LMFDB - Embedded newform 4.10.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4,10,Mod(1,4)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 10, names="a")

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

chi := DirichletCharacter("4.1");

S:= CuspForms(chi, 10);

N := Newforms(S);

Level:	$ N $	$=$	$ 4 = 2^{2} $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	4.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:	$2.06014334466$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	4.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+228.000 q^{3} -666.000 q^{5} -6328.00 q^{7} +32301.0 q^{9} +O(q^{10})$

$q+228.000 q^{3} -666.000 q^{5} -6328.00 q^{7} +32301.0 q^{9} -30420.0 q^{11} -32338.0 q^{13} -151848. q^{15} +590994. q^{17} +34676.0 q^{19} -1.44278e6 q^{21} +1.04854e6 q^{23} -1.50957e6 q^{25} +2.87690e6 q^{27} +4.40941e6 q^{29} -7.40118e6 q^{31} -6.93576e6 q^{33} +4.21445e6 q^{35} +1.02345e7 q^{37} -7.37306e6 q^{39} +1.83527e7 q^{41} -252340. q^{43} -2.15125e7 q^{45} -4.95171e7 q^{47} -310023. q^{49} +1.34747e8 q^{51} -6.63969e7 q^{53} +2.02597e7 q^{55} +7.90613e6 q^{57} -6.15237e7 q^{59} +3.56386e7 q^{61} -2.04401e8 q^{63} +2.15371e7 q^{65} +1.81742e8 q^{67} +2.39066e8 q^{69} +9.09050e7 q^{71} -2.62979e8 q^{73} -3.44182e8 q^{75} +1.92498e8 q^{77} -1.16503e8 q^{79} +2.01535e7 q^{81} -9.56372e6 q^{83} -3.93602e8 q^{85} +1.00534e9 q^{87} +6.11827e8 q^{89} +2.04635e8 q^{91} -1.68747e9 q^{93} -2.30942e7 q^{95} -2.59313e8 q^{97} -9.82596e8 q^{99} +O(q^{100})$

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
$3$	228.000	1.62513	0.812567	−	0.582868i	$-0.198069\pi$
$3$	228.000	1.62513	0.812567	+	0.582868i	$0.198069\pi$
$4$	0	0
$5$	−666.000	−0.476551	−0.238275	−	0.971198i	$-0.576582\pi$
$5$	−666.000	−0.476551	−0.238275	+	0.971198i	$0.576582\pi$
$6$	0	0
$7$	−6328.00	−0.996151	−0.498076	−	0.867134i	$-0.665960\pi$
$7$	−6328.00	−0.996151	−0.498076	+	0.867134i	$0.665960\pi$
$8$	0	0
$9$	32301.0	1.64106
$10$	0	0
$11$	−30420.0	−0.626458	−0.313229	−	0.949678i	$-0.601411\pi$
$11$	−30420.0	−0.626458	−0.313229	+	0.949678i	$0.601411\pi$
$12$	0	0
$13$	−32338.0	−0.314028	−0.157014	−	0.987596i	$-0.550187\pi$
$13$	−32338.0	−0.314028	−0.157014	+	0.987596i	$0.550187\pi$
$14$	0	0
$15$	−151848.	−0.774459
$16$	0	0
$17$	590994.	1.71618	0.858090	−	0.513499i	$-0.171651\pi$
$17$	590994.	1.71618	0.858090	+	0.513499i	$0.171651\pi$
$18$	0	0
$19$	34676.0	0.0610433	0.0305216	−	0.999534i	$-0.490283\pi$
$19$	34676.0	0.0610433	0.0305216	+	0.999534i	$0.490283\pi$
$20$	0	0
$21$	−1.44278e6	−1.61888
$22$	0	0
$23$	1.04854e6	0.781282	0.390641	−	0.920543i	$-0.372253\pi$
$23$	1.04854e6	0.781282	0.390641	+	0.920543i	$0.372253\pi$
$24$	0	0
$25$	−1.50957e6	−0.772899
$26$	0	0
$27$	2.87690e6	1.04181
$28$	0	0
$29$	4.40941e6	1.15768	0.578841	−	0.815441i	$-0.303505\pi$
$29$	4.40941e6	1.15768	0.578841	+	0.815441i	$0.303505\pi$
$30$	0	0
$31$	−7.40118e6	−1.43937	−0.719687	−	0.694299i	$-0.755715\pi$
$31$	−7.40118e6	−1.43937	−0.719687	+	0.694299i	$0.755715\pi$
$32$	0	0
$33$	−6.93576e6	−1.01808
$34$	0	0
$35$	4.21445e6	0.474717
$36$	0	0
$37$	1.02345e7	0.897757	0.448879	−	0.893593i	$-0.351824\pi$
$37$	1.02345e7	0.897757	0.448879	+	0.893593i	$0.351824\pi$
$38$	0	0
$39$	−7.37306e6	−0.510337
$40$	0	0
$41$	1.83527e7	1.01432	0.507158	−	0.861853i	$-0.330696\pi$
$41$	1.83527e7	1.01432	0.507158	+	0.861853i	$0.330696\pi$
$42$	0	0
$43$	−252340.	−0.0112558	−0.00562792	−	0.999984i	$-0.501791\pi$
$43$	−252340.	−0.0112558	−0.00562792	+	0.999984i	$0.501791\pi$
$44$	0	0
$45$	−2.15125e7	−0.782049
$46$	0	0
$47$	−4.95171e7	−1.48018	−0.740091	−	0.672507i	$-0.765218\pi$
$47$	−4.95171e7	−1.48018	−0.740091	+	0.672507i	$0.765218\pi$
$48$	0	0
$49$	−310023.	−0.00768266
$50$	0	0
$51$	1.34747e8	2.78902
$52$	0	0
$53$	−6.63969e7	−1.15586	−0.577932	−	0.816085i	$-0.696140\pi$
$53$	−6.63969e7	−1.15586	−0.577932	+	0.816085i	$0.696140\pi$
$54$	0	0
$55$	2.02597e7	0.298539
$56$	0	0
$57$	7.90613e6	0.0992035
$58$	0	0
$59$	−6.15237e7	−0.661011	−0.330506	−	0.943804i	$-0.607219\pi$
$59$	−6.15237e7	−0.661011	−0.330506	+	0.943804i	$0.607219\pi$
$60$	0	0
$61$	3.56386e7	0.329562	0.164781	−	0.986330i	$-0.447308\pi$
$61$	3.56386e7	0.329562	0.164781	+	0.986330i	$0.447308\pi$
$62$	0	0
$63$	−2.04401e8	−1.63474
$64$	0	0
$65$	2.15371e7	0.149650
$66$	0	0
$67$	1.81742e8	1.10184	0.550921	−	0.834557i	$-0.314276\pi$
$67$	1.81742e8	1.10184	0.550921	+	0.834557i	$0.314276\pi$
$68$	0	0
$69$	2.39066e8	1.26969
$70$	0	0
$71$	9.09050e7	0.424546	0.212273	−	0.977210i	$-0.431913\pi$
$71$	9.09050e7	0.424546	0.212273	+	0.977210i	$0.431913\pi$
$72$	0	0
$73$	−2.62979e8	−1.08385	−0.541923	−	0.840428i	$-0.682304\pi$
$73$	−2.62979e8	−1.08385	−0.541923	+	0.840428i	$0.682304\pi$
$74$	0	0
$75$	−3.44182e8	−1.25607
$76$	0	0
$77$	1.92498e8	0.624047
$78$	0	0
$79$	−1.16503e8	−0.336523	−0.168261	−	0.985742i	$-0.553815\pi$
$79$	−1.16503e8	−0.336523	−0.168261	+	0.985742i	$0.553815\pi$
$80$	0	0
$81$	2.01535e7	0.0520198
$82$	0	0
$83$	−9.56372e6	−0.0221195	−0.0110598	−	0.999939i	$-0.503521\pi$
$83$	−9.56372e6	−0.0221195	−0.0110598	+	0.999939i	$0.503521\pi$
$84$	0	0
$85$	−3.93602e8	−0.817847
$86$	0	0
$87$	1.00534e9	1.88139
$88$	0	0
$89$	6.11827e8	1.03365	0.516825	−	0.856091i	$-0.327114\pi$
$89$	6.11827e8	1.03365	0.516825	+	0.856091i	$0.327114\pi$
$90$	0	0
$91$	2.04635e8	0.312819
$92$	0	0
$93$	−1.68747e9	−2.33918
$94$	0	0
$95$	−2.30942e7	−0.0290902
$96$	0	0
$97$	−2.59313e8	−0.297407	−0.148703	−	0.988882i	$-0.547510\pi$
$97$	−2.59313e8	−0.297407	−0.148703	+	0.988882i	$0.547510\pi$
$98$	0	0
$99$	−9.82596e8	−1.02806
$100$	0	0
$101$	1.56555e9	1.49700	0.748498	−	0.663137i	$-0.230775\pi$
$101$	1.56555e9	1.49700	0.748498	+	0.663137i	$0.230775\pi$
$102$	0	0
$103$	3.77095e8	0.330129	0.165064	−	0.986283i	$-0.447217\pi$
$103$	3.77095e8	0.330129	0.165064	+	0.986283i	$0.447217\pi$
$104$	0	0
$105$	9.60894e8	0.771478
$106$	0	0
$107$	−2.17717e9	−1.60570	−0.802852	−	0.596178i	$-0.796685\pi$
$107$	−2.17717e9	−1.60570	−0.802852	+	0.596178i	$0.796685\pi$
$108$	0	0
$109$	1.50811e9	1.02333	0.511664	−	0.859185i	$-0.329029\pi$
$109$	1.50811e9	1.02333	0.511664	+	0.859185i	$0.329029\pi$
$110$	0	0
$111$	2.33347e9	1.45898
$112$	0	0
$113$	−1.45355e9	−0.838640	−0.419320	−	0.907838i	$-0.637731\pi$
$113$	−1.45355e9	−0.838640	−0.419320	+	0.907838i	$0.637731\pi$
$114$	0	0
$115$	−6.98325e8	−0.372321
$116$	0	0
$117$	−1.04455e9	−0.515339
$118$	0	0
$119$	−3.73981e9	−1.70958
$120$	0	0
$121$	−1.43257e9	−0.607550
$122$	0	0
$123$	4.18443e9	1.64840
$124$	0	0
$125$	2.30615e9	0.844877
$126$	0	0
$127$	2.43679e9	0.831193	0.415597	−	0.909549i	$-0.363573\pi$
$127$	2.43679e9	0.831193	0.415597	+	0.909549i	$0.363573\pi$
$128$	0	0
$129$	−5.75335e7	−0.0182923
$130$	0	0
$131$	−1.43358e9	−0.425305	−0.212653	−	0.977128i	$-0.568210\pi$
$131$	−1.43358e9	−0.425305	−0.212653	+	0.977128i	$0.568210\pi$
$132$	0	0
$133$	−2.19430e8	−0.0608083
$134$	0	0
$135$	−1.91602e9	−0.496475
$136$	0	0
$137$	9.30903e8	0.225768	0.112884	−	0.993608i	$-0.463991\pi$
$137$	9.30903e8	0.225768	0.112884	+	0.993608i	$0.463991\pi$
$138$	0	0
$139$	4.84316e9	1.10043	0.550215	−	0.835023i	$-0.314546\pi$
$139$	4.84316e9	1.10043	0.550215	+	0.835023i	$0.314546\pi$
$140$	0	0
$141$	−1.12899e10	−2.40549
$142$	0	0
$143$	9.83722e8	0.196725
$144$	0	0
$145$	−2.93666e9	−0.551694
$146$	0	0
$147$	−7.06852e7	−0.0124854
$148$	0	0
$149$	8.53269e9	1.41823	0.709117	−	0.705091i	$-0.249094\pi$
$149$	8.53269e9	1.41823	0.709117	+	0.705091i	$0.249094\pi$
$150$	0	0
$151$	−7.14515e9	−1.11845	−0.559223	−	0.829017i	$-0.688901\pi$
$151$	−7.14515e9	−1.11845	−0.559223	+	0.829017i	$0.688901\pi$
$152$	0	0
$153$	1.90897e10	2.81636
$154$	0	0
$155$	4.92919e9	0.685935
$156$	0	0
$157$	−3.38239e9	−0.444299	−0.222149	−	0.975013i	$-0.571307\pi$
$157$	−3.38239e9	−0.444299	−0.222149	+	0.975013i	$0.571307\pi$
$158$	0	0
$159$	−1.51385e10	−1.87843
$160$	0	0
$161$	−6.63514e9	−0.778276
$162$	0	0
$163$	−9.01515e8	−0.100030	−0.0500148	−	0.998748i	$-0.515927\pi$
$163$	−9.01515e8	−0.100030	−0.0500148	+	0.998748i	$0.515927\pi$
$164$	0	0
$165$	4.61922e9	0.485166
$166$	0	0
$167$	4.05605e9	0.403533	0.201767	−	0.979434i	$-0.435332\pi$
$167$	4.05605e9	0.403533	0.201767	+	0.979434i	$0.435332\pi$
$168$	0	0
$169$	−9.55875e9	−0.901387
$170$	0	0
$171$	1.12007e9	0.100176
$172$	0	0
$173$	−1.02760e9	−0.0872202	−0.0436101	−	0.999049i	$-0.513886\pi$
$173$	−1.02760e9	−0.0872202	−0.0436101	+	0.999049i	$0.513886\pi$
$174$	0	0
$175$	9.55255e9	0.769925
$176$	0	0
$177$	−1.40274e10	−1.07423
$178$	0	0
$179$	1.48472e10	1.08095	0.540476	−	0.841360i	$-0.318244\pi$
$179$	1.48472e10	1.08095	0.540476	+	0.841360i	$0.318244\pi$
$180$	0	0
$181$	2.53270e10	1.75400	0.877001	−	0.480488i	$-0.159541\pi$
$181$	2.53270e10	1.75400	0.877001	+	0.480488i	$0.159541\pi$
$182$	0	0
$183$	8.12561e9	0.535582
$184$	0	0
$185$	−6.81618e9	−0.427827
$186$	0	0
$187$	−1.79780e10	−1.07512
$188$	0	0
$189$	−1.82050e10	−1.03780
$190$	0	0
$191$	−1.61656e10	−0.878904	−0.439452	−	0.898266i	$-0.644827\pi$
$191$	−1.61656e10	−0.878904	−0.439452	+	0.898266i	$0.644827\pi$
$192$	0	0
$193$	−1.80189e9	−0.0934802	−0.0467401	−	0.998907i	$-0.514883\pi$
$193$	−1.80189e9	−0.0934802	−0.0467401	+	0.998907i	$0.514883\pi$
$194$	0	0
$195$	4.91046e9	0.243202
$196$	0	0
$197$	−1.86979e10	−0.884495	−0.442247	−	0.896893i	$-0.645819\pi$
$197$	−1.86979e10	−0.884495	−0.442247	+	0.896893i	$0.645819\pi$
$198$	0	0
$199$	2.89890e10	1.31037	0.655186	−	0.755468i	$-0.272590\pi$
$199$	2.89890e10	1.31037	0.655186	+	0.755468i	$0.272590\pi$
$200$	0	0
$201$	4.14373e10	1.79064
$202$	0	0
$203$	−2.79027e10	−1.15323
$204$	0	0
$205$	−1.22229e10	−0.483374
$206$	0	0
$207$	3.38688e10	1.28213
$208$	0	0
$209$	−1.05484e9	−0.0382411
$210$	0	0
$211$	−1.97990e10	−0.687657	−0.343828	−	0.939033i	$-0.611724\pi$
$211$	−1.97990e10	−0.687657	−0.343828	+	0.939033i	$0.611724\pi$
$212$	0	0
$213$	2.07263e10	0.689945
$214$	0	0
$215$	1.68058e8	0.00536398
$216$	0	0
$217$	4.68347e10	1.43383
$218$	0	0
$219$	−5.99591e10	−1.76140
$220$	0	0
$221$	−1.91116e10	−0.538928
$222$	0	0
$223$	−6.78768e10	−1.83802	−0.919009	−	0.394237i	$-0.871009\pi$
$223$	−6.78768e10	−1.83802	−0.919009	+	0.394237i	$0.871009\pi$
$224$	0	0
$225$	−4.87606e10	−1.26837
$226$	0	0
$227$	5.45606e10	1.36384	0.681919	−	0.731428i	$-0.261146\pi$
$227$	5.45606e10	1.36384	0.681919	+	0.731428i	$0.261146\pi$
$228$	0	0
$229$	4.63952e10	1.11484	0.557421	−	0.830230i	$-0.311791\pi$
$229$	4.63952e10	1.11484	0.557421	+	0.830230i	$0.311791\pi$
$230$	0	0
$231$	4.38895e10	1.01416
$232$	0	0
$233$	−3.91389e8	−0.00869975	−0.00434988	−	0.999991i	$-0.501385\pi$
$233$	−3.91389e8	−0.00869975	−0.00434988	+	0.999991i	$0.501385\pi$
$234$	0	0
$235$	3.29784e10	0.705382
$236$	0	0
$237$	−2.65626e10	−0.546895
$238$	0	0
$239$	9.06538e10	1.79720	0.898598	−	0.438772i	$-0.144586\pi$
$239$	9.06538e10	1.79720	0.898598	+	0.438772i	$0.144586\pi$
$240$	0	0
$241$	−6.77663e10	−1.29401	−0.647004	−	0.762486i	$-0.723978\pi$
$241$	−6.77663e10	−1.29401	−0.647004	+	0.762486i	$0.723978\pi$
$242$	0	0
$243$	−5.20311e10	−0.957271
$244$	0	0
$245$	2.06475e8	0.00366118
$246$	0	0
$247$	−1.12135e9	−0.0191693
$248$	0	0
$249$	−2.18053e9	−0.0359472
$250$	0	0
$251$	5.47163e10	0.870131	0.435066	−	0.900399i	$-0.356725\pi$
$251$	5.47163e10	0.870131	0.435066	+	0.900399i	$0.356725\pi$
$252$	0	0
$253$	−3.18965e10	−0.489441
$254$	0	0
$255$	−8.97413e10	−1.32911
$256$	0	0
$257$	−3.40900e10	−0.487447	−0.243724	−	0.969845i	$-0.578369\pi$
$257$	−3.40900e10	−0.487447	−0.243724	+	0.969845i	$0.578369\pi$
$258$	0	0
$259$	−6.47639e10	−0.894302
$260$	0	0
$261$	1.42428e11	1.89983
$262$	0	0
$263$	−7.17361e10	−0.924563	−0.462282	−	0.886733i	$-0.652969\pi$
$263$	−7.17361e10	−0.924563	−0.462282	+	0.886733i	$0.652969\pi$
$264$	0	0
$265$	4.42203e10	0.550828
$266$	0	0
$267$	1.39496e11	1.67982
$268$	0	0
$269$	2.31610e9	0.0269695	0.0134847	−	0.999909i	$-0.495708\pi$
$269$	2.31610e9	0.0269695	0.0134847	+	0.999909i	$0.495708\pi$
$270$	0	0
$271$	8.04662e10	0.906258	0.453129	−	0.891445i	$-0.350308\pi$
$271$	8.04662e10	0.906258	0.453129	+	0.891445i	$0.350308\pi$
$272$	0	0
$273$	4.66567e10	0.508373
$274$	0	0
$275$	4.59211e10	0.484189
$276$	0	0
$277$	−1.65644e11	−1.69051	−0.845253	−	0.534367i	$-0.820550\pi$
$277$	−1.65644e11	−1.69051	−0.845253	+	0.534367i	$0.820550\pi$
$278$	0	0
$279$	−2.39066e11	−2.36210
$280$	0	0
$281$	2.57177e10	0.246067	0.123034	−	0.992402i	$-0.460738\pi$
$281$	2.57177e10	0.246067	0.123034	+	0.992402i	$0.460738\pi$
$282$	0	0
$283$	4.33126e10	0.401398	0.200699	−	0.979653i	$-0.435679\pi$
$283$	4.33126e10	0.401398	0.200699	+	0.979653i	$0.435679\pi$
$284$	0	0
$285$	−5.26548e9	−0.0472755
$286$	0	0
$287$	−1.16136e11	−1.01041
$288$	0	0
$289$	2.30686e11	1.94528
$290$	0	0
$291$	−5.91233e10	−0.483326
$292$	0	0
$293$	−4.83473e10	−0.383238	−0.191619	−	0.981469i	$-0.561374\pi$
$293$	−4.83473e10	−0.383238	−0.191619	+	0.981469i	$0.561374\pi$
$294$	0	0
$295$	4.09748e10	0.315005
$296$	0	0
$297$	−8.75154e10	−0.652650
$298$	0	0
$299$	−3.39076e10	−0.245344
$300$	0	0
$301$	1.59681e9	0.0112125
$302$	0	0
$303$	3.56945e11	2.43282
$304$	0	0
$305$	−2.37353e10	−0.157053
$306$	0	0
$307$	1.37971e11	0.886470	0.443235	−	0.896406i	$-0.353831\pi$
$307$	1.37971e11	0.886470	0.443235	+	0.896406i	$0.353831\pi$
$308$	0	0
$309$	8.59776e10	0.536503
$310$	0	0
$311$	−2.04451e11	−1.23928	−0.619638	−	0.784887i	$-0.712721\pi$
$311$	−2.04451e11	−1.23928	−0.619638	+	0.784887i	$0.712721\pi$
$312$	0	0
$313$	−1.74184e11	−1.02579	−0.512897	−	0.858450i	$-0.671428\pi$
$313$	−1.74184e11	−1.02579	−0.512897	+	0.858450i	$0.671428\pi$
$314$	0	0
$315$	1.36131e11	0.779039
$316$	0	0
$317$	−8.42468e10	−0.468583	−0.234292	−	0.972166i	$-0.575277\pi$
$317$	−8.42468e10	−0.468583	−0.234292	+	0.972166i	$0.575277\pi$
$318$	0	0
$319$	−1.34134e11	−0.725239
$320$	0	0
$321$	−4.96395e11	−2.60949
$322$	0	0
$323$	2.04933e10	0.104761
$324$	0	0
$325$	4.88164e10	0.242712
$326$	0	0
$327$	3.43850e11	1.66305
$328$	0	0
$329$	3.13344e11	1.47449
$330$	0	0
$331$	−2.88777e11	−1.32232	−0.661160	−	0.750245i	$-0.729935\pi$
$331$	−2.88777e11	−1.32232	−0.661160	+	0.750245i	$0.729935\pi$
$332$	0	0
$333$	3.30585e11	1.47327
$334$	0	0
$335$	−1.21040e11	−0.525084
$336$	0	0
$337$	1.35030e11	0.570289	0.285144	−	0.958485i	$-0.407958\pi$
$337$	1.35030e11	0.570289	0.285144	+	0.958485i	$0.407958\pi$
$338$	0	0
$339$	−3.31408e11	−1.36290
$340$	0	0
$341$	2.25144e11	0.901708
$342$	0	0
$343$	2.57319e11	1.00380
$344$	0	0
$345$	−1.59218e11	−0.605071
$346$	0	0
$347$	−3.91903e10	−0.145110	−0.0725548	−	0.997364i	$-0.523115\pi$
$347$	−3.91903e10	−0.145110	−0.0725548	+	0.997364i	$0.523115\pi$
$348$	0	0
$349$	−4.58818e10	−0.165549	−0.0827744	−	0.996568i	$-0.526378\pi$
$349$	−4.58818e10	−0.165549	−0.0827744	+	0.996568i	$0.526378\pi$
$350$	0	0
$351$	−9.30333e10	−0.327157
$352$	0	0
$353$	5.29590e11	1.81532	0.907660	−	0.419706i	$-0.137867\pi$
$353$	5.29590e11	1.81532	0.907660	+	0.419706i	$0.137867\pi$
$354$	0	0
$355$	−6.05427e10	−0.202318
$356$	0	0
$357$	−8.52677e11	−2.77829
$358$	0	0
$359$	−4.54893e10	−0.144539	−0.0722693	−	0.997385i	$-0.523024\pi$
$359$	−4.54893e10	−0.144539	−0.0722693	+	0.997385i	$0.523024\pi$
$360$	0	0
$361$	−3.21485e11	−0.996274
$362$	0	0
$363$	−3.26626e11	−0.987350
$364$	0	0
$365$	1.75144e11	0.516508
$366$	0	0
$367$	2.45167e11	0.705447	0.352723	−	0.935728i	$-0.385256\pi$
$367$	2.45167e11	0.705447	0.352723	+	0.935728i	$0.385256\pi$
$368$	0	0
$369$	5.92812e11	1.66456
$370$	0	0
$371$	4.20160e11	1.15141
$372$	0	0
$373$	−1.60290e11	−0.428762	−0.214381	−	0.976750i	$-0.568773\pi$
$373$	−1.60290e11	−0.428762	−0.214381	+	0.976750i	$0.568773\pi$
$374$	0	0
$375$	5.25803e11	1.37304
$376$	0	0
$377$	−1.42591e11	−0.363544
$378$	0	0
$379$	−3.55772e11	−0.885719	−0.442859	−	0.896591i	$-0.646036\pi$
$379$	−3.55772e11	−0.885719	−0.442859	+	0.896591i	$0.646036\pi$
$380$	0	0
$381$	5.55589e11	1.35080
$382$	0	0
$383$	−4.97008e11	−1.18024	−0.590118	−	0.807317i	$-0.700919\pi$
$383$	−4.97008e11	−1.18024	−0.590118	+	0.807317i	$0.700919\pi$
$384$	0	0
$385$	−1.28204e11	−0.297390
$386$	0	0
$387$	−8.15083e9	−0.0184715
$388$	0	0
$389$	−5.94268e11	−1.31586	−0.657929	−	0.753080i	$-0.728568\pi$
$389$	−5.94268e11	−1.31586	−0.657929	+	0.753080i	$0.728568\pi$
$390$	0	0
$391$	6.19678e11	1.34082
$392$	0	0
$393$	−3.26856e11	−0.691178
$394$	0	0
$395$	7.75909e10	0.160370
$396$	0	0
$397$	−1.18575e11	−0.239572	−0.119786	−	0.992800i	$-0.538221\pi$
$397$	−1.18575e11	−0.239572	−0.119786	+	0.992800i	$0.538221\pi$
$398$	0	0
$399$	−5.00300e10	−0.0988217
$400$	0	0
$401$	−5.27598e11	−1.01895	−0.509475	−	0.860485i	$-0.670161\pi$
$401$	−5.27598e11	−1.01895	−0.509475	+	0.860485i	$0.670161\pi$
$402$	0	0
$403$	2.39339e11	0.452003
$404$	0	0
$405$	−1.34223e10	−0.0247901
$406$	0	0
$407$	−3.11334e11	−0.562407
$408$	0	0
$409$	−8.96872e10	−0.158480	−0.0792402	−	0.996856i	$-0.525249\pi$
$409$	−8.96872e10	−0.158480	−0.0792402	+	0.996856i	$0.525249\pi$
$410$	0	0
$411$	2.12246e11	0.366903
$412$	0	0
$413$	3.89322e11	0.658467
$414$	0	0
$415$	6.36944e9	0.0105411
$416$	0	0
$417$	1.10424e12	1.78835
$418$	0	0
$419$	9.26538e11	1.46859	0.734294	−	0.678831i	$-0.237513\pi$
$419$	9.26538e11	1.46859	0.734294	+	0.678831i	$0.237513\pi$
$420$	0	0
$421$	1.22692e12	1.90348	0.951740	−	0.306905i	$-0.0992934\pi$
$421$	1.22692e12	1.90348	0.951740	+	0.306905i	$0.0992934\pi$
$422$	0	0
$423$	−1.59945e12	−2.42907
$424$	0	0
$425$	−8.92146e11	−1.32643
$426$	0	0
$427$	−2.25521e11	−0.328293
$428$	0	0
$429$	2.24289e11	0.319705
$430$	0	0
$431$	−9.56151e11	−1.33469	−0.667343	−	0.744751i	$-0.732568\pi$
$431$	−9.56151e11	−1.33469	−0.667343	+	0.744751i	$0.732568\pi$
$432$	0	0
$433$	7.42841e10	0.101555	0.0507774	−	0.998710i	$-0.483830\pi$
$433$	7.42841e10	0.101555	0.0507774	+	0.998710i	$0.483830\pi$
$434$	0	0
$435$	−6.69559e11	−0.896577
$436$	0	0
$437$	3.63590e10	0.0476920
$438$	0	0
$439$	−1.66518e11	−0.213979	−0.106989	−	0.994260i	$-0.534121\pi$
$439$	−1.66518e11	−0.213979	−0.106989	+	0.994260i	$0.534121\pi$
$440$	0	0
$441$	−1.00141e10	−0.0126077
$442$	0	0
$443$	6.41581e11	0.791471	0.395735	−	0.918365i	$-0.370490\pi$
$443$	6.41581e11	0.791471	0.395735	+	0.918365i	$0.370490\pi$
$444$	0	0
$445$	−4.07477e11	−0.492587
$446$	0	0
$447$	1.94545e12	2.30482
$448$	0	0
$449$	−2.77233e11	−0.321911	−0.160956	−	0.986962i	$-0.551458\pi$
$449$	−2.77233e11	−0.321911	−0.160956	+	0.986962i	$0.551458\pi$
$450$	0	0
$451$	−5.58291e11	−0.635427
$452$	0	0
$453$	−1.62909e12	−1.81763
$454$	0	0
$455$	−1.36287e11	−0.149074
$456$	0	0
$457$	7.55228e11	0.809944	0.404972	−	0.914329i	$-0.367281\pi$
$457$	7.55228e11	0.809944	0.404972	+	0.914329i	$0.367281\pi$
$458$	0	0
$459$	1.70023e12	1.78793
$460$	0	0
$461$	9.15740e11	0.944318	0.472159	−	0.881514i	$-0.343475\pi$
$461$	9.15740e11	0.944318	0.472159	+	0.881514i	$0.343475\pi$
$462$	0	0
$463$	−6.35894e11	−0.643088	−0.321544	−	0.946895i	$-0.604202\pi$
$463$	−6.35894e11	−0.643088	−0.321544	+	0.946895i	$0.604202\pi$
$464$	0	0
$465$	1.12385e12	1.11474
$466$	0	0
$467$	6.17286e11	0.600566	0.300283	−	0.953850i	$-0.402919\pi$
$467$	6.17286e11	0.600566	0.300283	+	0.953850i	$0.402919\pi$
$468$	0	0
$469$	−1.15007e12	−1.09760
$470$	0	0
$471$	−7.71184e11	−0.722045
$472$	0	0
$473$	7.67618e9	0.00705132
$474$	0	0
$475$	−5.23458e10	−0.0471803
$476$	0	0
$477$	−2.14469e12	−1.89684
$478$	0	0
$479$	2.77942e11	0.241238	0.120619	−	0.992699i	$-0.461512\pi$
$479$	2.77942e11	0.241238	0.120619	+	0.992699i	$0.461512\pi$
$480$	0	0
$481$	−3.30963e11	−0.281921
$482$	0	0
$483$	−1.51281e12	−1.26480
$484$	0	0
$485$	1.72702e11	0.141730
$486$	0	0
$487$	−4.99400e11	−0.402317	−0.201158	−	0.979559i	$-0.564471\pi$
$487$	−4.99400e11	−0.402317	−0.201158	+	0.979559i	$0.564471\pi$
$488$	0	0
$489$	−2.05545e11	−0.162562
$490$	0	0
$491$	2.06241e12	1.60143	0.800715	−	0.599046i	$-0.204453\pi$
$491$	2.06241e12	1.60143	0.800715	+	0.599046i	$0.204453\pi$
$492$	0	0
$493$	2.60593e12	1.98679
$494$	0	0
$495$	6.54409e11	0.489921
$496$	0	0
$497$	−5.75247e11	−0.422912
$498$	0	0
$499$	−1.21912e12	−0.880227	−0.440113	−	0.897942i	$-0.645062\pi$
$499$	−1.21912e12	−0.880227	−0.440113	+	0.897942i	$0.645062\pi$
$500$	0	0
$501$	9.24780e11	0.655796
$502$	0	0
$503$	1.80430e12	1.25676	0.628380	−	0.777906i	$-0.283718\pi$
$503$	1.80430e12	1.25676	0.628380	+	0.777906i	$0.283718\pi$
$504$	0	0
$505$	−1.04266e12	−0.713395
$506$	0	0
$507$	−2.17940e12	−1.46487
$508$	0	0
$509$	−2.03239e11	−0.134208	−0.0671039	−	0.997746i	$-0.521376\pi$
$509$	−2.03239e11	−0.134208	−0.0671039	+	0.997746i	$0.521376\pi$
$510$	0	0
$511$	1.66413e12	1.07967
$512$	0	0
$513$	9.97595e10	0.0635955
$514$	0	0
$515$	−2.51145e11	−0.157323
$516$	0	0
$517$	1.50631e12	0.927272
$518$	0	0
$519$	−2.34293e11	−0.141745
$520$	0	0
$521$	−6.93093e11	−0.412118	−0.206059	−	0.978540i	$-0.566064\pi$
$521$	−6.93093e11	−0.412118	−0.206059	+	0.978540i	$0.566064\pi$
$522$	0	0
$523$	−1.97956e12	−1.15694	−0.578470	−	0.815704i	$-0.696350\pi$
$523$	−1.97956e12	−1.15694	−0.578470	+	0.815704i	$0.696350\pi$
$524$	0	0
$525$	2.17798e12	1.25123
$526$	0	0
$527$	−4.37406e12	−2.47022
$528$	0	0
$529$	−7.01725e11	−0.389598
$530$	0	0
$531$	−1.98728e12	−1.08476
$532$	0	0
$533$	−5.93491e11	−0.318524
$534$	0	0
$535$	1.45000e12	0.765200
$536$	0	0
$537$	3.38516e12	1.75669
$538$	0	0
$539$	9.43090e9	0.00481287
$540$	0	0
$541$	−2.95899e12	−1.48510	−0.742551	−	0.669790i	$-0.766384\pi$
$541$	−2.95899e12	−1.48510	−0.742551	+	0.669790i	$0.766384\pi$
$542$	0	0
$543$	5.77456e12	2.85049
$544$	0	0
$545$	−1.00440e12	−0.487668
$546$	0	0
$547$	3.27526e12	1.56424	0.782118	−	0.623130i	$-0.214139\pi$
$547$	3.27526e12	1.56424	0.782118	+	0.623130i	$0.214139\pi$
$548$	0	0
$549$	1.15116e12	0.540831
$550$	0	0
$551$	1.52901e11	0.0706687
$552$	0	0
$553$	7.37230e11	0.335228
$554$	0	0
$555$	−1.55409e12	−0.695276
$556$	0	0
$557$	−3.76405e12	−1.65694	−0.828470	−	0.560034i	$-0.810788\pi$
$557$	−3.76405e12	−1.65694	−0.828470	+	0.560034i	$0.810788\pi$
$558$	0	0
$559$	8.16017e9	0.00353465
$560$	0	0
$561$	−4.09899e12	−1.74721
$562$	0	0
$563$	2.34987e12	0.985725	0.492863	−	0.870107i	$-0.335951\pi$
$563$	2.34987e12	0.985725	0.492863	+	0.870107i	$0.335951\pi$
$564$	0	0
$565$	9.68061e11	0.399655
$566$	0	0
$567$	−1.27532e11	−0.0518196
$568$	0	0
$569$	2.66701e12	1.06664	0.533322	−	0.845912i	$-0.320943\pi$
$569$	2.66701e12	1.06664	0.533322	+	0.845912i	$0.320943\pi$
$570$	0	0
$571$	−1.72342e12	−0.678469	−0.339234	−	0.940702i	$-0.610168\pi$
$571$	−1.72342e12	−0.678469	−0.339234	+	0.940702i	$0.610168\pi$
$572$	0	0
$573$	−3.68576e12	−1.42834
$574$	0	0
$575$	−1.58284e12	−0.603853
$576$	0	0
$577$	1.55856e12	0.585374	0.292687	−	0.956208i	$-0.405451\pi$
$577$	1.55856e12	0.585374	0.292687	+	0.956208i	$0.405451\pi$
$578$	0	0
$579$	−4.10830e11	−0.151918
$580$	0	0
$581$	6.05192e10	0.0220344
$582$	0	0
$583$	2.01979e12	0.724100
$584$	0	0
$585$	6.95670e11	0.245585
$586$	0	0
$587$	−2.16623e12	−0.753065	−0.376533	−	0.926403i	$-0.622884\pi$
$587$	−2.16623e12	−0.753065	−0.376533	+	0.926403i	$0.622884\pi$
$588$	0	0
$589$	−2.56643e11	−0.0878641
$590$	0	0
$591$	−4.26313e12	−1.43742
$592$	0	0
$593$	3.56244e12	1.18304	0.591522	−	0.806289i	$-0.298527\pi$
$593$	3.56244e12	1.18304	0.591522	+	0.806289i	$0.298527\pi$
$594$	0	0
$595$	2.49071e12	0.814699
$596$	0	0
$597$	6.60949e12	2.12953
$598$	0	0
$599$	1.54407e12	0.490056	0.245028	−	0.969516i	$-0.421203\pi$
$599$	1.54407e12	0.490056	0.245028	+	0.969516i	$0.421203\pi$
$600$	0	0
$601$	−1.05277e12	−0.329155	−0.164577	−	0.986364i	$-0.552626\pi$
$601$	−1.05277e12	−0.329155	−0.164577	+	0.986364i	$0.552626\pi$
$602$	0	0
$603$	5.87046e12	1.80819
$604$	0	0
$605$	9.54092e11	0.289528
$606$	0	0
$607$	−4.47471e12	−1.33787	−0.668937	−	0.743319i	$-0.733251\pi$
$607$	−4.47471e12	−1.33787	−0.668937	+	0.743319i	$0.733251\pi$
$608$	0	0
$609$	−6.36182e12	−1.87415
$610$	0	0
$611$	1.60129e12	0.464818
$612$	0	0
$613$	−6.01862e12	−1.72157	−0.860785	−	0.508969i	$-0.830027\pi$
$613$	−6.01862e12	−1.72157	−0.860785	+	0.508969i	$0.830027\pi$
$614$	0	0
$615$	−2.78683e12	−0.785547
$616$	0	0
$617$	2.16191e12	0.600557	0.300278	−	0.953852i	$-0.402920\pi$
$617$	2.16191e12	0.600557	0.300278	+	0.953852i	$0.402920\pi$
$618$	0	0
$619$	−4.16924e12	−1.14143	−0.570714	−	0.821149i	$-0.693334\pi$
$619$	−4.16924e12	−1.14143	−0.570714	+	0.821149i	$0.693334\pi$
$620$	0	0
$621$	3.01654e12	0.813948
$622$	0	0
$623$	−3.87164e12	−1.02967
$624$	0	0
$625$	1.41248e12	0.370273
$626$	0	0
$627$	−2.40504e11	−0.0621468
$628$	0	0
$629$	6.04853e12	1.54071
$630$	0	0
$631$	4.10037e12	1.02965	0.514826	−	0.857295i	$-0.327857\pi$
$631$	4.10037e12	1.02965	0.514826	+	0.857295i	$0.327857\pi$
$632$	0	0
$633$	−4.51417e12	−1.11753
$634$	0	0
$635$	−1.62290e12	−0.396106
$636$	0	0
$637$	1.00255e10	0.00241257
$638$	0	0
$639$	2.93632e12	0.696706
$640$	0	0
$641$	1.87188e12	0.437942	0.218971	−	0.975731i	$-0.429730\pi$
$641$	1.87188e12	0.437942	0.218971	+	0.975731i	$0.429730\pi$
$642$	0	0
$643$	−1.34166e12	−0.309524	−0.154762	−	0.987952i	$-0.549461\pi$
$643$	−1.34166e12	−0.309524	−0.154762	+	0.987952i	$0.549461\pi$
$644$	0	0
$645$	3.83173e10	0.00871719
$646$	0	0
$647$	−4.94367e12	−1.10912	−0.554562	−	0.832142i	$-0.687114\pi$
$647$	−4.94367e12	−1.10912	−0.554562	+	0.832142i	$0.687114\pi$
$648$	0	0
$649$	1.87155e12	0.414096
$650$	0	0
$651$	1.06783e13	2.33017
$652$	0	0
$653$	2.67139e12	0.574947	0.287474	−	0.957789i	$-0.407185\pi$
$653$	2.67139e12	0.574947	0.287474	+	0.957789i	$0.407185\pi$
$654$	0	0
$655$	9.54763e11	0.202679
$656$	0	0
$657$	−8.49447e12	−1.77866
$658$	0	0
$659$	5.50089e12	1.13618	0.568092	−	0.822965i	$-0.307682\pi$
$659$	5.50089e12	1.13618	0.568092	+	0.822965i	$0.307682\pi$
$660$	0	0
$661$	1.06937e12	0.217881	0.108941	−	0.994048i	$-0.465254\pi$
$661$	1.06937e12	0.217881	0.108941	+	0.994048i	$0.465254\pi$
$662$	0	0
$663$	−4.35744e12	−0.875831
$664$	0	0
$665$	1.46140e11	0.0289783
$666$	0	0
$667$	4.62342e12	0.904476
$668$	0	0
$669$	−1.54759e13	−2.98702
$670$	0	0
$671$	−1.08413e12	−0.206457
$672$	0	0
$673$	−4.96567e12	−0.933062	−0.466531	−	0.884505i	$-0.654496\pi$
$673$	−4.96567e12	−0.933062	−0.466531	+	0.884505i	$0.654496\pi$
$674$	0	0
$675$	−4.34289e12	−0.805214
$676$	0	0
$677$	−2.75739e12	−0.504486	−0.252243	−	0.967664i	$-0.581168\pi$
$677$	−2.75739e12	−0.504486	−0.252243	+	0.967664i	$0.581168\pi$
$678$	0	0
$679$	1.64093e12	0.296262
$680$	0	0
$681$	1.24398e13	2.21642
$682$	0	0
$683$	−5.05528e12	−0.888898	−0.444449	−	0.895804i	$-0.646601\pi$
$683$	−5.05528e12	−0.888898	−0.444449	+	0.895804i	$0.646601\pi$
$684$	0	0
$685$	−6.19982e11	−0.107590
$686$	0	0
$687$	1.05781e13	1.81177
$688$	0	0
$689$	2.14714e12	0.362973
$690$	0	0
$691$	−2.55414e12	−0.426181	−0.213090	−	0.977033i	$-0.568353\pi$
$691$	−2.55414e12	−0.426181	−0.213090	+	0.977033i	$0.568353\pi$
$692$	0	0
$693$	6.21787e12	1.02410
$694$	0	0
$695$	−3.22554e12	−0.524410
$696$	0	0
$697$	1.08464e13	1.74075
$698$	0	0
$699$	−8.92367e10	−0.0141383
$700$	0	0
$701$	−8.11552e12	−1.26936	−0.634681	−	0.772774i	$-0.718868\pi$
$701$	−8.11552e12	−1.26936	−0.634681	+	0.772774i	$0.718868\pi$
$702$	0	0
$703$	3.54892e11	0.0548020
$704$	0	0
$705$	7.51908e12	1.14634
$706$	0	0
$707$	−9.90680e12	−1.49123
$708$	0	0
$709$	−2.04394e12	−0.303781	−0.151890	−	0.988397i	$-0.548536\pi$
$709$	−2.04394e12	−0.303781	−0.151890	+	0.988397i	$0.548536\pi$
$710$	0	0
$711$	−3.76316e12	−0.552254
$712$	0	0
$713$	−7.76041e12	−1.12456
$714$	0	0
$715$	−6.55159e11	−0.0937496
$716$	0	0
$717$	2.06691e13	2.92069
$718$	0	0
$719$	1.24231e13	1.73361	0.866804	−	0.498648i	$-0.166170\pi$
$719$	1.24231e13	1.73361	0.866804	+	0.498648i	$0.166170\pi$
$720$	0	0
$721$	−2.38626e12	−0.328858
$722$	0	0
$723$	−1.54507e13	−2.10294
$724$	0	0
$725$	−6.65630e12	−0.894771
$726$	0	0
$727$	−5.37434e12	−0.713543	−0.356771	−	0.934192i	$-0.616122\pi$
$727$	−5.37434e12	−0.713543	−0.356771	+	0.934192i	$0.616122\pi$
$728$	0	0
$729$	−1.22598e13	−1.60771
$730$	0	0
$731$	−1.49131e11	−0.0193171
$732$	0	0
$733$	1.28618e11	0.0164563	0.00822815	−	0.999966i	$-0.497381\pi$
$733$	1.28618e11	0.0164563	0.00822815	+	0.999966i	$0.497381\pi$
$734$	0	0
$735$	4.70764e10	0.00594990
$736$	0	0
$737$	−5.52860e12	−0.690258
$738$	0	0
$739$	1.36726e13	1.68636	0.843181	−	0.537630i	$-0.180680\pi$
$739$	1.36726e13	1.68636	0.843181	+	0.537630i	$0.180680\pi$
$740$	0	0
$741$	−2.55668e11	−0.0311527
$742$	0	0
$743$	1.31581e13	1.58396	0.791981	−	0.610546i	$-0.209050\pi$
$743$	1.31581e13	1.58396	0.791981	+	0.610546i	$0.209050\pi$
$744$	0	0
$745$	−5.68277e12	−0.675861
$746$	0	0
$747$	−3.08918e11	−0.0362995
$748$	0	0
$749$	1.37771e13	1.59952
$750$	0	0
$751$	−2.08682e12	−0.239389	−0.119695	−	0.992811i	$-0.538192\pi$
$751$	−2.08682e12	−0.239389	−0.119695	+	0.992811i	$0.538192\pi$
$752$	0	0
$753$	1.24753e13	1.41408
$754$	0	0
$755$	4.75867e12	0.532997
$756$	0	0
$757$	5.54660e12	0.613897	0.306948	−	0.951726i	$-0.400692\pi$
$757$	5.54660e12	0.613897	0.306948	+	0.951726i	$0.400692\pi$
$758$	0	0
$759$	−7.27239e12	−0.795407
$760$	0	0
$761$	−1.13451e12	−0.122625	−0.0613123	−	0.998119i	$-0.519529\pi$
$761$	−1.13451e12	−0.122625	−0.0613123	+	0.998119i	$0.519529\pi$
$762$	0	0
$763$	−9.54335e12	−1.01939
$764$	0	0
$765$	−1.27137e13	−1.34214
$766$	0	0
$767$	1.98955e12	0.207576
$768$	0	0
$769$	−2.61602e12	−0.269757	−0.134878	−	0.990862i	$-0.543064\pi$
$769$	−2.61602e12	−0.269757	−0.134878	+	0.990862i	$0.543064\pi$
$770$	0	0
$771$	−7.77252e12	−0.792167
$772$	0	0
$773$	−5.33154e10	−0.00537088	−0.00268544	−	0.999996i	$-0.500855\pi$
$773$	−5.33154e10	−0.00537088	−0.00268544	+	0.999996i	$0.500855\pi$
$774$	0	0
$775$	1.11726e13	1.11249
$776$	0	0
$777$	−1.47662e13	−1.45336
$778$	0	0
$779$	6.36400e11	0.0619172
$780$	0	0
$781$	−2.76533e12	−0.265961
$782$	0	0
$783$	1.26854e13	1.20608
$784$	0	0
$785$	2.25267e12	0.211731
$786$	0	0
$787$	−3.30783e12	−0.307367	−0.153683	−	0.988120i	$-0.549114\pi$
$787$	−3.30783e12	−0.307367	−0.153683	+	0.988120i	$0.549114\pi$
$788$	0	0
$789$	−1.63558e13	−1.50254
$790$	0	0
$791$	9.19804e12	0.835412
$792$	0	0
$793$	−1.15248e12	−0.103492
$794$	0	0
$795$	1.00822e13	0.895169
$796$	0	0
$797$	−3.86873e12	−0.339630	−0.169815	−	0.985476i	$-0.554317\pi$
$797$	−3.86873e12	−0.339630	−0.169815	+	0.985476i	$0.554317\pi$
$798$	0	0
$799$	−2.92643e13	−2.54026
$800$	0	0
$801$	1.97626e13	1.69628
$802$	0	0
$803$	7.99981e12	0.678984
$804$	0	0
$805$	4.41900e12	0.370888
$806$	0	0
$807$	5.28071e11	0.0438290
$808$	0	0
$809$	7.39526e12	0.606995	0.303497	−	0.952832i	$-0.401846\pi$
$809$	7.39526e12	0.606995	0.303497	+	0.952832i	$0.401846\pi$
$810$	0	0
$811$	8.92803e12	0.724706	0.362353	−	0.932041i	$-0.381974\pi$
$811$	8.92803e12	0.724706	0.362353	+	0.932041i	$0.381974\pi$
$812$	0	0
$813$	1.83463e13	1.47279
$814$	0	0
$815$	6.00409e11	0.0476692
$816$	0	0
$817$	−8.75014e9	−0.000687093 0
$818$	0	0
$819$	6.60991e12	0.513355
$820$	0	0
$821$	−1.05534e13	−0.810674	−0.405337	−	0.914167i	$-0.632846\pi$
$821$	−1.05534e13	−0.810674	−0.405337	+	0.914167i	$0.632846\pi$
$822$	0	0
$823$	−9.16030e12	−0.696002	−0.348001	−	0.937494i	$-0.613139\pi$
$823$	−9.16030e12	−0.696002	−0.348001	+	0.937494i	$0.613139\pi$
$824$	0	0
$825$	1.04700e13	0.786872
$826$	0	0
$827$	−2.44096e13	−1.81462	−0.907310	−	0.420462i	$-0.861868\pi$
$827$	−2.44096e13	−1.81462	−0.907310	+	0.420462i	$0.861868\pi$
$828$	0	0
$829$	9.12051e12	0.670693	0.335346	−	0.942095i	$-0.391147\pi$
$829$	9.12051e12	0.670693	0.335346	+	0.942095i	$0.391147\pi$
$830$	0	0
$831$	−3.77668e13	−2.74730
$832$	0	0
$833$	−1.83222e11	−0.0131848
$834$	0	0
$835$	−2.70133e12	−0.192304
$836$	0	0
$837$	−2.12925e13	−1.49955
$838$	0	0
$839$	6.07575e12	0.423322	0.211661	−	0.977343i	$-0.432113\pi$
$839$	6.07575e12	0.423322	0.211661	+	0.977343i	$0.432113\pi$
$840$	0	0
$841$	4.93572e12	0.340226
$842$	0	0
$843$	5.86364e12	0.399893
$844$	0	0
$845$	6.36613e12	0.429556
$846$	0	0
$847$	9.06531e12	0.605212
$848$	0	0
$849$	9.87527e12	0.652325
$850$	0	0
$851$	1.07312e13	0.701402
$852$	0	0
$853$	1.67917e13	1.08599	0.542993	−	0.839737i	$-0.317291\pi$
$853$	1.67917e13	1.08599	0.542993	+	0.839737i	$0.317291\pi$
$854$	0	0
$855$	−7.45966e11	−0.0477388
$856$	0	0
$857$	−2.77707e13	−1.75862	−0.879312	−	0.476246i	$-0.841997\pi$
$857$	−2.77707e13	−1.75862	−0.879312	+	0.476246i	$0.841997\pi$
$858$	0	0
$859$	1.85405e12	0.116186	0.0580928	−	0.998311i	$-0.481498\pi$
$859$	1.85405e12	0.116186	0.0580928	+	0.998311i	$0.481498\pi$
$860$	0	0
$861$	−2.64790e13	−1.64206
$862$	0	0
$863$	8.72142e12	0.535228	0.267614	−	0.963526i	$-0.413765\pi$
$863$	8.72142e12	0.535228	0.267614	+	0.963526i	$0.413765\pi$
$864$	0	0
$865$	6.84383e11	0.0415649
$866$	0	0
$867$	5.25964e13	3.16133
$868$	0	0
$869$	3.54402e12	0.210818
$870$	0	0
$871$	−5.87718e12	−0.346009
$872$	0	0
$873$	−8.37606e12	−0.488063
$874$	0	0
$875$	−1.45933e13	−0.841625
$876$	0	0
$877$	2.76222e13	1.57674	0.788369	−	0.615203i	$-0.210926\pi$
$877$	2.76222e13	1.57674	0.788369	+	0.615203i	$0.210926\pi$
$878$	0	0
$879$	−1.10232e13	−0.622813
$880$	0	0
$881$	−1.00186e13	−0.560295	−0.280147	−	0.959957i	$-0.590383\pi$
$881$	−1.00186e13	−0.560295	−0.280147	+	0.959957i	$0.590383\pi$
$882$	0	0
$883$	9.43702e12	0.522410	0.261205	−	0.965283i	$-0.415880\pi$
$883$	9.43702e12	0.522410	0.261205	+	0.965283i	$0.415880\pi$
$884$	0	0
$885$	9.34226e12	0.511926
$886$	0	0
$887$	−3.75635e12	−0.203756	−0.101878	−	0.994797i	$-0.532485\pi$
$887$	−3.75635e12	−0.203756	−0.101878	+	0.994797i	$0.532485\pi$
$888$	0	0
$889$	−1.54200e13	−0.827994
$890$	0	0
$891$	−6.13070e11	−0.0325882
$892$	0	0
$893$	−1.71706e12	−0.0903552
$894$	0	0
$895$	−9.88824e12	−0.515128
$896$	0	0
$897$	−7.73092e12	−0.398718
$898$	0	0
$899$	−3.26348e13	−1.66634
$900$	0	0
$901$	−3.92402e13	−1.98367
$902$	0	0
$903$	3.64072e11	0.0182219
$904$	0	0
$905$	−1.68678e13	−0.835871
$906$	0	0
$907$	2.36795e13	1.16182	0.580911	−	0.813967i	$-0.302696\pi$
$907$	2.36795e13	1.16182	0.580911	+	0.813967i	$0.302696\pi$
$908$	0	0
$909$	5.05688e13	2.45666
$910$	0	0
$911$	1.90030e13	0.914090	0.457045	−	0.889443i	$-0.348908\pi$
$911$	1.90030e13	0.914090	0.457045	+	0.889443i	$0.348908\pi$
$912$	0	0
$913$	2.90928e11	0.0138570
$914$	0	0
$915$	−5.41165e12	−0.255232
$916$	0	0
$917$	9.07168e12	0.423668
$918$	0	0
$919$	5.56992e12	0.257590	0.128795	−	0.991671i	$-0.458889\pi$
$919$	5.56992e12	0.257590	0.128795	+	0.991671i	$0.458889\pi$
$920$	0	0
$921$	3.14573e13	1.44063
$922$	0	0
$923$	−2.93968e12	−0.133319
$924$	0	0
$925$	−1.54497e13	−0.693876
$926$	0	0
$927$	1.21805e13	0.541761
$928$	0	0
$929$	3.54293e13	1.56060	0.780301	−	0.625404i	$-0.215066\pi$
$929$	3.54293e13	1.56060	0.780301	+	0.625404i	$0.215066\pi$
$930$	0	0
$931$	−1.07504e10	−0.000468975 0
$932$	0	0
$933$	−4.66149e13	−2.01399
$934$	0	0
$935$	1.19734e13	0.512347
$936$	0	0
$937$	−2.45592e13	−1.04084	−0.520422	−	0.853909i	$-0.674225\pi$
$937$	−2.45592e13	−1.04084	−0.520422	+	0.853909i	$0.674225\pi$
$938$	0	0
$939$	−3.97140e13	−1.66705
$940$	0	0
$941$	2.00516e13	0.833675	0.416838	−	0.908981i	$-0.363138\pi$
$941$	2.00516e13	0.833675	0.416838	+	0.908981i	$0.363138\pi$
$942$	0	0
$943$	1.92435e13	0.792468
$944$	0	0
$945$	1.21246e13	0.494564
$946$	0	0
$947$	−1.51295e13	−0.611294	−0.305647	−	0.952145i	$-0.598873\pi$
$947$	−1.51295e13	−0.611294	−0.305647	+	0.952145i	$0.598873\pi$
$948$	0	0
$949$	8.50420e12	0.340358
$950$	0	0
$951$	−1.92083e13	−0.761510
$952$	0	0
$953$	2.18751e13	0.859075	0.429538	−	0.903049i	$-0.358677\pi$
$953$	2.18751e13	0.859075	0.429538	+	0.903049i	$0.358677\pi$
$954$	0	0
$955$	1.07663e13	0.418843
$956$	0	0
$957$	−3.05826e13	−1.17861
$958$	0	0
$959$	−5.89076e12	−0.224899
$960$	0	0
$961$	2.83379e13	1.07180
$962$	0	0
$963$	−7.03248e13	−2.63506
$964$	0	0
$965$	1.20006e12	0.0445480
$966$	0	0
$967$	−3.32239e13	−1.22189	−0.610945	−	0.791673i	$-0.709210\pi$
$967$	−3.32239e13	−1.22189	−0.610945	+	0.791673i	$0.709210\pi$
$968$	0	0
$969$	4.67247e12	0.170251
$970$	0	0
$971$	3.71430e13	1.34088	0.670441	−	0.741963i	$-0.266105\pi$
$971$	3.71430e13	1.34088	0.670441	+	0.741963i	$0.266105\pi$
$972$	0	0
$973$	−3.06475e13	−1.09619
$974$	0	0
$975$	1.11301e13	0.394439
$976$	0	0
$977$	1.46872e13	0.515719	0.257860	−	0.966182i	$-0.416983\pi$
$977$	1.46872e13	0.515719	0.257860	+	0.966182i	$0.416983\pi$
$978$	0	0
$979$	−1.86118e13	−0.647538
$980$	0	0
$981$	4.87136e13	1.67934
$982$	0	0
$983$	2.18746e13	0.747220	0.373610	−	0.927586i	$-0.378120\pi$
$983$	2.18746e13	0.747220	0.373610	+	0.927586i	$0.378120\pi$
$984$	0	0
$985$	1.24528e13	0.421507
$986$	0	0
$987$	7.14425e13	2.39624
$988$	0	0
$989$	−2.64588e11	−0.00879399
$990$	0	0
$991$	4.20089e13	1.38360	0.691799	−	0.722091i	$-0.256819\pi$
$991$	4.20089e13	1.38360	0.691799	+	0.722091i	$0.256819\pi$
$992$	0	0
$993$	−6.58411e13	−2.14895
$994$	0	0
$995$	−1.93067e13	−0.624458
$996$	0	0
$997$	2.69704e12	0.0864489	0.0432245	−	0.999065i	$-0.486237\pi$
$997$	2.69704e12	0.0864489	0.0432245	+	0.999065i	$0.486237\pi$
$998$	0	0
$999$	2.94437e13	0.935292

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4.10.a.a.1.1	✓	1
3.2	odd	2		36.10.a.b.1.1		1
4.3	odd	2		16.10.a.a.1.1		1
5.2	odd	4		100.10.c.a.49.1		2
5.3	odd	4		100.10.c.a.49.2		2
5.4	even	2		100.10.a.a.1.1		1
7.2	even	3		196.10.e.a.165.1		2
7.3	odd	6		196.10.e.b.177.1		2
7.4	even	3		196.10.e.a.177.1		2
7.5	odd	6		196.10.e.b.165.1		2
7.6	odd	2		196.10.a.a.1.1		1
8.3	odd	2		64.10.a.i.1.1		1
8.5	even	2		64.10.a.a.1.1		1
9.2	odd	6		324.10.e.b.109.1		2
9.4	even	3		324.10.e.e.217.1		2
9.5	odd	6		324.10.e.b.217.1		2
9.7	even	3		324.10.e.e.109.1		2
12.11	even	2		144.10.a.j.1.1		1
16.3	odd	4		256.10.b.b.129.1		2
16.5	even	4		256.10.b.j.129.1		2
16.11	odd	4		256.10.b.b.129.2		2
16.13	even	4		256.10.b.j.129.2		2
20.3	even	4		400.10.c.a.49.1		2
20.7	even	4		400.10.c.a.49.2		2
20.19	odd	2		400.10.a.k.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4.10.a.a.1.1	✓	1	1.1	even	1	trivial
16.10.a.a.1.1		1	4.3	odd	2
36.10.a.b.1.1		1	3.2	odd	2
64.10.a.a.1.1		1	8.5	even	2
64.10.a.i.1.1		1	8.3	odd	2
100.10.a.a.1.1		1	5.4	even	2
100.10.c.a.49.1		2	5.2	odd	4
100.10.c.a.49.2		2	5.3	odd	4
144.10.a.j.1.1		1	12.11	even	2
196.10.a.a.1.1		1	7.6	odd	2
196.10.e.a.165.1		2	7.2	even	3
196.10.e.a.177.1		2	7.4	even	3
196.10.e.b.165.1		2	7.5	odd	6
196.10.e.b.177.1		2	7.3	odd	6
256.10.b.b.129.1		2	16.3	odd	4
256.10.b.b.129.2		2	16.11	odd	4
256.10.b.j.129.1		2	16.5	even	4
256.10.b.j.129.2		2	16.13	even	4
324.10.e.b.109.1		2	9.2	odd	6
324.10.e.b.217.1		2	9.5	odd	6
324.10.e.e.109.1		2	9.7	even	3
324.10.e.e.217.1		2	9.4	even	3
400.10.a.k.1.1		1	20.19	odd	2
400.10.c.a.49.1		2	20.3	even	4
400.10.c.a.49.2		2	20.7	even	4

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:	\( N \)	\(=\)	\( 4 = 2^{2} \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	4.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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