LMFDB - Embedded newform 42.12.a.f.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [42,12,Mod(1,42)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("42.1"); S:= CuspForms(chi, 12); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(42, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 12, names="a")

Level:	$ N $	$=$	$ 42 = 2 \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 12 $
Character orbit:	$[\chi]$	$=$	42.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,32,243,1024,-11880] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$32.2704135835$
Analytic rank:	$1$
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	$\chi$	$=$	42.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

\(q+32.0000 q^{2} +243.000 q^{3} +1024.00 q^{4} -11880.0 q^{5} +7776.00 q^{6} +16807.0 q^{7} +32768.0 q^{8} +59049.0 q^{9} -380160. q^{10} +727110. q^{11} +248832. q^{12} -1.89573e6 q^{13} +537824. q^{14} -2.88684e6 q^{15} +1.04858e6 q^{16} -1.02339e7 q^{17} +1.88957e6 q^{18} -1.27928e7 q^{19} -1.21651e7 q^{20} +4.08410e6 q^{21} +2.32675e7 q^{22} -2.74123e7 q^{23} +7.96262e6 q^{24} +9.23063e7 q^{25} -6.06635e7 q^{26} +1.43489e7 q^{27} +1.72104e7 q^{28} -1.08083e8 q^{29} -9.23789e7 q^{30} -2.02243e8 q^{31} +3.35544e7 q^{32} +1.76688e8 q^{33} -3.27485e8 q^{34} -1.99667e8 q^{35} +6.04662e7 q^{36} +4.12455e8 q^{37} -4.09369e8 q^{38} -4.60663e8 q^{39} -3.89284e8 q^{40} -2.45109e8 q^{41} +1.30691e8 q^{42} +5.09840e8 q^{43} +7.44561e8 q^{44} -7.01502e8 q^{45} -8.77193e8 q^{46} +6.99877e8 q^{47} +2.54804e8 q^{48} +2.82475e8 q^{49} +2.95380e9 q^{50} -2.48684e9 q^{51} -1.94123e9 q^{52} -4.83669e9 q^{53} +4.59165e8 q^{54} -8.63807e9 q^{55} +5.50732e8 q^{56} -3.10865e9 q^{57} -3.45865e9 q^{58} +2.46225e9 q^{59} -2.95612e9 q^{60} +9.05472e9 q^{61} -6.47179e9 q^{62} +9.92437e8 q^{63} +1.07374e9 q^{64} +2.25213e10 q^{65} +5.65401e9 q^{66} -2.92315e9 q^{67} -1.04795e10 q^{68} -6.66119e9 q^{69} -6.38935e9 q^{70} -1.12689e10 q^{71} +1.93492e9 q^{72} +2.80923e9 q^{73} +1.31986e10 q^{74} +2.24304e10 q^{75} -1.30998e10 q^{76} +1.22205e10 q^{77} -1.47412e10 q^{78} +2.55736e10 q^{79} -1.24571e10 q^{80} +3.48678e9 q^{81} -7.84348e9 q^{82} -3.87188e10 q^{83} +4.18212e9 q^{84} +1.21579e11 q^{85} +1.63149e10 q^{86} -2.62641e10 q^{87} +2.38259e10 q^{88} +9.07837e10 q^{89} -2.24481e10 q^{90} -3.18616e10 q^{91} -2.80702e10 q^{92} -4.91451e10 q^{93} +2.23961e10 q^{94} +1.51978e11 q^{95} +8.15373e9 q^{96} +1.30560e11 q^{97} +9.03921e9 q^{98} +4.29351e10 q^{99} +O(q^{100})\)

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
$5$	−11880.0	−1.70013	−0.850064	−	0.526680i	$-0.823437\pi$
$5$	−11880.0	−1.70013	−0.850064	+	0.526680i	$0.823437\pi$
$6$	7776.00	0.408248
$7$	16807.0	0.377964
$7$	16807.0	0.377964
$8$	32768.0	0.353553
$9$	59049.0	0.333333
$10$	−380160.	−1.20217
$11$	727110.	1.36126	0.680629	−	0.732628i	$-0.261707\pi$
$11$	727110.	1.36126	0.680629	+	0.732628i	$0.261707\pi$
$12$	248832.	0.288675
$13$	−1.89573e6	−1.41608	−0.708042	−	0.706170i	$-0.750421\pi$
$13$	−1.89573e6	−1.41608	−0.708042	+	0.706170i	$0.750421\pi$
$14$	537824.	0.267261
$15$	−2.88684e6	−0.981569
$16$	1.04858e6	0.250000
$17$	−1.02339e7	−1.74813	−0.874063	−	0.485813i	$-0.838524\pi$
$17$	−1.02339e7	−1.74813	−0.874063	+	0.485813i	$0.838524\pi$
$18$	1.88957e6	0.235702
$19$	−1.27928e7	−1.18528	−0.592640	−	0.805468i	$-0.701914\pi$
$19$	−1.27928e7	−1.18528	−0.592640	+	0.805468i	$0.701914\pi$
$20$	−1.21651e7	−0.850064
$21$	4.08410e6	0.218218
$22$	2.32675e7	0.962555
$23$	−2.74123e7	−0.888060	−0.444030	−	0.896012i	$-0.646452\pi$
$23$	−2.74123e7	−0.888060	−0.444030	+	0.896012i	$0.646452\pi$
$24$	7.96262e6	0.204124
$25$	9.23063e7	1.89043
$26$	−6.06635e7	−1.00132
$27$	1.43489e7	0.192450
$28$	1.72104e7	0.188982
$29$	−1.08083e8	−0.978516	−0.489258	−	0.872139i	$-0.662732\pi$
$29$	−1.08083e8	−0.978516	−0.489258	+	0.872139i	$0.662732\pi$
$30$	−9.23789e7	−0.694074
$31$	−2.02243e8	−1.26878	−0.634388	−	0.773015i	$-0.718748\pi$
$31$	−2.02243e8	−1.26878	−0.634388	+	0.773015i	$0.718748\pi$
$32$	3.35544e7	0.176777
$33$	1.76688e8	0.785922
$34$	−3.27485e8	−1.23611
$35$	−1.99667e8	−0.642588
$36$	6.04662e7	0.166667
$37$	4.12455e8	0.977838	0.488919	−	0.872329i	$-0.337391\pi$
$37$	4.12455e8	0.977838	0.488919	+	0.872329i	$0.337391\pi$
$38$	−4.09369e8	−0.838119
$39$	−4.60663e8	−0.817576
$40$	−3.89284e8	−0.601086
$41$	−2.45109e8	−0.330406	−0.165203	−	0.986260i	$-0.552828\pi$
$41$	−2.45109e8	−0.330406	−0.165203	+	0.986260i	$0.552828\pi$
$42$	1.30691e8	0.154303
$43$	5.09840e8	0.528880	0.264440	−	0.964402i	$-0.414813\pi$
$43$	5.09840e8	0.528880	0.264440	+	0.964402i	$0.414813\pi$
$44$	7.44561e8	0.680629
$45$	−7.01502e8	−0.566709
$46$	−8.77193e8	−0.627953
$47$	6.99877e8	0.445127	0.222563	−	0.974918i	$-0.428558\pi$
$47$	6.99877e8	0.445127	0.222563	+	0.974918i	$0.428558\pi$
$48$	2.54804e8	0.144338
$49$	2.82475e8	0.142857
$50$	2.95380e9	1.33674
$51$	−2.48684e9	−1.00928
$52$	−1.94123e9	−0.708042
$53$	−4.83669e9	−1.58866	−0.794330	−	0.607486i	$-0.792178\pi$
$53$	−4.83669e9	−1.58866	−0.794330	+	0.607486i	$0.792178\pi$
$54$	4.59165e8	0.136083
$55$	−8.63807e9	−2.31431
$56$	5.50732e8	0.133631
$57$	−3.10865e9	−0.684321
$58$	−3.45865e9	−0.691915
$59$	2.46225e9	0.448380	0.224190	−	0.974546i	$-0.428026\pi$
$59$	2.46225e9	0.448380	0.224190	+	0.974546i	$0.428026\pi$
$60$	−2.95612e9	−0.490784
$61$	9.05472e9	1.37265	0.686327	−	0.727294i	$-0.259222\pi$
$61$	9.05472e9	1.37265	0.686327	+	0.727294i	$0.259222\pi$
$62$	−6.47179e9	−0.897160
$63$	9.92437e8	0.125988
$64$	1.07374e9	0.125000
$65$	2.25213e10	2.40752
$66$	5.65401e9	0.555731
$67$	−2.92315e9	−0.264508	−0.132254	−	0.991216i	$-0.542222\pi$
$67$	−2.92315e9	−0.264508	−0.132254	+	0.991216i	$0.542222\pi$
$68$	−1.04795e10	−0.874063
$69$	−6.66119e9	−0.512722
$70$	−6.38935e9	−0.454378
$71$	−1.12689e10	−0.741241	−0.370620	−	0.928784i	$-0.620855\pi$
$71$	−1.12689e10	−0.741241	−0.370620	+	0.928784i	$0.620855\pi$
$72$	1.93492e9	0.117851
$73$	2.80923e9	0.158603	0.0793016	−	0.996851i	$-0.474731\pi$
$73$	2.80923e9	0.158603	0.0793016	+	0.996851i	$0.474731\pi$
$74$	1.31986e10	0.691436
$75$	2.24304e10	1.09144
$76$	−1.30998e10	−0.592640
$77$	1.22205e10	0.514507
$78$	−1.47412e10	−0.578114
$79$	2.55736e10	0.935066	0.467533	−	0.883976i	$-0.345143\pi$
$79$	2.55736e10	0.935066	0.467533	+	0.883976i	$0.345143\pi$
$80$	−1.24571e10	−0.425032
$81$	3.48678e9	0.111111
$82$	−7.84348e9	−0.233632
$83$	−3.87188e10	−1.07893	−0.539463	−	0.842009i	$-0.681373\pi$
$83$	−3.87188e10	−1.07893	−0.539463	+	0.842009i	$0.681373\pi$
$84$	4.18212e9	0.109109
$85$	1.21579e11	2.97204
$86$	1.63149e10	0.373975
$87$	−2.62641e10	−0.564947
$88$	2.38259e10	0.481277
$89$	9.07837e10	1.72331	0.861654	−	0.507496i	$-0.169429\pi$
$89$	9.07837e10	1.72331	0.861654	+	0.507496i	$0.169429\pi$
$90$	−2.24481e10	−0.400724
$91$	−3.18616e10	−0.535229
$92$	−2.80702e10	−0.444030
$93$	−4.91451e10	−0.732528
$94$	2.23961e10	0.314752
$95$	1.51978e11	2.01513
$96$	8.15373e9	0.102062
$97$	1.30560e11	1.54371	0.771854	−	0.635800i	$-0.219330\pi$
$97$	1.30560e11	1.54371	0.771854	+	0.635800i	$0.219330\pi$
$98$	9.03921e9	0.101015
$99$	4.29351e10	0.453753
$100$	9.45216e10	0.945216
$101$	1.03664e11	0.981430	0.490715	−	0.871320i	$-0.336736\pi$
$101$	1.03664e11	0.981430	0.490715	+	0.871320i	$0.336736\pi$
$102$	−7.95789e10	−0.713669
$103$	−9.81571e10	−0.834290	−0.417145	−	0.908840i	$-0.636969\pi$
$103$	−9.81571e10	−0.834290	−0.417145	+	0.908840i	$0.636969\pi$
$104$	−6.21194e10	−0.500661
$105$	−4.85191e10	−0.370998
$106$	−1.54774e11	−1.12335
$107$	−1.80148e11	−1.24170	−0.620852	−	0.783928i	$-0.713213\pi$
$107$	−1.80148e11	−1.24170	−0.620852	+	0.783928i	$0.713213\pi$
$108$	1.46933e10	0.0962250
$109$	−2.29481e11	−1.42857	−0.714284	−	0.699856i	$-0.753247\pi$
$109$	−2.29481e11	−1.42857	−0.714284	+	0.699856i	$0.753247\pi$
$110$	−2.76418e11	−1.63647
$111$	1.00227e11	0.564555
$112$	1.76234e10	0.0944911
$113$	−2.30038e11	−1.17454	−0.587272	−	0.809390i	$-0.699798\pi$
$113$	−2.30038e11	−1.17454	−0.587272	+	0.809390i	$0.699798\pi$
$114$	−9.94768e10	−0.483888
$115$	3.25658e11	1.50981
$116$	−1.10677e11	−0.489258
$117$	−1.11941e11	−0.472028
$118$	7.87919e10	0.317052
$119$	−1.72001e11	−0.660729
$120$	−9.45960e10	−0.347037
$121$	2.43377e11	0.853023
$122$	2.89751e11	0.970612
$123$	−5.95614e10	−0.190760
$124$	−2.07097e11	−0.634388
$125$	−5.16520e11	−1.51385
$126$	3.17580e10	0.0890871
$127$	9.30757e10	0.249986	0.124993	−	0.992158i	$-0.460109\pi$
$127$	9.30757e10	0.249986	0.124993	+	0.992158i	$0.460109\pi$
$128$	3.43597e10	0.0883883
$129$	1.23891e11	0.305349
$130$	7.20682e11	1.70238
$131$	−3.30891e10	−0.0749365	−0.0374683	−	0.999298i	$-0.511929\pi$
$131$	−3.30891e10	−0.0749365	−0.0374683	+	0.999298i	$0.511929\pi$
$132$	1.80928e11	0.392961
$133$	−2.15009e11	−0.447993
$134$	−9.35408e10	−0.187036
$135$	−1.70465e11	−0.327190
$136$	−3.35345e11	−0.618056
$137$	−1.97055e11	−0.348839	−0.174420	−	0.984671i	$-0.555805\pi$
$137$	−1.97055e11	−0.348839	−0.174420	+	0.984671i	$0.555805\pi$
$138$	−2.13158e11	−0.362549
$139$	9.13185e11	1.49272	0.746358	−	0.665545i	$-0.231801\pi$
$139$	9.13185e11	1.49272	0.746358	+	0.665545i	$0.231801\pi$
$140$	−2.04459e11	−0.321294
$141$	1.70070e11	0.256994
$142$	−3.60604e11	−0.524137
$143$	−1.37841e12	−1.92766
$144$	6.19174e10	0.0833333
$145$	1.28403e12	1.66360
$146$	8.98954e10	0.112149
$147$	6.86415e10	0.0824786
$148$	4.22354e11	0.488919
$149$	−1.18532e12	−1.32224	−0.661121	−	0.750280i	$-0.729919\pi$
$149$	−1.18532e12	−1.32224	−0.661121	+	0.750280i	$0.729919\pi$
$150$	7.17774e11	0.771766
$151$	−9.11227e11	−0.944612	−0.472306	−	0.881435i	$-0.656578\pi$
$151$	−9.11227e11	−0.944612	−0.472306	+	0.881435i	$0.656578\pi$
$152$	−4.19194e11	−0.419059
$153$	−6.04302e11	−0.582709
$154$	3.91057e11	0.363811
$155$	2.40265e12	2.15708
$156$	−4.71719e11	−0.408788
$157$	1.88387e12	1.57617	0.788085	−	0.615567i	$-0.211073\pi$
$157$	1.88387e12	1.57617	0.788085	+	0.615567i	$0.211073\pi$
$158$	8.18354e11	0.661192
$159$	−1.17532e12	−0.917213
$160$	−3.98627e11	−0.300543
$161$	−4.60718e11	−0.335655
$162$	1.11577e11	0.0785674
$163$	7.12107e11	0.484745	0.242372	−	0.970183i	$-0.422074\pi$
$163$	7.12107e11	0.484745	0.242372	+	0.970183i	$0.422074\pi$
$164$	−2.50991e11	−0.165203
$165$	−2.09905e12	−1.33617
$166$	−1.23900e12	−0.762917
$167$	−1.97324e12	−1.17555	−0.587773	−	0.809026i	$-0.699995\pi$
$167$	−1.97324e12	−1.17555	−0.587773	+	0.809026i	$0.699995\pi$
$168$	1.33828e11	0.0771517
$169$	1.80165e12	1.00529
$170$	3.89052e12	2.10155
$171$	−7.55402e11	−0.395093
$172$	5.22076e11	0.264440
$173$	2.51299e12	1.23293	0.616464	−	0.787383i	$-0.288564\pi$
$173$	2.51299e12	1.23293	0.616464	+	0.787383i	$0.288564\pi$
$174$	−8.40453e11	−0.399478
$175$	1.55139e12	0.714516
$176$	7.62430e11	0.340314
$177$	5.98326e11	0.258872
$178$	2.90508e12	1.21856
$179$	9.11742e11	0.370834	0.185417	−	0.982660i	$-0.440636\pi$
$179$	9.11742e11	0.370834	0.185417	+	0.982660i	$0.440636\pi$
$180$	−7.18338e11	−0.283355
$181$	−2.37570e12	−0.908990	−0.454495	−	0.890749i	$-0.650180\pi$
$181$	−2.37570e12	−0.908990	−0.454495	+	0.890749i	$0.650180\pi$
$182$	−1.01957e12	−0.378464
$183$	2.20030e12	0.792502
$184$	−8.98246e11	−0.313977
$185$	−4.89996e12	−1.66245
$186$	−1.57264e12	−0.517976
$187$	−7.44118e12	−2.37965
$188$	7.16674e11	0.222563
$189$	2.41162e11	0.0727393
$190$	4.86331e12	1.42491
$191$	3.21217e12	0.914355	0.457177	−	0.889376i	$-0.348860\pi$
$191$	3.21217e12	0.914355	0.457177	+	0.889376i	$0.348860\pi$
$192$	2.60919e11	0.0721688
$193$	1.04028e12	0.279630	0.139815	−	0.990178i	$-0.455349\pi$
$193$	1.04028e12	0.279630	0.139815	+	0.990178i	$0.455349\pi$
$194$	4.17792e12	1.09157
$195$	5.47268e12	1.38998
$196$	2.89255e11	0.0714286
$197$	−8.96811e11	−0.215346	−0.107673	−	0.994186i	$-0.534340\pi$
$197$	−8.96811e11	−0.215346	−0.107673	+	0.994186i	$0.534340\pi$
$198$	1.37392e12	0.320852
$199$	7.87544e11	0.178889	0.0894444	−	0.995992i	$-0.471491\pi$
$199$	7.87544e11	0.178889	0.0894444	+	0.995992i	$0.471491\pi$
$200$	3.02469e12	0.668369
$201$	−7.10325e11	−0.152714
$202$	3.31724e12	0.693976
$203$	−1.81655e12	−0.369844
$204$	−2.54652e12	−0.504640
$205$	2.91189e12	0.561731
$206$	−3.14103e12	−0.589932
$207$	−1.61867e12	−0.296020
$208$	−1.98782e12	−0.354021
$209$	−9.30177e12	−1.61347
$210$	−1.55261e12	−0.262335
$211$	1.75714e12	0.289237	0.144619	−	0.989487i	$-0.453805\pi$
$211$	1.75714e12	0.289237	0.144619	+	0.989487i	$0.453805\pi$
$212$	−4.95277e12	−0.794330
$213$	−2.73834e12	−0.427956
$214$	−5.76473e12	−0.878017
$215$	−6.05690e12	−0.899164
$216$	4.70185e11	0.0680414
$217$	−3.39910e12	−0.479552
$218$	−7.34339e12	−1.01015
$219$	6.82643e11	0.0915696
$220$	−8.84538e12	−1.15716
$221$	1.94008e13	2.47549
$222$	3.20725e12	0.399201
$223$	−7.56881e12	−0.919075	−0.459538	−	0.888158i	$-0.651985\pi$
$223$	−7.56881e12	−0.919075	−0.459538	+	0.888158i	$0.651985\pi$
$224$	5.63949e11	0.0668153
$225$	5.45059e12	0.630144
$226$	−7.36123e12	−0.830527
$227$	−3.28333e11	−0.0361553	−0.0180777	−	0.999837i	$-0.505755\pi$
$227$	−3.28333e11	−0.0361553	−0.0180777	+	0.999837i	$0.505755\pi$
$228$	−3.18326e12	−0.342161
$229$	−1.55478e12	−0.163145	−0.0815723	−	0.996667i	$-0.525994\pi$
$229$	−1.55478e12	−0.163145	−0.0815723	+	0.996667i	$0.525994\pi$
$230$	1.04211e13	1.06760
$231$	2.96959e12	0.297051
$232$	−3.54166e12	−0.345958
$233$	−1.08905e12	−0.103894	−0.0519469	−	0.998650i	$-0.516543\pi$
$233$	−1.08905e12	−0.103894	−0.0519469	+	0.998650i	$0.516543\pi$
$234$	−3.58212e12	−0.333774
$235$	−8.31454e12	−0.756772
$236$	2.52134e12	0.224190
$237$	6.21438e12	0.539861
$238$	−5.50404e12	−0.467206
$239$	−3.91266e12	−0.324552	−0.162276	−	0.986745i	$-0.551883\pi$
$239$	−3.91266e12	−0.324552	−0.162276	+	0.986745i	$0.551883\pi$
$240$	−3.02707e12	−0.245392
$241$	−1.32266e13	−1.04798	−0.523991	−	0.851724i	$-0.675557\pi$
$241$	−1.32266e13	−1.04798	−0.523991	+	0.851724i	$0.675557\pi$
$242$	7.78807e12	0.603178
$243$	8.47289e11	0.0641500
$244$	9.27203e12	0.686327
$245$	−3.35581e12	−0.242875
$246$	−1.90596e12	−0.134888
$247$	2.42517e13	1.67845
$248$	−6.62711e12	−0.448580
$249$	−9.40866e12	−0.622919
$250$	−1.65287e13	−1.07045
$251$	2.20325e13	1.39591	0.697957	−	0.716140i	$-0.254093\pi$
$251$	2.20325e13	1.39591	0.697957	+	0.716140i	$0.254093\pi$
$252$	1.01626e12	0.0629941
$253$	−1.99318e13	−1.20888
$254$	2.97842e12	0.176767
$255$	2.95437e13	1.71591
$256$	1.09951e12	0.0625000
$257$	−3.30770e13	−1.84032	−0.920161	−	0.391539i	$-0.871943\pi$
$257$	−3.30770e13	−1.84032	−0.920161	+	0.391539i	$0.871943\pi$
$258$	3.96451e12	0.215914
$259$	6.93213e12	0.369588
$260$	2.30618e13	1.20376
$261$	−6.38219e12	−0.326172
$262$	−1.05885e12	−0.0529881
$263$	−1.23692e13	−0.606157	−0.303078	−	0.952966i	$-0.598014\pi$
$263$	−1.23692e13	−0.606157	−0.303078	+	0.952966i	$0.598014\pi$
$264$	5.78970e12	0.277866
$265$	5.74599e13	2.70092
$266$	−6.88027e12	−0.316779
$267$	2.20604e13	0.994952
$268$	−2.99330e12	−0.132254
$269$	−3.16373e13	−1.36950	−0.684749	−	0.728779i	$-0.740088\pi$
$269$	−3.16373e13	−1.36950	−0.684749	+	0.728779i	$0.740088\pi$
$270$	−5.45488e12	−0.231358
$271$	−6.35915e12	−0.264282	−0.132141	−	0.991231i	$-0.542185\pi$
$271$	−6.35915e12	−0.264282	−0.132141	+	0.991231i	$0.542185\pi$
$272$	−1.07310e13	−0.437031
$273$	−7.74237e12	−0.309015
$274$	−6.30577e12	−0.246666
$275$	6.71168e13	2.57337
$276$	−6.82105e12	−0.256361
$277$	−1.15762e13	−0.426508	−0.213254	−	0.976997i	$-0.568406\pi$
$277$	−1.15762e13	−0.426508	−0.213254	+	0.976997i	$0.568406\pi$
$278$	2.92219e13	1.05551
$279$	−1.19423e13	−0.422925
$280$	−6.54269e12	−0.227189
$281$	−9.66264e12	−0.329011	−0.164506	−	0.986376i	$-0.552603\pi$
$281$	−9.66264e12	−0.329011	−0.164506	+	0.986376i	$0.552603\pi$
$282$	5.44224e12	0.181722
$283$	1.94524e13	0.637011	0.318505	−	0.947921i	$-0.396819\pi$
$283$	1.94524e13	0.637011	0.318505	+	0.947921i	$0.396819\pi$
$284$	−1.15393e13	−0.370620
$285$	3.69308e13	1.16343
$286$	−4.41090e13	−1.36306
$287$	−4.11954e12	−0.124882
$288$	1.98136e12	0.0589256
$289$	7.04611e13	2.05594
$290$	4.10888e13	1.17634
$291$	3.17261e13	0.891260
$292$	2.87665e12	0.0793016
$293$	2.95366e12	0.0799077	0.0399538	−	0.999202i	$-0.487279\pi$
$293$	2.95366e12	0.0799077	0.0399538	+	0.999202i	$0.487279\pi$
$294$	2.19653e12	0.0583212
$295$	−2.92515e13	−0.762302
$296$	1.35153e13	0.345718
$297$	1.04332e13	0.261974
$298$	−3.79302e13	−0.934966
$299$	5.19664e13	1.25757
$300$	2.29688e13	0.545721
$301$	8.56888e12	0.199898
$302$	−2.91593e13	−0.667942
$303$	2.51903e13	0.566629
$304$	−1.34142e13	−0.296320
$305$	−1.07570e14	−2.33368
$306$	−1.93377e13	−0.412037
$307$	−2.94743e13	−0.616855	−0.308427	−	0.951248i	$-0.599803\pi$
$307$	−2.94743e13	−0.616855	−0.308427	+	0.951248i	$0.599803\pi$
$308$	1.25138e13	0.257254
$309$	−2.38522e13	−0.481678
$310$	7.68848e13	1.52529
$311$	−8.57803e13	−1.67188	−0.835941	−	0.548819i	$-0.815078\pi$
$311$	−8.57803e13	−1.67188	−0.835941	+	0.548819i	$0.815078\pi$
$312$	−1.50950e13	−0.289057
$313$	−4.35562e13	−0.819514	−0.409757	−	0.912195i	$-0.634387\pi$
$313$	−4.35562e13	−0.819514	−0.409757	+	0.912195i	$0.634387\pi$
$314$	6.02838e13	1.11452
$315$	−1.17901e13	−0.214196
$316$	2.61873e13	0.467533
$317$	7.73948e12	0.135796	0.0678978	−	0.997692i	$-0.478371\pi$
$317$	7.73948e12	0.135796	0.0678978	+	0.997692i	$0.478371\pi$
$318$	−3.76101e13	−0.648568
$319$	−7.85882e13	−1.33201
$320$	−1.27561e13	−0.212516
$321$	−4.37759e13	−0.716898
$322$	−1.47430e13	−0.237344
$323$	1.30920e14	2.07202
$324$	3.57047e12	0.0555556
$325$	−1.74988e14	−2.67701
$326$	2.27874e13	0.342766
$327$	−5.57639e13	−0.824784
$328$	−8.03172e12	−0.116816
$329$	1.17628e13	0.168242
$330$	−6.71696e13	−0.944814
$331$	4.82151e13	0.667006	0.333503	−	0.942749i	$-0.391769\pi$
$331$	4.82151e13	0.667006	0.333503	+	0.942749i	$0.391769\pi$
$332$	−3.96480e13	−0.539463
$333$	2.43551e13	0.325946
$334$	−6.31437e13	−0.831237
$335$	3.47270e13	0.449698
$336$	4.28249e12	0.0545545
$337$	−4.41136e13	−0.552851	−0.276425	−	0.961035i	$-0.589150\pi$
$337$	−4.41136e13	−0.552851	−0.276425	+	0.961035i	$0.589150\pi$
$338$	5.76527e13	0.710850
$339$	−5.58993e13	−0.678123
$340$	1.24497e14	1.48602
$341$	−1.47053e14	−1.72713
$342$	−2.41729e13	−0.279373
$343$	4.74756e12	0.0539949
$344$	1.67064e13	0.186987
$345$	7.91349e13	0.871692
$346$	8.04158e13	0.871812
$347$	−1.59407e14	−1.70096	−0.850480	−	0.526008i	$-0.823688\pi$
$347$	−1.59407e14	−1.70096	−0.850480	+	0.526008i	$0.823688\pi$
$348$	−2.68945e13	−0.282473
$349$	9.46284e13	0.978321	0.489161	−	0.872194i	$-0.337303\pi$
$349$	9.46284e13	0.978321	0.489161	+	0.872194i	$0.337303\pi$
$350$	4.96445e13	0.505239
$351$	−2.72017e13	−0.272525
$352$	2.43978e13	0.240639
$353$	1.17261e14	1.13866	0.569330	−	0.822109i	$-0.307203\pi$
$353$	1.17261e14	1.13866	0.569330	+	0.822109i	$0.307203\pi$
$354$	1.91464e13	0.183050
$355$	1.33874e14	1.26020
$356$	9.29625e13	0.861654
$357$	−4.17963e13	−0.381472
$358$	2.91757e13	0.262220
$359$	−1.59450e14	−1.41126	−0.705628	−	0.708582i	$-0.749335\pi$
$359$	−1.59450e14	−1.41126	−0.705628	+	0.708582i	$0.749335\pi$
$360$	−2.29868e13	−0.200362
$361$	4.71654e13	0.404887
$362$	−7.60223e13	−0.642753
$363$	5.91407e13	0.492493
$364$	−3.26263e13	−0.267615
$365$	−3.33737e13	−0.269646
$366$	7.04095e13	0.560383
$367$	1.29687e14	1.01680	0.508398	−	0.861122i	$-0.330238\pi$
$367$	1.29687e14	1.01680	0.508398	+	0.861122i	$0.330238\pi$
$368$	−2.87439e13	−0.222015
$369$	−1.44734e13	−0.110135
$370$	−1.56799e14	−1.17553
$371$	−8.12903e13	−0.600457
$372$	−5.03246e13	−0.366264
$373$	−2.41174e13	−0.172954	−0.0864772	−	0.996254i	$-0.527561\pi$
$373$	−2.41174e13	−0.172954	−0.0864772	+	0.996254i	$0.527561\pi$
$374$	−2.38118e14	−1.68267
$375$	−1.25514e14	−0.874021
$376$	2.29336e13	0.157376
$377$	2.04896e14	1.38566
$378$	7.71719e12	0.0514344
$379$	9.25071e13	0.607658	0.303829	−	0.952727i	$-0.401735\pi$
$379$	9.25071e13	0.607658	0.303829	+	0.952727i	$0.401735\pi$
$380$	1.55626e14	1.00756
$381$	2.26174e13	0.144330
$382$	1.02789e14	0.646546
$383$	−1.09560e14	−0.679296	−0.339648	−	0.940553i	$-0.610308\pi$
$383$	−1.09560e14	−0.679296	−0.339648	+	0.940553i	$0.610308\pi$
$384$	8.34942e12	0.0510310
$385$	−1.45180e14	−0.874727
$386$	3.32888e13	0.197728
$387$	3.01055e13	0.176293
$388$	1.33693e14	0.771854
$389$	−5.16713e13	−0.294121	−0.147061	−	0.989127i	$-0.546981\pi$
$389$	−5.16713e13	−0.294121	−0.147061	+	0.989127i	$0.546981\pi$
$390$	1.75126e14	0.982867
$391$	2.80535e14	1.55244
$392$	9.25615e12	0.0505076
$393$	−8.04066e12	−0.0432646
$394$	−2.86980e13	−0.152273
$395$	−3.03814e14	−1.58973
$396$	4.39656e13	0.226876
$397$	1.26534e14	0.643962	0.321981	−	0.946746i	$-0.395651\pi$
$397$	1.26534e14	0.643962	0.321981	+	0.946746i	$0.395651\pi$
$398$	2.52014e13	0.126493
$399$	−5.22471e13	−0.258649
$400$	9.67901e13	0.472608
$401$	−4.13529e13	−0.199164	−0.0995821	−	0.995029i	$-0.531751\pi$
$401$	−4.13529e13	−0.199164	−0.0995821	+	0.995029i	$0.531751\pi$
$402$	−2.27304e13	−0.107985
$403$	3.83400e14	1.79669
$404$	1.06152e14	0.490715
$405$	−4.14230e13	−0.188903
$406$	−5.81296e13	−0.261519
$407$	2.99900e14	1.33109
$408$	−8.14888e13	−0.356835
$409$	−2.70220e14	−1.16745	−0.583727	−	0.811950i	$-0.698406\pi$
$409$	−2.70220e14	−1.16745	−0.583727	+	0.811950i	$0.698406\pi$
$410$	9.31805e13	0.397204
$411$	−4.78844e13	−0.201402
$412$	−1.00513e14	−0.417145
$413$	4.13830e13	0.169472
$414$	−5.17974e13	−0.209318
$415$	4.59979e14	1.83431
$416$	−6.36103e13	−0.250331
$417$	2.21904e14	0.861820
$418$	−2.97657e14	−1.14090
$419$	2.26272e14	0.855961	0.427981	−	0.903788i	$-0.359225\pi$
$419$	2.26272e14	0.855961	0.427981	+	0.903788i	$0.359225\pi$
$420$	−4.96836e13	−0.185499
$421$	−2.05280e14	−0.756475	−0.378237	−	0.925709i	$-0.623470\pi$
$421$	−2.05280e14	−0.756475	−0.378237	+	0.925709i	$0.623470\pi$
$422$	5.62286e13	0.204521
$423$	4.13270e13	0.148376
$424$	−1.58489e14	−0.561676
$425$	−9.44654e14	−3.30471
$426$	−8.76267e13	−0.302610
$427$	1.52183e14	0.518814
$428$	−1.84471e14	−0.620852
$429$	−3.34953e14	−1.11293
$430$	−1.93821e14	−0.635805
$431$	4.30539e14	1.39440	0.697201	−	0.716876i	$-0.254429\pi$
$431$	4.30539e14	1.39440	0.697201	+	0.716876i	$0.254429\pi$
$432$	1.50459e13	0.0481125
$433$	2.23687e14	0.706250	0.353125	−	0.935576i	$-0.385119\pi$
$433$	2.23687e14	0.706250	0.353125	+	0.935576i	$0.385119\pi$
$434$	−1.08771e14	−0.339095
$435$	3.12018e14	0.960481
$436$	−2.34988e14	−0.714284
$437$	3.50680e14	1.05260
$438$	2.18446e13	0.0647495
$439$	4.42276e14	1.29461	0.647304	−	0.762232i	$-0.275896\pi$
$439$	4.42276e14	1.29461	0.647304	+	0.762232i	$0.275896\pi$
$440$	−2.83052e14	−0.818233
$441$	1.66799e13	0.0476190
$442$	6.20825e14	1.75044
$443$	2.83290e14	0.788880	0.394440	−	0.918922i	$-0.370939\pi$
$443$	2.83290e14	0.788880	0.394440	+	0.918922i	$0.370939\pi$
$444$	1.02632e14	0.282278
$445$	−1.07851e15	−2.92984
$446$	−2.42202e14	−0.649884
$447$	−2.88032e14	−0.763396
$448$	1.80464e13	0.0472456
$449$	4.89597e14	1.26615	0.633073	−	0.774092i	$-0.281793\pi$
$449$	4.89597e14	1.26615	0.633073	+	0.774092i	$0.281793\pi$
$450$	1.74419e14	0.445579
$451$	−1.78221e14	−0.449767
$452$	−2.35559e14	−0.587272
$453$	−2.21428e14	−0.545372
$454$	−1.05067e13	−0.0255657
$455$	3.78516e14	0.909958
$456$	−1.01864e14	−0.241944
$457$	3.34142e14	0.784136	0.392068	−	0.919936i	$-0.371760\pi$
$457$	3.34142e14	0.784136	0.392068	+	0.919936i	$0.371760\pi$
$458$	−4.97528e13	−0.115361
$459$	−1.46845e14	−0.336427
$460$	3.33474e14	0.754907
$461$	6.71303e13	0.150163	0.0750817	−	0.997177i	$-0.476078\pi$
$461$	6.71303e13	0.150163	0.0750817	+	0.997177i	$0.476078\pi$
$462$	9.50269e13	0.210047
$463$	1.60328e14	0.350198	0.175099	−	0.984551i	$-0.443976\pi$
$463$	1.60328e14	0.350198	0.175099	+	0.984551i	$0.443976\pi$
$464$	−1.13333e14	−0.244629
$465$	5.83844e14	1.24539
$466$	−3.48496e13	−0.0734641
$467$	3.69287e14	0.769345	0.384673	−	0.923053i	$-0.374314\pi$
$467$	3.69287e14	0.769345	0.384673	+	0.923053i	$0.374314\pi$
$468$	−1.14628e14	−0.236014
$469$	−4.91294e13	−0.0999748
$470$	−2.66065e14	−0.535118
$471$	4.57780e14	0.910002
$472$	8.06830e13	0.158526
$473$	3.70710e14	0.719942
$474$	1.98860e14	0.381739
$475$	−1.18086e15	−2.24069
$476$	−1.76129e14	−0.330365
$477$	−2.85602e14	−0.529553
$478$	−1.25205e14	−0.229493
$479$	−2.72584e14	−0.493917	−0.246959	−	0.969026i	$-0.579431\pi$
$479$	−2.72584e14	−0.493917	−0.246959	+	0.969026i	$0.579431\pi$
$480$	−9.68663e13	−0.173519
$481$	−7.81905e14	−1.38470
$482$	−4.23250e14	−0.741035
$483$	−1.11955e14	−0.193791
$484$	2.49218e14	0.426511
$485$	−1.55105e15	−2.62450
$486$	2.71132e13	0.0453609
$487$	−1.20046e15	−1.98582	−0.992908	−	0.118882i	$-0.962069\pi$
$487$	−1.20046e15	−1.98582	−0.992908	+	0.118882i	$0.962069\pi$
$488$	2.96705e14	0.485306
$489$	1.73042e14	0.279868
$490$	−1.07386e14	−0.171739
$491$	3.66278e14	0.579245	0.289623	−	0.957141i	$-0.406470\pi$
$491$	3.66278e14	0.579245	0.289623	+	0.957141i	$0.406470\pi$
$492$	−6.09909e13	−0.0953799
$493$	1.10611e15	1.71057
$494$	7.76056e14	1.18685
$495$	−5.10069e14	−0.771437
$496$	−2.12068e14	−0.317194
$497$	−1.89396e14	−0.280163
$498$	−3.01077e14	−0.440470
$499$	−2.21334e14	−0.320255	−0.160128	−	0.987096i	$-0.551191\pi$
$499$	−2.21334e14	−0.320255	−0.160128	+	0.987096i	$0.551191\pi$
$500$	−5.28917e14	−0.756924
$501$	−4.79498e14	−0.678702
$502$	7.05040e14	0.987060
$503$	−1.20493e15	−1.66854	−0.834271	−	0.551355i	$-0.814111\pi$
$503$	−1.20493e15	−1.66854	−0.834271	+	0.551355i	$0.814111\pi$
$504$	3.25202e13	0.0445435
$505$	−1.23152e15	−1.66856
$506$	−6.37816e14	−0.854806
$507$	4.37800e14	0.580406
$508$	9.53095e13	0.124993
$509$	6.59225e14	0.855236	0.427618	−	0.903959i	$-0.359353\pi$
$509$	6.59225e14	0.855236	0.427618	+	0.903959i	$0.359353\pi$
$510$	9.45397e14	1.21333
$511$	4.72148e13	0.0599464
$512$	3.51844e13	0.0441942
$513$	−1.83563e14	−0.228107
$514$	−1.05846e15	−1.30130
$515$	1.16611e15	1.41840
$516$	1.26864e14	0.152675
$517$	5.08888e14	0.605932
$518$	2.21828e14	0.261338
$519$	6.10658e14	0.711832
$520$	7.37979e14	0.851188
$521$	5.58029e14	0.636868	0.318434	−	0.947945i	$-0.396843\pi$
$521$	5.58029e14	0.636868	0.318434	+	0.947945i	$0.396843\pi$
$522$	−2.04230e14	−0.230638
$523$	−1.74176e15	−1.94639	−0.973195	−	0.229981i	$-0.926134\pi$
$523$	−1.74176e15	−1.94639	−0.973195	+	0.229981i	$0.926134\pi$
$524$	−3.38833e13	−0.0374683
$525$	3.76988e14	0.412526
$526$	−3.95814e14	−0.428618
$527$	2.06974e15	2.21798
$528$	1.85271e14	0.196481
$529$	−2.01376e14	−0.211350
$530$	1.83872e15	1.90984
$531$	1.45393e14	0.149460
$532$	−2.20169e14	−0.223997
$533$	4.64661e14	0.467882
$534$	7.05934e14	0.703538
$535$	2.14016e15	2.11106
$536$	−9.57857e13	−0.0935178
$537$	2.21553e14	0.214101
$538$	−1.01239e15	−0.968382
$539$	2.05391e14	0.194465
$540$	−1.74556e14	−0.163595
$541$	−1.48330e15	−1.37608	−0.688041	−	0.725672i	$-0.741529\pi$
$541$	−1.48330e15	−1.37608	−0.688041	+	0.725672i	$0.741529\pi$
$542$	−2.03493e14	−0.186876
$543$	−5.77295e14	−0.524806
$544$	−3.43393e14	−0.309028
$545$	2.72623e15	2.42875
$546$	−2.47756e14	−0.218506
$547$	−3.46714e14	−0.302720	−0.151360	−	0.988479i	$-0.548365\pi$
$547$	−3.46714e14	−0.302720	−0.151360	+	0.988479i	$0.548365\pi$
$548$	−2.01785e14	−0.174420
$549$	5.34672e14	0.457551
$550$	2.14774e15	1.81964
$551$	1.38268e15	1.15982
$552$	−2.18274e14	−0.181274
$553$	4.29815e14	0.353422
$554$	−3.70438e14	−0.301586
$555$	−1.19069e15	−0.959816
$556$	9.35101e14	0.746358
$557$	1.20099e15	0.949150	0.474575	−	0.880215i	$-0.342602\pi$
$557$	1.20099e15	0.949150	0.474575	+	0.880215i	$0.342602\pi$
$558$	−3.82153e14	−0.299053
$559$	−9.66521e14	−0.748939
$560$	−2.09366e14	−0.160647
$561$	−1.80821e15	−1.37389
$562$	−3.09204e14	−0.232646
$563$	−1.44205e15	−1.07445	−0.537224	−	0.843440i	$-0.680527\pi$
$563$	−1.44205e15	−1.07445	−0.537224	+	0.843440i	$0.680527\pi$
$564$	1.74152e14	0.128497
$565$	2.73286e15	1.99687
$566$	6.22475e14	0.450435
$567$	5.86024e13	0.0419961
$568$	−3.69258e14	−0.262068
$569$	−2.07341e15	−1.45736	−0.728682	−	0.684852i	$-0.759867\pi$
$569$	−2.07341e15	−1.45736	−0.728682	+	0.684852i	$0.759867\pi$
$570$	1.18178e15	0.822672
$571$	−1.77668e15	−1.22493	−0.612463	−	0.790499i	$-0.709821\pi$
$571$	−1.77668e15	−1.22493	−0.612463	+	0.790499i	$0.709821\pi$
$572$	−1.41149e15	−0.963828
$573$	7.80557e14	0.527903
$574$	−1.31825e14	−0.0883046
$575$	−2.53033e15	−1.67882
$576$	6.34034e13	0.0416667
$577$	−3.41044e14	−0.221995	−0.110998	−	0.993821i	$-0.535405\pi$
$577$	−3.41044e14	−0.221995	−0.110998	+	0.993821i	$0.535405\pi$
$578$	2.25475e15	1.45377
$579$	2.52787e14	0.161444
$580$	1.31484e15	0.831801
$581$	−6.50746e14	−0.407796
$582$	1.01523e15	0.630216
$583$	−3.51681e15	−2.16258
$584$	9.20529e13	0.0560747
$585$	1.32986e15	0.802508
$586$	9.45171e13	0.0565033
$587$	3.92265e14	0.232311	0.116155	−	0.993231i	$-0.462943\pi$
$587$	3.92265e14	0.232311	0.116155	+	0.993231i	$0.462943\pi$
$588$	7.02889e13	0.0412393
$589$	2.58726e15	1.50385
$590$	−9.36048e14	−0.539029
$591$	−2.17925e14	−0.124330
$592$	4.32490e14	0.244460
$593$	−1.25539e14	−0.0703034	−0.0351517	−	0.999382i	$-0.511191\pi$
$593$	−1.25539e14	−0.0703034	−0.0351517	+	0.999382i	$0.511191\pi$
$594$	3.33863e14	0.185244
$595$	2.04338e15	1.12332
$596$	−1.21377e15	−0.661121
$597$	1.91373e14	0.103281
$598$	1.66293e15	0.889234
$599$	−3.25775e15	−1.72612	−0.863058	−	0.505105i	$-0.831454\pi$
$599$	−3.25775e15	−1.72612	−0.863058	+	0.505105i	$0.831454\pi$
$600$	7.35000e14	0.385883
$601$	6.02750e14	0.313565	0.156783	−	0.987633i	$-0.449888\pi$
$601$	6.02750e14	0.313565	0.156783	+	0.987633i	$0.449888\pi$
$602$	2.74204e14	0.141349
$603$	−1.72609e14	−0.0881695
$604$	−9.33097e14	−0.472306
$605$	−2.89132e15	−1.45025
$606$	8.06089e14	0.400667
$607$	2.75962e15	1.35929	0.679643	−	0.733543i	$-0.262135\pi$
$607$	2.75962e15	1.35929	0.679643	+	0.733543i	$0.262135\pi$
$608$	−4.29255e14	−0.209530
$609$	−4.41422e14	−0.213530
$610$	−3.44224e15	−1.65016
$611$	−1.32678e15	−0.630337
$612$	−6.18806e14	−0.291354
$613$	−4.01542e14	−0.187369	−0.0936846	−	0.995602i	$-0.529865\pi$
$613$	−4.01542e14	−0.187369	−0.0936846	+	0.995602i	$0.529865\pi$
$614$	−9.43179e14	−0.436182
$615$	7.07589e14	0.324316
$616$	4.00443e14	0.181906
$617$	3.13173e14	0.140999	0.0704994	−	0.997512i	$-0.477541\pi$
$617$	3.13173e14	0.140999	0.0704994	+	0.997512i	$0.477541\pi$
$618$	−7.63270e14	−0.340598
$619$	−3.78436e14	−0.167376	−0.0836882	−	0.996492i	$-0.526670\pi$
$619$	−3.78436e14	−0.167376	−0.0836882	+	0.996492i	$0.526670\pi$
$620$	2.46032e15	1.07854
$621$	−3.93336e14	−0.170907
$622$	−2.74497e15	−1.18220
$623$	1.52580e15	0.651349
$624$	−4.83041e14	−0.204394
$625$	1.62912e15	0.683303
$626$	−1.39380e15	−0.579484
$627$	−2.26033e15	−0.931538
$628$	1.92908e15	0.788085
$629$	−4.22103e15	−1.70938
$630$	−3.77285e14	−0.151459
$631$	3.27226e14	0.130223	0.0651113	−	0.997878i	$-0.479260\pi$
$631$	3.27226e14	0.130223	0.0651113	+	0.997878i	$0.479260\pi$
$632$	8.37995e14	0.330596
$633$	4.26986e14	0.166991
$634$	2.47663e14	0.0960220
$635$	−1.10574e15	−0.425008
$636$	−1.20352e15	−0.458607
$637$	−5.35498e14	−0.202298
$638$	−2.51482e15	−0.941875
$639$	−6.65415e14	−0.247080
$640$	−4.08194e14	−0.150271
$641$	1.30890e15	0.477737	0.238868	−	0.971052i	$-0.423224\pi$
$641$	1.30890e15	0.477737	0.238868	+	0.971052i	$0.423224\pi$
$642$	−1.40083e15	−0.506924
$643$	−4.99571e15	−1.79241	−0.896204	−	0.443642i	$-0.853686\pi$
$643$	−4.99571e15	−1.79241	−0.896204	+	0.443642i	$0.853686\pi$
$644$	−4.71776e14	−0.167828
$645$	−1.47183e15	−0.519132
$646$	4.18945e15	1.46514
$647$	4.61019e15	1.59862	0.799310	−	0.600919i	$-0.205199\pi$
$647$	4.61019e15	1.59862	0.799310	+	0.600919i	$0.205199\pi$
$648$	1.14255e14	0.0392837
$649$	1.79033e15	0.610360
$650$	−5.59962e15	−1.89293
$651$	−8.25982e14	−0.276870
$652$	7.29197e14	0.242372
$653$	−2.26916e15	−0.747897	−0.373948	−	0.927449i	$-0.621996\pi$
$653$	−2.26916e15	−0.747897	−0.373948	+	0.927449i	$0.621996\pi$
$654$	−1.78444e15	−0.583210
$655$	3.93099e14	0.127402
$656$	−2.57015e14	−0.0826014
$657$	1.65882e14	0.0528677
$658$	3.76411e14	0.118965
$659$	3.90658e15	1.22441	0.612206	−	0.790698i	$-0.290282\pi$
$659$	3.90658e15	1.22441	0.612206	+	0.790698i	$0.290282\pi$
$660$	−2.14943e15	−0.668084
$661$	2.21657e15	0.683240	0.341620	−	0.939838i	$-0.389024\pi$
$661$	2.21657e15	0.683240	0.341620	+	0.939838i	$0.389024\pi$
$662$	1.54288e15	0.471644
$663$	4.71439e15	1.42923
$664$	−1.26874e15	−0.381458
$665$	2.55430e15	0.761646
$666$	7.79362e14	0.230479
$667$	2.96280e15	0.868981
$668$	−2.02060e15	−0.587773
$669$	−1.83922e15	−0.530628
$670$	1.11126e15	0.317984
$671$	6.58378e15	1.86853
$672$	1.37040e14	0.0385758
$673$	−1.17493e15	−0.328041	−0.164020	−	0.986457i	$-0.552446\pi$
$673$	−1.17493e15	−0.328041	−0.164020	+	0.986457i	$0.552446\pi$
$674$	−1.41164e15	−0.390925
$675$	1.32449e15	0.363814
$676$	1.84489e15	0.502647
$677$	−9.32497e13	−0.0252005	−0.0126003	−	0.999921i	$-0.504011\pi$
$677$	−9.32497e13	−0.0252005	−0.0126003	+	0.999921i	$0.504011\pi$
$678$	−1.78878e15	−0.479505
$679$	2.19432e15	0.583467
$680$	3.98390e15	1.05077
$681$	−7.97849e13	−0.0208743
$682$	−4.70570e15	−1.22127
$683$	−5.24115e15	−1.34931	−0.674656	−	0.738132i	$-0.735708\pi$
$683$	−5.24115e15	−1.34931	−0.674656	+	0.738132i	$0.735708\pi$
$684$	−7.73531e14	−0.197547
$685$	2.34102e15	0.593071
$686$	1.51922e14	0.0381802
$687$	−3.77811e14	−0.0941916
$688$	5.34606e14	0.132220
$689$	9.16908e15	2.24968
$690$	2.53232e15	0.616379
$691$	−7.37295e15	−1.78038	−0.890189	−	0.455592i	$-0.849428\pi$
$691$	−7.37295e15	−1.78038	−0.890189	+	0.455592i	$0.849428\pi$
$692$	2.57331e15	0.616464
$693$	7.21611e14	0.171502
$694$	−5.10101e15	−1.20276
$695$	−1.08486e16	−2.53781
$696$	−8.60624e14	−0.199739
$697$	2.50842e15	0.577590
$698$	3.02811e15	0.691778
$699$	−2.64639e14	−0.0599832
$700$	1.58862e15	0.357258
$701$	5.62192e14	0.125440	0.0627199	−	0.998031i	$-0.480023\pi$
$701$	5.62192e14	0.125440	0.0627199	+	0.998031i	$0.480023\pi$
$702$	−8.70455e14	−0.192705
$703$	−5.27645e15	−1.15901
$704$	7.80728e14	0.170157
$705$	−2.02043e15	−0.436922
$706$	3.75236e15	0.805154
$707$	1.74228e15	0.370946
$708$	6.12686e14	0.129436
$709$	7.37200e15	1.54536	0.772682	−	0.634793i	$-0.218915\pi$
$709$	7.37200e15	1.54536	0.772682	+	0.634793i	$0.218915\pi$
$710$	4.28397e15	0.891099
$711$	1.51009e15	0.311689
$712$	2.97480e15	0.609281
$713$	5.54395e15	1.12675
$714$	−1.33748e15	−0.269742
$715$	1.63755e16	3.27726
$716$	9.33623e14	0.185417
$717$	−9.50777e14	−0.187380
$718$	−5.10241e15	−0.997909
$719$	1.30322e15	0.252935	0.126468	−	0.991971i	$-0.459636\pi$
$719$	1.30322e15	0.252935	0.126468	+	0.991971i	$0.459636\pi$
$720$	−7.35578e14	−0.141677
$721$	−1.64973e15	−0.315332
$722$	1.50929e15	0.286298
$723$	−3.21406e15	−0.605052
$724$	−2.43272e15	−0.454495
$725$	−9.97673e15	−1.84982
$726$	1.89250e15	0.348245
$727$	−6.80280e15	−1.24236	−0.621182	−	0.783667i	$-0.713347\pi$
$727$	−6.80280e15	−1.24236	−0.621182	+	0.783667i	$0.713347\pi$
$728$	−1.04404e15	−0.189232
$729$	2.05891e14	0.0370370
$730$	−1.06796e15	−0.190668
$731$	−5.21766e15	−0.924549
$732$	2.25310e15	0.396251
$733$	4.47676e15	0.781433	0.390716	−	0.920511i	$-0.372227\pi$
$733$	4.47676e15	0.781433	0.390716	+	0.920511i	$0.372227\pi$
$734$	4.14999e15	0.718983
$735$	−8.15461e14	−0.140224
$736$	−9.19804e14	−0.156988
$737$	−2.12545e15	−0.360064
$738$	−4.63149e14	−0.0778773
$739$	4.60655e15	0.768832	0.384416	−	0.923160i	$-0.374403\pi$
$739$	4.60655e15	0.768832	0.384416	+	0.923160i	$0.374403\pi$
$740$	−5.01756e15	−0.831225
$741$	5.89317e15	0.969056
$742$	−2.60129e15	−0.424587
$743$	−5.69608e15	−0.922864	−0.461432	−	0.887175i	$-0.652664\pi$
$743$	−5.69608e15	−0.922864	−0.461432	+	0.887175i	$0.652664\pi$
$744$	−1.61039e15	−0.258988
$745$	1.40816e16	2.24798
$746$	−7.71756e14	−0.122297
$747$	−2.28630e15	−0.359642
$748$	−7.61977e15	−1.18982
$749$	−3.02774e15	−0.469320
$750$	−4.01646e15	−0.618026
$751$	5.99691e14	0.0916026	0.0458013	−	0.998951i	$-0.485416\pi$
$751$	5.99691e14	0.0916026	0.0458013	+	0.998951i	$0.485416\pi$
$752$	7.33874e14	0.111282
$753$	5.35390e15	0.805931
$754$	6.55669e15	0.979810
$755$	1.08254e16	1.60596
$756$	2.46950e14	0.0363696
$757$	9.66233e15	1.41271	0.706357	−	0.707856i	$-0.250337\pi$
$757$	9.66233e15	1.41271	0.706357	+	0.707856i	$0.250337\pi$
$758$	2.96023e15	0.429679
$759$	−4.84342e15	−0.697946
$760$	4.98003e15	0.712454
$761$	−1.14206e16	−1.62208	−0.811041	−	0.584989i	$-0.801099\pi$
$761$	−1.14206e16	−1.62208	−0.811041	+	0.584989i	$0.801099\pi$
$762$	7.23757e14	0.102056
$763$	−3.85689e15	−0.539948
$764$	3.28926e15	0.457177
$765$	7.17911e15	0.990679
$766$	−3.50592e15	−0.480335
$767$	−4.66777e15	−0.634943
$768$	2.67181e14	0.0360844
$769$	−1.12027e16	−1.50220	−0.751098	−	0.660191i	$-0.770475\pi$
$769$	−1.12027e16	−1.50220	−0.751098	+	0.660191i	$0.770475\pi$
$770$	−4.64576e15	−0.618526
$771$	−8.03771e15	−1.06251
$772$	1.06524e15	0.139815
$773$	1.34319e16	1.75045	0.875226	−	0.483714i	$-0.160712\pi$
$773$	1.34319e16	1.75045	0.875226	+	0.483714i	$0.160712\pi$
$774$	9.63377e14	0.124658
$775$	−1.86683e16	−2.39854
$776$	4.27819e15	0.545783
$777$	1.68451e15	0.213382
$778$	−1.65348e15	−0.207975
$779$	3.13562e15	0.391623
$780$	5.60403e15	0.694992
$781$	−8.19371e15	−1.00902
$782$	8.97712e15	1.09774
$783$	−1.55087e15	−0.188316
$784$	2.96197e14	0.0357143
$785$	−2.23804e16	−2.67969
$786$	−2.57301e14	−0.0305927
$787$	8.66322e14	0.102287	0.0511433	−	0.998691i	$-0.483713\pi$
$787$	8.66322e14	0.102287	0.0511433	+	0.998691i	$0.483713\pi$
$788$	−9.18335e14	−0.107673
$789$	−3.00571e15	−0.349965
$790$	−9.72205e15	−1.12411
$791$	−3.86626e15	−0.443936
$792$	1.40690e15	0.160426
$793$	−1.71653e16	−1.94379
$794$	4.04909e15	0.455350
$795$	1.39628e16	1.55938
$796$	8.06445e14	0.0894444
$797$	−7.66872e14	−0.0844700	−0.0422350	−	0.999108i	$-0.513448\pi$
$797$	−7.66872e14	−0.0844700	−0.0422350	+	0.999108i	$0.513448\pi$
$798$	−1.67191e15	−0.182893
$799$	−7.16248e15	−0.778137
$800$	3.09728e15	0.334184
$801$	5.36069e15	0.574436
$802$	−1.32329e15	−0.140830
$803$	2.04262e15	0.215900
$804$	−7.27373e14	−0.0763570
$805$	5.47333e15	0.570656
$806$	1.22688e16	1.27045
$807$	−7.68786e15	−0.790680
$808$	3.39685e15	0.346988
$809$	8.15810e15	0.827699	0.413849	−	0.910345i	$-0.364184\pi$
$809$	8.15810e15	0.827699	0.413849	+	0.910345i	$0.364184\pi$
$810$	−1.32554e15	−0.133575
$811$	5.27977e15	0.528445	0.264223	−	0.964462i	$-0.414885\pi$
$811$	5.27977e15	0.528445	0.264223	+	0.964462i	$0.414885\pi$
$812$	−1.86015e15	−0.184922
$813$	−1.54527e15	−0.152584
$814$	9.59680e15	0.941223
$815$	−8.45983e15	−0.824128
$816$	−2.60764e15	−0.252320
$817$	−6.52228e15	−0.626871
$818$	−8.64705e15	−0.825515
$819$	−1.88140e15	−0.178410
$820$	2.98178e15	0.280866
$821$	1.67086e16	1.56334	0.781669	−	0.623694i	$-0.214369\pi$
$821$	1.67086e16	1.56334	0.781669	+	0.623694i	$0.214369\pi$
$822$	−1.53230e15	−0.142413
$823$	−9.88154e15	−0.912275	−0.456137	−	0.889909i	$-0.650767\pi$
$823$	−9.88154e15	−0.912275	−0.456137	+	0.889909i	$0.650767\pi$
$824$	−3.21641e15	−0.294966
$825$	1.63094e16	1.48573
$826$	1.32426e15	0.119834
$827$	1.12760e16	1.01362	0.506810	−	0.862058i	$-0.330825\pi$
$827$	1.12760e16	1.01362	0.506810	+	0.862058i	$0.330825\pi$
$828$	−1.65752e15	−0.148010
$829$	−1.73581e16	−1.53975	−0.769877	−	0.638192i	$-0.779682\pi$
$829$	−1.73581e16	−1.53975	−0.769877	+	0.638192i	$0.779682\pi$
$830$	1.47193e16	1.29706
$831$	−2.81301e15	−0.246244
$832$	−2.03553e15	−0.177010
$833$	−2.89083e15	−0.249732
$834$	7.10092e15	0.609399
$835$	2.34421e16	1.99858
$836$	−9.52501e15	−0.806735
$837$	−2.90197e15	−0.244176
$838$	7.24071e15	0.605256
$839$	6.05657e14	0.0502963	0.0251481	−	0.999684i	$-0.491994\pi$
$839$	6.05657e14	0.0502963	0.0251481	+	0.999684i	$0.491994\pi$
$840$	−1.58987e15	−0.131168
$841$	−5.18593e14	−0.0425059
$842$	−6.56895e15	−0.534909
$843$	−2.34802e15	−0.189955
$844$	1.79932e15	0.144619
$845$	−2.14036e16	−1.70913
$846$	1.32247e15	0.104917
$847$	4.09044e15	0.322412
$848$	−5.07164e15	−0.397165
$849$	4.72692e15	0.367778
$850$	−3.02289e16	−2.33679
$851$	−1.13063e16	−0.868379
$852$	−2.80406e15	−0.213978
$853$	−1.65022e16	−1.25119	−0.625594	−	0.780149i	$-0.715143\pi$
$853$	−1.65022e16	−1.25119	−0.625594	+	0.780149i	$0.715143\pi$
$854$	4.86984e15	0.366857
$855$	8.97417e15	0.671708
$856$	−5.90308e15	−0.439009
$857$	−1.00335e16	−0.741411	−0.370705	−	0.928751i	$-0.620884\pi$
$857$	−1.00335e16	−0.741411	−0.370705	+	0.928751i	$0.620884\pi$
$858$	−1.07185e16	−0.786962
$859$	4.67219e14	0.0340846	0.0170423	−	0.999855i	$-0.494575\pi$
$859$	4.67219e14	0.0340846	0.0170423	+	0.999855i	$0.494575\pi$
$860$	−6.20226e15	−0.449582
$861$	−1.00105e15	−0.0721004
$862$	1.37773e16	0.985991
$863$	−2.29922e15	−0.163501	−0.0817507	−	0.996653i	$-0.526051\pi$
$863$	−2.29922e15	−0.163501	−0.0817507	+	0.996653i	$0.526051\pi$
$864$	4.81469e14	0.0340207
$865$	−2.98544e16	−2.09614
$866$	7.15800e15	0.499394
$867$	1.71220e16	1.18700
$868$	−3.48068e15	−0.239776
$869$	1.85948e16	1.27287
$870$	9.98458e15	0.679163
$871$	5.54151e15	0.374566
$872$	−7.51963e15	−0.505075
$873$	7.70943e15	0.514569
$874$	1.12218e16	0.744300
$875$	−8.68116e15	−0.572181
$876$	6.99027e14	0.0457848
$877$	9.62704e15	0.626607	0.313303	−	0.949653i	$-0.398564\pi$
$877$	9.62704e15	0.626607	0.313303	+	0.949653i	$0.398564\pi$
$878$	1.41528e16	0.915427
$879$	7.17739e14	0.0461347
$880$	−9.05767e15	−0.578578
$881$	−1.96850e15	−0.124959	−0.0624795	−	0.998046i	$-0.519901\pi$
$881$	−1.96850e15	−0.124959	−0.0624795	+	0.998046i	$0.519901\pi$
$882$	5.33756e14	0.0336718
$883$	1.44941e16	0.908674	0.454337	−	0.890830i	$-0.349876\pi$
$883$	1.44941e16	0.908674	0.454337	+	0.890830i	$0.349876\pi$
$884$	1.98664e16	1.23775
$885$	−7.10812e15	−0.440115
$886$	9.06528e15	0.557822
$887$	−1.24437e15	−0.0760977	−0.0380488	−	0.999276i	$-0.512114\pi$
$887$	−1.24437e15	−0.0760977	−0.0380488	+	0.999276i	$0.512114\pi$
$888$	3.28422e15	0.199600
$889$	1.56432e15	0.0944859
$890$	−3.45123e16	−2.07171
$891$	2.53528e15	0.151251
$892$	−7.75046e15	−0.459538
$893$	−8.95338e15	−0.527599
$894$	−9.21704e15	−0.539803
$895$	−1.08315e16	−0.630466
$896$	5.77484e14	0.0334077
$897$	1.26278e16	0.726057
$898$	1.56671e16	0.895300
$899$	2.18591e16	1.24152
$900$	5.58141e15	0.315072
$901$	4.94983e16	2.77718
$902$	−5.70307e15	−0.318033
$903$	2.08224e15	0.115411
$904$	−7.53790e15	−0.415264
$905$	2.82233e16	1.54540
$906$	−7.08570e15	−0.385636
$907$	−1.98585e16	−1.07425	−0.537126	−	0.843502i	$-0.680490\pi$
$907$	−1.98585e16	−1.07425	−0.537126	+	0.843502i	$0.680490\pi$
$908$	−3.36213e14	−0.0180777
$909$	6.12124e15	0.327143
$910$	1.21125e16	0.643437
$911$	−1.70455e16	−0.900033	−0.450016	−	0.893020i	$-0.648582\pi$
$911$	−1.70455e16	−0.900033	−0.450016	+	0.893020i	$0.648582\pi$
$912$	−3.25966e15	−0.171080
$913$	−2.81528e16	−1.46870
$914$	1.06925e16	0.554468
$915$	−2.61395e16	−1.34735
$916$	−1.59209e15	−0.0815723
$917$	−5.56129e14	−0.0283233
$918$	−4.69905e15	−0.237890
$919$	2.89063e16	1.45465	0.727323	−	0.686296i	$-0.240764\pi$
$919$	2.89063e16	1.45465	0.727323	+	0.686296i	$0.240764\pi$
$920$	1.06712e16	0.533800
$921$	−7.16226e15	−0.356141
$922$	2.14817e15	0.106181
$923$	2.13628e16	1.04966
$924$	3.04086e15	0.148525
$925$	3.80722e16	1.84854
$926$	5.13049e15	0.247627
$927$	−5.79608e15	−0.278097
$928$	−3.62666e15	−0.172979
$929$	1.45121e16	0.688089	0.344044	−	0.938953i	$-0.388203\pi$
$929$	1.45121e16	0.688089	0.344044	+	0.938953i	$0.388203\pi$
$930$	1.86830e16	0.880625
$931$	−3.61365e15	−0.169326
$932$	−1.11519e15	−0.0519469
$933$	−2.08446e16	−0.965262
$934$	1.18172e16	0.544009
$935$	8.84012e16	4.04571
$936$	−3.66809e15	−0.166887
$937$	−6.48764e15	−0.293440	−0.146720	−	0.989178i	$-0.546872\pi$
$937$	−6.48764e15	−0.293440	−0.146720	+	0.989178i	$0.546872\pi$
$938$	−1.57214e15	−0.0706928
$939$	−1.05842e16	−0.473147
$940$	−8.51409e15	−0.378386
$941$	−2.22072e16	−0.981185	−0.490593	−	0.871389i	$-0.663220\pi$
$941$	−2.22072e16	−0.981185	−0.490593	+	0.871389i	$0.663220\pi$
$942$	1.46490e16	0.643468
$943$	6.71899e15	0.293420
$944$	2.58185e15	0.112095
$945$	−2.86501e15	−0.123666
$946$	1.18627e16	0.509076
$947$	1.75990e16	0.750867	0.375434	−	0.926849i	$-0.377494\pi$
$947$	1.75990e16	0.750867	0.375434	+	0.926849i	$0.377494\pi$
$948$	6.36352e15	0.269930
$949$	−5.32556e15	−0.224595
$950$	−3.77874e16	−1.58441
$951$	1.88069e15	0.0784016
$952$	−5.63614e15	−0.233603
$953$	−5.29841e15	−0.218340	−0.109170	−	0.994023i	$-0.534819\pi$
$953$	−5.29841e15	−0.218340	−0.109170	+	0.994023i	$0.534819\pi$
$954$	−9.13926e15	−0.374451
$955$	−3.81606e16	−1.55452
$956$	−4.00656e15	−0.162276
$957$	−1.90969e16	−0.769038
$958$	−8.72267e15	−0.349252
$959$	−3.31191e15	−0.131849
$960$	−3.09972e15	−0.122696
$961$	1.54939e16	0.609793
$962$	−2.50210e16	−0.979131
$963$	−1.06375e16	−0.413901
$964$	−1.35440e16	−0.523991
$965$	−1.23585e16	−0.475406
$966$	−3.58255e15	−0.137031
$967$	−5.31526e15	−0.202152	−0.101076	−	0.994879i	$-0.532229\pi$
$967$	−5.31526e15	−0.202152	−0.101076	+	0.994879i	$0.532229\pi$
$968$	7.97499e15	0.301589
$969$	3.18136e16	1.19628
$970$	−4.96336e16	−1.85580
$971$	3.87082e16	1.43912	0.719560	−	0.694431i	$-0.244344\pi$
$971$	3.87082e16	1.43912	0.719560	+	0.694431i	$0.244344\pi$
$972$	8.67624e14	0.0320750
$973$	1.53479e16	0.564194
$974$	−3.84148e16	−1.40418
$975$	−4.25221e16	−1.54557
$976$	9.49456e15	0.343163
$977$	−5.46075e16	−1.96260	−0.981301	−	0.192478i	$-0.938348\pi$
$977$	−5.46075e16	−1.96260	−0.981301	+	0.192478i	$0.938348\pi$
$978$	5.53734e15	0.197896
$979$	6.60097e16	2.34587
$980$	−3.43635e15	−0.121438
$981$	−1.35506e16	−0.476189
$982$	1.17209e16	0.409588
$983$	−1.13343e16	−0.393867	−0.196933	−	0.980417i	$-0.563098\pi$
$983$	−1.13343e16	−0.393867	−0.196933	+	0.980417i	$0.563098\pi$
$984$	−1.95171e15	−0.0674438
$985$	1.06541e16	0.366116
$986$	3.53956e16	1.20956
$987$	2.85837e15	0.0971346
$988$	2.48338e16	0.839227
$989$	−1.39759e16	−0.469677
$990$	−1.63222e16	−0.545488
$991$	−1.87120e16	−0.621891	−0.310946	−	0.950428i	$-0.600646\pi$
$991$	−1.87120e16	−0.621891	−0.310946	+	0.950428i	$0.600646\pi$
$992$	−6.78616e15	−0.224290
$993$	1.17163e16	0.385096
$994$	−6.06067e15	−0.198105
$995$	−9.35603e15	−0.304134
$996$	−9.63447e15	−0.311459
$997$	1.48619e15	0.0477807	0.0238903	−	0.999715i	$-0.492395\pi$
$997$	1.48619e15	0.0477807	0.0238903	+	0.999715i	$0.492395\pi$
$998$	−7.08270e15	−0.226454
$999$	5.91828e15	0.188185

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	42.12.a.f.1.1	✓	1
3.2	odd	2		126.12.a.c.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
42.12.a.f.1.1	✓	1	1.1	even	1	trivial
126.12.a.c.1.1		1	3.2	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:	\( N \)	\(=\)	\( 42 = 2 \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	42.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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