LMFDB - Embedded newform 528.8.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [528,8,Mod(1,528)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(528, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("528.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 528 = 2^{4} \cdot 3 \cdot 11 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	528.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:	$164.939293456$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 33)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	528.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-27.0000 q^{3} -410.000 q^{5} +1028.00 q^{7} +729.000 q^{9} +O(q^{10})$

$q-27.0000 q^{3} -410.000 q^{5} +1028.00 q^{7} +729.000 q^{9} +1331.00 q^{11} +12958.0 q^{13} +11070.0 q^{15} +17062.0 q^{17} +54168.0 q^{19} -27756.0 q^{21} +11488.0 q^{23} +89975.0 q^{25} -19683.0 q^{27} -186654. q^{29} +188672. q^{31} -35937.0 q^{33} -421480. q^{35} +395886. q^{37} -349866. q^{39} -47546.0 q^{41} -602088. q^{43} -298890. q^{45} +647200. q^{47} +233241. q^{49} -460674. q^{51} -1.31272e6 q^{53} -545710. q^{55} -1.46254e6 q^{57} +2.68114e6 q^{59} +551190. q^{61} +749412. q^{63} -5.31278e6 q^{65} -459260. q^{67} -310176. q^{69} +18072.0 q^{71} -426062. q^{73} -2.42932e6 q^{75} +1.36827e6 q^{77} -297764. q^{79} +531441. q^{81} -5.68403e6 q^{83} -6.99542e6 q^{85} +5.03966e6 q^{87} -6.34297e6 q^{89} +1.33208e7 q^{91} -5.09414e6 q^{93} -2.22089e7 q^{95} +1.66516e7 q^{97} +970299. q^{99} +O(q^{100})$

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
$3$	−27.0000	−0.577350
$3$	−27.0000	−0.577350
$4$	0	0
$5$	−410.000	−1.46686	−0.733430	−	0.679765i	$-0.762082\pi$
$5$	−410.000	−1.46686	−0.733430	+	0.679765i	$0.762082\pi$
$6$	0	0
$7$	1028.00	1.13279	0.566396	−	0.824133i	$-0.308337\pi$
$7$	1028.00	1.13279	0.566396	+	0.824133i	$0.308337\pi$
$8$	0	0
$9$	729.000	0.333333
$10$	0	0
$11$	1331.00	0.301511
$11$	1331.00	0.301511
$12$	0	0
$13$	12958.0	1.63582	0.817911	−	0.575344i	$-0.195132\pi$
$13$	12958.0	1.63582	0.817911	+	0.575344i	$0.195132\pi$
$14$	0	0
$15$	11070.0	0.846892
$16$	0	0
$17$	17062.0	0.842284	0.421142	−	0.906995i	$-0.361629\pi$
$17$	17062.0	0.842284	0.421142	+	0.906995i	$0.361629\pi$
$18$	0	0
$19$	54168.0	1.81178	0.905889	−	0.423514i	$-0.139204\pi$
$19$	54168.0	1.81178	0.905889	+	0.423514i	$0.139204\pi$
$20$	0	0
$21$	−27756.0	−0.654017
$22$	0	0
$23$	11488.0	0.196878	0.0984390	−	0.995143i	$-0.468615\pi$
$23$	11488.0	0.196878	0.0984390	+	0.995143i	$0.468615\pi$
$24$	0	0
$25$	89975.0	1.15168
$26$	0	0
$27$	−19683.0	−0.192450
$28$	0	0
$29$	−186654.	−1.42116	−0.710582	−	0.703614i	$-0.751568\pi$
$29$	−186654.	−1.42116	−0.710582	+	0.703614i	$0.751568\pi$
$30$	0	0
$31$	188672.	1.13747	0.568737	−	0.822519i	$-0.307432\pi$
$31$	188672.	1.13747	0.568737	+	0.822519i	$0.307432\pi$
$32$	0	0
$33$	−35937.0	−0.174078
$34$	0	0
$35$	−421480.	−1.66165
$36$	0	0
$37$	395886.	1.28488	0.642442	−	0.766334i	$-0.277921\pi$
$37$	395886.	1.28488	0.642442	+	0.766334i	$0.277921\pi$
$38$	0	0
$39$	−349866.	−0.944443
$40$	0	0
$41$	−47546.0	−0.107738	−0.0538692	−	0.998548i	$-0.517155\pi$
$41$	−47546.0	−0.107738	−0.0538692	+	0.998548i	$0.517155\pi$
$42$	0	0
$43$	−602088.	−1.15484	−0.577418	−	0.816449i	$-0.695940\pi$
$43$	−602088.	−1.15484	−0.577418	+	0.816449i	$0.695940\pi$
$44$	0	0
$45$	−298890.	−0.488954
$46$	0	0
$47$	647200.	0.909277	0.454638	−	0.890676i	$-0.349769\pi$
$47$	647200.	0.909277	0.454638	+	0.890676i	$0.349769\pi$
$48$	0	0
$49$	233241.	0.283217
$50$	0	0
$51$	−460674.	−0.486293
$52$	0	0
$53$	−1.31272e6	−1.21118	−0.605588	−	0.795778i	$-0.707062\pi$
$53$	−1.31272e6	−1.21118	−0.605588	+	0.795778i	$0.707062\pi$
$54$	0	0
$55$	−545710.	−0.442275
$56$	0	0
$57$	−1.46254e6	−1.04603
$58$	0	0
$59$	2.68114e6	1.69956	0.849782	−	0.527135i	$-0.176734\pi$
$59$	2.68114e6	1.69956	0.849782	+	0.527135i	$0.176734\pi$
$60$	0	0
$61$	551190.	0.310919	0.155459	−	0.987842i	$-0.450314\pi$
$61$	551190.	0.310919	0.155459	+	0.987842i	$0.450314\pi$
$62$	0	0
$63$	749412.	0.377597
$64$	0	0
$65$	−5.31278e6	−2.39952
$66$	0	0
$67$	−459260.	−0.186551	−0.0932753	−	0.995640i	$-0.529734\pi$
$67$	−459260.	−0.186551	−0.0932753	+	0.995640i	$0.529734\pi$
$68$	0	0
$69$	−310176.	−0.113668
$70$	0	0
$71$	18072.0	0.00599242	0.00299621	−	0.999996i	$-0.499046\pi$
$71$	18072.0	0.00599242	0.00299621	+	0.999996i	$0.499046\pi$
$72$	0	0
$73$	−426062.	−0.128187	−0.0640933	−	0.997944i	$-0.520416\pi$
$73$	−426062.	−0.128187	−0.0640933	+	0.997944i	$0.520416\pi$
$74$	0	0
$75$	−2.42932e6	−0.664923
$76$	0	0
$77$	1.36827e6	0.341549
$78$	0	0
$79$	−297764.	−0.0679481	−0.0339741	−	0.999423i	$-0.510816\pi$
$79$	−297764.	−0.0679481	−0.0339741	+	0.999423i	$0.510816\pi$
$80$	0	0
$81$	531441.	0.111111
$82$	0	0
$83$	−5.68403e6	−1.09115	−0.545573	−	0.838063i	$-0.683688\pi$
$83$	−5.68403e6	−1.09115	−0.545573	+	0.838063i	$0.683688\pi$
$84$	0	0
$85$	−6.99542e6	−1.23551
$86$	0	0
$87$	5.03966e6	0.820510
$88$	0	0
$89$	−6.34297e6	−0.953734	−0.476867	−	0.878975i	$-0.658228\pi$
$89$	−6.34297e6	−0.953734	−0.476867	+	0.878975i	$0.658228\pi$
$90$	0	0
$91$	1.33208e7	1.85305
$92$	0	0
$93$	−5.09414e6	−0.656721
$94$	0	0
$95$	−2.22089e7	−2.65763
$96$	0	0
$97$	1.66516e7	1.85248	0.926242	−	0.376929i	$-0.123020\pi$
$97$	1.66516e7	1.85248	0.926242	+	0.376929i	$0.123020\pi$
$98$	0	0
$99$	970299.	0.100504
$100$	0	0
$101$	−2.08327e6	−0.201197	−0.100598	−	0.994927i	$-0.532076\pi$
$101$	−2.08327e6	−0.201197	−0.100598	+	0.994927i	$0.532076\pi$
$102$	0	0
$103$	2.39046e6	0.215552	0.107776	−	0.994175i	$-0.465627\pi$
$103$	2.39046e6	0.215552	0.107776	+	0.994175i	$0.465627\pi$
$104$	0	0
$105$	1.13800e7	0.959352
$106$	0	0
$107$	1.40615e7	1.10965	0.554827	−	0.831966i	$-0.312785\pi$
$107$	1.40615e7	1.10965	0.554827	+	0.831966i	$0.312785\pi$
$108$	0	0
$109$	−1.11321e7	−0.823347	−0.411674	−	0.911331i	$-0.635056\pi$
$109$	−1.11321e7	−0.823347	−0.411674	+	0.911331i	$0.635056\pi$
$110$	0	0
$111$	−1.06889e7	−0.741828
$112$	0	0
$113$	5.66903e6	0.369602	0.184801	−	0.982776i	$-0.440836\pi$
$113$	5.66903e6	0.369602	0.184801	+	0.982776i	$0.440836\pi$
$114$	0	0
$115$	−4.71008e6	−0.288792
$116$	0	0
$117$	9.44638e6	0.545274
$118$	0	0
$119$	1.75397e7	0.954132
$120$	0	0
$121$	1.77156e6	0.0909091
$122$	0	0
$123$	1.28374e6	0.0622028
$124$	0	0
$125$	−4.85850e6	−0.222493
$126$	0	0
$127$	2.09170e7	0.906123	0.453061	−	0.891479i	$-0.350332\pi$
$127$	2.09170e7	0.906123	0.453061	+	0.891479i	$0.350332\pi$
$128$	0	0
$129$	1.62564e7	0.666745
$130$	0	0
$131$	1.12649e7	0.437802	0.218901	−	0.975747i	$-0.429753\pi$
$131$	1.12649e7	0.437802	0.218901	+	0.975747i	$0.429753\pi$
$132$	0	0
$133$	5.56847e7	2.05237
$134$	0	0
$135$	8.07003e6	0.282297
$136$	0	0
$137$	444290.	0.0147620	0.00738099	−	0.999973i	$-0.497651\pi$
$137$	444290.	0.0147620	0.00738099	+	0.999973i	$0.497651\pi$
$138$	0	0
$139$	−3.42613e7	−1.08206	−0.541030	−	0.841003i	$-0.681966\pi$
$139$	−3.42613e7	−1.08206	−0.541030	+	0.841003i	$0.681966\pi$
$140$	0	0
$141$	−1.74744e7	−0.524971
$142$	0	0
$143$	1.72471e7	0.493219
$144$	0	0
$145$	7.65281e7	2.08465
$146$	0	0
$147$	−6.29751e6	−0.163515
$148$	0	0
$149$	−4.82211e7	−1.19422	−0.597112	−	0.802158i	$-0.703685\pi$
$149$	−4.82211e7	−1.19422	−0.597112	+	0.802158i	$0.703685\pi$
$150$	0	0
$151$	4.48693e7	1.06055	0.530273	−	0.847827i	$-0.322089\pi$
$151$	4.48693e7	1.06055	0.530273	+	0.847827i	$0.322089\pi$
$152$	0	0
$153$	1.24382e7	0.280761
$154$	0	0
$155$	−7.73555e7	−1.66852
$156$	0	0
$157$	−5.38907e6	−0.111139	−0.0555693	−	0.998455i	$-0.517697\pi$
$157$	−5.38907e6	−0.111139	−0.0555693	+	0.998455i	$0.517697\pi$
$158$	0	0
$159$	3.54435e7	0.699273
$160$	0	0
$161$	1.18097e7	0.223022
$162$	0	0
$163$	−9.81674e7	−1.77546	−0.887730	−	0.460365i	$-0.847719\pi$
$163$	−9.81674e7	−1.77546	−0.887730	+	0.460365i	$0.847719\pi$
$164$	0	0
$165$	1.47342e7	0.255348
$166$	0	0
$167$	4.40611e7	0.732062	0.366031	−	0.930603i	$-0.380716\pi$
$167$	4.40611e7	0.732062	0.366031	+	0.930603i	$0.380716\pi$
$168$	0	0
$169$	1.05161e8	1.67592
$170$	0	0
$171$	3.94885e7	0.603926
$172$	0	0
$173$	6.71087e7	0.985411	0.492706	−	0.870196i	$-0.336008\pi$
$173$	6.71087e7	0.985411	0.492706	+	0.870196i	$0.336008\pi$
$174$	0	0
$175$	9.24943e7	1.30461
$176$	0	0
$177$	−7.23908e7	−0.981244
$178$	0	0
$179$	−4.34929e6	−0.0566804	−0.0283402	−	0.999598i	$-0.509022\pi$
$179$	−4.34929e6	−0.0566804	−0.0283402	+	0.999598i	$0.509022\pi$
$180$	0	0
$181$	−1.20238e7	−0.150719	−0.0753593	−	0.997156i	$-0.524010\pi$
$181$	−1.20238e7	−0.150719	−0.0753593	+	0.997156i	$0.524010\pi$
$182$	0	0
$183$	−1.48821e7	−0.179509
$184$	0	0
$185$	−1.62313e8	−1.88475
$186$	0	0
$187$	2.27095e7	0.253958
$188$	0	0
$189$	−2.02341e7	−0.218006
$190$	0	0
$191$	−5.96399e7	−0.619327	−0.309664	−	0.950846i	$-0.600216\pi$
$191$	−5.96399e7	−0.619327	−0.309664	+	0.950846i	$0.600216\pi$
$192$	0	0
$193$	−9.81036e7	−0.982278	−0.491139	−	0.871081i	$-0.663419\pi$
$193$	−9.81036e7	−0.982278	−0.491139	+	0.871081i	$0.663419\pi$
$194$	0	0
$195$	1.43445e8	1.38537
$196$	0	0
$197$	−1.09317e8	−1.01872	−0.509361	−	0.860553i	$-0.670118\pi$
$197$	−1.09317e8	−1.01872	−0.509361	+	0.860553i	$0.670118\pi$
$198$	0	0
$199$	3.64317e7	0.327713	0.163857	−	0.986484i	$-0.447607\pi$
$199$	3.64317e7	0.327713	0.163857	+	0.986484i	$0.447607\pi$
$200$	0	0
$201$	1.24000e7	0.107705
$202$	0	0
$203$	−1.91880e8	−1.60988
$204$	0	0
$205$	1.94939e7	0.158037
$206$	0	0
$207$	8.37475e6	0.0656260
$208$	0	0
$209$	7.20976e7	0.546272
$210$	0	0
$211$	−1.38637e7	−0.101599	−0.0507997	−	0.998709i	$-0.516177\pi$
$211$	−1.38637e7	−0.101599	−0.0507997	+	0.998709i	$0.516177\pi$
$212$	0	0
$213$	−487944.	−0.00345972
$214$	0	0
$215$	2.46856e8	1.69398
$216$	0	0
$217$	1.93955e8	1.28852
$218$	0	0
$219$	1.15037e7	0.0740086
$220$	0	0
$221$	2.21089e8	1.37783
$222$	0	0
$223$	−1.35935e8	−0.820850	−0.410425	−	0.911894i	$-0.634620\pi$
$223$	−1.35935e8	−0.820850	−0.410425	+	0.911894i	$0.634620\pi$
$224$	0	0
$225$	6.55918e7	0.383893
$226$	0	0
$227$	−2.82203e7	−0.160129	−0.0800646	−	0.996790i	$-0.525513\pi$
$227$	−2.82203e7	−0.160129	−0.0800646	+	0.996790i	$0.525513\pi$
$228$	0	0
$229$	−5.31215e7	−0.292312	−0.146156	−	0.989262i	$-0.546690\pi$
$229$	−5.31215e7	−0.292312	−0.146156	+	0.989262i	$0.546690\pi$
$230$	0	0
$231$	−3.69432e7	−0.197194
$232$	0	0
$233$	1.54589e8	0.800631	0.400316	−	0.916377i	$-0.368901\pi$
$233$	1.54589e8	0.800631	0.400316	+	0.916377i	$0.368901\pi$
$234$	0	0
$235$	−2.65352e8	−1.33378
$236$	0	0
$237$	8.03963e6	0.0392299
$238$	0	0
$239$	1.86143e8	0.881972	0.440986	−	0.897514i	$-0.354629\pi$
$239$	1.86143e8	0.881972	0.440986	+	0.897514i	$0.354629\pi$
$240$	0	0
$241$	2.62107e8	1.20620	0.603100	−	0.797666i	$-0.293932\pi$
$241$	2.62107e8	1.20620	0.603100	+	0.797666i	$0.293932\pi$
$242$	0	0
$243$	−1.43489e7	−0.0641500
$244$	0	0
$245$	−9.56288e7	−0.415439
$246$	0	0
$247$	7.01909e8	2.96375
$248$	0	0
$249$	1.53469e8	0.629973
$250$	0	0
$251$	2.75827e8	1.10098	0.550489	−	0.834842i	$-0.314441\pi$
$251$	2.75827e8	1.10098	0.550489	+	0.834842i	$0.314441\pi$
$252$	0	0
$253$	1.52905e7	0.0593609
$254$	0	0
$255$	1.88876e8	0.713324
$256$	0	0
$257$	1.06856e6	0.00392675	0.00196338	−	0.999998i	$-0.499375\pi$
$257$	1.06856e6	0.00392675	0.00196338	+	0.999998i	$0.499375\pi$
$258$	0	0
$259$	4.06971e8	1.45551
$260$	0	0
$261$	−1.36071e8	−0.473721
$262$	0	0
$263$	7.92924e7	0.268774	0.134387	−	0.990929i	$-0.457094\pi$
$263$	7.92924e7	0.268774	0.134387	+	0.990929i	$0.457094\pi$
$264$	0	0
$265$	5.38216e8	1.77663
$266$	0	0
$267$	1.71260e8	0.550639
$268$	0	0
$269$	2.10170e8	0.658321	0.329160	−	0.944274i	$-0.393234\pi$
$269$	2.10170e8	0.658321	0.329160	+	0.944274i	$0.393234\pi$
$270$	0	0
$271$	2.65510e8	0.810378	0.405189	−	0.914233i	$-0.367206\pi$
$271$	2.65510e8	0.810378	0.405189	+	0.914233i	$0.367206\pi$
$272$	0	0
$273$	−3.59662e8	−1.06986
$274$	0	0
$275$	1.19757e8	0.347245
$276$	0	0
$277$	−6.23529e8	−1.76270	−0.881349	−	0.472466i	$-0.843364\pi$
$277$	−6.23529e8	−1.76270	−0.881349	+	0.472466i	$0.843364\pi$
$278$	0	0
$279$	1.37542e8	0.379158
$280$	0	0
$281$	1.30611e8	0.351162	0.175581	−	0.984465i	$-0.443820\pi$
$281$	1.30611e8	0.351162	0.175581	+	0.984465i	$0.443820\pi$
$282$	0	0
$283$	2.20874e7	0.0579283	0.0289642	−	0.999580i	$-0.490779\pi$
$283$	2.20874e7	0.0579283	0.0289642	+	0.999580i	$0.490779\pi$
$284$	0	0
$285$	5.99640e8	1.53438
$286$	0	0
$287$	−4.88773e7	−0.122045
$288$	0	0
$289$	−1.19227e8	−0.290557
$290$	0	0
$291$	−4.49593e8	−1.06953
$292$	0	0
$293$	2.00188e8	0.464944	0.232472	−	0.972603i	$-0.425319\pi$
$293$	2.00188e8	0.464944	0.232472	+	0.972603i	$0.425319\pi$
$294$	0	0
$295$	−1.09927e9	−2.49302
$296$	0	0
$297$	−2.61981e7	−0.0580259
$298$	0	0
$299$	1.48862e8	0.322057
$300$	0	0
$301$	−6.18946e8	−1.30819
$302$	0	0
$303$	5.62483e7	0.116161
$304$	0	0
$305$	−2.25988e8	−0.456074
$306$	0	0
$307$	4.79736e8	0.946276	0.473138	−	0.880988i	$-0.343121\pi$
$307$	4.79736e8	0.946276	0.473138	+	0.880988i	$0.343121\pi$
$308$	0	0
$309$	−6.45425e7	−0.124449
$310$	0	0
$311$	5.19734e8	0.979761	0.489880	−	0.871790i	$-0.337040\pi$
$311$	5.19734e8	0.979761	0.489880	+	0.871790i	$0.337040\pi$
$312$	0	0
$313$	9.69759e8	1.78755	0.893776	−	0.448514i	$-0.148047\pi$
$313$	9.69759e8	1.78755	0.893776	+	0.448514i	$0.148047\pi$
$314$	0	0
$315$	−3.07259e8	−0.553882
$316$	0	0
$317$	7.56875e8	1.33450	0.667248	−	0.744836i	$-0.267472\pi$
$317$	7.56875e8	1.33450	0.667248	+	0.744836i	$0.267472\pi$
$318$	0	0
$319$	−2.48436e8	−0.428497
$320$	0	0
$321$	−3.79660e8	−0.640659
$322$	0	0
$323$	9.24214e8	1.52603
$324$	0	0
$325$	1.16590e9	1.88394
$326$	0	0
$327$	3.00566e8	0.475360
$328$	0	0
$329$	6.65322e8	1.03002
$330$	0	0
$331$	1.79867e8	0.272618	0.136309	−	0.990666i	$-0.456476\pi$
$331$	1.79867e8	0.272618	0.136309	+	0.990666i	$0.456476\pi$
$332$	0	0
$333$	2.88601e8	0.428295
$334$	0	0
$335$	1.88297e8	0.273644
$336$	0	0
$337$	−1.38092e9	−1.96546	−0.982728	−	0.185054i	$-0.940754\pi$
$337$	−1.38092e9	−1.96546	−0.982728	+	0.185054i	$0.940754\pi$
$338$	0	0
$339$	−1.53064e8	−0.213390
$340$	0	0
$341$	2.51122e8	0.342961
$342$	0	0
$343$	−6.06830e8	−0.811966
$344$	0	0
$345$	1.27172e8	0.166734
$346$	0	0
$347$	−7.66253e8	−0.984507	−0.492254	−	0.870452i	$-0.663827\pi$
$347$	−7.66253e8	−0.984507	−0.492254	+	0.870452i	$0.663827\pi$
$348$	0	0
$349$	−2.68852e8	−0.338552	−0.169276	−	0.985569i	$-0.554143\pi$
$349$	−2.68852e8	−0.338552	−0.169276	+	0.985569i	$0.554143\pi$
$350$	0	0
$351$	−2.55052e8	−0.314814
$352$	0	0
$353$	−3.95002e8	−0.477956	−0.238978	−	0.971025i	$-0.576812\pi$
$353$	−3.95002e8	−0.477956	−0.238978	+	0.971025i	$0.576812\pi$
$354$	0	0
$355$	−7.40952e6	−0.00879004
$356$	0	0
$357$	−4.73573e8	−0.550869
$358$	0	0
$359$	4.25768e7	0.0485671	0.0242836	−	0.999705i	$-0.492270\pi$
$359$	4.25768e7	0.0485671	0.0242836	+	0.999705i	$0.492270\pi$
$360$	0	0
$361$	2.04030e9	2.28254
$362$	0	0
$363$	−4.78321e7	−0.0524864
$364$	0	0
$365$	1.74685e8	0.188032
$366$	0	0
$367$	−1.85295e9	−1.95673	−0.978366	−	0.206882i	$-0.933668\pi$
$367$	−1.85295e9	−1.95673	−0.978366	+	0.206882i	$0.933668\pi$
$368$	0	0
$369$	−3.46610e7	−0.0359128
$370$	0	0
$371$	−1.34948e9	−1.37201
$372$	0	0
$373$	−4.83602e7	−0.0482511	−0.0241256	−	0.999709i	$-0.507680\pi$
$373$	−4.83602e7	−0.0482511	−0.0241256	+	0.999709i	$0.507680\pi$
$374$	0	0
$375$	1.31180e8	0.128457
$376$	0	0
$377$	−2.41866e9	−2.32477
$378$	0	0
$379$	2.26078e8	0.213315	0.106658	−	0.994296i	$-0.465985\pi$
$379$	2.26078e8	0.213315	0.106658	+	0.994296i	$0.465985\pi$
$380$	0	0
$381$	−5.64760e8	−0.523150
$382$	0	0
$383$	1.35198e9	1.22963	0.614815	−	0.788671i	$-0.289231\pi$
$383$	1.35198e9	1.22963	0.614815	+	0.788671i	$0.289231\pi$
$384$	0	0
$385$	−5.60990e8	−0.501005
$386$	0	0
$387$	−4.38922e8	−0.384945
$388$	0	0
$389$	−1.09107e9	−0.939789	−0.469894	−	0.882723i	$-0.655708\pi$
$389$	−1.09107e9	−0.939789	−0.469894	+	0.882723i	$0.655708\pi$
$390$	0	0
$391$	1.96008e8	0.165827
$392$	0	0
$393$	−3.04152e8	−0.252765
$394$	0	0
$395$	1.22083e8	0.0996704
$396$	0	0
$397$	−6.97868e8	−0.559766	−0.279883	−	0.960034i	$-0.590296\pi$
$397$	−6.97868e8	−0.559766	−0.279883	+	0.960034i	$0.590296\pi$
$398$	0	0
$399$	−1.50349e9	−1.18494
$400$	0	0
$401$	1.74689e9	1.35288	0.676441	−	0.736497i	$-0.263521\pi$
$401$	1.74689e9	1.35288	0.676441	+	0.736497i	$0.263521\pi$
$402$	0	0
$403$	2.44481e9	1.86071
$404$	0	0
$405$	−2.17891e8	−0.162985
$406$	0	0
$407$	5.26924e8	0.387407
$408$	0	0
$409$	−1.30304e9	−0.941729	−0.470865	−	0.882205i	$-0.656058\pi$
$409$	−1.30304e9	−0.941729	−0.470865	+	0.882205i	$0.656058\pi$
$410$	0	0
$411$	−1.19958e7	−0.00852283
$412$	0	0
$413$	2.75621e9	1.92525
$414$	0	0
$415$	2.33045e9	1.60056
$416$	0	0
$417$	9.25054e8	0.624728
$418$	0	0
$419$	−2.87139e9	−1.90697	−0.953484	−	0.301443i	$-0.902532\pi$
$419$	−2.87139e9	−1.90697	−0.953484	+	0.301443i	$0.902532\pi$
$420$	0	0
$421$	1.15946e9	0.757299	0.378650	−	0.925540i	$-0.376389\pi$
$421$	1.15946e9	0.757299	0.378650	+	0.925540i	$0.376389\pi$
$422$	0	0
$423$	4.71809e8	0.303092
$424$	0	0
$425$	1.53515e9	0.970042
$426$	0	0
$427$	5.66623e8	0.352206
$428$	0	0
$429$	−4.65672e8	−0.284760
$430$	0	0
$431$	1.66703e9	1.00294	0.501468	−	0.865176i	$-0.332793\pi$
$431$	1.66703e9	1.00294	0.501468	+	0.865176i	$0.332793\pi$
$432$	0	0
$433$	6.34094e8	0.375358	0.187679	−	0.982230i	$-0.439903\pi$
$433$	6.34094e8	0.375358	0.187679	+	0.982230i	$0.439903\pi$
$434$	0	0
$435$	−2.06626e9	−1.20357
$436$	0	0
$437$	6.22282e8	0.356699
$438$	0	0
$439$	1.22368e9	0.690307	0.345154	−	0.938546i	$-0.387827\pi$
$439$	1.22368e9	0.690307	0.345154	+	0.938546i	$0.387827\pi$
$440$	0	0
$441$	1.70033e8	0.0944055
$442$	0	0
$443$	1.23213e9	0.673355	0.336677	−	0.941620i	$-0.390697\pi$
$443$	1.23213e9	0.673355	0.336677	+	0.941620i	$0.390697\pi$
$444$	0	0
$445$	2.60062e9	1.39900
$446$	0	0
$447$	1.30197e9	0.689485
$448$	0	0
$449$	−3.07511e9	−1.60324	−0.801621	−	0.597833i	$-0.796029\pi$
$449$	−3.07511e9	−1.60324	−0.801621	+	0.597833i	$0.796029\pi$
$450$	0	0
$451$	−6.32837e7	−0.0324843
$452$	0	0
$453$	−1.21147e9	−0.612307
$454$	0	0
$455$	−5.46154e9	−2.71816
$456$	0	0
$457$	−2.44730e9	−1.19945	−0.599723	−	0.800207i	$-0.704723\pi$
$457$	−2.44730e9	−1.19945	−0.599723	+	0.800207i	$0.704723\pi$
$458$	0	0
$459$	−3.35831e8	−0.162098
$460$	0	0
$461$	9.52419e8	0.452767	0.226383	−	0.974038i	$-0.427310\pi$
$461$	9.52419e8	0.452767	0.226383	+	0.974038i	$0.427310\pi$
$462$	0	0
$463$	6.05200e8	0.283378	0.141689	−	0.989911i	$-0.454747\pi$
$463$	6.05200e8	0.283378	0.141689	+	0.989911i	$0.454747\pi$
$464$	0	0
$465$	2.08860e9	0.963318
$466$	0	0
$467$	1.37708e9	0.625676	0.312838	−	0.949806i	$-0.398720\pi$
$467$	1.37708e9	0.625676	0.312838	+	0.949806i	$0.398720\pi$
$468$	0	0
$469$	−4.72119e8	−0.211323
$470$	0	0
$471$	1.45505e8	0.0641659
$472$	0	0
$473$	−8.01379e8	−0.348196
$474$	0	0
$475$	4.87377e9	2.08659
$476$	0	0
$477$	−9.56974e8	−0.403725
$478$	0	0
$479$	4.00222e9	1.66390	0.831949	−	0.554851i	$-0.187225\pi$
$479$	4.00222e9	1.66390	0.831949	+	0.554851i	$0.187225\pi$
$480$	0	0
$481$	5.12989e9	2.10184
$482$	0	0
$483$	−3.18861e8	−0.128762
$484$	0	0
$485$	−6.82715e9	−2.71734
$486$	0	0
$487$	2.88677e9	1.13256	0.566279	−	0.824214i	$-0.308383\pi$
$487$	2.88677e9	1.13256	0.566279	+	0.824214i	$0.308383\pi$
$488$	0	0
$489$	2.65052e9	1.02506
$490$	0	0
$491$	−1.19743e8	−0.0456525	−0.0228262	−	0.999739i	$-0.507266\pi$
$491$	−1.19743e8	−0.0456525	−0.0228262	+	0.999739i	$0.507266\pi$
$492$	0	0
$493$	−3.18469e9	−1.19702
$494$	0	0
$495$	−3.97823e8	−0.147425
$496$	0	0
$497$	1.85780e7	0.00678816
$498$	0	0
$499$	4.78950e9	1.72559	0.862796	−	0.505552i	$-0.168711\pi$
$499$	4.78950e9	1.72559	0.862796	+	0.505552i	$0.168711\pi$
$500$	0	0
$501$	−1.18965e9	−0.422656
$502$	0	0
$503$	−3.83047e9	−1.34203	−0.671017	−	0.741442i	$-0.734142\pi$
$503$	−3.83047e9	−1.34203	−0.671017	+	0.741442i	$0.734142\pi$
$504$	0	0
$505$	8.54141e8	0.295127
$506$	0	0
$507$	−2.83935e9	−0.967591
$508$	0	0
$509$	−2.34385e9	−0.787803	−0.393902	−	0.919153i	$-0.628875\pi$
$509$	−2.34385e9	−0.787803	−0.393902	+	0.919153i	$0.628875\pi$
$510$	0	0
$511$	−4.37992e8	−0.145209
$512$	0	0
$513$	−1.06619e9	−0.348677
$514$	0	0
$515$	−9.80090e8	−0.316185
$516$	0	0
$517$	8.61423e8	0.274157
$518$	0	0
$519$	−1.81194e9	−0.568928
$520$	0	0
$521$	5.77085e9	1.78775	0.893877	−	0.448313i	$-0.147975\pi$
$521$	5.77085e9	1.78775	0.893877	+	0.448313i	$0.147975\pi$
$522$	0	0
$523$	3.49411e8	0.106802	0.0534012	−	0.998573i	$-0.482994\pi$
$523$	3.49411e8	0.106802	0.0534012	+	0.998573i	$0.482994\pi$
$524$	0	0
$525$	−2.49735e9	−0.753219
$526$	0	0
$527$	3.21912e9	0.958077
$528$	0	0
$529$	−3.27285e9	−0.961239
$530$	0	0
$531$	1.95455e9	0.566521
$532$	0	0
$533$	−6.16101e8	−0.176241
$534$	0	0
$535$	−5.76520e9	−1.62771
$536$	0	0
$537$	1.17431e8	0.0327244
$538$	0	0
$539$	3.10444e8	0.0853930
$540$	0	0
$541$	−5.10025e9	−1.38484	−0.692422	−	0.721493i	$-0.743456\pi$
$541$	−5.10025e9	−1.38484	−0.692422	+	0.721493i	$0.743456\pi$
$542$	0	0
$543$	3.24643e8	0.0870175
$544$	0	0
$545$	4.56415e9	1.20774
$546$	0	0
$547$	−4.96217e9	−1.29633	−0.648166	−	0.761499i	$-0.724464\pi$
$547$	−4.96217e9	−1.29633	−0.648166	+	0.761499i	$0.724464\pi$
$548$	0	0
$549$	4.01818e8	0.103640
$550$	0	0
$551$	−1.01107e10	−2.57484
$552$	0	0
$553$	−3.06101e8	−0.0769710
$554$	0	0
$555$	4.38246e9	1.08816
$556$	0	0
$557$	1.42590e9	0.349620	0.174810	−	0.984602i	$-0.444069\pi$
$557$	1.42590e9	0.349620	0.174810	+	0.984602i	$0.444069\pi$
$558$	0	0
$559$	−7.80186e9	−1.88911
$560$	0	0
$561$	−6.13157e8	−0.146623
$562$	0	0
$563$	−5.96929e9	−1.40975	−0.704876	−	0.709330i	$-0.748998\pi$
$563$	−5.96929e9	−1.40975	−0.704876	+	0.709330i	$0.748998\pi$
$564$	0	0
$565$	−2.32430e9	−0.542155
$566$	0	0
$567$	5.46321e8	0.125866
$568$	0	0
$569$	−3.51616e9	−0.800158	−0.400079	−	0.916481i	$-0.631017\pi$
$569$	−3.51616e9	−0.800158	−0.400079	+	0.916481i	$0.631017\pi$
$570$	0	0
$571$	−6.44706e8	−0.144922	−0.0724611	−	0.997371i	$-0.523085\pi$
$571$	−6.44706e8	−0.144922	−0.0724611	+	0.997371i	$0.523085\pi$
$572$	0	0
$573$	1.61028e9	0.357569
$574$	0	0
$575$	1.03363e9	0.226740
$576$	0	0
$577$	−2.63322e9	−0.570652	−0.285326	−	0.958430i	$-0.592102\pi$
$577$	−2.63322e9	−0.570652	−0.285326	+	0.958430i	$0.592102\pi$
$578$	0	0
$579$	2.64880e9	0.567118
$580$	0	0
$581$	−5.84318e9	−1.23604
$582$	0	0
$583$	−1.74723e9	−0.365183
$584$	0	0
$585$	−3.87302e9	−0.799841
$586$	0	0
$587$	−6.76347e9	−1.38018	−0.690090	−	0.723723i	$-0.742429\pi$
$587$	−6.76347e9	−1.38018	−0.690090	+	0.723723i	$0.742429\pi$
$588$	0	0
$589$	1.02200e10	2.06085
$590$	0	0
$591$	2.95156e9	0.588159
$592$	0	0
$593$	−4.22718e9	−0.832452	−0.416226	−	0.909261i	$-0.636648\pi$
$593$	−4.22718e9	−0.832452	−0.416226	+	0.909261i	$0.636648\pi$
$594$	0	0
$595$	−7.19129e9	−1.39958
$596$	0	0
$597$	−9.83657e8	−0.189205
$598$	0	0
$599$	−4.00299e9	−0.761010	−0.380505	−	0.924779i	$-0.624250\pi$
$599$	−4.00299e9	−0.761010	−0.380505	+	0.924779i	$0.624250\pi$
$600$	0	0
$601$	6.67554e9	1.25437	0.627185	−	0.778870i	$-0.284207\pi$
$601$	6.67554e9	1.25437	0.627185	+	0.778870i	$0.284207\pi$
$602$	0	0
$603$	−3.34801e8	−0.0621836
$604$	0	0
$605$	−7.26340e8	−0.133351
$606$	0	0
$607$	5.30634e9	0.963018	0.481509	−	0.876441i	$-0.340089\pi$
$607$	5.30634e9	0.963018	0.481509	+	0.876441i	$0.340089\pi$
$608$	0	0
$609$	5.18077e9	0.929466
$610$	0	0
$611$	8.38642e9	1.48742
$612$	0	0
$613$	−8.65802e9	−1.51812	−0.759061	−	0.651019i	$-0.774342\pi$
$613$	−8.65802e9	−1.51812	−0.759061	+	0.651019i	$0.774342\pi$
$614$	0	0
$615$	−5.26334e8	−0.0912428
$616$	0	0
$617$	−7.38891e9	−1.26643	−0.633217	−	0.773974i	$-0.718266\pi$
$617$	−7.38891e9	−1.26643	−0.633217	+	0.773974i	$0.718266\pi$
$618$	0	0
$619$	−9.99141e9	−1.69321	−0.846603	−	0.532225i	$-0.821356\pi$
$619$	−9.99141e9	−1.69321	−0.846603	+	0.532225i	$0.821356\pi$
$620$	0	0
$621$	−2.26118e8	−0.0378892
$622$	0	0
$623$	−6.52057e9	−1.08038
$624$	0	0
$625$	−5.03731e9	−0.825313
$626$	0	0
$627$	−1.94664e9	−0.315390
$628$	0	0
$629$	6.75461e9	1.08224
$630$	0	0
$631$	3.29834e9	0.522628	0.261314	−	0.965254i	$-0.415844\pi$
$631$	3.29834e9	0.522628	0.261314	+	0.965254i	$0.415844\pi$
$632$	0	0
$633$	3.74320e8	0.0586584
$634$	0	0
$635$	−8.57598e9	−1.32916
$636$	0	0
$637$	3.02234e9	0.463292
$638$	0	0
$639$	1.31745e7	0.00199747
$640$	0	0
$641$	−9.76971e9	−1.46514	−0.732569	−	0.680692i	$-0.761679\pi$
$641$	−9.76971e9	−1.46514	−0.732569	+	0.680692i	$0.761679\pi$
$642$	0	0
$643$	4.18444e9	0.620724	0.310362	−	0.950618i	$-0.399550\pi$
$643$	4.18444e9	0.620724	0.310362	+	0.950618i	$0.399550\pi$
$644$	0	0
$645$	−6.66511e9	−0.978022
$646$	0	0
$647$	6.96085e8	0.101041	0.0505204	−	0.998723i	$-0.483912\pi$
$647$	6.96085e8	0.101041	0.0505204	+	0.998723i	$0.483912\pi$
$648$	0	0
$649$	3.56860e9	0.512438
$650$	0	0
$651$	−5.23678e9	−0.743928
$652$	0	0
$653$	6.20046e9	0.871420	0.435710	−	0.900087i	$-0.356497\pi$
$653$	6.20046e9	0.871420	0.435710	+	0.900087i	$0.356497\pi$
$654$	0	0
$655$	−4.61861e9	−0.642194
$656$	0	0
$657$	−3.10599e8	−0.0427289
$658$	0	0
$659$	1.11404e10	1.51636	0.758178	−	0.652047i	$-0.226090\pi$
$659$	1.11404e10	1.51636	0.758178	+	0.652047i	$0.226090\pi$
$660$	0	0
$661$	4.56096e9	0.614258	0.307129	−	0.951668i	$-0.400632\pi$
$661$	4.56096e9	0.614258	0.307129	+	0.951668i	$0.400632\pi$
$662$	0	0
$663$	−5.96941e9	−0.795489
$664$	0	0
$665$	−2.28307e10	−3.01054
$666$	0	0
$667$	−2.14428e9	−0.279796
$668$	0	0
$669$	3.67024e9	0.473918
$670$	0	0
$671$	7.33634e8	0.0937455
$672$	0	0
$673$	5.82879e9	0.737099	0.368550	−	0.929608i	$-0.379854\pi$
$673$	5.82879e9	0.737099	0.368550	+	0.929608i	$0.379854\pi$
$674$	0	0
$675$	−1.77098e9	−0.221641
$676$	0	0
$677$	4.99624e9	0.618846	0.309423	−	0.950924i	$-0.399864\pi$
$677$	4.99624e9	0.618846	0.309423	+	0.950924i	$0.399864\pi$
$678$	0	0
$679$	1.71178e10	2.09848
$680$	0	0
$681$	7.61947e8	0.0924507
$682$	0	0
$683$	−1.21371e10	−1.45762	−0.728808	−	0.684718i	$-0.759925\pi$
$683$	−1.21371e10	−1.45762	−0.728808	+	0.684718i	$0.759925\pi$
$684$	0	0
$685$	−1.82159e8	−0.0216538
$686$	0	0
$687$	1.43428e9	0.168766
$688$	0	0
$689$	−1.70103e10	−1.98127
$690$	0	0
$691$	9.23403e9	1.06468	0.532339	−	0.846531i	$-0.321313\pi$
$691$	9.23403e9	1.06468	0.532339	+	0.846531i	$0.321313\pi$
$692$	0	0
$693$	9.97467e8	0.113850
$694$	0	0
$695$	1.40471e10	1.58723
$696$	0	0
$697$	−8.11230e8	−0.0907464
$698$	0	0
$699$	−4.17390e9	−0.462245
$700$	0	0
$701$	4.74530e9	0.520296	0.260148	−	0.965569i	$-0.416229\pi$
$701$	4.74530e9	0.520296	0.260148	+	0.965569i	$0.416229\pi$
$702$	0	0
$703$	2.14444e10	2.32793
$704$	0	0
$705$	7.16450e9	0.770059
$706$	0	0
$707$	−2.14160e9	−0.227914
$708$	0	0
$709$	1.34547e10	1.41779	0.708894	−	0.705315i	$-0.249195\pi$
$709$	1.34547e10	1.41779	0.708894	+	0.705315i	$0.249195\pi$
$710$	0	0
$711$	−2.17070e8	−0.0226494
$712$	0	0
$713$	2.16746e9	0.223944
$714$	0	0
$715$	−7.07131e9	−0.723484
$716$	0	0
$717$	−5.02587e9	−0.509207
$718$	0	0
$719$	−2.63976e9	−0.264858	−0.132429	−	0.991192i	$-0.542278\pi$
$719$	−2.63976e9	−0.264858	−0.132429	+	0.991192i	$0.542278\pi$
$720$	0	0
$721$	2.45740e9	0.244175
$722$	0	0
$723$	−7.07689e9	−0.696400
$724$	0	0
$725$	−1.67942e10	−1.63673
$726$	0	0
$727$	3.52707e9	0.340442	0.170221	−	0.985406i	$-0.445552\pi$
$727$	3.52707e9	0.340442	0.170221	+	0.985406i	$0.445552\pi$
$728$	0	0
$729$	3.87420e8	0.0370370
$730$	0	0
$731$	−1.02728e10	−0.972700
$732$	0	0
$733$	−1.03828e10	−0.973760	−0.486880	−	0.873469i	$-0.661865\pi$
$733$	−1.03828e10	−0.973760	−0.486880	+	0.873469i	$0.661865\pi$
$734$	0	0
$735$	2.58198e9	0.239854
$736$	0	0
$737$	−6.11275e8	−0.0562471
$738$	0	0
$739$	−2.05418e9	−0.187233	−0.0936164	−	0.995608i	$-0.529843\pi$
$739$	−2.05418e9	−0.187233	−0.0936164	+	0.995608i	$0.529843\pi$
$740$	0	0
$741$	−1.89515e10	−1.71112
$742$	0	0
$743$	−4.87476e9	−0.436006	−0.218003	−	0.975948i	$-0.569954\pi$
$743$	−4.87476e9	−0.436006	−0.218003	+	0.975948i	$0.569954\pi$
$744$	0	0
$745$	1.97707e10	1.75176
$746$	0	0
$747$	−4.14366e9	−0.363715
$748$	0	0
$749$	1.44552e10	1.25701
$750$	0	0
$751$	−1.15809e10	−0.997705	−0.498853	−	0.866687i	$-0.666245\pi$
$751$	−1.15809e10	−0.997705	−0.498853	+	0.866687i	$0.666245\pi$
$752$	0	0
$753$	−7.44733e9	−0.635650
$754$	0	0
$755$	−1.83964e10	−1.55567
$756$	0	0
$757$	3.46735e9	0.290511	0.145255	−	0.989394i	$-0.453600\pi$
$757$	3.46735e9	0.290511	0.145255	+	0.989394i	$0.453600\pi$
$758$	0	0
$759$	−4.12844e8	−0.0342720
$760$	0	0
$761$	−1.14023e10	−0.937877	−0.468938	−	0.883231i	$-0.655363\pi$
$761$	−1.14023e10	−0.937877	−0.468938	+	0.883231i	$0.655363\pi$
$762$	0	0
$763$	−1.14438e10	−0.932681
$764$	0	0
$765$	−5.09966e9	−0.411838
$766$	0	0
$767$	3.47422e10	2.78018
$768$	0	0
$769$	2.30715e10	1.82951	0.914754	−	0.404012i	$-0.132384\pi$
$769$	2.30715e10	1.82951	0.914754	+	0.404012i	$0.132384\pi$
$770$	0	0
$771$	−2.88512e7	−0.00226711
$772$	0	0
$773$	2.15091e10	1.67492	0.837461	−	0.546497i	$-0.184039\pi$
$773$	2.15091e10	1.67492	0.837461	+	0.546497i	$0.184039\pi$
$774$	0	0
$775$	1.69758e10	1.31001
$776$	0	0
$777$	−1.09882e10	−0.840337
$778$	0	0
$779$	−2.57547e9	−0.195198
$780$	0	0
$781$	2.40538e7	0.00180678
$782$	0	0
$783$	3.67391e9	0.273503
$784$	0	0
$785$	2.20952e9	0.163025
$786$	0	0
$787$	4.46678e9	0.326651	0.163325	−	0.986572i	$-0.447778\pi$
$787$	4.46678e9	0.326651	0.163325	+	0.986572i	$0.447778\pi$
$788$	0	0
$789$	−2.14090e9	−0.155176
$790$	0	0
$791$	5.82777e9	0.418682
$792$	0	0
$793$	7.14232e9	0.508608
$794$	0	0
$795$	−1.45318e10	−1.02574
$796$	0	0
$797$	2.43899e10	1.70650	0.853248	−	0.521505i	$-0.174629\pi$
$797$	2.43899e10	1.70650	0.853248	+	0.521505i	$0.174629\pi$
$798$	0	0
$799$	1.10425e10	0.765869
$800$	0	0
$801$	−4.62402e9	−0.317911
$802$	0	0
$803$	−5.67089e8	−0.0386497
$804$	0	0
$805$	−4.84196e9	−0.327142
$806$	0	0
$807$	−5.67459e9	−0.380082
$808$	0	0
$809$	9.88857e9	0.656620	0.328310	−	0.944570i	$-0.393521\pi$
$809$	9.88857e9	0.656620	0.328310	+	0.944570i	$0.393521\pi$
$810$	0	0
$811$	1.15204e10	0.758395	0.379198	−	0.925316i	$-0.376200\pi$
$811$	1.15204e10	0.758395	0.379198	+	0.925316i	$0.376200\pi$
$812$	0	0
$813$	−7.16876e9	−0.467872
$814$	0	0
$815$	4.02487e10	2.60435
$816$	0	0
$817$	−3.26139e10	−2.09231
$818$	0	0
$819$	9.71088e9	0.617682
$820$	0	0
$821$	2.63516e9	0.166191	0.0830953	−	0.996542i	$-0.473519\pi$
$821$	2.63516e9	0.166191	0.0830953	+	0.996542i	$0.473519\pi$
$822$	0	0
$823$	1.27039e10	0.794400	0.397200	−	0.917732i	$-0.369982\pi$
$823$	1.27039e10	0.794400	0.397200	+	0.917732i	$0.369982\pi$
$824$	0	0
$825$	−3.23343e9	−0.200482
$826$	0	0
$827$	1.11339e10	0.684504	0.342252	−	0.939608i	$-0.388810\pi$
$827$	1.11339e10	0.684504	0.342252	+	0.939608i	$0.388810\pi$
$828$	0	0
$829$	2.79852e10	1.70604	0.853018	−	0.521881i	$-0.174770\pi$
$829$	2.79852e10	1.70604	0.853018	+	0.521881i	$0.174770\pi$
$830$	0	0
$831$	1.68353e10	1.01769
$832$	0	0
$833$	3.97956e9	0.238549
$834$	0	0
$835$	−1.80651e10	−1.07383
$836$	0	0
$837$	−3.71363e9	−0.218907
$838$	0	0
$839$	−2.71170e9	−0.158517	−0.0792583	−	0.996854i	$-0.525255\pi$
$839$	−2.71170e9	−0.158517	−0.0792583	+	0.996854i	$0.525255\pi$
$840$	0	0
$841$	1.75898e10	1.01971
$842$	0	0
$843$	−3.52650e9	−0.202744
$844$	0	0
$845$	−4.31161e10	−2.45834
$846$	0	0
$847$	1.82116e9	0.102981
$848$	0	0
$849$	−5.96359e8	−0.0334449
$850$	0	0
$851$	4.54794e9	0.252965
$852$	0	0
$853$	1.97175e10	1.08775	0.543877	−	0.839165i	$-0.316956\pi$
$853$	1.97175e10	1.08775	0.543877	+	0.839165i	$0.316956\pi$
$854$	0	0
$855$	−1.61903e10	−0.885876
$856$	0	0
$857$	1.89411e10	1.02795	0.513976	−	0.857804i	$-0.328172\pi$
$857$	1.89411e10	1.02795	0.513976	+	0.857804i	$0.328172\pi$
$858$	0	0
$859$	6.77637e9	0.364772	0.182386	−	0.983227i	$-0.441618\pi$
$859$	6.77637e9	0.364772	0.182386	+	0.983227i	$0.441618\pi$
$860$	0	0
$861$	1.31969e9	0.0704628
$862$	0	0
$863$	−2.80635e10	−1.48629	−0.743146	−	0.669129i	$-0.766667\pi$
$863$	−2.80635e10	−1.48629	−0.743146	+	0.669129i	$0.766667\pi$
$864$	0	0
$865$	−2.75146e10	−1.44546
$866$	0	0
$867$	3.21912e9	0.167753
$868$	0	0
$869$	−3.96324e8	−0.0204871
$870$	0	0
$871$	−5.95109e9	−0.305164
$872$	0	0
$873$	1.21390e10	0.617495
$874$	0	0
$875$	−4.99454e9	−0.252039
$876$	0	0
$877$	−1.01559e10	−0.508418	−0.254209	−	0.967149i	$-0.581815\pi$
$877$	−1.01559e10	−0.508418	−0.254209	+	0.967149i	$0.581815\pi$
$878$	0	0
$879$	−5.40507e9	−0.268436
$880$	0	0
$881$	−2.78023e10	−1.36982	−0.684912	−	0.728626i	$-0.740159\pi$
$881$	−2.78023e10	−1.36982	−0.684912	+	0.728626i	$0.740159\pi$
$882$	0	0
$883$	−2.15199e10	−1.05191	−0.525954	−	0.850513i	$-0.676291\pi$
$883$	−2.15199e10	−1.05191	−0.525954	+	0.850513i	$0.676291\pi$
$884$	0	0
$885$	2.96802e10	1.43935
$886$	0	0
$887$	3.13376e10	1.50776	0.753881	−	0.657011i	$-0.228180\pi$
$887$	3.13376e10	1.50776	0.753881	+	0.657011i	$0.228180\pi$
$888$	0	0
$889$	2.15027e10	1.02645
$890$	0	0
$891$	7.07348e8	0.0335013
$892$	0	0
$893$	3.50575e10	1.64741
$894$	0	0
$895$	1.78321e9	0.0831423
$896$	0	0
$897$	−4.01926e9	−0.185940
$898$	0	0
$899$	−3.52164e10	−1.61654
$900$	0	0
$901$	−2.23977e10	−1.02015
$902$	0	0
$903$	1.67116e10	0.755283
$904$	0	0
$905$	4.92976e9	0.221083
$906$	0	0
$907$	1.62459e10	0.722966	0.361483	−	0.932379i	$-0.382270\pi$
$907$	1.62459e10	0.722966	0.361483	+	0.932379i	$0.382270\pi$
$908$	0	0
$909$	−1.51870e9	−0.0670656
$910$	0	0
$911$	3.15726e9	0.138356	0.0691778	−	0.997604i	$-0.477962\pi$
$911$	3.15726e9	0.138356	0.0691778	+	0.997604i	$0.477962\pi$
$912$	0	0
$913$	−7.56544e9	−0.328993
$914$	0	0
$915$	6.10167e9	0.263315
$916$	0	0
$917$	1.15803e10	0.495938
$918$	0	0
$919$	−2.53655e9	−0.107805	−0.0539026	−	0.998546i	$-0.517166\pi$
$919$	−2.53655e9	−0.107805	−0.0539026	+	0.998546i	$0.517166\pi$
$920$	0	0
$921$	−1.29529e10	−0.546333
$922$	0	0
$923$	2.34177e8	0.00980253
$924$	0	0
$925$	3.56198e10	1.47978
$926$	0	0
$927$	1.74265e9	0.0718506
$928$	0	0
$929$	−2.10282e10	−0.860494	−0.430247	−	0.902711i	$-0.641574\pi$
$929$	−2.10282e10	−0.860494	−0.430247	+	0.902711i	$0.641574\pi$
$930$	0	0
$931$	1.26342e10	0.513126
$932$	0	0
$933$	−1.40328e10	−0.565665
$934$	0	0
$935$	−9.31090e9	−0.372521
$936$	0	0
$937$	3.12893e10	1.24253	0.621265	−	0.783601i	$-0.286619\pi$
$937$	3.12893e10	1.24253	0.621265	+	0.783601i	$0.286619\pi$
$938$	0	0
$939$	−2.61835e10	−1.03204
$940$	0	0
$941$	7.91706e9	0.309742	0.154871	−	0.987935i	$-0.450504\pi$
$941$	7.91706e9	0.309742	0.154871	+	0.987935i	$0.450504\pi$
$942$	0	0
$943$	−5.46208e8	−0.0212113
$944$	0	0
$945$	8.29599e9	0.319784
$946$	0	0
$947$	−8.55849e9	−0.327471	−0.163735	−	0.986504i	$-0.552354\pi$
$947$	−8.55849e9	−0.327471	−0.163735	+	0.986504i	$0.552354\pi$
$948$	0	0
$949$	−5.52091e9	−0.209691
$950$	0	0
$951$	−2.04356e10	−0.770471
$952$	0	0
$953$	−8.49661e9	−0.317995	−0.158998	−	0.987279i	$-0.550826\pi$
$953$	−8.49661e9	−0.317995	−0.158998	+	0.987279i	$0.550826\pi$
$954$	0	0
$955$	2.44524e10	0.908466
$956$	0	0
$957$	6.70778e9	0.247393
$958$	0	0
$959$	4.56730e8	0.0167222
$960$	0	0
$961$	8.08451e9	0.293847
$962$	0	0
$963$	1.02508e10	0.369885
$964$	0	0
$965$	4.02225e10	1.44086
$966$	0	0
$967$	1.47988e10	0.526300	0.263150	−	0.964755i	$-0.415239\pi$
$967$	1.47988e10	0.526300	0.263150	+	0.964755i	$0.415239\pi$
$968$	0	0
$969$	−2.49538e10	−0.881056
$970$	0	0
$971$	2.86157e10	1.00308	0.501542	−	0.865133i	$-0.332766\pi$
$971$	2.86157e10	1.00308	0.501542	+	0.865133i	$0.332766\pi$
$972$	0	0
$973$	−3.52206e10	−1.22575
$974$	0	0
$975$	−3.14792e10	−1.08770
$976$	0	0
$977$	3.37991e10	1.15951	0.579755	−	0.814791i	$-0.303148\pi$
$977$	3.37991e10	1.15951	0.579755	+	0.814791i	$0.303148\pi$
$978$	0	0
$979$	−8.44249e9	−0.287562
$980$	0	0
$981$	−8.11528e9	−0.274449
$982$	0	0
$983$	−1.03134e9	−0.0346308	−0.0173154	−	0.999850i	$-0.505512\pi$
$983$	−1.03134e9	−0.0346308	−0.0173154	+	0.999850i	$0.505512\pi$
$984$	0	0
$985$	4.48199e10	1.49432
$986$	0	0
$987$	−1.79637e10	−0.594683
$988$	0	0
$989$	−6.91679e9	−0.227362
$990$	0	0
$991$	5.63139e10	1.83805	0.919027	−	0.394195i	$-0.128977\pi$
$991$	5.63139e10	1.83805	0.919027	+	0.394195i	$0.128977\pi$
$992$	0	0
$993$	−4.85642e9	−0.157396
$994$	0	0
$995$	−1.49370e10	−0.480710
$996$	0	0
$997$	2.55531e10	0.816603	0.408301	−	0.912847i	$-0.366121\pi$
$997$	2.55531e10	0.816603	0.408301	+	0.912847i	$0.366121\pi$
$998$	0	0
$999$	−7.79222e9	−0.247276

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	528.8.a.a.1.1		1
4.3	odd	2		33.8.a.a.1.1	✓	1
12.11	even	2		99.8.a.a.1.1		1
44.43	even	2		363.8.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.8.a.a.1.1	✓	1	4.3	odd	2
99.8.a.a.1.1		1	12.11	even	2
363.8.a.a.1.1		1	44.43	even	2
528.8.a.a.1.1		1	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 528 = 2^{4} \cdot 3 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	528.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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