LMFDB - Embedded newform 150.12.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [150,12,Mod(1,150)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(150, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("150.1");

S:= CuspForms(chi, 12);

N := Newforms(S);

Level:	$ N $	$=$	$ 150 = 2 \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 12 $
Character orbit:	$[\chi]$	$=$	150.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:	$115.251477084$
Analytic rank:	$0$
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	$\chi$	$=$	150.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

\(q-32.0000 q^{2} -243.000 q^{3} +1024.00 q^{4} +7776.00 q^{6} -32936.0 q^{7} -32768.0 q^{8} +59049.0 q^{9} -758748. q^{11} -248832. q^{12} +2.48286e6 q^{13} +1.05395e6 q^{14} +1.04858e6 q^{16} -8.29039e6 q^{17} -1.88957e6 q^{18} -1.08673e7 q^{19} +8.00345e6 q^{21} +2.42799e7 q^{22} -2.05393e7 q^{23} +7.96262e6 q^{24} -7.94515e7 q^{26} -1.43489e7 q^{27} -3.37265e7 q^{28} +2.88146e7 q^{29} +1.50501e8 q^{31} -3.35544e7 q^{32} +1.84376e8 q^{33} +2.65292e8 q^{34} +6.04662e7 q^{36} +3.19892e8 q^{37} +3.47754e8 q^{38} -6.03334e8 q^{39} -3.68009e8 q^{41} -2.56110e8 q^{42} -6.20470e8 q^{43} -7.76958e8 q^{44} +6.57257e8 q^{46} -2.76311e9 q^{47} -2.54804e8 q^{48} -8.92547e8 q^{49} +2.01456e9 q^{51} +2.54245e9 q^{52} +2.68284e8 q^{53} +4.59165e8 q^{54} +1.07925e9 q^{56} +2.64075e9 q^{57} -9.22066e8 q^{58} +1.67289e9 q^{59} -7.78720e9 q^{61} -4.81604e9 q^{62} -1.94484e9 q^{63} +1.07374e9 q^{64} -5.90002e9 q^{66} -1.87067e10 q^{67} -8.48936e9 q^{68} +4.99104e9 q^{69} -8.34699e9 q^{71} -1.93492e9 q^{72} -1.96417e10 q^{73} -1.02365e10 q^{74} -1.11281e10 q^{76} +2.49901e10 q^{77} +1.93067e10 q^{78} -5.87381e9 q^{79} +3.48678e9 q^{81} +1.17763e10 q^{82} -8.49256e9 q^{83} +8.19553e9 q^{84} +1.98550e10 q^{86} -7.00194e9 q^{87} +2.48627e10 q^{88} +7.55279e10 q^{89} -8.17754e10 q^{91} -2.10322e10 q^{92} -3.65718e10 q^{93} +8.84195e10 q^{94} +8.15373e9 q^{96} +8.23568e10 q^{97} +2.85615e10 q^{98} -4.48033e10 q^{99} +O(q^{100})\)

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 $

$2$	−32.0000	−0.707107
$2$	−32.0000	−0.707107
$3$	−243.000	−0.577350
$3$	−243.000	−0.577350
$4$	1024.00	0.500000
$5$	0	0
$5$	0	0
$6$	7776.00	0.408248
$7$	−32936.0	−0.740682	−0.370341	−	0.928896i	$-0.620759\pi$
$7$	−32936.0	−0.740682	−0.370341	+	0.928896i	$0.620759\pi$
$8$	−32768.0	−0.353553
$9$	59049.0	0.333333
$10$	0	0
$11$	−758748.	−1.42049	−0.710244	−	0.703955i	$-0.751416\pi$
$11$	−758748.	−1.42049	−0.710244	+	0.703955i	$0.751416\pi$
$12$	−248832.	−0.288675
$13$	2.48286e6	1.85466	0.927328	−	0.374249i	$-0.122100\pi$
$13$	2.48286e6	1.85466	0.927328	+	0.374249i	$0.122100\pi$
$14$	1.05395e6	0.523741
$15$	0	0
$16$	1.04858e6	0.250000
$17$	−8.29039e6	−1.41614	−0.708069	−	0.706143i	$-0.750434\pi$
$17$	−8.29039e6	−1.41614	−0.708069	+	0.706143i	$0.750434\pi$
$18$	−1.88957e6	−0.235702
$19$	−1.08673e7	−1.00688	−0.503439	−	0.864031i	$-0.667932\pi$
$19$	−1.08673e7	−1.00688	−0.503439	+	0.864031i	$0.667932\pi$
$20$	0	0
$21$	8.00345e6	0.427633
$22$	2.42799e7	1.00444
$23$	−2.05393e7	−0.665399	−0.332699	−	0.943033i	$-0.607959\pi$
$23$	−2.05393e7	−0.665399	−0.332699	+	0.943033i	$0.607959\pi$
$24$	7.96262e6	0.204124
$25$	0	0
$26$	−7.94515e7	−1.31144
$27$	−1.43489e7	−0.192450
$28$	−3.37265e7	−0.370341
$29$	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	$-0.343501\pi$
$31$	1.50501e8	0.944172	0.472086	+	0.881552i	$0.343501\pi$
$32$	−3.35544e7	−0.176777
$33$	1.84376e8	0.820120
$34$	2.65292e8	1.00136
$35$	0	0
$36$	6.04662e7	0.166667
$37$	3.19892e8	0.758392	0.379196	−	0.925316i	$-0.376201\pi$
$37$	3.19892e8	0.758392	0.379196	+	0.925316i	$0.376201\pi$
$38$	3.47754e8	0.711970
$39$	−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	$0.579786\pi$
$42$	−2.56110e8	−0.302382
$43$	−6.20470e8	−0.643641	−0.321821	−	0.946801i	$-0.604295\pi$
$43$	−6.20470e8	−0.643641	−0.321821	+	0.946801i	$0.604295\pi$
$44$	−7.76958e8	−0.710244
$45$	0	0
$46$	6.57257e8	0.470508
$47$	−2.76311e9	−1.75736	−0.878679	−	0.477414i	$-0.841574\pi$
$47$	−2.76311e9	−1.75736	−0.878679	+	0.477414i	$0.841574\pi$
$48$	−2.54804e8	−0.144338
$49$	−8.92547e8	−0.451391
$50$	0	0
$51$	2.01456e9	0.817608
$52$	2.54245e9	0.927328
$53$	2.68284e8	0.0881207	0.0440603	−	0.999029i	$-0.485971\pi$
$53$	2.68284e8	0.0881207	0.0440603	+	0.999029i	$0.485971\pi$
$54$	4.59165e8	0.136083
$55$	0	0
$56$	1.07925e9	0.261871
$57$	2.64075e9	0.581321
$58$	−9.22066e8	−0.184462
$59$	1.67289e9	0.304637	0.152318	−	0.988331i	$-0.451326\pi$
$59$	1.67289e9	0.304637	0.152318	+	0.988331i	$0.451326\pi$
$60$	0	0
$61$	−7.78720e9	−1.18050	−0.590252	−	0.807219i	$-0.700971\pi$
$61$	−7.78720e9	−1.18050	−0.590252	+	0.807219i	$0.700971\pi$
$62$	−4.81604e9	−0.667631
$63$	−1.94484e9	−0.246894
$64$	1.07374e9	0.125000
$65$	0	0
$66$	−5.90002e9	−0.579912
$67$	−1.87067e10	−1.69272	−0.846361	−	0.532610i	$-0.821211\pi$
$67$	−1.87067e10	−1.69272	−0.846361	+	0.532610i	$0.821211\pi$
$68$	−8.48936e9	−0.708069
$69$	4.99104e9	0.384168
$70$	0	0
$71$	−8.34699e9	−0.549046	−0.274523	−	0.961580i	$-0.588520\pi$
$71$	−8.34699e9	−0.549046	−0.274523	+	0.961580i	$0.588520\pi$
$72$	−1.93492e9	−0.117851
$73$	−1.96417e10	−1.10893	−0.554465	−	0.832207i	$-0.687077\pi$
$73$	−1.96417e10	−1.10893	−0.554465	+	0.832207i	$0.687077\pi$
$74$	−1.02365e10	−0.536264
$75$	0	0
$76$	−1.11281e10	−0.503439
$77$	2.49901e10	1.05213
$78$	1.93067e10	0.757160
$79$	−5.87381e9	−0.214769	−0.107384	−	0.994218i	$-0.534248\pi$
$79$	−5.87381e9	−0.214769	−0.107384	+	0.994218i	$0.534248\pi$
$80$	0	0
$81$	3.48678e9	0.111111
$82$	1.17763e10	0.350778
$83$	−8.49256e9	−0.236651	−0.118326	−	0.992975i	$-0.537753\pi$
$83$	−8.49256e9	−0.236651	−0.118326	+	0.992975i	$0.537753\pi$
$84$	8.19553e9	0.213816
$85$	0	0
$86$	1.98550e10	0.455123
$87$	−7.00194e9	−0.150613
$88$	2.48627e10	0.502219
$89$	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
$92$	−2.10322e10	−0.332699
$93$	−3.65718e10	−0.545118
$94$	8.84195e10	1.24264
$95$	0	0
$96$	8.15373e9	0.102062
$97$	8.23568e10	0.973767	0.486883	−	0.873467i	$-0.338134\pi$
$97$	8.23568e10	0.973767	0.486883	+	0.873467i	$0.338134\pi$
$98$	2.85615e10	0.319181
$99$	−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	$0.562655\pi$
$102$	−6.44660e10	−0.578136
$103$	6.45183e10	0.548376	0.274188	−	0.961676i	$-0.411591\pi$
$103$	6.45183e10	0.548376	0.274188	+	0.961676i	$0.411591\pi$
$104$	−8.13583e10	−0.655720
$105$	0	0
$106$	−8.58510e9	−0.0623107
$107$	−1.10219e11	−0.759706	−0.379853	−	0.925047i	$-0.624025\pi$
$107$	−1.10219e11	−0.759706	−0.379853	+	0.925047i	$0.624025\pi$
$108$	−1.46933e10	−0.0962250
$109$	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
$112$	−3.45359e10	−0.185170
$113$	−7.89206e10	−0.402957	−0.201479	−	0.979493i	$-0.564575\pi$
$113$	−7.89206e10	−0.402957	−0.201479	+	0.979493i	$0.564575\pi$
$114$	−8.45041e10	−0.411056
$115$	0	0
$116$	2.95061e10	0.130435
$117$	1.46610e11	0.618219
$118$	−5.35326e10	−0.215411
$119$	2.73052e11	1.04891
$120$	0	0
$121$	2.90387e11	1.01779
$122$	2.49190e11	0.834742
$123$	8.94262e10	0.286409
$124$	1.54113e11	0.472086
$125$	0	0
$126$	6.22348e10	0.174580
$127$	2.76649e11	0.743035	0.371518	−	0.928426i	$-0.378838\pi$
$127$	2.76649e11	0.743035	0.371518	+	0.928426i	$0.378838\pi$
$128$	−3.43597e10	−0.0883883
$129$	1.50774e11	0.371607
$130$	0	0
$131$	−3.40872e11	−0.771968	−0.385984	−	0.922505i	$-0.626138\pi$
$131$	−3.40872e11	−0.771968	−0.385984	+	0.922505i	$0.626138\pi$
$132$	1.88801e11	0.410060
$133$	3.57925e11	0.745776
$134$	5.98614e11	1.19694
$135$	0	0
$136$	2.71659e11	0.500681
$137$	9.44784e11	1.67251	0.836256	−	0.548339i	$-0.184740\pi$
$137$	9.44784e11	1.67251	0.836256	+	0.548339i	$0.184740\pi$
$138$	−1.59713e11	−0.271648
$139$	1.03770e11	0.169626	0.0848128	−	0.996397i	$-0.472971\pi$
$139$	1.03770e11	0.169626	0.0848128	+	0.996397i	$0.472971\pi$
$140$	0	0
$141$	6.71436e11	1.01461
$142$	2.67104e11	0.388234
$143$	−1.88386e12	−2.63452
$144$	6.19174e10	0.0833333
$145$	0	0
$146$	6.28536e11	0.784132
$147$	2.16889e11	0.260610
$148$	3.27569e11	0.379196
$149$	−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	$-0.464875\pi$
$151$	2.12466e11	0.220250	0.110125	+	0.993918i	$0.464875\pi$
$152$	3.56100e11	0.355985
$153$	−4.89539e11	−0.472046
$154$	−7.99684e11	−0.743968
$155$	0	0
$156$	−6.17815e11	−0.535393
$157$	1.58632e12	1.32722	0.663612	−	0.748077i	$-0.269023\pi$
$157$	1.58632e12	1.32722	0.663612	+	0.748077i	$0.269023\pi$
$158$	1.87962e11	0.151864
$159$	−6.51931e10	−0.0508765
$160$	0	0
$161$	6.76481e11	0.492849
$162$	−1.11577e11	−0.0785674
$163$	−8.65020e11	−0.588836	−0.294418	−	0.955677i	$-0.595126\pi$
$163$	−8.65020e11	−0.588836	−0.294418	+	0.955677i	$0.595126\pi$
$164$	−3.76841e11	−0.248037
$165$	0	0
$166$	2.71762e11	0.167338
$167$	2.25802e12	1.34520	0.672600	−	0.740006i	$-0.265177\pi$
$167$	2.25802e12	1.34520	0.672600	+	0.740006i	$0.265177\pi$
$168$	−2.62257e11	−0.151191
$169$	4.37242e12	2.43975
$170$	0	0
$171$	−6.41703e11	−0.335626
$172$	−6.35361e11	−0.321821
$173$	2.15960e12	1.05955	0.529773	−	0.848139i	$-0.322277\pi$
$173$	2.15960e12	1.05955	0.529773	+	0.848139i	$0.322277\pi$
$174$	2.24062e11	0.106499
$175$	0	0
$176$	−7.95605e11	−0.355122
$177$	−4.06513e11	−0.175882
$178$	−2.41689e12	−1.01379
$179$	4.50786e12	1.83349	0.916745	−	0.399472i	$-0.130807\pi$
$179$	4.50786e12	1.83349	0.916745	+	0.399472i	$0.130807\pi$
$180$	0	0
$181$	4.34053e11	0.166077	0.0830387	−	0.996546i	$-0.473537\pi$
$181$	4.34053e11	0.166077	0.0830387	+	0.996546i	$0.473537\pi$
$182$	2.61681e12	0.971360
$183$	1.89229e12	0.681564
$184$	6.73031e11	0.235254
$185$	0	0
$186$	1.17030e12	0.385457
$187$	6.29031e12	2.01161
$188$	−2.82942e12	−0.878679
$189$	4.72596e11	0.142544
$190$	0	0
$191$	8.67702e11	0.246994	0.123497	−	0.992345i	$-0.460589\pi$
$191$	8.67702e11	0.246994	0.123497	+	0.992345i	$0.460589\pi$
$192$	−2.60919e11	−0.0721688
$193$	−1.86946e11	−0.0502516	−0.0251258	−	0.999684i	$-0.507999\pi$
$193$	−1.86946e11	−0.0502516	−0.0251258	+	0.999684i	$0.507999\pi$
$194$	−2.63542e12	−0.688557
$195$	0	0
$196$	−9.13968e11	−0.225695
$197$	−3.32798e12	−0.799129	−0.399565	−	0.916705i	$-0.630839\pi$
$197$	−3.32798e12	−0.799129	−0.399565	+	0.916705i	$0.630839\pi$
$198$	1.43371e12	0.334812
$199$	−1.58581e12	−0.360212	−0.180106	−	0.983647i	$-0.557644\pi$
$199$	−1.58581e12	−0.360212	−0.180106	+	0.983647i	$0.557644\pi$
$200$	0	0
$201$	4.54573e12	0.977293
$202$	1.32205e12	0.276577
$203$	−9.49036e11	−0.193221
$204$	2.06291e12	0.408804
$205$	0	0
$206$	−2.06459e12	−0.387760
$207$	−1.21282e12	−0.221800
$208$	2.60347e12	0.463664
$209$	8.24554e12	1.43026
$210$	0	0
$211$	3.54958e12	0.584283	0.292142	−	0.956375i	$-0.405632\pi$
$211$	3.54958e12	0.584283	0.292142	+	0.956375i	$0.405632\pi$
$212$	2.74723e11	0.0440603
$213$	2.02832e12	0.316992
$214$	3.52701e12	0.537193
$215$	0	0
$216$	4.70185e11	0.0680414
$217$	−4.95691e12	−0.699331
$218$	−5.68890e12	−0.782560
$219$	4.77294e12	0.640241
$220$	0	0
$221$	−2.05839e13	−2.62645
$222$	2.48748e12	0.309612
$223$	5.18022e12	0.629030	0.314515	−	0.949252i	$-0.398158\pi$
$223$	5.18022e12	0.629030	0.314515	+	0.949252i	$0.398158\pi$
$224$	1.10515e12	0.130935
$225$	0	0
$226$	2.52546e12	0.284934
$227$	1.33665e13	1.47189	0.735943	−	0.677044i	$-0.236739\pi$
$227$	1.33665e13	1.47189	0.735943	+	0.677044i	$0.236739\pi$
$228$	2.70413e12	0.290661
$229$	−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
$232$	−9.44195e11	−0.0922312
$233$	−7.13322e12	−0.680500	−0.340250	−	0.940335i	$-0.610512\pi$
$233$	−7.13322e12	−0.680500	−0.340250	+	0.940335i	$0.610512\pi$
$234$	−4.69153e12	−0.437147
$235$	0	0
$236$	1.71304e12	0.152318
$237$	1.42734e12	0.123997
$238$	−8.73767e12	−0.741690
$239$	3.30363e12	0.274033	0.137016	−	0.990569i	$-0.456249\pi$
$239$	3.30363e12	0.274033	0.137016	+	0.990569i	$0.456249\pi$
$240$	0	0
$241$	4.16938e11	0.0330353	0.0165176	−	0.999864i	$-0.494742\pi$
$241$	4.16938e11	0.0330353	0.0165176	+	0.999864i	$0.494742\pi$
$242$	−9.29238e12	−0.719685
$243$	−8.47289e11	−0.0641500
$244$	−7.97409e12	−0.590252
$245$	0	0
$246$	−2.86164e12	−0.202522
$247$	−2.69820e13	−1.86741
$248$	−4.93163e12	−0.333815
$249$	2.06369e12	0.136631
$250$	0	0
$251$	−1.91970e12	−0.121626	−0.0608131	−	0.998149i	$-0.519369\pi$
$251$	−1.91970e12	−0.121626	−0.0608131	+	0.998149i	$0.519369\pi$
$252$	−1.99151e12	−0.123447
$253$	1.55841e13	0.945191
$254$	−8.85278e12	−0.525405
$255$	0	0
$256$	1.09951e12	0.0625000
$257$	−9.76031e12	−0.543040	−0.271520	−	0.962433i	$-0.587526\pi$
$257$	−9.76031e12	−0.543040	−0.271520	+	0.962433i	$0.587526\pi$
$258$	−4.82477e12	−0.262766
$259$	−1.05360e13	−0.561727
$260$	0	0
$261$	1.70147e12	0.0869564
$262$	1.09079e13	0.545864
$263$	4.04845e12	0.198396	0.0991980	−	0.995068i	$-0.468372\pi$
$263$	4.04845e12	0.198396	0.0991980	+	0.995068i	$0.468372\pi$
$264$	−6.04163e12	−0.289956
$265$	0	0
$266$	−1.14536e13	−0.527343
$267$	−1.83533e13	−0.827754
$268$	−1.91557e13	−0.846361
$269$	−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	$-0.515567\pi$
$271$	−2.35252e12	−0.0977691	−0.0488846	+	0.998804i	$0.515567\pi$
$272$	−8.69310e12	−0.354035
$273$	1.98714e13	0.793112
$274$	−3.02331e13	−1.18264
$275$	0	0
$276$	5.11083e12	0.192084
$277$	4.36327e13	1.60758	0.803791	−	0.594912i	$-0.202813\pi$
$277$	4.36327e13	1.60758	0.803791	+	0.594912i	$0.202813\pi$
$278$	−3.32065e12	−0.119943
$279$	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	$0.344847\pi$
$282$	−2.14859e13	−0.717438
$283$	2.57815e13	0.844273	0.422136	−	0.906532i	$-0.361280\pi$
$283$	2.57815e13	0.844273	0.422136	+	0.906532i	$0.361280\pi$
$284$	−8.54732e12	−0.274523
$285$	0	0
$286$	6.02836e13	1.86289
$287$	1.21207e13	0.367434
$288$	−1.98136e12	−0.0589256
$289$	3.44586e13	1.00545
$290$	0	0
$291$	−2.00127e13	−0.562204
$292$	−2.01131e13	−0.554465
$293$	−5.59324e13	−1.51318	−0.756591	−	0.653888i	$-0.773137\pi$
$293$	−5.59324e13	−1.51318	−0.756591	+	0.653888i	$0.773137\pi$
$294$	−6.94044e12	−0.184279
$295$	0	0
$296$	−1.04822e13	−0.268132
$297$	1.08872e13	0.273373
$298$	3.93126e13	0.969041
$299$	−5.09961e13	−1.23409
$300$	0	0
$301$	2.04358e13	0.476733
$302$	−6.79890e12	−0.155740
$303$	1.00393e13	0.225824
$304$	−1.13952e13	−0.251719
$305$	0	0
$306$	1.56652e13	0.333787
$307$	−5.68056e13	−1.18886	−0.594429	−	0.804148i	$-0.702622\pi$
$307$	−5.68056e13	−1.18886	−0.594429	+	0.804148i	$0.702622\pi$
$308$	2.55899e13	0.526065
$309$	−1.56779e13	−0.316605
$310$	0	0
$311$	−6.71193e13	−1.30817	−0.654087	−	0.756419i	$-0.726947\pi$
$311$	−6.71193e13	−1.30817	−0.654087	+	0.756419i	$0.726947\pi$
$312$	1.97701e13	0.378580
$313$	3.66685e13	0.689920	0.344960	−	0.938617i	$-0.387892\pi$
$313$	3.66685e13	0.689920	0.344960	+	0.938617i	$0.387892\pi$
$314$	−5.07624e13	−0.938488
$315$	0	0
$316$	−6.01478e12	−0.107384
$317$	−1.01210e14	−1.77581	−0.887905	−	0.460027i	$-0.847840\pi$
$317$	−1.01210e14	−1.77581	−0.887905	+	0.460027i	$0.847840\pi$
$318$	2.08618e12	0.0359751
$319$	−2.18630e13	−0.370562
$320$	0	0
$321$	2.67832e13	0.438616
$322$	−2.16474e13	−0.348497
$323$	9.00941e13	1.42588
$324$	3.57047e12	0.0555556
$325$	0	0
$326$	2.76806e13	0.416370
$327$	−4.32001e13	−0.638957
$328$	1.20589e13	0.175389
$329$	9.10058e13	1.30164
$330$	0	0
$331$	4.96614e13	0.687013	0.343506	−	0.939150i	$-0.388385\pi$
$331$	4.96614e13	0.687013	0.343506	+	0.939150i	$0.388385\pi$
$332$	−8.69638e12	−0.118326
$333$	1.88893e13	0.252797
$334$	−7.22566e13	−0.951200
$335$	0	0
$336$	8.39222e12	0.106908
$337$	8.40508e13	1.05336	0.526681	−	0.850063i	$-0.323436\pi$
$337$	8.40508e13	1.05336	0.526681	+	0.850063i	$0.323436\pi$
$338$	−1.39918e14	−1.72516
$339$	1.91777e13	0.232647
$340$	0	0
$341$	−1.14193e14	−1.34119
$342$	2.05345e13	0.237323
$343$	9.45221e13	1.07502
$344$	2.03315e13	0.227562
$345$	0	0
$346$	−6.91072e13	−0.749212
$347$	−1.74097e13	−0.185772	−0.0928858	−	0.995677i	$-0.529609\pi$
$347$	−1.74097e13	−0.185772	−0.0928858	+	0.995677i	$0.529609\pi$
$348$	−7.16998e12	−0.0753065
$349$	−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
$352$	2.54594e13	0.251109
$353$	1.74556e14	1.69502	0.847508	−	0.530783i	$-0.178102\pi$
$353$	1.74556e14	1.69502	0.847508	+	0.530783i	$0.178102\pi$
$354$	1.30084e13	0.124368
$355$	0	0
$356$	7.73405e13	0.716856
$357$	−6.63517e13	−0.605587
$358$	−1.44251e14	−1.29647
$359$	3.60488e13	0.319059	0.159530	−	0.987193i	$-0.449002\pi$
$359$	3.60488e13	0.319059	0.159530	+	0.987193i	$0.449002\pi$
$360$	0	0
$361$	1.60795e12	0.0138033
$362$	−1.38897e13	−0.117434
$363$	−7.05640e13	−0.587620
$364$	−8.37380e13	−0.686855
$365$	0	0
$366$	−6.05532e13	−0.481938
$367$	5.04151e13	0.395273	0.197636	−	0.980275i	$-0.436673\pi$
$367$	5.04151e13	0.395273	0.197636	+	0.980275i	$0.436673\pi$
$368$	−2.15370e13	−0.166350
$369$	−2.17306e13	−0.165358
$370$	0	0
$371$	−8.83621e12	−0.0652694
$372$	−3.74496e13	−0.272559
$373$	−2.09367e14	−1.50144	−0.750722	−	0.660619i	$-0.770294\pi$
$373$	−2.09367e14	−1.50144	−0.750722	+	0.660619i	$0.770294\pi$
$374$	−2.01290e14	−1.42242
$375$	0	0
$376$	9.05416e13	0.621320
$377$	7.15424e13	0.483823
$378$	−1.51231e13	−0.100794
$379$	2.95197e14	1.93908	0.969540	−	0.244931i	$-0.0787655\pi$
$379$	2.95197e14	1.93908	0.969540	+	0.244931i	$0.0787655\pi$
$380$	0	0
$381$	−6.72258e13	−0.428992
$382$	−2.77664e13	−0.174651
$383$	−1.28924e14	−0.799356	−0.399678	−	0.916656i	$-0.630878\pi$
$383$	−1.28924e14	−0.799356	−0.399678	+	0.916656i	$0.630878\pi$
$384$	8.34942e12	0.0510310
$385$	0	0
$386$	5.98226e12	0.0355333
$387$	−3.66381e13	−0.214547
$388$	8.43333e13	0.486883
$389$	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
$392$	2.92470e13	0.159591
$393$	8.28319e13	0.445696
$394$	1.06496e14	0.565070
$395$	0	0
$396$	−4.58786e13	−0.236748
$397$	1.15408e14	0.587338	0.293669	−	0.955907i	$-0.405124\pi$
$397$	1.15408e14	0.587338	0.293669	+	0.955907i	$0.405124\pi$
$398$	5.07458e13	0.254709
$399$	−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	$0.678493\pi$
$402$	−1.45463e14	−0.691051
$403$	3.73674e14	1.75111
$404$	−4.23056e13	−0.195569
$405$	0	0
$406$	3.03692e13	0.136628
$407$	−2.42717e14	−1.07729
$408$	−6.60132e13	−0.289068
$409$	−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
$412$	6.60667e13	0.274188
$413$	−5.50985e13	−0.225639
$414$	3.88104e13	0.156836
$415$	0	0
$416$	−8.33109e13	−0.327860
$417$	−2.52162e13	−0.0979334
$418$	−2.63857e14	−1.01135
$419$	2.58945e14	0.979560	0.489780	−	0.871846i	$-0.337077\pi$
$419$	2.58945e14	0.979560	0.489780	+	0.871846i	$0.337077\pi$
$420$	0	0
$421$	1.09351e14	0.402968	0.201484	−	0.979492i	$-0.435424\pi$
$421$	1.09351e14	0.402968	0.201484	+	0.979492i	$0.435424\pi$
$422$	−1.13587e14	−0.413151
$423$	−1.63159e14	−0.585786
$424$	−8.79114e12	−0.0311554
$425$	0	0
$426$	−6.49062e13	−0.224147
$427$	2.56479e14	0.874377
$428$	−1.12864e14	−0.379853
$429$	4.57779e14	1.52104
$430$	0	0
$431$	2.92180e14	0.946292	0.473146	−	0.880984i	$-0.343118\pi$
$431$	2.92180e14	0.946292	0.473146	+	0.880984i	$0.343118\pi$
$432$	−1.50459e13	−0.0481125
$433$	−9.53988e13	−0.301203	−0.150602	−	0.988595i	$-0.548121\pi$
$433$	−9.53988e13	−0.301203	−0.150602	+	0.988595i	$0.548121\pi$
$434$	1.58621e14	0.494502
$435$	0	0
$436$	1.82045e14	0.553353
$437$	2.23206e14	0.669975
$438$	−1.52734e14	−0.452719
$439$	−4.80574e14	−1.40671	−0.703357	−	0.710837i	$-0.748316\pi$
$439$	−4.80574e14	−1.40671	−0.703357	+	0.710837i	$0.748316\pi$
$440$	0	0
$441$	−5.27040e13	−0.150464
$442$	6.58683e14	1.85718
$443$	3.15148e14	0.877596	0.438798	−	0.898586i	$-0.355404\pi$
$443$	3.15148e14	0.877596	0.438798	+	0.898586i	$0.355404\pi$
$444$	−7.95993e13	−0.218929
$445$	0	0
$446$	−1.65767e14	−0.444792
$447$	2.98530e14	0.791219
$448$	−3.53648e13	−0.0925852
$449$	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
$452$	−8.08147e13	−0.201479
$453$	−5.16292e13	−0.127161
$454$	−4.27726e14	−1.04078
$455$	0	0
$456$	−8.65322e13	−0.205528
$457$	2.96302e14	0.695338	0.347669	−	0.937617i	$-0.386973\pi$
$457$	2.96302e14	0.695338	0.347669	+	0.937617i	$0.386973\pi$
$458$	6.49857e13	0.150681
$459$	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	$0.464222\pi$
$462$	1.94323e14	0.429530
$463$	−2.49998e14	−0.546061	−0.273030	−	0.962005i	$-0.588026\pi$
$463$	−2.49998e14	−0.546061	−0.273030	+	0.962005i	$0.588026\pi$
$464$	3.02142e13	0.0652173
$465$	0	0
$466$	2.28263e14	0.481186
$467$	7.16272e13	0.149223	0.0746114	−	0.997213i	$-0.476228\pi$
$467$	7.16272e13	0.149223	0.0746114	+	0.997213i	$0.476228\pi$
$468$	1.50129e14	0.309109
$469$	6.16124e14	1.25377
$470$	0	0
$471$	−3.85477e14	−0.766273
$472$	−5.48174e13	−0.107705
$473$	4.70780e14	0.914285
$474$	−4.56747e13	−0.0876789
$475$	0	0
$476$	2.79605e14	0.524454
$477$	1.58419e13	0.0293736
$478$	−1.05716e14	−0.193771
$479$	2.34866e14	0.425574	0.212787	−	0.977099i	$-0.431746\pi$
$479$	2.34866e14	0.425574	0.212787	+	0.977099i	$0.431746\pi$
$480$	0	0
$481$	7.94246e14	1.40656
$482$	−1.33420e13	−0.0233595
$483$	−1.64385e14	−0.284546
$484$	2.97356e14	0.508894
$485$	0	0
$486$	2.71132e13	0.0453609
$487$	9.89828e14	1.63738	0.818692	−	0.574233i	$-0.194700\pi$
$487$	9.89828e14	1.63738	0.818692	+	0.574233i	$0.194700\pi$
$488$	2.55171e14	0.417371
$489$	2.10200e14	0.339965
$490$	0	0
$491$	−8.70345e14	−1.37639	−0.688197	−	0.725523i	$-0.741598\pi$
$491$	−8.70345e14	−1.37639	−0.688197	+	0.725523i	$0.741598\pi$
$492$	9.15724e13	0.143204
$493$	−2.38884e14	−0.369427
$494$	8.63423e14	1.32046
$495$	0	0
$496$	1.57812e14	0.236043
$497$	2.74916e14	0.406669
$498$	−6.60381e13	−0.0966125
$499$	−5.38314e14	−0.778902	−0.389451	−	0.921047i	$-0.627335\pi$
$499$	−5.38314e14	−0.778902	−0.389451	+	0.921047i	$0.627335\pi$
$500$	0	0
$501$	−5.48698e14	−0.776652
$502$	6.14303e13	0.0860027
$503$	−8.38818e14	−1.16157	−0.580783	−	0.814059i	$-0.697253\pi$
$503$	−8.38818e14	−1.16157	−0.580783	+	0.814059i	$0.697253\pi$
$504$	6.37284e13	0.0872902
$505$	0	0
$506$	−4.98692e14	−0.668351
$507$	−1.06250e15	−1.40859
$508$	2.83289e14	0.371518
$509$	−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
$512$	−3.51844e13	−0.0441942
$513$	1.55934e14	0.193774
$514$	3.12330e14	0.383987
$515$	0	0
$516$	1.54393e14	0.185803
$517$	2.09650e15	2.49631
$518$	3.37151e14	0.397201
$519$	−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	$0.215151\pi$
$522$	−5.44471e13	−0.0614875
$523$	−6.90497e14	−0.771619	−0.385809	−	0.922579i	$-0.626078\pi$
$523$	−6.90497e14	−0.771619	−0.385809	+	0.922579i	$0.626078\pi$
$524$	−3.49053e14	−0.385984
$525$	0	0
$526$	−1.29551e14	−0.140287
$527$	−1.24771e15	−1.33708
$528$	1.93332e14	0.205030
$529$	−5.30948e14	−0.557245
$530$	0	0
$531$	9.87828e13	0.101546
$532$	3.66516e14	0.372888
$533$	−9.13714e14	−0.920048
$534$	5.87305e14	0.585311
$535$	0	0
$536$	6.12981e14	0.598468
$537$	−1.09541e15	−1.05857
$538$	3.63056e14	0.347273
$539$	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	$0.566645\pi$
$542$	7.52806e13	0.0691332
$543$	−1.05475e14	−0.0958848
$544$	2.78179e14	0.250340
$545$	0	0
$546$	−6.35886e14	−0.560815
$547$	−8.45293e14	−0.738035	−0.369017	−	0.929423i	$-0.620306\pi$
$547$	−8.45293e14	−0.738035	−0.369017	+	0.929423i	$0.620306\pi$
$548$	9.67459e14	0.836256
$549$	−4.59826e14	−0.393501
$550$	0	0
$551$	−3.13136e14	−0.262663
$552$	−1.63547e14	−0.135824
$553$	1.93460e14	0.159075
$554$	−1.39625e15	−1.13673
$555$	0	0
$556$	1.06261e14	0.0848128
$557$	6.21109e14	0.490868	0.245434	−	0.969413i	$-0.421070\pi$
$557$	6.21109e14	0.490868	0.245434	+	0.969413i	$0.421070\pi$
$558$	−2.84383e14	−0.222544
$559$	−1.54054e15	−1.19373
$560$	0	0
$561$	−1.52855e15	−1.16140
$562$	−8.80316e14	−0.662352
$563$	2.31809e15	1.72717	0.863585	−	0.504203i	$-0.168214\pi$
$563$	2.31809e15	1.72717	0.863585	+	0.504203i	$0.168214\pi$
$564$	6.87550e14	0.507305
$565$	0	0
$566$	−8.25008e14	−0.596991
$567$	−1.14841e14	−0.0822980
$568$	2.73514e14	0.194117
$569$	−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	$-0.314812\pi$
$571$	1.59408e15	1.09903	0.549516	+	0.835483i	$0.314812\pi$
$572$	−1.92908e15	−1.31726
$573$	−2.10851e14	−0.142602
$574$	−3.87864e14	−0.259815
$575$	0	0
$576$	6.34034e13	0.0416667
$577$	−1.77831e15	−1.15755	−0.578777	−	0.815486i	$-0.696470\pi$
$577$	−1.77831e15	−1.15755	−0.578777	+	0.815486i	$0.696470\pi$
$578$	−1.10268e15	−0.710959
$579$	4.54278e13	0.0290128
$580$	0	0
$581$	2.79711e14	0.175283
$582$	6.40406e14	0.397539
$583$	−2.03560e14	−0.125174
$584$	6.43621e14	0.392066
$585$	0	0
$586$	1.78984e15	1.06998
$587$	−2.98810e14	−0.176964	−0.0884821	−	0.996078i	$-0.528202\pi$
$587$	−2.98810e14	−0.176964	−0.0884821	+	0.996078i	$0.528202\pi$
$588$	2.22094e14	0.130305
$589$	−1.63554e15	−0.950666
$590$	0	0
$591$	8.08700e14	0.461378
$592$	3.35431e14	0.189598
$593$	−2.13060e15	−1.19317	−0.596585	−	0.802550i	$-0.703476\pi$
$593$	−2.13060e15	−1.19317	−0.596585	+	0.802550i	$0.703476\pi$
$594$	−3.48391e14	−0.193304
$595$	0	0
$596$	−1.25800e15	−0.685216
$597$	3.85351e14	0.207969
$598$	1.63188e15	0.872631
$599$	6.91981e14	0.366646	0.183323	−	0.983053i	$-0.441315\pi$
$599$	6.91981e14	0.366646	0.183323	+	0.983053i	$0.441315\pi$
$600$	0	0
$601$	1.90167e15	0.989296	0.494648	−	0.869094i	$-0.335297\pi$
$601$	1.90167e15	0.989296	0.494648	+	0.869094i	$0.335297\pi$
$602$	−6.53945e14	−0.337101
$603$	−1.10461e15	−0.564241
$604$	2.17565e14	0.110125
$605$	0	0
$606$	−3.21258e14	−0.159682
$607$	−2.60368e14	−0.128248	−0.0641238	−	0.997942i	$-0.520425\pi$
$607$	−2.60368e14	−0.128248	−0.0641238	+	0.997942i	$0.520425\pi$
$608$	3.64646e14	0.177993
$609$	2.30616e14	0.111556
$610$	0	0
$611$	−6.86041e15	−3.25929
$612$	−5.01288e14	−0.236023
$613$	3.20683e15	1.49638	0.748192	−	0.663482i	$-0.230922\pi$
$613$	3.20683e15	1.49638	0.748192	+	0.663482i	$0.230922\pi$
$614$	1.81778e15	0.840649
$615$	0	0
$616$	−8.18876e14	−0.371984
$617$	−3.99520e15	−1.79875	−0.899374	−	0.437180i	$-0.855977\pi$
$617$	−3.99520e15	−1.79875	−0.899374	+	0.437180i	$0.855977\pi$
$618$	5.01694e14	0.223873
$619$	−1.00277e15	−0.443508	−0.221754	−	0.975103i	$-0.571178\pi$
$619$	−1.00277e15	−0.443508	−0.221754	+	0.975103i	$0.571178\pi$
$620$	0	0
$621$	2.94716e14	0.128056
$622$	2.14782e15	0.925019
$623$	−2.48759e15	−1.06192
$624$	−6.32642e14	−0.267697
$625$	0	0
$626$	−1.17339e15	−0.487847
$627$	−2.00367e15	−0.825760
$628$	1.62440e15	0.663612
$629$	−2.65203e15	−1.07399
$630$	0	0
$631$	−2.06951e15	−0.823579	−0.411790	−	0.911279i	$-0.635096\pi$
$631$	−2.06951e15	−0.823579	−0.411790	+	0.911279i	$0.635096\pi$
$632$	1.92473e14	0.0759321
$633$	−8.62548e14	−0.337336
$634$	3.23871e15	1.25569
$635$	0	0
$636$	−6.67577e13	−0.0254383
$637$	−2.21607e15	−0.837174
$638$	6.99615e14	0.262027
$639$	−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	$0.505811\pi$
$642$	−8.57063e14	−0.310149
$643$	3.03786e14	0.108995	0.0544976	−	0.998514i	$-0.482644\pi$
$643$	3.03786e14	0.108995	0.0544976	+	0.998514i	$0.482644\pi$
$644$	6.92717e14	0.246424
$645$	0	0
$646$	−2.88301e15	−1.00825
$647$	−1.90746e14	−0.0661426	−0.0330713	−	0.999453i	$-0.510529\pi$
$647$	−1.90746e14	−0.0661426	−0.0330713	+	0.999453i	$0.510529\pi$
$648$	−1.14255e14	−0.0392837
$649$	−1.26931e15	−0.432733
$650$	0	0
$651$	1.20453e15	0.403759
$652$	−8.85780e14	−0.294418
$653$	−1.64793e15	−0.543145	−0.271573	−	0.962418i	$-0.587544\pi$
$653$	−1.64793e15	−0.543145	−0.271573	+	0.962418i	$0.587544\pi$
$654$	1.38240e15	0.451811
$655$	0	0
$656$	−3.85885e14	−0.124019
$657$	−1.15983e15	−0.369644
$658$	−2.91219e15	−0.920400
$659$	−2.62861e15	−0.823866	−0.411933	−	0.911214i	$-0.635146\pi$
$659$	−2.62861e15	−0.823866	−0.411933	+	0.911214i	$0.635146\pi$
$660$	0	0
$661$	4.80241e15	1.48031	0.740153	−	0.672439i	$-0.234753\pi$
$661$	4.80241e15	1.48031	0.740153	+	0.672439i	$0.234753\pi$
$662$	−1.58916e15	−0.485791
$663$	5.00188e15	1.51638
$664$	2.78284e14	0.0836689
$665$	0	0
$666$	−6.04457e14	−0.178755
$667$	−5.91830e14	−0.173582
$668$	2.31221e15	0.672600
$669$	−1.25879e15	−0.363171
$670$	0	0
$671$	5.90852e15	1.67689
$672$	−2.68551e14	−0.0755955
$673$	1.25325e15	0.349909	0.174954	−	0.984577i	$-0.444022\pi$
$673$	1.25325e15	0.349909	0.174954	+	0.984577i	$0.444022\pi$
$674$	−2.68963e15	−0.744839
$675$	0	0
$676$	4.47736e15	1.21988
$677$	1.92035e15	0.518969	0.259485	−	0.965747i	$-0.416447\pi$
$677$	1.92035e15	0.518969	0.259485	+	0.965747i	$0.416447\pi$
$678$	−6.13686e14	−0.164507
$679$	−2.71250e15	−0.721251
$680$	0	0
$681$	−3.24805e15	−0.849793
$682$	3.65416e15	0.948362
$683$	−6.12548e14	−0.157698	−0.0788490	−	0.996887i	$-0.525125\pi$
$683$	−6.12548e14	−0.157698	−0.0788490	+	0.996887i	$0.525125\pi$
$684$	−6.57104e14	−0.167813
$685$	0	0
$686$	−3.02471e15	−0.760153
$687$	4.93485e14	0.123030
$688$	−6.50610e14	−0.160910
$689$	6.66112e14	0.163434
$690$	0	0
$691$	5.08990e15	1.22908	0.614539	−	0.788886i	$-0.289342\pi$
$691$	5.08990e15	1.22908	0.614539	+	0.788886i	$0.289342\pi$
$692$	2.21143e15	0.529773
$693$	1.47564e15	0.350710
$694$	5.57111e14	0.131360
$695$	0	0
$696$	2.29439e14	0.0532497
$697$	3.05094e15	0.702511
$698$	2.11494e15	0.483163
$699$	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	$0.563874\pi$
$702$	1.14004e15	0.252387
$703$	−3.47636e15	−0.763608
$704$	−8.14699e14	−0.177561
$705$	0	0
$706$	−5.58579e15	−1.19856
$707$	1.36072e15	0.289709
$708$	−4.16270e14	−0.0879411
$709$	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
$712$	−2.47490e15	−0.506894
$713$	−3.09119e15	−0.628251
$714$	2.12325e15	0.428215
$715$	0	0
$716$	4.61605e15	0.916745
$717$	−8.02782e14	−0.158213
$718$	−1.15356e15	−0.225609
$719$	8.02554e15	1.55763	0.778817	−	0.627251i	$-0.215820\pi$
$719$	8.02554e15	1.55763	0.778817	+	0.627251i	$0.215820\pi$
$720$	0	0
$721$	−2.12497e15	−0.406172
$722$	−5.14544e13	−0.00976041
$723$	−1.01316e14	−0.0190729
$724$	4.44470e14	0.0830387
$725$	0	0
$726$	2.25805e15	0.415510
$727$	2.93804e15	0.536560	0.268280	−	0.963341i	$-0.413545\pi$
$727$	2.93804e15	0.536560	0.268280	+	0.963341i	$0.413545\pi$
$728$	2.67962e15	0.485680
$729$	2.05891e14	0.0370370
$730$	0	0
$731$	5.14393e15	0.911485
$732$	1.93770e15	0.340782
$733$	4.42319e15	0.772083	0.386042	−	0.922481i	$-0.373842\pi$
$733$	4.42319e15	0.772083	0.386042	+	0.922481i	$0.373842\pi$
$734$	−1.61328e15	−0.279500
$735$	0	0
$736$	6.89184e14	0.117627
$737$	1.41937e16	2.40449
$738$	6.95378e14	0.116926
$739$	−1.11732e16	−1.86481	−0.932405	−	0.361416i	$-0.882293\pi$
$739$	−1.11732e16	−1.86481	−0.932405	+	0.361416i	$0.882293\pi$
$740$	0	0
$741$	6.55662e15	1.07815
$742$	2.82759e14	0.0461524
$743$	−1.98624e15	−0.321805	−0.160903	−	0.986970i	$-0.551441\pi$
$743$	−1.98624e15	−0.321805	−0.160903	+	0.986970i	$0.551441\pi$
$744$	1.19839e15	0.192728
$745$	0	0
$746$	6.69973e15	1.06168
$747$	−5.01477e14	−0.0788838
$748$	6.44128e15	1.00580
$749$	3.63017e15	0.562700
$750$	0	0
$751$	−7.63503e15	−1.16625	−0.583125	−	0.812383i	$-0.698170\pi$
$751$	−7.63503e15	−1.16625	−0.583125	+	0.812383i	$0.698170\pi$
$752$	−2.89733e15	−0.439339
$753$	4.66486e14	0.0702209
$754$	−2.28936e15	−0.342114
$755$	0	0
$756$	4.83938e14	0.0712721
$757$	−7.89241e15	−1.15394	−0.576969	−	0.816766i	$-0.695765\pi$
$757$	−7.89241e15	−1.15394	−0.576969	+	0.816766i	$0.695765\pi$
$758$	−9.44630e15	−1.37114
$759$	−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	$0.787566\pi$
$762$	2.15123e15	0.303343
$763$	−5.85530e15	−0.819717
$764$	8.88526e14	0.123497
$765$	0	0
$766$	4.12557e15	0.565230
$767$	4.15356e15	0.564997
$768$	−2.67181e14	−0.0360844
$769$	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
$772$	−1.91432e14	−0.0251258
$773$	−6.92568e15	−0.902558	−0.451279	−	0.892383i	$-0.649032\pi$
$773$	−6.92568e15	−0.902558	−0.451279	+	0.892383i	$0.649032\pi$
$774$	1.17242e15	0.151708
$775$	0	0
$776$	−2.69867e15	−0.344278
$777$	2.56024e15	0.324313
$778$	−3.33264e15	−0.419180
$779$	3.99926e15	0.499487
$780$	0	0
$781$	6.33326e15	0.779914
$782$	−5.44891e15	−0.666304
$783$	−4.13457e14	−0.0502043
$784$	−9.35903e14	−0.112848
$785$	0	0
$786$	−2.65062e15	−0.315155
$787$	2.99770e15	0.353939	0.176969	−	0.984216i	$-0.443371\pi$
$787$	2.99770e15	0.353939	0.176969	+	0.984216i	$0.443371\pi$
$788$	−3.40786e15	−0.399565
$789$	−9.83775e14	−0.114544
$790$	0	0
$791$	2.59933e15	0.298463
$792$	1.46811e15	0.167406
$793$	−1.93345e16	−2.18943
$794$	−3.69306e15	−0.415311
$795$	0	0
$796$	−1.62387e15	−0.180106
$797$	5.67794e15	0.625417	0.312709	−	0.949849i	$-0.398764\pi$
$797$	5.67794e15	0.625417	0.312709	+	0.949849i	$0.398764\pi$
$798$	2.78323e15	0.304462
$799$	2.29073e16	2.48866
$800$	0	0
$801$	4.45984e15	0.477904
$802$	7.06711e15	0.752113
$803$	1.49031e16	1.57522
$804$	4.65482e15	0.488647
$805$	0	0
$806$	−1.19576e16	−1.23823
$807$	2.75696e15	0.283547
$808$	1.35378e15	0.138288
$809$	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	$-0.648598\pi$
$811$	−8.99323e15	−0.900121	−0.450060	+	0.892998i	$0.648598\pi$
$812$	−9.71813e14	−0.0966105
$813$	5.71662e14	0.0564470
$814$	7.76695e15	0.761757
$815$	0	0
$816$	2.11242e15	0.204402
$817$	6.74283e15	0.648068
$818$	2.74193e15	0.261766
$819$	−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	$0.360348\pi$
$822$	7.34664e15	0.682800
$823$	1.29654e16	1.19698	0.598488	−	0.801132i	$-0.295768\pi$
$823$	1.29654e16	1.19698	0.598488	+	0.801132i	$0.295768\pi$
$824$	−2.11414e15	−0.193880
$825$	0	0
$826$	1.76315e15	0.159551
$827$	5.40262e15	0.485651	0.242826	−	0.970070i	$-0.421926\pi$
$827$	5.40262e15	0.485651	0.242826	+	0.970070i	$0.421926\pi$
$828$	−1.24193e15	−0.110900
$829$	−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
$832$	2.66595e15	0.231832
$833$	7.39956e15	0.639231
$834$	8.06918e14	0.0692494
$835$	0	0
$836$	8.44344e15	0.715129
$837$	−2.15953e15	−0.181706
$838$	−8.28625e15	−0.692653
$839$	1.02313e16	0.849646	0.424823	−	0.905276i	$-0.360336\pi$
$839$	1.02313e16	0.849646	0.424823	+	0.905276i	$0.360336\pi$
$840$	0	0
$841$	−1.13702e16	−0.931947
$842$	−3.49922e15	−0.284941
$843$	−6.68490e15	−0.540808
$844$	3.63477e15	0.292142
$845$	0	0
$846$	5.22108e15	0.414213
$847$	−9.56418e15	−0.753857
$848$	2.81316e14	0.0220302
$849$	−6.26490e15	−0.487441
$850$	0	0
$851$	−6.57034e15	−0.504633
$852$	2.07700e15	0.158496
$853$	1.33010e16	1.00847	0.504237	−	0.863565i	$-0.331774\pi$
$853$	1.33010e16	1.00847	0.504237	+	0.863565i	$0.331774\pi$
$854$	−8.20733e15	−0.618278
$855$	0	0
$856$	3.61166e15	0.268597
$857$	−3.38483e15	−0.250116	−0.125058	−	0.992149i	$-0.539912\pi$
$857$	−3.38483e15	−0.250116	−0.125058	+	0.992149i	$0.539912\pi$
$858$	−1.46489e16	−1.07554
$859$	2.32376e16	1.69523	0.847615	−	0.530612i	$-0.178038\pi$
$859$	2.32376e16	1.69523	0.847615	+	0.530612i	$0.178038\pi$
$860$	0	0
$861$	−2.94534e15	−0.212138
$862$	−9.34976e15	−0.669130
$863$	−2.44743e16	−1.74041	−0.870203	−	0.492693i	$-0.836013\pi$
$863$	−2.44743e16	−1.74041	−0.870203	+	0.492693i	$0.836013\pi$
$864$	4.81469e14	0.0340207
$865$	0	0
$866$	3.05276e15	0.212983
$867$	−8.37344e15	−0.580496
$868$	−5.07588e15	−0.349666
$869$	4.45674e15	0.305076
$870$	0	0
$871$	−4.64461e16	−3.13942
$872$	−5.82543e15	−0.391280
$873$	4.86309e15	0.324589
$874$	−7.14261e15	−0.473744
$875$	0	0
$876$	4.88750e15	0.320121
$877$	−2.15305e16	−1.40138	−0.700689	−	0.713467i	$-0.747124\pi$
$877$	−2.15305e16	−1.40138	−0.700689	+	0.713467i	$0.747124\pi$
$878$	1.53784e16	0.994697
$879$	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	$0.157885\pi$
$882$	1.68653e15	0.106394
$883$	8.61677e15	0.540208	0.270104	−	0.962831i	$-0.412942\pi$
$883$	8.61677e15	0.540208	0.270104	+	0.962831i	$0.412942\pi$
$884$	−2.10779e16	−1.31322
$885$	0	0
$886$	−1.00848e16	−0.620554
$887$	3.22651e16	1.97312	0.986558	−	0.163409i	$-0.0522490\pi$
$887$	3.22651e16	1.97312	0.986558	+	0.163409i	$0.0522490\pi$
$888$	2.54718e15	0.154806
$889$	−9.11173e15	−0.550353
$890$	0	0
$891$	−2.64559e15	−0.157832
$892$	5.30455e15	0.314515
$893$	3.00275e16	1.76944
$894$	−9.55296e15	−0.559476
$895$	0	0
$896$	1.13167e15	0.0654676
$897$	1.23921e16	0.712500
$898$	−1.61181e16	−0.921075
$899$	4.33663e15	0.246305
$900$	0	0
$901$	−2.22418e15	−0.124791
$902$	−8.93523e15	−0.498276
$903$	−4.96590e15	−0.275242
$904$	2.58607e15	0.142467
$905$	0	0
$906$	1.65213e15	0.0899166
$907$	−3.10436e16	−1.67932	−0.839658	−	0.543115i	$-0.817245\pi$
$907$	−3.10436e16	−1.67932	−0.839658	+	0.543115i	$0.817245\pi$
$908$	1.36872e16	0.735943
$909$	−2.43955e15	−0.130379
$910$	0	0
$911$	1.78173e16	0.940786	0.470393	−	0.882457i	$-0.344112\pi$
$911$	1.78173e16	0.940786	0.470393	+	0.882457i	$0.344112\pi$
$912$	2.76903e15	0.145330
$913$	6.44371e15	0.336161
$914$	−9.48167e15	−0.491678
$915$	0	0
$916$	−2.07954e15	−0.106547
$917$	1.12270e16	0.571783
$918$	−3.80666e15	−0.192712
$919$	2.90046e16	1.45959	0.729797	−	0.683664i	$-0.239615\pi$
$919$	2.90046e16	1.45959	0.729797	+	0.683664i	$0.239615\pi$
$920$	0	0
$921$	1.38038e16	0.686387
$922$	−3.20910e15	−0.158622
$923$	−2.07244e16	−1.01829
$924$	−6.21834e15	−0.303724
$925$	0	0
$926$	7.99993e15	0.386123
$927$	3.80974e15	0.182792
$928$	−9.66856e14	−0.0461156
$929$	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
$932$	−7.30442e15	−0.340250
$933$	1.63100e16	0.755274
$934$	−2.29207e15	−0.105516
$935$	0	0
$936$	−4.80413e15	−0.218573
$937$	−9.13874e15	−0.413351	−0.206675	−	0.978410i	$-0.566264\pi$
$937$	−9.13874e15	−0.413351	−0.206675	+	0.978410i	$0.566264\pi$
$938$	−1.97160e16	−0.886548
$939$	−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	$0.495097\pi$
$942$	1.23353e16	0.541837
$943$	7.55864e15	0.330088
$944$	1.75416e15	0.0761592
$945$	0	0
$946$	−1.50650e16	−0.646497
$947$	2.17555e15	0.0928205	0.0464103	−	0.998922i	$-0.485222\pi$
$947$	2.17555e15	0.0928205	0.0464103	+	0.998922i	$0.485222\pi$
$948$	1.46159e15	0.0619983
$949$	−4.87677e16	−2.05669
$950$	0	0
$951$	2.45940e16	1.02526
$952$	−8.94737e15	−0.370845
$953$	3.59675e16	1.48217	0.741087	−	0.671409i	$-0.234310\pi$
$953$	3.59675e16	1.48217	0.741087	+	0.671409i	$0.234310\pi$
$954$	−5.06941e14	−0.0207702
$955$	0	0
$956$	3.38291e15	0.137016
$957$	5.31270e15	0.213944
$958$	−7.51572e15	−0.300926
$959$	−3.11174e16	−1.23880
$960$	0	0
$961$	−2.75781e15	−0.108539
$962$	−2.54159e16	−0.994585
$963$	−6.50832e15	−0.253235
$964$	4.26945e14	0.0165176
$965$	0	0
$966$	5.26032e15	0.201205
$967$	−3.64625e16	−1.38676	−0.693379	−	0.720573i	$-0.743879\pi$
$967$	−3.64625e16	−1.38676	−0.693379	+	0.720573i	$0.743879\pi$
$968$	−9.51540e15	−0.359842
$969$	−2.18929e16	−0.823231
$970$	0	0
$971$	−1.01490e16	−0.377328	−0.188664	−	0.982042i	$-0.560416\pi$
$971$	−1.01490e16	−0.377328	−0.188664	+	0.982042i	$0.560416\pi$
$972$	−8.67624e14	−0.0320750
$973$	−3.41778e15	−0.125639
$974$	−3.16745e16	−1.15781
$975$	0	0
$976$	−8.16547e15	−0.295126
$977$	−2.99278e16	−1.07561	−0.537804	−	0.843070i	$-0.680746\pi$
$977$	−2.99278e16	−1.07561	−0.537804	+	0.843070i	$0.680746\pi$
$978$	−6.72639e15	−0.240391
$979$	−5.73066e16	−2.03657
$980$	0	0
$981$	1.04976e16	0.368902
$982$	2.78510e16	0.973258
$983$	3.23592e16	1.12448	0.562242	−	0.826973i	$-0.309939\pi$
$983$	3.23592e16	1.12448	0.562242	+	0.826973i	$0.309939\pi$
$984$	−2.93032e15	−0.101261
$985$	0	0
$986$	7.64428e15	0.261224
$987$	−2.21144e16	−0.751503
$988$	−2.76295e16	−0.933706
$989$	1.27440e16	0.428278
$990$	0	0
$991$	−2.28589e16	−0.759716	−0.379858	−	0.925045i	$-0.624027\pi$
$991$	−2.28589e16	−0.759716	−0.379858	+	0.925045i	$0.624027\pi$
$992$	−5.04999e15	−0.166908
$993$	−1.20677e16	−0.396647
$994$	−8.79733e15	−0.287558
$995$	0	0
$996$	2.11322e15	0.0683154
$997$	1.99784e16	0.642298	0.321149	−	0.947029i	$-0.395931\pi$
$997$	1.99784e16	0.642298	0.321149	+	0.947029i	$0.395931\pi$
$998$	1.72260e16	0.550767
$999$	−4.59010e15	−0.145953

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	150.12.a.a.1.1		1
5.2	odd	4		150.12.c.e.49.1		2
5.3	odd	4		150.12.c.e.49.2		2
5.4	even	2		6.12.a.c.1.1	✓	1
15.14	odd	2		18.12.a.a.1.1		1
20.19	odd	2		48.12.a.d.1.1		1
40.19	odd	2		192.12.a.m.1.1		1
40.29	even	2		192.12.a.c.1.1		1
45.4	even	6		162.12.c.b.55.1		2
45.14	odd	6		162.12.c.i.55.1		2
45.29	odd	6		162.12.c.i.109.1		2
45.34	even	6		162.12.c.b.109.1		2
60.59	even	2		144.12.a.e.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6.12.a.c.1.1	✓	1	5.4	even	2
18.12.a.a.1.1		1	15.14	odd	2
48.12.a.d.1.1		1	20.19	odd	2
144.12.a.e.1.1		1	60.59	even	2
150.12.a.a.1.1		1	1.1	even	1	trivial
150.12.c.e.49.1		2	5.2	odd	4
150.12.c.e.49.2		2	5.3	odd	4
162.12.c.b.55.1		2	45.4	even	6
162.12.c.b.109.1		2	45.34	even	6
162.12.c.i.55.1		2	45.14	odd	6
162.12.c.i.109.1		2	45.29	odd	6
192.12.a.c.1.1		1	40.29	even	2
192.12.a.m.1.1		1	40.19	odd	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 150 = 2 \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	150.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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