LMFDB - Embedded newform 48.22.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [48,22,Mod(1,48)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

N = Newforms(chi, 22, names="a")

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

chi := DirichletCharacter("48.1");

S:= CuspForms(chi, 22);

N := Newforms(S);

Level:	$ N $	$=$	$ 48 = 2^{4} \cdot 3 $
Weight:	$ k $	$=$	$ 22 $
Character orbit:	$[\chi]$	$=$	48.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$134.149125258$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 12)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

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

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-59049.0 q^{3} -1.12681e7 q^{5} -2.81914e8 q^{7} +3.48678e9 q^{9} +O(q^{10})$

$q-59049.0 q^{3} -1.12681e7 q^{5} -2.81914e8 q^{7} +3.48678e9 q^{9} +3.61721e10 q^{11} -4.49099e11 q^{13} +6.65369e11 q^{15} +2.12186e12 q^{17} +4.60941e12 q^{19} +1.66467e13 q^{21} -9.50953e13 q^{23} -3.49867e14 q^{25} -2.05891e14 q^{27} -2.24574e15 q^{29} +3.15569e15 q^{31} -2.13593e15 q^{33} +3.17663e15 q^{35} -1.81785e16 q^{37} +2.65188e16 q^{39} -1.69650e17 q^{41} +1.58969e17 q^{43} -3.92894e16 q^{45} +1.34697e17 q^{47} -4.79070e17 q^{49} -1.25294e17 q^{51} -1.56374e16 q^{53} -4.07590e17 q^{55} -2.72181e17 q^{57} -2.97724e18 q^{59} +3.60386e18 q^{61} -9.82974e17 q^{63} +5.06048e18 q^{65} -2.10662e19 q^{67} +5.61528e18 q^{69} -2.19801e19 q^{71} -1.70544e19 q^{73} +2.06593e19 q^{75} -1.01974e19 q^{77} +1.15020e20 q^{79} +1.21577e19 q^{81} +9.66285e19 q^{83} -2.39093e19 q^{85} +1.32609e20 q^{87} +6.04276e19 q^{89} +1.26607e20 q^{91} -1.86341e20 q^{93} -5.19392e19 q^{95} -4.07820e20 q^{97} +1.26124e20 q^{99} +O(q^{100})$

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
$3$	−59049.0	−0.577350
$3$	−59049.0	−0.577350
$4$	0	0
$5$	−1.12681e7	−0.516018	−0.258009	−	0.966142i	$-0.583067\pi$
$5$	−1.12681e7	−0.516018	−0.258009	+	0.966142i	$0.583067\pi$
$6$	0	0
$7$	−2.81914e8	−0.377214	−0.188607	−	0.982053i	$-0.560397\pi$
$7$	−2.81914e8	−0.377214	−0.188607	+	0.982053i	$0.560397\pi$
$8$	0	0
$9$	3.48678e9	0.333333
$10$	0	0
$11$	3.61721e10	0.420485	0.210242	−	0.977649i	$-0.432575\pi$
$11$	3.61721e10	0.420485	0.210242	+	0.977649i	$0.432575\pi$
$12$	0	0
$13$	−4.49099e11	−0.903517	−0.451759	−	0.892140i	$-0.649203\pi$
$13$	−4.49099e11	−0.903517	−0.451759	+	0.892140i	$0.649203\pi$
$14$	0	0
$15$	6.65369e11	0.297923
$16$	0	0
$17$	2.12186e12	0.255272	0.127636	−	0.991821i	$-0.459261\pi$
$17$	2.12186e12	0.255272	0.127636	+	0.991821i	$0.459261\pi$
$18$	0	0
$19$	4.60941e12	0.172477	0.0862387	−	0.996275i	$-0.472515\pi$
$19$	4.60941e12	0.172477	0.0862387	+	0.996275i	$0.472515\pi$
$20$	0	0
$21$	1.66467e13	0.217784
$22$	0	0
$23$	−9.50953e13	−0.478648	−0.239324	−	0.970940i	$-0.576926\pi$
$23$	−9.50953e13	−0.478648	−0.239324	+	0.970940i	$0.576926\pi$
$24$	0	0
$25$	−3.49867e14	−0.733725
$26$	0	0
$27$	−2.05891e14	−0.192450
$28$	0	0
$29$	−2.24574e15	−0.991245	−0.495622	−	0.868538i	$-0.665060\pi$
$29$	−2.24574e15	−0.991245	−0.495622	+	0.868538i	$0.665060\pi$
$30$	0	0
$31$	3.15569e15	0.691508	0.345754	−	0.938325i	$-0.387623\pi$
$31$	3.15569e15	0.691508	0.345754	+	0.938325i	$0.387623\pi$
$32$	0	0
$33$	−2.13593e15	−0.242767
$34$	0	0
$35$	3.17663e15	0.194649
$36$	0	0
$37$	−1.81785e16	−0.621498	−0.310749	−	0.950492i	$-0.600580\pi$
$37$	−1.81785e16	−0.621498	−0.310749	+	0.950492i	$0.600580\pi$
$38$	0	0
$39$	2.65188e16	0.521646
$40$	0	0
$41$	−1.69650e17	−1.97389	−0.986944	−	0.161061i	$-0.948508\pi$
$41$	−1.69650e17	−1.97389	−0.986944	+	0.161061i	$0.948508\pi$
$42$	0	0
$43$	1.58969e17	1.12174	0.560870	−	0.827904i	$-0.310467\pi$
$43$	1.58969e17	1.12174	0.560870	+	0.827904i	$0.310467\pi$
$44$	0	0
$45$	−3.92894e16	−0.172006
$46$	0	0
$47$	1.34697e17	0.373536	0.186768	−	0.982404i	$-0.440199\pi$
$47$	1.34697e17	0.373536	0.186768	+	0.982404i	$0.440199\pi$
$48$	0	0
$49$	−4.79070e17	−0.857710
$50$	0	0
$51$	−1.25294e17	−0.147381
$52$	0	0
$53$	−1.56374e16	−0.0122819	−0.00614097	−	0.999981i	$-0.501955\pi$
$53$	−1.56374e16	−0.0122819	−0.00614097	+	0.999981i	$0.501955\pi$
$54$	0	0
$55$	−4.07590e17	−0.216978
$56$	0	0
$57$	−2.72181e17	−0.0995799
$58$	0	0
$59$	−2.97724e18	−0.758347	−0.379173	−	0.925326i	$-0.623792\pi$
$59$	−2.97724e18	−0.758347	−0.379173	+	0.925326i	$0.623792\pi$
$60$	0	0
$61$	3.60386e18	0.646851	0.323425	−	0.946254i	$-0.395166\pi$
$61$	3.60386e18	0.646851	0.323425	+	0.946254i	$0.395166\pi$
$62$	0	0
$63$	−9.82974e17	−0.125738
$64$	0	0
$65$	5.06048e18	0.466232
$66$	0	0
$67$	−2.10662e19	−1.41189	−0.705945	−	0.708267i	$-0.749477\pi$
$67$	−2.10662e19	−1.41189	−0.705945	+	0.708267i	$0.749477\pi$
$68$	0	0
$69$	5.61528e18	0.276348
$70$	0	0
$71$	−2.19801e19	−0.801340	−0.400670	−	0.916222i	$-0.631223\pi$
$71$	−2.19801e19	−0.801340	−0.400670	+	0.916222i	$0.631223\pi$
$72$	0	0
$73$	−1.70544e19	−0.464458	−0.232229	−	0.972661i	$-0.574602\pi$
$73$	−1.70544e19	−0.464458	−0.232229	+	0.972661i	$0.574602\pi$
$74$	0	0
$75$	2.06593e19	0.423616
$76$	0	0
$77$	−1.01974e19	−0.158613
$78$	0	0
$79$	1.15020e20	1.36675	0.683375	−	0.730067i	$-0.260511\pi$
$79$	1.15020e20	1.36675	0.683375	+	0.730067i	$0.260511\pi$
$80$	0	0
$81$	1.21577e19	0.111111
$82$	0	0
$83$	9.66285e19	0.683574	0.341787	−	0.939777i	$-0.388968\pi$
$83$	9.66285e19	0.683574	0.341787	+	0.939777i	$0.388968\pi$
$84$	0	0
$85$	−2.39093e19	−0.131725
$86$	0	0
$87$	1.32609e20	0.572295
$88$	0	0
$89$	6.04276e19	0.205419	0.102709	−	0.994711i	$-0.467249\pi$
$89$	6.04276e19	0.205419	0.102709	+	0.994711i	$0.467249\pi$
$90$	0	0
$91$	1.26607e20	0.340819
$92$	0	0
$93$	−1.86341e20	−0.399242
$94$	0	0
$95$	−5.19392e19	−0.0890015
$96$	0	0
$97$	−4.07820e20	−0.561520	−0.280760	−	0.959778i	$-0.590587\pi$
$97$	−4.07820e20	−0.561520	−0.280760	+	0.959778i	$0.590587\pi$
$98$	0	0
$99$	1.26124e20	0.140162
$100$	0	0
$101$	1.95076e21	1.75724	0.878618	−	0.477525i	$-0.158466\pi$
$101$	1.95076e21	1.75724	0.878618	+	0.477525i	$0.158466\pi$
$102$	0	0
$103$	−8.98058e20	−0.658436	−0.329218	−	0.944254i	$-0.606785\pi$
$103$	−8.98058e20	−0.658436	−0.329218	+	0.944254i	$0.606785\pi$
$104$	0	0
$105$	−1.87577e20	−0.112381
$106$	0	0
$107$	−3.22013e21	−1.58250	−0.791250	−	0.611493i	$-0.790569\pi$
$107$	−3.22013e21	−1.58250	−0.791250	+	0.611493i	$0.790569\pi$
$108$	0	0
$109$	−5.55319e20	−0.224680	−0.112340	−	0.993670i	$-0.535835\pi$
$109$	−5.55319e20	−0.224680	−0.112340	+	0.993670i	$0.535835\pi$
$110$	0	0
$111$	1.07342e21	0.358822
$112$	0	0
$113$	4.00790e21	1.11069	0.555346	−	0.831620i	$-0.312586\pi$
$113$	4.00790e21	1.11069	0.555346	+	0.831620i	$0.312586\pi$
$114$	0	0
$115$	1.07154e21	0.246991
$116$	0	0
$117$	−1.56591e21	−0.301172
$118$	0	0
$119$	−5.98182e20	−0.0962920
$120$	0	0
$121$	−6.09183e21	−0.823193
$122$	0	0
$123$	1.00176e22	1.13963
$124$	0	0
$125$	9.31538e21	0.894634
$126$	0	0
$127$	1.78249e22	1.44906	0.724532	−	0.689241i	$-0.242056\pi$
$127$	1.78249e22	1.44906	0.724532	+	0.689241i	$0.242056\pi$
$128$	0	0
$129$	−9.38693e21	−0.647637
$130$	0	0
$131$	−5.43009e21	−0.318756	−0.159378	−	0.987218i	$-0.550949\pi$
$131$	−5.43009e21	−0.318756	−0.159378	+	0.987218i	$0.550949\pi$
$132$	0	0
$133$	−1.29946e21	−0.0650608
$134$	0	0
$135$	2.32000e21	0.0993078
$136$	0	0
$137$	2.64038e22	0.968502	0.484251	−	0.874929i	$-0.339092\pi$
$137$	2.64038e22	0.968502	0.484251	+	0.874929i	$0.339092\pi$
$138$	0	0
$139$	4.41795e22	1.39176	0.695882	−	0.718156i	$-0.255014\pi$
$139$	4.41795e22	1.39176	0.695882	+	0.718156i	$0.255014\pi$
$140$	0	0
$141$	−7.95375e21	−0.215661
$142$	0	0
$143$	−1.62448e22	−0.379915
$144$	0	0
$145$	2.53052e22	0.511501
$146$	0	0
$147$	2.82886e22	0.495199
$148$	0	0
$149$	1.19466e23	1.81464	0.907319	−	0.420444i	$-0.138126\pi$
$149$	1.19466e23	1.81464	0.907319	+	0.420444i	$0.138126\pi$
$150$	0	0
$151$	−5.87257e21	−0.0775479	−0.0387739	−	0.999248i	$-0.512345\pi$
$151$	−5.87257e21	−0.0775479	−0.0387739	+	0.999248i	$0.512345\pi$
$152$	0	0
$153$	7.39846e21	0.0850906
$154$	0	0
$155$	−3.55586e22	−0.356831
$156$	0	0
$157$	1.96848e23	1.72657	0.863286	−	0.504716i	$-0.168403\pi$
$157$	1.96848e23	1.72657	0.863286	+	0.504716i	$0.168403\pi$
$158$	0	0
$159$	9.23371e20	0.00709099
$160$	0	0
$161$	2.68087e22	0.180553
$162$	0	0
$163$	1.93738e23	1.14616	0.573079	−	0.819500i	$-0.305749\pi$
$163$	1.93738e23	1.14616	0.573079	+	0.819500i	$0.305749\pi$
$164$	0	0
$165$	2.40678e22	0.125272
$166$	0	0
$167$	2.08785e23	0.957585	0.478792	−	0.877928i	$-0.341075\pi$
$167$	2.08785e23	0.957585	0.478792	+	0.877928i	$0.341075\pi$
$168$	0	0
$169$	−4.53750e22	−0.183656
$170$	0	0
$171$	1.60720e22	0.0574925
$172$	0	0
$173$	−2.48568e23	−0.786975	−0.393488	−	0.919330i	$-0.628732\pi$
$173$	−2.48568e23	−0.786975	−0.393488	+	0.919330i	$0.628732\pi$
$174$	0	0
$175$	9.86325e22	0.276771
$176$	0	0
$177$	1.75803e23	0.437832
$178$	0	0
$179$	7.15674e23	1.58401	0.792006	−	0.610513i	$-0.209037\pi$
$179$	7.15674e23	1.58401	0.792006	+	0.610513i	$0.209037\pi$
$180$	0	0
$181$	2.74196e22	0.0540052	0.0270026	−	0.999635i	$-0.491404\pi$
$181$	2.74196e22	0.0540052	0.0270026	+	0.999635i	$0.491404\pi$
$182$	0	0
$183$	−2.12804e23	−0.373459
$184$	0	0
$185$	2.04837e23	0.320705
$186$	0	0
$187$	7.67521e22	0.107338
$188$	0	0
$189$	5.80436e22	0.0725948
$190$	0	0
$191$	1.35303e24	1.51516	0.757580	−	0.652742i	$-0.226382\pi$
$191$	1.35303e24	1.51516	0.757580	+	0.652742i	$0.226382\pi$
$192$	0	0
$193$	1.27470e24	1.27955	0.639774	−	0.768563i	$-0.279028\pi$
$193$	1.27470e24	1.27955	0.639774	+	0.768563i	$0.279028\pi$
$194$	0	0
$195$	−2.98816e23	−0.269179
$196$	0	0
$197$	−1.36579e23	−0.110532	−0.0552659	−	0.998472i	$-0.517601\pi$
$197$	−1.36579e23	−0.110532	−0.0552659	+	0.998472i	$0.517601\pi$
$198$	0	0
$199$	−1.10960e24	−0.807625	−0.403813	−	0.914842i	$-0.632315\pi$
$199$	−1.10960e24	−0.807625	−0.403813	+	0.914842i	$0.632315\pi$
$200$	0	0
$201$	1.24394e24	0.815155
$202$	0	0
$203$	6.33107e23	0.373911
$204$	0	0
$205$	1.91163e24	1.01856
$206$	0	0
$207$	−3.31577e23	−0.159549
$208$	0	0
$209$	1.66732e23	0.0725241
$210$	0	0
$211$	4.25779e23	0.167579	0.0837893	−	0.996483i	$-0.473298\pi$
$211$	4.25779e23	0.167579	0.0837893	+	0.996483i	$0.473298\pi$
$212$	0	0
$213$	1.29790e24	0.462654
$214$	0	0
$215$	−1.79127e24	−0.578839
$216$	0	0
$217$	−8.89635e23	−0.260846
$218$	0	0
$219$	1.00705e24	0.268155
$220$	0	0
$221$	−9.52924e23	−0.230642
$222$	0	0
$223$	1.06482e23	0.0234463	0.0117231	−	0.999931i	$-0.496268\pi$
$223$	1.06482e23	0.0234463	0.0117231	+	0.999931i	$0.496268\pi$
$224$	0	0
$225$	−1.21991e24	−0.244575
$226$	0	0
$227$	5.78572e24	1.05703	0.528513	−	0.848925i	$-0.322750\pi$
$227$	5.78572e24	1.05703	0.528513	+	0.848925i	$0.322750\pi$
$228$	0	0
$229$	−7.95470e24	−1.32541	−0.662707	−	0.748879i	$-0.730592\pi$
$229$	−7.95470e24	−1.32541	−0.662707	+	0.748879i	$0.730592\pi$
$230$	0	0
$231$	6.02148e23	0.0915750
$232$	0	0
$233$	−6.84067e24	−0.950302	−0.475151	−	0.879904i	$-0.657607\pi$
$233$	−6.84067e24	−0.950302	−0.475151	+	0.879904i	$0.657607\pi$
$234$	0	0
$235$	−1.51778e24	−0.192751
$236$	0	0
$237$	−6.79182e24	−0.789094
$238$	0	0
$239$	−1.71539e25	−1.82467	−0.912335	−	0.409445i	$-0.865722\pi$
$239$	−1.71539e25	−1.82467	−0.912335	+	0.409445i	$0.865722\pi$
$240$	0	0
$241$	−6.00591e24	−0.585329	−0.292664	−	0.956215i	$-0.594542\pi$
$241$	−6.00591e24	−0.585329	−0.292664	+	0.956215i	$0.594542\pi$
$242$	0	0
$243$	−7.17898e23	−0.0641500
$244$	0	0
$245$	5.39821e24	0.442594
$246$	0	0
$247$	−2.07008e24	−0.155836
$248$	0	0
$249$	−5.70582e24	−0.394661
$250$	0	0
$251$	−1.48883e25	−0.946829	−0.473415	−	0.880840i	$-0.656979\pi$
$251$	−1.48883e25	−0.946829	−0.473415	+	0.880840i	$0.656979\pi$
$252$	0	0
$253$	−3.43979e24	−0.201264
$254$	0	0
$255$	1.41182e24	0.0760514
$256$	0	0
$257$	1.76815e25	0.877445	0.438723	−	0.898622i	$-0.355431\pi$
$257$	1.76815e25	0.877445	0.438723	+	0.898622i	$0.355431\pi$
$258$	0	0
$259$	5.12478e24	0.234438
$260$	0	0
$261$	−7.83042e24	−0.330415
$262$	0	0
$263$	−3.25801e25	−1.26887	−0.634434	−	0.772977i	$-0.718767\pi$
$263$	−3.25801e25	−1.26887	−0.634434	+	0.772977i	$0.718767\pi$
$264$	0	0
$265$	1.76203e23	0.00633771
$266$	0	0
$267$	−3.56819e24	−0.118599
$268$	0	0
$269$	2.09117e25	0.642674	0.321337	−	0.946965i	$-0.395868\pi$
$269$	2.09117e25	0.642674	0.321337	+	0.946965i	$0.395868\pi$
$270$	0	0
$271$	−1.57378e25	−0.447474	−0.223737	−	0.974650i	$-0.571826\pi$
$271$	−1.57378e25	−0.447474	−0.223737	+	0.974650i	$0.571826\pi$
$272$	0	0
$273$	−7.47603e24	−0.196772
$274$	0	0
$275$	−1.26554e25	−0.308520
$276$	0	0
$277$	−1.23463e25	−0.278934	−0.139467	−	0.990227i	$-0.544539\pi$
$277$	−1.23463e25	−0.278934	−0.139467	+	0.990227i	$0.544539\pi$
$278$	0	0
$279$	1.10032e25	0.230503
$280$	0	0
$281$	−5.51786e25	−1.07239	−0.536197	−	0.844093i	$-0.680140\pi$
$281$	−5.51786e25	−1.07239	−0.536197	+	0.844093i	$0.680140\pi$
$282$	0	0
$283$	−1.45585e25	−0.262639	−0.131320	−	0.991340i	$-0.541921\pi$
$283$	−1.45585e25	−0.262639	−0.131320	+	0.991340i	$0.541921\pi$
$284$	0	0
$285$	3.06696e24	0.0513851
$286$	0	0
$287$	4.78267e25	0.744578
$288$	0	0
$289$	−6.45896e25	−0.934836
$290$	0	0
$291$	2.40814e25	0.324194
$292$	0	0
$293$	−2.86675e25	−0.359153	−0.179576	−	0.983744i	$-0.557473\pi$
$293$	−2.86675e25	−0.359153	−0.179576	+	0.983744i	$0.557473\pi$
$294$	0	0
$295$	3.35478e25	0.391321
$296$	0	0
$297$	−7.44751e24	−0.0809223
$298$	0	0
$299$	4.27072e25	0.432467
$300$	0	0
$301$	−4.48155e25	−0.423136
$302$	0	0
$303$	−1.15191e26	−1.01454
$304$	0	0
$305$	−4.06086e25	−0.333787
$306$	0	0
$307$	−8.31398e25	−0.638052	−0.319026	−	0.947746i	$-0.603356\pi$
$307$	−8.31398e25	−0.638052	−0.319026	+	0.947746i	$0.603356\pi$
$308$	0	0
$309$	5.30294e25	0.380148
$310$	0	0
$311$	1.80388e26	1.20843	0.604217	−	0.796820i	$-0.293486\pi$
$311$	1.80388e26	1.20843	0.604217	+	0.796820i	$0.293486\pi$
$312$	0	0
$313$	2.33157e26	1.46027	0.730133	−	0.683305i	$-0.239458\pi$
$313$	2.33157e26	1.46027	0.730133	+	0.683305i	$0.239458\pi$
$314$	0	0
$315$	1.10762e25	0.0648831
$316$	0	0
$317$	8.51585e25	0.466773	0.233387	−	0.972384i	$-0.425019\pi$
$317$	8.51585e25	0.466773	0.233387	+	0.972384i	$0.425019\pi$
$318$	0	0
$319$	−8.12332e25	−0.416803
$320$	0	0
$321$	1.90145e26	0.913657
$322$	0	0
$323$	9.78051e24	0.0440286
$324$	0	0
$325$	1.57125e26	0.662933
$326$	0	0
$327$	3.27910e25	0.129719
$328$	0	0
$329$	−3.79731e25	−0.140903
$330$	0	0
$331$	−5.68513e25	−0.197946	−0.0989730	−	0.995090i	$-0.531556\pi$
$331$	−5.68513e25	−0.197946	−0.0989730	+	0.995090i	$0.531556\pi$
$332$	0	0
$333$	−6.33845e25	−0.207166
$334$	0	0
$335$	2.37376e26	0.728561
$336$	0	0
$337$	2.08109e26	0.600034	0.300017	−	0.953934i	$-0.403008\pi$
$337$	2.08109e26	0.600034	0.300017	+	0.953934i	$0.403008\pi$
$338$	0	0
$339$	−2.36663e26	−0.641258
$340$	0	0
$341$	1.14148e26	0.290768
$342$	0	0
$343$	2.92519e26	0.700754
$344$	0	0
$345$	−6.32735e25	−0.142601
$346$	0	0
$347$	5.50754e26	1.16815	0.584075	−	0.811700i	$-0.301457\pi$
$347$	5.50754e26	1.16815	0.584075	+	0.811700i	$0.301457\pi$
$348$	0	0
$349$	−2.04674e25	−0.0408692	−0.0204346	−	0.999791i	$-0.506505\pi$
$349$	−2.04674e25	−0.0408692	−0.0204346	+	0.999791i	$0.506505\pi$
$350$	0	0
$351$	9.24654e25	0.173882
$352$	0	0
$353$	5.92034e26	1.04885	0.524424	−	0.851458i	$-0.324281\pi$
$353$	5.92034e26	1.04885	0.524424	+	0.851458i	$0.324281\pi$
$354$	0	0
$355$	2.47674e26	0.413506
$356$	0	0
$357$	3.53220e25	0.0555942
$358$	0	0
$359$	5.22516e26	0.775547	0.387773	−	0.921755i	$-0.373244\pi$
$359$	5.22516e26	0.775547	0.387773	+	0.921755i	$0.373244\pi$
$360$	0	0
$361$	−6.92963e26	−0.970252
$362$	0	0
$363$	3.59716e26	0.475270
$364$	0	0
$365$	1.92171e26	0.239669
$366$	0	0
$367$	1.15652e27	1.36195	0.680974	−	0.732308i	$-0.261557\pi$
$367$	1.15652e27	1.36195	0.680974	+	0.732308i	$0.261557\pi$
$368$	0	0
$369$	−5.91532e26	−0.657963
$370$	0	0
$371$	4.40840e24	0.00463292
$372$	0	0
$373$	2.94328e26	0.292340	0.146170	−	0.989259i	$-0.453305\pi$
$373$	2.94328e26	0.292340	0.146170	+	0.989259i	$0.453305\pi$
$374$	0	0
$375$	−5.50064e26	−0.516517
$376$	0	0
$377$	1.00856e27	0.895607
$378$	0	0
$379$	−1.24279e27	−1.04397	−0.521984	−	0.852955i	$-0.674808\pi$
$379$	−1.24279e27	−1.04397	−0.521984	+	0.852955i	$0.674808\pi$
$380$	0	0
$381$	−1.05254e27	−0.836618
$382$	0	0
$383$	−2.18301e27	−1.64236	−0.821181	−	0.570668i	$-0.806684\pi$
$383$	−2.18301e27	−1.64236	−0.821181	+	0.570668i	$0.806684\pi$
$384$	0	0
$385$	1.14905e26	0.0818470
$386$	0	0
$387$	5.54289e26	0.373914
$388$	0	0
$389$	1.52211e27	0.972690	0.486345	−	0.873767i	$-0.338330\pi$
$389$	1.52211e27	0.972690	0.486345	+	0.873767i	$0.338330\pi$
$390$	0	0
$391$	−2.01779e26	−0.122185
$392$	0	0
$393$	3.20642e26	0.184034
$394$	0	0
$395$	−1.29606e27	−0.705269
$396$	0	0
$397$	−6.32566e26	−0.326441	−0.163221	−	0.986590i	$-0.552188\pi$
$397$	−6.32566e26	−0.326441	−0.163221	+	0.986590i	$0.552188\pi$
$398$	0	0
$399$	7.67316e25	0.0375629
$400$	0	0
$401$	−3.76087e26	−0.174692	−0.0873459	−	0.996178i	$-0.527839\pi$
$401$	−3.76087e26	−0.174692	−0.0873459	+	0.996178i	$0.527839\pi$
$402$	0	0
$403$	−1.41722e27	−0.624789
$404$	0	0
$405$	−1.36994e26	−0.0573354
$406$	0	0
$407$	−6.57554e26	−0.261331
$408$	0	0
$409$	−3.84109e27	−1.44997	−0.724986	−	0.688764i	$-0.758154\pi$
$409$	−3.84109e27	−1.44997	−0.724986	+	0.688764i	$0.758154\pi$
$410$	0	0
$411$	−1.55912e27	−0.559165
$412$	0	0
$413$	8.39326e26	0.286059
$414$	0	0
$415$	−1.08882e27	−0.352737
$416$	0	0
$417$	−2.60876e27	−0.803535
$418$	0	0
$419$	2.15090e27	0.630048	0.315024	−	0.949084i	$-0.397987\pi$
$419$	2.15090e27	0.630048	0.315024	+	0.949084i	$0.397987\pi$
$420$	0	0
$421$	−1.87739e27	−0.523109	−0.261554	−	0.965189i	$-0.584235\pi$
$421$	−1.87739e27	−0.523109	−0.261554	+	0.965189i	$0.584235\pi$
$422$	0	0
$423$	4.69661e26	0.124512
$424$	0	0
$425$	−7.42369e26	−0.187299
$426$	0	0
$427$	−1.01598e27	−0.244001
$428$	0	0
$429$	9.59241e26	0.219344
$430$	0	0
$431$	5.68016e27	1.23694	0.618471	−	0.785807i	$-0.287752\pi$
$431$	5.68016e27	1.23694	0.618471	+	0.785807i	$0.287752\pi$
$432$	0	0
$433$	−2.55781e27	−0.530574	−0.265287	−	0.964170i	$-0.585467\pi$
$433$	−2.55781e27	−0.530574	−0.265287	+	0.964170i	$0.585467\pi$
$434$	0	0
$435$	−1.49425e27	−0.295315
$436$	0	0
$437$	−4.38333e26	−0.0825560
$438$	0	0
$439$	−4.04971e27	−0.727020	−0.363510	−	0.931590i	$-0.618422\pi$
$439$	−4.04971e27	−0.727020	−0.363510	+	0.931590i	$0.618422\pi$
$440$	0	0
$441$	−1.67041e27	−0.285903
$442$	0	0
$443$	1.15784e28	1.88978	0.944890	−	0.327388i	$-0.106169\pi$
$443$	1.15784e28	1.88978	0.944890	+	0.327388i	$0.106169\pi$
$444$	0	0
$445$	−6.80903e26	−0.106000
$446$	0	0
$447$	−7.05437e27	−1.04768
$448$	0	0
$449$	−1.09657e27	−0.155399	−0.0776997	−	0.996977i	$-0.524758\pi$
$449$	−1.09657e27	−0.155399	−0.0776997	+	0.996977i	$0.524758\pi$
$450$	0	0
$451$	−6.13658e27	−0.829990
$452$	0	0
$453$	3.46769e26	0.0447723
$454$	0	0
$455$	−1.42662e27	−0.175869
$456$	0	0
$457$	−1.08169e27	−0.127345	−0.0636725	−	0.997971i	$-0.520281\pi$
$457$	−1.08169e27	−0.127345	−0.0636725	+	0.997971i	$0.520281\pi$
$458$	0	0
$459$	−4.36872e26	−0.0491271
$460$	0	0
$461$	1.20305e27	0.129248	0.0646238	−	0.997910i	$-0.479415\pi$
$461$	1.20305e27	0.129248	0.0646238	+	0.997910i	$0.479415\pi$
$462$	0	0
$463$	−9.71985e27	−0.997836	−0.498918	−	0.866649i	$-0.666269\pi$
$463$	−9.71985e27	−0.997836	−0.498918	+	0.866649i	$0.666269\pi$
$464$	0	0
$465$	2.09970e27	0.206016
$466$	0	0
$467$	−1.01684e27	−0.0953729	−0.0476865	−	0.998862i	$-0.515185\pi$
$467$	−1.01684e27	−0.0953729	−0.0476865	+	0.998862i	$0.515185\pi$
$468$	0	0
$469$	5.93886e27	0.532584
$470$	0	0
$471$	−1.16237e28	−0.996836
$472$	0	0
$473$	5.75022e27	0.471675
$474$	0	0
$475$	−1.61268e27	−0.126551
$476$	0	0
$477$	−5.45242e25	−0.00409398
$478$	0	0
$479$	5.80013e27	0.416788	0.208394	−	0.978045i	$-0.433176\pi$
$479$	5.80013e27	0.416788	0.208394	+	0.978045i	$0.433176\pi$
$480$	0	0
$481$	8.16394e27	0.561535
$482$	0	0
$483$	−1.58303e27	−0.104242
$484$	0	0
$485$	4.59535e27	0.289755
$486$	0	0
$487$	7.92907e27	0.478815	0.239408	−	0.970919i	$-0.423047\pi$
$487$	7.92907e27	0.478815	0.239408	+	0.970919i	$0.423047\pi$
$488$	0	0
$489$	−1.14400e28	−0.661735
$490$	0	0
$491$	1.94270e28	1.07659	0.538294	−	0.842757i	$-0.319069\pi$
$491$	1.94270e28	1.07659	0.538294	+	0.842757i	$0.319069\pi$
$492$	0	0
$493$	−4.76515e27	−0.253037
$494$	0	0
$495$	−1.42118e27	−0.0723260
$496$	0	0
$497$	6.19650e27	0.302276
$498$	0	0
$499$	7.12364e27	0.333155	0.166578	−	0.986028i	$-0.446728\pi$
$499$	7.12364e27	0.333155	0.166578	+	0.986028i	$0.446728\pi$
$500$	0	0
$501$	−1.23286e28	−0.552862
$502$	0	0
$503$	3.58499e28	1.54179	0.770893	−	0.636964i	$-0.219810\pi$
$503$	3.58499e28	1.54179	0.770893	+	0.636964i	$0.219810\pi$
$504$	0	0
$505$	−2.19814e28	−0.906766
$506$	0	0
$507$	2.67935e27	0.106034
$508$	0	0
$509$	4.53859e28	1.72339	0.861696	−	0.507425i	$-0.169402\pi$
$509$	4.53859e28	1.72339	0.861696	+	0.507425i	$0.169402\pi$
$510$	0	0
$511$	4.80788e27	0.175200
$512$	0	0
$513$	−9.49036e26	−0.0331933
$514$	0	0
$515$	1.01194e28	0.339765
$516$	0	0
$517$	4.87229e27	0.157066
$518$	0	0
$519$	1.46777e28	0.454360
$520$	0	0
$521$	4.13279e28	1.22870	0.614352	−	0.789032i	$-0.289417\pi$
$521$	4.13279e28	1.22870	0.614352	+	0.789032i	$0.289417\pi$
$522$	0	0
$523$	2.57605e28	0.735675	0.367837	−	0.929890i	$-0.380098\pi$
$523$	2.57605e28	0.735675	0.367837	+	0.929890i	$0.380098\pi$
$524$	0	0
$525$	−5.82415e27	−0.159794
$526$	0	0
$527$	6.69594e27	0.176522
$528$	0	0
$529$	−3.04285e28	−0.770896
$530$	0	0
$531$	−1.03810e28	−0.252782
$532$	0	0
$533$	7.61895e28	1.78344
$534$	0	0
$535$	3.62847e28	0.816599
$536$	0	0
$537$	−4.22598e28	−0.914530
$538$	0	0
$539$	−1.73290e28	−0.360654
$540$	0	0
$541$	5.00618e28	1.00216	0.501078	−	0.865402i	$-0.332937\pi$
$541$	5.00618e28	1.00216	0.501078	+	0.865402i	$0.332937\pi$
$542$	0	0
$543$	−1.61910e27	−0.0311799
$544$	0	0
$545$	6.25739e27	0.115939
$546$	0	0
$547$	−7.69499e28	−1.37196	−0.685979	−	0.727621i	$-0.740626\pi$
$547$	−7.69499e28	−1.37196	−0.685979	+	0.727621i	$0.740626\pi$
$548$	0	0
$549$	1.25659e28	0.215617
$550$	0	0
$551$	−1.03515e28	−0.170967
$552$	0	0
$553$	−3.24258e28	−0.515557
$554$	0	0
$555$	−1.20954e28	−0.185159
$556$	0	0
$557$	9.27070e28	1.36657	0.683287	−	0.730150i	$-0.260550\pi$
$557$	9.27070e28	1.36657	0.683287	+	0.730150i	$0.260550\pi$
$558$	0	0
$559$	−7.13926e28	−1.01351
$560$	0	0
$561$	−4.53213e27	−0.0619716
$562$	0	0
$563$	3.62203e28	0.477104	0.238552	−	0.971130i	$-0.423327\pi$
$563$	3.62203e28	0.477104	0.238552	+	0.971130i	$0.423327\pi$
$564$	0	0
$565$	−4.51614e28	−0.573137
$566$	0	0
$567$	−3.42742e27	−0.0419126
$568$	0	0
$569$	−1.10807e28	−0.130583	−0.0652917	−	0.997866i	$-0.520798\pi$
$569$	−1.10807e28	−0.130583	−0.0652917	+	0.997866i	$0.520798\pi$
$570$	0	0
$571$	−1.89713e28	−0.215485	−0.107743	−	0.994179i	$-0.534362\pi$
$571$	−1.89713e28	−0.215485	−0.107743	+	0.994179i	$0.534362\pi$
$572$	0	0
$573$	−7.98953e28	−0.874778
$574$	0	0
$575$	3.32707e28	0.351196
$576$	0	0
$577$	−1.05096e29	−1.06965	−0.534825	−	0.844963i	$-0.679623\pi$
$577$	−1.05096e29	−1.06965	−0.534825	+	0.844963i	$0.679623\pi$
$578$	0	0
$579$	−7.52698e28	−0.738748
$580$	0	0
$581$	−2.72409e28	−0.257853
$582$	0	0
$583$	−5.65636e26	−0.00516437
$584$	0	0
$585$	1.76448e28	0.155411
$586$	0	0
$587$	1.07068e29	0.909825	0.454912	−	0.890536i	$-0.349671\pi$
$587$	1.07068e29	0.909825	0.454912	+	0.890536i	$0.349671\pi$
$588$	0	0
$589$	1.45459e28	0.119269
$590$	0	0
$591$	8.06483e27	0.0638155
$592$	0	0
$593$	−2.26566e29	−1.73029	−0.865147	−	0.501519i	$-0.832775\pi$
$593$	−2.26566e29	−1.73029	−0.865147	+	0.501519i	$0.832775\pi$
$594$	0	0
$595$	6.74037e27	0.0496885
$596$	0	0
$597$	6.55209e28	0.466283
$598$	0	0
$599$	1.58466e29	1.08882	0.544408	−	0.838821i	$-0.316755\pi$
$599$	1.58466e29	1.08882	0.544408	+	0.838821i	$0.316755\pi$
$600$	0	0
$601$	−1.75299e29	−1.16304	−0.581522	−	0.813530i	$-0.697543\pi$
$601$	−1.75299e29	−1.16304	−0.581522	+	0.813530i	$0.697543\pi$
$602$	0	0
$603$	−7.34533e28	−0.470630
$604$	0	0
$605$	6.86433e28	0.424783
$606$	0	0
$607$	−1.39496e29	−0.833834	−0.416917	−	0.908945i	$-0.636889\pi$
$607$	−1.39496e29	−0.833834	−0.416917	+	0.908945i	$0.636889\pi$
$608$	0	0
$609$	−3.73843e28	−0.215878
$610$	0	0
$611$	−6.04924e28	−0.337496
$612$	0	0
$613$	1.79163e29	0.965858	0.482929	−	0.875660i	$-0.339573\pi$
$613$	1.79163e29	0.965858	0.482929	+	0.875660i	$0.339573\pi$
$614$	0	0
$615$	−1.12880e29	−0.588068
$616$	0	0
$617$	−1.93960e29	−0.976604	−0.488302	−	0.872675i	$-0.662384\pi$
$617$	−1.93960e29	−0.976604	−0.488302	+	0.872675i	$0.662384\pi$
$618$	0	0
$619$	−1.29272e29	−0.629150	−0.314575	−	0.949233i	$-0.601862\pi$
$619$	−1.29272e29	−0.629150	−0.314575	+	0.949233i	$0.601862\pi$
$620$	0	0
$621$	1.95793e28	0.0921159
$622$	0	0
$623$	−1.70354e28	−0.0774868
$624$	0	0
$625$	6.18632e28	0.272077
$626$	0	0
$627$	−9.84535e27	−0.0418718
$628$	0	0
$629$	−3.85722e28	−0.158651
$630$	0	0
$631$	−1.99938e29	−0.795401	−0.397701	−	0.917515i	$-0.630192\pi$
$631$	−1.99938e29	−0.795401	−0.397701	+	0.917515i	$0.630192\pi$
$632$	0	0
$633$	−2.51418e28	−0.0967516
$634$	0	0
$635$	−2.00852e29	−0.747744
$636$	0	0
$637$	2.15150e29	0.774956
$638$	0	0
$639$	−7.66398e28	−0.267113
$640$	0	0
$641$	−1.98808e29	−0.670539	−0.335269	−	0.942122i	$-0.608827\pi$
$641$	−1.98808e29	−0.670539	−0.335269	+	0.942122i	$0.608827\pi$
$642$	0	0
$643$	1.68097e29	0.548712	0.274356	−	0.961628i	$-0.411535\pi$
$643$	1.68097e29	0.548712	0.274356	+	0.961628i	$0.411535\pi$
$644$	0	0
$645$	1.05773e29	0.334193
$646$	0	0
$647$	−8.67978e27	−0.0265469	−0.0132735	−	0.999912i	$-0.504225\pi$
$647$	−8.67978e27	−0.0265469	−0.0132735	+	0.999912i	$0.504225\pi$
$648$	0	0
$649$	−1.07693e29	−0.318873
$650$	0	0
$651$	5.25320e28	0.150600
$652$	0	0
$653$	5.07947e29	1.41003	0.705017	−	0.709190i	$-0.250939\pi$
$653$	5.07947e29	1.41003	0.705017	+	0.709190i	$0.250939\pi$
$654$	0	0
$655$	6.11868e28	0.164484
$656$	0	0
$657$	−5.94651e28	−0.154819
$658$	0	0
$659$	−3.42808e28	−0.0864477	−0.0432239	−	0.999065i	$-0.513763\pi$
$659$	−3.42808e28	−0.0864477	−0.0432239	+	0.999065i	$0.513763\pi$
$660$	0	0
$661$	5.03536e28	0.123003	0.0615015	−	0.998107i	$-0.480411\pi$
$661$	5.03536e28	0.123003	0.0615015	+	0.998107i	$0.480411\pi$
$662$	0	0
$663$	5.62692e28	0.133161
$664$	0	0
$665$	1.46424e28	0.0335726
$666$	0	0
$667$	2.13559e29	0.474458
$668$	0	0
$669$	−6.28764e27	−0.0135367
$670$	0	0
$671$	1.30359e29	0.271991
$672$	0	0
$673$	−4.22880e29	−0.855182	−0.427591	−	0.903972i	$-0.640638\pi$
$673$	−4.22880e29	−0.855182	−0.427591	+	0.903972i	$0.640638\pi$
$674$	0	0
$675$	7.20346e28	0.141205
$676$	0	0
$677$	−2.94132e27	−0.00558936	−0.00279468	−	0.999996i	$-0.500890\pi$
$677$	−2.94132e27	−0.00558936	−0.00279468	+	0.999996i	$0.500890\pi$
$678$	0	0
$679$	1.14970e29	0.211813
$680$	0	0
$681$	−3.41641e29	−0.610274
$682$	0	0
$683$	2.09159e29	0.362293	0.181147	−	0.983456i	$-0.442019\pi$
$683$	2.09159e29	0.362293	0.181147	+	0.983456i	$0.442019\pi$
$684$	0	0
$685$	−2.97520e29	−0.499765
$686$	0	0
$687$	4.69717e29	0.765228
$688$	0	0
$689$	7.02272e27	0.0110970
$690$	0	0
$691$	7.10396e29	1.08888	0.544442	−	0.838799i	$-0.316742\pi$
$691$	7.10396e29	1.08888	0.544442	+	0.838799i	$0.316742\pi$
$692$	0	0
$693$	−3.55562e28	−0.0528709
$694$	0	0
$695$	−4.97819e29	−0.718176
$696$	0	0
$697$	−3.59973e29	−0.503878
$698$	0	0
$699$	4.03935e29	0.548657
$700$	0	0
$701$	8.10616e29	1.06850	0.534252	−	0.845325i	$-0.320593\pi$
$701$	8.10616e29	1.06850	0.534252	+	0.845325i	$0.320593\pi$
$702$	0	0
$703$	−8.37921e28	−0.107194
$704$	0	0
$705$	8.96236e28	0.111285
$706$	0	0
$707$	−5.49948e29	−0.662854
$708$	0	0
$709$	1.31827e30	1.54248	0.771241	−	0.636544i	$-0.219637\pi$
$709$	1.31827e30	1.54248	0.771241	+	0.636544i	$0.219637\pi$
$710$	0	0
$711$	4.01050e29	0.455584
$712$	0	0
$713$	−3.00092e29	−0.330989
$714$	0	0
$715$	1.83048e29	0.196043
$716$	0	0
$717$	1.01292e30	1.05347
$718$	0	0
$719$	−1.04204e30	−1.05252	−0.526262	−	0.850323i	$-0.676407\pi$
$719$	−1.04204e30	−1.05252	−0.526262	+	0.850323i	$0.676407\pi$
$720$	0	0
$721$	2.53175e29	0.248371
$722$	0	0
$723$	3.54643e29	0.337940
$724$	0	0
$725$	7.85712e29	0.727301
$726$	0	0
$727$	1.06635e30	0.958933	0.479467	−	0.877560i	$-0.340830\pi$
$727$	1.06635e30	0.958933	0.479467	+	0.877560i	$0.340830\pi$
$728$	0	0
$729$	4.23912e28	0.0370370
$730$	0	0
$731$	3.37309e29	0.286349
$732$	0	0
$733$	2.08122e30	1.71682	0.858412	−	0.512961i	$-0.171451\pi$
$733$	2.08122e30	1.71682	0.858412	+	0.512961i	$0.171451\pi$
$734$	0	0
$735$	−3.18759e29	−0.255532
$736$	0	0
$737$	−7.62008e29	−0.593678
$738$	0	0
$739$	−1.80814e29	−0.136920	−0.0684598	−	0.997654i	$-0.521808\pi$
$739$	−1.80814e29	−0.136920	−0.0684598	+	0.997654i	$0.521808\pi$
$740$	0	0
$741$	1.22236e29	0.0899721
$742$	0	0
$743$	−4.87745e29	−0.348988	−0.174494	−	0.984658i	$-0.555829\pi$
$743$	−4.87745e29	−0.348988	−0.174494	+	0.984658i	$0.555829\pi$
$744$	0	0
$745$	−1.34616e30	−0.936386
$746$	0	0
$747$	3.36923e29	0.227858
$748$	0	0
$749$	9.07800e29	0.596941
$750$	0	0
$751$	9.22610e29	0.589928	0.294964	−	0.955508i	$-0.404692\pi$
$751$	9.22610e29	0.589928	0.294964	+	0.955508i	$0.404692\pi$
$752$	0	0
$753$	8.79141e29	0.546652
$754$	0	0
$755$	6.61726e28	0.0400161
$756$	0	0
$757$	1.92184e30	1.13035	0.565173	−	0.824972i	$-0.308809\pi$
$757$	1.92184e30	1.13035	0.565173	+	0.824972i	$0.308809\pi$
$758$	0	0
$759$	2.03116e29	0.116200
$760$	0	0
$761$	−3.51722e30	−1.95732	−0.978658	−	0.205496i	$-0.934119\pi$
$761$	−3.51722e30	−1.95732	−0.978658	+	0.205496i	$0.934119\pi$
$762$	0	0
$763$	1.56552e29	0.0847524
$764$	0	0
$765$	−8.33666e28	−0.0439083
$766$	0	0
$767$	1.33707e30	0.685180
$768$	0	0
$769$	1.78079e29	0.0887946	0.0443973	−	0.999014i	$-0.485863\pi$
$769$	1.78079e29	0.0887946	0.0443973	+	0.999014i	$0.485863\pi$
$770$	0	0
$771$	−1.04407e30	−0.506593
$772$	0	0
$773$	−4.21079e30	−1.98829	−0.994143	−	0.108071i	$-0.965533\pi$
$773$	−4.21079e30	−1.98829	−0.994143	+	0.108071i	$0.965533\pi$
$774$	0	0
$775$	−1.10407e30	−0.507376
$776$	0	0
$777$	−3.02613e29	−0.135353
$778$	0	0
$779$	−7.81985e29	−0.340451
$780$	0	0
$781$	−7.95066e29	−0.336951
$782$	0	0
$783$	4.62378e29	0.190765
$784$	0	0
$785$	−2.21810e30	−0.890943
$786$	0	0
$787$	−3.65263e29	−0.142847	−0.0714235	−	0.997446i	$-0.522754\pi$
$787$	−3.65263e29	−0.142847	−0.0714235	+	0.997446i	$0.522754\pi$
$788$	0	0
$789$	1.92382e30	0.732581
$790$	0	0
$791$	−1.12988e30	−0.418968
$792$	0	0
$793$	−1.61849e30	−0.584441
$794$	0	0
$795$	−1.04046e28	−0.00365908
$796$	0	0
$797$	−3.15540e30	−1.08079	−0.540396	−	0.841411i	$-0.681726\pi$
$797$	−3.15540e30	−1.08079	−0.540396	+	0.841411i	$0.681726\pi$
$798$	0	0
$799$	2.85809e29	0.0953531
$800$	0	0
$801$	2.10698e29	0.0684729
$802$	0	0
$803$	−6.16894e29	−0.195298
$804$	0	0
$805$	−3.02083e29	−0.0931686
$806$	0	0
$807$	−1.23482e30	−0.371048
$808$	0	0
$809$	−1.77681e30	−0.520214	−0.260107	−	0.965580i	$-0.583758\pi$
$809$	−1.77681e30	−0.520214	−0.260107	+	0.965580i	$0.583758\pi$
$810$	0	0
$811$	−3.55440e30	−1.01402	−0.507010	−	0.861940i	$-0.669249\pi$
$811$	−3.55440e30	−1.01402	−0.507010	+	0.861940i	$0.669249\pi$
$812$	0	0
$813$	9.29304e29	0.258349
$814$	0	0
$815$	−2.18305e30	−0.591439
$816$	0	0
$817$	7.32751e29	0.193475
$818$	0	0
$819$	4.41452e29	0.113606
$820$	0	0
$821$	2.53502e30	0.635884	0.317942	−	0.948110i	$-0.397008\pi$
$821$	2.53502e30	0.635884	0.317942	+	0.948110i	$0.397008\pi$
$822$	0	0
$823$	1.71174e30	0.418543	0.209272	−	0.977858i	$-0.432891\pi$
$823$	1.71174e30	0.418543	0.209272	+	0.977858i	$0.432891\pi$
$824$	0	0
$825$	7.47290e29	0.178124
$826$	0	0
$827$	−6.61940e30	−1.53819	−0.769097	−	0.639132i	$-0.779294\pi$
$827$	−6.61940e30	−1.53819	−0.769097	+	0.639132i	$0.779294\pi$
$828$	0	0
$829$	6.52629e30	1.47858	0.739288	−	0.673389i	$-0.235162\pi$
$829$	6.52629e30	1.47858	0.739288	+	0.673389i	$0.235162\pi$
$830$	0	0
$831$	7.29039e29	0.161042
$832$	0	0
$833$	−1.01652e30	−0.218949
$834$	0	0
$835$	−2.35261e30	−0.494131
$836$	0	0
$837$	−6.49729e29	−0.133081
$838$	0	0
$839$	−3.94428e30	−0.787894	−0.393947	−	0.919133i	$-0.628891\pi$
$839$	−3.94428e30	−0.787894	−0.393947	+	0.919133i	$0.628891\pi$
$840$	0	0
$841$	−8.94839e28	−0.0174336
$842$	0	0
$843$	3.25824e30	0.619147
$844$	0	0
$845$	5.11290e29	0.0947701
$846$	0	0
$847$	1.71737e30	0.310519
$848$	0	0
$849$	8.59666e29	0.151635
$850$	0	0
$851$	1.72869e30	0.297479
$852$	0	0
$853$	−1.15149e31	−1.93329	−0.966644	−	0.256124i	$-0.917554\pi$
$853$	−1.15149e31	−1.93329	−0.966644	+	0.256124i	$0.917554\pi$
$854$	0	0
$855$	−1.81101e29	−0.0296672
$856$	0	0
$857$	2.63232e29	0.0420766	0.0210383	−	0.999779i	$-0.493303\pi$
$857$	2.63232e29	0.0420766	0.0210383	+	0.999779i	$0.493303\pi$
$858$	0	0
$859$	−2.57894e30	−0.402266	−0.201133	−	0.979564i	$-0.564462\pi$
$859$	−2.57894e30	−0.402266	−0.201133	+	0.979564i	$0.564462\pi$
$860$	0	0
$861$	−2.82412e30	−0.429882
$862$	0	0
$863$	3.33890e30	0.496009	0.248005	−	0.968759i	$-0.420225\pi$
$863$	3.33890e30	0.496009	0.248005	+	0.968759i	$0.420225\pi$
$864$	0	0
$865$	2.80089e30	0.406094
$866$	0	0
$867$	3.81395e30	0.539728
$868$	0	0
$869$	4.16052e30	0.574698
$870$	0	0
$871$	9.46080e30	1.27567
$872$	0	0
$873$	−1.42198e30	−0.187173
$874$	0	0
$875$	−2.62614e30	−0.337468
$876$	0	0
$877$	−1.94355e30	−0.243838	−0.121919	−	0.992540i	$-0.538905\pi$
$877$	−1.94355e30	−0.243838	−0.121919	+	0.992540i	$0.538905\pi$
$878$	0	0
$879$	1.69278e30	0.207357
$880$	0	0
$881$	3.05871e30	0.365840	0.182920	−	0.983128i	$-0.441445\pi$
$881$	3.05871e30	0.365840	0.182920	+	0.983128i	$0.441445\pi$
$882$	0	0
$883$	1.12080e31	1.30901	0.654503	−	0.756060i	$-0.272878\pi$
$883$	1.12080e31	1.30901	0.654503	+	0.756060i	$0.272878\pi$
$884$	0	0
$885$	−1.98097e30	−0.225929
$886$	0	0
$887$	−5.65586e30	−0.629942	−0.314971	−	0.949101i	$-0.601995\pi$
$887$	−5.65586e30	−0.629942	−0.314971	+	0.949101i	$0.601995\pi$
$888$	0	0
$889$	−5.02509e30	−0.546607
$890$	0	0
$891$	4.39768e29	0.0467205
$892$	0	0
$893$	6.20875e29	0.0644264
$894$	0	0
$895$	−8.06428e30	−0.817380
$896$	0	0
$897$	−2.52181e30	−0.249685
$898$	0	0
$899$	−7.08687e30	−0.685453
$900$	0	0
$901$	−3.31803e28	−0.00313523
$902$	0	0
$903$	2.64631e30	0.244298
$904$	0	0
$905$	−3.08966e29	−0.0278677
$906$	0	0
$907$	1.28972e31	1.13663	0.568314	−	0.822812i	$-0.307596\pi$
$907$	1.28972e31	1.13663	0.568314	+	0.822812i	$0.307596\pi$
$908$	0	0
$909$	6.80189e30	0.585745
$910$	0	0
$911$	1.11830e31	0.941058	0.470529	−	0.882384i	$-0.344063\pi$
$911$	1.11830e31	0.941058	0.470529	+	0.882384i	$0.344063\pi$
$912$	0	0
$913$	3.49525e30	0.287432
$914$	0	0
$915$	2.39790e30	0.192712
$916$	0	0
$917$	1.53082e30	0.120239
$918$	0	0
$919$	−1.99044e31	−1.52804	−0.764022	−	0.645190i	$-0.776778\pi$
$919$	−1.99044e31	−1.52804	−0.764022	+	0.645190i	$0.776778\pi$
$920$	0	0
$921$	4.90932e30	0.368379
$922$	0	0
$923$	9.87123e30	0.724025
$924$	0	0
$925$	6.36006e30	0.456009
$926$	0	0
$927$	−3.13134e30	−0.219479
$928$	0	0
$929$	2.46924e31	1.69199	0.845995	−	0.533190i	$-0.179007\pi$
$929$	2.46924e31	1.69199	0.845995	+	0.533190i	$0.179007\pi$
$930$	0	0
$931$	−2.20823e30	−0.147936
$932$	0	0
$933$	−1.06517e31	−0.697690
$934$	0	0
$935$	−8.64849e29	−0.0553883
$936$	0	0
$937$	2.80473e31	1.75641	0.878204	−	0.478286i	$-0.158742\pi$
$937$	2.80473e31	1.75641	0.878204	+	0.478286i	$0.158742\pi$
$938$	0	0
$939$	−1.37677e31	−0.843086
$940$	0	0
$941$	8.99005e30	0.538358	0.269179	−	0.963090i	$-0.413248\pi$
$941$	8.99005e30	0.538358	0.269179	+	0.963090i	$0.413248\pi$
$942$	0	0
$943$	1.61329e31	0.944799
$944$	0	0
$945$	−6.54041e29	−0.0374603
$946$	0	0
$947$	2.54170e31	1.42380	0.711901	−	0.702280i	$-0.247834\pi$
$947$	2.54170e31	1.42380	0.711901	+	0.702280i	$0.247834\pi$
$948$	0	0
$949$	7.65911e30	0.419646
$950$	0	0
$951$	−5.02852e30	−0.269492
$952$	0	0
$953$	−9.08338e30	−0.476181	−0.238090	−	0.971243i	$-0.576521\pi$
$953$	−9.08338e30	−0.476181	−0.238090	+	0.971243i	$0.576521\pi$
$954$	0	0
$955$	−1.52461e31	−0.781851
$956$	0	0
$957$	4.79674e30	0.240642
$958$	0	0
$959$	−7.44360e30	−0.365332
$960$	0	0
$961$	−1.08671e31	−0.521817
$962$	0	0
$963$	−1.12279e31	−0.527500
$964$	0	0
$965$	−1.43635e31	−0.660271
$966$	0	0
$967$	1.32705e31	0.596909	0.298455	−	0.954424i	$-0.403529\pi$
$967$	1.32705e31	0.596909	0.298455	+	0.954424i	$0.403529\pi$
$968$	0	0
$969$	−5.77529e29	−0.0254199
$970$	0	0
$971$	3.95025e31	1.70146	0.850732	−	0.525600i	$-0.176159\pi$
$971$	3.95025e31	1.70146	0.850732	+	0.525600i	$0.176159\pi$
$972$	0	0
$973$	−1.24548e31	−0.524992
$974$	0	0
$975$	−9.27807e30	−0.382745
$976$	0	0
$977$	1.04812e31	0.423172	0.211586	−	0.977359i	$-0.432137\pi$
$977$	1.04812e31	0.423172	0.211586	+	0.977359i	$0.432137\pi$
$978$	0	0
$979$	2.18579e30	0.0863755
$980$	0	0
$981$	−1.93628e30	−0.0748933
$982$	0	0
$983$	4.33201e31	1.64013	0.820063	−	0.572273i	$-0.193938\pi$
$983$	4.33201e31	1.64013	0.820063	+	0.572273i	$0.193938\pi$
$984$	0	0
$985$	1.53898e30	0.0570364
$986$	0	0
$987$	2.24227e30	0.0813502
$988$	0	0
$989$	−1.51172e31	−0.536919
$990$	0	0
$991$	−2.71476e31	−0.943968	−0.471984	−	0.881607i	$-0.656462\pi$
$991$	−2.71476e31	−0.943968	−0.471984	+	0.881607i	$0.656462\pi$
$992$	0	0
$993$	3.35701e30	0.114284
$994$	0	0
$995$	1.25031e31	0.416750
$996$	0	0
$997$	−5.18699e31	−1.69284	−0.846421	−	0.532515i	$-0.821247\pi$
$997$	−5.18699e31	−1.69284	−0.846421	+	0.532515i	$0.821247\pi$
$998$	0	0
$999$	3.74279e30	0.119607

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	48.22.a.b.1.1		1
4.3	odd	2		12.22.a.a.1.1	✓	1
12.11	even	2		36.22.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
12.22.a.a.1.1	✓	1	4.3	odd	2
36.22.a.a.1.1		1	12.11	even	2
48.22.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:	\( N \)	\(=\)	\( 48 = 2^{4} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 22 \)
Character orbit:	\([\chi]\)	\(=\)	48.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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