LMFDB - Embedded newform 16.42.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [16,42,Mod(1,16)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(16, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 42, names="a")

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

chi := DirichletCharacter("16.1");

S:= CuspForms(chi, 42);

N := Newforms(S);

Level:	$ N $	$=$	$ 16 = 2^{4} $
Weight:	$ k $	$=$	$ 42 $
Character orbit:	$[\chi]$	$=$	16.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:	$170.354672730$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	16.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-5.04352e9 q^{3} -4.85042e13 q^{5} +1.19392e17 q^{7} -1.10359e19 q^{9} +O(q^{10})$

$q-5.04352e9 q^{3} -4.85042e13 q^{5} +1.19392e17 q^{7} -1.10359e19 q^{9} -3.15381e21 q^{11} -1.14103e22 q^{13} +2.44632e23 q^{15} -2.67238e25 q^{17} -6.79752e25 q^{19} -6.02158e26 q^{21} +1.35051e28 q^{23} -4.31221e28 q^{25} +2.39612e29 q^{27} +1.36715e29 q^{29} -3.06142e30 q^{31} +1.59063e31 q^{33} -5.79103e30 q^{35} -2.21949e32 q^{37} +5.75480e31 q^{39} -5.01985e32 q^{41} +3.11848e33 q^{43} +5.35289e32 q^{45} -1.31555e34 q^{47} -3.03131e34 q^{49} +1.34782e35 q^{51} -3.23999e35 q^{53} +1.52973e35 q^{55} +3.42834e35 q^{57} -3.45957e36 q^{59} -9.78043e35 q^{61} -1.31761e36 q^{63} +5.53447e35 q^{65} -1.66275e37 q^{67} -6.81131e37 q^{69} -1.16969e38 q^{71} +1.90709e38 q^{73} +2.17487e38 q^{75} -3.76541e38 q^{77} +5.61362e38 q^{79} -8.05974e38 q^{81} +6.05771e38 q^{83} +1.29621e39 q^{85} -6.89527e38 q^{87} +1.19154e40 q^{89} -1.36230e39 q^{91} +1.54403e40 q^{93} +3.29708e39 q^{95} -6.35760e40 q^{97} +3.48052e40 q^{99} +O(q^{100})$

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
$3$	−5.04352e9	−0.835118	−0.417559	−	0.908650i	$-0.637114\pi$
$3$	−5.04352e9	−0.835118	−0.417559	+	0.908650i	$0.637114\pi$
$4$	0	0
$5$	−4.85042e13	−0.227454	−0.113727	−	0.993512i	$-0.536279\pi$
$5$	−4.85042e13	−0.227454	−0.113727	+	0.993512i	$0.536279\pi$
$6$	0	0
$7$	1.19392e17	0.565545	0.282772	−	0.959187i	$-0.408746\pi$
$7$	1.19392e17	0.565545	0.282772	+	0.959187i	$0.408746\pi$
$8$	0	0
$9$	−1.10359e19	−0.302578
$10$	0	0
$11$	−3.15381e21	−1.41347	−0.706733	−	0.707480i	$-0.749832\pi$
$11$	−3.15381e21	−1.41347	−0.706733	+	0.707480i	$0.749832\pi$
$12$	0	0
$13$	−1.14103e22	−0.166517	−0.0832585	−	0.996528i	$-0.526533\pi$
$13$	−1.14103e22	−0.166517	−0.0832585	+	0.996528i	$0.526533\pi$
$14$	0	0
$15$	2.44632e23	0.189951
$16$	0	0
$17$	−2.67238e25	−1.59476	−0.797379	−	0.603479i	$-0.793781\pi$
$17$	−2.67238e25	−1.59476	−0.797379	+	0.603479i	$0.793781\pi$
$18$	0	0
$19$	−6.79752e25	−0.414860	−0.207430	−	0.978250i	$-0.566510\pi$
$19$	−6.79752e25	−0.414860	−0.207430	+	0.978250i	$0.566510\pi$
$20$	0	0
$21$	−6.02158e26	−0.472297
$22$	0	0
$23$	1.35051e28	1.64088	0.820439	−	0.571734i	$-0.193729\pi$
$23$	1.35051e28	1.64088	0.820439	+	0.571734i	$0.193729\pi$
$24$	0	0
$25$	−4.31221e28	−0.948265
$26$	0	0
$27$	2.39612e29	1.08781
$28$	0	0
$29$	1.36715e29	0.143436	0.0717181	−	0.997425i	$-0.477152\pi$
$29$	1.36715e29	0.143436	0.0717181	+	0.997425i	$0.477152\pi$
$30$	0	0
$31$	−3.06142e30	−0.818481	−0.409241	−	0.912427i	$-0.634206\pi$
$31$	−3.06142e30	−0.818481	−0.409241	+	0.912427i	$0.634206\pi$
$32$	0	0
$33$	1.59063e31	1.18041
$34$	0	0
$35$	−5.79103e30	−0.128636
$36$	0	0
$37$	−2.21949e32	−1.57804	−0.789021	−	0.614366i	$-0.789412\pi$
$37$	−2.21949e32	−1.57804	−0.789021	+	0.614366i	$0.789412\pi$
$38$	0	0
$39$	5.75480e31	0.139061
$40$	0	0
$41$	−5.01985e32	−0.435133	−0.217566	−	0.976046i	$-0.569812\pi$
$41$	−5.01985e32	−0.435133	−0.217566	+	0.976046i	$0.569812\pi$
$42$	0	0
$43$	3.11848e33	1.01821	0.509107	−	0.860703i	$-0.329976\pi$
$43$	3.11848e33	1.01821	0.509107	+	0.860703i	$0.329976\pi$
$44$	0	0
$45$	5.35289e32	0.0688227
$46$	0	0
$47$	−1.31555e34	−0.693588	−0.346794	−	0.937941i	$-0.612730\pi$
$47$	−1.31555e34	−0.693588	−0.346794	+	0.937941i	$0.612730\pi$
$48$	0	0
$49$	−3.03131e34	−0.680159
$50$	0	0
$51$	1.34782e35	1.33181
$52$	0	0
$53$	−3.23999e35	−1.45508	−0.727542	−	0.686064i	$-0.759337\pi$
$53$	−3.23999e35	−1.45508	−0.727542	+	0.686064i	$0.759337\pi$
$54$	0	0
$55$	1.52973e35	0.321499
$56$	0	0
$57$	3.42834e35	0.346457
$58$	0	0
$59$	−3.45957e36	−1.72408	−0.862038	−	0.506844i	$-0.830812\pi$
$59$	−3.45957e36	−1.72408	−0.862038	+	0.506844i	$0.830812\pi$
$60$	0	0
$61$	−9.78043e35	−0.246091	−0.123046	−	0.992401i	$-0.539266\pi$
$61$	−9.78043e35	−0.246091	−0.123046	+	0.992401i	$0.539266\pi$
$62$	0	0
$63$	−1.31761e36	−0.171122
$64$	0	0
$65$	5.53447e35	0.0378750
$66$	0	0
$67$	−1.66275e37	−0.611357	−0.305678	−	0.952135i	$-0.598883\pi$
$67$	−1.66275e37	−0.611357	−0.305678	+	0.952135i	$0.598883\pi$
$68$	0	0
$69$	−6.81131e37	−1.37033
$70$	0	0
$71$	−1.16969e38	−1.31001	−0.655004	−	0.755625i	$-0.727333\pi$
$71$	−1.16969e38	−1.31001	−0.655004	+	0.755625i	$0.727333\pi$
$72$	0	0
$73$	1.90709e38	1.20851	0.604257	−	0.796790i	$-0.293470\pi$
$73$	1.90709e38	1.20851	0.604257	+	0.796790i	$0.293470\pi$
$74$	0	0
$75$	2.17487e38	0.791913
$76$	0	0
$77$	−3.76541e38	−0.799378
$78$	0	0
$79$	5.61362e38	0.704511	0.352256	−	0.935904i	$-0.385415\pi$
$79$	5.61362e38	0.704511	0.352256	+	0.935904i	$0.385415\pi$
$80$	0	0
$81$	−8.05974e38	−0.605868
$82$	0	0
$83$	6.05771e38	0.276190	0.138095	−	0.990419i	$-0.455902\pi$
$83$	6.05771e38	0.276190	0.138095	+	0.990419i	$0.455902\pi$
$84$	0	0
$85$	1.29621e39	0.362735
$86$	0	0
$87$	−6.89527e38	−0.119786
$88$	0	0
$89$	1.19154e40	1.29902	0.649508	−	0.760355i	$-0.274975\pi$
$89$	1.19154e40	1.29902	0.649508	+	0.760355i	$0.274975\pi$
$90$	0	0
$91$	−1.36230e39	−0.0941728
$92$	0	0
$93$	1.54403e40	0.683528
$94$	0	0
$95$	3.29708e39	0.0943618
$96$	0	0
$97$	−6.35760e40	−1.18706	−0.593530	−	0.804812i	$-0.702266\pi$
$97$	−6.35760e40	−1.18706	−0.593530	+	0.804812i	$0.702266\pi$
$98$	0	0
$99$	3.48052e40	0.427684
$100$	0	0
$101$	6.20987e39	0.0506401	0.0253200	−	0.999679i	$-0.491940\pi$
$101$	6.20987e39	0.0506401	0.0253200	+	0.999679i	$0.491940\pi$
$102$	0	0
$103$	−1.73412e41	−0.946056	−0.473028	−	0.881047i	$-0.656839\pi$
$103$	−1.73412e41	−0.946056	−0.473028	+	0.881047i	$0.656839\pi$
$104$	0	0
$105$	2.92072e40	0.107426
$106$	0	0
$107$	−1.77847e41	−0.444304	−0.222152	−	0.975012i	$-0.571308\pi$
$107$	−1.77847e41	−0.444304	−0.222152	+	0.975012i	$0.571308\pi$
$108$	0	0
$109$	−1.02737e42	−1.75583	−0.877916	−	0.478815i	$-0.841066\pi$
$109$	−1.02737e42	−1.75583	−0.877916	+	0.478815i	$0.841066\pi$
$110$	0	0
$111$	1.11941e42	1.31785
$112$	0	0
$113$	1.36076e42	1.11089	0.555447	−	0.831552i	$-0.312547\pi$
$113$	1.36076e42	1.11089	0.555447	+	0.831552i	$0.312547\pi$
$114$	0	0
$115$	−6.55053e41	−0.373225
$116$	0	0
$117$	1.25923e41	0.0503844
$118$	0	0
$119$	−3.19062e42	−0.901907
$120$	0	0
$121$	4.96799e42	0.997886
$122$	0	0
$123$	2.53177e42	0.363387
$124$	0	0
$125$	4.29732e42	0.443141
$126$	0	0
$127$	1.25701e43	0.936185	0.468092	−	0.883679i	$-0.344941\pi$
$127$	1.25701e43	0.936185	0.468092	+	0.883679i	$0.344941\pi$
$128$	0	0
$129$	−1.57281e43	−0.850329
$130$	0	0
$131$	1.14997e43	0.453551	0.226776	−	0.973947i	$-0.427182\pi$
$131$	1.14997e43	0.453551	0.226776	+	0.973947i	$0.427182\pi$
$132$	0	0
$133$	−8.11573e42	−0.234622
$134$	0	0
$135$	−1.16222e43	−0.247426
$136$	0	0
$137$	−1.99120e43	−0.313577	−0.156788	−	0.987632i	$-0.550114\pi$
$137$	−1.99120e43	−0.313577	−0.156788	+	0.987632i	$0.550114\pi$
$138$	0	0
$139$	−8.57436e43	−1.00322	−0.501612	−	0.865093i	$-0.667260\pi$
$139$	−8.57436e43	−1.00322	−0.501612	+	0.865093i	$0.667260\pi$
$140$	0	0
$141$	6.63501e43	0.579228
$142$	0	0
$143$	3.59859e43	0.235366
$144$	0	0
$145$	−6.63127e42	−0.0326252
$146$	0	0
$147$	1.52885e44	0.568013
$148$	0	0
$149$	2.85509e43	0.0804083	0.0402041	−	0.999191i	$-0.487199\pi$
$149$	2.85509e43	0.0804083	0.0402041	+	0.999191i	$0.487199\pi$
$150$	0	0
$151$	−1.89816e44	−0.406729	−0.203365	−	0.979103i	$-0.565188\pi$
$151$	−1.89816e44	−0.406729	−0.203365	+	0.979103i	$0.565188\pi$
$152$	0	0
$153$	2.94922e44	0.482539
$154$	0	0
$155$	1.48492e44	0.186167
$156$	0	0
$157$	3.84758e44	0.370890	0.185445	−	0.982655i	$-0.440627\pi$
$157$	3.84758e44	0.370890	0.185445	+	0.982655i	$0.440627\pi$
$158$	0	0
$159$	1.63409e45	1.21517
$160$	0	0
$161$	1.61240e45	0.927990
$162$	0	0
$163$	4.75006e44	0.212253	0.106126	−	0.994353i	$-0.466155\pi$
$163$	4.75006e44	0.212253	0.106126	+	0.994353i	$0.466155\pi$
$164$	0	0
$165$	−7.71522e44	−0.268489
$166$	0	0
$167$	4.32345e45	1.17528	0.587642	−	0.809121i	$-0.300056\pi$
$167$	4.32345e45	1.17528	0.587642	+	0.809121i	$0.300056\pi$
$168$	0	0
$169$	−4.56526e45	−0.972272
$170$	0	0
$171$	7.50170e44	0.125528
$172$	0	0
$173$	1.44924e46	1.91072	0.955358	−	0.295452i	$-0.0954703\pi$
$173$	1.44924e46	1.91072	0.955358	+	0.295452i	$0.0954703\pi$
$174$	0	0
$175$	−5.14845e45	−0.536286
$176$	0	0
$177$	1.74484e46	1.43981
$178$	0	0
$179$	−5.67669e45	−0.372056	−0.186028	−	0.982544i	$-0.559561\pi$
$179$	−5.67669e45	−0.372056	−0.186028	+	0.982544i	$0.559561\pi$
$180$	0	0
$181$	−2.20895e46	−1.15286	−0.576429	−	0.817147i	$-0.695554\pi$
$181$	−2.20895e46	−1.15286	−0.576429	+	0.817147i	$0.695554\pi$
$182$	0	0
$183$	4.93278e45	0.205515
$184$	0	0
$185$	1.07655e46	0.358932
$186$	0	0
$187$	8.42816e46	2.25414
$188$	0	0
$189$	2.86079e46	0.615203
$190$	0	0
$191$	6.23395e46	1.08039	0.540194	−	0.841541i	$-0.318351\pi$
$191$	6.23395e46	1.08039	0.540194	+	0.841541i	$0.318351\pi$
$192$	0	0
$193$	1.06448e47	1.49009	0.745043	−	0.667017i	$-0.232429\pi$
$193$	1.06448e47	1.49009	0.745043	+	0.667017i	$0.232429\pi$
$194$	0	0
$195$	−2.79132e45	−0.0316301
$196$	0	0
$197$	8.49296e46	0.780731	0.390365	−	0.920660i	$-0.372349\pi$
$197$	8.49296e46	0.780731	0.390365	+	0.920660i	$0.372349\pi$
$198$	0	0
$199$	1.82716e47	1.36549	0.682746	−	0.730656i	$-0.260786\pi$
$199$	1.82716e47	1.36549	0.682746	+	0.730656i	$0.260786\pi$
$200$	0	0
$201$	8.38613e46	0.510555
$202$	0	0
$203$	1.63228e46	0.0811196
$204$	0	0
$205$	2.43484e46	0.0989728
$206$	0	0
$207$	−1.49041e47	−0.496494
$208$	0	0
$209$	2.14381e47	0.586391
$210$	0	0
$211$	7.87226e47	1.77137	0.885684	−	0.464288i	$-0.153690\pi$
$211$	7.87226e47	1.77137	0.885684	+	0.464288i	$0.153690\pi$
$212$	0	0
$213$	5.89934e47	1.09401
$214$	0	0
$215$	−1.51259e47	−0.231597
$216$	0	0
$217$	−3.65511e47	−0.462888
$218$	0	0
$219$	−9.61842e47	−1.00925
$220$	0	0
$221$	3.04926e47	0.265554
$222$	0	0
$223$	1.32160e48	0.956862	0.478431	−	0.878125i	$-0.341206\pi$
$223$	1.32160e48	0.956862	0.478431	+	0.878125i	$0.341206\pi$
$224$	0	0
$225$	4.75893e47	0.286924
$226$	0	0
$227$	1.43456e48	0.721419	0.360710	−	0.932678i	$-0.382535\pi$
$227$	1.43456e48	0.721419	0.360710	+	0.932678i	$0.382535\pi$
$228$	0	0
$229$	−3.68923e48	−1.54991	−0.774957	−	0.632014i	$-0.782229\pi$
$229$	−3.68923e48	−1.54991	−0.774957	+	0.632014i	$0.782229\pi$
$230$	0	0
$231$	1.89909e48	0.667575
$232$	0	0
$233$	6.32561e48	1.86340	0.931700	−	0.363228i	$-0.118325\pi$
$233$	6.32561e48	1.86340	0.931700	+	0.363228i	$0.118325\pi$
$234$	0	0
$235$	6.38098e47	0.157760
$236$	0	0
$237$	−2.83124e48	−0.588350
$238$	0	0
$239$	−8.39385e48	−1.46826	−0.734132	−	0.679006i	$-0.762411\pi$
$239$	−8.39385e48	−1.46826	−0.734132	+	0.679006i	$0.762411\pi$
$240$	0	0
$241$	2.76168e48	0.407215	0.203608	−	0.979053i	$-0.434733\pi$
$241$	2.76168e48	0.407215	0.203608	+	0.979053i	$0.434733\pi$
$242$	0	0
$243$	−4.67443e48	−0.581835
$244$	0	0
$245$	1.47031e48	0.154705
$246$	0	0
$247$	7.75618e47	0.0690812
$248$	0	0
$249$	−3.05522e48	−0.230651
$250$	0	0
$251$	−5.47506e48	−0.350814	−0.175407	−	0.984496i	$-0.556124\pi$
$251$	−5.47506e48	−0.350814	−0.175407	+	0.984496i	$0.556124\pi$
$252$	0	0
$253$	−4.25924e49	−2.31933
$254$	0	0
$255$	−6.53748e48	−0.302926
$256$	0	0
$257$	−2.41403e49	−0.953045	−0.476522	−	0.879162i	$-0.658103\pi$
$257$	−2.41403e49	−0.953045	−0.476522	+	0.879162i	$0.658103\pi$
$258$	0	0
$259$	−2.64991e49	−0.892453
$260$	0	0
$261$	−1.50878e48	−0.0434007
$262$	0	0
$263$	1.00767e49	0.247871	0.123936	−	0.992290i	$-0.460448\pi$
$263$	1.00767e49	0.247871	0.123936	+	0.992290i	$0.460448\pi$
$264$	0	0
$265$	1.57153e49	0.330965
$266$	0	0
$267$	−6.00957e49	−1.08483
$268$	0	0
$269$	−3.47580e49	−0.538435	−0.269217	−	0.963079i	$-0.586765\pi$
$269$	−3.47580e49	−0.538435	−0.269217	+	0.963079i	$0.586765\pi$
$270$	0	0
$271$	−9.20219e49	−1.22468	−0.612338	−	0.790596i	$-0.709771\pi$
$271$	−9.20219e49	−1.22468	−0.612338	+	0.790596i	$0.709771\pi$
$272$	0	0
$273$	6.87080e48	0.0786454
$274$	0	0
$275$	1.35999e50	1.34034
$276$	0	0
$277$	−1.12019e50	−0.951605	−0.475803	−	0.879552i	$-0.657842\pi$
$277$	−1.12019e50	−0.951605	−0.475803	+	0.879552i	$0.657842\pi$
$278$	0	0
$279$	3.37857e49	0.247655
$280$	0	0
$281$	1.13941e50	0.721441	0.360720	−	0.932674i	$-0.382531\pi$
$281$	1.13941e50	0.721441	0.360720	+	0.932674i	$0.382531\pi$
$282$	0	0
$283$	−2.06091e50	−1.12833	−0.564167	−	0.825661i	$-0.690803\pi$
$283$	−2.06091e50	−1.12833	−0.564167	+	0.825661i	$0.690803\pi$
$284$	0	0
$285$	−1.66289e49	−0.0788032
$286$	0	0
$287$	−5.99332e49	−0.246087
$288$	0	0
$289$	4.33354e50	1.54325
$290$	0	0
$291$	3.20647e50	0.991335
$292$	0	0
$293$	−3.36281e50	−0.903475	−0.451738	−	0.892151i	$-0.649196\pi$
$293$	−3.36281e50	−0.903475	−0.451738	+	0.892151i	$0.649196\pi$
$294$	0	0
$295$	1.67804e50	0.392149
$296$	0	0
$297$	−7.55691e50	−1.53758
$298$	0	0
$299$	−1.54097e50	−0.273234
$300$	0	0
$301$	3.72323e50	0.575846
$302$	0	0
$303$	−3.13196e49	−0.0422904
$304$	0	0
$305$	4.74392e49	0.0559746
$306$	0	0
$307$	6.52252e50	0.673099	0.336550	−	0.941666i	$-0.390740\pi$
$307$	6.52252e50	0.673099	0.336550	+	0.941666i	$0.390740\pi$
$308$	0	0
$309$	8.74608e50	0.790068
$310$	0	0
$311$	7.73007e50	0.611781	0.305890	−	0.952067i	$-0.401046\pi$
$311$	7.73007e50	0.611781	0.305890	+	0.952067i	$0.401046\pi$
$312$	0	0
$313$	2.76877e51	1.92144	0.960722	−	0.277511i	$-0.0895095\pi$
$313$	2.76877e51	1.92144	0.960722	+	0.277511i	$0.0895095\pi$
$314$	0	0
$315$	6.39095e49	0.0389223
$316$	0	0
$317$	7.05058e50	0.377146	0.188573	−	0.982059i	$-0.439614\pi$
$317$	7.05058e50	0.377146	0.188573	+	0.982059i	$0.439614\pi$
$318$	0	0
$319$	−4.31174e50	−0.202742
$320$	0	0
$321$	8.96976e50	0.371046
$322$	0	0
$323$	1.81655e51	0.661601
$324$	0	0
$325$	4.92036e50	0.157902
$326$	0	0
$327$	5.18155e51	1.46633
$328$	0	0
$329$	−1.57067e51	−0.392255
$330$	0	0
$331$	2.05137e51	0.452449	0.226224	−	0.974075i	$-0.427362\pi$
$331$	2.05137e51	0.452449	0.226224	+	0.974075i	$0.427362\pi$
$332$	0	0
$333$	2.44942e51	0.477481
$334$	0	0
$335$	8.06506e50	0.139056
$336$	0	0
$337$	−1.03725e52	−1.58296	−0.791479	−	0.611196i	$-0.790689\pi$
$337$	−1.03725e52	−1.58296	−0.791479	+	0.611196i	$0.790689\pi$
$338$	0	0
$339$	−6.86301e51	−0.927728
$340$	0	0
$341$	9.65514e51	1.15689
$342$	0	0
$343$	−8.94019e51	−0.950205
$344$	0	0
$345$	3.30377e51	0.311687
$346$	0	0
$347$	−2.85107e50	−0.0238921	−0.0119461	−	0.999929i	$-0.503803\pi$
$347$	−2.85107e50	−0.0238921	−0.0119461	+	0.999929i	$0.503803\pi$
$348$	0	0
$349$	−9.31673e51	−0.693973	−0.346986	−	0.937870i	$-0.612795\pi$
$349$	−9.31673e51	−0.693973	−0.346986	+	0.937870i	$0.612795\pi$
$350$	0	0
$351$	−2.73405e51	−0.181138
$352$	0	0
$353$	−1.81721e52	−1.07158	−0.535788	−	0.844353i	$-0.679985\pi$
$353$	−1.81721e52	−1.07158	−0.535788	+	0.844353i	$0.679985\pi$
$354$	0	0
$355$	5.67348e51	0.297967
$356$	0	0
$357$	1.60919e52	0.753198
$358$	0	0
$359$	−1.90791e52	−0.796384	−0.398192	−	0.917302i	$-0.630362\pi$
$359$	−1.90791e52	−0.796384	−0.398192	+	0.917302i	$0.630362\pi$
$360$	0	0
$361$	−2.22265e52	−0.827891
$362$	0	0
$363$	−2.50562e52	−0.833352
$364$	0	0
$365$	−9.25016e51	−0.274882
$366$	0	0
$367$	2.11760e52	0.562586	0.281293	−	0.959622i	$-0.409237\pi$
$367$	2.11760e52	0.562586	0.281293	+	0.959622i	$0.409237\pi$
$368$	0	0
$369$	5.53988e51	0.131662
$370$	0	0
$371$	−3.86830e52	−0.822915
$372$	0	0
$373$	−2.64566e52	−0.504083	−0.252042	−	0.967716i	$-0.581102\pi$
$373$	−2.64566e52	−0.504083	−0.252042	+	0.967716i	$0.581102\pi$
$374$	0	0
$375$	−2.16736e52	−0.370075
$376$	0	0
$377$	−1.55996e51	−0.0238845
$378$	0	0
$379$	6.28388e52	0.863225	0.431613	−	0.902059i	$-0.357945\pi$
$379$	6.28388e52	0.863225	0.431613	+	0.902059i	$0.357945\pi$
$380$	0	0
$381$	−6.33974e52	−0.781825
$382$	0	0
$383$	7.96221e52	0.881980	0.440990	−	0.897512i	$-0.354627\pi$
$383$	7.96221e52	0.881980	0.440990	+	0.897512i	$0.354627\pi$
$384$	0	0
$385$	1.82638e52	0.181822
$386$	0	0
$387$	−3.44153e52	−0.308090
$388$	0	0
$389$	−1.09705e53	−0.883607	−0.441803	−	0.897112i	$-0.645661\pi$
$389$	−1.09705e53	−0.883607	−0.441803	+	0.897112i	$0.645661\pi$
$390$	0	0
$391$	−3.60906e53	−2.61680
$392$	0	0
$393$	−5.79991e52	−0.378769
$394$	0	0
$395$	−2.72284e52	−0.160244
$396$	0	0
$397$	3.48161e53	1.84746	0.923729	−	0.383046i	$-0.125125\pi$
$397$	3.48161e53	1.84746	0.923729	+	0.383046i	$0.125125\pi$
$398$	0	0
$399$	4.09318e52	0.195937
$400$	0	0
$401$	2.78180e52	0.120190	0.0600948	−	0.998193i	$-0.480860\pi$
$401$	2.78180e52	0.120190	0.0600948	+	0.998193i	$0.480860\pi$
$402$	0	0
$403$	3.49318e52	0.136291
$404$	0	0
$405$	3.90931e52	0.137807
$406$	0	0
$407$	6.99986e53	2.23051
$408$	0	0
$409$	4.34541e52	0.125228	0.0626141	−	0.998038i	$-0.480056\pi$
$409$	4.34541e52	0.125228	0.0626141	+	0.998038i	$0.480056\pi$
$410$	0	0
$411$	1.00427e53	0.261873
$412$	0	0
$413$	−4.13047e53	−0.975042
$414$	0	0
$415$	−2.93824e52	−0.0628206
$416$	0	0
$417$	4.32449e53	0.837811
$418$	0	0
$419$	5.48344e53	0.963087	0.481544	−	0.876422i	$-0.340076\pi$
$419$	5.48344e53	0.963087	0.481544	+	0.876422i	$0.340076\pi$
$420$	0	0
$421$	3.35909e53	0.535105	0.267552	−	0.963543i	$-0.413785\pi$
$421$	3.35909e53	0.535105	0.267552	+	0.963543i	$0.413785\pi$
$422$	0	0
$423$	1.45183e53	0.209865
$424$	0	0
$425$	1.15238e54	1.51225
$426$	0	0
$427$	−1.16771e53	−0.139176
$428$	0	0
$429$	−1.81496e53	−0.196558
$430$	0	0
$431$	7.52113e53	0.740456	0.370228	−	0.928941i	$-0.379280\pi$
$431$	7.52113e53	0.740456	0.370228	+	0.928941i	$0.379280\pi$
$432$	0	0
$433$	1.01171e54	0.905848	0.452924	−	0.891549i	$-0.350381\pi$
$433$	1.01171e54	0.905848	0.452924	+	0.891549i	$0.350381\pi$
$434$	0	0
$435$	3.34449e52	0.0272459
$436$	0	0
$437$	−9.18010e53	−0.680735
$438$	0	0
$439$	−1.17795e54	−0.795436	−0.397718	−	0.917508i	$-0.630198\pi$
$439$	−1.17795e54	−0.795436	−0.397718	+	0.917508i	$0.630198\pi$
$440$	0	0
$441$	3.34533e53	0.205801
$442$	0	0
$443$	−5.72043e53	−0.320739	−0.160370	−	0.987057i	$-0.551269\pi$
$443$	−5.72043e53	−0.320739	−0.160370	+	0.987057i	$0.551269\pi$
$444$	0	0
$445$	−5.77948e53	−0.295467
$446$	0	0
$447$	−1.43997e53	−0.0671504
$448$	0	0
$449$	−2.89270e54	−1.23099	−0.615493	−	0.788142i	$-0.711043\pi$
$449$	−2.89270e54	−1.23099	−0.615493	+	0.788142i	$0.711043\pi$
$450$	0	0
$451$	1.58316e54	0.615045
$452$	0	0
$453$	9.57340e53	0.339667
$454$	0	0
$455$	6.60774e52	0.0214200
$456$	0	0
$457$	−3.35307e53	−0.0993483	−0.0496742	−	0.998765i	$-0.515818\pi$
$457$	−3.35307e53	−0.0993483	−0.0496742	+	0.998765i	$0.515818\pi$
$458$	0	0
$459$	−6.40334e54	−1.73479
$460$	0	0
$461$	6.94064e53	0.172001	0.0860004	−	0.996295i	$-0.472591\pi$
$461$	6.94064e53	0.172001	0.0860004	+	0.996295i	$0.472591\pi$
$462$	0	0
$463$	8.64109e53	0.195956	0.0979779	−	0.995189i	$-0.468763\pi$
$463$	8.64109e53	0.195956	0.0979779	+	0.995189i	$0.468763\pi$
$464$	0	0
$465$	−7.48921e53	−0.155471
$466$	0	0
$467$	5.75866e54	1.09478	0.547388	−	0.836879i	$-0.315622\pi$
$467$	5.75866e54	1.09478	0.547388	+	0.836879i	$0.315622\pi$
$468$	0	0
$469$	−1.98520e54	−0.345750
$470$	0	0
$471$	−1.94053e54	−0.309737
$472$	0	0
$473$	−9.83508e54	−1.43921
$474$	0	0
$475$	2.93123e54	0.393397
$476$	0	0
$477$	3.57563e54	0.440277
$478$	0	0
$479$	−1.06241e55	−1.20065	−0.600323	−	0.799757i	$-0.704961\pi$
$479$	−1.06241e55	−1.20065	−0.600323	+	0.799757i	$0.704961\pi$
$480$	0	0
$481$	2.53251e54	0.262771
$482$	0	0
$483$	−8.13219e54	−0.774981
$484$	0	0
$485$	3.08370e54	0.270002
$486$	0	0
$487$	−2.53731e54	−0.204188	−0.102094	−	0.994775i	$-0.532554\pi$
$487$	−2.53731e54	−0.204188	−0.102094	+	0.994775i	$0.532554\pi$
$488$	0	0
$489$	−2.39570e54	−0.177256
$490$	0	0
$491$	−1.74013e55	−1.18416	−0.592080	−	0.805880i	$-0.701693\pi$
$491$	−1.74013e55	−1.18416	−0.592080	+	0.805880i	$0.701693\pi$
$492$	0	0
$493$	−3.65355e54	−0.228746
$494$	0	0
$495$	−1.68820e54	−0.0972786
$496$	0	0
$497$	−1.39652e55	−0.740868
$498$	0	0
$499$	−1.71532e54	−0.0838079	−0.0419039	−	0.999122i	$-0.513342\pi$
$499$	−1.71532e54	−0.0838079	−0.0419039	+	0.999122i	$0.513342\pi$
$500$	0	0
$501$	−2.18054e55	−0.981501
$502$	0	0
$503$	3.16291e55	1.31203	0.656014	−	0.754749i	$-0.272241\pi$
$503$	3.16291e55	1.31203	0.656014	+	0.754749i	$0.272241\pi$
$504$	0	0
$505$	−3.01205e53	−0.0115183
$506$	0	0
$507$	2.30250e55	0.811962
$508$	0	0
$509$	3.59116e55	1.16821	0.584104	−	0.811679i	$-0.301446\pi$
$509$	3.59116e55	1.16821	0.584104	+	0.811679i	$0.301446\pi$
$510$	0	0
$511$	2.27692e55	0.683469
$512$	0	0
$513$	−1.62877e55	−0.451288
$514$	0	0
$515$	8.41122e54	0.215185
$516$	0	0
$517$	4.14900e55	0.980364
$518$	0	0
$519$	−7.30928e55	−1.59567
$520$	0	0
$521$	1.03534e55	0.208885	0.104443	−	0.994531i	$-0.466694\pi$
$521$	1.03534e55	0.208885	0.104443	+	0.994531i	$0.466694\pi$
$522$	0	0
$523$	−1.16796e54	−0.0217843	−0.0108922	−	0.999941i	$-0.503467\pi$
$523$	−1.16796e54	−0.0217843	−0.0108922	+	0.999941i	$0.503467\pi$
$524$	0	0
$525$	2.59663e55	0.447862
$526$	0	0
$527$	8.18127e55	1.30528
$528$	0	0
$529$	1.14648e56	1.69248
$530$	0	0
$531$	3.81796e55	0.521668
$532$	0	0
$533$	5.72780e54	0.0724569
$534$	0	0
$535$	8.62635e54	0.101059
$536$	0	0
$537$	2.86305e55	0.310710
$538$	0	0
$539$	9.56017e55	0.961382
$540$	0	0
$541$	−1.27601e56	−1.18935	−0.594677	−	0.803964i	$-0.702720\pi$
$541$	−1.27601e56	−1.18935	−0.594677	+	0.803964i	$0.702720\pi$
$542$	0	0
$543$	1.11409e56	0.962772
$544$	0	0
$545$	4.98317e55	0.399372
$546$	0	0
$547$	−1.90287e56	−1.41472	−0.707359	−	0.706854i	$-0.750114\pi$
$547$	−1.90287e56	−1.41472	−0.707359	+	0.706854i	$0.750114\pi$
$548$	0	0
$549$	1.07936e55	0.0744619
$550$	0	0
$551$	−9.29326e54	−0.0595060
$552$	0	0
$553$	6.70224e55	0.398433
$554$	0	0
$555$	−5.42958e55	−0.299751
$556$	0	0
$557$	1.52473e55	0.0781921	0.0390960	−	0.999235i	$-0.487552\pi$
$557$	1.52473e55	0.0781921	0.0390960	+	0.999235i	$0.487552\pi$
$558$	0	0
$559$	−3.55828e55	−0.169550
$560$	0	0
$561$	−4.25076e56	−1.88247
$562$	0	0
$563$	8.99832e55	0.370458	0.185229	−	0.982695i	$-0.440697\pi$
$563$	8.99832e55	0.370458	0.185229	+	0.982695i	$0.440697\pi$
$564$	0	0
$565$	−6.60025e55	−0.252678
$566$	0	0
$567$	−9.62272e55	−0.342646
$568$	0	0
$569$	3.81740e56	1.26464	0.632319	−	0.774708i	$-0.282103\pi$
$569$	3.81740e56	1.26464	0.632319	+	0.774708i	$0.282103\pi$
$570$	0	0
$571$	2.86684e56	0.883819	0.441910	−	0.897060i	$-0.354301\pi$
$571$	2.86684e56	0.883819	0.441910	+	0.897060i	$0.354301\pi$
$572$	0	0
$573$	−3.14410e56	−0.902251
$574$	0	0
$575$	−5.82367e56	−1.55599
$576$	0	0
$577$	−2.66238e55	−0.0662469	−0.0331235	−	0.999451i	$-0.510545\pi$
$577$	−2.66238e55	−0.0662469	−0.0331235	+	0.999451i	$0.510545\pi$
$578$	0	0
$579$	−5.36870e56	−1.24440
$580$	0	0
$581$	7.23245e55	0.156198
$582$	0	0
$583$	1.02183e57	2.05671
$584$	0	0
$585$	−6.10781e54	−0.0114602
$586$	0	0
$587$	−7.89073e56	−1.38050	−0.690250	−	0.723571i	$-0.742499\pi$
$587$	−7.89073e56	−1.38050	−0.690250	+	0.723571i	$0.742499\pi$
$588$	0	0
$589$	2.08101e56	0.339555
$590$	0	0
$591$	−4.28344e56	−0.652002
$592$	0	0
$593$	2.06394e56	0.293141	0.146570	−	0.989200i	$-0.453177\pi$
$593$	2.06394e56	0.293141	0.146570	+	0.989200i	$0.453177\pi$
$594$	0	0
$595$	1.54758e56	0.205143
$596$	0	0
$597$	−9.21532e56	−1.14035
$598$	0	0
$599$	−2.76254e56	−0.319196	−0.159598	−	0.987182i	$-0.551020\pi$
$599$	−2.76254e56	−0.319196	−0.159598	+	0.987182i	$0.551020\pi$
$600$	0	0
$601$	−8.31600e56	−0.897404	−0.448702	−	0.893681i	$-0.648113\pi$
$601$	−8.31600e56	−0.897404	−0.448702	+	0.893681i	$0.648113\pi$
$602$	0	0
$603$	1.83501e56	0.184983
$604$	0	0
$605$	−2.40968e56	−0.226973
$606$	0	0
$607$	2.29755e56	0.202254	0.101127	−	0.994874i	$-0.467755\pi$
$607$	2.29755e56	0.202254	0.101127	+	0.994874i	$0.467755\pi$
$608$	0	0
$609$	−8.23243e55	−0.0677444
$610$	0	0
$611$	1.50108e56	0.115494
$612$	0	0
$613$	−2.25026e57	−1.61917	−0.809585	−	0.587002i	$-0.800308\pi$
$613$	−2.25026e57	−1.61917	−0.809585	+	0.587002i	$0.800308\pi$
$614$	0	0
$615$	−1.22801e56	−0.0826540
$616$	0	0
$617$	−3.86090e56	−0.243132	−0.121566	−	0.992583i	$-0.538792\pi$
$617$	−3.86090e56	−0.243132	−0.121566	+	0.992583i	$0.538792\pi$
$618$	0	0
$619$	−1.66300e57	−0.980017	−0.490008	−	0.871718i	$-0.663006\pi$
$619$	−1.66300e57	−0.980017	−0.490008	+	0.871718i	$0.663006\pi$
$620$	0	0
$621$	3.23598e57	1.78496
$622$	0	0
$623$	1.42261e57	0.734652
$624$	0	0
$625$	1.75253e57	0.847470
$626$	0	0
$627$	−1.08123e57	−0.489705
$628$	0	0
$629$	5.93132e57	2.51659
$630$	0	0
$631$	1.11322e57	0.442569	0.221285	−	0.975209i	$-0.428975\pi$
$631$	1.11322e57	0.442569	0.221285	+	0.975209i	$0.428975\pi$
$632$	0	0
$633$	−3.97039e57	−1.47930
$634$	0	0
$635$	−6.09701e56	−0.212939
$636$	0	0
$637$	3.45881e56	0.113258
$638$	0	0
$639$	1.29086e57	0.396380
$640$	0	0
$641$	−4.23006e57	−1.21831	−0.609153	−	0.793053i	$-0.708491\pi$
$641$	−4.23006e57	−1.21831	−0.609153	+	0.793053i	$0.708491\pi$
$642$	0	0
$643$	3.65416e57	0.987330	0.493665	−	0.869652i	$-0.335657\pi$
$643$	3.65416e57	0.987330	0.493665	+	0.869652i	$0.335657\pi$
$644$	0	0
$645$	7.62879e56	0.193411
$646$	0	0
$647$	8.11040e57	1.92976	0.964882	−	0.262682i	$-0.0846070\pi$
$647$	8.11040e57	1.92976	0.964882	+	0.262682i	$0.0846070\pi$
$648$	0	0
$649$	1.09108e58	2.43692
$650$	0	0
$651$	1.84346e57	0.386566
$652$	0	0
$653$	1.52356e57	0.300013	0.150006	−	0.988685i	$-0.452071\pi$
$653$	1.52356e57	0.300013	0.150006	+	0.988685i	$0.452071\pi$
$654$	0	0
$655$	−5.57786e56	−0.103162
$656$	0	0
$657$	−2.10465e57	−0.365670
$658$	0	0
$659$	−4.79803e57	−0.783271	−0.391635	−	0.920120i	$-0.628091\pi$
$659$	−4.79803e57	−0.783271	−0.391635	+	0.920120i	$0.628091\pi$
$660$	0	0
$661$	−7.23138e56	−0.110941	−0.0554704	−	0.998460i	$-0.517666\pi$
$661$	−7.23138e56	−0.110941	−0.0554704	+	0.998460i	$0.517666\pi$
$662$	0	0
$663$	−1.53790e57	−0.221769
$664$	0	0
$665$	3.93647e56	0.0533658
$666$	0	0
$667$	1.84635e57	0.235361
$668$	0	0
$669$	−6.66549e57	−0.799092
$670$	0	0
$671$	3.08456e57	0.347842
$672$	0	0
$673$	4.08444e57	0.433334	0.216667	−	0.976246i	$-0.430481\pi$
$673$	4.08444e57	0.433334	0.216667	+	0.976246i	$0.430481\pi$
$674$	0	0
$675$	−1.03326e58	−1.03153
$676$	0	0
$677$	−5.75968e57	−0.541166	−0.270583	−	0.962697i	$-0.587217\pi$
$677$	−5.75968e57	−0.541166	−0.270583	+	0.962697i	$0.587217\pi$
$678$	0	0
$679$	−7.59050e57	−0.671336
$680$	0	0
$681$	−7.23521e57	−0.602470
$682$	0	0
$683$	4.27246e56	0.0335007	0.0167503	−	0.999860i	$-0.494668\pi$
$683$	4.27246e56	0.0335007	0.0167503	+	0.999860i	$0.494668\pi$
$684$	0	0
$685$	9.65817e56	0.0713244
$686$	0	0
$687$	1.86067e58	1.29436
$688$	0	0
$689$	3.69692e57	0.242296
$690$	0	0
$691$	−2.15615e58	−1.33161	−0.665807	−	0.746124i	$-0.731912\pi$
$691$	−2.15615e58	−1.33161	−0.665807	+	0.746124i	$0.731912\pi$
$692$	0	0
$693$	4.15548e57	0.241875
$694$	0	0
$695$	4.15892e57	0.228188
$696$	0	0
$697$	1.34149e58	0.693931
$698$	0	0
$699$	−3.19033e58	−1.55616
$700$	0	0
$701$	−1.34583e58	−0.619113	−0.309557	−	0.950881i	$-0.600181\pi$
$701$	−1.34583e58	−0.619113	−0.309557	+	0.950881i	$0.600181\pi$
$702$	0	0
$703$	1.50871e58	0.654667
$704$	0	0
$705$	−3.21826e57	−0.131748
$706$	0	0
$707$	7.41412e56	0.0286392
$708$	0	0
$709$	−4.49893e58	−1.64006	−0.820032	−	0.572317i	$-0.806045\pi$
$709$	−4.49893e58	−1.64006	−0.820032	+	0.572317i	$0.806045\pi$
$710$	0	0
$711$	−6.19516e57	−0.213170
$712$	0	0
$713$	−4.13447e58	−1.34303
$714$	0	0
$715$	−1.74547e57	−0.0535350
$716$	0	0
$717$	4.23345e58	1.22617
$718$	0	0
$719$	3.60264e58	0.985550	0.492775	−	0.870157i	$-0.335983\pi$
$719$	3.60264e58	0.985550	0.492775	+	0.870157i	$0.335983\pi$
$720$	0	0
$721$	−2.07041e58	−0.535037
$722$	0	0
$723$	−1.39286e58	−0.340073
$724$	0	0
$725$	−5.89546e57	−0.136015
$726$	0	0
$727$	1.21072e58	0.263991	0.131995	−	0.991250i	$-0.457862\pi$
$727$	1.21072e58	0.263991	0.131995	+	0.991250i	$0.457862\pi$
$728$	0	0
$729$	5.29718e58	1.09177
$730$	0	0
$731$	−8.33375e58	−1.62381
$732$	0	0
$733$	−4.56071e58	−0.840234	−0.420117	−	0.907470i	$-0.638011\pi$
$733$	−4.56071e58	−0.840234	−0.420117	+	0.907470i	$0.638011\pi$
$734$	0	0
$735$	−7.41554e57	−0.129197
$736$	0	0
$737$	5.24401e58	0.864132
$738$	0	0
$739$	−5.98997e58	−0.933713	−0.466857	−	0.884333i	$-0.654614\pi$
$739$	−5.98997e58	−0.933713	−0.466857	+	0.884333i	$0.654614\pi$
$740$	0	0
$741$	−3.91184e57	−0.0576910
$742$	0	0
$743$	1.60800e58	0.224395	0.112198	−	0.993686i	$-0.464211\pi$
$743$	1.60800e58	0.224395	0.112198	+	0.993686i	$0.464211\pi$
$744$	0	0
$745$	−1.38484e57	−0.0182892
$746$	0	0
$747$	−6.68525e57	−0.0835690
$748$	0	0
$749$	−2.12336e58	−0.251274
$750$	0	0
$751$	1.29410e59	1.44994	0.724970	−	0.688781i	$-0.241854\pi$
$751$	1.29410e59	1.44994	0.724970	+	0.688781i	$0.241854\pi$
$752$	0	0
$753$	2.76135e58	0.292971
$754$	0	0
$755$	9.20687e57	0.0925123
$756$	0	0
$757$	1.04679e59	0.996312	0.498156	−	0.867088i	$-0.334011\pi$
$757$	1.04679e59	0.996312	0.498156	+	0.867088i	$0.334011\pi$
$758$	0	0
$759$	2.14816e59	1.93691
$760$	0	0
$761$	−5.22023e58	−0.445969	−0.222984	−	0.974822i	$-0.571580\pi$
$761$	−5.22023e58	−0.445969	−0.222984	+	0.974822i	$0.571580\pi$
$762$	0	0
$763$	−1.22660e59	−0.993002
$764$	0	0
$765$	−1.43049e58	−0.109756
$766$	0	0
$767$	3.94748e58	0.287088
$768$	0	0
$769$	−1.14958e59	−0.792592	−0.396296	−	0.918123i	$-0.629705\pi$
$769$	−1.14958e59	−0.792592	−0.396296	+	0.918123i	$0.629705\pi$
$770$	0	0
$771$	1.21752e59	0.795904
$772$	0	0
$773$	2.16390e59	1.34140	0.670699	−	0.741729i	$-0.265994\pi$
$773$	2.16390e59	1.34140	0.670699	+	0.741729i	$0.265994\pi$
$774$	0	0
$775$	1.32015e59	0.776137
$776$	0	0
$777$	1.33649e59	0.745304
$778$	0	0
$779$	3.41225e58	0.180519
$780$	0	0
$781$	3.68897e59	1.85165
$782$	0	0
$783$	3.27587e58	0.156031
$784$	0	0
$785$	−1.86624e58	−0.0843605
$786$	0	0
$787$	−3.26265e59	−1.39987	−0.699937	−	0.714204i	$-0.746789\pi$
$787$	−3.26265e59	−1.39987	−0.699937	+	0.714204i	$0.746789\pi$
$788$	0	0
$789$	−5.08221e58	−0.207002
$790$	0	0
$791$	1.62464e59	0.628261
$792$	0	0
$793$	1.11598e58	0.0409784
$794$	0	0
$795$	−7.92603e58	−0.276395
$796$	0	0
$797$	−5.37864e59	−1.78146	−0.890731	−	0.454530i	$-0.849807\pi$
$797$	−5.37864e59	−1.78146	−0.890731	+	0.454530i	$0.849807\pi$
$798$	0	0
$799$	3.51565e59	1.10611
$800$	0	0
$801$	−1.31498e59	−0.393054
$802$	0	0
$803$	−6.01458e59	−1.70819
$804$	0	0
$805$	−7.82083e58	−0.211075
$806$	0	0
$807$	1.75303e59	0.449657
$808$	0	0
$809$	−7.04425e58	−0.171747	−0.0858737	−	0.996306i	$-0.527368\pi$
$809$	−7.04425e58	−0.171747	−0.0858737	+	0.996306i	$0.527368\pi$
$810$	0	0
$811$	−6.75289e59	−1.56517	−0.782586	−	0.622543i	$-0.786100\pi$
$811$	−6.75289e59	−1.56517	−0.782586	+	0.622543i	$0.786100\pi$
$812$	0	0
$813$	4.64114e59	1.02275
$814$	0	0
$815$	−2.30398e58	−0.0482778
$816$	0	0
$817$	−2.11979e59	−0.422417
$818$	0	0
$819$	1.50343e58	0.0284946
$820$	0	0
$821$	1.04795e59	0.188932	0.0944659	−	0.995528i	$-0.469886\pi$
$821$	1.04795e59	0.188932	0.0944659	+	0.995528i	$0.469886\pi$
$822$	0	0
$823$	−2.14107e57	−0.00367227	−0.00183614	−	0.999998i	$-0.500584\pi$
$823$	−2.14107e57	−0.00367227	−0.00183614	+	0.999998i	$0.500584\pi$
$824$	0	0
$825$	−6.85912e59	−1.11934
$826$	0	0
$827$	3.28295e58	0.0509802	0.0254901	−	0.999675i	$-0.491885\pi$
$827$	3.28295e58	0.0509802	0.0254901	+	0.999675i	$0.491885\pi$
$828$	0	0
$829$	−7.20495e59	−1.06479	−0.532394	−	0.846497i	$-0.678707\pi$
$829$	−7.20495e59	−1.06479	−0.532394	+	0.846497i	$0.678707\pi$
$830$	0	0
$831$	5.64970e59	0.794703
$832$	0	0
$833$	8.10080e59	1.08469
$834$	0	0
$835$	−2.09706e59	−0.267324
$836$	0	0
$837$	−7.33554e59	−0.890349
$838$	0	0
$839$	2.37196e59	0.274149	0.137074	−	0.990561i	$-0.456230\pi$
$839$	2.37196e59	0.274149	0.137074	+	0.990561i	$0.456230\pi$
$840$	0	0
$841$	−8.89794e59	−0.979426
$842$	0	0
$843$	−5.74664e59	−0.602488
$844$	0	0
$845$	2.21434e59	0.221148
$846$	0	0
$847$	5.93141e59	0.564349
$848$	0	0
$849$	1.03942e60	0.942292
$850$	0	0
$851$	−2.99744e60	−2.58937
$852$	0	0
$853$	1.03258e59	0.0850096	0.0425048	−	0.999096i	$-0.486466\pi$
$853$	1.03258e59	0.0850096	0.0425048	+	0.999096i	$0.486466\pi$
$854$	0	0
$855$	−3.63864e58	−0.0285518
$856$	0	0
$857$	1.28058e60	0.957852	0.478926	−	0.877855i	$-0.341026\pi$
$857$	1.28058e60	0.957852	0.478926	+	0.877855i	$0.341026\pi$
$858$	0	0
$859$	1.16067e60	0.827657	0.413828	−	0.910355i	$-0.364191\pi$
$859$	1.16067e60	0.827657	0.413828	+	0.910355i	$0.364191\pi$
$860$	0	0
$861$	3.02274e59	0.205512
$862$	0	0
$863$	1.19897e60	0.777296	0.388648	−	0.921386i	$-0.372942\pi$
$863$	1.19897e60	0.777296	0.388648	+	0.921386i	$0.372942\pi$
$864$	0	0
$865$	−7.02943e59	−0.434600
$866$	0	0
$867$	−2.18563e60	−1.28880
$868$	0	0
$869$	−1.77043e60	−0.995803
$870$	0	0
$871$	1.89725e59	0.101801
$872$	0	0
$873$	7.01621e59	0.359179
$874$	0	0
$875$	5.13067e59	0.250616
$876$	0	0
$877$	−1.62618e60	−0.758014	−0.379007	−	0.925394i	$-0.623734\pi$
$877$	−1.62618e60	−0.758014	−0.379007	+	0.925394i	$0.623734\pi$
$878$	0	0
$879$	1.69604e60	0.754508
$880$	0	0
$881$	5.20975e59	0.221213	0.110607	−	0.993864i	$-0.464721\pi$
$881$	5.20975e59	0.221213	0.110607	+	0.993864i	$0.464721\pi$
$882$	0	0
$883$	1.27377e60	0.516292	0.258146	−	0.966106i	$-0.416888\pi$
$883$	1.27377e60	0.516292	0.258146	+	0.966106i	$0.416888\pi$
$884$	0	0
$885$	−8.46322e59	−0.327490
$886$	0	0
$887$	1.40271e60	0.518243	0.259122	−	0.965845i	$-0.416567\pi$
$887$	1.40271e60	0.518243	0.259122	+	0.965845i	$0.416567\pi$
$888$	0	0
$889$	1.50077e60	0.529455
$890$	0	0
$891$	2.54189e60	0.856374
$892$	0	0
$893$	8.94249e59	0.287742
$894$	0	0
$895$	2.75343e59	0.0846256
$896$	0	0
$897$	7.77190e59	0.228183
$898$	0	0
$899$	−4.18544e59	−0.117400
$900$	0	0
$901$	8.65846e60	2.32050
$902$	0	0
$903$	−1.87782e60	−0.480899
$904$	0	0
$905$	1.07143e60	0.262222
$906$	0	0
$907$	−1.32324e60	−0.309519	−0.154760	−	0.987952i	$-0.549460\pi$
$907$	−1.32324e60	−0.309519	−0.154760	+	0.987952i	$0.549460\pi$
$908$	0	0
$909$	−6.85317e58	−0.0153226
$910$	0	0
$911$	4.51973e60	0.966019	0.483009	−	0.875615i	$-0.339544\pi$
$911$	4.51973e60	0.966019	0.483009	+	0.875615i	$0.339544\pi$
$912$	0	0
$913$	−1.91049e60	−0.390385
$914$	0	0
$915$	−2.39260e59	−0.0467454
$916$	0	0
$917$	1.37298e60	0.256504
$918$	0	0
$919$	3.08144e60	0.550536	0.275268	−	0.961368i	$-0.411233\pi$
$919$	3.08144e60	0.550536	0.275268	+	0.961368i	$0.411233\pi$
$920$	0	0
$921$	−3.28965e60	−0.562117
$922$	0	0
$923$	1.33465e60	0.218139
$924$	0	0
$925$	9.57092e60	1.49640
$926$	0	0
$927$	1.91377e60	0.286256
$928$	0	0
$929$	−3.19320e60	−0.456987	−0.228494	−	0.973545i	$-0.573380\pi$
$929$	−3.19320e60	−0.456987	−0.228494	+	0.973545i	$0.573380\pi$
$930$	0	0
$931$	2.06054e60	0.282171
$932$	0	0
$933$	−3.89868e60	−0.510909
$934$	0	0
$935$	−4.08801e60	−0.512713
$936$	0	0
$937$	−1.07427e60	−0.128959	−0.0644794	−	0.997919i	$-0.520539\pi$
$937$	−1.07427e60	−0.128959	−0.0644794	+	0.997919i	$0.520539\pi$
$938$	0	0
$939$	−1.39643e61	−1.60463
$940$	0	0
$941$	−1.47841e61	−1.62633	−0.813164	−	0.582035i	$-0.802257\pi$
$941$	−1.47841e61	−1.62633	−0.813164	+	0.582035i	$0.802257\pi$
$942$	0	0
$943$	−6.77934e60	−0.714000
$944$	0	0
$945$	−1.38760e60	−0.139931
$946$	0	0
$947$	8.72067e60	0.842122	0.421061	−	0.907032i	$-0.361658\pi$
$947$	8.72067e60	0.842122	0.421061	+	0.907032i	$0.361658\pi$
$948$	0	0
$949$	−2.17604e60	−0.201238
$950$	0	0
$951$	−3.55597e60	−0.314962
$952$	0	0
$953$	6.02037e59	0.0510762	0.0255381	−	0.999674i	$-0.491870\pi$
$953$	6.02037e59	0.0510762	0.0255381	+	0.999674i	$0.491870\pi$
$954$	0	0
$955$	−3.02373e60	−0.245739
$956$	0	0
$957$	2.17464e60	0.169314
$958$	0	0
$959$	−2.37735e60	−0.177342
$960$	0	0
$961$	−4.61807e60	−0.330089
$962$	0	0
$963$	1.96271e60	0.134437
$964$	0	0
$965$	−5.16316e60	−0.338926
$966$	0	0
$967$	−1.85730e61	−1.16853	−0.584266	−	0.811562i	$-0.698617\pi$
$967$	−1.85730e61	−1.16853	−0.584266	+	0.811562i	$0.698617\pi$
$968$	0	0
$969$	−9.16182e60	−0.552515
$970$	0	0
$971$	2.80101e61	1.61928	0.809638	−	0.586929i	$-0.199663\pi$
$971$	2.80101e61	1.61928	0.809638	+	0.586929i	$0.199663\pi$
$972$	0	0
$973$	−1.02371e61	−0.567369
$974$	0	0
$975$	−2.48159e60	−0.131867
$976$	0	0
$977$	2.94713e60	0.150162	0.0750812	−	0.997177i	$-0.476078\pi$
$977$	2.94713e60	0.150162	0.0750812	+	0.997177i	$0.476078\pi$
$978$	0	0
$979$	−3.75790e61	−1.83611
$980$	0	0
$981$	1.13380e61	0.531277
$982$	0	0
$983$	−1.11718e61	−0.502084	−0.251042	−	0.967976i	$-0.580773\pi$
$983$	−1.11718e61	−0.502084	−0.251042	+	0.967976i	$0.580773\pi$
$984$	0	0
$985$	−4.11944e60	−0.177581
$986$	0	0
$987$	7.92170e60	0.327579
$988$	0	0
$989$	4.21153e61	1.67077
$990$	0	0
$991$	9.78682e60	0.372505	0.186252	−	0.982502i	$-0.440366\pi$
$991$	9.78682e60	0.372505	0.186252	+	0.982502i	$0.440366\pi$
$992$	0	0
$993$	−1.03461e61	−0.377848
$994$	0	0
$995$	−8.86250e60	−0.310587
$996$	0	0
$997$	−1.62230e61	−0.545609	−0.272805	−	0.962069i	$-0.587951\pi$
$997$	−1.62230e61	−0.545609	−0.272805	+	0.962069i	$0.587951\pi$
$998$	0	0
$999$	−5.31817e61	−1.71660

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	16.42.a.a.1.1		1
4.3	odd	2		2.42.a.a.1.1	✓	1
12.11	even	2		18.42.a.b.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2.42.a.a.1.1	✓	1	4.3	odd	2
16.42.a.a.1.1		1	1.1	even	1	trivial
18.42.a.b.1.1		1	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 16 = 2^{4} \)
Weight:	\( k \)	\(=\)	\( 42 \)
Character orbit:	\([\chi]\)	\(=\)	16.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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