LMFDB - Embedded newform 150.16.a.f.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [150,16,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, 16, names="a")

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

chi := DirichletCharacter("150.1");

S:= CuspForms(chi, 16);

N := Newforms(S);

Level:	$ N $	$=$	$ 150 = 2 \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 16 $
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:	$214.040257650$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 6)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	150.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-128.000 q^{2} +2187.00 q^{3} +16384.0 q^{4} -279936. q^{6} +3.03453e6 q^{7} -2.09715e6 q^{8} +4.78297e6 q^{9} +O(q^{10})$

\(q-128.000 q^{2} +2187.00 q^{3} +16384.0 q^{4} -279936. q^{6} +3.03453e6 q^{7} -2.09715e6 q^{8} +4.78297e6 q^{9} -1.03452e8 q^{11} +3.58318e7 q^{12} +1.04366e8 q^{13} -3.88420e8 q^{14} +2.68435e8 q^{16} -9.97690e8 q^{17} -6.12220e8 q^{18} +4.93402e9 q^{19} +6.63651e9 q^{21} +1.32418e10 q^{22} -8.32492e9 q^{23} -4.58647e9 q^{24} -1.33588e10 q^{26} +1.04604e10 q^{27} +4.97177e10 q^{28} +1.04128e11 q^{29} -2.96697e11 q^{31} -3.43597e10 q^{32} -2.26249e11 q^{33} +1.27704e11 q^{34} +7.83642e10 q^{36} +1.78337e11 q^{37} -6.31554e11 q^{38} +2.28248e11 q^{39} -1.79088e12 q^{41} -8.49474e11 q^{42} +2.86346e12 q^{43} -1.69495e12 q^{44} +1.06559e12 q^{46} -4.33291e12 q^{47} +5.87068e11 q^{48} +4.46080e12 q^{49} -2.18195e12 q^{51} +1.70993e12 q^{52} -9.73232e12 q^{53} -1.33893e12 q^{54} -6.36387e12 q^{56} +1.07907e13 q^{57} -1.33284e13 q^{58} -1.35148e13 q^{59} +5.35266e12 q^{61} +3.79772e13 q^{62} +1.45141e13 q^{63} +4.39805e12 q^{64} +2.89599e13 q^{66} +5.32339e13 q^{67} -1.63461e13 q^{68} -1.82066e13 q^{69} -2.02297e13 q^{71} -1.00306e13 q^{72} -2.62642e13 q^{73} -2.28272e13 q^{74} +8.08389e13 q^{76} -3.13927e14 q^{77} -2.92158e13 q^{78} -3.39031e14 q^{79} +2.28768e13 q^{81} +2.29233e14 q^{82} -1.31685e14 q^{83} +1.08733e14 q^{84} -3.66523e14 q^{86} +2.27728e14 q^{87} +2.16954e14 q^{88} -3.93521e13 q^{89} +3.16701e14 q^{91} -1.36395e14 q^{92} -6.48876e14 q^{93} +5.54612e14 q^{94} -7.51447e13 q^{96} -1.12875e15 q^{97} -5.70982e14 q^{98} -4.94806e14 q^{99} +O(q^{100})\)

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 $

$2$	−128.000	−0.707107
$2$	−128.000	−0.707107
$3$	2187.00	0.577350
$3$	2187.00	0.577350
$4$	16384.0	0.500000
$5$	0	0
$5$	0	0
$6$	−279936.	−0.408248
$7$	3.03453e6	1.39269	0.696347	−	0.717705i	$-0.254807\pi$
$7$	3.03453e6	1.39269	0.696347	+	0.717705i	$0.254807\pi$
$8$	−2.09715e6	−0.353553
$9$	4.78297e6	0.333333
$10$	0	0
$11$	−1.03452e8	−1.60064	−0.800318	−	0.599576i	$-0.795336\pi$
$11$	−1.03452e8	−1.60064	−0.800318	+	0.599576i	$0.795336\pi$
$12$	3.58318e7	0.288675
$13$	1.04366e8	0.461300	0.230650	−	0.973037i	$-0.425915\pi$
$13$	1.04366e8	0.461300	0.230650	+	0.973037i	$0.425915\pi$
$14$	−3.88420e8	−0.984784
$15$	0	0
$16$	2.68435e8	0.250000
$17$	−9.97690e8	−0.589697	−0.294848	−	0.955544i	$-0.595269\pi$
$17$	−9.97690e8	−0.589697	−0.294848	+	0.955544i	$0.595269\pi$
$18$	−6.12220e8	−0.235702
$19$	4.93402e9	1.26633	0.633167	−	0.774015i	$-0.281754\pi$
$19$	4.93402e9	1.26633	0.633167	+	0.774015i	$0.281754\pi$
$20$	0	0
$21$	6.63651e9	0.804073
$22$	1.32418e10	1.13182
$23$	−8.32492e9	−0.509825	−0.254913	−	0.966964i	$-0.582047\pi$
$23$	−8.32492e9	−0.509825	−0.254913	+	0.966964i	$0.582047\pi$
$24$	−4.58647e9	−0.204124
$25$	0	0
$26$	−1.33588e10	−0.326188
$27$	1.04604e10	0.192450
$28$	4.97177e10	0.696347
$29$	1.04128e11	1.12094	0.560472	−	0.828174i	$-0.310620\pi$
$29$	1.04128e11	1.12094	0.560472	+	0.828174i	$0.310620\pi$
$30$	0	0
$31$	−2.96697e11	−1.93687	−0.968434	−	0.249270i	$-0.919809\pi$
$31$	−2.96697e11	−1.93687	−0.968434	+	0.249270i	$0.919809\pi$
$32$	−3.43597e10	−0.176777
$33$	−2.26249e11	−0.924127
$34$	1.27704e11	0.416978
$35$	0	0
$36$	7.83642e10	0.166667
$37$	1.78337e11	0.308837	0.154419	−	0.988006i	$-0.450650\pi$
$37$	1.78337e11	0.308837	0.154419	+	0.988006i	$0.450650\pi$
$38$	−6.31554e11	−0.895434
$39$	2.28248e11	0.266332
$40$	0	0
$41$	−1.79088e12	−1.43611	−0.718056	−	0.695986i	$-0.754968\pi$
$41$	−1.79088e12	−1.43611	−0.718056	+	0.695986i	$0.754968\pi$
$42$	−8.49474e11	−0.568565
$43$	2.86346e12	1.60649	0.803244	−	0.595650i	$-0.203105\pi$
$43$	2.86346e12	1.60649	0.803244	+	0.595650i	$0.203105\pi$
$44$	−1.69495e12	−0.800318
$45$	0	0
$46$	1.06559e12	0.360501
$47$	−4.33291e12	−1.24751	−0.623757	−	0.781618i	$-0.714395\pi$
$47$	−4.33291e12	−1.24751	−0.623757	+	0.781618i	$0.714395\pi$
$48$	5.87068e11	0.144338
$49$	4.46080e12	0.939598
$50$	0	0
$51$	−2.18195e12	−0.340461
$52$	1.70993e12	0.230650
$53$	−9.73232e12	−1.13801	−0.569007	−	0.822333i	$-0.692672\pi$
$53$	−9.73232e12	−1.13801	−0.569007	+	0.822333i	$0.692672\pi$
$54$	−1.33893e12	−0.136083
$55$	0	0
$56$	−6.36387e12	−0.492392
$57$	1.07907e13	0.731119
$58$	−1.33284e13	−0.792626
$59$	−1.35148e13	−0.707002	−0.353501	−	0.935434i	$-0.615009\pi$
$59$	−1.35148e13	−0.707002	−0.353501	+	0.935434i	$0.615009\pi$
$60$	0	0
$61$	5.35266e12	0.218070	0.109035	−	0.994038i	$-0.465224\pi$
$61$	5.35266e12	0.218070	0.109035	+	0.994038i	$0.465224\pi$
$62$	3.79772e13	1.36957
$63$	1.45141e13	0.464231
$64$	4.39805e12	0.125000
$65$	0	0
$66$	2.89599e13	0.653457
$67$	5.32339e13	1.07307	0.536534	−	0.843879i	$-0.319733\pi$
$67$	5.32339e13	1.07307	0.536534	+	0.843879i	$0.319733\pi$
$68$	−1.63461e13	−0.294848
$69$	−1.82066e13	−0.294348
$70$	0	0
$71$	−2.02297e13	−0.263968	−0.131984	−	0.991252i	$-0.542135\pi$
$71$	−2.02297e13	−0.263968	−0.131984	+	0.991252i	$0.542135\pi$
$72$	−1.00306e13	−0.117851
$73$	−2.62642e13	−0.278254	−0.139127	−	0.990275i	$-0.544430\pi$
$73$	−2.62642e13	−0.278254	−0.139127	+	0.990275i	$0.544430\pi$
$74$	−2.28272e13	−0.218381
$75$	0	0
$76$	8.08389e13	0.633167
$77$	−3.13927e14	−2.22920
$78$	−2.92158e13	−0.188325
$79$	−3.39031e14	−1.98626	−0.993131	−	0.117005i	$-0.962671\pi$
$79$	−3.39031e14	−1.98626	−0.993131	+	0.117005i	$0.962671\pi$
$80$	0	0
$81$	2.28768e13	0.111111
$82$	2.29233e14	1.01548
$83$	−1.31685e14	−0.532660	−0.266330	−	0.963882i	$-0.585811\pi$
$83$	−1.31685e14	−0.532660	−0.266330	+	0.963882i	$0.585811\pi$
$84$	1.08733e14	0.402036
$85$	0	0
$86$	−3.66523e14	−1.13596
$87$	2.27728e14	0.647177
$88$	2.16954e14	0.565910
$89$	−3.93521e13	−0.0943069	−0.0471534	−	0.998888i	$-0.515015\pi$
$89$	−3.93521e13	−0.0943069	−0.0471534	+	0.998888i	$0.515015\pi$
$90$	0	0
$91$	3.16701e14	0.642450
$92$	−1.36395e14	−0.254913
$93$	−6.48876e14	−1.11825
$94$	5.54612e14	0.882126
$95$	0	0
$96$	−7.51447e13	−0.102062
$97$	−1.12875e15	−1.41844	−0.709219	−	0.704989i	$-0.750952\pi$
$97$	−1.12875e15	−1.41844	−0.709219	+	0.704989i	$0.750952\pi$
$98$	−5.70982e14	−0.664396
$99$	−4.94806e14	−0.533545
$100$	0	0
$101$	3.79528e13	0.0352236	0.0176118	−	0.999845i	$-0.494394\pi$
$101$	3.79528e13	0.0352236	0.0176118	+	0.999845i	$0.494394\pi$
$102$	2.79289e14	0.240743
$103$	2.07297e14	0.166079	0.0830393	−	0.996546i	$-0.473537\pi$
$103$	2.07297e14	0.166079	0.0830393	+	0.996546i	$0.473537\pi$
$104$	−2.18871e14	−0.163094
$105$	0	0
$106$	1.24574e15	0.804697
$107$	1.99692e15	1.20221	0.601107	−	0.799169i	$-0.294727\pi$
$107$	1.99692e15	1.20221	0.601107	+	0.799169i	$0.294727\pi$
$108$	1.71382e14	0.0962250
$109$	1.35603e13	0.00710510	0.00355255	−	0.999994i	$-0.498869\pi$
$109$	1.35603e13	0.00710510	0.00355255	+	0.999994i	$0.498869\pi$
$110$	0	0
$111$	3.90024e14	0.178307
$112$	8.14575e14	0.348174
$113$	7.08794e14	0.283421	0.141710	−	0.989908i	$-0.454740\pi$
$113$	7.08794e14	0.283421	0.141710	+	0.989908i	$0.454740\pi$
$114$	−1.38121e15	−0.516979
$115$	0	0
$116$	1.70604e15	0.560472
$117$	4.99179e14	0.153767
$118$	1.72990e15	0.499926
$119$	−3.02752e15	−0.821267
$120$	0	0
$121$	6.52501e15	1.56203
$122$	−6.85141e14	−0.154199
$123$	−3.91666e15	−0.829139
$124$	−4.86108e15	−0.968434
$125$	0	0
$126$	−1.85780e15	−0.328261
$127$	1.23021e15	0.204858	0.102429	−	0.994740i	$-0.467339\pi$
$127$	1.23021e15	0.204858	0.102429	+	0.994740i	$0.467339\pi$
$128$	−5.62950e14	−0.0883883
$129$	6.26239e15	0.927507
$130$	0	0
$131$	7.94836e15	1.04892	0.524460	−	0.851435i	$-0.324267\pi$
$131$	7.94836e15	1.04892	0.524460	+	0.851435i	$0.324267\pi$
$132$	−3.70686e15	−0.462064
$133$	1.49724e16	1.76362
$134$	−6.81394e15	−0.758774
$135$	0	0
$136$	2.09231e15	0.208489
$137$	1.78131e16	1.68010	0.840050	−	0.542508i	$-0.182525\pi$
$137$	1.78131e16	1.68010	0.840050	+	0.542508i	$0.182525\pi$
$138$	2.33044e15	0.208135
$139$	3.89941e15	0.329904	0.164952	−	0.986302i	$-0.447253\pi$
$139$	3.89941e15	0.329904	0.164952	+	0.986302i	$0.447253\pi$
$140$	0	0
$141$	−9.47607e15	−0.720253
$142$	2.58940e15	0.186653
$143$	−1.07968e16	−0.738373
$144$	1.28392e15	0.0833333
$145$	0	0
$146$	3.36181e15	0.196756
$147$	9.75577e15	0.542477
$148$	2.92188e15	0.154419
$149$	−3.48726e15	−0.175222	−0.0876108	−	0.996155i	$-0.527923\pi$
$149$	−3.48726e15	−0.175222	−0.0876108	+	0.996155i	$0.527923\pi$
$150$	0	0
$151$	6.85712e15	0.311756	0.155878	−	0.987776i	$-0.450179\pi$
$151$	6.85712e15	0.311756	0.155878	+	0.987776i	$0.450179\pi$
$152$	−1.03474e16	−0.447717
$153$	−4.77192e15	−0.196566
$154$	4.01827e16	1.57628
$155$	0	0
$156$	3.73962e15	0.133166
$157$	−3.69836e16	−1.25534	−0.627670	−	0.778480i	$-0.715991\pi$
$157$	−3.69836e16	−1.25534	−0.627670	+	0.778480i	$0.715991\pi$
$158$	4.33960e16	1.40450
$159$	−2.12846e16	−0.657032
$160$	0	0
$161$	−2.52622e16	−0.710031
$162$	−2.92823e15	−0.0785674
$163$	−7.42535e15	−0.190244	−0.0951218	−	0.995466i	$-0.530324\pi$
$163$	−7.42535e15	−0.190244	−0.0951218	+	0.995466i	$0.530324\pi$
$164$	−2.93418e16	−0.718056
$165$	0	0
$166$	1.68557e16	0.376647
$167$	1.47365e16	0.314789	0.157395	−	0.987536i	$-0.449691\pi$
$167$	1.47365e16	0.314789	0.157395	+	0.987536i	$0.449691\pi$
$168$	−1.39178e16	−0.284283
$169$	−4.02937e16	−0.787203
$170$	0	0
$171$	2.35992e16	0.422112
$172$	4.69149e16	0.803244
$173$	−3.40039e16	−0.557421	−0.278710	−	0.960375i	$-0.589907\pi$
$173$	−3.40039e16	−0.557421	−0.278710	+	0.960375i	$0.589907\pi$
$174$	−2.91492e16	−0.457623
$175$	0	0
$176$	−2.77701e16	−0.400159
$177$	−2.95569e16	−0.408188
$178$	5.03707e15	0.0666850
$179$	3.81276e16	0.483996	0.241998	−	0.970277i	$-0.422197\pi$
$179$	3.81276e16	0.483996	0.241998	+	0.970277i	$0.422197\pi$
$180$	0	0
$181$	−5.14124e16	−0.600452	−0.300226	−	0.953868i	$-0.597062\pi$
$181$	−5.14124e16	−0.600452	−0.300226	+	0.953868i	$0.597062\pi$
$182$	−4.05377e16	−0.454280
$183$	1.17063e16	0.125903
$184$	1.74586e16	0.180250
$185$	0	0
$186$	8.30561e16	0.790723
$187$	1.03213e17	0.943889
$188$	−7.09904e16	−0.623757
$189$	3.17422e16	0.268024
$190$	0	0
$191$	6.75568e16	0.527131	0.263566	−	0.964641i	$-0.415101\pi$
$191$	6.75568e16	0.527131	0.263566	+	0.964641i	$0.415101\pi$
$192$	9.61853e15	0.0721688
$193$	2.07235e16	0.149549	0.0747746	−	0.997200i	$-0.476176\pi$
$193$	2.07235e16	0.149549	0.0747746	+	0.997200i	$0.476176\pi$
$194$	1.44480e17	1.00299
$195$	0	0
$196$	7.30857e16	0.469799
$197$	−1.71244e16	−0.105954	−0.0529772	−	0.998596i	$-0.516871\pi$
$197$	−1.71244e16	−0.105954	−0.0529772	+	0.998596i	$0.516871\pi$
$198$	6.33352e16	0.377273
$199$	−1.05743e17	−0.606533	−0.303267	−	0.952906i	$-0.598077\pi$
$199$	−1.05743e17	−0.606533	−0.303267	+	0.952906i	$0.598077\pi$
$200$	0	0
$201$	1.16423e17	0.619536
$202$	−4.85796e15	−0.0249069
$203$	3.15980e17	1.56113
$204$	−3.57490e16	−0.170231
$205$	0	0
$206$	−2.65340e16	−0.117435
$207$	−3.98178e16	−0.169942
$208$	2.80155e16	0.115325
$209$	−5.10432e17	−2.02694
$210$	0	0
$211$	−3.13093e17	−1.15759	−0.578795	−	0.815473i	$-0.696477\pi$
$211$	−3.13093e17	−1.15759	−0.578795	+	0.815473i	$0.696477\pi$
$212$	−1.59454e17	−0.569007
$213$	−4.42423e16	−0.152402
$214$	−2.55605e17	−0.850093
$215$	0	0
$216$	−2.19370e16	−0.0680414
$217$	−9.00334e17	−2.69747
$218$	−1.73572e15	−0.00502406
$219$	−5.74397e16	−0.160650
$220$	0	0
$221$	−1.04125e17	−0.272027
$222$	−4.99231e16	−0.126082
$223$	−3.35017e17	−0.818052	−0.409026	−	0.912523i	$-0.634131\pi$
$223$	−3.35017e17	−0.818052	−0.409026	+	0.912523i	$0.634131\pi$
$224$	−1.04266e17	−0.246196
$225$	0	0
$226$	−9.07256e16	−0.200409
$227$	−4.67560e17	−0.999180	−0.499590	−	0.866262i	$-0.666516\pi$
$227$	−4.67560e17	−0.999180	−0.499590	+	0.866262i	$0.666516\pi$
$228$	1.76795e17	0.365559
$229$	3.61186e17	0.722711	0.361355	−	0.932428i	$-0.382314\pi$
$229$	3.61186e17	0.722711	0.361355	+	0.932428i	$0.382314\pi$
$230$	0	0
$231$	−6.86559e17	−1.28703
$232$	−2.18373e17	−0.396313
$233$	−3.61787e17	−0.635747	−0.317873	−	0.948133i	$-0.602969\pi$
$233$	−3.61787e17	−0.635747	−0.317873	+	0.948133i	$0.602969\pi$
$234$	−6.38949e16	−0.108729
$235$	0	0
$236$	−2.21427e17	−0.353501
$237$	−7.41462e17	−1.14677
$238$	3.87522e17	0.580724
$239$	−1.06842e18	−1.55152	−0.775762	−	0.631026i	$-0.782634\pi$
$239$	−1.06842e18	−1.55152	−0.775762	+	0.631026i	$0.782634\pi$
$240$	0	0
$241$	−1.00588e18	−1.37220	−0.686102	−	0.727505i	$-0.740680\pi$
$241$	−1.00588e18	−1.37220	−0.686102	+	0.727505i	$0.740680\pi$
$242$	−8.35201e17	−1.10453
$243$	5.00315e16	0.0641500
$244$	8.76980e16	0.109035
$245$	0	0
$246$	5.01332e17	0.586290
$247$	5.14943e17	0.584160
$248$	6.22218e17	0.684786
$249$	−2.87995e17	−0.307531
$250$	0	0
$251$	1.74766e18	1.75754	0.878768	−	0.477249i	$-0.158366\pi$
$251$	1.74766e18	1.75754	0.878768	+	0.477249i	$0.158366\pi$
$252$	2.37798e17	0.232116
$253$	8.61227e17	0.816044
$254$	−1.57467e17	−0.144856
$255$	0	0
$256$	7.20576e16	0.0625000
$257$	2.77264e17	0.233558	0.116779	−	0.993158i	$-0.462743\pi$
$257$	2.77264e17	0.233558	0.116779	+	0.993158i	$0.462743\pi$
$258$	−8.01585e17	−0.655846
$259$	5.41170e17	0.430116
$260$	0	0
$261$	4.98042e17	0.373648
$262$	−1.01739e18	−0.741699
$263$	−1.73303e18	−1.22783	−0.613914	−	0.789373i	$-0.710406\pi$
$263$	−1.73303e18	−1.22783	−0.613914	+	0.789373i	$0.710406\pi$
$264$	4.74478e17	0.326728
$265$	0	0
$266$	−1.91647e18	−1.24707
$267$	−8.60631e16	−0.0544481
$268$	8.72184e17	0.536534
$269$	−5.11271e16	−0.0305850	−0.0152925	−	0.999883i	$-0.504868\pi$
$269$	−5.11271e16	−0.0305850	−0.0152925	+	0.999883i	$0.504868\pi$
$270$	0	0
$271$	2.08455e17	0.117962	0.0589812	−	0.998259i	$-0.481215\pi$
$271$	2.08455e17	0.117962	0.0589812	+	0.998259i	$0.481215\pi$
$272$	−2.67815e17	−0.147424
$273$	6.92625e17	0.370918
$274$	−2.28008e18	−1.18801
$275$	0	0
$276$	−2.98297e17	−0.147174
$277$	−3.29723e18	−1.58326	−0.791628	−	0.611003i	$-0.790766\pi$
$277$	−3.29723e18	−1.58326	−0.791628	+	0.611003i	$0.790766\pi$
$278$	−4.99124e17	−0.233277
$279$	−1.41909e18	−0.645623
$280$	0	0
$281$	−3.98328e18	−1.71768	−0.858841	−	0.512242i	$-0.828815\pi$
$281$	−3.98328e18	−1.71768	−0.858841	+	0.512242i	$0.828815\pi$
$282$	1.21294e18	0.509296
$283$	2.19051e18	0.895668	0.447834	−	0.894117i	$-0.352196\pi$
$283$	2.19051e18	0.895668	0.447834	+	0.894117i	$0.352196\pi$
$284$	−3.31443e17	−0.131984
$285$	0	0
$286$	1.38199e18	0.522108
$287$	−5.43448e18	−2.00006
$288$	−1.64342e17	−0.0589256
$289$	−1.86704e18	−0.652258
$290$	0	0
$291$	−2.46858e18	−0.818935
$292$	−4.30312e17	−0.139127
$293$	5.27183e18	1.66132	0.830662	−	0.556778i	$-0.187962\pi$
$293$	5.27183e18	1.66132	0.830662	+	0.556778i	$0.187962\pi$
$294$	−1.24874e18	−0.383589
$295$	0	0
$296$	−3.74001e17	−0.109190
$297$	−1.08214e18	−0.308042
$298$	4.46369e17	0.123900
$299$	−8.68837e17	−0.235182
$300$	0	0
$301$	8.68925e18	2.23735
$302$	−8.77712e17	−0.220445
$303$	8.30029e16	0.0203364
$304$	1.32446e18	0.316584
$305$	0	0
$306$	6.10806e17	0.138993
$307$	−4.77067e18	−1.05935	−0.529677	−	0.848199i	$-0.677687\pi$
$307$	−4.77067e18	−1.05935	−0.529677	+	0.848199i	$0.677687\pi$
$308$	−5.14338e18	−1.11460
$309$	4.53359e17	0.0958856
$310$	0	0
$311$	−6.46666e18	−1.30310	−0.651549	−	0.758607i	$-0.725880\pi$
$311$	−6.46666e18	−1.30310	−0.651549	+	0.758607i	$0.725880\pi$
$312$	−4.78671e17	−0.0941624
$313$	6.00129e18	1.15256	0.576278	−	0.817254i	$-0.304504\pi$
$313$	6.00129e18	1.15256	0.576278	+	0.817254i	$0.304504\pi$
$314$	4.73389e18	0.887659
$315$	0	0
$316$	−5.55469e18	−0.993131
$317$	9.53943e17	0.166563	0.0832814	−	0.996526i	$-0.473460\pi$
$317$	9.53943e17	0.166563	0.0832814	+	0.996526i	$0.473460\pi$
$318$	2.72443e18	0.464592
$319$	−1.07722e19	−1.79422
$320$	0	0
$321$	4.36726e18	0.694098
$322$	3.23356e18	0.502068
$323$	−4.92262e18	−0.746753
$324$	3.74813e17	0.0555556
$325$	0	0
$326$	9.50445e17	0.134523
$327$	2.96563e16	0.00410213
$328$	3.75575e18	0.507742
$329$	−1.31483e19	−1.73741
$330$	0	0
$331$	−5.29071e18	−0.668042	−0.334021	−	0.942566i	$-0.608406\pi$
$331$	−5.29071e18	−0.668042	−0.334021	+	0.942566i	$0.608406\pi$
$332$	−2.15752e18	−0.266330
$333$	8.52983e17	0.102946
$334$	−1.88627e18	−0.222590
$335$	0	0
$336$	1.78148e18	0.201018
$337$	−6.22409e18	−0.686834	−0.343417	−	0.939183i	$-0.611584\pi$
$337$	−6.22409e18	−0.686834	−0.343417	+	0.939183i	$0.611584\pi$
$338$	5.15759e18	0.556636
$339$	1.55013e18	0.163633
$340$	0	0
$341$	3.06938e19	3.10022
$342$	−3.02070e18	−0.298478
$343$	−8.70190e17	−0.0841217
$344$	−6.00511e18	−0.567979
$345$	0	0
$346$	4.35250e18	0.394156
$347$	3.22035e18	0.285386	0.142693	−	0.989767i	$-0.454424\pi$
$347$	3.22035e18	0.285386	0.142693	+	0.989767i	$0.454424\pi$
$348$	3.73110e18	0.323588
$349$	−6.17407e18	−0.524060	−0.262030	−	0.965060i	$-0.584392\pi$
$349$	−6.17407e18	−0.524060	−0.262030	+	0.965060i	$0.584392\pi$
$350$	0	0
$351$	1.09170e18	0.0887772
$352$	3.55457e18	0.282955
$353$	6.14267e18	0.478681	0.239341	−	0.970936i	$-0.423069\pi$
$353$	6.14267e18	0.478681	0.239341	+	0.970936i	$0.423069\pi$
$354$	3.78329e18	0.288632
$355$	0	0
$356$	−6.44746e17	−0.0471534
$357$	−6.62118e18	−0.474159
$358$	−4.88033e18	−0.342237
$359$	−5.81103e18	−0.399066	−0.199533	−	0.979891i	$-0.563943\pi$
$359$	−5.81103e18	−0.399066	−0.199533	+	0.979891i	$0.563943\pi$
$360$	0	0
$361$	9.16338e18	0.603603
$362$	6.58079e18	0.424584
$363$	1.42702e19	0.901841
$364$	5.18883e18	0.321225
$365$	0	0
$366$	−1.49840e18	−0.0890268
$367$	2.77147e17	0.0161330	0.00806650	−	0.999967i	$-0.497432\pi$
$367$	2.77147e17	0.0161330	0.00806650	+	0.999967i	$0.497432\pi$
$368$	−2.23470e18	−0.127456
$369$	−8.56574e18	−0.478704
$370$	0	0
$371$	−2.95330e19	−1.58490
$372$	−1.06312e19	−0.559126
$373$	−2.30498e19	−1.18809	−0.594047	−	0.804430i	$-0.702471\pi$
$373$	−2.30498e19	−1.18809	−0.594047	+	0.804430i	$0.702471\pi$
$374$	−1.32112e19	−0.667431
$375$	0	0
$376$	9.08677e18	0.441063
$377$	1.08674e19	0.517091
$378$	−4.06301e18	−0.189522
$379$	−1.34397e19	−0.614604	−0.307302	−	0.951612i	$-0.599426\pi$
$379$	−1.34397e19	−0.614604	−0.307302	+	0.951612i	$0.599426\pi$
$380$	0	0
$381$	2.69048e18	0.118275
$382$	−8.64727e18	−0.372738
$383$	−2.64151e19	−1.11651	−0.558254	−	0.829670i	$-0.688529\pi$
$383$	−2.64151e19	−1.11651	−0.558254	+	0.829670i	$0.688529\pi$
$384$	−1.23117e18	−0.0510310
$385$	0	0
$386$	−2.65261e18	−0.105747
$387$	1.36958e19	0.535496
$388$	−1.84935e19	−0.709219
$389$	3.95275e19	1.48688	0.743442	−	0.668800i	$-0.233192\pi$
$389$	3.95275e19	1.48688	0.743442	+	0.668800i	$0.233192\pi$
$390$	0	0
$391$	8.30569e18	0.300642
$392$	−9.35497e18	−0.332198
$393$	1.73831e19	0.605595
$394$	2.19192e18	0.0749210
$395$	0	0
$396$	−8.10691e18	−0.266773
$397$	−2.82890e18	−0.0913457	−0.0456729	−	0.998956i	$-0.514543\pi$
$397$	−2.82890e18	−0.0913457	−0.0456729	+	0.998956i	$0.514543\pi$
$398$	1.35351e19	0.428884
$399$	3.27447e19	1.01822
$400$	0	0
$401$	−4.63681e19	−1.38879	−0.694395	−	0.719594i	$-0.744328\pi$
$401$	−4.63681e19	−1.38879	−0.694395	+	0.719594i	$0.744328\pi$
$402$	−1.49021e19	−0.438078
$403$	−3.09650e19	−0.893477
$404$	6.21819e17	0.0176118
$405$	0	0
$406$	−4.04454e19	−1.10389
$407$	−1.84493e19	−0.494336
$408$	4.57588e18	0.120371
$409$	−1.17092e19	−0.302415	−0.151207	−	0.988502i	$-0.548316\pi$
$409$	−1.17092e19	−0.302415	−0.151207	+	0.988502i	$0.548316\pi$
$410$	0	0
$411$	3.89573e19	0.970007
$412$	3.39635e18	0.0830393
$413$	−4.10112e19	−0.984638
$414$	5.09668e18	0.120167
$415$	0	0
$416$	−3.58598e18	−0.0815470
$417$	8.52801e18	0.190470
$418$	6.53353e19	1.43326
$419$	−4.20102e19	−0.905210	−0.452605	−	0.891711i	$-0.649505\pi$
$419$	−4.20102e19	−0.905210	−0.452605	+	0.891711i	$0.649505\pi$
$420$	0	0
$421$	−1.59718e19	−0.332077	−0.166039	−	0.986119i	$-0.553098\pi$
$421$	−1.59718e19	−0.332077	−0.166039	+	0.986119i	$0.553098\pi$
$422$	4.00759e19	0.818540
$423$	−2.07242e19	−0.415838
$424$	2.04101e19	0.402348
$425$	0	0
$426$	5.66301e18	0.107764
$427$	1.62428e19	0.303705
$428$	3.27175e19	0.601107
$429$	−2.36127e19	−0.426300
$430$	0	0
$431$	−9.76365e19	−1.70229	−0.851143	−	0.524934i	$-0.824090\pi$
$431$	−9.76365e19	−1.70229	−0.851143	+	0.524934i	$0.824090\pi$
$432$	2.80793e18	0.0481125
$433$	6.91455e19	1.16441	0.582204	−	0.813043i	$-0.302191\pi$
$433$	6.91455e19	1.16441	0.582204	+	0.813043i	$0.302191\pi$
$434$	1.15243e20	1.90740
$435$	0	0
$436$	2.22172e17	0.00355255
$437$	−4.10753e19	−0.645609
$438$	7.35229e18	0.113597
$439$	5.48990e19	0.833836	0.416918	−	0.908944i	$-0.363110\pi$
$439$	5.48990e19	0.833836	0.416918	+	0.908944i	$0.363110\pi$
$440$	0	0
$441$	2.13359e19	0.313199
$442$	1.33280e19	0.192352
$443$	−3.80504e19	−0.539923	−0.269961	−	0.962871i	$-0.587011\pi$
$443$	−3.80504e19	−0.539923	−0.269961	+	0.962871i	$0.587011\pi$
$444$	6.39015e18	0.0891536
$445$	0	0
$446$	4.28822e19	0.578450
$447$	−7.62664e18	−0.101164
$448$	1.33460e19	0.174087
$449$	1.11994e18	0.0143663	0.00718317	−	0.999974i	$-0.497714\pi$
$449$	1.11994e18	0.0143663	0.00718317	+	0.999974i	$0.497714\pi$
$450$	0	0
$451$	1.85270e20	2.29869
$452$	1.16129e19	0.141710
$453$	1.49965e19	0.179992
$454$	5.98477e19	0.706527
$455$	0	0
$456$	−2.26297e19	−0.258489
$457$	9.70734e19	1.09076	0.545379	−	0.838189i	$-0.316386\pi$
$457$	9.70734e19	1.09076	0.545379	+	0.838189i	$0.316386\pi$
$458$	−4.62318e19	−0.511034
$459$	−1.04362e19	−0.113487
$460$	0	0
$461$	−9.78165e19	−1.02957	−0.514784	−	0.857320i	$-0.672128\pi$
$461$	−9.78165e19	−1.02957	−0.514784	+	0.857320i	$0.672128\pi$
$462$	8.78795e19	0.910066
$463$	2.17159e19	0.221269	0.110634	−	0.993861i	$-0.464712\pi$
$463$	2.17159e19	0.221269	0.110634	+	0.993861i	$0.464712\pi$
$464$	2.79517e19	0.280236
$465$	0	0
$466$	4.63088e19	0.449541
$467$	1.98443e20	1.89565	0.947826	−	0.318788i	$-0.103276\pi$
$467$	1.98443e20	1.89565	0.947826	+	0.318788i	$0.103276\pi$
$468$	8.17854e18	0.0768833
$469$	1.61540e20	1.49446
$470$	0	0
$471$	−8.08830e19	−0.724771
$472$	2.83427e19	0.249963
$473$	−2.96230e20	−2.57140
$474$	9.49071e19	0.810888
$475$	0	0
$476$	−4.96028e19	−0.410634
$477$	−4.65494e19	−0.379338
$478$	1.36758e20	1.09709
$479$	−3.78661e19	−0.299043	−0.149522	−	0.988758i	$-0.547773\pi$
$479$	−3.78661e19	−0.299043	−0.149522	+	0.988758i	$0.547773\pi$
$480$	0	0
$481$	1.86123e19	0.142467
$482$	1.28753e20	0.970295
$483$	−5.52484e19	−0.409936
$484$	1.06906e20	0.781017
$485$	0	0
$486$	−6.40404e18	−0.0453609
$487$	−3.89265e19	−0.271505	−0.135752	−	0.990743i	$-0.543345\pi$
$487$	−3.89265e19	−0.271505	−0.135752	+	0.990743i	$0.543345\pi$
$488$	−1.12253e19	−0.0770994
$489$	−1.62392e19	−0.109837
$490$	0	0
$491$	1.40516e20	0.921753	0.460876	−	0.887464i	$-0.347535\pi$
$491$	1.40516e20	0.921753	0.460876	+	0.887464i	$0.347535\pi$
$492$	−6.41706e19	−0.414570
$493$	−1.03888e20	−0.661016
$494$	−6.59127e19	−0.413063
$495$	0	0
$496$	−7.96439e19	−0.484217
$497$	−6.13875e19	−0.367627
$498$	3.68633e19	0.217457
$499$	−1.53703e20	−0.893157	−0.446578	−	0.894745i	$-0.647358\pi$
$499$	−1.53703e20	−0.893157	−0.446578	+	0.894745i	$0.647358\pi$
$500$	0	0
$501$	3.22287e19	0.181744
$502$	−2.23701e20	−1.24277
$503$	−5.62265e18	−0.0307738	−0.0153869	−	0.999882i	$-0.504898\pi$
$503$	−5.62265e18	−0.0307738	−0.0153869	+	0.999882i	$0.504898\pi$
$504$	−3.04382e19	−0.164131
$505$	0	0
$506$	−1.10237e20	−0.577031
$507$	−8.81222e19	−0.454492
$508$	2.01558e19	0.102429
$509$	3.29307e20	1.64899	0.824495	−	0.565869i	$-0.191459\pi$
$509$	3.29307e20	1.64899	0.824495	+	0.565869i	$0.191459\pi$
$510$	0	0
$511$	−7.96993e19	−0.387523
$512$	−9.22337e18	−0.0441942
$513$	5.16115e19	0.243706
$514$	−3.54898e19	−0.165151
$515$	0	0
$516$	1.02603e20	0.463753
$517$	4.48247e20	1.99682
$518$	−6.92698e19	−0.304138
$519$	−7.43666e19	−0.321827
$520$	0	0
$521$	1.71775e20	0.722231	0.361116	−	0.932521i	$-0.382396\pi$
$521$	1.71775e20	0.722231	0.361116	+	0.932521i	$0.382396\pi$
$522$	−6.37494e19	−0.264209
$523$	1.59414e20	0.651275	0.325637	−	0.945495i	$-0.394421\pi$
$523$	1.59414e20	0.651275	0.325637	+	0.945495i	$0.394421\pi$
$524$	1.30226e20	0.524460
$525$	0	0
$526$	2.21828e20	0.868205
$527$	2.96011e20	1.14216
$528$	−6.07332e19	−0.231032
$529$	−1.97331e20	−0.740078
$530$	0	0
$531$	−6.46410e19	−0.235667
$532$	2.45308e20	0.881809
$533$	−1.86907e20	−0.662478
$534$	1.10161e19	0.0385006
$535$	0	0
$536$	−1.11640e20	−0.379387
$537$	8.33851e19	0.279435
$538$	6.54427e18	0.0216269
$539$	−4.61477e20	−1.50395
$540$	0	0
$541$	4.74121e19	0.150283	0.0751415	−	0.997173i	$-0.476059\pi$
$541$	4.74121e19	0.150283	0.0751415	+	0.997173i	$0.476059\pi$
$542$	−2.66823e19	−0.0834120
$543$	−1.12439e20	−0.346671
$544$	3.42804e19	0.104245
$545$	0	0
$546$	−8.86560e19	−0.262279
$547$	2.80986e20	0.819934	0.409967	−	0.912100i	$-0.365540\pi$
$547$	2.80986e20	0.819934	0.409967	+	0.912100i	$0.365540\pi$
$548$	2.91850e20	0.840050
$549$	2.56016e19	0.0726900
$550$	0	0
$551$	5.13770e20	1.41949
$552$	3.81820e19	0.104068
$553$	−1.02880e21	−2.76626
$554$	4.22046e20	1.11953
$555$	0	0
$556$	6.38879e19	0.164952
$557$	3.49642e20	0.890654	0.445327	−	0.895368i	$-0.353087\pi$
$557$	3.49642e20	0.890654	0.445327	+	0.895368i	$0.353087\pi$
$558$	1.81644e20	0.456524
$559$	2.98847e20	0.741073
$560$	0	0
$561$	2.25726e20	0.544955
$562$	5.09859e20	1.21459
$563$	−2.68215e20	−0.630479	−0.315240	−	0.949012i	$-0.602085\pi$
$563$	−2.68215e20	−0.630479	−0.315240	+	0.949012i	$0.602085\pi$
$564$	−1.55256e20	−0.360126
$565$	0	0
$566$	−2.80385e20	−0.633333
$567$	6.94203e19	0.154744
$568$	4.24247e19	0.0933267
$569$	−7.95691e20	−1.72744	−0.863719	−	0.503974i	$-0.831871\pi$
$569$	−7.95691e20	−1.72744	−0.863719	+	0.503974i	$0.831871\pi$
$570$	0	0
$571$	8.84470e20	1.87030	0.935152	−	0.354246i	$-0.115262\pi$
$571$	8.84470e20	1.87030	0.935152	+	0.354246i	$0.115262\pi$
$572$	−1.76895e20	−0.369186
$573$	1.47747e20	0.304339
$574$	6.95614e20	1.41426
$575$	0	0
$576$	2.10357e19	0.0416667
$577$	−3.15722e20	−0.617286	−0.308643	−	0.951178i	$-0.599875\pi$
$577$	−3.15722e20	−0.617286	−0.308643	+	0.951178i	$0.599875\pi$
$578$	2.38981e20	0.461216
$579$	4.53224e19	0.0863423
$580$	0	0
$581$	−3.99601e20	−0.741832
$582$	3.15978e20	0.579075
$583$	1.00682e21	1.82154
$584$	5.50799e19	0.0983778
$585$	0	0
$586$	−6.74794e20	−1.17473
$587$	−7.33397e20	−1.26053	−0.630265	−	0.776380i	$-0.717054\pi$
$587$	−7.33397e20	−1.26053	−0.630265	+	0.776380i	$0.717054\pi$
$588$	1.59838e20	0.271239
$589$	−1.46391e21	−2.45272
$590$	0	0
$591$	−3.74511e19	−0.0611728
$592$	4.78721e19	0.0772093
$593$	3.25571e20	0.518485	0.259242	−	0.965812i	$-0.416527\pi$
$593$	3.25571e20	0.518485	0.259242	+	0.965812i	$0.416527\pi$
$594$	1.38514e20	0.217819
$595$	0	0
$596$	−5.71353e19	−0.0876108
$597$	−2.31261e20	−0.350182
$598$	1.11211e20	0.166299
$599$	6.61173e20	0.976368	0.488184	−	0.872741i	$-0.337659\pi$
$599$	6.61173e20	0.976368	0.488184	+	0.872741i	$0.337659\pi$
$600$	0	0
$601$	−2.49467e20	−0.359297	−0.179649	−	0.983731i	$-0.557496\pi$
$601$	−2.49467e20	−0.359297	−0.179649	+	0.983731i	$0.557496\pi$
$602$	−1.11222e21	−1.58204
$603$	2.54616e20	0.357689
$604$	1.12347e20	0.155878
$605$	0	0
$606$	−1.06244e19	−0.0143800
$607$	−1.08917e21	−1.45606	−0.728031	−	0.685544i	$-0.759564\pi$
$607$	−1.08917e21	−1.45606	−0.728031	+	0.685544i	$0.759564\pi$
$608$	−1.69531e20	−0.223858
$609$	6.91048e20	0.901320
$610$	0	0
$611$	−4.52208e20	−0.575478
$612$	−7.81831e19	−0.0982828
$613$	−1.99450e20	−0.247674	−0.123837	−	0.992303i	$-0.539520\pi$
$613$	−1.99450e20	−0.247674	−0.123837	+	0.992303i	$0.539520\pi$
$614$	6.10645e20	0.749076
$615$	0	0
$616$	6.58353e20	0.788140
$617$	−4.21127e20	−0.498052	−0.249026	−	0.968497i	$-0.580110\pi$
$617$	−4.21127e20	−0.498052	−0.249026	+	0.968497i	$0.580110\pi$
$618$	−5.80299e19	−0.0678013
$619$	−1.27272e21	−1.46910	−0.734552	−	0.678552i	$-0.762608\pi$
$619$	−1.27272e21	−1.46910	−0.734552	+	0.678552i	$0.762608\pi$
$620$	0	0
$621$	−8.70816e19	−0.0981159
$622$	8.27732e20	0.921429
$623$	−1.19415e20	−0.131341
$624$	6.12699e19	0.0665829
$625$	0	0
$626$	−7.68165e20	−0.814980
$627$	−1.11632e21	−1.17025
$628$	−6.05938e20	−0.627670
$629$	−1.77925e20	−0.182120
$630$	0	0
$631$	−3.42884e20	−0.342711	−0.171355	−	0.985209i	$-0.554815\pi$
$631$	−3.42884e20	−0.342711	−0.171355	+	0.985209i	$0.554815\pi$
$632$	7.11000e20	0.702250
$633$	−6.84735e20	−0.668335
$634$	−1.22105e20	−0.117778
$635$	0	0
$636$	−3.48727e20	−0.328516
$637$	4.65555e20	0.433436
$638$	1.37885e21	1.26871
$639$	−9.67578e19	−0.0879893
$640$	0	0
$641$	1.00075e21	0.888980	0.444490	−	0.895784i	$-0.353385\pi$
$641$	1.00075e21	0.888980	0.444490	+	0.895784i	$0.353385\pi$
$642$	−5.59009e20	−0.490802
$643$	1.35729e20	0.117785	0.0588927	−	0.998264i	$-0.481243\pi$
$643$	1.35729e20	0.117785	0.0588927	+	0.998264i	$0.481243\pi$
$644$	−4.13896e20	−0.355015
$645$	0	0
$646$	6.30095e20	0.528034
$647$	−2.15462e21	−1.78479	−0.892397	−	0.451251i	$-0.850978\pi$
$647$	−2.15462e21	−1.78479	−0.892397	+	0.451251i	$0.850978\pi$
$648$	−4.79761e19	−0.0392837
$649$	1.39813e21	1.13165
$650$	0	0
$651$	−1.96903e21	−1.55738
$652$	−1.21657e20	−0.0951218
$653$	2.06101e21	1.59306	0.796529	−	0.604601i	$-0.206667\pi$
$653$	2.06101e21	1.59306	0.796529	+	0.604601i	$0.206667\pi$
$654$	−3.79601e18	−0.00290064
$655$	0	0
$656$	−4.80736e20	−0.359028
$657$	−1.25621e20	−0.0927515
$658$	1.68299e21	1.22853
$659$	−1.61309e21	−1.16418	−0.582089	−	0.813125i	$-0.697764\pi$
$659$	−1.61309e21	−1.16418	−0.582089	+	0.813125i	$0.697764\pi$
$660$	0	0
$661$	1.60888e21	1.13504	0.567521	−	0.823359i	$-0.307903\pi$
$661$	1.60888e21	1.13504	0.567521	+	0.823359i	$0.307903\pi$
$662$	6.77211e20	0.472377
$663$	−2.27721e20	−0.157055
$664$	2.76163e20	0.188324
$665$	0	0
$666$	−1.09182e20	−0.0727936
$667$	−8.66859e20	−0.571485
$668$	2.41442e20	0.157395
$669$	−7.32683e20	−0.472303
$670$	0	0
$671$	−5.53742e20	−0.349051
$672$	−2.28029e20	−0.142141
$673$	−1.67294e21	−1.03126	−0.515628	−	0.856812i	$-0.672441\pi$
$673$	−1.67294e21	−1.03126	−0.515628	+	0.856812i	$0.672441\pi$
$674$	7.96684e20	0.485665
$675$	0	0
$676$	−6.60171e20	−0.393601
$677$	−7.27000e20	−0.428666	−0.214333	−	0.976761i	$-0.568758\pi$
$677$	−7.27000e20	−0.428666	−0.214333	+	0.976761i	$0.568758\pi$
$678$	−1.98417e20	−0.115706
$679$	−3.42523e21	−1.97545
$680$	0	0
$681$	−1.02255e21	−0.576877
$682$	−3.92880e21	−2.19219
$683$	−2.85909e21	−1.57787	−0.788936	−	0.614475i	$-0.789368\pi$
$683$	−2.85909e21	−1.57787	−0.788936	+	0.614475i	$0.789368\pi$
$684$	3.86650e20	0.211056
$685$	0	0
$686$	1.11384e20	0.0594830
$687$	7.89913e20	0.417257
$688$	7.68654e20	0.401622
$689$	−1.01572e21	−0.524965
$690$	0	0
$691$	1.71274e21	0.866178	0.433089	−	0.901351i	$-0.357424\pi$
$691$	1.71274e21	0.866178	0.433089	+	0.901351i	$0.357424\pi$
$692$	−5.57120e20	−0.278710
$693$	−1.50150e21	−0.743065
$694$	−4.12205e20	−0.201798
$695$	0	0
$696$	−4.77581e20	−0.228812
$697$	1.78675e21	0.846870
$698$	7.90281e20	0.370566
$699$	−7.91229e20	−0.367049
$700$	0	0
$701$	−8.95519e20	−0.406621	−0.203311	−	0.979114i	$-0.565170\pi$
$701$	−8.95519e20	−0.406621	−0.203311	+	0.979114i	$0.565170\pi$
$702$	−1.39738e20	−0.0627749
$703$	8.79920e20	0.391091
$704$	−4.54985e20	−0.200079
$705$	0	0
$706$	−7.86261e20	−0.338479
$707$	1.15169e20	0.0490557
$708$	−4.84261e20	−0.204094
$709$	−3.99642e21	−1.66657	−0.833286	−	0.552842i	$-0.813543\pi$
$709$	−3.99642e21	−1.66657	−0.833286	+	0.552842i	$0.813543\pi$
$710$	0	0
$711$	−1.62158e21	−0.662088
$712$	8.25274e19	0.0333425
$713$	2.46998e21	0.987464
$714$	8.47511e20	0.335281
$715$	0	0
$716$	6.24683e20	0.241998
$717$	−2.33664e21	−0.895772
$718$	7.43812e20	0.282183
$719$	−2.66669e20	−0.100117	−0.0500583	−	0.998746i	$-0.515941\pi$
$719$	−2.66669e20	−0.100117	−0.0500583	+	0.998746i	$0.515941\pi$
$720$	0	0
$721$	6.29049e20	0.231297
$722$	−1.17291e21	−0.426812
$723$	−2.19986e21	−0.792242
$724$	−8.42341e20	−0.300226
$725$	0	0
$726$	−1.82658e21	−0.637698
$727$	1.69447e21	0.585499	0.292750	−	0.956189i	$-0.405430\pi$
$727$	1.69447e21	0.585499	0.292750	+	0.956189i	$0.405430\pi$
$728$	−6.64170e20	−0.227140
$729$	1.09419e20	0.0370370
$730$	0	0
$731$	−2.85684e21	−0.947341
$732$	1.91796e20	0.0629514
$733$	−3.16239e21	−1.02739	−0.513695	−	0.857973i	$-0.671724\pi$
$733$	−3.16239e21	−1.02739	−0.513695	+	0.857973i	$0.671724\pi$
$734$	−3.54749e19	−0.0114078
$735$	0	0
$736$	2.86042e20	0.0901252
$737$	−5.50714e21	−1.71759
$738$	1.09641e21	0.338495
$739$	2.01381e21	0.615440	0.307720	−	0.951477i	$-0.400434\pi$
$739$	2.01381e21	0.615440	0.307720	+	0.951477i	$0.400434\pi$
$740$	0	0
$741$	1.12618e21	0.337265
$742$	3.78022e21	1.12070
$743$	−2.11411e21	−0.620458	−0.310229	−	0.950662i	$-0.600406\pi$
$743$	−2.11411e21	−0.620458	−0.310229	+	0.950662i	$0.600406\pi$
$744$	1.36079e21	0.395361
$745$	0	0
$746$	2.95038e21	0.840110
$747$	−6.29844e20	−0.177553
$748$	1.69104e21	0.471945
$749$	6.05970e21	1.67432
$750$	0	0
$751$	4.95087e21	1.34086	0.670428	−	0.741974i	$-0.266110\pi$
$751$	4.95087e21	1.34086	0.670428	+	0.741974i	$0.266110\pi$
$752$	−1.16311e21	−0.311879
$753$	3.82213e21	1.01471
$754$	−1.39103e21	−0.365638
$755$	0	0
$756$	5.20065e20	0.134012
$757$	4.85879e21	1.23968	0.619839	−	0.784729i	$-0.287198\pi$
$757$	4.85879e21	1.23968	0.619839	+	0.784729i	$0.287198\pi$
$758$	1.72028e21	0.434590
$759$	1.88350e21	0.471143
$760$	0	0
$761$	−3.69092e21	−0.905211	−0.452605	−	0.891711i	$-0.649505\pi$
$761$	−3.69092e21	−0.905211	−0.452605	+	0.891711i	$0.649505\pi$
$762$	−3.44381e20	−0.0836329
$763$	4.11491e19	0.00989523
$764$	1.10685e21	0.263566
$765$	0	0
$766$	3.38114e21	0.789491
$767$	−1.41049e21	−0.326140
$768$	1.57590e20	0.0360844
$769$	−2.33815e21	−0.530181	−0.265091	−	0.964224i	$-0.585402\pi$
$769$	−2.33815e21	−0.530181	−0.265091	+	0.964224i	$0.585402\pi$
$770$	0	0
$771$	6.06376e20	0.134845
$772$	3.39534e20	0.0747746
$773$	5.20682e21	1.13560	0.567801	−	0.823166i	$-0.307794\pi$
$773$	5.20682e21	1.13560	0.567801	+	0.823166i	$0.307794\pi$
$774$	−1.75307e21	−0.378653
$775$	0	0
$776$	2.36716e21	0.501493
$777$	1.18354e21	0.248328
$778$	−5.05952e21	−1.05139
$779$	−8.83624e21	−1.81860
$780$	0	0
$781$	2.09279e21	0.422516
$782$	−1.06313e21	−0.212586
$783$	1.08922e21	0.215726
$784$	1.19744e21	0.234899
$785$	0	0
$786$	−2.22503e21	−0.428220
$787$	1.92196e21	0.366381	0.183191	−	0.983077i	$-0.441357\pi$
$787$	1.92196e21	0.366381	0.183191	+	0.983077i	$0.441357\pi$
$788$	−2.80566e20	−0.0529772
$789$	−3.79013e21	−0.708887
$790$	0	0
$791$	2.15085e21	0.394718
$792$	1.03768e21	0.188637
$793$	5.58635e20	0.100596
$794$	3.62099e20	0.0645912
$795$	0	0
$796$	−1.73250e21	−0.303267
$797$	6.23880e20	0.108184	0.0540921	−	0.998536i	$-0.482774\pi$
$797$	6.23880e20	0.108184	0.0540921	+	0.998536i	$0.482774\pi$
$798$	−4.19132e21	−0.719994
$799$	4.32290e21	0.735655
$800$	0	0
$801$	−1.88220e20	−0.0314356
$802$	5.93511e21	0.982022
$803$	2.71707e21	0.445384
$804$	1.90747e21	0.309768
$805$	0	0
$806$	3.96352e21	0.631783
$807$	−1.11815e20	−0.0176583
$808$	−7.95929e19	−0.0124534
$809$	5.72858e21	0.888042	0.444021	−	0.896016i	$-0.353552\pi$
$809$	5.72858e21	0.888042	0.444021	+	0.896016i	$0.353552\pi$
$810$	0	0
$811$	4.80014e21	0.730462	0.365231	−	0.930917i	$-0.380990\pi$
$811$	4.80014e21	0.730462	0.365231	+	0.930917i	$0.380990\pi$
$812$	5.17702e21	0.780566
$813$	4.55892e20	0.0681056
$814$	2.36151e21	0.349548
$815$	0	0
$816$	−5.85712e20	−0.0851154
$817$	1.41284e22	2.03435
$818$	1.49878e21	0.213840
$819$	1.51477e21	0.214150
$820$	0	0
$821$	8.89775e21	1.23511	0.617556	−	0.786527i	$-0.288123\pi$
$821$	8.89775e21	1.23511	0.617556	+	0.786527i	$0.288123\pi$
$822$	−4.98653e21	−0.685898
$823$	1.39946e21	0.190749	0.0953744	−	0.995441i	$-0.469595\pi$
$823$	1.39946e21	0.190749	0.0953744	+	0.995441i	$0.469595\pi$
$824$	−4.34733e20	−0.0587177
$825$	0	0
$826$	5.24943e21	0.696244
$827$	−6.61343e21	−0.869232	−0.434616	−	0.900616i	$-0.643116\pi$
$827$	−6.61343e21	−0.869232	−0.434616	+	0.900616i	$0.643116\pi$
$828$	−6.52375e20	−0.0849709
$829$	4.14735e21	0.535318	0.267659	−	0.963514i	$-0.413750\pi$
$829$	4.14735e21	0.535318	0.267659	+	0.963514i	$0.413750\pi$
$830$	0	0
$831$	−7.21105e21	−0.914093
$832$	4.59006e20	0.0576625
$833$	−4.45049e21	−0.554078
$834$	−1.09158e21	−0.134683
$835$	0	0
$836$	−8.36292e21	−1.01347
$837$	−3.10355e21	−0.372750
$838$	5.37730e21	0.640080
$839$	9.57811e21	1.12997	0.564983	−	0.825103i	$-0.308883\pi$
$839$	9.57811e21	1.12997	0.564983	+	0.825103i	$0.308883\pi$
$840$	0	0
$841$	2.21350e21	0.256513
$842$	2.04440e21	0.234814
$843$	−8.71142e21	−0.991705
$844$	−5.12972e21	−0.578795
$845$	0	0
$846$	2.65269e21	0.294042
$847$	1.98003e22	2.17544
$848$	−2.61250e21	−0.284503
$849$	4.79065e21	0.517114
$850$	0	0
$851$	−1.48465e21	−0.157453
$852$	−7.24865e20	−0.0762009
$853$	8.15361e21	0.849634	0.424817	−	0.905279i	$-0.360338\pi$
$853$	8.15361e21	0.849634	0.424817	+	0.905279i	$0.360338\pi$
$854$	−2.07908e21	−0.214752
$855$	0	0
$856$	−4.18784e21	−0.425047
$857$	1.53552e22	1.54490	0.772449	−	0.635076i	$-0.219031\pi$
$857$	1.53552e22	1.54490	0.772449	+	0.635076i	$0.219031\pi$
$858$	3.02242e21	0.301439
$859$	−4.87617e21	−0.482092	−0.241046	−	0.970514i	$-0.577490\pi$
$859$	−4.87617e21	−0.482092	−0.241046	+	0.970514i	$0.577490\pi$
$860$	0	0
$861$	−1.18852e22	−1.15474
$862$	1.24975e22	1.20370
$863$	4.76667e21	0.455128	0.227564	−	0.973763i	$-0.426924\pi$
$863$	4.76667e21	0.455128	0.227564	+	0.973763i	$0.426924\pi$
$864$	−3.59415e20	−0.0340207
$865$	0	0
$866$	−8.85063e21	−0.823360
$867$	−4.08321e21	−0.376581
$868$	−1.47511e22	−1.34873
$869$	3.50734e22	3.17928
$870$	0	0
$871$	5.55580e21	0.495006
$872$	−2.84380e19	−0.00251203
$873$	−5.39878e21	−0.472812
$874$	5.25764e21	0.456515
$875$	0	0
$876$	−9.41093e20	−0.0803252
$877$	−1.57944e22	−1.33661	−0.668306	−	0.743887i	$-0.732980\pi$
$877$	−1.57944e22	−1.33661	−0.668306	+	0.743887i	$0.732980\pi$
$878$	−7.02707e21	−0.589611
$879$	1.15295e22	0.959165
$880$	0	0
$881$	−1.12789e22	−0.922461	−0.461230	−	0.887280i	$-0.652592\pi$
$881$	−1.12789e22	−0.922461	−0.461230	+	0.887280i	$0.652592\pi$
$882$	−2.73099e21	−0.221465
$883$	−4.48307e21	−0.360471	−0.180236	−	0.983623i	$-0.557686\pi$
$883$	−4.48307e21	−0.360471	−0.180236	+	0.983623i	$0.557686\pi$
$884$	−1.70598e21	−0.136013
$885$	0	0
$886$	4.87045e21	0.381783
$887$	8.56246e21	0.665536	0.332768	−	0.943009i	$-0.392017\pi$
$887$	8.56246e21	0.665536	0.332768	+	0.943009i	$0.392017\pi$
$888$	−8.17940e20	−0.0630411
$889$	3.73312e21	0.285304
$890$	0	0
$891$	−2.36664e21	−0.177848
$892$	−5.48893e21	−0.409026
$893$	−2.13786e22	−1.57977
$894$	9.76210e20	0.0715339
$895$	0	0
$896$	−1.70829e21	−0.123098
$897$	−1.90015e21	−0.135783
$898$	−1.43352e20	−0.0101585
$899$	−3.08945e22	−2.17112
$900$	0	0
$901$	9.70983e21	0.671082
$902$	−2.37145e22	−1.62542
$903$	1.90034e22	1.29173
$904$	−1.48645e21	−0.100204
$905$	0	0
$906$	−1.91956e21	−0.127274
$907$	−7.34638e21	−0.483079	−0.241540	−	0.970391i	$-0.577652\pi$
$907$	−7.34638e21	−0.483079	−0.241540	+	0.970391i	$0.577652\pi$
$908$	−7.66051e21	−0.499590
$909$	1.81527e20	0.0117412
$910$	0	0
$911$	−1.18999e22	−0.757106	−0.378553	−	0.925580i	$-0.623578\pi$
$911$	−1.18999e22	−0.757106	−0.378553	+	0.925580i	$0.623578\pi$
$912$	2.89660e21	0.182780
$913$	1.36230e22	0.852594
$914$	−1.24254e22	−0.771283
$915$	0	0
$916$	5.91767e21	0.361355
$917$	2.41195e22	1.46083
$918$	1.33583e21	0.0802475
$919$	1.92629e22	1.14777	0.573886	−	0.818935i	$-0.305435\pi$
$919$	1.92629e22	1.14777	0.573886	+	0.818935i	$0.305435\pi$
$920$	0	0
$921$	−1.04334e22	−0.611618
$922$	1.25205e22	0.728015
$923$	−2.11129e21	−0.121768
$924$	−1.12486e22	−0.643514
$925$	0	0
$926$	−2.77964e21	−0.156461
$927$	9.91495e20	0.0553596
$928$	−3.57782e21	−0.198157
$929$	8.78109e21	0.482426	0.241213	−	0.970472i	$-0.422455\pi$
$929$	8.78109e21	0.482426	0.241213	+	0.970472i	$0.422455\pi$
$930$	0	0
$931$	2.20096e22	1.18985
$932$	−5.92753e21	−0.317873
$933$	−1.41426e22	−0.752343
$934$	−2.54007e22	−1.34043
$935$	0	0
$936$	−1.04685e21	−0.0543647
$937$	2.35424e22	1.21284	0.606421	−	0.795144i	$-0.292605\pi$
$937$	2.35424e22	1.21284	0.606421	+	0.795144i	$0.292605\pi$
$938$	−2.06771e22	−1.05674
$939$	1.31248e22	0.665429
$940$	0	0
$941$	−2.31660e22	−1.15592	−0.577961	−	0.816064i	$-0.696152\pi$
$941$	−2.31660e22	−1.15592	−0.577961	+	0.816064i	$0.696152\pi$
$942$	1.03530e22	0.512490
$943$	1.49090e22	0.732166
$944$	−3.62786e21	−0.176751
$945$	0	0
$946$	3.79174e22	1.81826
$947$	−2.02578e22	−0.963755	−0.481877	−	0.876239i	$-0.660045\pi$
$947$	−2.02578e22	−0.963755	−0.481877	+	0.876239i	$0.660045\pi$
$948$	−1.21481e22	−0.573385
$949$	−2.74108e21	−0.128359
$950$	0	0
$951$	2.08627e21	0.0961651
$952$	6.34916e21	0.290362
$953$	1.66351e22	0.754794	0.377397	−	0.926052i	$-0.376819\pi$
$953$	1.66351e22	0.754794	0.377397	+	0.926052i	$0.376819\pi$
$954$	5.95832e21	0.268232
$955$	0	0
$956$	−1.75050e22	−0.775762
$957$	−2.35589e22	−1.03589
$958$	4.84686e21	0.211456
$959$	5.40544e22	2.33987
$960$	0	0
$961$	6.45637e22	2.75146
$962$	−2.38238e21	−0.100739
$963$	9.55119e21	0.400738
$964$	−1.64804e22	−0.686102
$965$	0	0
$966$	7.07180e21	0.289869
$967$	2.10794e22	0.857355	0.428677	−	0.903458i	$-0.358980\pi$
$967$	2.10794e22	0.857355	0.428677	+	0.903458i	$0.358980\pi$
$968$	−1.36839e22	−0.552263
$969$	−1.07658e22	−0.431138
$970$	0	0
$971$	−3.09261e22	−1.21950	−0.609749	−	0.792594i	$-0.708730\pi$
$971$	−3.09261e22	−1.21950	−0.609749	+	0.792594i	$0.708730\pi$
$972$	8.19717e20	0.0320750
$973$	1.18329e22	0.459456
$974$	4.98259e21	0.191983
$975$	0	0
$976$	1.43684e21	0.0545175
$977$	−7.85725e21	−0.295843	−0.147922	−	0.988999i	$-0.547258\pi$
$977$	−7.85725e21	−0.295843	−0.147922	+	0.988999i	$0.547258\pi$
$978$	2.07862e21	0.0776666
$979$	4.07105e21	0.150951
$980$	0	0
$981$	6.48584e19	0.00236837
$982$	−1.79860e22	−0.651778
$983$	−1.27981e22	−0.460251	−0.230125	−	0.973161i	$-0.573914\pi$
$983$	−1.27981e22	−0.460251	−0.230125	+	0.973161i	$0.573914\pi$
$984$	8.21383e21	0.293145
$985$	0	0
$986$	1.32976e22	0.467409
$987$	−2.87554e22	−1.00309
$988$	8.43682e21	0.292080
$989$	−2.38381e22	−0.819028
$990$	0	0
$991$	−5.39385e22	−1.82535	−0.912675	−	0.408686i	$-0.865987\pi$
$991$	−5.39385e22	−1.82535	−0.912675	+	0.408686i	$0.865987\pi$
$992$	1.01944e22	0.342393
$993$	−1.15708e22	−0.385694
$994$	7.85760e21	0.259951
$995$	0	0
$996$	−4.71850e21	−0.153766
$997$	1.99678e22	0.645828	0.322914	−	0.946428i	$-0.395337\pi$
$997$	1.99678e22	0.645828	0.322914	+	0.946428i	$0.395337\pi$
$998$	1.96740e22	0.631557
$999$	1.86547e21	0.0594358

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	150.16.a.f.1.1		1
5.2	odd	4		150.16.c.a.49.1		2
5.3	odd	4		150.16.c.a.49.2		2
5.4	even	2		6.16.a.b.1.1	✓	1
15.14	odd	2		18.16.a.b.1.1		1
20.19	odd	2		48.16.a.d.1.1		1
60.59	even	2		144.16.a.j.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6.16.a.b.1.1	✓	1	5.4	even	2
18.16.a.b.1.1		1	15.14	odd	2
48.16.a.d.1.1		1	20.19	odd	2
144.16.a.j.1.1		1	60.59	even	2
150.16.a.f.1.1		1	1.1	even	1	trivial
150.16.c.a.49.1		2	5.2	odd	4
150.16.c.a.49.2		2	5.3	odd	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:	\( N \)	\(=\)	\( 150 = 2 \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 16 \)
Character orbit:	\([\chi]\)	\(=\)	150.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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