LMFDB - Embedded newform 50.10.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [50,10,Mod(1,50)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(50, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("50.1");

S:= CuspForms(chi, 10);

N := Newforms(S);

Level:	$ N $	$=$	$ 50 = 2 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	50.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:	$25.7517918082$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 10)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	50.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-16.0000 q^{2} -174.000 q^{3} +256.000 q^{4} +2784.00 q^{6} -4658.00 q^{7} -4096.00 q^{8} +10593.0 q^{9} +O(q^{10})$

\(q-16.0000 q^{2} -174.000 q^{3} +256.000 q^{4} +2784.00 q^{6} -4658.00 q^{7} -4096.00 q^{8} +10593.0 q^{9} +28992.0 q^{11} -44544.0 q^{12} +164446. q^{13} +74528.0 q^{14} +65536.0 q^{16} +594822. q^{17} -169488. q^{18} -295780. q^{19} +810492. q^{21} -463872. q^{22} -2.54453e6 q^{23} +712704. q^{24} -2.63114e6 q^{26} +1.58166e6 q^{27} -1.19245e6 q^{28} -3.72297e6 q^{29} +2.33577e6 q^{31} -1.04858e6 q^{32} -5.04461e6 q^{33} -9.51715e6 q^{34} +2.71181e6 q^{36} -1.08404e7 q^{37} +4.73248e6 q^{38} -2.86136e7 q^{39} +2.15939e7 q^{41} -1.29679e7 q^{42} -1.08323e7 q^{43} +7.42195e6 q^{44} +4.07125e7 q^{46} -5.17214e6 q^{47} -1.14033e7 q^{48} -1.86566e7 q^{49} -1.03499e8 q^{51} +4.20982e7 q^{52} -9.81797e7 q^{53} -2.53066e7 q^{54} +1.90792e7 q^{56} +5.14657e7 q^{57} +5.95675e7 q^{58} +1.61629e7 q^{59} -4.39282e7 q^{61} -3.73724e7 q^{62} -4.93422e7 q^{63} +1.67772e7 q^{64} +8.07137e7 q^{66} +8.15574e7 q^{67} +1.52274e8 q^{68} +4.42749e8 q^{69} +1.61308e8 q^{71} -4.33889e7 q^{72} +2.47148e8 q^{73} +1.73447e8 q^{74} -7.57197e7 q^{76} -1.35045e8 q^{77} +4.57818e8 q^{78} -5.83346e8 q^{79} -4.83711e8 q^{81} -3.45502e8 q^{82} +1.45718e7 q^{83} +2.07486e8 q^{84} +1.73317e8 q^{86} +6.47797e8 q^{87} -1.18751e8 q^{88} +4.70134e8 q^{89} -7.65989e8 q^{91} -6.51401e8 q^{92} -4.06424e8 q^{93} +8.27542e7 q^{94} +1.82452e8 q^{96} +1.17838e8 q^{97} +2.98506e8 q^{98} +3.07112e8 q^{99} +O(q^{100})\)

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 $

$2$	−16.0000	−0.707107
$2$	−16.0000	−0.707107
$3$	−174.000	−1.24023	−0.620117	−	0.784509i	$-0.712915\pi$
$3$	−174.000	−1.24023	−0.620117	+	0.784509i	$0.712915\pi$
$4$	256.000	0.500000
$5$	0	0
$5$	0	0
$6$	2784.00	0.876978
$7$	−4658.00	−0.733261	−0.366630	−	0.930367i	$-0.619489\pi$
$7$	−4658.00	−0.733261	−0.366630	+	0.930367i	$0.619489\pi$
$8$	−4096.00	−0.353553
$9$	10593.0	0.538180
$10$	0	0
$11$	28992.0	0.597051	0.298525	−	0.954402i	$-0.403505\pi$
$11$	28992.0	0.597051	0.298525	+	0.954402i	$0.403505\pi$
$12$	−44544.0	−0.620117
$13$	164446.	1.59690	0.798451	−	0.602060i	$-0.205653\pi$
$13$	164446.	1.59690	0.798451	+	0.602060i	$0.205653\pi$
$14$	74528.0	0.518493
$15$	0	0
$16$	65536.0	0.250000
$17$	594822.	1.72730	0.863648	−	0.504095i	$-0.168174\pi$
$17$	594822.	1.72730	0.863648	+	0.504095i	$0.168174\pi$
$18$	−169488.	−0.380551
$19$	−295780.	−0.520688	−0.260344	−	0.965516i	$-0.583836\pi$
$19$	−295780.	−0.520688	−0.260344	+	0.965516i	$0.583836\pi$
$20$	0	0
$21$	810492.	0.909415
$22$	−463872.	−0.422178
$23$	−2.54453e6	−1.89598	−0.947988	−	0.318305i	$-0.896886\pi$
$23$	−2.54453e6	−1.89598	−0.947988	+	0.318305i	$0.896886\pi$
$24$	712704.	0.438489
$25$	0	0
$26$	−2.63114e6	−1.12918
$27$	1.58166e6	0.572765
$28$	−1.19245e6	−0.366630
$29$	−3.72297e6	−0.977459	−0.488729	−	0.872435i	$-0.662539\pi$
$29$	−3.72297e6	−0.977459	−0.488729	+	0.872435i	$0.662539\pi$
$30$	0	0
$31$	2.33577e6	0.454258	0.227129	−	0.973865i	$-0.427066\pi$
$31$	2.33577e6	0.454258	0.227129	+	0.973865i	$0.427066\pi$
$32$	−1.04858e6	−0.176777
$33$	−5.04461e6	−0.740482
$34$	−9.51715e6	−1.22138
$35$	0	0
$36$	2.71181e6	0.269090
$37$	−1.08404e7	−0.950907	−0.475454	−	0.879741i	$-0.657716\pi$
$37$	−1.08404e7	−0.950907	−0.475454	+	0.879741i	$0.657716\pi$
$38$	4.73248e6	0.368182
$39$	−2.86136e7	−1.98053
$40$	0	0
$41$	2.15939e7	1.19345	0.596723	−	0.802447i	$-0.296469\pi$
$41$	2.15939e7	1.19345	0.596723	+	0.802447i	$0.296469\pi$
$42$	−1.29679e7	−0.643053
$43$	−1.08323e7	−0.483184	−0.241592	−	0.970378i	$-0.577670\pi$
$43$	−1.08323e7	−0.483184	−0.241592	+	0.970378i	$0.577670\pi$
$44$	7.42195e6	0.298525
$45$	0	0
$46$	4.07125e7	1.34066
$47$	−5.17214e6	−0.154607	−0.0773036	−	0.997008i	$-0.524631\pi$
$47$	−5.17214e6	−0.154607	−0.0773036	+	0.997008i	$0.524631\pi$
$48$	−1.14033e7	−0.310058
$49$	−1.86566e7	−0.462329
$50$	0	0
$51$	−1.03499e8	−2.14225
$52$	4.20982e7	0.798451
$53$	−9.81797e7	−1.70915	−0.854575	−	0.519328i	$-0.826182\pi$
$53$	−9.81797e7	−1.70915	−0.854575	+	0.519328i	$0.826182\pi$
$54$	−2.53066e7	−0.405006
$55$	0	0
$56$	1.90792e7	0.259247
$57$	5.14657e7	0.645775
$58$	5.95675e7	0.691168
$59$	1.61629e7	0.173654	0.0868269	−	0.996223i	$-0.472327\pi$
$59$	1.61629e7	0.173654	0.0868269	+	0.996223i	$0.472327\pi$
$60$	0	0
$61$	−4.39282e7	−0.406218	−0.203109	−	0.979156i	$-0.565105\pi$
$61$	−4.39282e7	−0.406218	−0.203109	+	0.979156i	$0.565105\pi$
$62$	−3.73724e7	−0.321209
$63$	−4.93422e7	−0.394626
$64$	1.67772e7	0.125000
$65$	0	0
$66$	8.07137e7	0.523600
$67$	8.15574e7	0.494455	0.247228	−	0.968957i	$-0.420480\pi$
$67$	8.15574e7	0.494455	0.247228	+	0.968957i	$0.420480\pi$
$68$	1.52274e8	0.863648
$69$	4.42749e8	2.35145
$70$	0	0
$71$	1.61308e8	0.753343	0.376671	−	0.926347i	$-0.377069\pi$
$71$	1.61308e8	0.753343	0.376671	+	0.926347i	$0.377069\pi$
$72$	−4.33889e7	−0.190275
$73$	2.47148e8	1.01860	0.509301	−	0.860589i	$-0.329904\pi$
$73$	2.47148e8	1.01860	0.509301	+	0.860589i	$0.329904\pi$
$74$	1.73447e8	0.672393
$75$	0	0
$76$	−7.57197e7	−0.260344
$77$	−1.35045e8	−0.437794
$78$	4.57818e8	1.40045
$79$	−5.83346e8	−1.68502	−0.842508	−	0.538684i	$-0.818922\pi$
$79$	−5.83346e8	−1.68502	−0.842508	+	0.538684i	$0.818922\pi$
$80$	0	0
$81$	−4.83711e8	−1.24854
$82$	−3.45502e8	−0.843894
$83$	1.45718e7	0.0337024	0.0168512	−	0.999858i	$-0.494636\pi$
$83$	1.45718e7	0.0337024	0.0168512	+	0.999858i	$0.494636\pi$
$84$	2.07486e8	0.454707
$85$	0	0
$86$	1.73317e8	0.341663
$87$	6.47797e8	1.21228
$88$	−1.18751e8	−0.211089
$89$	4.70134e8	0.794267	0.397133	−	0.917761i	$-0.370005\pi$
$89$	4.70134e8	0.794267	0.397133	+	0.917761i	$0.370005\pi$
$90$	0	0
$91$	−7.65989e8	−1.17095
$92$	−6.51401e8	−0.947988
$93$	−4.06424e8	−0.563386
$94$	8.27542e7	0.109324
$95$	0	0
$96$	1.82452e8	0.219244
$97$	1.17838e8	0.135149	0.0675747	−	0.997714i	$-0.478474\pi$
$97$	1.17838e8	0.135149	0.0675747	+	0.997714i	$0.478474\pi$
$98$	2.98506e8	0.326916
$99$	3.07112e8	0.321321
$100$	0	0
$101$	−8.60927e7	−0.0823228	−0.0411614	−	0.999153i	$-0.513106\pi$
$101$	−8.60927e7	−0.0823228	−0.0411614	+	0.999153i	$0.513106\pi$
$102$	1.65598e9	1.51480
$103$	−1.92872e9	−1.68850	−0.844252	−	0.535947i	$-0.819955\pi$
$103$	−1.92872e9	−1.68850	−0.844252	+	0.535947i	$0.819955\pi$
$104$	−6.73571e8	−0.564590
$105$	0	0
$106$	1.57087e9	1.20855
$107$	−1.39685e9	−1.03020	−0.515100	−	0.857130i	$-0.672245\pi$
$107$	−1.39685e9	−1.03020	−0.515100	+	0.857130i	$0.672245\pi$
$108$	4.04905e8	0.286382
$109$	−6.04327e8	−0.410065	−0.205033	−	0.978755i	$-0.565730\pi$
$109$	−6.04327e8	−0.410065	−0.205033	+	0.978755i	$0.565730\pi$
$110$	0	0
$111$	1.88623e9	1.17935
$112$	−3.05267e8	−0.183315
$113$	1.68580e9	0.972643	0.486322	−	0.873780i	$-0.338338\pi$
$113$	1.68580e9	0.972643	0.486322	+	0.873780i	$0.338338\pi$
$114$	−8.23452e8	−0.456632
$115$	0	0
$116$	−9.53080e8	−0.488729
$117$	1.74198e9	0.859421
$118$	−2.58606e8	−0.122792
$119$	−2.77068e9	−1.26656
$120$	0	0
$121$	−1.51741e9	−0.643531
$122$	7.02851e8	0.287239
$123$	−3.75733e9	−1.48015
$124$	5.97958e8	0.227129
$125$	0	0
$126$	7.89475e8	0.279043
$127$	−3.70716e9	−1.26452	−0.632258	−	0.774758i	$-0.717872\pi$
$127$	−3.70716e9	−1.26452	−0.632258	+	0.774758i	$0.717872\pi$
$128$	−2.68435e8	−0.0883883
$129$	1.88482e9	0.599261
$130$	0	0
$131$	2.54080e9	0.753789	0.376895	−	0.926256i	$-0.376992\pi$
$131$	2.54080e9	0.753789	0.376895	+	0.926256i	$0.376992\pi$
$132$	−1.29142e9	−0.370241
$133$	1.37774e9	0.381800
$134$	−1.30492e9	−0.349633
$135$	0	0
$136$	−2.43639e9	−0.610692
$137$	1.15390e9	0.279849	0.139925	−	0.990162i	$-0.455314\pi$
$137$	1.15390e9	0.279849	0.139925	+	0.990162i	$0.455314\pi$
$138$	−7.08398e9	−1.66273
$139$	−5.62721e9	−1.27858	−0.639288	−	0.768968i	$-0.720771\pi$
$139$	−5.62721e9	−1.27858	−0.639288	+	0.768968i	$0.720771\pi$
$140$	0	0
$141$	8.99952e8	0.191749
$142$	−2.58092e9	−0.532694
$143$	4.76762e9	0.953431
$144$	6.94223e8	0.134545
$145$	0	0
$146$	−3.95437e9	−0.720260
$147$	3.24626e9	0.573396
$148$	−2.77515e9	−0.475454
$149$	−2.13688e9	−0.355174	−0.177587	−	0.984105i	$-0.556829\pi$
$149$	−2.13688e9	−0.355174	−0.177587	+	0.984105i	$0.556829\pi$
$150$	0	0
$151$	−9.67515e6	−0.00151447	−0.000757236	−	1.00000i	$-0.500241\pi$
$151$	−9.67515e6	−0.00151447	−0.000757236		1.00000i	$0.500241\pi$
$152$	1.21151e9	0.184091
$153$	6.30095e9	0.929597
$154$	2.16072e9	0.309567
$155$	0	0
$156$	−7.32508e9	−0.990266
$157$	6.88488e8	0.0904373	0.0452187	−	0.998977i	$-0.485602\pi$
$157$	6.88488e8	0.0904373	0.0452187	+	0.998977i	$0.485602\pi$
$158$	9.33353e9	1.19149
$159$	1.70833e10	2.11975
$160$	0	0
$161$	1.18524e10	1.39024
$162$	7.73937e9	0.882853
$163$	1.43082e10	1.58759	0.793797	−	0.608182i	$-0.208101\pi$
$163$	1.43082e10	1.58759	0.793797	+	0.608182i	$0.208101\pi$
$164$	5.52803e9	0.596723
$165$	0	0
$166$	−2.33149e8	−0.0238312
$167$	−9.98735e9	−0.993633	−0.496817	−	0.867856i	$-0.665498\pi$
$167$	−9.98735e9	−0.993633	−0.496817	+	0.867856i	$0.665498\pi$
$168$	−3.31978e9	−0.321527
$169$	1.64380e10	1.55010
$170$	0	0
$171$	−3.13320e9	−0.280224
$172$	−2.77307e9	−0.241592
$173$	−3.51396e9	−0.298256	−0.149128	−	0.988818i	$-0.547647\pi$
$173$	−3.51396e9	−0.298256	−0.149128	+	0.988818i	$0.547647\pi$
$174$	−1.03647e10	−0.857210
$175$	0	0
$176$	1.90002e9	0.149263
$177$	−2.81234e9	−0.215371
$178$	−7.52214e9	−0.561631
$179$	1.19502e9	0.0870038	0.0435019	−	0.999053i	$-0.486149\pi$
$179$	1.19502e9	0.0870038	0.0435019	+	0.999053i	$0.486149\pi$
$180$	0	0
$181$	−9.12053e9	−0.631635	−0.315818	−	0.948820i	$-0.602279\pi$
$181$	−9.12053e9	−0.631635	−0.315818	+	0.948820i	$0.602279\pi$
$182$	1.22558e10	0.827983
$183$	7.64350e9	0.503805
$184$	1.04224e10	0.670329
$185$	0	0
$186$	6.50279e9	0.398374
$187$	1.72451e10	1.03128
$188$	−1.32407e9	−0.0773036
$189$	−7.36737e9	−0.419986
$190$	0	0
$191$	−9.37431e9	−0.509670	−0.254835	−	0.966985i	$-0.582021\pi$
$191$	−9.37431e9	−0.509670	−0.254835	+	0.966985i	$0.582021\pi$
$192$	−2.91924e9	−0.155029
$193$	−2.40000e10	−1.24510	−0.622550	−	0.782580i	$-0.713903\pi$
$193$	−2.40000e10	−1.24510	−0.622550	+	0.782580i	$0.713903\pi$
$194$	−1.88542e9	−0.0955651
$195$	0	0
$196$	−4.77610e9	−0.231165
$197$	5.56124e8	0.0263071	0.0131536	−	0.999913i	$-0.495813\pi$
$197$	5.56124e8	0.0263071	0.0131536	+	0.999913i	$0.495813\pi$
$198$	−4.91380e9	−0.227208
$199$	−2.51255e10	−1.13573	−0.567866	−	0.823121i	$-0.692231\pi$
$199$	−2.51255e10	−1.13573	−0.567866	+	0.823121i	$0.692231\pi$
$200$	0	0
$201$	−1.41910e10	−0.613240
$202$	1.37748e9	0.0582110
$203$	1.73416e10	0.716732
$204$	−2.64958e10	−1.07113
$205$	0	0
$206$	3.08595e10	1.19395
$207$	−2.69542e10	−1.02038
$208$	1.07771e10	0.399225
$209$	−8.57525e9	−0.310877
$210$	0	0
$211$	−1.63915e10	−0.569309	−0.284654	−	0.958630i	$-0.591879\pi$
$211$	−1.63915e10	−0.569309	−0.284654	+	0.958630i	$0.591879\pi$
$212$	−2.51340e10	−0.854575
$213$	−2.80675e10	−0.934321
$214$	2.23495e10	0.728461
$215$	0	0
$216$	−6.47848e9	−0.202503
$217$	−1.08800e10	−0.333090
$218$	9.66924e9	0.289960
$219$	−4.30037e10	−1.26330
$220$	0	0
$221$	9.78161e10	2.75832
$222$	−3.01797e10	−0.833925
$223$	−4.65257e10	−1.25986	−0.629928	−	0.776654i	$-0.716916\pi$
$223$	−4.65257e10	−1.25986	−0.629928	+	0.776654i	$0.716916\pi$
$224$	4.88427e9	0.129623
$225$	0	0
$226$	−2.69728e10	−0.687763
$227$	1.91415e9	0.0478475	0.0239237	−	0.999714i	$-0.492384\pi$
$227$	1.91415e9	0.0478475	0.0239237	+	0.999714i	$0.492384\pi$
$228$	1.31752e10	0.322887
$229$	−1.45825e10	−0.350406	−0.175203	−	0.984532i	$-0.556058\pi$
$229$	−1.45825e10	−0.350406	−0.175203	+	0.984532i	$0.556058\pi$
$230$	0	0
$231$	2.34978e10	0.542966
$232$	1.52493e10	0.345584
$233$	−4.12790e10	−0.917545	−0.458773	−	0.888554i	$-0.651711\pi$
$233$	−4.12790e10	−0.917545	−0.458773	+	0.888554i	$0.651711\pi$
$234$	−2.78716e10	−0.607702
$235$	0	0
$236$	4.13769e9	0.0868269
$237$	1.01502e11	2.08981
$238$	4.43309e10	0.895592
$239$	3.65502e10	0.724602	0.362301	−	0.932061i	$-0.381991\pi$
$239$	3.65502e10	0.724602	0.362301	+	0.932061i	$0.381991\pi$
$240$	0	0
$241$	−8.66070e10	−1.65377	−0.826887	−	0.562368i	$-0.809890\pi$
$241$	−8.66070e10	−1.65377	−0.826887	+	0.562368i	$0.809890\pi$
$242$	2.42786e10	0.455045
$243$	5.30339e10	0.975720
$244$	−1.12456e10	−0.203109
$245$	0	0
$246$	6.01173e10	1.04663
$247$	−4.86398e10	−0.831488
$248$	−9.56732e9	−0.160605
$249$	−2.53549e9	−0.0417989
$250$	0	0
$251$	−7.20769e10	−1.14621	−0.573105	−	0.819482i	$-0.694261\pi$
$251$	−7.20769e10	−1.14621	−0.573105	+	0.819482i	$0.694261\pi$
$252$	−1.26316e10	−0.197313
$253$	−7.37711e10	−1.13199
$254$	5.93145e10	0.894148
$255$	0	0
$256$	4.29497e9	0.0625000
$257$	1.12729e11	1.61190	0.805948	−	0.591986i	$-0.201656\pi$
$257$	1.12729e11	1.61190	0.805948	+	0.591986i	$0.201656\pi$
$258$	−3.01571e10	−0.423741
$259$	5.04947e10	0.697263
$260$	0	0
$261$	−3.94374e10	−0.526049
$262$	−4.06528e10	−0.533009
$263$	−5.55225e10	−0.715596	−0.357798	−	0.933799i	$-0.616472\pi$
$263$	−5.55225e10	−0.715596	−0.357798	+	0.933799i	$0.616472\pi$
$264$	2.06627e10	0.261800
$265$	0	0
$266$	−2.20439e10	−0.269973
$267$	−8.18033e10	−0.985076
$268$	2.08787e10	0.247228
$269$	2.79726e10	0.325722	0.162861	−	0.986649i	$-0.447928\pi$
$269$	2.79726e10	0.325722	0.162861	+	0.986649i	$0.447928\pi$
$270$	0	0
$271$	3.32884e10	0.374914	0.187457	−	0.982273i	$-0.439975\pi$
$271$	3.32884e10	0.374914	0.187457	+	0.982273i	$0.439975\pi$
$272$	3.89823e10	0.431824
$273$	1.33282e11	1.45225
$274$	−1.84623e10	−0.197883
$275$	0	0
$276$	1.13344e11	1.17573
$277$	−5.46240e10	−0.557474	−0.278737	−	0.960367i	$-0.589916\pi$
$277$	−5.46240e10	−0.557474	−0.278737	+	0.960367i	$0.589916\pi$
$278$	9.00353e10	0.904090
$279$	2.47428e10	0.244473
$280$	0	0
$281$	−8.37818e10	−0.801625	−0.400813	−	0.916160i	$-0.631272\pi$
$281$	−8.37818e10	−0.801625	−0.400813	+	0.916160i	$0.631272\pi$
$282$	−1.43992e10	−0.135587
$283$	8.36086e10	0.774840	0.387420	−	0.921903i	$-0.373366\pi$
$283$	8.36086e10	0.774840	0.387420	+	0.921903i	$0.373366\pi$
$284$	4.12948e10	0.376671
$285$	0	0
$286$	−7.62819e10	−0.674178
$287$	−1.00584e11	−0.875107
$288$	−1.11076e10	−0.0951377
$289$	2.35225e11	1.98355
$290$	0	0
$291$	−2.05039e10	−0.167617
$292$	6.32699e10	0.509301
$293$	2.28547e11	1.81164	0.905819	−	0.423666i	$-0.139257\pi$
$293$	2.28547e11	1.81164	0.905819	+	0.423666i	$0.139257\pi$
$294$	−5.19401e10	−0.405452
$295$	0	0
$296$	4.44024e10	0.336197
$297$	4.58555e10	0.341969
$298$	3.41900e10	0.251146
$299$	−4.18438e11	−3.02769
$300$	0	0
$301$	5.04568e10	0.354300
$302$	1.54802e8	0.00107089
$303$	1.49801e10	0.102100
$304$	−1.93842e10	−0.130172
$305$	0	0
$306$	−1.00815e11	−0.657324
$307$	−1.95064e11	−1.25330	−0.626648	−	0.779302i	$-0.715574\pi$
$307$	−1.95064e11	−1.25330	−0.626648	+	0.779302i	$0.715574\pi$
$308$	−3.45715e10	−0.218897
$309$	3.35598e11	2.09414
$310$	0	0
$311$	2.15637e11	1.30708	0.653540	−	0.756892i	$-0.273283\pi$
$311$	2.15637e11	1.30708	0.653540	+	0.756892i	$0.273283\pi$
$312$	1.17201e11	0.700224
$313$	−1.91755e11	−1.12927	−0.564635	−	0.825341i	$-0.690983\pi$
$313$	−1.91755e11	−1.12927	−0.564635	+	0.825341i	$0.690983\pi$
$314$	−1.10158e10	−0.0639488
$315$	0	0
$316$	−1.49337e11	−0.842508
$317$	3.38886e11	1.88489	0.942446	−	0.334358i	$-0.108519\pi$
$317$	3.38886e11	1.88489	0.942446	+	0.334358i	$0.108519\pi$
$318$	−2.73332e11	−1.49889
$319$	−1.07936e11	−0.583592
$320$	0	0
$321$	2.43051e11	1.27769
$322$	−1.89639e11	−0.983052
$323$	−1.75936e11	−0.899383
$324$	−1.23830e11	−0.624271
$325$	0	0
$326$	−2.28931e11	−1.12260
$327$	1.05153e11	0.508577
$328$	−8.84485e10	−0.421947
$329$	2.40918e10	0.113367
$330$	0	0
$331$	−1.78427e11	−0.817022	−0.408511	−	0.912753i	$-0.633952\pi$
$331$	−1.78427e11	−0.817022	−0.408511	+	0.912753i	$0.633952\pi$
$332$	3.73038e9	0.0168512
$333$	−1.14833e11	−0.511760
$334$	1.59798e11	0.702605
$335$	0	0
$336$	5.31164e10	0.227354
$337$	−2.64693e11	−1.11791	−0.558957	−	0.829196i	$-0.688798\pi$
$337$	−2.64693e11	−1.11791	−0.558957	+	0.829196i	$0.688798\pi$
$338$	−2.63008e11	−1.09608
$339$	−2.93330e11	−1.20631
$340$	0	0
$341$	6.77187e10	0.271215
$342$	5.01312e10	0.198148
$343$	2.74870e11	1.07227
$344$	4.43691e10	0.170831
$345$	0	0
$346$	5.62233e10	0.210899
$347$	8.11912e10	0.300626	0.150313	−	0.988638i	$-0.451972\pi$
$347$	8.11912e10	0.300626	0.150313	+	0.988638i	$0.451972\pi$
$348$	1.65836e11	0.606139
$349$	−2.26688e11	−0.817926	−0.408963	−	0.912551i	$-0.634110\pi$
$349$	−2.26688e11	−0.817926	−0.408963	+	0.912551i	$0.634110\pi$
$350$	0	0
$351$	2.60098e11	0.914649
$352$	−3.04003e10	−0.105545
$353$	−3.28733e10	−0.112683	−0.0563413	−	0.998412i	$-0.517944\pi$
$353$	−3.28733e10	−0.112683	−0.0563413	+	0.998412i	$0.517944\pi$
$354$	4.49974e10	0.152290
$355$	0	0
$356$	1.20354e11	0.397133
$357$	4.82098e11	1.57083
$358$	−1.91204e10	−0.0615210
$359$	−3.12096e11	−0.991661	−0.495831	−	0.868419i	$-0.665136\pi$
$359$	−3.12096e11	−0.991661	−0.495831	+	0.868419i	$0.665136\pi$
$360$	0	0
$361$	−2.35202e11	−0.728884
$362$	1.45928e11	0.446634
$363$	2.64030e11	0.798129
$364$	−1.96093e11	−0.585473
$365$	0	0
$366$	−1.22296e11	−0.356244
$367$	−6.28209e11	−1.80762	−0.903810	−	0.427934i	$-0.859241\pi$
$367$	−6.28209e11	−1.80762	−0.903810	+	0.427934i	$0.859241\pi$
$368$	−1.66759e11	−0.473994
$369$	2.28744e11	0.642289
$370$	0	0
$371$	4.57321e11	1.25325
$372$	−1.04045e11	−0.281693
$373$	9.84770e10	0.263418	0.131709	−	0.991288i	$-0.457954\pi$
$373$	9.84770e10	0.263418	0.131709	+	0.991288i	$0.457954\pi$
$374$	−2.75921e11	−0.729227
$375$	0	0
$376$	2.11851e10	0.0546619
$377$	−6.12228e11	−1.56091
$378$	1.17878e11	0.296975
$379$	2.89024e11	0.719544	0.359772	−	0.933040i	$-0.382855\pi$
$379$	2.89024e11	0.719544	0.359772	+	0.933040i	$0.382855\pi$
$380$	0	0
$381$	6.45046e11	1.56830
$382$	1.49989e11	0.360391
$383$	−2.01350e11	−0.478141	−0.239071	−	0.971002i	$-0.576843\pi$
$383$	−2.01350e11	−0.478141	−0.239071	+	0.971002i	$0.576843\pi$
$384$	4.67078e10	0.109622
$385$	0	0
$386$	3.84001e11	0.880418
$387$	−1.14746e11	−0.260040
$388$	3.01666e10	0.0675747
$389$	3.93769e11	0.871904	0.435952	−	0.899970i	$-0.356412\pi$
$389$	3.93769e11	0.871904	0.435952	+	0.899970i	$0.356412\pi$
$390$	0	0
$391$	−1.51354e12	−3.27491
$392$	7.64176e10	0.163458
$393$	−4.42099e11	−0.934875
$394$	−8.89798e9	−0.0186019
$395$	0	0
$396$	7.86207e10	0.160660
$397$	5.15625e11	1.04178	0.520891	−	0.853623i	$-0.325600\pi$
$397$	5.15625e11	1.04178	0.520891	+	0.853623i	$0.325600\pi$
$398$	4.02008e11	0.803084
$399$	−2.39727e11	−0.473521
$400$	0	0
$401$	−8.43309e11	−1.62869	−0.814343	−	0.580384i	$-0.802902\pi$
$401$	−8.43309e11	−1.62869	−0.814343	+	0.580384i	$0.802902\pi$
$402$	2.27056e11	0.433626
$403$	3.84108e11	0.725406
$404$	−2.20397e10	−0.0411614
$405$	0	0
$406$	−2.77466e11	−0.506806
$407$	−3.14285e11	−0.567740
$408$	4.23932e11	0.757400
$409$	5.64549e11	0.997577	0.498789	−	0.866724i	$-0.333778\pi$
$409$	5.64549e11	0.997577	0.498789	+	0.866724i	$0.333778\pi$
$410$	0	0
$411$	−2.00778e11	−0.347078
$412$	−4.93753e11	−0.844252
$413$	−7.52866e10	−0.127333
$414$	4.31268e11	0.721516
$415$	0	0
$416$	−1.72434e11	−0.282295
$417$	9.79134e11	1.58573
$418$	1.37204e11	0.219823
$419$	6.92475e11	1.09759	0.548796	−	0.835956i	$-0.315086\pi$
$419$	6.92475e11	1.09759	0.548796	+	0.835956i	$0.315086\pi$
$420$	0	0
$421$	4.08125e11	0.633176	0.316588	−	0.948563i	$-0.397463\pi$
$421$	4.08125e11	0.633176	0.316588	+	0.948563i	$0.397463\pi$
$422$	2.62264e11	0.402562
$423$	−5.47885e10	−0.0832065
$424$	4.02144e11	0.604276
$425$	0	0
$426$	4.49081e11	0.660665
$427$	2.04617e11	0.297863
$428$	−3.57592e11	−0.515100
$429$	−8.29566e11	−1.18248
$430$	0	0
$431$	4.41055e11	0.615665	0.307833	−	0.951441i	$-0.400396\pi$
$431$	4.41055e11	0.615665	0.307833	+	0.951441i	$0.400396\pi$
$432$	1.03656e11	0.143191
$433$	7.15390e10	0.0978019	0.0489009	−	0.998804i	$-0.484428\pi$
$433$	7.15390e10	0.0978019	0.0489009	+	0.998804i	$0.484428\pi$
$434$	1.74080e11	0.235530
$435$	0	0
$436$	−1.54708e11	−0.205033
$437$	7.52622e11	0.987212
$438$	6.88060e11	0.893291
$439$	4.74967e11	0.610342	0.305171	−	0.952298i	$-0.401286\pi$
$439$	4.74967e11	0.610342	0.305171	+	0.952298i	$0.401286\pi$
$440$	0	0
$441$	−1.97630e11	−0.248816
$442$	−1.56506e12	−1.95043
$443$	1.97072e11	0.243113	0.121556	−	0.992585i	$-0.461211\pi$
$443$	1.97072e11	0.243113	0.121556	+	0.992585i	$0.461211\pi$
$444$	4.82876e11	0.589674
$445$	0	0
$446$	7.44411e11	0.890853
$447$	3.71817e11	0.440499
$448$	−7.81483e10	−0.0916576
$449$	−6.20156e11	−0.720100	−0.360050	−	0.932933i	$-0.617240\pi$
$449$	−6.20156e11	−0.720100	−0.360050	+	0.932933i	$0.617240\pi$
$450$	0	0
$451$	6.26049e11	0.712548
$452$	4.31565e11	0.486322
$453$	1.68348e9	0.00187830
$454$	−3.06264e10	−0.0338333
$455$	0	0
$456$	−2.10804e11	−0.228316
$457$	−7.88006e11	−0.845097	−0.422549	−	0.906340i	$-0.638864\pi$
$457$	−7.88006e11	−0.845097	−0.422549	+	0.906340i	$0.638864\pi$
$458$	2.33319e11	0.247774
$459$	9.40806e11	0.989334
$460$	0	0
$461$	1.53496e10	0.0158286	0.00791429	−	0.999969i	$-0.497481\pi$
$461$	1.53496e10	0.0158286	0.00791429	+	0.999969i	$0.497481\pi$
$462$	−3.75965e11	−0.383935
$463$	−1.94494e11	−0.196694	−0.0983471	−	0.995152i	$-0.531356\pi$
$463$	−1.94494e11	−0.196694	−0.0983471	+	0.995152i	$0.531356\pi$
$464$	−2.43989e11	−0.244365
$465$	0	0
$466$	6.60464e11	0.648802
$467$	−1.06503e11	−0.103618	−0.0518089	−	0.998657i	$-0.516499\pi$
$467$	−1.06503e11	−0.103618	−0.0518089	+	0.998657i	$0.516499\pi$
$468$	4.45946e11	0.429710
$469$	−3.79894e11	−0.362564
$470$	0	0
$471$	−1.19797e11	−0.112163
$472$	−6.62031e10	−0.0613959
$473$	−3.14050e11	−0.288485
$474$	−1.62403e12	−1.47772
$475$	0	0
$476$	−7.09294e11	−0.633279
$477$	−1.04002e12	−0.919831
$478$	−5.84803e11	−0.512371
$479$	−8.31146e11	−0.721386	−0.360693	−	0.932685i	$-0.617460\pi$
$479$	−8.31146e11	−0.721386	−0.360693	+	0.932685i	$0.617460\pi$
$480$	0	0
$481$	−1.78266e12	−1.51851
$482$	1.38571e12	1.16939
$483$	−2.06232e12	−1.72423
$484$	−3.88457e11	−0.321765
$485$	0	0
$486$	−8.48542e11	−0.689938
$487$	1.38566e12	1.11629	0.558146	−	0.829743i	$-0.311513\pi$
$487$	1.38566e12	1.11629	0.558146	+	0.829743i	$0.311513\pi$
$488$	1.79930e11	0.143620
$489$	−2.48962e12	−1.96899
$490$	0	0
$491$	3.92804e11	0.305007	0.152503	−	0.988303i	$-0.451266\pi$
$491$	3.92804e11	0.305007	0.152503	+	0.988303i	$0.451266\pi$
$492$	−9.61877e11	−0.740076
$493$	−2.21450e12	−1.68836
$494$	7.78237e11	0.587951
$495$	0	0
$496$	1.53077e11	0.113565
$497$	−7.51371e11	−0.552397
$498$	4.05679e10	0.0295563
$499$	−7.52029e11	−0.542978	−0.271489	−	0.962442i	$-0.587516\pi$
$499$	−7.52029e11	−0.542978	−0.271489	+	0.962442i	$0.587516\pi$
$500$	0	0
$501$	1.73780e12	1.23234
$502$	1.15323e12	0.810493
$503$	1.83429e12	1.27765	0.638826	−	0.769351i	$-0.279420\pi$
$503$	1.83429e12	1.27765	0.638826	+	0.769351i	$0.279420\pi$
$504$	2.02106e11	0.139521
$505$	0	0
$506$	1.18034e12	0.800441
$507$	−2.86021e12	−1.92248
$508$	−9.49033e11	−0.632258
$509$	−6.19864e11	−0.409323	−0.204662	−	0.978833i	$-0.565609\pi$
$509$	−6.19864e11	−0.409323	−0.204662	+	0.978833i	$0.565609\pi$
$510$	0	0
$511$	−1.15122e12	−0.746900
$512$	−6.87195e10	−0.0441942
$513$	−4.67823e11	−0.298232
$514$	−1.80367e12	−1.13978
$515$	0	0
$516$	4.82514e11	0.299630
$517$	−1.49951e11	−0.0923083
$518$	−8.07915e11	−0.493039
$519$	6.11428e11	0.369907
$520$	0	0
$521$	5.25683e11	0.312575	0.156287	−	0.987712i	$-0.450047\pi$
$521$	5.25683e11	0.312575	0.156287	+	0.987712i	$0.450047\pi$
$522$	6.30999e11	0.371973
$523$	−1.68426e11	−0.0984353	−0.0492177	−	0.998788i	$-0.515673\pi$
$523$	−1.68426e11	−0.0984353	−0.0492177	+	0.998788i	$0.515673\pi$
$524$	6.50445e11	0.376895
$525$	0	0
$526$	8.88360e11	0.506003
$527$	1.38937e12	0.784639
$528$	−3.30603e11	−0.185121
$529$	4.67350e12	2.59473
$530$	0	0
$531$	1.71213e11	0.0934570
$532$	3.52702e11	0.190900
$533$	3.55102e12	1.90582
$534$	1.30885e12	0.696554
$535$	0	0
$536$	−3.34059e11	−0.174816
$537$	−2.07934e11	−0.107905
$538$	−4.47561e11	−0.230320
$539$	−5.40893e11	−0.276034
$540$	0	0
$541$	2.20612e12	1.10724	0.553620	−	0.832769i	$-0.313246\pi$
$541$	2.20612e12	1.10724	0.553620	+	0.832769i	$0.313246\pi$
$542$	−5.32615e11	−0.265104
$543$	1.58697e12	0.783375
$544$	−6.23716e11	−0.305346
$545$	0	0
$546$	−2.13251e12	−1.02689
$547$	3.51639e12	1.67940	0.839701	−	0.543049i	$-0.182730\pi$
$547$	3.51639e12	1.67940	0.839701	+	0.543049i	$0.182730\pi$
$548$	2.95397e11	0.139925
$549$	−4.65331e11	−0.218618
$550$	0	0
$551$	1.10118e12	0.508951
$552$	−1.81350e12	−0.831365
$553$	2.71722e12	1.23556
$554$	8.73985e11	0.394194
$555$	0	0
$556$	−1.44057e12	−0.639288
$557$	−6.29996e11	−0.277325	−0.138663	−	0.990340i	$-0.544280\pi$
$557$	−6.29996e11	−0.277325	−0.138663	+	0.990340i	$0.544280\pi$
$558$	−3.95885e11	−0.172868
$559$	−1.78133e12	−0.771597
$560$	0	0
$561$	−3.00064e12	−1.27903
$562$	1.34051e12	0.566835
$563$	−4.02619e11	−0.168891	−0.0844455	−	0.996428i	$-0.526912\pi$
$563$	−4.02619e11	−0.168891	−0.0844455	+	0.996428i	$0.526912\pi$
$564$	2.30388e11	0.0958746
$565$	0	0
$566$	−1.33774e12	−0.547894
$567$	2.25313e12	0.915507
$568$	−6.60716e11	−0.266347
$569$	2.55482e12	1.02177	0.510887	−	0.859648i	$-0.329317\pi$
$569$	2.55482e12	1.02177	0.510887	+	0.859648i	$0.329317\pi$
$570$	0	0
$571$	1.16243e12	0.457620	0.228810	−	0.973471i	$-0.426516\pi$
$571$	1.16243e12	0.457620	0.228810	+	0.973471i	$0.426516\pi$
$572$	1.22051e12	0.476716
$573$	1.63113e12	0.632110
$574$	1.60935e12	0.618794
$575$	0	0
$576$	1.77721e11	0.0672725
$577$	−4.66332e12	−1.75148	−0.875738	−	0.482787i	$-0.839625\pi$
$577$	−4.66332e12	−1.75148	−0.875738	+	0.482787i	$0.839625\pi$
$578$	−3.76361e12	−1.40258
$579$	4.17601e12	1.54421
$580$	0	0
$581$	−6.78754e10	−0.0247127
$582$	3.28062e11	0.118523
$583$	−2.84643e12	−1.02045
$584$	−1.01232e12	−0.360130
$585$	0	0
$586$	−3.65675e12	−1.28102
$587$	−3.88422e12	−1.35030	−0.675152	−	0.737678i	$-0.735922\pi$
$587$	−3.88422e12	−1.35030	−0.675152	+	0.737678i	$0.735922\pi$
$588$	8.31042e11	0.286698
$589$	−6.90875e11	−0.236527
$590$	0	0
$591$	−9.67655e10	−0.0326270
$592$	−7.10438e11	−0.237727
$593$	2.01341e12	0.668631	0.334315	−	0.942461i	$-0.391495\pi$
$593$	2.01341e12	0.668631	0.334315	+	0.942461i	$0.391495\pi$
$594$	−7.33688e11	−0.241809
$595$	0	0
$596$	−5.47041e11	−0.177587
$597$	4.37184e12	1.40857
$598$	6.69502e12	2.14090
$599$	2.10578e12	0.668333	0.334166	−	0.942514i	$-0.391545\pi$
$599$	2.10578e12	0.668333	0.334166	+	0.942514i	$0.391545\pi$
$600$	0	0
$601$	4.74058e12	1.48216	0.741082	−	0.671415i	$-0.234313\pi$
$601$	4.74058e12	1.48216	0.741082	+	0.671415i	$0.234313\pi$
$602$	−8.07309e11	−0.250528
$603$	8.63938e11	0.266106
$604$	−2.47684e9	−0.000757236 0
$605$	0	0
$606$	−2.39682e11	−0.0721953
$607$	4.01476e12	1.20036	0.600179	−	0.799866i	$-0.295096\pi$
$607$	4.01476e12	1.20036	0.600179	+	0.799866i	$0.295096\pi$
$608$	3.10148e11	0.0920455
$609$	−3.01744e12	−0.888915
$610$	0	0
$611$	−8.50537e11	−0.246893
$612$	1.61304e12	0.464798
$613$	−1.30399e12	−0.372993	−0.186496	−	0.982456i	$-0.559713\pi$
$613$	−1.30399e12	−0.372993	−0.186496	+	0.982456i	$0.559713\pi$
$614$	3.12102e12	0.886214
$615$	0	0
$616$	5.53143e11	0.154783
$617$	−5.10164e12	−1.41719	−0.708593	−	0.705618i	$-0.750670\pi$
$617$	−5.10164e12	−1.41719	−0.708593	+	0.705618i	$0.750670\pi$
$618$	−5.36956e12	−1.48078
$619$	−4.50630e11	−0.123371	−0.0616854	−	0.998096i	$-0.519648\pi$
$619$	−4.50630e11	−0.123371	−0.0616854	+	0.998096i	$0.519648\pi$
$620$	0	0
$621$	−4.02459e12	−1.08595
$622$	−3.45020e12	−0.924246
$623$	−2.18988e12	−0.582404
$624$	−1.87522e12	−0.495133
$625$	0	0
$626$	3.06808e12	0.798514
$627$	1.49209e12	0.385560
$628$	1.76253e11	0.0452187
$629$	−6.44812e12	−1.64250
$630$	0	0
$631$	7.97105e10	0.0200163	0.0100081	−	0.999950i	$-0.496814\pi$
$631$	7.97105e10	0.0200163	0.0100081	+	0.999950i	$0.496814\pi$
$632$	2.38938e12	0.595743
$633$	2.85212e12	0.706076
$634$	−5.42217e12	−1.33282
$635$	0	0
$636$	4.37332e12	1.05987
$637$	−3.06801e12	−0.738294
$638$	1.72698e12	0.412662
$639$	1.70873e12	0.405434
$640$	0	0
$641$	−3.12648e12	−0.731468	−0.365734	−	0.930719i	$-0.619182\pi$
$641$	−3.12648e12	−0.731468	−0.365734	+	0.930719i	$0.619182\pi$
$642$	−3.88882e12	−0.903462
$643$	5.94982e12	1.37263	0.686316	−	0.727303i	$-0.259227\pi$
$643$	5.94982e12	1.37263	0.686316	+	0.727303i	$0.259227\pi$
$644$	3.03422e12	0.695122
$645$	0	0
$646$	2.81498e12	0.635960
$647$	−3.56187e12	−0.799115	−0.399557	−	0.916708i	$-0.630836\pi$
$647$	−3.56187e12	−0.799115	−0.399557	+	0.916708i	$0.630836\pi$
$648$	1.98128e12	0.441426
$649$	4.68594e11	0.103680
$650$	0	0
$651$	1.89312e12	0.413109
$652$	3.66289e12	0.793797
$653$	1.37883e11	0.0296757	0.0148379	−	0.999890i	$-0.495277\pi$
$653$	1.37883e11	0.0296757	0.0148379	+	0.999890i	$0.495277\pi$
$654$	−1.68245e12	−0.359618
$655$	0	0
$656$	1.41518e12	0.298362
$657$	2.61804e12	0.548191
$658$	−3.85469e11	−0.0801628
$659$	−4.16654e12	−0.860581	−0.430290	−	0.902691i	$-0.641589\pi$
$659$	−4.16654e12	−0.860581	−0.430290	+	0.902691i	$0.641589\pi$
$660$	0	0
$661$	−3.29083e12	−0.670500	−0.335250	−	0.942129i	$-0.608821\pi$
$661$	−3.29083e12	−0.670500	−0.335250	+	0.942129i	$0.608821\pi$
$662$	2.85483e12	0.577722
$663$	−1.70200e13	−3.42097
$664$	−5.96860e10	−0.0119156
$665$	0	0
$666$	1.83732e12	0.361869
$667$	9.47322e12	1.85324
$668$	−2.55676e12	−0.496817
$669$	8.09547e12	1.56252
$670$	0	0
$671$	−1.27357e12	−0.242532
$672$	−8.49862e11	−0.160763
$673$	−5.74732e12	−1.07993	−0.539967	−	0.841686i	$-0.681563\pi$
$673$	−5.74732e12	−1.07993	−0.539967	+	0.841686i	$0.681563\pi$
$674$	4.23510e12	0.790485
$675$	0	0
$676$	4.20812e12	0.775048
$677$	−9.31245e12	−1.70379	−0.851893	−	0.523716i	$-0.824545\pi$
$677$	−9.31245e12	−1.70379	−0.851893	+	0.523716i	$0.824545\pi$
$678$	4.69327e12	0.852987
$679$	−5.48892e11	−0.0990998
$680$	0	0
$681$	−3.33062e11	−0.0593421
$682$	−1.08350e12	−0.191778
$683$	−9.19967e12	−1.61763	−0.808815	−	0.588063i	$-0.799891\pi$
$683$	−9.19967e12	−1.61763	−0.808815	+	0.588063i	$0.799891\pi$
$684$	−8.02099e11	−0.140112
$685$	0	0
$686$	−4.39792e12	−0.758208
$687$	2.53735e12	0.434585
$688$	−7.09905e11	−0.120796
$689$	−1.61453e13	−2.72934
$690$	0	0
$691$	1.03125e13	1.72074	0.860369	−	0.509672i	$-0.170233\pi$
$691$	1.03125e13	1.72074	0.860369	+	0.509672i	$0.170233\pi$
$692$	−8.99573e11	−0.149128
$693$	−1.43053e12	−0.235612
$694$	−1.29906e12	−0.212574
$695$	0	0
$696$	−2.65338e12	−0.428605
$697$	1.28445e13	2.06144
$698$	3.62701e12	0.578361
$699$	7.18255e12	1.13797
$700$	0	0
$701$	−9.52566e12	−1.48992	−0.744961	−	0.667108i	$-0.767532\pi$
$701$	−9.52566e12	−1.48992	−0.744961	+	0.667108i	$0.767532\pi$
$702$	−4.16156e12	−0.646754
$703$	3.20638e12	0.495126
$704$	4.86405e11	0.0746313
$705$	0	0
$706$	5.25973e11	0.0796787
$707$	4.01020e11	0.0603641
$708$	−7.19958e11	−0.107686
$709$	−8.38537e12	−1.24628	−0.623138	−	0.782112i	$-0.714142\pi$
$709$	−8.38537e12	−1.24628	−0.623138	+	0.782112i	$0.714142\pi$
$710$	0	0
$711$	−6.17938e12	−0.906842
$712$	−1.92567e12	−0.280816
$713$	−5.94345e12	−0.861263
$714$	−7.71358e12	−1.11074
$715$	0	0
$716$	3.05926e11	0.0435019
$717$	−6.35974e12	−0.898676
$718$	4.99354e12	0.701211
$719$	6.94013e12	0.968473	0.484236	−	0.874937i	$-0.339097\pi$
$719$	6.94013e12	0.968473	0.484236	+	0.874937i	$0.339097\pi$
$720$	0	0
$721$	8.98398e12	1.23811
$722$	3.76323e12	0.515399
$723$	1.50696e13	2.05107
$724$	−2.33485e12	−0.315818
$725$	0	0
$726$	−4.22447e12	−0.564362
$727$	−7.74984e12	−1.02893	−0.514467	−	0.857510i	$-0.672010\pi$
$727$	−7.74984e12	−1.02893	−0.514467	+	0.857510i	$0.672010\pi$
$728$	3.13749e12	0.413992
$729$	2.92986e11	0.0384214
$730$	0	0
$731$	−6.44329e12	−0.834602
$732$	1.95674e12	0.251902
$733$	−4.12555e12	−0.527854	−0.263927	−	0.964543i	$-0.585018\pi$
$733$	−4.12555e12	−0.527854	−0.263927	+	0.964543i	$0.585018\pi$
$734$	1.00513e13	1.27818
$735$	0	0
$736$	2.66814e12	0.335164
$737$	2.36451e12	0.295215
$738$	−3.65990e12	−0.454167
$739$	3.33931e12	0.411867	0.205933	−	0.978566i	$-0.433977\pi$
$739$	3.33931e12	0.411867	0.205933	+	0.978566i	$0.433977\pi$
$740$	0	0
$741$	8.46333e12	1.03124
$742$	−7.31713e12	−0.886183
$743$	1.26471e13	1.52245	0.761224	−	0.648489i	$-0.224599\pi$
$743$	1.26471e13	1.52245	0.761224	+	0.648489i	$0.224599\pi$
$744$	1.66471e12	0.199187
$745$	0	0
$746$	−1.57563e12	−0.186265
$747$	1.54359e11	0.0181380
$748$	4.41474e12	0.515642
$749$	6.50651e12	0.755405
$750$	0	0
$751$	1.56552e12	0.179588	0.0897941	−	0.995960i	$-0.471379\pi$
$751$	1.56552e12	0.179588	0.0897941	+	0.995960i	$0.471379\pi$
$752$	−3.38961e11	−0.0386518
$753$	1.25414e13	1.42157
$754$	9.79564e12	1.10373
$755$	0	0
$756$	−1.88605e12	−0.209993
$757$	−9.54781e11	−0.105675	−0.0528375	−	0.998603i	$-0.516827\pi$
$757$	−9.54781e11	−0.105675	−0.0528375	+	0.998603i	$0.516827\pi$
$758$	−4.62438e12	−0.508794
$759$	1.28362e13	1.40394
$760$	0	0
$761$	5.27330e12	0.569969	0.284985	−	0.958532i	$-0.408012\pi$
$761$	5.27330e12	0.569969	0.284985	+	0.958532i	$0.408012\pi$
$762$	−1.03207e13	−1.10895
$763$	2.81496e12	0.300685
$764$	−2.39982e12	−0.254835
$765$	0	0
$766$	3.22159e12	0.338097
$767$	2.65792e12	0.277308
$768$	−7.47324e11	−0.0775146
$769$	−6.96923e12	−0.718648	−0.359324	−	0.933213i	$-0.616993\pi$
$769$	−6.96923e12	−0.718648	−0.359324	+	0.933213i	$0.616993\pi$
$770$	0	0
$771$	−1.96149e13	−1.99913
$772$	−6.14401e12	−0.622550
$773$	3.79383e12	0.382182	0.191091	−	0.981572i	$-0.438797\pi$
$773$	3.79383e12	0.382182	0.191091	+	0.981572i	$0.438797\pi$
$774$	1.83594e12	0.183876
$775$	0	0
$776$	−4.82666e11	−0.0477825
$777$	−8.78607e12	−0.864769
$778$	−6.30031e12	−0.616530
$779$	−6.38703e12	−0.621413
$780$	0	0
$781$	4.67663e12	0.449784
$782$	2.42167e13	2.31571
$783$	−5.88847e12	−0.559854
$784$	−1.22268e12	−0.115582
$785$	0	0
$786$	7.07359e12	0.661056
$787$	2.05939e13	1.91361	0.956803	−	0.290736i	$-0.0939000\pi$
$787$	2.05939e13	1.91361	0.956803	+	0.290736i	$0.0939000\pi$
$788$	1.42368e11	0.0131536
$789$	9.66092e12	0.887507
$790$	0	0
$791$	−7.85247e12	−0.713201
$792$	−1.25793e12	−0.113604
$793$	−7.22381e12	−0.648690
$794$	−8.25000e12	−0.736651
$795$	0	0
$796$	−6.43213e12	−0.567866
$797$	−1.45507e12	−0.127738	−0.0638690	−	0.997958i	$-0.520344\pi$
$797$	−1.45507e12	−0.127738	−0.0638690	+	0.997958i	$0.520344\pi$
$798$	3.83564e12	0.334830
$799$	−3.07650e12	−0.267052
$800$	0	0
$801$	4.98013e12	0.427459
$802$	1.34929e13	1.15165
$803$	7.16531e12	0.608156
$804$	−3.63289e12	−0.306620
$805$	0	0
$806$	−6.14573e12	−0.512939
$807$	−4.86722e12	−0.403971
$808$	3.52636e11	0.0291055
$809$	1.31749e13	1.08138	0.540690	−	0.841222i	$-0.318163\pi$
$809$	1.31749e13	1.08138	0.540690	+	0.841222i	$0.318163\pi$
$810$	0	0
$811$	9.97833e12	0.809961	0.404980	−	0.914325i	$-0.367278\pi$
$811$	9.97833e12	0.809961	0.404980	+	0.914325i	$0.367278\pi$
$812$	4.43945e12	0.358366
$813$	−5.79219e12	−0.464981
$814$	5.02857e12	0.401453
$815$	0	0
$816$	−6.78291e12	−0.535563
$817$	3.20398e12	0.251588
$818$	−9.03278e12	−0.705394
$819$	−8.11413e12	−0.630179
$820$	0	0
$821$	−1.80913e13	−1.38972	−0.694859	−	0.719146i	$-0.744533\pi$
$821$	−1.80913e13	−1.38972	−0.694859	+	0.719146i	$0.744533\pi$
$822$	3.21245e12	0.245422
$823$	−1.39801e12	−0.106221	−0.0531107	−	0.998589i	$-0.516914\pi$
$823$	−1.39801e12	−0.106221	−0.0531107	+	0.998589i	$0.516914\pi$
$824$	7.90004e12	0.596976
$825$	0	0
$826$	1.20459e12	0.0900383
$827$	−2.26201e13	−1.68159	−0.840794	−	0.541354i	$-0.817912\pi$
$827$	−2.26201e13	−1.68159	−0.840794	+	0.541354i	$0.817912\pi$
$828$	−6.90029e12	−0.510189
$829$	1.72646e13	1.26959	0.634793	−	0.772682i	$-0.281085\pi$
$829$	1.72646e13	1.26959	0.634793	+	0.772682i	$0.281085\pi$
$830$	0	0
$831$	9.50458e12	0.691398
$832$	2.75895e12	0.199613
$833$	−1.10974e13	−0.798579
$834$	−1.56662e13	−1.12128
$835$	0	0
$836$	−2.19526e12	−0.155439
$837$	3.69440e12	0.260183
$838$	−1.10796e13	−0.776115
$839$	−1.76607e12	−0.123049	−0.0615245	−	0.998106i	$-0.519596\pi$
$839$	−1.76607e12	−0.123049	−0.0615245	+	0.998106i	$0.519596\pi$
$840$	0	0
$841$	−6.46640e11	−0.0445739
$842$	−6.53001e12	−0.447723
$843$	1.45780e13	0.994203
$844$	−4.19622e12	−0.284654
$845$	0	0
$846$	8.76615e11	0.0588359
$847$	7.06810e12	0.471876
$848$	−6.43430e12	−0.427287
$849$	−1.45479e13	−0.960983
$850$	0	0
$851$	2.75838e13	1.80290
$852$	−7.18529e12	−0.467161
$853$	2.37844e13	1.53823	0.769116	−	0.639109i	$-0.220697\pi$
$853$	2.37844e13	1.53823	0.769116	+	0.639109i	$0.220697\pi$
$854$	−3.27388e12	−0.210621
$855$	0	0
$856$	5.72148e12	0.364230
$857$	−7.61941e12	−0.482511	−0.241256	−	0.970462i	$-0.577559\pi$
$857$	−7.61941e12	−0.482511	−0.241256	+	0.970462i	$0.577559\pi$
$858$	1.32730e13	0.836138
$859$	−1.27695e13	−0.800212	−0.400106	−	0.916469i	$-0.631027\pi$
$859$	−1.27695e13	−0.800212	−0.400106	+	0.916469i	$0.631027\pi$
$860$	0	0
$861$	1.75017e13	1.08534
$862$	−7.05687e12	−0.435341
$863$	−2.63038e13	−1.61425	−0.807124	−	0.590382i	$-0.798977\pi$
$863$	−2.63038e13	−1.61425	−0.807124	+	0.590382i	$0.798977\pi$
$864$	−1.65849e12	−0.101251
$865$	0	0
$866$	−1.14462e12	−0.0691564
$867$	−4.09292e13	−2.46007
$868$	−2.78529e12	−0.166545
$869$	−1.69124e13	−1.00604
$870$	0	0
$871$	1.34118e13	0.789596
$872$	2.47532e12	0.144980
$873$	1.24826e12	0.0727347
$874$	−1.20420e13	−0.698065
$875$	0	0
$876$	−1.10090e13	−0.631652
$877$	−4.76832e11	−0.0272187	−0.0136093	−	0.999907i	$-0.504332\pi$
$877$	−4.76832e11	−0.0272187	−0.0136093	+	0.999907i	$0.504332\pi$
$878$	−7.59947e12	−0.431577
$879$	−3.97672e13	−2.24685
$880$	0	0
$881$	1.74839e13	0.977791	0.488896	−	0.872342i	$-0.337400\pi$
$881$	1.74839e13	0.977791	0.488896	+	0.872342i	$0.337400\pi$
$882$	3.16208e12	0.175940
$883$	6.48321e12	0.358894	0.179447	−	0.983768i	$-0.442569\pi$
$883$	6.48321e12	0.358894	0.179447	+	0.983768i	$0.442569\pi$
$884$	2.50409e13	1.37916
$885$	0	0
$886$	−3.15315e12	−0.171907
$887$	9.92450e12	0.538335	0.269167	−	0.963093i	$-0.413252\pi$
$887$	9.92450e12	0.538335	0.269167	+	0.963093i	$0.413252\pi$
$888$	−7.72601e12	−0.416962
$889$	1.72679e13	0.927220
$890$	0	0
$891$	−1.40237e13	−0.745443
$892$	−1.19106e13	−0.629928
$893$	1.52981e12	0.0805021
$894$	−5.94907e12	−0.311480
$895$	0	0
$896$	1.25037e12	0.0648117
$897$	7.28083e13	3.75504
$898$	9.92250e12	0.509187
$899$	−8.69601e12	−0.444019
$900$	0	0
$901$	−5.83994e13	−2.95221
$902$	−1.00168e13	−0.503847
$903$	−8.77949e12	−0.439414
$904$	−6.90505e12	−0.343881
$905$	0	0
$906$	−2.69356e10	−0.00132816
$907$	−1.40768e12	−0.0690671	−0.0345335	−	0.999404i	$-0.510995\pi$
$907$	−1.40768e12	−0.0690671	−0.0345335	+	0.999404i	$0.510995\pi$
$908$	4.90022e11	0.0239237
$909$	−9.11980e11	−0.0443045
$910$	0	0
$911$	6.91343e12	0.332553	0.166277	−	0.986079i	$-0.446826\pi$
$911$	6.91343e12	0.332553	0.166277	+	0.986079i	$0.446826\pi$
$912$	3.37286e12	0.161444
$913$	4.22465e11	0.0201221
$914$	1.26081e13	0.597574
$915$	0	0
$916$	−3.73311e12	−0.175203
$917$	−1.18351e13	−0.552724
$918$	−1.50529e13	−0.699565
$919$	−8.50189e12	−0.393184	−0.196592	−	0.980485i	$-0.562987\pi$
$919$	−8.50189e12	−0.393184	−0.196592	+	0.980485i	$0.562987\pi$
$920$	0	0
$921$	3.39411e13	1.55438
$922$	−2.45593e11	−0.0111925
$923$	2.65264e13	1.20301
$924$	6.01543e12	0.271483
$925$	0	0
$926$	3.11190e12	0.139084
$927$	−2.04309e13	−0.908719
$928$	3.90382e12	0.172792
$929$	8.90639e12	0.392311	0.196156	−	0.980573i	$-0.437154\pi$
$929$	8.90639e12	0.392311	0.196156	+	0.980573i	$0.437154\pi$
$930$	0	0
$931$	5.51826e12	0.240729
$932$	−1.05674e13	−0.458773
$933$	−3.75209e13	−1.62109
$934$	1.70404e12	0.0732689
$935$	0	0
$936$	−7.13514e12	−0.303851
$937$	−7.40064e12	−0.313647	−0.156824	−	0.987627i	$-0.550125\pi$
$937$	−7.40064e12	−0.313647	−0.156824	+	0.987627i	$0.550125\pi$
$938$	6.07831e12	0.256372
$939$	3.33654e13	1.40056
$940$	0	0
$941$	−8.13867e12	−0.338377	−0.169188	−	0.985584i	$-0.554115\pi$
$941$	−8.13867e12	−0.338377	−0.169188	+	0.985584i	$0.554115\pi$
$942$	1.91675e12	0.0793115
$943$	−5.49463e13	−2.26275
$944$	1.05925e12	0.0434134
$945$	0	0
$946$	5.02480e12	0.203990
$947$	1.16975e13	0.472627	0.236313	−	0.971677i	$-0.424061\pi$
$947$	1.16975e13	0.472627	0.236313	+	0.971677i	$0.424061\pi$
$948$	2.59846e13	1.04491
$949$	4.06425e13	1.62661
$950$	0	0
$951$	−5.89661e13	−2.33771
$952$	1.13487e13	0.447796
$953$	2.50410e13	0.983406	0.491703	−	0.870763i	$-0.336375\pi$
$953$	2.50410e13	0.983406	0.491703	+	0.870763i	$0.336375\pi$
$954$	1.66403e13	0.650418
$955$	0	0
$956$	9.35686e12	0.362301
$957$	1.87809e13	0.723791
$958$	1.32983e13	0.510097
$959$	−5.37485e12	−0.205202
$960$	0	0
$961$	−2.09838e13	−0.793649
$962$	2.85226e13	1.07375
$963$	−1.47968e13	−0.554433
$964$	−2.21714e13	−0.826887
$965$	0	0
$966$	3.29972e13	1.21921
$967$	−2.68904e13	−0.988960	−0.494480	−	0.869189i	$-0.664641\pi$
$967$	−2.68904e13	−0.988960	−0.494480	+	0.869189i	$0.664641\pi$
$968$	6.21532e12	0.227522
$969$	3.06129e13	1.11544
$970$	0	0
$971$	3.28780e13	1.18691	0.593456	−	0.804866i	$-0.297763\pi$
$971$	3.28780e13	1.18691	0.593456	+	0.804866i	$0.297763\pi$
$972$	1.35767e13	0.487860
$973$	2.62115e13	0.937529
$974$	−2.21706e13	−0.789338
$975$	0	0
$976$	−2.87888e12	−0.101554
$977$	8.13505e11	0.0285650	0.0142825	−	0.999898i	$-0.495454\pi$
$977$	8.13505e11	0.0285650	0.0142825	+	0.999898i	$0.495454\pi$
$978$	3.98339e13	1.39229
$979$	1.36301e13	0.474217
$980$	0	0
$981$	−6.40164e12	−0.220689
$982$	−6.28487e12	−0.215672
$983$	4.13240e13	1.41160	0.705800	−	0.708412i	$-0.250588\pi$
$983$	4.13240e13	1.41160	0.705800	+	0.708412i	$0.250588\pi$
$984$	1.53900e13	0.523313
$985$	0	0
$986$	3.54321e13	1.19385
$987$	−4.19198e12	−0.140602
$988$	−1.24518e13	−0.415744
$989$	2.75631e13	0.916105
$990$	0	0
$991$	−3.61227e13	−1.18973	−0.594866	−	0.803825i	$-0.702795\pi$
$991$	−3.61227e13	−1.18973	−0.594866	+	0.803825i	$0.702795\pi$
$992$	−2.44923e12	−0.0803023
$993$	3.10462e13	1.01330
$994$	1.20219e13	0.390603
$995$	0	0
$996$	−6.49086e11	−0.0208995
$997$	3.07166e13	0.984565	0.492283	−	0.870435i	$-0.336163\pi$
$997$	3.07166e13	0.984565	0.492283	+	0.870435i	$0.336163\pi$
$998$	1.20325e13	0.383944
$999$	−1.71459e13	−0.544646

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	50.10.a.a.1.1		1
4.3	odd	2		400.10.a.j.1.1		1
5.2	odd	4		50.10.b.d.49.1		2
5.3	odd	4		50.10.b.d.49.2		2
5.4	even	2		10.10.a.c.1.1	✓	1
15.14	odd	2		90.10.a.e.1.1		1
20.3	even	4		400.10.c.c.49.2		2
20.7	even	4		400.10.c.c.49.1		2
20.19	odd	2		80.10.a.a.1.1		1
40.19	odd	2		320.10.a.i.1.1		1
40.29	even	2		320.10.a.b.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
10.10.a.c.1.1	✓	1	5.4	even	2
50.10.a.a.1.1		1	1.1	even	1	trivial
50.10.b.d.49.1		2	5.2	odd	4
50.10.b.d.49.2		2	5.3	odd	4
80.10.a.a.1.1		1	20.19	odd	2
90.10.a.e.1.1		1	15.14	odd	2
320.10.a.b.1.1		1	40.29	even	2
320.10.a.i.1.1		1	40.19	odd	2
400.10.a.j.1.1		1	4.3	odd	2
400.10.c.c.49.1		2	20.7	even	4
400.10.c.c.49.2		2	20.3	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 50 = 2 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	50.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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