LMFDB - Embedded newform 48.12.a.d.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [48,12,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, 12, names="a")

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

chi := DirichletCharacter("48.1");

S:= CuspForms(chi, 12);

N := Newforms(S);

Level:	$ N $	$=$	$ 48 = 2^{4} \cdot 3 $
Weight:	$ k $	$=$	$ 12 $
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:	$36.8804726669$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 6)
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)

$f(q)$

$=$

$q-243.000 q^{3} +3630.00 q^{5} -32936.0 q^{7} +59049.0 q^{9} +O(q^{10})$

$q-243.000 q^{3} +3630.00 q^{5} -32936.0 q^{7} +59049.0 q^{9} +758748. q^{11} -2.48286e6 q^{13} -882090. q^{15} +8.29039e6 q^{17} +1.08673e7 q^{19} +8.00345e6 q^{21} -2.05393e7 q^{23} -3.56512e7 q^{25} -1.43489e7 q^{27} +2.88146e7 q^{29} -1.50501e8 q^{31} -1.84376e8 q^{33} -1.19558e8 q^{35} -3.19892e8 q^{37} +6.03334e8 q^{39} -3.68009e8 q^{41} -6.20470e8 q^{43} +2.14348e8 q^{45} -2.76311e9 q^{47} -8.92547e8 q^{49} -2.01456e9 q^{51} -2.68284e8 q^{53} +2.75426e9 q^{55} -2.64075e9 q^{57} -1.67289e9 q^{59} -7.78720e9 q^{61} -1.94484e9 q^{63} -9.01277e9 q^{65} -1.87067e10 q^{67} +4.99104e9 q^{69} +8.34699e9 q^{71} +1.96417e10 q^{73} +8.66325e9 q^{75} -2.49901e10 q^{77} +5.87381e9 q^{79} +3.48678e9 q^{81} -8.49256e9 q^{83} +3.00941e10 q^{85} -7.00194e9 q^{87} +7.55279e10 q^{89} +8.17754e10 q^{91} +3.65718e10 q^{93} +3.94483e10 q^{95} -8.23568e10 q^{97} +4.48033e10 q^{99} +O(q^{100})$

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
$3$	−243.000	−0.577350
$3$	−243.000	−0.577350
$4$	0	0
$5$	3630.00	0.519483	0.259742	−	0.965678i	$-0.416363\pi$
$5$	3630.00	0.519483	0.259742	+	0.965678i	$0.416363\pi$
$6$	0	0
$7$	−32936.0	−0.740682	−0.370341	−	0.928896i	$-0.620759\pi$
$7$	−32936.0	−0.740682	−0.370341	+	0.928896i	$0.620759\pi$
$8$	0	0
$9$	59049.0	0.333333
$10$	0	0
$11$	758748.	1.42049	0.710244	−	0.703955i	$-0.248584\pi$
$11$	758748.	1.42049	0.710244	+	0.703955i	$0.248584\pi$
$12$	0	0
$13$	−2.48286e6	−1.85466	−0.927328	−	0.374249i	$-0.877900\pi$
$13$	−2.48286e6	−1.85466	−0.927328	+	0.374249i	$0.877900\pi$
$14$	0	0
$15$	−882090.	−0.299924
$16$	0	0
$17$	8.29039e6	1.41614	0.708069	−	0.706143i	$-0.249566\pi$
$17$	8.29039e6	1.41614	0.708069	+	0.706143i	$0.249566\pi$
$18$	0	0
$19$	1.08673e7	1.00688	0.503439	−	0.864031i	$-0.332068\pi$
$19$	1.08673e7	1.00688	0.503439	+	0.864031i	$0.332068\pi$
$20$	0	0
$21$	8.00345e6	0.427633
$22$	0	0
$23$	−2.05393e7	−0.665399	−0.332699	−	0.943033i	$-0.607959\pi$
$23$	−2.05393e7	−0.665399	−0.332699	+	0.943033i	$0.607959\pi$
$24$	0	0
$25$	−3.56512e7	−0.730137
$26$	0	0
$27$	−1.43489e7	−0.192450
$28$	0	0
$29$	2.88146e7	0.260869	0.130435	−	0.991457i	$-0.458363\pi$
$29$	2.88146e7	0.260869	0.130435	+	0.991457i	$0.458363\pi$
$30$	0	0
$31$	−1.50501e8	−0.944172	−0.472086	−	0.881552i	$-0.656499\pi$
$31$	−1.50501e8	−0.944172	−0.472086	+	0.881552i	$0.656499\pi$
$32$	0	0
$33$	−1.84376e8	−0.820120
$34$	0	0
$35$	−1.19558e8	−0.384772
$36$	0	0
$37$	−3.19892e8	−0.758392	−0.379196	−	0.925316i	$-0.623799\pi$
$37$	−3.19892e8	−0.758392	−0.379196	+	0.925316i	$0.623799\pi$
$38$	0	0
$39$	6.03334e8	1.07079
$40$	0	0
$41$	−3.68009e8	−0.496075	−0.248037	−	0.968750i	$-0.579786\pi$
$41$	−3.68009e8	−0.496075	−0.248037	+	0.968750i	$0.579786\pi$
$42$	0	0
$43$	−6.20470e8	−0.643641	−0.321821	−	0.946801i	$-0.604295\pi$
$43$	−6.20470e8	−0.643641	−0.321821	+	0.946801i	$0.604295\pi$
$44$	0	0
$45$	2.14348e8	0.173161
$46$	0	0
$47$	−2.76311e9	−1.75736	−0.878679	−	0.477414i	$-0.841574\pi$
$47$	−2.76311e9	−1.75736	−0.878679	+	0.477414i	$0.841574\pi$
$48$	0	0
$49$	−8.92547e8	−0.451391
$50$	0	0
$51$	−2.01456e9	−0.817608
$52$	0	0
$53$	−2.68284e8	−0.0881207	−0.0440603	−	0.999029i	$-0.514029\pi$
$53$	−2.68284e8	−0.0881207	−0.0440603	+	0.999029i	$0.514029\pi$
$54$	0	0
$55$	2.75426e9	0.737920
$56$	0	0
$57$	−2.64075e9	−0.581321
$58$	0	0
$59$	−1.67289e9	−0.304637	−0.152318	−	0.988331i	$-0.548674\pi$
$59$	−1.67289e9	−0.304637	−0.152318	+	0.988331i	$0.548674\pi$
$60$	0	0
$61$	−7.78720e9	−1.18050	−0.590252	−	0.807219i	$-0.700971\pi$
$61$	−7.78720e9	−1.18050	−0.590252	+	0.807219i	$0.700971\pi$
$62$	0	0
$63$	−1.94484e9	−0.246894
$64$	0	0
$65$	−9.01277e9	−0.963463
$66$	0	0
$67$	−1.87067e10	−1.69272	−0.846361	−	0.532610i	$-0.821211\pi$
$67$	−1.87067e10	−1.69272	−0.846361	+	0.532610i	$0.821211\pi$
$68$	0	0
$69$	4.99104e9	0.384168
$70$	0	0
$71$	8.34699e9	0.549046	0.274523	−	0.961580i	$-0.411480\pi$
$71$	8.34699e9	0.549046	0.274523	+	0.961580i	$0.411480\pi$
$72$	0	0
$73$	1.96417e10	1.10893	0.554465	−	0.832207i	$-0.312923\pi$
$73$	1.96417e10	1.10893	0.554465	+	0.832207i	$0.312923\pi$
$74$	0	0
$75$	8.66325e9	0.421545
$76$	0	0
$77$	−2.49901e10	−1.05213
$78$	0	0
$79$	5.87381e9	0.214769	0.107384	−	0.994218i	$-0.465752\pi$
$79$	5.87381e9	0.214769	0.107384	+	0.994218i	$0.465752\pi$
$80$	0	0
$81$	3.48678e9	0.111111
$82$	0	0
$83$	−8.49256e9	−0.236651	−0.118326	−	0.992975i	$-0.537753\pi$
$83$	−8.49256e9	−0.236651	−0.118326	+	0.992975i	$0.537753\pi$
$84$	0	0
$85$	3.00941e10	0.735660
$86$	0	0
$87$	−7.00194e9	−0.150613
$88$	0	0
$89$	7.55279e10	1.43371	0.716856	−	0.697221i	$-0.245580\pi$
$89$	7.55279e10	1.43371	0.716856	+	0.697221i	$0.245580\pi$
$90$	0	0
$91$	8.17754e10	1.37371
$92$	0	0
$93$	3.65718e10	0.545118
$94$	0	0
$95$	3.94483e10	0.523056
$96$	0	0
$97$	−8.23568e10	−0.973767	−0.486883	−	0.873467i	$-0.661866\pi$
$97$	−8.23568e10	−0.973767	−0.486883	+	0.873467i	$0.661866\pi$
$98$	0	0
$99$	4.48033e10	0.473496
$100$	0	0
$101$	−4.13141e10	−0.391138	−0.195569	−	0.980690i	$-0.562655\pi$
$101$	−4.13141e10	−0.391138	−0.195569	+	0.980690i	$0.562655\pi$
$102$	0	0
$103$	6.45183e10	0.548376	0.274188	−	0.961676i	$-0.411591\pi$
$103$	6.45183e10	0.548376	0.274188	+	0.961676i	$0.411591\pi$
$104$	0	0
$105$	2.90525e10	0.222148
$106$	0	0
$107$	−1.10219e11	−0.759706	−0.379853	−	0.925047i	$-0.624025\pi$
$107$	−1.10219e11	−0.759706	−0.379853	+	0.925047i	$0.624025\pi$
$108$	0	0
$109$	1.77778e11	1.10671	0.553353	−	0.832947i	$-0.313348\pi$
$109$	1.77778e11	1.10671	0.553353	+	0.832947i	$0.313348\pi$
$110$	0	0
$111$	7.77337e10	0.437858
$112$	0	0
$113$	7.89206e10	0.402957	0.201479	−	0.979493i	$-0.435425\pi$
$113$	7.89206e10	0.402957	0.201479	+	0.979493i	$0.435425\pi$
$114$	0	0
$115$	−7.45576e10	−0.345664
$116$	0	0
$117$	−1.46610e11	−0.618219
$118$	0	0
$119$	−2.73052e11	−1.04891
$120$	0	0
$121$	2.90387e11	1.01779
$122$	0	0
$123$	8.94262e10	0.286409
$124$	0	0
$125$	−3.06660e11	−0.898777
$126$	0	0
$127$	2.76649e11	0.743035	0.371518	−	0.928426i	$-0.378838\pi$
$127$	2.76649e11	0.743035	0.371518	+	0.928426i	$0.378838\pi$
$128$	0	0
$129$	1.50774e11	0.371607
$130$	0	0
$131$	3.40872e11	0.771968	0.385984	−	0.922505i	$-0.373862\pi$
$131$	3.40872e11	0.771968	0.385984	+	0.922505i	$0.373862\pi$
$132$	0	0
$133$	−3.57925e11	−0.745776
$134$	0	0
$135$	−5.20865e10	−0.0999746
$136$	0	0
$137$	−9.44784e11	−1.67251	−0.836256	−	0.548339i	$-0.815260\pi$
$137$	−9.44784e11	−1.67251	−0.836256	+	0.548339i	$0.815260\pi$
$138$	0	0
$139$	−1.03770e11	−0.169626	−0.0848128	−	0.996397i	$-0.527029\pi$
$139$	−1.03770e11	−0.169626	−0.0848128	+	0.996397i	$0.527029\pi$
$140$	0	0
$141$	6.71436e11	1.01461
$142$	0	0
$143$	−1.88386e12	−2.63452
$144$	0	0
$145$	1.04597e11	0.135517
$146$	0	0
$147$	2.16889e11	0.260610
$148$	0	0
$149$	−1.22852e12	−1.37043	−0.685216	−	0.728340i	$-0.740292\pi$
$149$	−1.22852e12	−1.37043	−0.685216	+	0.728340i	$0.740292\pi$
$150$	0	0
$151$	−2.12466e11	−0.220250	−0.110125	−	0.993918i	$-0.535125\pi$
$151$	−2.12466e11	−0.220250	−0.110125	+	0.993918i	$0.535125\pi$
$152$	0	0
$153$	4.89539e11	0.472046
$154$	0	0
$155$	−5.46320e11	−0.490482
$156$	0	0
$157$	−1.58632e12	−1.32722	−0.663612	−	0.748077i	$-0.730977\pi$
$157$	−1.58632e12	−1.32722	−0.663612	+	0.748077i	$0.730977\pi$
$158$	0	0
$159$	6.51931e10	0.0508765
$160$	0	0
$161$	6.76481e11	0.492849
$162$	0	0
$163$	−8.65020e11	−0.588836	−0.294418	−	0.955677i	$-0.595126\pi$
$163$	−8.65020e11	−0.588836	−0.294418	+	0.955677i	$0.595126\pi$
$164$	0	0
$165$	−6.69284e11	−0.426038
$166$	0	0
$167$	2.25802e12	1.34520	0.672600	−	0.740006i	$-0.265177\pi$
$167$	2.25802e12	1.34520	0.672600	+	0.740006i	$0.265177\pi$
$168$	0	0
$169$	4.37242e12	2.43975
$170$	0	0
$171$	6.41703e11	0.335626
$172$	0	0
$173$	−2.15960e12	−1.05955	−0.529773	−	0.848139i	$-0.677723\pi$
$173$	−2.15960e12	−1.05955	−0.529773	+	0.848139i	$0.677723\pi$
$174$	0	0
$175$	1.17421e12	0.540799
$176$	0	0
$177$	4.06513e11	0.175882
$178$	0	0
$179$	−4.50786e12	−1.83349	−0.916745	−	0.399472i	$-0.869193\pi$
$179$	−4.50786e12	−1.83349	−0.916745	+	0.399472i	$0.869193\pi$
$180$	0	0
$181$	4.34053e11	0.166077	0.0830387	−	0.996546i	$-0.473537\pi$
$181$	4.34053e11	0.166077	0.0830387	+	0.996546i	$0.473537\pi$
$182$	0	0
$183$	1.89229e12	0.681564
$184$	0	0
$185$	−1.16121e12	−0.393972
$186$	0	0
$187$	6.29031e12	2.01161
$188$	0	0
$189$	4.72596e11	0.142544
$190$	0	0
$191$	−8.67702e11	−0.246994	−0.123497	−	0.992345i	$-0.539411\pi$
$191$	−8.67702e11	−0.246994	−0.123497	+	0.992345i	$0.539411\pi$
$192$	0	0
$193$	1.86946e11	0.0502516	0.0251258	−	0.999684i	$-0.492001\pi$
$193$	1.86946e11	0.0502516	0.0251258	+	0.999684i	$0.492001\pi$
$194$	0	0
$195$	2.19010e12	0.556256
$196$	0	0
$197$	3.32798e12	0.799129	0.399565	−	0.916705i	$-0.369161\pi$
$197$	3.32798e12	0.799129	0.399565	+	0.916705i	$0.369161\pi$
$198$	0	0
$199$	1.58581e12	0.360212	0.180106	−	0.983647i	$-0.442356\pi$
$199$	1.58581e12	0.360212	0.180106	+	0.983647i	$0.442356\pi$
$200$	0	0
$201$	4.54573e12	0.977293
$202$	0	0
$203$	−9.49036e11	−0.193221
$204$	0	0
$205$	−1.33587e12	−0.257703
$206$	0	0
$207$	−1.21282e12	−0.221800
$208$	0	0
$209$	8.24554e12	1.43026
$210$	0	0
$211$	−3.54958e12	−0.584283	−0.292142	−	0.956375i	$-0.594368\pi$
$211$	−3.54958e12	−0.584283	−0.292142	+	0.956375i	$0.594368\pi$
$212$	0	0
$213$	−2.02832e12	−0.316992
$214$	0	0
$215$	−2.25230e12	−0.334361
$216$	0	0
$217$	4.95691e12	0.699331
$218$	0	0
$219$	−4.77294e12	−0.640241
$220$	0	0
$221$	−2.05839e13	−2.62645
$222$	0	0
$223$	5.18022e12	0.629030	0.314515	−	0.949252i	$-0.398158\pi$
$223$	5.18022e12	0.629030	0.314515	+	0.949252i	$0.398158\pi$
$224$	0	0
$225$	−2.10517e12	−0.243379
$226$	0	0
$227$	1.33665e13	1.47189	0.735943	−	0.677044i	$-0.236739\pi$
$227$	1.33665e13	1.47189	0.735943	+	0.677044i	$0.236739\pi$
$228$	0	0
$229$	−2.03080e12	−0.213095	−0.106547	−	0.994308i	$-0.533980\pi$
$229$	−2.03080e12	−0.213095	−0.106547	+	0.994308i	$0.533980\pi$
$230$	0	0
$231$	6.07260e12	0.607448
$232$	0	0
$233$	7.13322e12	0.680500	0.340250	−	0.940335i	$-0.389488\pi$
$233$	7.13322e12	0.680500	0.340250	+	0.940335i	$0.389488\pi$
$234$	0	0
$235$	−1.00301e13	−0.912918
$236$	0	0
$237$	−1.42734e12	−0.123997
$238$	0	0
$239$	−3.30363e12	−0.274033	−0.137016	−	0.990569i	$-0.543751\pi$
$239$	−3.30363e12	−0.274033	−0.137016	+	0.990569i	$0.543751\pi$
$240$	0	0
$241$	4.16938e11	0.0330353	0.0165176	−	0.999864i	$-0.494742\pi$
$241$	4.16938e11	0.0330353	0.0165176	+	0.999864i	$0.494742\pi$
$242$	0	0
$243$	−8.47289e11	−0.0641500
$244$	0	0
$245$	−3.23994e12	−0.234490
$246$	0	0
$247$	−2.69820e13	−1.86741
$248$	0	0
$249$	2.06369e12	0.136631
$250$	0	0
$251$	1.91970e12	0.121626	0.0608131	−	0.998149i	$-0.480631\pi$
$251$	1.91970e12	0.121626	0.0608131	+	0.998149i	$0.480631\pi$
$252$	0	0
$253$	−1.55841e13	−0.945191
$254$	0	0
$255$	−7.31287e12	−0.424734
$256$	0	0
$257$	9.76031e12	0.543040	0.271520	−	0.962433i	$-0.412474\pi$
$257$	9.76031e12	0.543040	0.271520	+	0.962433i	$0.412474\pi$
$258$	0	0
$259$	1.05360e13	0.561727
$260$	0	0
$261$	1.70147e12	0.0869564
$262$	0	0
$263$	4.04845e12	0.198396	0.0991980	−	0.995068i	$-0.468372\pi$
$263$	4.04845e12	0.198396	0.0991980	+	0.995068i	$0.468372\pi$
$264$	0	0
$265$	−9.73872e11	−0.0457772
$266$	0	0
$267$	−1.83533e13	−0.827754
$268$	0	0
$269$	−1.13455e13	−0.491118	−0.245559	−	0.969382i	$-0.578972\pi$
$269$	−1.13455e13	−0.491118	−0.245559	+	0.969382i	$0.578972\pi$
$270$	0	0
$271$	2.35252e12	0.0977691	0.0488846	−	0.998804i	$-0.484433\pi$
$271$	2.35252e12	0.0977691	0.0488846	+	0.998804i	$0.484433\pi$
$272$	0	0
$273$	−1.98714e13	−0.793112
$274$	0	0
$275$	−2.70503e13	−1.03715
$276$	0	0
$277$	−4.36327e13	−1.60758	−0.803791	−	0.594912i	$-0.797187\pi$
$277$	−4.36327e13	−1.60758	−0.803791	+	0.594912i	$0.797187\pi$
$278$	0	0
$279$	−8.88696e12	−0.314724
$280$	0	0
$281$	2.75099e13	0.936707	0.468353	−	0.883541i	$-0.344847\pi$
$281$	2.75099e13	0.936707	0.468353	+	0.883541i	$0.344847\pi$
$282$	0	0
$283$	2.57815e13	0.844273	0.422136	−	0.906532i	$-0.361280\pi$
$283$	2.57815e13	0.844273	0.422136	+	0.906532i	$0.361280\pi$
$284$	0	0
$285$	−9.58594e12	−0.301987
$286$	0	0
$287$	1.21207e13	0.367434
$288$	0	0
$289$	3.44586e13	1.00545
$290$	0	0
$291$	2.00127e13	0.562204
$292$	0	0
$293$	5.59324e13	1.51318	0.756591	−	0.653888i	$-0.226863\pi$
$293$	5.59324e13	1.51318	0.756591	+	0.653888i	$0.226863\pi$
$294$	0	0
$295$	−6.07261e12	−0.158254
$296$	0	0
$297$	−1.08872e13	−0.273373
$298$	0	0
$299$	5.09961e13	1.23409
$300$	0	0
$301$	2.04358e13	0.476733
$302$	0	0
$303$	1.00393e13	0.225824
$304$	0	0
$305$	−2.82675e13	−0.613252
$306$	0	0
$307$	−5.68056e13	−1.18886	−0.594429	−	0.804148i	$-0.702622\pi$
$307$	−5.68056e13	−1.18886	−0.594429	+	0.804148i	$0.702622\pi$
$308$	0	0
$309$	−1.56779e13	−0.316605
$310$	0	0
$311$	6.71193e13	1.30817	0.654087	−	0.756419i	$-0.273053\pi$
$311$	6.71193e13	1.30817	0.654087	+	0.756419i	$0.273053\pi$
$312$	0	0
$313$	−3.66685e13	−0.689920	−0.344960	−	0.938617i	$-0.612108\pi$
$313$	−3.66685e13	−0.689920	−0.344960	+	0.938617i	$0.612108\pi$
$314$	0	0
$315$	−7.05976e12	−0.128257
$316$	0	0
$317$	1.01210e14	1.77581	0.887905	−	0.460027i	$-0.152160\pi$
$317$	1.01210e14	1.77581	0.887905	+	0.460027i	$0.152160\pi$
$318$	0	0
$319$	2.18630e13	0.370562
$320$	0	0
$321$	2.67832e13	0.438616
$322$	0	0
$323$	9.00941e13	1.42588
$324$	0	0
$325$	8.85169e13	1.35415
$326$	0	0
$327$	−4.32001e13	−0.638957
$328$	0	0
$329$	9.10058e13	1.30164
$330$	0	0
$331$	−4.96614e13	−0.687013	−0.343506	−	0.939150i	$-0.611615\pi$
$331$	−4.96614e13	−0.687013	−0.343506	+	0.939150i	$0.611615\pi$
$332$	0	0
$333$	−1.88893e13	−0.252797
$334$	0	0
$335$	−6.79053e13	−0.879341
$336$	0	0
$337$	−8.40508e13	−1.05336	−0.526681	−	0.850063i	$-0.676564\pi$
$337$	−8.40508e13	−1.05336	−0.526681	+	0.850063i	$0.676564\pi$
$338$	0	0
$339$	−1.91777e13	−0.232647
$340$	0	0
$341$	−1.14193e14	−1.34119
$342$	0	0
$343$	9.45221e13	1.07502
$344$	0	0
$345$	1.81175e13	0.199569
$346$	0	0
$347$	−1.74097e13	−0.185772	−0.0928858	−	0.995677i	$-0.529609\pi$
$347$	−1.74097e13	−0.185772	−0.0928858	+	0.995677i	$0.529609\pi$
$348$	0	0
$349$	−6.60920e13	−0.683296	−0.341648	−	0.939828i	$-0.610985\pi$
$349$	−6.60920e13	−0.683296	−0.341648	+	0.939828i	$0.610985\pi$
$350$	0	0
$351$	3.56263e13	0.356929
$352$	0	0
$353$	−1.74556e14	−1.69502	−0.847508	−	0.530783i	$-0.821898\pi$
$353$	−1.74556e14	−1.69502	−0.847508	+	0.530783i	$0.821898\pi$
$354$	0	0
$355$	3.02996e13	0.285220
$356$	0	0
$357$	6.63517e13	0.605587
$358$	0	0
$359$	−3.60488e13	−0.319059	−0.159530	−	0.987193i	$-0.550998\pi$
$359$	−3.60488e13	−0.319059	−0.159530	+	0.987193i	$0.550998\pi$
$360$	0	0
$361$	1.60795e12	0.0138033
$362$	0	0
$363$	−7.05640e13	−0.587620
$364$	0	0
$365$	7.12995e13	0.576071
$366$	0	0
$367$	5.04151e13	0.395273	0.197636	−	0.980275i	$-0.436673\pi$
$367$	5.04151e13	0.395273	0.197636	+	0.980275i	$0.436673\pi$
$368$	0	0
$369$	−2.17306e13	−0.165358
$370$	0	0
$371$	8.83621e12	0.0652694
$372$	0	0
$373$	2.09367e14	1.50144	0.750722	−	0.660619i	$-0.229706\pi$
$373$	2.09367e14	1.50144	0.750722	+	0.660619i	$0.229706\pi$
$374$	0	0
$375$	7.45184e13	0.518909
$376$	0	0
$377$	−7.15424e13	−0.483823
$378$	0	0
$379$	−2.95197e14	−1.93908	−0.969540	−	0.244931i	$-0.921235\pi$
$379$	−2.95197e14	−1.93908	−0.969540	+	0.244931i	$0.921235\pi$
$380$	0	0
$381$	−6.72258e13	−0.428992
$382$	0	0
$383$	−1.28924e14	−0.799356	−0.399678	−	0.916656i	$-0.630878\pi$
$383$	−1.28924e14	−0.799356	−0.399678	+	0.916656i	$0.630878\pi$
$384$	0	0
$385$	−9.07142e13	−0.546564
$386$	0	0
$387$	−3.66381e13	−0.214547
$388$	0	0
$389$	1.04145e14	0.592810	0.296405	−	0.955062i	$-0.404212\pi$
$389$	1.04145e14	0.592810	0.296405	+	0.955062i	$0.404212\pi$
$390$	0	0
$391$	−1.70278e14	−0.942297
$392$	0	0
$393$	−8.28319e13	−0.445696
$394$	0	0
$395$	2.13219e13	0.111569
$396$	0	0
$397$	−1.15408e14	−0.587338	−0.293669	−	0.955907i	$-0.594876\pi$
$397$	−1.15408e14	−0.587338	−0.293669	+	0.955907i	$0.594876\pi$
$398$	0	0
$399$	8.69759e13	0.430574
$400$	0	0
$401$	−2.20847e14	−1.06365	−0.531824	−	0.846855i	$-0.678493\pi$
$401$	−2.20847e14	−1.06365	−0.531824	+	0.846855i	$0.678493\pi$
$402$	0	0
$403$	3.73674e14	1.75111
$404$	0	0
$405$	1.26570e13	0.0577204
$406$	0	0
$407$	−2.42717e14	−1.07729
$408$	0	0
$409$	−8.56852e13	−0.370193	−0.185096	−	0.982720i	$-0.559260\pi$
$409$	−8.56852e13	−0.370193	−0.185096	+	0.982720i	$0.559260\pi$
$410$	0	0
$411$	2.29582e14	0.965626
$412$	0	0
$413$	5.50985e13	0.225639
$414$	0	0
$415$	−3.08280e13	−0.122936
$416$	0	0
$417$	2.52162e13	0.0979334
$418$	0	0
$419$	−2.58945e14	−0.979560	−0.489780	−	0.871846i	$-0.662923\pi$
$419$	−2.58945e14	−0.979560	−0.489780	+	0.871846i	$0.662923\pi$
$420$	0	0
$421$	1.09351e14	0.402968	0.201484	−	0.979492i	$-0.435424\pi$
$421$	1.09351e14	0.402968	0.201484	+	0.979492i	$0.435424\pi$
$422$	0	0
$423$	−1.63159e14	−0.585786
$424$	0	0
$425$	−2.95562e14	−1.03398
$426$	0	0
$427$	2.56479e14	0.874377
$428$	0	0
$429$	4.57779e14	1.52104
$430$	0	0
$431$	−2.92180e14	−0.946292	−0.473146	−	0.880984i	$-0.656882\pi$
$431$	−2.92180e14	−0.946292	−0.473146	+	0.880984i	$0.656882\pi$
$432$	0	0
$433$	9.53988e13	0.301203	0.150602	−	0.988595i	$-0.451879\pi$
$433$	9.53988e13	0.301203	0.150602	+	0.988595i	$0.451879\pi$
$434$	0	0
$435$	−2.54170e13	−0.0782409
$436$	0	0
$437$	−2.23206e14	−0.669975
$438$	0	0
$439$	4.80574e14	1.40671	0.703357	−	0.710837i	$-0.251684\pi$
$439$	4.80574e14	1.40671	0.703357	+	0.710837i	$0.251684\pi$
$440$	0	0
$441$	−5.27040e13	−0.150464
$442$	0	0
$443$	3.15148e14	0.877596	0.438798	−	0.898586i	$-0.355404\pi$
$443$	3.15148e14	0.877596	0.438798	+	0.898586i	$0.355404\pi$
$444$	0	0
$445$	2.74166e14	0.744790
$446$	0	0
$447$	2.98530e14	0.791219
$448$	0	0
$449$	5.03692e14	1.30260	0.651299	−	0.758821i	$-0.274224\pi$
$449$	5.03692e14	1.30260	0.651299	+	0.758821i	$0.274224\pi$
$450$	0	0
$451$	−2.79226e14	−0.704669
$452$	0	0
$453$	5.16292e13	0.127161
$454$	0	0
$455$	2.96845e14	0.713619
$456$	0	0
$457$	−2.96302e14	−0.695338	−0.347669	−	0.937617i	$-0.613027\pi$
$457$	−2.96302e14	−0.695338	−0.347669	+	0.937617i	$0.613027\pi$
$458$	0	0
$459$	−1.18958e14	−0.272536
$460$	0	0
$461$	1.00284e14	0.224325	0.112163	−	0.993690i	$-0.464222\pi$
$461$	1.00284e14	0.224325	0.112163	+	0.993690i	$0.464222\pi$
$462$	0	0
$463$	−2.49998e14	−0.546061	−0.273030	−	0.962005i	$-0.588026\pi$
$463$	−2.49998e14	−0.546061	−0.273030	+	0.962005i	$0.588026\pi$
$464$	0	0
$465$	1.32756e14	0.283180
$466$	0	0
$467$	7.16272e13	0.149223	0.0746114	−	0.997213i	$-0.476228\pi$
$467$	7.16272e13	0.149223	0.0746114	+	0.997213i	$0.476228\pi$
$468$	0	0
$469$	6.16124e14	1.25377
$470$	0	0
$471$	3.85477e14	0.766273
$472$	0	0
$473$	−4.70780e14	−0.914285
$474$	0	0
$475$	−3.87433e14	−0.735159
$476$	0	0
$477$	−1.58419e13	−0.0293736
$478$	0	0
$479$	−2.34866e14	−0.425574	−0.212787	−	0.977099i	$-0.568254\pi$
$479$	−2.34866e14	−0.425574	−0.212787	+	0.977099i	$0.568254\pi$
$480$	0	0
$481$	7.94246e14	1.40656
$482$	0	0
$483$	−1.64385e14	−0.284546
$484$	0	0
$485$	−2.98955e14	−0.505855
$486$	0	0
$487$	9.89828e14	1.63738	0.818692	−	0.574233i	$-0.194700\pi$
$487$	9.89828e14	1.63738	0.818692	+	0.574233i	$0.194700\pi$
$488$	0	0
$489$	2.10200e14	0.339965
$490$	0	0
$491$	8.70345e14	1.37639	0.688197	−	0.725523i	$-0.258402\pi$
$491$	8.70345e14	1.37639	0.688197	+	0.725523i	$0.258402\pi$
$492$	0	0
$493$	2.38884e14	0.369427
$494$	0	0
$495$	1.62636e14	0.245973
$496$	0	0
$497$	−2.74916e14	−0.406669
$498$	0	0
$499$	5.38314e14	0.778902	0.389451	−	0.921047i	$-0.372665\pi$
$499$	5.38314e14	0.778902	0.389451	+	0.921047i	$0.372665\pi$
$500$	0	0
$501$	−5.48698e14	−0.776652
$502$	0	0
$503$	−8.38818e14	−1.16157	−0.580783	−	0.814059i	$-0.697253\pi$
$503$	−8.38818e14	−1.16157	−0.580783	+	0.814059i	$0.697253\pi$
$504$	0	0
$505$	−1.49970e14	−0.203190
$506$	0	0
$507$	−1.06250e15	−1.40859
$508$	0	0
$509$	−5.97231e14	−0.774809	−0.387404	−	0.921910i	$-0.626628\pi$
$509$	−5.97231e14	−0.774809	−0.387404	+	0.921910i	$0.626628\pi$
$510$	0	0
$511$	−6.46921e14	−0.821365
$512$	0	0
$513$	−1.55934e14	−0.193774
$514$	0	0
$515$	2.34201e14	0.284872
$516$	0	0
$517$	−2.09650e15	−2.49631
$518$	0	0
$519$	5.24783e14	0.611729
$520$	0	0
$521$	1.36712e15	1.56027	0.780133	−	0.625613i	$-0.215151\pi$
$521$	1.36712e15	1.56027	0.780133	+	0.625613i	$0.215151\pi$
$522$	0	0
$523$	−6.90497e14	−0.771619	−0.385809	−	0.922579i	$-0.626078\pi$
$523$	−6.90497e14	−0.771619	−0.385809	+	0.922579i	$0.626078\pi$
$524$	0	0
$525$	−2.85333e14	−0.312231
$526$	0	0
$527$	−1.24771e15	−1.33708
$528$	0	0
$529$	−5.30948e14	−0.557245
$530$	0	0
$531$	−9.87828e13	−0.101546
$532$	0	0
$533$	9.13714e14	0.920048
$534$	0	0
$535$	−4.00095e14	−0.394655
$536$	0	0
$537$	1.09541e15	1.05857
$538$	0	0
$539$	−6.77218e14	−0.641195
$540$	0	0
$541$	−4.48077e14	−0.415688	−0.207844	−	0.978162i	$-0.566645\pi$
$541$	−4.48077e14	−0.415688	−0.207844	+	0.978162i	$0.566645\pi$
$542$	0	0
$543$	−1.05475e14	−0.0958848
$544$	0	0
$545$	6.45334e14	0.574915
$546$	0	0
$547$	−8.45293e14	−0.738035	−0.369017	−	0.929423i	$-0.620306\pi$
$547$	−8.45293e14	−0.738035	−0.369017	+	0.929423i	$0.620306\pi$
$548$	0	0
$549$	−4.59826e14	−0.393501
$550$	0	0
$551$	3.13136e14	0.262663
$552$	0	0
$553$	−1.93460e14	−0.159075
$554$	0	0
$555$	2.82173e14	0.227460
$556$	0	0
$557$	−6.21109e14	−0.490868	−0.245434	−	0.969413i	$-0.578930\pi$
$557$	−6.21109e14	−0.490868	−0.245434	+	0.969413i	$0.578930\pi$
$558$	0	0
$559$	1.54054e15	1.19373
$560$	0	0
$561$	−1.52855e15	−1.16140
$562$	0	0
$563$	2.31809e15	1.72717	0.863585	−	0.504203i	$-0.168214\pi$
$563$	2.31809e15	1.72717	0.863585	+	0.504203i	$0.168214\pi$
$564$	0	0
$565$	2.86482e14	0.209330
$566$	0	0
$567$	−1.14841e14	−0.0822980
$568$	0	0
$569$	−1.54963e15	−1.08921	−0.544605	−	0.838693i	$-0.683320\pi$
$569$	−1.54963e15	−1.08921	−0.544605	+	0.838693i	$0.683320\pi$
$570$	0	0
$571$	−1.59408e15	−1.09903	−0.549516	−	0.835483i	$-0.685188\pi$
$571$	−1.59408e15	−1.09903	−0.549516	+	0.835483i	$0.685188\pi$
$572$	0	0
$573$	2.10851e14	0.142602
$574$	0	0
$575$	7.32250e14	0.485832
$576$	0	0
$577$	1.77831e15	1.15755	0.578777	−	0.815486i	$-0.303530\pi$
$577$	1.77831e15	1.15755	0.578777	+	0.815486i	$0.303530\pi$
$578$	0	0
$579$	−4.54278e13	−0.0290128
$580$	0	0
$581$	2.79711e14	0.175283
$582$	0	0
$583$	−2.03560e14	−0.125174
$584$	0	0
$585$	−5.32195e14	−0.321154
$586$	0	0
$587$	−2.98810e14	−0.176964	−0.0884821	−	0.996078i	$-0.528202\pi$
$587$	−2.98810e14	−0.176964	−0.0884821	+	0.996078i	$0.528202\pi$
$588$	0	0
$589$	−1.63554e15	−0.950666
$590$	0	0
$591$	−8.08700e14	−0.461378
$592$	0	0
$593$	2.13060e15	1.19317	0.596585	−	0.802550i	$-0.296524\pi$
$593$	2.13060e15	1.19317	0.596585	+	0.802550i	$0.296524\pi$
$594$	0	0
$595$	−9.91179e14	−0.544890
$596$	0	0
$597$	−3.85351e14	−0.207969
$598$	0	0
$599$	−6.91981e14	−0.366646	−0.183323	−	0.983053i	$-0.558685\pi$
$599$	−6.91981e14	−0.366646	−0.183323	+	0.983053i	$0.558685\pi$
$600$	0	0
$601$	1.90167e15	0.989296	0.494648	−	0.869094i	$-0.335297\pi$
$601$	1.90167e15	0.989296	0.494648	+	0.869094i	$0.335297\pi$
$602$	0	0
$603$	−1.10461e15	−0.564241
$604$	0	0
$605$	1.05410e15	0.528724
$606$	0	0
$607$	−2.60368e14	−0.128248	−0.0641238	−	0.997942i	$-0.520425\pi$
$607$	−2.60368e14	−0.128248	−0.0641238	+	0.997942i	$0.520425\pi$
$608$	0	0
$609$	2.30616e14	0.111556
$610$	0	0
$611$	6.86041e15	3.25929
$612$	0	0
$613$	−3.20683e15	−1.49638	−0.748192	−	0.663482i	$-0.769078\pi$
$613$	−3.20683e15	−1.49638	−0.748192	+	0.663482i	$0.769078\pi$
$614$	0	0
$615$	3.24617e14	0.148785
$616$	0	0
$617$	3.99520e15	1.79875	0.899374	−	0.437180i	$-0.144023\pi$
$617$	3.99520e15	1.79875	0.899374	+	0.437180i	$0.144023\pi$
$618$	0	0
$619$	1.00277e15	0.443508	0.221754	−	0.975103i	$-0.428822\pi$
$619$	1.00277e15	0.443508	0.221754	+	0.975103i	$0.428822\pi$
$620$	0	0
$621$	2.94716e14	0.128056
$622$	0	0
$623$	−2.48759e15	−1.06192
$624$	0	0
$625$	6.27607e14	0.263237
$626$	0	0
$627$	−2.00367e15	−0.825760
$628$	0	0
$629$	−2.65203e15	−1.07399
$630$	0	0
$631$	2.06951e15	0.823579	0.411790	−	0.911279i	$-0.364904\pi$
$631$	2.06951e15	0.823579	0.411790	+	0.911279i	$0.364904\pi$
$632$	0	0
$633$	8.62548e14	0.337336
$634$	0	0
$635$	1.00424e15	0.385994
$636$	0	0
$637$	2.21607e15	0.837174
$638$	0	0
$639$	4.92881e14	0.183015
$640$	0	0
$641$	−1.00036e14	−0.0365122	−0.0182561	−	0.999833i	$-0.505811\pi$
$641$	−1.00036e14	−0.0365122	−0.0182561	+	0.999833i	$0.505811\pi$
$642$	0	0
$643$	3.03786e14	0.108995	0.0544976	−	0.998514i	$-0.482644\pi$
$643$	3.03786e14	0.108995	0.0544976	+	0.998514i	$0.482644\pi$
$644$	0	0
$645$	5.47310e14	0.193043
$646$	0	0
$647$	−1.90746e14	−0.0661426	−0.0330713	−	0.999453i	$-0.510529\pi$
$647$	−1.90746e14	−0.0661426	−0.0330713	+	0.999453i	$0.510529\pi$
$648$	0	0
$649$	−1.26931e15	−0.432733
$650$	0	0
$651$	−1.20453e15	−0.403759
$652$	0	0
$653$	1.64793e15	0.543145	0.271573	−	0.962418i	$-0.412456\pi$
$653$	1.64793e15	0.543145	0.271573	+	0.962418i	$0.412456\pi$
$654$	0	0
$655$	1.23737e15	0.401025
$656$	0	0
$657$	1.15983e15	0.369644
$658$	0	0
$659$	2.62861e15	0.823866	0.411933	−	0.911214i	$-0.364854\pi$
$659$	2.62861e15	0.823866	0.411933	+	0.911214i	$0.364854\pi$
$660$	0	0
$661$	4.80241e15	1.48031	0.740153	−	0.672439i	$-0.234753\pi$
$661$	4.80241e15	1.48031	0.740153	+	0.672439i	$0.234753\pi$
$662$	0	0
$663$	5.00188e15	1.51638
$664$	0	0
$665$	−1.29927e15	−0.387418
$666$	0	0
$667$	−5.91830e14	−0.173582
$668$	0	0
$669$	−1.25879e15	−0.363171
$670$	0	0
$671$	−5.90852e15	−1.67689
$672$	0	0
$673$	−1.25325e15	−0.349909	−0.174954	−	0.984577i	$-0.555978\pi$
$673$	−1.25325e15	−0.349909	−0.174954	+	0.984577i	$0.555978\pi$
$674$	0	0
$675$	5.11556e14	0.140515
$676$	0	0
$677$	−1.92035e15	−0.518969	−0.259485	−	0.965747i	$-0.583553\pi$
$677$	−1.92035e15	−0.518969	−0.259485	+	0.965747i	$0.583553\pi$
$678$	0	0
$679$	2.71250e15	0.721251
$680$	0	0
$681$	−3.24805e15	−0.849793
$682$	0	0
$683$	−6.12548e14	−0.157698	−0.0788490	−	0.996887i	$-0.525125\pi$
$683$	−6.12548e14	−0.157698	−0.0788490	+	0.996887i	$0.525125\pi$
$684$	0	0
$685$	−3.42957e15	−0.868842
$686$	0	0
$687$	4.93485e14	0.123030
$688$	0	0
$689$	6.66112e14	0.163434
$690$	0	0
$691$	−5.08990e15	−1.22908	−0.614539	−	0.788886i	$-0.710658\pi$
$691$	−5.08990e15	−1.22908	−0.614539	+	0.788886i	$0.710658\pi$
$692$	0	0
$693$	−1.47564e15	−0.350710
$694$	0	0
$695$	−3.76686e14	−0.0881177
$696$	0	0
$697$	−3.05094e15	−0.702511
$698$	0	0
$699$	−1.73337e15	−0.392887
$700$	0	0
$701$	−1.78664e15	−0.398647	−0.199323	−	0.979934i	$-0.563874\pi$
$701$	−1.78664e15	−0.398647	−0.199323	+	0.979934i	$0.563874\pi$
$702$	0	0
$703$	−3.47636e15	−0.763608
$704$	0	0
$705$	2.43731e15	0.527073
$706$	0	0
$707$	1.36072e15	0.289709
$708$	0	0
$709$	4.18187e15	0.876631	0.438315	−	0.898821i	$-0.355575\pi$
$709$	4.18187e15	0.876631	0.438315	+	0.898821i	$0.355575\pi$
$710$	0	0
$711$	3.46842e14	0.0715895
$712$	0	0
$713$	3.09119e15	0.628251
$714$	0	0
$715$	−6.83842e15	−1.36859
$716$	0	0
$717$	8.02782e14	0.158213
$718$	0	0
$719$	−8.02554e15	−1.55763	−0.778817	−	0.627251i	$-0.784180\pi$
$719$	−8.02554e15	−1.55763	−0.778817	+	0.627251i	$0.784180\pi$
$720$	0	0
$721$	−2.12497e15	−0.406172
$722$	0	0
$723$	−1.01316e14	−0.0190729
$724$	0	0
$725$	−1.02727e15	−0.190470
$726$	0	0
$727$	2.93804e15	0.536560	0.268280	−	0.963341i	$-0.413545\pi$
$727$	2.93804e15	0.536560	0.268280	+	0.963341i	$0.413545\pi$
$728$	0	0
$729$	2.05891e14	0.0370370
$730$	0	0
$731$	−5.14393e15	−0.911485
$732$	0	0
$733$	−4.42319e15	−0.772083	−0.386042	−	0.922481i	$-0.626158\pi$
$733$	−4.42319e15	−0.772083	−0.386042	+	0.922481i	$0.626158\pi$
$734$	0	0
$735$	7.87306e14	0.135383
$736$	0	0
$737$	−1.41937e16	−2.40449
$738$	0	0
$739$	1.11732e16	1.86481	0.932405	−	0.361416i	$-0.117707\pi$
$739$	1.11732e16	1.86481	0.932405	+	0.361416i	$0.117707\pi$
$740$	0	0
$741$	6.55662e15	1.07815
$742$	0	0
$743$	−1.98624e15	−0.321805	−0.160903	−	0.986970i	$-0.551441\pi$
$743$	−1.98624e15	−0.321805	−0.160903	+	0.986970i	$0.551441\pi$
$744$	0	0
$745$	−4.45952e15	−0.711916
$746$	0	0
$747$	−5.01477e14	−0.0788838
$748$	0	0
$749$	3.63017e15	0.562700
$750$	0	0
$751$	7.63503e15	1.16625	0.583125	−	0.812383i	$-0.301830\pi$
$751$	7.63503e15	1.16625	0.583125	+	0.812383i	$0.301830\pi$
$752$	0	0
$753$	−4.66486e14	−0.0702209
$754$	0	0
$755$	−7.71251e14	−0.114416
$756$	0	0
$757$	7.89241e15	1.15394	0.576969	−	0.816766i	$-0.304235\pi$
$757$	7.89241e15	1.15394	0.576969	+	0.816766i	$0.304235\pi$
$758$	0	0
$759$	3.78694e15	0.545706
$760$	0	0
$761$	−1.10602e16	−1.57089	−0.785446	−	0.618930i	$-0.787566\pi$
$761$	−1.10602e16	−1.57089	−0.785446	+	0.618930i	$0.787566\pi$
$762$	0	0
$763$	−5.85530e15	−0.819717
$764$	0	0
$765$	1.77703e15	0.245220
$766$	0	0
$767$	4.15356e15	0.564997
$768$	0	0
$769$	1.66698e15	0.223529	0.111765	−	0.993735i	$-0.464350\pi$
$769$	1.66698e15	0.223529	0.111765	+	0.993735i	$0.464350\pi$
$770$	0	0
$771$	−2.37176e15	−0.313524
$772$	0	0
$773$	6.92568e15	0.902558	0.451279	−	0.892383i	$-0.350968\pi$
$773$	6.92568e15	0.902558	0.451279	+	0.892383i	$0.350968\pi$
$774$	0	0
$775$	5.36556e15	0.689375
$776$	0	0
$777$	−2.56024e15	−0.324313
$778$	0	0
$779$	−3.99926e15	−0.499487
$780$	0	0
$781$	6.33326e15	0.779914
$782$	0	0
$783$	−4.13457e14	−0.0502043
$784$	0	0
$785$	−5.75836e15	−0.689470
$786$	0	0
$787$	2.99770e15	0.353939	0.176969	−	0.984216i	$-0.443371\pi$
$787$	2.99770e15	0.353939	0.176969	+	0.984216i	$0.443371\pi$
$788$	0	0
$789$	−9.83775e14	−0.114544
$790$	0	0
$791$	−2.59933e15	−0.298463
$792$	0	0
$793$	1.93345e16	2.18943
$794$	0	0
$795$	2.36651e14	0.0264295
$796$	0	0
$797$	−5.67794e15	−0.625417	−0.312709	−	0.949849i	$-0.601236\pi$
$797$	−5.67794e15	−0.625417	−0.312709	+	0.949849i	$0.601236\pi$
$798$	0	0
$799$	−2.29073e16	−2.48866
$800$	0	0
$801$	4.45984e15	0.477904
$802$	0	0
$803$	1.49031e16	1.57522
$804$	0	0
$805$	2.45563e15	0.256027
$806$	0	0
$807$	2.75696e15	0.283547
$808$	0	0
$809$	4.43159e15	0.449617	0.224809	−	0.974403i	$-0.427824\pi$
$809$	4.43159e15	0.449617	0.224809	+	0.974403i	$0.427824\pi$
$810$	0	0
$811$	8.99323e15	0.900121	0.450060	−	0.892998i	$-0.351402\pi$
$811$	8.99323e15	0.900121	0.450060	+	0.892998i	$0.351402\pi$
$812$	0	0
$813$	−5.71662e14	−0.0564470
$814$	0	0
$815$	−3.14002e15	−0.305890
$816$	0	0
$817$	−6.74283e15	−0.648068
$818$	0	0
$819$	4.82876e15	0.457903
$820$	0	0
$821$	9.08013e15	0.849581	0.424790	−	0.905292i	$-0.360348\pi$
$821$	9.08013e15	0.849581	0.424790	+	0.905292i	$0.360348\pi$
$822$	0	0
$823$	1.29654e16	1.19698	0.598488	−	0.801132i	$-0.295768\pi$
$823$	1.29654e16	1.19698	0.598488	+	0.801132i	$0.295768\pi$
$824$	0	0
$825$	6.57322e15	0.598800
$826$	0	0
$827$	5.40262e15	0.485651	0.242826	−	0.970070i	$-0.421926\pi$
$827$	5.40262e15	0.485651	0.242826	+	0.970070i	$0.421926\pi$
$828$	0	0
$829$	−7.97388e14	−0.0707326	−0.0353663	−	0.999374i	$-0.511260\pi$
$829$	−7.97388e14	−0.0707326	−0.0353663	+	0.999374i	$0.511260\pi$
$830$	0	0
$831$	1.06027e16	0.928138
$832$	0	0
$833$	−7.39956e15	−0.639231
$834$	0	0
$835$	8.19661e15	0.698809
$836$	0	0
$837$	2.15953e15	0.181706
$838$	0	0
$839$	−1.02313e16	−0.849646	−0.424823	−	0.905276i	$-0.639664\pi$
$839$	−1.02313e16	−0.849646	−0.424823	+	0.905276i	$0.639664\pi$
$840$	0	0
$841$	−1.13702e16	−0.931947
$842$	0	0
$843$	−6.68490e15	−0.540808
$844$	0	0
$845$	1.58719e16	1.26741
$846$	0	0
$847$	−9.56418e15	−0.753857
$848$	0	0
$849$	−6.26490e15	−0.487441
$850$	0	0
$851$	6.57034e15	0.504633
$852$	0	0
$853$	−1.33010e16	−1.00847	−0.504237	−	0.863565i	$-0.668226\pi$
$853$	−1.33010e16	−1.00847	−0.504237	+	0.863565i	$0.668226\pi$
$854$	0	0
$855$	2.32938e15	0.174352
$856$	0	0
$857$	3.38483e15	0.250116	0.125058	−	0.992149i	$-0.460088\pi$
$857$	3.38483e15	0.250116	0.125058	+	0.992149i	$0.460088\pi$
$858$	0	0
$859$	−2.32376e16	−1.69523	−0.847615	−	0.530612i	$-0.821962\pi$
$859$	−2.32376e16	−1.69523	−0.847615	+	0.530612i	$0.821962\pi$
$860$	0	0
$861$	−2.94534e15	−0.212138
$862$	0	0
$863$	−2.44743e16	−1.74041	−0.870203	−	0.492693i	$-0.836013\pi$
$863$	−2.44743e16	−1.74041	−0.870203	+	0.492693i	$0.836013\pi$
$864$	0	0
$865$	−7.83935e15	−0.550417
$866$	0	0
$867$	−8.37344e15	−0.580496
$868$	0	0
$869$	4.45674e15	0.305076
$870$	0	0
$871$	4.64461e16	3.13942
$872$	0	0
$873$	−4.86309e15	−0.324589
$874$	0	0
$875$	1.01002e16	0.665708
$876$	0	0
$877$	2.15305e16	1.40138	0.700689	−	0.713467i	$-0.252876\pi$
$877$	2.15305e16	1.40138	0.700689	+	0.713467i	$0.252876\pi$
$878$	0	0
$879$	−1.35916e16	−0.873636
$880$	0	0
$881$	2.77094e16	1.75898	0.879488	−	0.475922i	$-0.157885\pi$
$881$	2.77094e16	1.75898	0.879488	+	0.475922i	$0.157885\pi$
$882$	0	0
$883$	8.61677e15	0.540208	0.270104	−	0.962831i	$-0.412942\pi$
$883$	8.61677e15	0.540208	0.270104	+	0.962831i	$0.412942\pi$
$884$	0	0
$885$	1.47564e15	0.0913679
$886$	0	0
$887$	3.22651e16	1.97312	0.986558	−	0.163409i	$-0.0522490\pi$
$887$	3.22651e16	1.97312	0.986558	+	0.163409i	$0.0522490\pi$
$888$	0	0
$889$	−9.11173e15	−0.550353
$890$	0	0
$891$	2.64559e15	0.157832
$892$	0	0
$893$	−3.00275e16	−1.76944
$894$	0	0
$895$	−1.63635e16	−0.952468
$896$	0	0
$897$	−1.23921e16	−0.712500
$898$	0	0
$899$	−4.33663e15	−0.246305
$900$	0	0
$901$	−2.22418e15	−0.124791
$902$	0	0
$903$	−4.96590e15	−0.275242
$904$	0	0
$905$	1.57561e15	0.0862744
$906$	0	0
$907$	−3.10436e16	−1.67932	−0.839658	−	0.543115i	$-0.817245\pi$
$907$	−3.10436e16	−1.67932	−0.839658	+	0.543115i	$0.817245\pi$
$908$	0	0
$909$	−2.43955e15	−0.130379
$910$	0	0
$911$	−1.78173e16	−0.940786	−0.470393	−	0.882457i	$-0.655888\pi$
$911$	−1.78173e16	−0.940786	−0.470393	+	0.882457i	$0.655888\pi$
$912$	0	0
$913$	−6.44371e15	−0.336161
$914$	0	0
$915$	6.86901e15	0.354061
$916$	0	0
$917$	−1.12270e16	−0.571783
$918$	0	0
$919$	−2.90046e16	−1.45959	−0.729797	−	0.683664i	$-0.760385\pi$
$919$	−2.90046e16	−1.45959	−0.729797	+	0.683664i	$0.760385\pi$
$920$	0	0
$921$	1.38038e16	0.686387
$922$	0	0
$923$	−2.07244e16	−1.01829
$924$	0	0
$925$	1.14045e16	0.553730
$926$	0	0
$927$	3.80974e15	0.182792
$928$	0	0
$929$	1.09610e16	0.519713	0.259856	−	0.965647i	$-0.416325\pi$
$929$	1.09610e16	0.519713	0.259856	+	0.965647i	$0.416325\pi$
$930$	0	0
$931$	−9.69957e15	−0.454495
$932$	0	0
$933$	−1.63100e16	−0.755274
$934$	0	0
$935$	2.28338e16	1.04500
$936$	0	0
$937$	9.13874e15	0.413351	0.206675	−	0.978410i	$-0.433736\pi$
$937$	9.13874e15	0.413351	0.206675	+	0.978410i	$0.433736\pi$
$938$	0	0
$939$	8.91043e15	0.398325
$940$	0	0
$941$	6.97287e14	0.0308083	0.0154042	−	0.999881i	$-0.495097\pi$
$941$	6.97287e14	0.0308083	0.0154042	+	0.999881i	$0.495097\pi$
$942$	0	0
$943$	7.55864e15	0.330088
$944$	0	0
$945$	1.71552e15	0.0740494
$946$	0	0
$947$	2.17555e15	0.0928205	0.0464103	−	0.998922i	$-0.485222\pi$
$947$	2.17555e15	0.0928205	0.0464103	+	0.998922i	$0.485222\pi$
$948$	0	0
$949$	−4.87677e16	−2.05669
$950$	0	0
$951$	−2.45940e16	−1.02526
$952$	0	0
$953$	−3.59675e16	−1.48217	−0.741087	−	0.671409i	$-0.765690\pi$
$953$	−3.59675e16	−1.48217	−0.741087	+	0.671409i	$0.765690\pi$
$954$	0	0
$955$	−3.14976e15	−0.128309
$956$	0	0
$957$	−5.31270e15	−0.213944
$958$	0	0
$959$	3.11174e16	1.23880
$960$	0	0
$961$	−2.75781e15	−0.108539
$962$	0	0
$963$	−6.50832e15	−0.253235
$964$	0	0
$965$	6.78613e14	0.0261049
$966$	0	0
$967$	−3.64625e16	−1.38676	−0.693379	−	0.720573i	$-0.743879\pi$
$967$	−3.64625e16	−1.38676	−0.693379	+	0.720573i	$0.743879\pi$
$968$	0	0
$969$	−2.18929e16	−0.823231
$970$	0	0
$971$	1.01490e16	0.377328	0.188664	−	0.982042i	$-0.439584\pi$
$971$	1.01490e16	0.377328	0.188664	+	0.982042i	$0.439584\pi$
$972$	0	0
$973$	3.41778e15	0.125639
$974$	0	0
$975$	−2.15096e16	−0.781821
$976$	0	0
$977$	2.99278e16	1.07561	0.537804	−	0.843070i	$-0.319254\pi$
$977$	2.99278e16	1.07561	0.537804	+	0.843070i	$0.319254\pi$
$978$	0	0
$979$	5.73066e16	2.03657
$980$	0	0
$981$	1.04976e16	0.368902
$982$	0	0
$983$	3.23592e16	1.12448	0.562242	−	0.826973i	$-0.309939\pi$
$983$	3.23592e16	1.12448	0.562242	+	0.826973i	$0.309939\pi$
$984$	0	0
$985$	1.20806e16	0.415134
$986$	0	0
$987$	−2.21144e16	−0.751503
$988$	0	0
$989$	1.27440e16	0.428278
$990$	0	0
$991$	2.28589e16	0.759716	0.379858	−	0.925045i	$-0.375973\pi$
$991$	2.28589e16	0.759716	0.379858	+	0.925045i	$0.375973\pi$
$992$	0	0
$993$	1.20677e16	0.396647
$994$	0	0
$995$	5.75648e15	0.187124
$996$	0	0
$997$	−1.99784e16	−0.642298	−0.321149	−	0.947029i	$-0.604069\pi$
$997$	−1.99784e16	−0.642298	−0.321149	+	0.947029i	$0.604069\pi$
$998$	0	0
$999$	4.59010e15	0.145953

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	48.12.a.d.1.1		1
3.2	odd	2		144.12.a.e.1.1		1
4.3	odd	2		6.12.a.c.1.1	✓	1
8.3	odd	2		192.12.a.c.1.1		1
8.5	even	2		192.12.a.m.1.1		1
12.11	even	2		18.12.a.a.1.1		1
20.3	even	4		150.12.c.e.49.1		2
20.7	even	4		150.12.c.e.49.2		2
20.19	odd	2		150.12.a.a.1.1		1
36.7	odd	6		162.12.c.b.109.1		2
36.11	even	6		162.12.c.i.109.1		2
36.23	even	6		162.12.c.i.55.1		2
36.31	odd	6		162.12.c.b.55.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6.12.a.c.1.1	✓	1	4.3	odd	2
18.12.a.a.1.1		1	12.11	even	2
48.12.a.d.1.1		1	1.1	even	1	trivial
144.12.a.e.1.1		1	3.2	odd	2
150.12.a.a.1.1		1	20.19	odd	2
150.12.c.e.49.1		2	20.3	even	4
150.12.c.e.49.2		2	20.7	even	4
162.12.c.b.55.1		2	36.31	odd	6
162.12.c.b.109.1		2	36.7	odd	6
162.12.c.i.55.1		2	36.23	even	6
162.12.c.i.109.1		2	36.11	even	6
192.12.a.c.1.1		1	8.3	odd	2
192.12.a.m.1.1		1	8.5	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 48 = 2^{4} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	48.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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