0.250007	+	0.968244i	$0.580433\pi$
$38$	0	0
$39$	−8.37192e9	−0.380982
$40$	0	0
$41$	5.28826e10	1.73867	0.869337	−	0.494220i	$-0.164546\pi$
$41$	5.28826e10	1.73867	0.869337	+	0.494220i	$0.164546\pi$
$42$	0	0
$43$	2.60184e10	0.627676	0.313838	−	0.949476i	$-0.398385\pi$
$43$	2.60184e10	0.627676	0.313838	+	0.949476i	$0.398385\pi$
$44$	0	0
$45$	−7.89190e9	−0.141677
$46$	0	0
$47$	1.42371e11	1.92657	0.963285	−	0.268482i	$-0.0865220\pi$
$47$	1.42371e11	1.92657	0.963285	+	0.268482i	$0.0865220\pi$
$48$	0	0
$49$	−9.29331e10	−0.959171
$50$	0	0
$51$	−8.74544e10	−0.695946
$52$	0	0
$53$	1.37700e10	0.0853379	0.0426689	−	0.999089i	$-0.486414\pi$
$53$	1.37700e10	0.0853379	0.0426689	+	0.999089i	$0.486414\pi$
$54$	0	0
$55$	−7.58068e10	−0.369276
$56$	0	0
$57$	−2.42466e11	−0.936407
$58$	0	0
$59$	3.36465e11	1.03849	0.519244	−	0.854626i	$-0.326213\pi$
$59$	3.36465e11	1.03849	0.519244	+	0.854626i	$0.326213\pi$
$60$	0	0
$61$	−6.77261e11	−1.68311	−0.841554	−	0.540173i	$-0.818359\pi$
$61$	−6.77261e11	−1.68311	−0.841554	+	0.540173i	$0.818359\pi$
$62$	0	0
$63$	−3.34255e10	−0.0673542
$64$	0	0
$65$	−1.70539e11	−0.280470
$66$	0	0
$67$	2.62302e11	0.354254	0.177127	−	0.984188i	$-0.443320\pi$
$67$	2.62302e11	0.354254	0.177127	+	0.984188i	$0.443320\pi$
$68$	0	0
$69$	−2.55824e11	−0.285379
$70$	0	0
$71$	1.59496e12	1.47765	0.738824	−	0.673899i	$-0.235382\pi$
$71$	1.59496e12	1.47765	0.738824	+	0.673899i	$0.235382\pi$
$72$	0	0
$73$	5.78813e11	0.447651	0.223825	−	0.974629i	$-0.428145\pi$
$73$	5.78813e11	0.447651	0.223825	+	0.974629i	$0.428145\pi$
$74$	0	0
$75$	7.29132e11	0.473051
$76$	0	0
$77$	−3.21074e11	−0.175556
$78$	0	0
$79$	2.49582e12	1.15515	0.577573	−	0.816339i	$-0.304000\pi$
$79$	2.49582e12	1.15515	0.577573	+	0.816339i	$0.304000\pi$
$80$	0	0
$81$	2.82430e11	0.111111
$82$	0	0
$83$	−2.69324e12	−0.904205	−0.452102	−	0.891966i	$-0.649326\pi$
$83$	−2.69324e12	−0.904205	−0.452102	+	0.891966i	$0.649326\pi$
$84$	0	0
$85$	−1.78148e12	−0.512339
$86$	0	0
$87$	1.28384e12	0.317422
$88$	0	0
$89$	−7.93554e12	−1.69255	−0.846275	−	0.532747i	$-0.821160\pi$
$89$	−7.93554e12	−1.69255	−0.846275	+	0.532747i	$0.821160\pi$
$90$	0	0
$91$	−7.22305e11	−0.133337
$92$	0	0
$93$	2.86805e12	0.459671
$94$	0	0
$95$	−4.93913e12	−0.689361
$96$	0	0
$97$	−7.85860e9	−0.000957919 0	−0.000478960	−	1.00000i	$-0.500152\pi$
$97$	−7.85860e9	−0.000957919 0	−0.000478960		1.00000i	$0.500152\pi$
$98$	0	0
$99$	2.71292e12	0.289606
$100$	0	0
$101$	−5.65694e12	−0.530265	−0.265132	−	0.964212i	$-0.585416\pi$
$101$	−5.65694e12	−0.530265	−0.265132	+	0.964212i	$0.585416\pi$
$102$	0	0
$103$	9.23368e12	0.761961	0.380981	−	0.924583i	$-0.375586\pi$
$103$	9.23368e12	0.761961	0.380981	+	0.924583i	$0.375586\pi$
$104$	0	0
$105$	−6.80890e11	−0.0495846
$106$	0	0
$107$	−5.05786e12	−0.325816	−0.162908	−	0.986641i	$-0.552087\pi$
$107$	−5.05786e12	−0.325816	−0.162908	+	0.986641i	$0.552087\pi$
$108$	0	0
$109$	1.56956e13	0.896407	0.448203	−	0.893932i	$-0.352064\pi$
$109$	1.56956e13	0.896407	0.448203	+	0.893932i	$0.352064\pi$
$110$	0	0
$111$	5.68880e12	0.288683
$112$	0	0
$113$	3.92295e13	1.77257	0.886283	−	0.463144i	$-0.153279\pi$
$113$	3.92295e13	1.77257	0.886283	+	0.463144i	$0.153279\pi$
$114$	0	0
$115$	−5.21122e12	−0.210089
$116$	0	0
$117$	6.10313e12	0.219960
$118$	0	0
$119$	−7.54531e12	−0.243569
$120$	0	0
$121$	−8.46336e12	−0.245153
$122$	0	0
$123$	−3.85515e13	−1.00382
$124$	0	0
$125$	3.29801e13	0.773281
$126$	0	0
$127$	−7.32993e13	−1.55016	−0.775078	−	0.631866i	$-0.782289\pi$
$127$	−7.32993e13	−1.55016	−0.775078	+	0.631866i	$0.782289\pi$
$128$	0	0
$129$	−1.89674e13	−0.362389
$130$	0	0
$131$	−3.69422e13	−0.638645	−0.319322	−	0.947646i	$-0.603455\pi$
$131$	−3.69422e13	−0.638645	−0.319322	+	0.947646i	$0.603455\pi$
$132$	0	0
$133$	−2.09193e13	−0.327726
$134$	0	0
$135$	5.75319e12	0.0817974
$136$	0	0
$137$	8.09482e13	1.04598	0.522990	−	0.852339i	$-0.324816\pi$
$137$	8.09482e13	1.04598	0.522990	+	0.852339i	$0.324816\pi$
$138$	0	0
$139$	−1.06615e13	−0.125379	−0.0626893	−	0.998033i	$-0.519968\pi$
$139$	−1.06615e13	−0.125379	−0.0626893	+	0.998033i	$0.519968\pi$
$140$	0	0
$141$	−1.03788e14	−1.11231
$142$	0	0
$143$	5.86245e13	0.573317
$144$	0	0
$145$	2.61524e13	0.233679
$146$	0	0
$147$	6.77482e13	0.553777
$148$	0	0
$149$	6.66354e13	0.498878	0.249439	−	0.968391i	$-0.419754\pi$
$149$	6.66354e13	0.498878	0.249439	+	0.968391i	$0.419754\pi$
$150$	0	0
$151$	5.39592e13	0.370438	0.185219	−	0.982697i	$-0.440701\pi$
$151$	5.39592e13	0.370438	0.185219	+	0.982697i	$0.440701\pi$
$152$	0	0
$153$	6.37542e13	0.401804
$154$	0	0
$155$	5.84232e13	0.338399
$156$	0	0
$157$	−4.23104e13	−0.225475	−0.112738	−	0.993625i	$-0.535962\pi$
$157$	−4.23104e13	−0.225475	−0.112738	+	0.993625i	$0.535962\pi$
$158$	0	0
$159$	−1.00384e13	−0.0492699
$160$	0	0
$161$	−2.20717e13	−0.0998776
$162$	0	0
$163$	3.25135e14	1.35783	0.678914	−	0.734218i	$-0.262451\pi$
$163$	3.25135e14	1.35783	0.678914	+	0.734218i	$0.262451\pi$
$164$	0	0
$165$	5.52632e13	0.213201
$166$	0	0
$167$	−4.29305e14	−1.53147	−0.765735	−	0.643156i	$-0.777625\pi$
$167$	−4.29305e14	−1.53147	−0.765735	+	0.643156i	$0.777625\pi$
$168$	0	0
$169$	−1.70990e14	−0.564557
$170$	0	0
$171$	1.76758e14	0.540635
$172$	0	0
$173$	−3.92694e14	−1.11367	−0.556833	−	0.830625i	$-0.687984\pi$
$173$	−3.92694e14	−1.11367	−0.556833	+	0.830625i	$0.687984\pi$
$174$	0	0
$175$	6.29074e13	0.165560
$176$	0	0
$177$	−2.45283e14	−0.599572
$178$	0	0
$179$	−5.72645e14	−1.30119	−0.650595	−	0.759425i	$-0.725480\pi$
$179$	−5.72645e14	−1.30119	−0.650595	+	0.759425i	$0.725480\pi$
$180$	0	0
$181$	−5.19989e14	−1.09922	−0.549608	−	0.835422i	$-0.685223\pi$
$181$	−5.19989e14	−1.09922	−0.549608	+	0.835422i	$0.685223\pi$
$182$	0	0
$183$	4.93723e14	0.971743
$184$	0	0
$185$	1.15883e14	0.212522
$186$	0	0
$187$	6.12401e14	1.04729
$188$	0	0
$189$	2.43672e13	0.0388870
$190$	0	0
$191$	2.63577e13	0.0392817	0.0196408	−	0.999807i	$-0.493748\pi$
$191$	2.63577e13	0.0392817	0.0196408	+	0.999807i	$0.493748\pi$
$192$	0	0
$193$	3.77456e14	0.525706	0.262853	−	0.964836i	$-0.415336\pi$
$193$	3.77456e14	0.525706	0.262853	+	0.964836i	$0.415336\pi$
$194$	0	0
$195$	1.24323e14	0.161930
$196$	0	0
$197$	4.93892e14	0.602007	0.301004	−	0.953623i	$-0.402678\pi$
$197$	4.93892e14	0.602007	0.301004	+	0.953623i	$0.402678\pi$
$198$	0	0
$199$	6.29123e14	0.718109	0.359055	−	0.933317i	$-0.383099\pi$
$199$	6.29123e14	0.718109	0.359055	+	0.933317i	$0.383099\pi$
$200$	0	0
$201$	−1.91218e14	−0.204529
$202$	0	0
$203$	1.10766e14	0.111092
$204$	0	0
$205$	−7.85307e14	−0.738992
$206$	0	0
$207$	1.86495e14	0.164764
$208$	0	0
$209$	1.69787e15	1.40914
$210$	0	0
$211$	−1.72575e13	−0.0134630	−0.00673149	−	0.999977i	$-0.502143\pi$
$211$	−1.72575e13	−0.0134630	−0.00673149	+	0.999977i	$0.502143\pi$
$212$	0	0
$213$	−1.16273e15	−0.853120
$214$	0	0
$215$	−3.86373e14	−0.266782
$216$	0	0
$217$	2.47447e14	0.160877
$218$	0	0
$219$	−4.21955e14	−0.258451
$220$	0	0
$221$	1.37769e15	0.795429
$222$	0	0
$223$	−2.19976e15	−1.19783	−0.598913	−	0.800814i	$-0.704400\pi$
$223$	−2.19976e15	−1.19783	−0.598913	+	0.800814i	$0.704400\pi$
$224$	0	0
$225$	−5.31537e14	−0.273116
$226$	0	0
$227$	−2.29969e15	−1.11558	−0.557790	−	0.829982i	$-0.688351\pi$
$227$	−2.29969e15	−1.11558	−0.557790	+	0.829982i	$0.688351\pi$
$228$	0	0
$229$	−2.64886e15	−1.21375	−0.606875	−	0.794797i	$-0.707577\pi$
$229$	−2.64886e15	−1.21375	−0.606875	+	0.794797i	$0.707577\pi$
$230$	0	0
$231$	2.34063e14	0.101357
$232$	0	0
$233$	3.99553e15	1.63592	0.817958	−	0.575278i	$-0.195106\pi$
$233$	3.99553e15	1.63592	0.817958	+	0.575278i	$0.195106\pi$
$234$	0	0
$235$	−2.11421e15	−0.818853
$236$	0	0
$237$	−1.81945e15	−0.666924
$238$	0	0
$239$	−1.38016e15	−0.479009	−0.239505	−	0.970895i	$-0.576985\pi$
$239$	−1.38016e15	−0.479009	−0.239505	+	0.970895i	$0.576985\pi$
$240$	0	0
$241$	2.99230e15	0.983772	0.491886	−	0.870660i	$-0.336308\pi$
$241$	2.99230e15	0.983772	0.491886	+	0.870660i	$0.336308\pi$
$242$	0	0
$243$	−2.05891e14	−0.0641500
$244$	0	0
$245$	1.38006e15	0.407678
$246$	0	0
$247$	3.81963e15	1.07026
$248$	0	0
$249$	1.96337e15	0.522043
$250$	0	0
$251$	−9.23660e14	−0.233149	−0.116574	−	0.993182i	$-0.537191\pi$
$251$	−9.23660e14	−0.233149	−0.116574	+	0.993182i	$0.537191\pi$
$252$	0	0
$253$	1.79141e15	0.429449
$254$	0	0
$255$	1.29870e15	0.295799
$256$	0	0
$257$	5.54095e15	1.19955	0.599776	−	0.800168i	$-0.295256\pi$
$257$	5.54095e15	1.19955	0.599776	+	0.800168i	$0.295256\pi$
$258$	0	0
$259$	4.90813e14	0.101034
$260$	0	0
$261$	−9.35922e14	−0.183264
$262$	0	0
$263$	−4.04352e15	−0.753437	−0.376719	−	0.926328i	$-0.622948\pi$
$263$	−4.04352e15	−0.753437	−0.376719	+	0.926328i	$0.622948\pi$
$264$	0	0
$265$	−2.04485e14	−0.0362713
$266$	0	0
$267$	5.78501e15	0.977194
$268$	0	0
$269$	−1.14344e16	−1.84002	−0.920009	−	0.391896i	$-0.871819\pi$
$269$	−1.14344e16	−1.84002	−0.920009	+	0.391896i	$0.871819\pi$
$270$	0	0
$271$	6.47216e15	0.992541	0.496271	−	0.868168i	$-0.334702\pi$
$271$	6.47216e15	0.992541	0.496271	+	0.868168i	$0.334702\pi$
$272$	0	0
$273$	5.26560e14	0.0769823
$274$	0	0
$275$	−5.10576e15	−0.711865
$276$	0	0
$277$	−1.16730e16	−1.55262	−0.776309	−	0.630352i	$-0.782910\pi$
$277$	−1.16730e16	−1.55262	−0.776309	+	0.630352i	$0.782910\pi$
$278$	0	0
$279$	−2.09081e15	−0.265391
$280$	0	0
$281$	−8.34626e14	−0.101135	−0.0505674	−	0.998721i	$-0.516103\pi$
$281$	−8.34626e14	−0.101135	−0.0505674	+	0.998721i	$0.516103\pi$
$282$	0	0
$283$	3.98360e15	0.460960	0.230480	−	0.973077i	$-0.425970\pi$
$283$	3.98360e15	0.460960	0.230480	+	0.973077i	$0.425970\pi$
$284$	0	0
$285$	3.60062e15	0.398003
$286$	0	0
$287$	−3.32611e15	−0.351321
$288$	0	0
$289$	4.48698e15	0.453021
$290$	0	0
$291$	5.72892e12	0.000553055 0
$292$	0	0
$293$	−1.53058e16	−1.41324	−0.706620	−	0.707593i	$-0.749781\pi$
$293$	−1.53058e16	−1.41324	−0.706620	+	0.707593i	$0.749781\pi$
$294$	0	0
$295$	−4.99651e15	−0.441391
$296$	0	0
$297$	−1.97772e15	−0.167204
$298$	0	0
$299$	4.03005e15	0.326173
$300$	0	0
$301$	−1.63645e15	−0.126830
$302$	0	0
$303$	4.12391e15	0.306148
$304$	0	0
$305$	1.00573e16	0.715374
$306$	0	0
$307$	1.03151e16	0.703194	0.351597	−	0.936151i	$-0.385639\pi$
$307$	1.03151e16	0.703194	0.351597	+	0.936151i	$0.385639\pi$
$308$	0	0
$309$	−6.73135e15	−0.439919
$310$	0	0
$311$	1.98138e16	1.24172	0.620861	−	0.783920i	$-0.286783\pi$
$311$	1.98138e16	1.24172	0.620861	+	0.783920i	$0.286783\pi$
$312$	0	0
$313$	1.19327e16	0.717302	0.358651	−	0.933472i	$-0.383237\pi$
$313$	1.19327e16	0.717302	0.358651	+	0.933472i	$0.383237\pi$
$314$	0	0
$315$	4.96369e14	0.0286277
$316$	0	0
$317$	8.23776e15	0.455957	0.227979	−	0.973666i	$-0.426788\pi$
$317$	8.23776e15	0.455957	0.227979	+	0.973666i	$0.426788\pi$
$318$	0	0
$319$	−8.99014e15	−0.477669
$320$	0	0
$321$	3.68718e15	0.188110
$322$	0	0
$323$	3.99004e16	1.95506
$324$	0	0
$325$	−1.14862e16	−0.540672
$326$	0	0
$327$	−1.14421e16	−0.517541
$328$	0	0
$329$	−8.95455e15	−0.389287
$330$	0	0
$331$	−2.63775e16	−1.10243	−0.551217	−	0.834362i	$-0.685836\pi$
$331$	−2.63775e16	−1.10243	−0.551217	+	0.834362i	$0.685836\pi$
$332$	0	0
$333$	−4.14714e15	−0.166671
$334$	0	0
$335$	−3.89518e15	−0.150569
$336$	0	0
$337$	9.50245e15	0.353379	0.176690	−	0.984267i	$-0.443461\pi$
$337$	9.50245e15	0.353379	0.176690	+	0.984267i	$0.443461\pi$
$338$	0	0
$339$	−2.85983e16	−1.02339
$340$	0	0
$341$	−2.00836e16	−0.691731
$342$	0	0
$343$	1.19391e16	0.395875
$344$	0	0
$345$	3.79898e15	0.121295
$346$	0	0
$347$	2.18727e15	0.0672606	0.0336303	−	0.999434i	$-0.489293\pi$
$347$	2.18727e15	0.0672606	0.0336303	+	0.999434i	$0.489293\pi$
$348$	0	0
$349$	−5.90444e16	−1.74909	−0.874547	−	0.484940i	$-0.838841\pi$
$349$	−5.90444e16	−1.74909	−0.874547	+	0.484940i	$0.838841\pi$
$350$	0	0
$351$	−4.44918e15	−0.126994
$352$	0	0
$353$	3.46542e16	0.953279	0.476639	−	0.879099i	$-0.341855\pi$
$353$	3.46542e16	0.953279	0.476639	+	0.879099i	$0.341855\pi$
$354$	0	0
$355$	−2.36852e16	−0.628047
$356$	0	0
$357$	5.50053e15	0.140625
$358$	0	0
$359$	−2.37891e16	−0.586496	−0.293248	−	0.956036i	$-0.594736\pi$
$359$	−2.37891e16	−0.586496	−0.293248	+	0.956036i	$0.594736\pi$
$360$	0	0
$361$	6.85705e16	1.63057
$362$	0	0
$363$	6.16979e15	0.141539
$364$	0	0
$365$	−8.59537e15	−0.190266
$366$	0	0
$367$	−4.84086e16	−1.03417	−0.517086	−	0.855933i	$-0.672983\pi$
$367$	−4.84086e16	−1.03417	−0.517086	+	0.855933i	$0.672983\pi$
$368$	0	0
$369$	2.81040e16	0.579558
$370$	0	0
$371$	−8.66080e14	−0.0172436
$372$	0	0
$373$	9.59492e16	1.84474	0.922368	−	0.386313i	$-0.126252\pi$
$373$	9.59492e16	1.84474	0.922368	+	0.386313i	$0.126252\pi$
$374$	0	0
$375$	−2.40425e16	−0.446454
$376$	0	0
$377$	−2.02247e16	−0.362796
$378$	0	0
$379$	2.98016e15	0.0516517	0.0258259	−	0.999666i	$-0.491778\pi$
$379$	2.98016e15	0.0516517	0.0258259	+	0.999666i	$0.491778\pi$
$380$	0	0
$381$	5.34352e16	0.894983
$382$	0	0
$383$	5.97285e16	0.966917	0.483459	−	0.875367i	$-0.339380\pi$
$383$	5.97285e16	0.966917	0.483459	+	0.875367i	$0.339380\pi$
$384$	0	0
$385$	4.76795e15	0.0746168
$386$	0	0
$387$	1.38273e16	0.209225
$388$	0	0
$389$	−9.41541e16	−1.37774	−0.688869	−	0.724886i	$-0.741893\pi$
$389$	−9.41541e16	−1.37774	−0.688869	+	0.724886i	$0.741893\pi$
$390$	0	0
$391$	4.20986e16	0.595824
$392$	0	0
$393$	2.69309e16	0.368722
$394$	0	0
$395$	−3.70629e16	−0.490974
$396$	0	0
$397$	1.15887e17	1.48558	0.742792	−	0.669522i	$-0.233501\pi$
$397$	1.15887e17	1.48558	0.742792	+	0.669522i	$0.233501\pi$
$398$	0	0
$399$	1.52502e16	0.189213
$400$	0	0
$401$	−6.07329e16	−0.729433	−0.364717	−	0.931119i	$-0.618834\pi$
$401$	−6.07329e16	−0.729433	−0.364717	+	0.931119i	$0.618834\pi$
$402$	0	0
$403$	−4.51811e16	−0.525380
$404$	0	0
$405$	−4.19408e15	−0.0472258
$406$	0	0
$407$	−3.98359e16	−0.434422
$408$	0	0
$409$	−1.52632e17	−1.61230	−0.806148	−	0.591714i	$-0.798452\pi$
$409$	−1.52632e17	−1.61230	−0.806148	+	0.591714i	$0.798452\pi$
$410$	0	0
$411$	−5.90112e16	−0.603897
$412$	0	0
$413$	−2.11623e16	−0.209840
$414$	0	0
$415$	3.99945e16	0.384316
$416$	0	0
$417$	7.77225e15	0.0723873
$418$	0	0
$419$	−5.80860e16	−0.524421	−0.262211	−	0.965011i	$-0.584452\pi$
$419$	−5.80860e16	−0.524421	−0.262211	+	0.965011i	$0.584452\pi$
$420$	0	0
$421$	−1.52831e17	−1.33775	−0.668877	−	0.743373i	$-0.733225\pi$
$421$	−1.52831e17	−1.33775	−0.668877	+	0.743373i	$0.733225\pi$
$422$	0	0
$423$	7.56616e16	0.642190
$424$	0	0
$425$	−1.19987e17	−0.987653
$426$	0	0
$427$	4.25970e16	0.340093
$428$	0	0
$429$	−4.27373e16	−0.331005
$430$	0	0
$431$	2.05480e17	1.54407	0.772034	−	0.635581i	$-0.219239\pi$
$431$	2.05480e17	1.54407	0.772034	+	0.635581i	$0.219239\pi$
$432$	0	0
$433$	1.51022e17	1.10120	0.550602	−	0.834768i	$-0.314398\pi$
$433$	1.51022e17	1.10120	0.550602	+	0.834768i	$0.314398\pi$
$434$	0	0
$435$	−1.90651e16	−0.134914
$436$	0	0
$437$	1.16718e17	0.801692
$438$	0	0
$439$	−2.02468e17	−1.35001	−0.675004	−	0.737814i	$-0.735858\pi$
$439$	−2.02468e17	−1.35001	−0.675004	+	0.737814i	$0.735858\pi$
$440$	0	0
$441$	−4.93885e16	−0.319724
$442$	0	0
$443$	2.82951e17	1.77864	0.889319	−	0.457288i	$-0.151179\pi$
$443$	2.82951e17	1.77864	0.889319	+	0.457288i	$0.151179\pi$
$444$	0	0
$445$	1.17843e17	0.719387
$446$	0	0
$447$	−4.85772e16	−0.288027
$448$	0	0
$449$	−1.02746e17	−0.591783	−0.295891	−	0.955222i	$-0.595617\pi$
$449$	−1.02746e17	−0.591783	−0.295891	+	0.955222i	$0.595617\pi$
$450$	0	0
$451$	2.69957e17	1.51059
$452$	0	0
$453$	−3.93362e16	−0.213872
$454$	0	0
$455$	1.07262e16	0.0566725
$456$	0	0
$457$	−3.33812e17	−1.71414	−0.857072	−	0.515197i	$-0.827719\pi$
$457$	−3.33812e17	−1.71414	−0.857072	+	0.515197i	$0.827719\pi$
$458$	0	0
$459$	−4.64768e16	−0.231982
$460$	0	0
$461$	3.53590e17	1.71571	0.857855	−	0.513891i	$-0.171797\pi$
$461$	3.53590e17	1.71571	0.857855	+	0.513891i	$0.171797\pi$
$462$	0	0
$463$	1.10554e17	0.521552	0.260776	−	0.965399i	$-0.416022\pi$
$463$	1.10554e17	0.521552	0.260776	+	0.965399i	$0.416022\pi$
$464$	0	0
$465$	−4.25905e16	−0.195375
$466$	0	0
$467$	1.26768e17	0.565521	0.282761	−	0.959191i	$-0.408750\pi$
$467$	1.26768e17	0.565521	0.282761	+	0.959191i	$0.408750\pi$
$468$	0	0
$469$	−1.64977e16	−0.0715814
$470$	0	0
$471$	3.08443e16	0.130178
$472$	0	0
$473$	1.32820e17	0.545337
$474$	0	0
$475$	−3.32661e17	−1.32890
$476$	0	0
$477$	7.31796e15	0.0284460
$478$	0	0
$479$	−3.85822e17	−1.45951	−0.729754	−	0.683709i	$-0.760366\pi$
$479$	−3.85822e17	−1.45951	−0.729754	+	0.683709i	$0.760366\pi$
$480$	0	0
$481$	−8.96170e16	−0.329950
$482$	0	0
$483$	1.60903e16	0.0576644
$484$	0	0
$485$	1.16700e14	0.000407146 0
$486$	0	0
$487$	2.37010e17	0.805061	0.402530	−	0.915407i	$-0.368131\pi$
$487$	2.37010e17	0.805061	0.402530	+	0.915407i	$0.368131\pi$
$488$	0	0
$489$	−2.37024e17	−0.783942
$490$	0	0
$491$	−1.82105e17	−0.586532	−0.293266	−	0.956031i	$-0.594742\pi$
$491$	−1.82105e17	−0.586532	−0.293266	+	0.956031i	$0.594742\pi$
$492$	0	0
$493$	−2.11270e17	−0.662725
$494$	0	0
$495$	−4.02868e16	−0.123092
$496$	0	0
$497$	−1.00317e17	−0.298577
$498$	0	0
$499$	−1.89109e17	−0.548351	−0.274175	−	0.961680i	$-0.588405\pi$
$499$	−1.89109e17	−0.548351	−0.274175	+	0.961680i	$0.588405\pi$
$500$	0	0
$501$	3.12963e17	0.884195
$502$	0	0
$503$	−3.64373e17	−1.00312	−0.501561	−	0.865122i	$-0.667241\pi$
$503$	−3.64373e17	−1.00312	−0.501561	+	0.865122i	$0.667241\pi$
$504$	0	0
$505$	8.40056e16	0.225379
$506$	0	0
$507$	1.24652e17	0.325947
$508$	0	0
$509$	2.81987e16	0.0718727	0.0359363	−	0.999354i	$-0.488559\pi$
$509$	2.81987e16	0.0718727	0.0359363	+	0.999354i	$0.488559\pi$
$510$	0	0
$511$	−3.64050e16	−0.0904535
$512$	0	0
$513$	−1.28856e17	−0.312136
$514$	0	0
$515$	−1.37120e17	−0.323858
$516$	0	0
$517$	7.26779e17	1.67384
$518$	0	0
$519$	2.86274e17	0.642975
$520$	0	0
$521$	2.17349e17	0.476116	0.238058	−	0.971251i	$-0.423489\pi$
$521$	2.17349e17	0.476116	0.238058	+	0.971251i	$0.423489\pi$
$522$	0	0
$523$	8.28792e17	1.77086	0.885430	−	0.464772i	$-0.153864\pi$
$523$	8.28792e17	1.77086	0.885430	+	0.464772i	$0.153864\pi$
$524$	0	0
$525$	−4.58595e16	−0.0955858
$526$	0	0
$527$	−4.71969e17	−0.959719
$528$	0	0
$529$	−3.80889e17	−0.755677
$530$	0	0
$531$	1.78811e17	0.346163
$532$	0	0
$533$	6.07310e17	1.14732
$534$	0	0
$535$	7.51092e16	0.138482
$536$	0	0
$537$	4.17458e17	0.751242
$538$	0	0
$539$	−4.74408e17	−0.833346
$540$	0	0
$541$	8.20076e17	1.40628	0.703140	−	0.711052i	$-0.251781\pi$
$541$	8.20076e17	1.40628	0.703140	+	0.711052i	$0.251781\pi$
$542$	0	0
$543$	3.79072e17	0.634633
$544$	0	0
$545$	−2.33079e17	−0.381001
$546$	0	0
$547$	−1.07695e18	−1.71902	−0.859508	−	0.511122i	$-0.829230\pi$
$547$	−1.07695e18	−1.71902	−0.859508	+	0.511122i	$0.829230\pi$
$548$	0	0
$549$	−3.59924e17	−0.561036
$550$	0	0
$551$	−5.85744e17	−0.891708
$552$	0	0
$553$	−1.56977e17	−0.233412
$554$	0	0
$555$	−8.44787e16	−0.122699
$556$	0	0
$557$	4.98702e17	0.707591	0.353796	−	0.935323i	$-0.384891\pi$
$557$	4.98702e17	0.707591	0.353796	+	0.935323i	$0.384891\pi$
$558$	0	0
$559$	2.98798e17	0.414192
$560$	0	0
$561$	−4.46440e17	−0.604651
$562$	0	0
$563$	8.27432e16	0.109503	0.0547517	−	0.998500i	$-0.482563\pi$
$563$	8.27432e16	0.109503	0.0547517	+	0.998500i	$0.482563\pi$
$564$	0	0
$565$	−5.82557e17	−0.753397
$566$	0	0
$567$	−1.77637e16	−0.0224514
$568$	0	0
$569$	−3.61100e17	−0.446065	−0.223032	−	0.974811i	$-0.571596\pi$
$569$	−3.61100e17	−0.446065	−0.223032	+	0.974811i	$0.571596\pi$
$570$	0	0
$571$	1.90289e17	0.229763	0.114881	−	0.993379i	$-0.463351\pi$
$571$	1.90289e17	0.229763	0.114881	+	0.993379i	$0.463351\pi$
$572$	0	0
$573$	−1.92147e16	−0.0226793
$574$	0	0
$575$	−3.50988e17	−0.404996
$576$	0	0
$577$	−3.12413e17	−0.352441	−0.176221	−	0.984351i	$-0.556387\pi$
$577$	−3.12413e17	−0.352441	−0.176221	+	0.984351i	$0.556387\pi$
$578$	0	0
$579$	−2.75165e17	−0.303517
$580$	0	0
$581$	1.69394e17	0.182706
$582$	0	0
$583$	7.02938e16	0.0741432
$584$	0	0
$585$	−9.06314e16	−0.0934901
$586$	0	0
$587$	5.24758e17	0.529433	0.264717	−	0.964326i	$-0.414722\pi$
$587$	5.24758e17	0.529433	0.264717	+	0.964326i	$0.414722\pi$
$588$	0	0
$589$	−1.30853e18	−1.29132
$590$	0	0
$591$	−3.60048e17	−0.347569
$592$	0	0
$593$	6.77574e17	0.639884	0.319942	−	0.947437i	$-0.396337\pi$
$593$	6.77574e17	0.639884	0.319942	+	0.947437i	$0.396337\pi$
$594$	0	0
$595$	1.12048e17	0.103525
$596$	0	0
$597$	−4.58631e17	−0.414601
$598$	0	0
$599$	1.72875e18	1.52918	0.764590	−	0.644517i	$-0.222941\pi$
$599$	1.72875e18	1.52918	0.764590	+	0.644517i	$0.222941\pi$
$600$	0	0
$601$	−1.15191e18	−0.997094	−0.498547	−	0.866863i	$-0.666133\pi$
$601$	−1.15191e18	−0.997094	−0.498547	+	0.866863i	$0.666133\pi$
$602$	0	0
$603$	1.39398e17	0.118085
$604$	0	0
$605$	1.25681e17	0.104198
$606$	0	0
$607$	1.70378e18	1.38257	0.691283	−	0.722584i	$-0.257046\pi$
$607$	1.70378e18	1.38257	0.691283	+	0.722584i	$0.257046\pi$
$608$	0	0
$609$	−8.07486e16	−0.0641391
$610$	0	0
$611$	1.63500e18	1.27131
$612$	0	0
$613$	8.29456e17	0.631394	0.315697	−	0.948860i	$-0.397762\pi$
$613$	8.29456e17	0.631394	0.315697	+	0.948860i	$0.397762\pi$
$614$	0	0
$615$	5.72489e17	0.426657
$616$	0	0
$617$	6.26750e17	0.457341	0.228671	−	0.973504i	$-0.426562\pi$
$617$	6.26750e17	0.457341	0.228671	+	0.973504i	$0.426562\pi$
$618$	0	0
$619$	−2.45472e17	−0.175393	−0.0876967	−	0.996147i	$-0.527951\pi$
$619$	−2.45472e17	−0.175393	−0.0876967	+	0.996147i	$0.527951\pi$
$620$	0	0
$621$	−1.35955e17	−0.0951263
$622$	0	0
$623$	4.99114e17	0.342001
$624$	0	0
$625$	7.31169e17	0.490679
$626$	0	0
$627$	−1.23775e18	−0.813568
$628$	0	0
$629$	−9.36154e17	−0.602723
$630$	0	0
$631$	−2.12636e17	−0.134106	−0.0670528	−	0.997749i	$-0.521360\pi$
$631$	−2.12636e17	−0.134106	−0.0670528	+	0.997749i	$0.521360\pi$
$632$	0	0
$633$	1.25807e16	0.00777285
$634$	0	0
$635$	1.08849e18	0.658866
$636$	0	0
$637$	−1.06725e18	−0.632938
$638$	0	0
$639$	8.47628e17	0.492549
$640$	0	0
$641$	−7.88201e17	−0.448807	−0.224404	−	0.974496i	$-0.572043\pi$
$641$	−7.88201e17	−0.448807	−0.224404	+	0.974496i	$0.572043\pi$
$642$	0	0
$643$	−1.88627e18	−1.05253	−0.526263	−	0.850322i	$-0.676407\pi$
$643$	−1.88627e18	−1.05253	−0.526263	+	0.850322i	$0.676407\pi$
$644$	0	0
$645$	2.81666e17	0.154027
$646$	0	0
$647$	−3.52142e18	−1.88730	−0.943649	−	0.330949i	$-0.892631\pi$
$647$	−3.52142e18	−1.88730	−0.943649	+	0.330949i	$0.892631\pi$
$648$	0	0
$649$	1.71760e18	0.902259
$650$	0	0
$651$	−1.80389e17	−0.0928824
$652$	0	0
$653$	−2.46611e18	−1.24474	−0.622368	−	0.782725i	$-0.713829\pi$
$653$	−2.46611e18	−1.24474	−0.622368	+	0.782725i	$0.713829\pi$
$654$	0	0
$655$	5.48592e17	0.271444
$656$	0	0
$657$	3.07605e17	0.149217
$658$	0	0
$659$	7.21384e17	0.343093	0.171546	−	0.985176i	$-0.445124\pi$
$659$	7.21384e17	0.343093	0.171546	+	0.985176i	$0.445124\pi$
$660$	0	0
$661$	−1.02233e18	−0.476741	−0.238370	−	0.971174i	$-0.576613\pi$
$661$	−1.02233e18	−0.476741	−0.238370	+	0.971174i	$0.576613\pi$
$662$	0	0
$663$	−1.00434e18	−0.459241
$664$	0	0
$665$	3.10651e17	0.139294
$666$	0	0
$667$	−6.18013e17	−0.271756
$668$	0	0
$669$	1.60362e18	0.691565
$670$	0	0
$671$	−3.45731e18	−1.46232
$672$	0	0
$673$	1.79394e18	0.744234	0.372117	−	0.928186i	$-0.378632\pi$
$673$	1.79394e18	0.744234	0.372117	+	0.928186i	$0.378632\pi$
$674$	0	0
$675$	3.87490e17	0.157684
$676$	0	0
$677$	−1.70461e18	−0.680454	−0.340227	−	0.940343i	$-0.610504\pi$
$677$	−1.70461e18	−0.680454	−0.340227	+	0.940343i	$0.610504\pi$
$678$	0	0
$679$	4.94275e14	0.000193560 0
$680$	0	0
$681$	1.67647e18	0.644081
$682$	0	0
$683$	2.57535e18	0.970739	0.485369	−	0.874309i	$-0.338685\pi$
$683$	2.57535e18	0.970739	0.485369	+	0.874309i	$0.338685\pi$
$684$	0	0
$685$	−1.20208e18	−0.444575
$686$	0	0
$687$	1.93102e18	0.700759
$688$	0	0
$689$	1.58137e17	0.0563128
$690$	0	0
$691$	9.02467e17	0.315373	0.157686	−	0.987489i	$-0.449597\pi$
$691$	9.02467e17	0.315373	0.157686	+	0.987489i	$0.449597\pi$
$692$	0	0
$693$	−1.70632e17	−0.0585186
$694$	0	0
$695$	1.58324e17	0.0532899
$696$	0	0
$697$	6.34406e18	2.09582
$698$	0	0
$699$	−2.91274e18	−0.944496
$700$	0	0
$701$	−2.68879e18	−0.855833	−0.427917	−	0.903818i	$-0.640752\pi$
$701$	−2.68879e18	−0.855833	−0.427917	+	0.903818i	$0.640752\pi$
$702$	0	0
$703$	−2.59547e18	−0.810974
$704$	0	0
$705$	1.54126e18	0.472765
$706$	0	0
$707$	3.55799e17	0.107147
$708$	0	0
$709$	6.07998e18	1.79764	0.898818	−	0.438323i	$-0.144427\pi$
$709$	6.07998e18	1.79764	0.898818	+	0.438323i	$0.144427\pi$
$710$	0	0
$711$	1.32638e18	0.385049
$712$	0	0
$713$	−1.38061e18	−0.393541
$714$	0	0
$715$	−8.70574e17	−0.243678
$716$	0	0
$717$	1.00614e18	0.276556
$718$	0	0
$719$	4.58875e18	1.23867	0.619337	−	0.785125i	$-0.287402\pi$
$719$	4.58875e18	1.23867	0.619337	+	0.785125i	$0.287402\pi$
$720$	0	0
$721$	−5.80762e17	−0.153964
$722$	0	0
$723$	−2.18139e18	−0.567981
$724$	0	0
$725$	1.76142e18	0.450470
$726$	0	0
$727$	−2.34576e18	−0.589264	−0.294632	−	0.955611i	$-0.595197\pi$
$727$	−2.34576e18	−0.589264	−0.294632	+	0.955611i	$0.595197\pi$
$728$	0	0
$729$	1.50095e17	0.0370370
$730$	0	0
$731$	3.12129e18	0.756610
$732$	0	0
$733$	−5.54485e18	−1.32043	−0.660213	−	0.751078i	$-0.729534\pi$
$733$	−5.54485e18	−1.32043	−0.660213	+	0.751078i	$0.729534\pi$
$734$	0	0
$735$	−1.00606e18	−0.235373
$736$	0	0
$737$	1.33901e18	0.307783
$738$	0	0
$739$	−1.13021e18	−0.255253	−0.127626	−	0.991822i	$-0.540736\pi$
$739$	−1.13021e18	−0.255253	−0.127626	+	0.991822i	$0.540736\pi$
$740$	0	0
$741$	−2.78451e18	−0.617917
$742$	0	0
$743$	−7.12111e18	−1.55282	−0.776408	−	0.630230i	$-0.782960\pi$
$743$	−7.12111e18	−1.55282	−0.776408	+	0.630230i	$0.782960\pi$
$744$	0	0
$745$	−9.89535e17	−0.212039
$746$	0	0
$747$	−1.43130e18	−0.301402
$748$	0	0
$749$	3.18119e17	0.0658351
$750$	0	0
$751$	−1.80740e18	−0.367616	−0.183808	−	0.982962i	$-0.558842\pi$
$751$	−1.80740e18	−0.367616	−0.183808	+	0.982962i	$0.558842\pi$
$752$	0	0
$753$	6.73348e17	0.134608
$754$	0	0
$755$	−8.01293e17	−0.157448
$756$	0	0
$757$	−6.87816e18	−1.32846	−0.664231	−	0.747527i	$-0.731241\pi$
$757$	−6.87816e18	−1.32846	−0.664231	+	0.747527i	$0.731241\pi$
$758$	0	0
$759$	−1.30594e18	−0.247943
$760$	0	0
$761$	5.30513e18	0.990137	0.495069	−	0.868854i	$-0.335143\pi$
$761$	5.30513e18	0.990137	0.495069	+	0.868854i	$0.335143\pi$
$762$	0	0
$763$	−9.87189e17	−0.181130
$764$	0	0
$765$	−9.46750e17	−0.170780
$766$	0	0
$767$	3.86400e18	0.685279
$768$	0	0
$769$	−8.65718e17	−0.150958	−0.0754789	−	0.997147i	$-0.524049\pi$
$769$	−8.65718e17	−0.150958	−0.0754789	+	0.997147i	$0.524049\pi$
$770$	0	0
$771$	−4.03935e18	−0.692562
$772$	0	0
$773$	−5.31929e18	−0.896782	−0.448391	−	0.893838i	$-0.648003\pi$
$773$	−5.31929e18	−0.896782	−0.448391	+	0.893838i	$0.648003\pi$
$774$	0	0
$775$	3.93494e18	0.652344
$776$	0	0
$777$	−3.57803e17	−0.0583320
$778$	0	0
$779$	1.75888e19	2.81996
$780$	0	0
$781$	8.14202e18	1.28381
$782$	0	0
$783$	6.82287e17	0.105807
$784$	0	0
$785$	6.28309e17	0.0958342
$786$	0	0
$787$	−3.62340e18	−0.543601	−0.271801	−	0.962354i	$-0.587619\pi$
$787$	−3.62340e18	−0.543601	−0.271801	+	0.962354i	$0.587619\pi$
$788$	0	0
$789$	2.94773e18	0.434997
$790$	0	0
$791$	−2.46738e18	−0.358169
$792$	0	0
$793$	−7.77774e18	−1.11065
$794$	0	0
$795$	1.49070e17	0.0209413
$796$	0	0
$797$	−5.72313e18	−0.790960	−0.395480	−	0.918474i	$-0.629422\pi$
$797$	−5.72313e18	−0.790960	−0.395480	+	0.918474i	$0.629422\pi$
$798$	0	0
$799$	1.70795e19	2.32231
$800$	0	0
$801$	−4.21727e18	−0.564183
$802$	0	0
$803$	2.95474e18	0.388928
$804$	0	0
$805$	3.27765e17	0.0424512
$806$	0	0
$807$	8.33565e18	1.06234
$808$	0	0
$809$	8.89222e18	1.11518	0.557590	−	0.830116i	$-0.311726\pi$
$809$	8.89222e18	1.11518	0.557590	+	0.830116i	$0.311726\pi$
$810$	0	0
$811$	−1.04870e19	−1.29424	−0.647121	−	0.762387i	$-0.724027\pi$
$811$	−1.04870e19	−1.29424	−0.647121	+	0.762387i	$0.724027\pi$
$812$	0	0
$813$	−4.71820e18	−0.573044
$814$	0	0
$815$	−4.82826e18	−0.577120
$816$	0	0
$817$	8.65375e18	1.01803
$818$	0	0
$819$	−3.83862e17	−0.0444457
$820$	0	0
$821$	−3.16350e18	−0.360527	−0.180263	−	0.983618i	$-0.557695\pi$
$821$	−3.16350e18	−0.360527	−0.180263	+	0.983618i	$0.557695\pi$
$822$	0	0
$823$	4.40532e18	0.494172	0.247086	−	0.968994i	$-0.420527\pi$
$823$	4.40532e18	0.494172	0.247086	+	0.968994i	$0.420527\pi$
$824$	0	0
$825$	3.72210e18	0.410996
$826$	0	0
$827$	−2.12463e18	−0.230940	−0.115470	−	0.993311i	$-0.536837\pi$
$827$	−2.12463e18	−0.230940	−0.115470	+	0.993311i	$0.536837\pi$
$828$	0	0
$829$	−9.29946e18	−0.995069	−0.497534	−	0.867444i	$-0.665761\pi$
$829$	−9.29946e18	−0.995069	−0.497534	+	0.867444i	$0.665761\pi$
$830$	0	0
$831$	8.50963e18	0.896405
$832$	0	0
$833$	−1.11487e19	−1.15620
$834$	0	0
$835$	6.37517e18	0.650924
$836$	0	0
$837$	1.52420e18	0.153224
$838$	0	0
$839$	−1.64176e19	−1.62501	−0.812504	−	0.582955i	$-0.801896\pi$
$839$	−1.64176e19	−1.62501	−0.812504	+	0.582955i	$0.801896\pi$
$840$	0	0
$841$	−7.15915e18	−0.697730
$842$	0	0
$843$	6.08442e17	0.0583902
$844$	0	0
$845$	2.53921e18	0.239955
$846$	0	0
$847$	5.32312e17	0.0495363
$848$	0	0
$849$	−2.90404e18	−0.266136
$850$	0	0
$851$	−2.73846e18	−0.247152
$852$	0	0
$853$	−4.43448e17	−0.0394161	−0.0197080	−	0.999806i	$-0.506274\pi$
$853$	−4.43448e17	−0.0394161	−0.0197080	+	0.999806i	$0.506274\pi$
$854$	0	0
$855$	−2.62485e18	−0.229787
$856$	0	0
$857$	−1.76085e19	−1.51826	−0.759132	−	0.650937i	$-0.774376\pi$
$857$	−1.76085e19	−1.51826	−0.759132	+	0.650937i	$0.774376\pi$
$858$	0	0
$859$	7.41595e17	0.0629813	0.0314906	−	0.999504i	$-0.489975\pi$
$859$	7.41595e17	0.0629813	0.0314906	+	0.999504i	$0.489975\pi$
$860$	0	0
$861$	2.42473e18	0.202835
$862$	0	0
$863$	−1.39368e19	−1.14840	−0.574200	−	0.818715i	$-0.694687\pi$
$863$	−1.39368e19	−1.14840	−0.574200	+	0.818715i	$0.694687\pi$
$864$	0	0
$865$	5.83151e18	0.473343
$866$	0	0
$867$	−3.27101e18	−0.261552
$868$	0	0
$869$	1.27407e19	1.00361
$870$	0	0
$871$	3.01230e18	0.233765
$872$	0	0
$873$	−4.17638e15	−0.000319306 0
$874$	0	0
$875$	−2.07432e18	−0.156251
$876$	0	0
$877$	1.09505e19	0.812710	0.406355	−	0.913715i	$-0.366800\pi$
$877$	1.09505e19	0.812710	0.406355	+	0.913715i	$0.366800\pi$
$878$	0	0
$879$	1.11579e19	0.815934
$880$	0	0
$881$	2.52895e18	0.182220	0.0911100	−	0.995841i	$-0.470959\pi$
$881$	2.52895e18	0.182220	0.0911100	+	0.995841i	$0.470959\pi$
$882$	0	0
$883$	−1.90684e19	−1.35384	−0.676922	−	0.736054i	$-0.736687\pi$
$883$	−1.90684e19	−1.35384	−0.676922	+	0.736054i	$0.736687\pi$
$884$	0	0
$885$	3.64245e18	0.254837
$886$	0	0
$887$	−1.64311e19	−1.13282	−0.566412	−	0.824122i	$-0.691669\pi$
$887$	−1.64311e19	−1.13282	−0.566412	+	0.824122i	$0.691669\pi$
$888$	0	0
$889$	4.61023e18	0.313228
$890$	0	0
$891$	1.44176e18	0.0965355
$892$	0	0
$893$	4.73527e19	3.12471
$894$	0	0
$895$	8.50378e18	0.553047
$896$	0	0
$897$	−2.93791e18	−0.188316
$898$	0	0
$899$	6.92857e18	0.437729
$900$	0	0
$901$	1.65192e18	0.102867
$902$	0	0
$903$	1.19298e18	0.0732253
$904$	0	0
$905$	7.72184e18	0.467202
$906$	0	0
$907$	2.93372e19	1.74973	0.874864	−	0.484368i	$-0.160950\pi$
$907$	2.93372e19	1.74973	0.874864	+	0.484368i	$0.160950\pi$
$908$	0	0
$909$	−3.00633e18	−0.176755
$910$	0	0
$911$	−1.56962e19	−0.909756	−0.454878	−	0.890554i	$-0.650317\pi$
$911$	−1.56962e19	−0.909756	−0.454878	+	0.890554i	$0.650317\pi$
$912$	0	0
$913$	−1.37485e19	−0.785590
$914$	0	0
$915$	−7.33179e18	−0.413022
$916$	0	0
$917$	2.32352e18	0.129046
$918$	0	0
$919$	−1.53970e19	−0.843113	−0.421556	−	0.906802i	$-0.638516\pi$
$919$	−1.53970e19	−0.843113	−0.421556	+	0.906802i	$0.638516\pi$
$920$	0	0
$921$	−7.51973e18	−0.405989
$922$	0	0
$923$	1.83167e19	0.975071
$924$	0	0
$925$	7.80498e18	0.409685
$926$	0	0
$927$	4.90716e18	0.253987
$928$	0	0
$929$	2.14122e19	1.09285	0.546423	−	0.837509i	$-0.315989\pi$
$929$	2.14122e19	1.09285	0.546423	+	0.837509i	$0.315989\pi$
$930$	0	0
$931$	−3.09096e19	−1.55568
$932$	0	0
$933$	−1.44442e19	−0.716909
$934$	0	0
$935$	−9.09415e18	−0.445130
$936$	0	0
$937$	−3.34334e19	−1.61389	−0.806944	−	0.590628i	$-0.798880\pi$
$937$	−3.34334e19	−1.61389	−0.806944	+	0.590628i	$0.798880\pi$
$938$	0	0
$939$	−8.69897e18	−0.414135
$940$	0	0
$941$	2.87989e17	0.0135221	0.00676103	−	0.999977i	$-0.497848\pi$
$941$	2.87989e17	0.0135221	0.00676103	+	0.999977i	$0.497848\pi$
$942$	0	0
$943$	1.85578e19	0.859410
$944$	0	0
$945$	−3.61853e17	−0.0165282
$946$	0	0
$947$	6.93297e17	0.0312352	0.0156176	−	0.999878i	$-0.495029\pi$
$947$	6.93297e17	0.0312352	0.0156176	+	0.999878i	$0.495029\pi$
$948$	0	0
$949$	6.64715e18	0.295396
$950$	0	0
$951$	−6.00533e18	−0.263247
$952$	0	0
$953$	−7.21492e18	−0.311981	−0.155990	−	0.987759i	$-0.549857\pi$
$953$	−7.21492e18	−0.311981	−0.155990	+	0.987759i	$0.549857\pi$
$954$	0	0
$955$	−3.91411e17	−0.0166960
$956$	0	0
$957$	6.55381e18	0.275782
$958$	0	0
$959$	−5.09132e18	−0.211353
$960$	0	0
$961$	−8.93942e18	−0.366107
$962$	0	0
$963$	−2.68795e18	−0.108605
$964$	0	0
$965$	−5.60521e18	−0.223442
$966$	0	0
$967$	−1.31073e19	−0.515514	−0.257757	−	0.966210i	$-0.582983\pi$
$967$	−1.31073e19	−0.515514	−0.257757	+	0.966210i	$0.582983\pi$
$968$	0	0
$969$	−2.90874e19	−1.12876
$970$	0	0
$971$	3.77088e19	1.44384	0.721918	−	0.691979i	$-0.243261\pi$
$971$	3.77088e19	1.44384	0.721918	+	0.691979i	$0.243261\pi$
$972$	0	0
$973$	6.70567e17	0.0253343
$974$	0	0
$975$	8.37343e18	0.312157
$976$	0	0
$977$	−5.80521e18	−0.213552	−0.106776	−	0.994283i	$-0.534053\pi$
$977$	−5.80521e18	−0.213552	−0.106776	+	0.994283i	$0.534053\pi$
$978$	0	0
$979$	−4.05096e19	−1.47052
$980$	0	0
$981$	8.34128e18	0.298802
$982$	0	0
$983$	−1.34366e19	−0.474998	−0.237499	−	0.971388i	$-0.576328\pi$
$983$	−1.34366e19	−0.474998	−0.237499	+	0.971388i	$0.576328\pi$
$984$	0	0
$985$	−7.33430e18	−0.255872
$986$	0	0
$987$	6.52787e18	0.224755
$988$	0	0
$989$	9.13049e18	0.310255
$990$	0	0
$991$	5.21589e19	1.74924	0.874621	−	0.484808i	$-0.161111\pi$
$991$	5.21589e19	1.74924	0.874621	+	0.484808i	$0.161111\pi$
$992$	0	0
$993$	1.92292e19	0.636490
$994$	0	0
$995$	−9.34248e18	−0.305219
$996$	0	0
$997$	1.54725e19	0.498934	0.249467	−	0.968383i	$-0.419745\pi$
$997$	1.54725e19	0.498934	0.249467	+	0.968383i	$0.419745\pi$
$998$	0	0
$999$	3.02326e18	0.0962277

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	12.14.a.a.1.1	✓	1
3.2	odd	2		36.14.a.c.1.1		1
4.3	odd	2		48.14.a.d.1.1		1
8.3	odd	2		192.14.a.b.1.1		1
8.5	even	2		192.14.a.g.1.1		1
12.11	even	2		144.14.a.g.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
12.14.a.a.1.1	✓	1	1.1	even	1	trivial
36.14.a.c.1.1		1	3.2	odd	2
48.14.a.d.1.1		1	4.3	odd	2
144.14.a.g.1.1		1	12.11	even	2
192.14.a.b.1.1		1	8.3	odd	2
192.14.a.g.1.1		1	8.5	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 12 = 2^{2} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 14 \)
Character orbit:	\([\chi]\)	\(=\)	12.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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