LMFDB - Embedded newform 18.42.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [18,42,Mod(1,18)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(18, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("18.1");

S:= CuspForms(chi, 42);

N := Newforms(S);

Level:	$ N $	$=$	$ 18 = 2 \cdot 3^{2} $
Weight:	$ k $	$=$	$ 42 $
Character orbit:	$[\chi]$	$=$	18.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:	$191.649006822$
Analytic rank:	$1$
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	$\chi$	$=$	18.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.04858e6 q^{2} +1.09951e12 q^{4} +4.85042e13 q^{5} -1.19392e17 q^{7} +1.15292e18 q^{8} +O(q^{10})$

\(q+1.04858e6 q^{2} +1.09951e12 q^{4} +4.85042e13 q^{5} -1.19392e17 q^{7} +1.15292e18 q^{8} +5.08603e19 q^{10} -3.15381e21 q^{11} -1.14103e22 q^{13} -1.25192e23 q^{14} +1.20893e24 q^{16} +2.67238e25 q^{17} +6.79752e25 q^{19} +5.33309e25 q^{20} -3.30701e27 q^{22} +1.35051e28 q^{23} -4.31221e28 q^{25} -1.19646e28 q^{26} -1.31273e29 q^{28} -1.36715e29 q^{29} +3.06142e30 q^{31} +1.26765e30 q^{32} +2.80219e31 q^{34} -5.79103e30 q^{35} -2.21949e32 q^{37} +7.12772e31 q^{38} +5.59215e31 q^{40} +5.01985e32 q^{41} -3.11848e33 q^{43} -3.46765e33 q^{44} +1.41611e34 q^{46} -1.31555e34 q^{47} -3.03131e34 q^{49} -4.52168e34 q^{50} -1.25458e34 q^{52} +3.23999e35 q^{53} -1.52973e35 q^{55} -1.37650e35 q^{56} -1.43357e35 q^{58} -3.45957e36 q^{59} -9.78043e35 q^{61} +3.21013e36 q^{62} +1.32923e36 q^{64} -5.53447e35 q^{65} +1.66275e37 q^{67} +2.93831e37 q^{68} -6.07234e36 q^{70} -1.16969e38 q^{71} +1.90709e38 q^{73} -2.32731e38 q^{74} +7.47395e37 q^{76} +3.76541e38 q^{77} -5.61362e38 q^{79} +5.86380e37 q^{80} +5.26369e38 q^{82} +6.05771e38 q^{83} +1.29621e39 q^{85} -3.26996e39 q^{86} -3.63609e39 q^{88} -1.19154e40 q^{89} +1.36230e39 q^{91} +1.48490e40 q^{92} -1.37946e40 q^{94} +3.29708e39 q^{95} -6.35760e40 q^{97} -3.17856e40 q^{98} +O(q^{100})\)

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 $

$2$	1.04858e6	0.707107
$2$	1.04858e6	0.707107
$3$	0	0
$3$	0	0
$4$	1.09951e12	0.500000
$5$	4.85042e13	0.227454	0.113727	−	0.993512i	$-0.463721\pi$
$5$	4.85042e13	0.227454	0.113727	+	0.993512i	$0.463721\pi$
$6$	0	0
$7$	−1.19392e17	−0.565545	−0.282772	−	0.959187i	$-0.591254\pi$
$7$	−1.19392e17	−0.565545	−0.282772	+	0.959187i	$0.591254\pi$
$8$	1.15292e18	0.353553
$9$	0	0
$10$	5.08603e19	0.160835
$11$	−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	$0.526533\pi$
$14$	−1.25192e23	−0.399901
$15$	0	0
$16$	1.20893e24	0.250000
$17$	2.67238e25	1.59476	0.797379	−	0.603479i	$-0.206219\pi$
$17$	2.67238e25	1.59476	0.797379	+	0.603479i	$0.206219\pi$
$18$	0	0
$19$	6.79752e25	0.414860	0.207430	−	0.978250i	$-0.433490\pi$
$19$	6.79752e25	0.414860	0.207430	+	0.978250i	$0.433490\pi$
$20$	5.33309e25	0.113727
$21$	0	0
$22$	−3.30701e27	−0.999471
$23$	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
$26$	−1.19646e28	−0.117745
$27$	0	0
$28$	−1.31273e29	−0.282772
$29$	−1.36715e29	−0.143436	−0.0717181	−	0.997425i	$-0.522848\pi$
$29$	−1.36715e29	−0.143436	−0.0717181	+	0.997425i	$0.522848\pi$
$30$	0	0
$31$	3.06142e30	0.818481	0.409241	−	0.912427i	$-0.365794\pi$
$31$	3.06142e30	0.818481	0.409241	+	0.912427i	$0.365794\pi$
$32$	1.26765e30	0.176777
$33$	0	0
$34$	2.80219e31	1.12766
$35$	−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	$0.789412\pi$
$38$	7.12772e31	0.293350
$39$	0	0
$40$	5.59215e31	0.0804173
$41$	5.01985e32	0.435133	0.217566	−	0.976046i	$-0.430188\pi$
$41$	5.01985e32	0.435133	0.217566	+	0.976046i	$0.430188\pi$
$42$	0	0
$43$	−3.11848e33	−1.01821	−0.509107	−	0.860703i	$-0.670024\pi$
$43$	−3.11848e33	−1.01821	−0.509107	+	0.860703i	$0.670024\pi$
$44$	−3.46765e33	−0.706733
$45$	0	0
$46$	1.41611e34	1.16028
$47$	−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
$50$	−4.52168e34	−0.670524
$51$	0	0
$52$	−1.25458e34	−0.0832585
$53$	3.23999e35	1.45508	0.727542	−	0.686064i	$-0.240663\pi$
$53$	3.23999e35	1.45508	0.727542	+	0.686064i	$0.240663\pi$
$54$	0	0
$55$	−1.52973e35	−0.321499
$56$	−1.37650e35	−0.199950
$57$	0	0
$58$	−1.43357e35	−0.101425
$59$	−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	$0.539266\pi$
$62$	3.21013e36	0.578753
$63$	0	0
$64$	1.32923e36	0.125000
$65$	−5.53447e35	−0.0378750
$66$	0	0
$67$	1.66275e37	0.611357	0.305678	−	0.952135i	$-0.401117\pi$
$67$	1.66275e37	0.611357	0.305678	+	0.952135i	$0.401117\pi$
$68$	2.93831e37	0.797379
$69$	0	0
$70$	−6.07234e36	−0.0909591
$71$	−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	$0.293470\pi$
$74$	−2.32731e38	−1.11584
$75$	0	0
$76$	7.47395e37	0.207430
$77$	3.76541e38	0.799378
$78$	0	0
$79$	−5.61362e38	−0.704511	−0.352256	−	0.935904i	$-0.614585\pi$
$79$	−5.61362e38	−0.704511	−0.352256	+	0.935904i	$0.614585\pi$
$80$	5.86380e37	0.0568636
$81$	0	0
$82$	5.26369e38	0.307685
$83$	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
$86$	−3.26996e39	−0.719987
$87$	0	0
$88$	−3.63609e39	−0.499736
$89$	−1.19154e40	−1.29902	−0.649508	−	0.760355i	$-0.725025\pi$
$89$	−1.19154e40	−1.29902	−0.649508	+	0.760355i	$0.725025\pi$
$90$	0	0
$91$	1.36230e39	0.0941728
$92$	1.48490e40	0.820439
$93$	0	0
$94$	−1.37946e40	−0.490441
$95$	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	$0.702266\pi$
$98$	−3.17856e40	−0.480945
$99$	0	0
$100$	−4.74132e40	−0.474132
$101$	−6.20987e39	−0.0506401	−0.0253200	−	0.999679i	$-0.508060\pi$
$101$	−6.20987e39	−0.0506401	−0.0253200	+	0.999679i	$0.508060\pi$
$102$	0	0
$103$	1.73412e41	0.946056	0.473028	−	0.881047i	$-0.343161\pi$
$103$	1.73412e41	0.946056	0.473028	+	0.881047i	$0.343161\pi$
$104$	−1.31552e40	−0.0588726
$105$	0	0
$106$	3.39737e41	1.02890
$107$	−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	$0.841066\pi$
$110$	−1.60404e41	−0.227334
$111$	0	0
$112$	−1.44337e41	−0.141386
$113$	−1.36076e42	−1.11089	−0.555447	−	0.831552i	$-0.687453\pi$
$113$	−1.36076e42	−1.11089	−0.555447	+	0.831552i	$0.687453\pi$
$114$	0	0
$115$	6.55053e41	0.373225
$116$	−1.50320e41	−0.0717181
$117$	0	0
$118$	−3.62763e42	−1.21911
$119$	−3.19062e42	−0.901907
$120$	0	0
$121$	4.96799e42	0.997886
$122$	−1.02555e42	−0.174013
$123$	0	0
$124$	3.36607e42	0.409241
$125$	−4.29732e42	−0.443141
$126$	0	0
$127$	−1.25701e43	−0.936185	−0.468092	−	0.883679i	$-0.655059\pi$
$127$	−1.25701e43	−0.936185	−0.468092	+	0.883679i	$0.655059\pi$
$128$	1.39380e42	0.0883883
$129$	0	0
$130$	−5.80332e41	−0.0267817
$131$	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
$134$	1.74352e43	0.432295
$135$	0	0
$136$	3.08104e43	0.563832
$137$	1.99120e43	0.313577	0.156788	−	0.987632i	$-0.449886\pi$
$137$	1.99120e43	0.313577	0.156788	+	0.987632i	$0.449886\pi$
$138$	0	0
$139$	8.57436e43	1.00322	0.501612	−	0.865093i	$-0.332740\pi$
$139$	8.57436e43	1.00322	0.501612	+	0.865093i	$0.332740\pi$
$140$	−6.36731e42	−0.0643178
$141$	0	0
$142$	−1.22651e44	−0.926316
$143$	3.59859e43	0.235366
$144$	0	0
$145$	−6.63127e42	−0.0326252
$146$	1.99972e44	0.854548
$147$	0	0
$148$	−2.44036e44	−0.789021
$149$	−2.85509e43	−0.0804083	−0.0402041	−	0.999191i	$-0.512801\pi$
$149$	−2.85509e43	−0.0804083	−0.0402041	+	0.999191i	$0.512801\pi$
$150$	0	0
$151$	1.89816e44	0.406729	0.203365	−	0.979103i	$-0.434812\pi$
$151$	1.89816e44	0.406729	0.203365	+	0.979103i	$0.434812\pi$
$152$	7.83701e43	0.146675
$153$	0	0
$154$	3.94832e44	0.565246
$155$	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	$0.440627\pi$
$158$	−5.88631e44	−0.498165
$159$	0	0
$160$	6.14864e43	0.0402086
$161$	−1.61240e45	−0.927990
$162$	0	0
$163$	−4.75006e44	−0.212253	−0.106126	−	0.994353i	$-0.533845\pi$
$163$	−4.75006e44	−0.212253	−0.106126	+	0.994353i	$0.533845\pi$
$164$	5.51938e44	0.217566
$165$	0	0
$166$	6.35197e44	0.195296
$167$	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
$170$	1.35918e45	0.256492
$171$	0	0
$172$	−3.42880e45	−0.509107
$173$	−1.44924e46	−1.91072	−0.955358	−	0.295452i	$-0.904530\pi$
$173$	−1.44924e46	−1.91072	−0.955358	+	0.295452i	$0.904530\pi$
$174$	0	0
$175$	5.14845e45	0.536286
$176$	−3.81272e45	−0.353366
$177$	0	0
$178$	−1.24942e46	−0.918543
$179$	−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	$0.695554\pi$
$182$	1.42848e45	0.0665902
$183$	0	0
$184$	1.55703e46	0.580138
$185$	−1.07655e46	−0.358932
$186$	0	0
$187$	−8.42816e46	−2.25414
$188$	−1.44646e46	−0.346794
$189$	0	0
$190$	3.45724e45	0.0667238
$191$	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	$0.232429\pi$
$194$	−6.66643e46	−0.839378
$195$	0	0
$196$	−3.33296e46	−0.340080
$197$	−8.49296e46	−0.780731	−0.390365	−	0.920660i	$-0.627651\pi$
$197$	−8.49296e46	−0.780731	−0.390365	+	0.920660i	$0.627651\pi$
$198$	0	0
$199$	−1.82716e47	−1.36549	−0.682746	−	0.730656i	$-0.739214\pi$
$199$	−1.82716e47	−1.36549	−0.682746	+	0.730656i	$0.739214\pi$
$200$	−4.97164e46	−0.335262
$201$	0	0
$202$	−6.51152e45	−0.0358079
$203$	1.63228e46	0.0811196
$204$	0	0
$205$	2.43484e46	0.0989728
$206$	1.81836e47	0.668963
$207$	0	0
$208$	−1.37942e46	−0.0416292
$209$	−2.14381e47	−0.586391
$210$	0	0
$211$	−7.87226e47	−1.77137	−0.885684	−	0.464288i	$-0.846310\pi$
$211$	−7.87226e47	−1.77137	−0.885684	+	0.464288i	$0.846310\pi$
$212$	3.56240e47	0.727542
$213$	0	0
$214$	−1.86487e47	−0.314170
$215$	−1.51259e47	−0.231597
$216$	0	0
$217$	−3.65511e47	−0.462888
$218$	−1.07727e48	−1.24156
$219$	0	0
$220$	−1.68196e47	−0.160749
$221$	−3.04926e47	−0.265554
$222$	0	0
$223$	−1.32160e48	−0.956862	−0.478431	−	0.878125i	$-0.658794\pi$
$223$	−1.32160e48	−0.956862	−0.478431	+	0.878125i	$0.658794\pi$
$224$	−1.51348e47	−0.0999751
$225$	0	0
$226$	−1.42686e48	−0.785521
$227$	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	$0.782229\pi$
$230$	6.86873e47	0.263910
$231$	0	0
$232$	−1.57622e47	−0.0507123
$233$	−6.32561e48	−1.86340	−0.931700	−	0.363228i	$-0.881675\pi$
$233$	−6.32561e48	−1.86340	−0.931700	+	0.363228i	$0.881675\pi$
$234$	0	0
$235$	−6.38098e47	−0.157760
$236$	−3.80384e48	−0.862038
$237$	0	0
$238$	−3.34560e48	−0.637744
$239$	−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	$0.434733\pi$
$242$	5.20932e48	0.705612
$243$	0	0
$244$	−1.07537e48	−0.123046
$245$	−1.47031e48	−0.154705
$246$	0	0
$247$	−7.75618e47	−0.0690812
$248$	3.52958e48	0.289377
$249$	0	0
$250$	−4.50606e48	−0.313348
$251$	−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
$254$	−1.31807e49	−0.661983
$255$	0	0
$256$	1.46150e48	0.0625000
$257$	2.41403e49	0.953045	0.476522	−	0.879162i	$-0.341897\pi$
$257$	2.41403e49	0.953045	0.476522	+	0.879162i	$0.341897\pi$
$258$	0	0
$259$	2.64991e49	0.892453
$260$	−6.08522e47	−0.0189375
$261$	0	0
$262$	1.20583e49	0.320709
$263$	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
$266$	−8.50996e48	−0.165903
$267$	0	0
$268$	1.82822e49	0.305678
$269$	3.47580e49	0.538435	0.269217	−	0.963079i	$-0.413235\pi$
$269$	3.47580e49	0.538435	0.269217	+	0.963079i	$0.413235\pi$
$270$	0	0
$271$	9.20219e49	1.22468	0.612338	−	0.790596i	$-0.290229\pi$
$271$	9.20219e49	1.22468	0.612338	+	0.790596i	$0.290229\pi$
$272$	3.23070e49	0.398689
$273$	0	0
$274$	2.08793e49	0.221732
$275$	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	$0.657842\pi$
$278$	8.99087e49	0.709387
$279$	0	0
$280$	−6.67661e48	−0.0454796
$281$	−1.13941e50	−0.721441	−0.360720	−	0.932674i	$-0.617469\pi$
$281$	−1.13941e50	−0.721441	−0.360720	+	0.932674i	$0.617469\pi$
$282$	0	0
$283$	2.06091e50	1.12833	0.564167	−	0.825661i	$-0.309197\pi$
$283$	2.06091e50	1.12833	0.564167	+	0.825661i	$0.309197\pi$
$284$	−1.28609e50	−0.655004
$285$	0	0
$286$	3.77340e49	0.166429
$287$	−5.99332e49	−0.246087
$288$	0	0
$289$	4.33354e50	1.54325
$290$	−6.95339e48	−0.0230695
$291$	0	0
$292$	2.09686e50	0.604257
$293$	3.36281e50	0.903475	0.451738	−	0.892151i	$-0.350804\pi$
$293$	3.36281e50	0.903475	0.451738	+	0.892151i	$0.350804\pi$
$294$	0	0
$295$	−1.67804e50	−0.392149
$296$	−2.55890e50	−0.557922
$297$	0	0
$298$	−2.99378e49	−0.0568573
$299$	−1.54097e50	−0.273234
$300$	0	0
$301$	3.72323e50	0.575846
$302$	1.99037e50	0.287601
$303$	0	0
$304$	8.21770e49	0.103715
$305$	−4.74392e49	−0.0559746
$306$	0	0
$307$	−6.52252e50	−0.673099	−0.336550	−	0.941666i	$-0.609260\pi$
$307$	−6.52252e50	−0.673099	−0.336550	+	0.941666i	$0.609260\pi$
$308$	4.14011e50	0.399689
$309$	0	0
$310$	1.55705e50	0.131640
$311$	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	$0.0895095\pi$
$314$	4.03448e50	0.262259
$315$	0	0
$316$	−6.17225e50	−0.352256
$317$	−7.05058e50	−0.377146	−0.188573	−	0.982059i	$-0.560386\pi$
$317$	−7.05058e50	−0.377146	−0.188573	+	0.982059i	$0.560386\pi$
$318$	0	0
$319$	4.31174e50	0.202742
$320$	6.44731e49	0.0284318
$321$	0	0
$322$	−1.69073e51	−0.656188
$323$	1.81655e51	0.661601
$324$	0	0
$325$	4.92036e50	0.157902
$326$	−4.98079e50	−0.150085
$327$	0	0
$328$	5.78749e50	0.153843
$329$	1.57067e51	0.392255
$330$	0	0
$331$	−2.05137e51	−0.452449	−0.226224	−	0.974075i	$-0.572638\pi$
$331$	−2.05137e51	−0.452449	−0.226224	+	0.974075i	$0.572638\pi$
$332$	6.66052e50	0.138095
$333$	0	0
$334$	4.53347e51	0.831052
$335$	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	$0.790689\pi$
$338$	−4.78702e51	−0.687500
$339$	0	0
$340$	1.42520e51	0.181367
$341$	−9.65514e51	−1.15689
$342$	0	0
$343$	8.94019e51	0.950205
$344$	−3.59536e51	−0.359993
$345$	0	0
$346$	−1.51964e52	−1.35108
$347$	−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	$0.612795\pi$
$350$	5.39854e51	0.379212
$351$	0	0
$352$	−3.99793e51	−0.249868
$353$	1.81721e52	1.07158	0.535788	−	0.844353i	$-0.320015\pi$
$353$	1.81721e52	1.07158	0.535788	+	0.844353i	$0.320015\pi$
$354$	0	0
$355$	−5.67348e51	−0.297967
$356$	−1.31012e52	−0.649508
$357$	0	0
$358$	−5.95244e51	−0.263083
$359$	−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
$362$	−2.31626e52	−0.815193
$363$	0	0
$364$	1.49787e51	0.0470864
$365$	9.25016e51	0.274882
$366$	0	0
$367$	−2.11760e52	−0.562586	−0.281293	−	0.959622i	$-0.590763\pi$
$367$	−2.11760e52	−0.562586	−0.281293	+	0.959622i	$0.590763\pi$
$368$	1.63266e52	0.410220
$369$	0	0
$370$	−1.12884e52	−0.253804
$371$	−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	$0.581102\pi$
$374$	−8.83757e52	−1.59391
$375$	0	0
$376$	−1.51673e52	−0.245221
$377$	1.55996e51	0.0238845
$378$	0	0
$379$	−6.28388e52	−0.863225	−0.431613	−	0.902059i	$-0.642055\pi$
$379$	−6.28388e52	−0.863225	−0.431613	+	0.902059i	$0.642055\pi$
$380$	3.62518e51	0.0471809
$381$	0	0
$382$	6.53677e52	0.763949
$383$	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
$386$	1.11618e53	1.05365
$387$	0	0
$388$	−6.99026e52	−0.593530
$389$	1.09705e53	0.883607	0.441803	−	0.897112i	$-0.354339\pi$
$389$	1.09705e53	0.883607	0.441803	+	0.897112i	$0.354339\pi$
$390$	0	0
$391$	3.60906e53	2.61680
$392$	−3.49486e52	−0.240473
$393$	0	0
$394$	−8.90551e52	−0.552060
$395$	−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	$0.125125\pi$
$398$	−1.91592e53	−0.965548
$399$	0	0
$400$	−5.21314e52	−0.237066
$401$	−2.78180e52	−0.120190	−0.0600948	−	0.998193i	$-0.519140\pi$
$401$	−2.78180e52	−0.120190	−0.0600948	+	0.998193i	$0.519140\pi$
$402$	0	0
$403$	−3.49318e52	−0.136291
$404$	−6.82782e51	−0.0253200
$405$	0	0
$406$	1.71157e52	0.0573602
$407$	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	$0.480056\pi$
$410$	2.55311e52	0.0699843
$411$	0	0
$412$	1.90669e53	0.473028
$413$	4.13047e53	0.975042
$414$	0	0
$415$	2.93824e52	0.0628206
$416$	−1.44643e52	−0.0294363
$417$	0	0
$418$	−2.24795e53	−0.414641
$419$	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	$0.413785\pi$
$422$	−8.25466e53	−1.25255
$423$	0	0
$424$	3.73545e53	0.514450
$425$	−1.15238e54	−1.51225
$426$	0	0
$427$	1.16771e53	0.139176
$428$	−1.95545e53	−0.222152
$429$	0	0
$430$	−1.58607e53	−0.163764
$431$	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	$0.350381\pi$
$434$	−3.83266e53	−0.327311
$435$	0	0
$436$	−1.12960e54	−0.877916
$437$	9.18010e53	0.680735
$438$	0	0
$439$	1.17795e54	0.795436	0.397718	−	0.917508i	$-0.369802\pi$
$439$	1.17795e54	0.795436	0.397718	+	0.917508i	$0.369802\pi$
$440$	−1.76366e53	−0.113667
$441$	0	0
$442$	−3.19738e53	−0.187775
$443$	−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
$446$	−1.38579e54	−0.676604
$447$	0	0
$448$	−1.58700e53	−0.0706931
$449$	2.89270e54	1.23099	0.615493	−	0.788142i	$-0.288957\pi$
$449$	2.89270e54	1.23099	0.615493	+	0.788142i	$0.288957\pi$
$450$	0	0
$451$	−1.58316e54	−0.615045
$452$	−1.49617e54	−0.555447
$453$	0	0
$454$	1.50424e54	0.510121
$455$	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	$0.515818\pi$
$458$	−3.86843e54	−1.09596
$459$	0	0
$460$	7.20238e53	0.186612
$461$	−6.94064e53	−0.172001	−0.0860004	−	0.996295i	$-0.527409\pi$
$461$	−6.94064e53	−0.172001	−0.0860004	+	0.996295i	$0.527409\pi$
$462$	0	0
$463$	−8.64109e53	−0.195956	−0.0979779	−	0.995189i	$-0.531237\pi$
$463$	−8.64109e53	−0.195956	−0.0979779	+	0.995189i	$0.531237\pi$
$464$	−1.65279e53	−0.0358590
$465$	0	0
$466$	−6.63289e54	−1.31762
$467$	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
$470$	−6.69094e53	−0.111553
$471$	0	0
$472$	−3.98862e54	−0.609553
$473$	9.83508e54	1.43921
$474$	0	0
$475$	−2.93123e54	−0.393397
$476$	−3.50812e54	−0.450953
$477$	0	0
$478$	−8.80159e54	−1.03822
$479$	−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
$482$	2.89583e54	0.287945
$483$	0	0
$484$	5.46237e54	0.498943
$485$	−3.08370e54	−0.270002
$486$	0	0
$487$	2.53731e54	0.204188	0.102094	−	0.994775i	$-0.467446\pi$
$487$	2.53731e54	0.204188	0.102094	+	0.994775i	$0.467446\pi$
$488$	−1.12761e54	−0.0870065
$489$	0	0
$490$	−1.54173e54	−0.109393
$491$	−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
$494$	−8.13294e53	−0.0488478
$495$	0	0
$496$	3.70103e54	0.204620
$497$	1.39652e55	0.740868
$498$	0	0
$499$	1.71532e54	0.0838079	0.0419039	−	0.999122i	$-0.486658\pi$
$499$	1.71532e54	0.0838079	0.0419039	+	0.999122i	$0.486658\pi$
$500$	−4.72495e54	−0.221571
$501$	0	0
$502$	−5.74101e54	−0.248063
$503$	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
$506$	−4.46614e55	−1.64001
$507$	0	0
$508$	−1.38209e55	−0.468092
$509$	−3.59116e55	−1.16821	−0.584104	−	0.811679i	$-0.698554\pi$
$509$	−3.59116e55	−1.16821	−0.584104	+	0.811679i	$0.698554\pi$
$510$	0	0
$511$	−2.27692e55	−0.683469
$512$	1.53250e54	0.0441942
$513$	0	0
$514$	2.53129e55	0.673904
$515$	8.41122e54	0.215185
$516$	0	0
$517$	4.14900e55	0.980364
$518$	2.77863e55	0.631060
$519$	0	0
$520$	−6.38081e53	−0.0133908
$521$	−1.03534e55	−0.208885	−0.104443	−	0.994531i	$-0.533306\pi$
$521$	−1.03534e55	−0.208885	−0.104443	+	0.994531i	$0.533306\pi$
$522$	0	0
$523$	1.16796e54	0.0217843	0.0108922	−	0.999941i	$-0.496533\pi$
$523$	1.16796e54	0.0217843	0.0108922	+	0.999941i	$0.496533\pi$
$524$	1.26441e55	0.226776
$525$	0	0
$526$	1.05662e55	0.175272
$527$	8.18127e55	1.30528
$528$	0	0
$529$	1.14648e56	1.69248
$530$	1.64787e55	0.234028
$531$	0	0
$532$	−8.92334e54	−0.117311
$533$	−5.72780e54	−0.0724569
$534$	0	0
$535$	−8.62635e54	−0.101059
$536$	1.91703e55	0.216147
$537$	0	0
$538$	3.64464e55	0.380731
$539$	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	$0.702720\pi$
$542$	9.64920e55	0.865976
$543$	0	0
$544$	3.38764e55	0.281916
$545$	−4.98317e55	−0.399372
$546$	0	0
$547$	1.90287e56	1.41472	0.707359	−	0.706854i	$-0.249886\pi$
$547$	1.90287e56	1.41472	0.707359	+	0.706854i	$0.249886\pi$
$548$	2.18935e55	0.156788
$549$	0	0
$550$	1.42605e56	0.947763
$551$	−9.29326e54	−0.0595060
$552$	0	0
$553$	6.70224e55	0.398433
$554$	−1.17461e56	−0.672887
$555$	0	0
$556$	9.42761e55	0.501612
$557$	−1.52473e55	−0.0781921	−0.0390960	−	0.999235i	$-0.512448\pi$
$557$	−1.52473e55	−0.0781921	−0.0390960	+	0.999235i	$0.512448\pi$
$558$	0	0
$559$	3.55828e55	0.169550
$560$	−7.00093e54	−0.0321589
$561$	0	0
$562$	−1.19476e56	−0.510136
$563$	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
$566$	2.16102e56	0.797853
$567$	0	0
$568$	−1.34856e56	−0.463158
$569$	−3.81740e56	−1.26464	−0.632319	−	0.774708i	$-0.717897\pi$
$569$	−3.81740e56	−1.26464	−0.632319	+	0.774708i	$0.717897\pi$
$570$	0	0
$571$	−2.86684e56	−0.883819	−0.441910	−	0.897060i	$-0.645699\pi$
$571$	−2.86684e56	−0.883819	−0.441910	+	0.897060i	$0.645699\pi$
$572$	3.95669e55	0.117683
$573$	0	0
$574$	−6.28445e55	−0.174010
$575$	−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	$0.510545\pi$
$578$	4.54404e56	1.09124
$579$	0	0
$580$	−7.29116e54	−0.0163126
$581$	−7.23245e55	−0.156198
$582$	0	0
$583$	−1.02183e57	−2.05671
$584$	2.19872e56	0.427274
$585$	0	0
$586$	3.52616e56	0.638853
$587$	−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
$590$	−1.75955e56	−0.277291
$591$	0	0
$592$	−2.68320e56	−0.394510
$593$	−2.06394e56	−0.293141	−0.146570	−	0.989200i	$-0.546823\pi$
$593$	−2.06394e56	−0.293141	−0.146570	+	0.989200i	$0.546823\pi$
$594$	0	0
$595$	−1.54758e56	−0.205143
$596$	−3.13920e55	−0.0402041
$597$	0	0
$598$	−1.61582e56	−0.193206
$599$	−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	$0.648113\pi$
$602$	3.90409e56	0.407185
$603$	0	0
$604$	2.08705e56	0.203365
$605$	2.40968e56	0.226973
$606$	0	0
$607$	−2.29755e56	−0.202254	−0.101127	−	0.994874i	$-0.532245\pi$
$607$	−2.29755e56	−0.202254	−0.101127	+	0.994874i	$0.532245\pi$
$608$	8.61688e55	0.0733376
$609$	0	0
$610$	−4.97436e55	−0.0395800
$611$	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	$0.800308\pi$
$614$	−6.83936e56	−0.475953
$615$	0	0
$616$	4.34122e56	0.282623
$617$	3.86090e56	0.243132	0.121566	−	0.992583i	$-0.461208\pi$
$617$	3.86090e56	0.243132	0.121566	+	0.992583i	$0.461208\pi$
$618$	0	0
$619$	1.66300e57	0.980017	0.490008	−	0.871718i	$-0.336994\pi$
$619$	1.66300e57	0.980017	0.490008	+	0.871718i	$0.336994\pi$
$620$	1.63269e56	0.0930835
$621$	0	0
$622$	8.10557e56	0.432594
$623$	1.42261e57	0.734652
$624$	0	0
$625$	1.75253e57	0.847470
$626$	2.90327e57	1.35867
$627$	0	0
$628$	4.23046e56	0.185445
$629$	−5.93132e57	−2.51659
$630$	0	0
$631$	−1.11322e57	−0.442569	−0.221285	−	0.975209i	$-0.571025\pi$
$631$	−1.11322e57	−0.442569	−0.221285	+	0.975209i	$0.571025\pi$
$632$	−6.47207e56	−0.249082
$633$	0	0
$634$	−7.39307e56	−0.266683
$635$	−6.09701e56	−0.212939
$636$	0	0
$637$	3.45881e56	0.113258
$638$	4.52119e56	0.143360
$639$	0	0
$640$	6.76050e55	0.0201043
$641$	4.23006e57	1.21831	0.609153	−	0.793053i	$-0.291509\pi$
$641$	4.23006e57	1.21831	0.609153	+	0.793053i	$0.291509\pi$
$642$	0	0
$643$	−3.65416e57	−0.987330	−0.493665	−	0.869652i	$-0.664343\pi$
$643$	−3.65416e57	−0.987330	−0.493665	+	0.869652i	$0.664343\pi$
$644$	−1.77286e57	−0.463995
$645$	0	0
$646$	1.90479e57	0.467823
$647$	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
$650$	5.15937e56	0.111654
$651$	0	0
$652$	−5.22274e56	−0.106126
$653$	−1.52356e57	−0.300013	−0.150006	−	0.988685i	$-0.547929\pi$
$653$	−1.52356e57	−0.300013	−0.150006	+	0.988685i	$0.547929\pi$
$654$	0	0
$655$	5.57786e56	0.103162
$656$	6.06863e56	0.108783
$657$	0	0
$658$	1.64697e57	0.277366
$659$	−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	$0.517666\pi$
$662$	−2.15101e57	−0.319930
$663$	0	0
$664$	6.98406e56	0.0976478
$665$	−3.93647e56	−0.0533658
$666$	0	0
$667$	−1.84635e57	−0.235361
$668$	4.75369e57	0.587642
$669$	0	0
$670$	8.45683e56	0.0983273
$671$	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	$0.430481\pi$
$674$	−1.08763e58	−1.11932
$675$	0	0
$676$	−5.01955e57	−0.486136
$677$	5.75968e57	0.541166	0.270583	−	0.962697i	$-0.412783\pi$
$677$	5.75968e57	0.541166	0.270583	+	0.962697i	$0.412783\pi$
$678$	0	0
$679$	7.59050e57	0.671336
$680$	1.49443e57	0.128246
$681$	0	0
$682$	−1.01242e58	−0.818048
$683$	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
$686$	9.37447e57	0.671897
$687$	0	0
$688$	−3.77001e57	−0.254554
$689$	−3.69692e57	−0.242296
$690$	0	0
$691$	2.15615e58	1.33161	0.665807	−	0.746124i	$-0.268088\pi$
$691$	2.15615e58	1.33161	0.665807	+	0.746124i	$0.268088\pi$
$692$	−1.59346e58	−0.955358
$693$	0	0
$694$	−2.98957e56	−0.0168943
$695$	4.15892e57	0.228188
$696$	0	0
$697$	1.34149e58	0.693931
$698$	−9.76930e57	−0.490713
$699$	0	0
$700$	5.66078e57	0.268143
$701$	1.34583e58	0.619113	0.309557	−	0.950881i	$-0.399819\pi$
$701$	1.34583e58	0.619113	0.309557	+	0.950881i	$0.399819\pi$
$702$	0	0
$703$	−1.50871e58	−0.654667
$704$	−4.19213e57	−0.176683
$705$	0	0
$706$	1.90548e58	0.757719
$707$	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	$0.806045\pi$
$710$	−5.94908e57	−0.210695
$711$	0	0
$712$	−1.37376e58	−0.459271
$713$	4.13447e58	1.34303
$714$	0	0
$715$	1.74547e57	0.0535350
$716$	−6.24158e57	−0.186028
$717$	0	0
$718$	−2.00059e58	−0.563128
$719$	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
$722$	−2.33062e58	−0.585407
$723$	0	0
$724$	−2.42877e58	−0.576429
$725$	5.89546e57	0.136015
$726$	0	0
$727$	−1.21072e58	−0.263991	−0.131995	−	0.991250i	$-0.542138\pi$
$727$	−1.21072e58	−0.263991	−0.131995	+	0.991250i	$0.542138\pi$
$728$	1.57063e57	0.0332951
$729$	0	0
$730$	9.69950e57	0.194371
$731$	−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	$0.638011\pi$
$734$	−2.22046e58	−0.397808
$735$	0	0
$736$	1.71197e58	0.290069
$737$	−5.24401e58	−0.864132
$738$	0	0
$739$	5.98997e58	0.933713	0.466857	−	0.884333i	$-0.345386\pi$
$739$	5.98997e58	0.933713	0.466857	+	0.884333i	$0.345386\pi$
$740$	−1.18368e58	−0.179466
$741$	0	0
$742$	−4.05620e58	−0.581889
$743$	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
$746$	−2.77417e58	−0.356441
$747$	0	0
$748$	−9.26686e58	−1.12707
$749$	2.12336e58	0.251274
$750$	0	0
$751$	−1.29410e59	−1.44994	−0.724970	−	0.688781i	$-0.758146\pi$
$751$	−1.29410e59	−1.44994	−0.724970	+	0.688781i	$0.758146\pi$
$752$	−1.59040e58	−0.173397
$753$	0	0
$754$	1.63574e57	0.0168889
$755$	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	$0.334011\pi$
$758$	−6.58913e58	−0.610392
$759$	0	0
$760$	3.80128e57	0.0333619
$761$	5.22023e58	0.445969	0.222984	−	0.974822i	$-0.428420\pi$
$761$	5.22023e58	0.445969	0.222984	+	0.974822i	$0.428420\pi$
$762$	0	0
$763$	1.22660e59	0.993002
$764$	6.85430e58	0.540194
$765$	0	0
$766$	8.34899e58	0.623654
$767$	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	$0.629705\pi$
$770$	1.91510e58	0.128568
$771$	0	0
$772$	1.17040e59	0.745043
$773$	−2.16390e59	−1.34140	−0.670699	−	0.741729i	$-0.734006\pi$
$773$	−2.16390e59	−1.34140	−0.670699	+	0.741729i	$0.734006\pi$
$774$	0	0
$775$	−1.32015e59	−0.776137
$776$	−7.32981e58	−0.419689
$777$	0	0
$778$	1.15034e59	0.624804
$779$	3.41225e58	0.180519
$780$	0	0
$781$	3.68897e59	1.85165
$782$	3.78438e59	1.85036
$783$	0	0
$784$	−3.66463e58	−0.170040
$785$	1.86624e58	0.0843605
$786$	0	0
$787$	3.26265e59	1.39987	0.699937	−	0.714204i	$-0.253211\pi$
$787$	3.26265e59	1.39987	0.699937	+	0.714204i	$0.253211\pi$
$788$	−9.33811e58	−0.390365
$789$	0	0
$790$	−2.85511e58	−0.113310
$791$	1.62464e59	0.628261
$792$	0	0
$793$	1.11598e58	0.0409784
$794$	3.65073e59	1.30635
$795$	0	0
$796$	−2.00899e59	−0.682746
$797$	5.37864e59	1.78146	0.890731	−	0.454530i	$-0.150193\pi$
$797$	5.37864e59	1.78146	0.890731	+	0.454530i	$0.150193\pi$
$798$	0	0
$799$	−3.51565e59	−1.10611
$800$	−5.46637e58	−0.167631
$801$	0	0
$802$	−2.91693e58	−0.0849868
$803$	−6.01458e59	−1.70819
$804$	0	0
$805$	−7.82083e58	−0.211075
$806$	−3.66286e58	−0.0963722
$807$	0	0
$808$	−7.15949e57	−0.0179040
$809$	7.04425e58	0.171747	0.0858737	−	0.996306i	$-0.472632\pi$
$809$	7.04425e58	0.171747	0.0858737	+	0.996306i	$0.472632\pi$
$810$	0	0
$811$	6.75289e59	1.56517	0.782586	−	0.622543i	$-0.213900\pi$
$811$	6.75289e59	1.56517	0.782586	+	0.622543i	$0.213900\pi$
$812$	1.79471e58	0.0405598
$813$	0	0
$814$	7.33988e59	1.57721
$815$	−2.30398e58	−0.0482778
$816$	0	0
$817$	−2.11979e59	−0.422417
$818$	4.55649e58	0.0885498
$819$	0	0
$820$	2.67713e58	0.0494864
$821$	−1.04795e59	−0.188932	−0.0944659	−	0.995528i	$-0.530114\pi$
$821$	−1.04795e59	−0.188932	−0.0944659	+	0.995528i	$0.530114\pi$
$822$	0	0
$823$	2.14107e57	0.00367227	0.00183614	−	0.999998i	$-0.499416\pi$
$823$	2.14107e57	0.00367227	0.00183614	+	0.999998i	$0.499416\pi$
$824$	1.99931e59	0.334481
$825$	0	0
$826$	4.33111e59	0.689459
$827$	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	$0.678707\pi$
$830$	3.08097e58	0.0444208
$831$	0	0
$832$	−1.51669e58	−0.0208146
$833$	−8.10080e59	−1.08469
$834$	0	0
$835$	2.09706e59	0.267324
$836$	−2.35714e59	−0.293195
$837$	0	0
$838$	5.74980e59	0.681006
$839$	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
$842$	3.52226e59	0.378376
$843$	0	0
$844$	−8.65564e59	−0.885684
$845$	−2.21434e59	−0.221148
$846$	0	0
$847$	−5.93141e59	−0.564349
$848$	3.91690e59	0.363771
$849$	0	0
$850$	−1.20836e60	−1.06932
$851$	−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	$0.486466\pi$
$854$	1.22443e59	0.0984121
$855$	0	0
$856$	−2.05044e59	−0.157085
$857$	−1.28058e60	−0.957852	−0.478926	−	0.877855i	$-0.658974\pi$
$857$	−1.28058e60	−0.957852	−0.478926	+	0.877855i	$0.658974\pi$
$858$	0	0
$859$	−1.16067e60	−0.827657	−0.413828	−	0.910355i	$-0.635809\pi$
$859$	−1.16067e60	−0.827657	−0.413828	+	0.910355i	$0.635809\pi$
$860$	−1.66311e59	−0.115799
$861$	0	0
$862$	7.88648e59	0.523581
$863$	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
$866$	1.06086e60	0.640531
$867$	0	0
$868$	−4.01883e59	−0.231444
$869$	1.77043e60	0.995803
$870$	0	0
$871$	−1.89725e59	−0.101801
$872$	−1.18447e60	−0.620780
$873$	0	0
$874$	9.62604e59	0.481352
$875$	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	$0.623734\pi$
$878$	1.23517e60	0.562459
$879$	0	0
$880$	−1.84933e59	−0.0803747
$881$	−5.20975e59	−0.221213	−0.110607	−	0.993864i	$-0.535279\pi$
$881$	−5.20975e59	−0.221213	−0.110607	+	0.993864i	$0.535279\pi$
$882$	0	0
$883$	−1.27377e60	−0.516292	−0.258146	−	0.966106i	$-0.583112\pi$
$883$	−1.27377e60	−0.516292	−0.258146	+	0.966106i	$0.583112\pi$
$884$	−3.35270e59	−0.132777
$885$	0	0
$886$	−5.99831e59	−0.226797
$887$	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
$890$	−6.06023e59	−0.208927
$891$	0	0
$892$	−1.45311e60	−0.478431
$893$	−8.94249e59	−0.287742
$894$	0	0
$895$	−2.75343e59	−0.0846256
$896$	−1.66409e59	−0.0499876
$897$	0	0
$898$	3.03321e60	0.870439
$899$	−4.18544e59	−0.117400
$900$	0	0
$901$	8.65846e60	2.32050
$902$	−1.66007e60	−0.434903
$903$	0	0
$904$	−1.56885e60	−0.392761
$905$	−1.07143e60	−0.262222
$906$	0	0
$907$	1.32324e60	0.309519	0.154760	−	0.987952i	$-0.450540\pi$
$907$	1.32324e60	0.309519	0.154760	+	0.987952i	$0.450540\pi$
$908$	1.57731e60	0.360710
$909$	0	0
$910$	6.92872e58	0.0151462
$911$	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
$914$	−3.51595e59	−0.0702499
$915$	0	0
$916$	−4.05635e60	−0.774957
$917$	−1.37298e60	−0.256504
$918$	0	0
$919$	−3.08144e60	−0.550536	−0.275268	−	0.961368i	$-0.588767\pi$
$919$	−3.08144e60	−0.550536	−0.275268	+	0.961368i	$0.588767\pi$
$920$	7.55224e59	0.131955
$921$	0	0
$922$	−7.27779e59	−0.121623
$923$	1.33465e60	0.218139
$924$	0	0
$925$	9.57092e60	1.49640
$926$	−9.06084e59	−0.138562
$927$	0	0
$928$	−1.73307e59	−0.0253562
$929$	3.19320e60	0.456987	0.228494	−	0.973545i	$-0.426620\pi$
$929$	3.19320e60	0.456987	0.228494	+	0.973545i	$0.426620\pi$
$930$	0	0
$931$	−2.06054e60	−0.282171
$932$	−6.95509e60	−0.931700
$933$	0	0
$934$	6.03839e60	0.774124
$935$	−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	$0.520539\pi$
$938$	−2.08164e60	−0.244482
$939$	0	0
$940$	−7.01596e59	−0.0788799
$941$	1.47841e61	1.62633	0.813164	−	0.582035i	$-0.197743\pi$
$941$	1.47841e61	1.62633	0.813164	+	0.582035i	$0.197743\pi$
$942$	0	0
$943$	6.77934e60	0.714000
$944$	−4.18237e60	−0.431019
$945$	0	0
$946$	1.03128e61	1.01768
$947$	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
$950$	−3.07362e60	−0.278174
$951$	0	0
$952$	−3.67853e60	−0.318872
$953$	−6.02037e59	−0.0510762	−0.0255381	−	0.999674i	$-0.508130\pi$
$953$	−6.02037e59	−0.0510762	−0.0255381	+	0.999674i	$0.508130\pi$
$954$	0	0
$955$	3.02373e60	0.245739
$956$	−9.22914e60	−0.734132
$957$	0	0
$958$	−1.11402e61	−0.848986
$959$	−2.37735e60	−0.177342
$960$	0	0
$961$	−4.61807e60	−0.330089
$962$	2.65553e60	0.185807
$963$	0	0
$964$	3.03650e60	0.203608
$965$	5.16316e60	0.338926
$966$	0	0
$967$	1.85730e61	1.16853	0.584266	−	0.811562i	$-0.301383\pi$
$967$	1.85730e61	1.16853	0.584266	+	0.811562i	$0.301383\pi$
$968$	5.72771e60	0.352806
$969$	0	0
$970$	−3.23350e60	−0.190920
$971$	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
$974$	2.66057e60	0.144383
$975$	0	0
$976$	−1.18238e60	−0.0615229
$977$	−2.94713e60	−0.150162	−0.0750812	−	0.997177i	$-0.523922\pi$
$977$	−2.94713e60	−0.150162	−0.0750812	+	0.997177i	$0.523922\pi$
$978$	0	0
$979$	3.75790e61	1.83611
$980$	−1.61662e60	−0.0773526
$981$	0	0
$982$	−1.82465e61	−0.837327
$983$	−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
$986$	−3.83103e60	−0.161748
$987$	0	0
$988$	−8.52801e59	−0.0345406
$989$	−4.21153e61	−1.67077
$990$	0	0
$991$	−9.78682e60	−0.372505	−0.186252	−	0.982502i	$-0.559634\pi$
$991$	−9.78682e60	−0.372505	−0.186252	+	0.982502i	$0.559634\pi$
$992$	3.88081e60	0.144688
$993$	0	0
$994$	1.46436e61	0.523873
$995$	−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	$0.587951\pi$
$998$	1.79865e60	0.0592611
$999$	0	0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2.42.a.a.1.1	✓	1	3.2	odd	2
16.42.a.a.1.1		1	12.11	even	2
18.42.a.b.1.1		1	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 18 = 2 \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 42 \)
Character orbit:	\([\chi]\)	\(=\)	18.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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