LMFDB - Embedded newform 400.10.a.j.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [400,10,Mod(1,400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(400, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("400.1");

S:= CuspForms(chi, 10);

N := Newforms(S);

Level:	$ N $	$=$	$ 400 = 2^{4} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	400.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:	$206.014334466$
Analytic rank:	$0$
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	$\chi$	$=$	400.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+174.000 q^{3} +4658.00 q^{7} +10593.0 q^{9} +O(q^{10})$

$q+174.000 q^{3} +4658.00 q^{7} +10593.0 q^{9} -28992.0 q^{11} +164446. q^{13} +594822. q^{17} +295780. q^{19} +810492. q^{21} +2.54453e6 q^{23} -1.58166e6 q^{27} -3.72297e6 q^{29} -2.33577e6 q^{31} -5.04461e6 q^{33} -1.08404e7 q^{37} +2.86136e7 q^{39} +2.15939e7 q^{41} +1.08323e7 q^{43} +5.17214e6 q^{47} -1.86566e7 q^{49} +1.03499e8 q^{51} -9.81797e7 q^{53} +5.14657e7 q^{57} -1.61629e7 q^{59} -4.39282e7 q^{61} +4.93422e7 q^{63} -8.15574e7 q^{67} +4.42749e8 q^{69} -1.61308e8 q^{71} +2.47148e8 q^{73} -1.35045e8 q^{77} +5.83346e8 q^{79} -4.83711e8 q^{81} -1.45718e7 q^{83} -6.47797e8 q^{87} +4.70134e8 q^{89} +7.65989e8 q^{91} -4.06424e8 q^{93} +1.17838e8 q^{97} -3.07112e8 q^{99} +O(q^{100})$

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 $

$2$	0	0
$2$	0	0
$3$	174.000	1.24023	0.620117	−	0.784509i	$-0.287085\pi$
$3$	174.000	1.24023	0.620117	+	0.784509i	$0.287085\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	4658.00	0.733261	0.366630	−	0.930367i	$-0.380511\pi$
$7$	4658.00	0.733261	0.366630	+	0.930367i	$0.380511\pi$
$8$	0	0
$9$	10593.0	0.538180
$10$	0	0
$11$	−28992.0	−0.597051	−0.298525	−	0.954402i	$-0.596495\pi$
$11$	−28992.0	−0.597051	−0.298525	+	0.954402i	$0.596495\pi$
$12$	0	0
$13$	164446.	1.59690	0.798451	−	0.602060i	$-0.205653\pi$
$13$	164446.	1.59690	0.798451	+	0.602060i	$0.205653\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	594822.	1.72730	0.863648	−	0.504095i	$-0.168174\pi$
$17$	594822.	1.72730	0.863648	+	0.504095i	$0.168174\pi$
$18$	0	0
$19$	295780.	0.520688	0.260344	−	0.965516i	$-0.416164\pi$
$19$	295780.	0.520688	0.260344	+	0.965516i	$0.416164\pi$
$20$	0	0
$21$	810492.	0.909415
$22$	0	0
$23$	2.54453e6	1.89598	0.947988	−	0.318305i	$-0.103114\pi$
$23$	2.54453e6	1.89598	0.947988	+	0.318305i	$0.103114\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−1.58166e6	−0.572765
$28$	0	0
$29$	−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	$-0.572934\pi$
$31$	−2.33577e6	−0.454258	−0.227129	+	0.973865i	$0.572934\pi$
$32$	0	0
$33$	−5.04461e6	−0.740482
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−1.08404e7	−0.950907	−0.475454	−	0.879741i	$-0.657716\pi$
$37$	−1.08404e7	−0.950907	−0.475454	+	0.879741i	$0.657716\pi$
$38$	0	0
$39$	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	$0.296469\pi$
$42$	0	0
$43$	1.08323e7	0.483184	0.241592	−	0.970378i	$-0.422330\pi$
$43$	1.08323e7	0.483184	0.241592	+	0.970378i	$0.422330\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	5.17214e6	0.154607	0.0773036	−	0.997008i	$-0.475369\pi$
$47$	5.17214e6	0.154607	0.0773036	+	0.997008i	$0.475369\pi$
$48$	0	0
$49$	−1.86566e7	−0.462329
$50$	0	0
$51$	1.03499e8	2.14225
$52$	0	0
$53$	−9.81797e7	−1.70915	−0.854575	−	0.519328i	$-0.826182\pi$
$53$	−9.81797e7	−1.70915	−0.854575	+	0.519328i	$0.826182\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	5.14657e7	0.645775
$58$	0	0
$59$	−1.61629e7	−0.173654	−0.0868269	−	0.996223i	$-0.527673\pi$
$59$	−1.61629e7	−0.173654	−0.0868269	+	0.996223i	$0.527673\pi$
$60$	0	0
$61$	−4.39282e7	−0.406218	−0.203109	−	0.979156i	$-0.565105\pi$
$61$	−4.39282e7	−0.406218	−0.203109	+	0.979156i	$0.565105\pi$
$62$	0	0
$63$	4.93422e7	0.394626
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−8.15574e7	−0.494455	−0.247228	−	0.968957i	$-0.579520\pi$
$67$	−8.15574e7	−0.494455	−0.247228	+	0.968957i	$0.579520\pi$
$68$	0	0
$69$	4.42749e8	2.35145
$70$	0	0
$71$	−1.61308e8	−0.753343	−0.376671	−	0.926347i	$-0.622931\pi$
$71$	−1.61308e8	−0.753343	−0.376671	+	0.926347i	$0.622931\pi$
$72$	0	0
$73$	2.47148e8	1.01860	0.509301	−	0.860589i	$-0.329904\pi$
$73$	2.47148e8	1.01860	0.509301	+	0.860589i	$0.329904\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−1.35045e8	−0.437794
$78$	0	0
$79$	5.83346e8	1.68502	0.842508	−	0.538684i	$-0.181078\pi$
$79$	5.83346e8	1.68502	0.842508	+	0.538684i	$0.181078\pi$
$80$	0	0
$81$	−4.83711e8	−1.24854
$82$	0	0
$83$	−1.45718e7	−0.0337024	−0.0168512	−	0.999858i	$-0.505364\pi$
$83$	−1.45718e7	−0.0337024	−0.0168512	+	0.999858i	$0.505364\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−6.47797e8	−1.21228
$88$	0	0
$89$	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
$92$	0	0
$93$	−4.06424e8	−0.563386
$94$	0	0
$95$	0	0
$96$	0	0
$97$	1.17838e8	0.135149	0.0675747	−	0.997714i	$-0.478474\pi$
$97$	1.17838e8	0.135149	0.0675747	+	0.997714i	$0.478474\pi$
$98$	0	0
$99$	−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	$0.513106\pi$
$102$	0	0
$103$	1.92872e9	1.68850	0.844252	−	0.535947i	$-0.180045\pi$
$103$	1.92872e9	1.68850	0.844252	+	0.535947i	$0.180045\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1.39685e9	1.03020	0.515100	−	0.857130i	$-0.327755\pi$
$107$	1.39685e9	1.03020	0.515100	+	0.857130i	$0.327755\pi$
$108$	0	0
$109$	−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
$112$	0	0
$113$	1.68580e9	0.972643	0.486322	−	0.873780i	$-0.338338\pi$
$113$	1.68580e9	0.972643	0.486322	+	0.873780i	$0.338338\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	1.74198e9	0.859421
$118$	0	0
$119$	2.77068e9	1.26656
$120$	0	0
$121$	−1.51741e9	−0.643531
$122$	0	0
$123$	3.75733e9	1.48015
$124$	0	0
$125$	0	0
$126$	0	0
$127$	3.70716e9	1.26452	0.632258	−	0.774758i	$-0.282128\pi$
$127$	3.70716e9	1.26452	0.632258	+	0.774758i	$0.282128\pi$
$128$	0	0
$129$	1.88482e9	0.599261
$130$	0	0
$131$	−2.54080e9	−0.753789	−0.376895	−	0.926256i	$-0.623008\pi$
$131$	−2.54080e9	−0.753789	−0.376895	+	0.926256i	$0.623008\pi$
$132$	0	0
$133$	1.37774e9	0.381800
$134$	0	0
$135$	0	0
$136$	0	0
$137$	1.15390e9	0.279849	0.139925	−	0.990162i	$-0.455314\pi$
$137$	1.15390e9	0.279849	0.139925	+	0.990162i	$0.455314\pi$
$138$	0	0
$139$	5.62721e9	1.27858	0.639288	−	0.768968i	$-0.279229\pi$
$139$	5.62721e9	1.27858	0.639288	+	0.768968i	$0.279229\pi$
$140$	0	0
$141$	8.99952e8	0.191749
$142$	0	0
$143$	−4.76762e9	−0.953431
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−3.24626e9	−0.573396
$148$	0	0
$149$	−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	$-0.499759\pi$
$151$	9.67515e6	0.00151447	0.000757236		1.00000i	$0.499759\pi$
$152$	0	0
$153$	6.30095e9	0.929597
$154$	0	0
$155$	0	0
$156$	0	0
$157$	6.88488e8	0.0904373	0.0452187	−	0.998977i	$-0.485602\pi$
$157$	6.88488e8	0.0904373	0.0452187	+	0.998977i	$0.485602\pi$
$158$	0	0
$159$	−1.70833e10	−2.11975
$160$	0	0
$161$	1.18524e10	1.39024
$162$	0	0
$163$	−1.43082e10	−1.58759	−0.793797	−	0.608182i	$-0.791899\pi$
$163$	−1.43082e10	−1.58759	−0.793797	+	0.608182i	$0.791899\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	9.98735e9	0.993633	0.496817	−	0.867856i	$-0.334502\pi$
$167$	9.98735e9	0.993633	0.496817	+	0.867856i	$0.334502\pi$
$168$	0	0
$169$	1.64380e10	1.55010
$170$	0	0
$171$	3.13320e9	0.280224
$172$	0	0
$173$	−3.51396e9	−0.298256	−0.149128	−	0.988818i	$-0.547647\pi$
$173$	−3.51396e9	−0.298256	−0.149128	+	0.988818i	$0.547647\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−2.81234e9	−0.215371
$178$	0	0
$179$	−1.19502e9	−0.0870038	−0.0435019	−	0.999053i	$-0.513851\pi$
$179$	−1.19502e9	−0.0870038	−0.0435019	+	0.999053i	$0.513851\pi$
$180$	0	0
$181$	−9.12053e9	−0.631635	−0.315818	−	0.948820i	$-0.602279\pi$
$181$	−9.12053e9	−0.631635	−0.315818	+	0.948820i	$0.602279\pi$
$182$	0	0
$183$	−7.64350e9	−0.503805
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−1.72451e10	−1.03128
$188$	0	0
$189$	−7.36737e9	−0.419986
$190$	0	0
$191$	9.37431e9	0.509670	0.254835	−	0.966985i	$-0.417979\pi$
$191$	9.37431e9	0.509670	0.254835	+	0.966985i	$0.417979\pi$
$192$	0	0
$193$	−2.40000e10	−1.24510	−0.622550	−	0.782580i	$-0.713903\pi$
$193$	−2.40000e10	−1.24510	−0.622550	+	0.782580i	$0.713903\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	5.56124e8	0.0263071	0.0131536	−	0.999913i	$-0.495813\pi$
$197$	5.56124e8	0.0263071	0.0131536	+	0.999913i	$0.495813\pi$
$198$	0	0
$199$	2.51255e10	1.13573	0.567866	−	0.823121i	$-0.307769\pi$
$199$	2.51255e10	1.13573	0.567866	+	0.823121i	$0.307769\pi$
$200$	0	0
$201$	−1.41910e10	−0.613240
$202$	0	0
$203$	−1.73416e10	−0.716732
$204$	0	0
$205$	0	0
$206$	0	0
$207$	2.69542e10	1.02038
$208$	0	0
$209$	−8.57525e9	−0.310877
$210$	0	0
$211$	1.63915e10	0.569309	0.284654	−	0.958630i	$-0.408121\pi$
$211$	1.63915e10	0.569309	0.284654	+	0.958630i	$0.408121\pi$
$212$	0	0
$213$	−2.80675e10	−0.934321
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−1.08800e10	−0.333090
$218$	0	0
$219$	4.30037e10	1.26330
$220$	0	0
$221$	9.78161e10	2.75832
$222$	0	0
$223$	4.65257e10	1.25986	0.629928	−	0.776654i	$-0.283084\pi$
$223$	4.65257e10	1.25986	0.629928	+	0.776654i	$0.283084\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−1.91415e9	−0.0478475	−0.0239237	−	0.999714i	$-0.507616\pi$
$227$	−1.91415e9	−0.0478475	−0.0239237	+	0.999714i	$0.507616\pi$
$228$	0	0
$229$	−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
$232$	0	0
$233$	−4.12790e10	−0.917545	−0.458773	−	0.888554i	$-0.651711\pi$
$233$	−4.12790e10	−0.917545	−0.458773	+	0.888554i	$0.651711\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	1.01502e11	2.08981
$238$	0	0
$239$	−3.65502e10	−0.724602	−0.362301	−	0.932061i	$-0.618009\pi$
$239$	−3.65502e10	−0.724602	−0.362301	+	0.932061i	$0.618009\pi$
$240$	0	0
$241$	−8.66070e10	−1.65377	−0.826887	−	0.562368i	$-0.809890\pi$
$241$	−8.66070e10	−1.65377	−0.826887	+	0.562368i	$0.809890\pi$
$242$	0	0
$243$	−5.30339e10	−0.975720
$244$	0	0
$245$	0	0
$246$	0	0
$247$	4.86398e10	0.831488
$248$	0	0
$249$	−2.53549e9	−0.0417989
$250$	0	0
$251$	7.20769e10	1.14621	0.573105	−	0.819482i	$-0.305739\pi$
$251$	7.20769e10	1.14621	0.573105	+	0.819482i	$0.305739\pi$
$252$	0	0
$253$	−7.37711e10	−1.13199
$254$	0	0
$255$	0	0
$256$	0	0
$257$	1.12729e11	1.61190	0.805948	−	0.591986i	$-0.201656\pi$
$257$	1.12729e11	1.61190	0.805948	+	0.591986i	$0.201656\pi$
$258$	0	0
$259$	−5.04947e10	−0.697263
$260$	0	0
$261$	−3.94374e10	−0.526049
$262$	0	0
$263$	5.55225e10	0.715596	0.357798	−	0.933799i	$-0.383528\pi$
$263$	5.55225e10	0.715596	0.357798	+	0.933799i	$0.383528\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	8.18033e10	0.985076
$268$	0	0
$269$	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	$-0.560025\pi$
$271$	−3.32884e10	−0.374914	−0.187457	+	0.982273i	$0.560025\pi$
$272$	0	0
$273$	1.33282e11	1.45225
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−5.46240e10	−0.557474	−0.278737	−	0.960367i	$-0.589916\pi$
$277$	−5.46240e10	−0.557474	−0.278737	+	0.960367i	$0.589916\pi$
$278$	0	0
$279$	−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	$0.631272\pi$
$282$	0	0
$283$	−8.36086e10	−0.774840	−0.387420	−	0.921903i	$-0.626634\pi$
$283$	−8.36086e10	−0.774840	−0.387420	+	0.921903i	$0.626634\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	1.00584e11	0.875107
$288$	0	0
$289$	2.35225e11	1.98355
$290$	0	0
$291$	2.05039e10	0.167617
$292$	0	0
$293$	2.28547e11	1.81164	0.905819	−	0.423666i	$-0.139257\pi$
$293$	2.28547e11	1.81164	0.905819	+	0.423666i	$0.139257\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	4.58555e10	0.341969
$298$	0	0
$299$	4.18438e11	3.02769
$300$	0	0
$301$	5.04568e10	0.354300
$302$	0	0
$303$	−1.49801e10	−0.102100
$304$	0	0
$305$	0	0
$306$	0	0
$307$	1.95064e11	1.25330	0.626648	−	0.779302i	$-0.284426\pi$
$307$	1.95064e11	1.25330	0.626648	+	0.779302i	$0.284426\pi$
$308$	0	0
$309$	3.35598e11	2.09414
$310$	0	0
$311$	−2.15637e11	−1.30708	−0.653540	−	0.756892i	$-0.726717\pi$
$311$	−2.15637e11	−1.30708	−0.653540	+	0.756892i	$0.726717\pi$
$312$	0	0
$313$	−1.91755e11	−1.12927	−0.564635	−	0.825341i	$-0.690983\pi$
$313$	−1.91755e11	−1.12927	−0.564635	+	0.825341i	$0.690983\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	3.38886e11	1.88489	0.942446	−	0.334358i	$-0.108519\pi$
$317$	3.38886e11	1.88489	0.942446	+	0.334358i	$0.108519\pi$
$318$	0	0
$319$	1.07936e11	0.583592
$320$	0	0
$321$	2.43051e11	1.27769
$322$	0	0
$323$	1.75936e11	0.899383
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−1.05153e11	−0.508577
$328$	0	0
$329$	2.40918e10	0.113367
$330$	0	0
$331$	1.78427e11	0.817022	0.408511	−	0.912753i	$-0.366048\pi$
$331$	1.78427e11	0.817022	0.408511	+	0.912753i	$0.366048\pi$
$332$	0	0
$333$	−1.14833e11	−0.511760
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−2.64693e11	−1.11791	−0.558957	−	0.829196i	$-0.688798\pi$
$337$	−2.64693e11	−1.11791	−0.558957	+	0.829196i	$0.688798\pi$
$338$	0	0
$339$	2.93330e11	1.20631
$340$	0	0
$341$	6.77187e10	0.271215
$342$	0	0
$343$	−2.74870e11	−1.07227
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−8.11912e10	−0.300626	−0.150313	−	0.988638i	$-0.548028\pi$
$347$	−8.11912e10	−0.300626	−0.150313	+	0.988638i	$0.548028\pi$
$348$	0	0
$349$	−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
$352$	0	0
$353$	−3.28733e10	−0.112683	−0.0563413	−	0.998412i	$-0.517944\pi$
$353$	−3.28733e10	−0.112683	−0.0563413	+	0.998412i	$0.517944\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	4.82098e11	1.57083
$358$	0	0
$359$	3.12096e11	0.991661	0.495831	−	0.868419i	$-0.334864\pi$
$359$	3.12096e11	0.991661	0.495831	+	0.868419i	$0.334864\pi$
$360$	0	0
$361$	−2.35202e11	−0.728884
$362$	0	0
$363$	−2.64030e11	−0.798129
$364$	0	0
$365$	0	0
$366$	0	0
$367$	6.28209e11	1.80762	0.903810	−	0.427934i	$-0.140759\pi$
$367$	6.28209e11	1.80762	0.903810	+	0.427934i	$0.140759\pi$
$368$	0	0
$369$	2.28744e11	0.642289
$370$	0	0
$371$	−4.57321e11	−1.25325
$372$	0	0
$373$	9.84770e10	0.263418	0.131709	−	0.991288i	$-0.457954\pi$
$373$	9.84770e10	0.263418	0.131709	+	0.991288i	$0.457954\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−6.12228e11	−1.56091
$378$	0	0
$379$	−2.89024e11	−0.719544	−0.359772	−	0.933040i	$-0.617145\pi$
$379$	−2.89024e11	−0.719544	−0.359772	+	0.933040i	$0.617145\pi$
$380$	0	0
$381$	6.45046e11	1.56830
$382$	0	0
$383$	2.01350e11	0.478141	0.239071	−	0.971002i	$-0.423157\pi$
$383$	2.01350e11	0.478141	0.239071	+	0.971002i	$0.423157\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	1.14746e11	0.260040
$388$	0	0
$389$	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
$392$	0	0
$393$	−4.42099e11	−0.934875
$394$	0	0
$395$	0	0
$396$	0	0
$397$	5.15625e11	1.04178	0.520891	−	0.853623i	$-0.325600\pi$
$397$	5.15625e11	1.04178	0.520891	+	0.853623i	$0.325600\pi$
$398$	0	0
$399$	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	$0.802902\pi$
$402$	0	0
$403$	−3.84108e11	−0.725406
$404$	0	0
$405$	0	0
$406$	0	0
$407$	3.14285e11	0.567740
$408$	0	0
$409$	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
$412$	0	0
$413$	−7.52866e10	−0.127333
$414$	0	0
$415$	0	0
$416$	0	0
$417$	9.79134e11	1.58573
$418$	0	0
$419$	−6.92475e11	−1.09759	−0.548796	−	0.835956i	$-0.684914\pi$
$419$	−6.92475e11	−1.09759	−0.548796	+	0.835956i	$0.684914\pi$
$420$	0	0
$421$	4.08125e11	0.633176	0.316588	−	0.948563i	$-0.397463\pi$
$421$	4.08125e11	0.633176	0.316588	+	0.948563i	$0.397463\pi$
$422$	0	0
$423$	5.47885e10	0.0832065
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−2.04617e11	−0.297863
$428$	0	0
$429$	−8.29566e11	−1.18248
$430$	0	0
$431$	−4.41055e11	−0.615665	−0.307833	−	0.951441i	$-0.599604\pi$
$431$	−4.41055e11	−0.615665	−0.307833	+	0.951441i	$0.599604\pi$
$432$	0	0
$433$	7.15390e10	0.0978019	0.0489009	−	0.998804i	$-0.484428\pi$
$433$	7.15390e10	0.0978019	0.0489009	+	0.998804i	$0.484428\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	7.52622e11	0.987212
$438$	0	0
$439$	−4.74967e11	−0.610342	−0.305171	−	0.952298i	$-0.598714\pi$
$439$	−4.74967e11	−0.610342	−0.305171	+	0.952298i	$0.598714\pi$
$440$	0	0
$441$	−1.97630e11	−0.248816
$442$	0	0
$443$	−1.97072e11	−0.243113	−0.121556	−	0.992585i	$-0.538789\pi$
$443$	−1.97072e11	−0.243113	−0.121556	+	0.992585i	$0.538789\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−3.71817e11	−0.440499
$448$	0	0
$449$	−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
$452$	0	0
$453$	1.68348e9	0.00187830
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−7.88006e11	−0.845097	−0.422549	−	0.906340i	$-0.638864\pi$
$457$	−7.88006e11	−0.845097	−0.422549	+	0.906340i	$0.638864\pi$
$458$	0	0
$459$	−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	$0.497481\pi$
$462$	0	0
$463$	1.94494e11	0.196694	0.0983471	−	0.995152i	$-0.468644\pi$
$463$	1.94494e11	0.196694	0.0983471	+	0.995152i	$0.468644\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	1.06503e11	0.103618	0.0518089	−	0.998657i	$-0.483501\pi$
$467$	1.06503e11	0.103618	0.0518089	+	0.998657i	$0.483501\pi$
$468$	0	0
$469$	−3.79894e11	−0.362564
$470$	0	0
$471$	1.19797e11	0.112163
$472$	0	0
$473$	−3.14050e11	−0.288485
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−1.04002e12	−0.919831
$478$	0	0
$479$	8.31146e11	0.721386	0.360693	−	0.932685i	$-0.382540\pi$
$479$	8.31146e11	0.721386	0.360693	+	0.932685i	$0.382540\pi$
$480$	0	0
$481$	−1.78266e12	−1.51851
$482$	0	0
$483$	2.06232e12	1.72423
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−1.38566e12	−1.11629	−0.558146	−	0.829743i	$-0.688487\pi$
$487$	−1.38566e12	−1.11629	−0.558146	+	0.829743i	$0.688487\pi$
$488$	0	0
$489$	−2.48962e12	−1.96899
$490$	0	0
$491$	−3.92804e11	−0.305007	−0.152503	−	0.988303i	$-0.548734\pi$
$491$	−3.92804e11	−0.305007	−0.152503	+	0.988303i	$0.548734\pi$
$492$	0	0
$493$	−2.21450e12	−1.68836
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−7.51371e11	−0.552397
$498$	0	0
$499$	7.52029e11	0.542978	0.271489	−	0.962442i	$-0.412484\pi$
$499$	7.52029e11	0.542978	0.271489	+	0.962442i	$0.412484\pi$
$500$	0	0
$501$	1.73780e12	1.23234
$502$	0	0
$503$	−1.83429e12	−1.27765	−0.638826	−	0.769351i	$-0.720580\pi$
$503$	−1.83429e12	−1.27765	−0.638826	+	0.769351i	$0.720580\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	2.86021e12	1.92248
$508$	0	0
$509$	−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
$512$	0	0
$513$	−4.67823e11	−0.298232
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−1.49951e11	−0.0923083
$518$	0	0
$519$	−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	$0.450047\pi$
$522$	0	0
$523$	1.68426e11	0.0984353	0.0492177	−	0.998788i	$-0.484327\pi$
$523$	1.68426e11	0.0984353	0.0492177	+	0.998788i	$0.484327\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−1.38937e12	−0.784639
$528$	0	0
$529$	4.67350e12	2.59473
$530$	0	0
$531$	−1.71213e11	−0.0934570
$532$	0	0
$533$	3.55102e12	1.90582
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−2.07934e11	−0.107905
$538$	0	0
$539$	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	$0.313246\pi$
$542$	0	0
$543$	−1.58697e12	−0.783375
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−3.51639e12	−1.67940	−0.839701	−	0.543049i	$-0.817270\pi$
$547$	−3.51639e12	−1.67940	−0.839701	+	0.543049i	$0.817270\pi$
$548$	0	0
$549$	−4.65331e11	−0.218618
$550$	0	0
$551$	−1.10118e12	−0.508951
$552$	0	0
$553$	2.71722e12	1.23556
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−6.29996e11	−0.277325	−0.138663	−	0.990340i	$-0.544280\pi$
$557$	−6.29996e11	−0.277325	−0.138663	+	0.990340i	$0.544280\pi$
$558$	0	0
$559$	1.78133e12	0.771597
$560$	0	0
$561$	−3.00064e12	−1.27903
$562$	0	0
$563$	4.02619e11	0.168891	0.0844455	−	0.996428i	$-0.473088\pi$
$563$	4.02619e11	0.168891	0.0844455	+	0.996428i	$0.473088\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−2.25313e12	−0.915507
$568$	0	0
$569$	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	$-0.573484\pi$
$571$	−1.16243e12	−0.457620	−0.228810	+	0.973471i	$0.573484\pi$
$572$	0	0
$573$	1.63113e12	0.632110
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−4.66332e12	−1.75148	−0.875738	−	0.482787i	$-0.839625\pi$
$577$	−4.66332e12	−1.75148	−0.875738	+	0.482787i	$0.839625\pi$
$578$	0	0
$579$	−4.17601e12	−1.54421
$580$	0	0
$581$	−6.78754e10	−0.0247127
$582$	0	0
$583$	2.84643e12	1.02045
$584$	0	0
$585$	0	0
$586$	0	0
$587$	3.88422e12	1.35030	0.675152	−	0.737678i	$-0.264078\pi$
$587$	3.88422e12	1.35030	0.675152	+	0.737678i	$0.264078\pi$
$588$	0	0
$589$	−6.90875e11	−0.236527
$590$	0	0
$591$	9.67655e10	0.0326270
$592$	0	0
$593$	2.01341e12	0.668631	0.334315	−	0.942461i	$-0.391495\pi$
$593$	2.01341e12	0.668631	0.334315	+	0.942461i	$0.391495\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	4.37184e12	1.40857
$598$	0	0
$599$	−2.10578e12	−0.668333	−0.334166	−	0.942514i	$-0.608455\pi$
$599$	−2.10578e12	−0.668333	−0.334166	+	0.942514i	$0.608455\pi$
$600$	0	0
$601$	4.74058e12	1.48216	0.741082	−	0.671415i	$-0.234313\pi$
$601$	4.74058e12	1.48216	0.741082	+	0.671415i	$0.234313\pi$
$602$	0	0
$603$	−8.63938e11	−0.266106
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−4.01476e12	−1.20036	−0.600179	−	0.799866i	$-0.704904\pi$
$607$	−4.01476e12	−1.20036	−0.600179	+	0.799866i	$0.704904\pi$
$608$	0	0
$609$	−3.01744e12	−0.888915
$610$	0	0
$611$	8.50537e11	0.246893
$612$	0	0
$613$	−1.30399e12	−0.372993	−0.186496	−	0.982456i	$-0.559713\pi$
$613$	−1.30399e12	−0.372993	−0.186496	+	0.982456i	$0.559713\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−5.10164e12	−1.41719	−0.708593	−	0.705618i	$-0.750670\pi$
$617$	−5.10164e12	−1.41719	−0.708593	+	0.705618i	$0.750670\pi$
$618$	0	0
$619$	4.50630e11	0.123371	0.0616854	−	0.998096i	$-0.480352\pi$
$619$	4.50630e11	0.123371	0.0616854	+	0.998096i	$0.480352\pi$
$620$	0	0
$621$	−4.02459e12	−1.08595
$622$	0	0
$623$	2.18988e12	0.582404
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−1.49209e12	−0.385560
$628$	0	0
$629$	−6.44812e12	−1.64250
$630$	0	0
$631$	−7.97105e10	−0.0200163	−0.0100081	−	0.999950i	$-0.503186\pi$
$631$	−7.97105e10	−0.0200163	−0.0100081	+	0.999950i	$0.503186\pi$
$632$	0	0
$633$	2.85212e12	0.706076
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−3.06801e12	−0.738294
$638$	0	0
$639$	−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	$0.619182\pi$
$642$	0	0
$643$	−5.94982e12	−1.37263	−0.686316	−	0.727303i	$-0.740773\pi$
$643$	−5.94982e12	−1.37263	−0.686316	+	0.727303i	$0.740773\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	3.56187e12	0.799115	0.399557	−	0.916708i	$-0.369164\pi$
$647$	3.56187e12	0.799115	0.399557	+	0.916708i	$0.369164\pi$
$648$	0	0
$649$	4.68594e11	0.103680
$650$	0	0
$651$	−1.89312e12	−0.413109
$652$	0	0
$653$	1.37883e11	0.0296757	0.0148379	−	0.999890i	$-0.495277\pi$
$653$	1.37883e11	0.0296757	0.0148379	+	0.999890i	$0.495277\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	2.61804e12	0.548191
$658$	0	0
$659$	4.16654e12	0.860581	0.430290	−	0.902691i	$-0.358411\pi$
$659$	4.16654e12	0.860581	0.430290	+	0.902691i	$0.358411\pi$
$660$	0	0
$661$	−3.29083e12	−0.670500	−0.335250	−	0.942129i	$-0.608821\pi$
$661$	−3.29083e12	−0.670500	−0.335250	+	0.942129i	$0.608821\pi$
$662$	0	0
$663$	1.70200e13	3.42097
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−9.47322e12	−1.85324
$668$	0	0
$669$	8.09547e12	1.56252
$670$	0	0
$671$	1.27357e12	0.242532
$672$	0	0
$673$	−5.74732e12	−1.07993	−0.539967	−	0.841686i	$-0.681563\pi$
$673$	−5.74732e12	−1.07993	−0.539967	+	0.841686i	$0.681563\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−9.31245e12	−1.70379	−0.851893	−	0.523716i	$-0.824545\pi$
$677$	−9.31245e12	−1.70379	−0.851893	+	0.523716i	$0.824545\pi$
$678$	0	0
$679$	5.48892e11	0.0990998
$680$	0	0
$681$	−3.33062e11	−0.0593421
$682$	0	0
$683$	9.19967e12	1.61763	0.808815	−	0.588063i	$-0.200109\pi$
$683$	9.19967e12	1.61763	0.808815	+	0.588063i	$0.200109\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−2.53735e12	−0.434585
$688$	0	0
$689$	−1.61453e13	−2.72934
$690$	0	0
$691$	−1.03125e13	−1.72074	−0.860369	−	0.509672i	$-0.829767\pi$
$691$	−1.03125e13	−1.72074	−0.860369	+	0.509672i	$0.829767\pi$
$692$	0	0
$693$	−1.43053e12	−0.235612
$694$	0	0
$695$	0	0
$696$	0	0
$697$	1.28445e13	2.06144
$698$	0	0
$699$	−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	$0.767532\pi$
$702$	0	0
$703$	−3.20638e12	−0.495126
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−4.01020e11	−0.0603641
$708$	0	0
$709$	−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
$712$	0	0
$713$	−5.94345e12	−0.861263
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−6.35974e12	−0.898676
$718$	0	0
$719$	−6.94013e12	−0.968473	−0.484236	−	0.874937i	$-0.660903\pi$
$719$	−6.94013e12	−0.968473	−0.484236	+	0.874937i	$0.660903\pi$
$720$	0	0
$721$	8.98398e12	1.23811
$722$	0	0
$723$	−1.50696e13	−2.05107
$724$	0	0
$725$	0	0
$726$	0	0
$727$	7.74984e12	1.02893	0.514467	−	0.857510i	$-0.327990\pi$
$727$	7.74984e12	1.02893	0.514467	+	0.857510i	$0.327990\pi$
$728$	0	0
$729$	2.92986e11	0.0384214
$730$	0	0
$731$	6.44329e12	0.834602
$732$	0	0
$733$	−4.12555e12	−0.527854	−0.263927	−	0.964543i	$-0.585018\pi$
$733$	−4.12555e12	−0.527854	−0.263927	+	0.964543i	$0.585018\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	2.36451e12	0.295215
$738$	0	0
$739$	−3.33931e12	−0.411867	−0.205933	−	0.978566i	$-0.566023\pi$
$739$	−3.33931e12	−0.411867	−0.205933	+	0.978566i	$0.566023\pi$
$740$	0	0
$741$	8.46333e12	1.03124
$742$	0	0
$743$	−1.26471e13	−1.52245	−0.761224	−	0.648489i	$-0.775401\pi$
$743$	−1.26471e13	−1.52245	−0.761224	+	0.648489i	$0.775401\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−1.54359e11	−0.0181380
$748$	0	0
$749$	6.50651e12	0.755405
$750$	0	0
$751$	−1.56552e12	−0.179588	−0.0897941	−	0.995960i	$-0.528621\pi$
$751$	−1.56552e12	−0.179588	−0.0897941	+	0.995960i	$0.528621\pi$
$752$	0	0
$753$	1.25414e13	1.42157
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−9.54781e11	−0.105675	−0.0528375	−	0.998603i	$-0.516827\pi$
$757$	−9.54781e11	−0.105675	−0.0528375	+	0.998603i	$0.516827\pi$
$758$	0	0
$759$	−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	$0.408012\pi$
$762$	0	0
$763$	−2.81496e12	−0.300685
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−2.65792e12	−0.277308
$768$	0	0
$769$	−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
$772$	0	0
$773$	3.79383e12	0.382182	0.191091	−	0.981572i	$-0.438797\pi$
$773$	3.79383e12	0.382182	0.191091	+	0.981572i	$0.438797\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−8.78607e12	−0.864769
$778$	0	0
$779$	6.38703e12	0.621413
$780$	0	0
$781$	4.67663e12	0.449784
$782$	0	0
$783$	5.88847e12	0.559854
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−2.05939e13	−1.91361	−0.956803	−	0.290736i	$-0.906100\pi$
$787$	−2.05939e13	−1.91361	−0.956803	+	0.290736i	$0.906100\pi$
$788$	0	0
$789$	9.66092e12	0.887507
$790$	0	0
$791$	7.85247e12	0.713201
$792$	0	0
$793$	−7.22381e12	−0.648690
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−1.45507e12	−0.127738	−0.0638690	−	0.997958i	$-0.520344\pi$
$797$	−1.45507e12	−0.127738	−0.0638690	+	0.997958i	$0.520344\pi$
$798$	0	0
$799$	3.07650e12	0.267052
$800$	0	0
$801$	4.98013e12	0.427459
$802$	0	0
$803$	−7.16531e12	−0.608156
$804$	0	0
$805$	0	0
$806$	0	0
$807$	4.86722e12	0.403971
$808$	0	0
$809$	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	$-0.632722\pi$
$811$	−9.97833e12	−0.809961	−0.404980	+	0.914325i	$0.632722\pi$
$812$	0	0
$813$	−5.79219e12	−0.464981
$814$	0	0
$815$	0	0
$816$	0	0
$817$	3.20398e12	0.251588
$818$	0	0
$819$	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	$0.744533\pi$
$822$	0	0
$823$	1.39801e12	0.106221	0.0531107	−	0.998589i	$-0.483086\pi$
$823$	1.39801e12	0.106221	0.0531107	+	0.998589i	$0.483086\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	2.26201e13	1.68159	0.840794	−	0.541354i	$-0.182088\pi$
$827$	2.26201e13	1.68159	0.840794	+	0.541354i	$0.182088\pi$
$828$	0	0
$829$	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
$832$	0	0
$833$	−1.10974e13	−0.798579
$834$	0	0
$835$	0	0
$836$	0	0
$837$	3.69440e12	0.260183
$838$	0	0
$839$	1.76607e12	0.123049	0.0615245	−	0.998106i	$-0.480404\pi$
$839$	1.76607e12	0.123049	0.0615245	+	0.998106i	$0.480404\pi$
$840$	0	0
$841$	−6.46640e11	−0.0445739
$842$	0	0
$843$	−1.45780e13	−0.994203
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−7.06810e12	−0.471876
$848$	0	0
$849$	−1.45479e13	−0.960983
$850$	0	0
$851$	−2.75838e13	−1.80290
$852$	0	0
$853$	2.37844e13	1.53823	0.769116	−	0.639109i	$-0.220697\pi$
$853$	2.37844e13	1.53823	0.769116	+	0.639109i	$0.220697\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−7.61941e12	−0.482511	−0.241256	−	0.970462i	$-0.577559\pi$
$857$	−7.61941e12	−0.482511	−0.241256	+	0.970462i	$0.577559\pi$
$858$	0	0
$859$	1.27695e13	0.800212	0.400106	−	0.916469i	$-0.368973\pi$
$859$	1.27695e13	0.800212	0.400106	+	0.916469i	$0.368973\pi$
$860$	0	0
$861$	1.75017e13	1.08534
$862$	0	0
$863$	2.63038e13	1.61425	0.807124	−	0.590382i	$-0.201023\pi$
$863$	2.63038e13	1.61425	0.807124	+	0.590382i	$0.201023\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	4.09292e13	2.46007
$868$	0	0
$869$	−1.69124e13	−1.00604
$870$	0	0
$871$	−1.34118e13	−0.789596
$872$	0	0
$873$	1.24826e12	0.0727347
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−4.76832e11	−0.0272187	−0.0136093	−	0.999907i	$-0.504332\pi$
$877$	−4.76832e11	−0.0272187	−0.0136093	+	0.999907i	$0.504332\pi$
$878$	0	0
$879$	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	$0.337400\pi$
$882$	0	0
$883$	−6.48321e12	−0.358894	−0.179447	−	0.983768i	$-0.557431\pi$
$883$	−6.48321e12	−0.358894	−0.179447	+	0.983768i	$0.557431\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−9.92450e12	−0.538335	−0.269167	−	0.963093i	$-0.586748\pi$
$887$	−9.92450e12	−0.538335	−0.269167	+	0.963093i	$0.586748\pi$
$888$	0	0
$889$	1.72679e13	0.927220
$890$	0	0
$891$	1.40237e13	0.745443
$892$	0	0
$893$	1.52981e12	0.0805021
$894$	0	0
$895$	0	0
$896$	0	0
$897$	7.28083e13	3.75504
$898$	0	0
$899$	8.69601e12	0.444019
$900$	0	0
$901$	−5.83994e13	−2.95221
$902$	0	0
$903$	8.77949e12	0.439414
$904$	0	0
$905$	0	0
$906$	0	0
$907$	1.40768e12	0.0690671	0.0345335	−	0.999404i	$-0.489005\pi$
$907$	1.40768e12	0.0690671	0.0345335	+	0.999404i	$0.489005\pi$
$908$	0	0
$909$	−9.11980e11	−0.0443045
$910$	0	0
$911$	−6.91343e12	−0.332553	−0.166277	−	0.986079i	$-0.553174\pi$
$911$	−6.91343e12	−0.332553	−0.166277	+	0.986079i	$0.553174\pi$
$912$	0	0
$913$	4.22465e11	0.0201221
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−1.18351e13	−0.552724
$918$	0	0
$919$	8.50189e12	0.393184	0.196592	−	0.980485i	$-0.437013\pi$
$919$	8.50189e12	0.393184	0.196592	+	0.980485i	$0.437013\pi$
$920$	0	0
$921$	3.39411e13	1.55438
$922$	0	0
$923$	−2.65264e13	−1.20301
$924$	0	0
$925$	0	0
$926$	0	0
$927$	2.04309e13	0.908719
$928$	0	0
$929$	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
$932$	0	0
$933$	−3.75209e13	−1.62109
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−7.40064e12	−0.313647	−0.156824	−	0.987627i	$-0.550125\pi$
$937$	−7.40064e12	−0.313647	−0.156824	+	0.987627i	$0.550125\pi$
$938$	0	0
$939$	−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	$0.554115\pi$
$942$	0	0
$943$	5.49463e13	2.26275
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−1.16975e13	−0.472627	−0.236313	−	0.971677i	$-0.575939\pi$
$947$	−1.16975e13	−0.472627	−0.236313	+	0.971677i	$0.575939\pi$
$948$	0	0
$949$	4.06425e13	1.62661
$950$	0	0
$951$	5.89661e13	2.33771
$952$	0	0
$953$	2.50410e13	0.983406	0.491703	−	0.870763i	$-0.336375\pi$
$953$	2.50410e13	0.983406	0.491703	+	0.870763i	$0.336375\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	1.87809e13	0.723791
$958$	0	0
$959$	5.37485e12	0.205202
$960$	0	0
$961$	−2.09838e13	−0.793649
$962$	0	0
$963$	1.47968e13	0.554433
$964$	0	0
$965$	0	0
$966$	0	0
$967$	2.68904e13	0.988960	0.494480	−	0.869189i	$-0.335359\pi$
$967$	2.68904e13	0.988960	0.494480	+	0.869189i	$0.335359\pi$
$968$	0	0
$969$	3.06129e13	1.11544
$970$	0	0
$971$	−3.28780e13	−1.18691	−0.593456	−	0.804866i	$-0.702237\pi$
$971$	−3.28780e13	−1.18691	−0.593456	+	0.804866i	$0.702237\pi$
$972$	0	0
$973$	2.62115e13	0.937529
$974$	0	0
$975$	0	0
$976$	0	0
$977$	8.13505e11	0.0285650	0.0142825	−	0.999898i	$-0.495454\pi$
$977$	8.13505e11	0.0285650	0.0142825	+	0.999898i	$0.495454\pi$
$978$	0	0
$979$	−1.36301e13	−0.474217
$980$	0	0
$981$	−6.40164e12	−0.220689
$982$	0	0
$983$	−4.13240e13	−1.41160	−0.705800	−	0.708412i	$-0.749412\pi$
$983$	−4.13240e13	−1.41160	−0.705800	+	0.708412i	$0.749412\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	4.19198e12	0.140602
$988$	0	0
$989$	2.75631e13	0.916105
$990$	0	0
$991$	3.61227e13	1.18973	0.594866	−	0.803825i	$-0.297205\pi$
$991$	3.61227e13	1.18973	0.594866	+	0.803825i	$0.297205\pi$
$992$	0	0
$993$	3.10462e13	1.01330
$994$	0	0
$995$	0	0
$996$	0	0
$997$	3.07166e13	0.984565	0.492283	−	0.870435i	$-0.336163\pi$
$997$	3.07166e13	0.984565	0.492283	+	0.870435i	$0.336163\pi$
$998$	0	0
$999$	1.71459e13	0.544646

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
10.10.a.c.1.1	✓	1	20.19	odd	2
50.10.a.a.1.1		1	4.3	odd	2
50.10.b.d.49.1		2	20.7	even	4
50.10.b.d.49.2		2	20.3	even	4
80.10.a.a.1.1		1	5.4	even	2
90.10.a.e.1.1		1	60.59	even	2
320.10.a.b.1.1		1	40.19	odd	2
320.10.a.i.1.1		1	40.29	even	2
400.10.a.j.1.1		1	1.1	even	1	trivial
400.10.c.c.49.1		2	5.2	odd	4
400.10.c.c.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:	\( N \)	\(=\)	\( 400 = 2^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	400.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more