LMFDB - Embedded newform 192.8.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [192,8,Mod(1,192)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(192, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("192.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 192 = 2^{6} \cdot 3 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	192.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:	$59.9779248930$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 24)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	192.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-27.0000 q^{3} -110.000 q^{5} +504.000 q^{7} +729.000 q^{9} +O(q^{10})$

$q-27.0000 q^{3} -110.000 q^{5} +504.000 q^{7} +729.000 q^{9} -3812.00 q^{11} -9574.00 q^{13} +2970.00 q^{15} +26098.0 q^{17} +38308.0 q^{19} -13608.0 q^{21} -71128.0 q^{23} -66025.0 q^{25} -19683.0 q^{27} -74262.0 q^{29} -275680. q^{31} +102924. q^{33} -55440.0 q^{35} +266610. q^{37} +258498. q^{39} +684762. q^{41} -245956. q^{43} -80190.0 q^{45} +478800. q^{47} -569527. q^{49} -704646. q^{51} +569410. q^{53} +419320. q^{55} -1.03432e6 q^{57} +1.52532e6 q^{59} +2.64046e6 q^{61} +367416. q^{63} +1.05314e6 q^{65} -1.41624e6 q^{67} +1.92046e6 q^{69} -3.51130e6 q^{71} +4.73862e6 q^{73} +1.78268e6 q^{75} -1.92125e6 q^{77} +4.66149e6 q^{79} +531441. q^{81} +5.72925e6 q^{83} -2.87078e6 q^{85} +2.00507e6 q^{87} +1.19935e7 q^{89} -4.82530e6 q^{91} +7.44336e6 q^{93} -4.21388e6 q^{95} +7.15075e6 q^{97} -2.77895e6 q^{99} +O(q^{100})$

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
$5$	−110.000	−0.393548	−0.196774	−	0.980449i	$-0.563047\pi$
$5$	−110.000	−0.393548	−0.196774	+	0.980449i	$0.563047\pi$
$6$	0	0
$7$	504.000	0.555376	0.277688	−	0.960671i	$-0.410432\pi$
$7$	504.000	0.555376	0.277688	+	0.960671i	$0.410432\pi$
$8$	0	0
$9$	729.000	0.333333
$10$	0	0
$11$	−3812.00	−0.863532	−0.431766	−	0.901986i	$-0.642109\pi$
$11$	−3812.00	−0.863532	−0.431766	+	0.901986i	$0.642109\pi$
$12$	0	0
$13$	−9574.00	−1.20863	−0.604313	−	0.796747i	$-0.706552\pi$
$13$	−9574.00	−1.20863	−0.604313	+	0.796747i	$0.706552\pi$
$14$	0	0
$15$	2970.00	0.227215
$16$	0	0
$17$	26098.0	1.28836	0.644178	−	0.764875i	$-0.277200\pi$
$17$	26098.0	1.28836	0.644178	+	0.764875i	$0.277200\pi$
$18$	0	0
$19$	38308.0	1.28130	0.640652	−	0.767832i	$-0.278664\pi$
$19$	38308.0	1.28130	0.640652	+	0.767832i	$0.278664\pi$
$20$	0	0
$21$	−13608.0	−0.320647
$22$	0	0
$23$	−71128.0	−1.21897	−0.609485	−	0.792797i	$-0.708624\pi$
$23$	−71128.0	−1.21897	−0.609485	+	0.792797i	$0.708624\pi$
$24$	0	0
$25$	−66025.0	−0.845120
$26$	0	0
$27$	−19683.0	−0.192450
$28$	0	0
$29$	−74262.0	−0.565423	−0.282712	−	0.959205i	$-0.591234\pi$
$29$	−74262.0	−0.565423	−0.282712	+	0.959205i	$0.591234\pi$
$30$	0	0
$31$	−275680.	−1.66203	−0.831016	−	0.556249i	$-0.812240\pi$
$31$	−275680.	−1.66203	−0.831016	+	0.556249i	$0.812240\pi$
$32$	0	0
$33$	102924.	0.498560
$34$	0	0
$35$	−55440.0	−0.218567
$36$	0	0
$37$	266610.	0.865307	0.432654	−	0.901560i	$-0.357577\pi$
$37$	266610.	0.865307	0.432654	+	0.901560i	$0.357577\pi$
$38$	0	0
$39$	258498.	0.697800
$40$	0	0
$41$	684762.	1.55166	0.775829	−	0.630943i	$-0.217332\pi$
$41$	684762.	1.55166	0.775829	+	0.630943i	$0.217332\pi$
$42$	0	0
$43$	−245956.	−0.471756	−0.235878	−	0.971783i	$-0.575797\pi$
$43$	−245956.	−0.471756	−0.235878	+	0.971783i	$0.575797\pi$
$44$	0	0
$45$	−80190.0	−0.131183
$46$	0	0
$47$	478800.	0.672685	0.336342	−	0.941740i	$-0.390810\pi$
$47$	478800.	0.672685	0.336342	+	0.941740i	$0.390810\pi$
$48$	0	0
$49$	−569527.	−0.691557
$50$	0	0
$51$	−704646.	−0.743833
$52$	0	0
$53$	569410.	0.525363	0.262682	−	0.964883i	$-0.415393\pi$
$53$	569410.	0.525363	0.262682	+	0.964883i	$0.415393\pi$
$54$	0	0
$55$	419320.	0.339841
$56$	0	0
$57$	−1.03432e6	−0.739761
$58$	0	0
$59$	1.52532e6	0.966897	0.483448	−	0.875373i	$-0.339384\pi$
$59$	1.52532e6	0.966897	0.483448	+	0.875373i	$0.339384\pi$
$60$	0	0
$61$	2.64046e6	1.48945	0.744723	−	0.667374i	$-0.232582\pi$
$61$	2.64046e6	1.48945	0.744723	+	0.667374i	$0.232582\pi$
$62$	0	0
$63$	367416.	0.185125
$64$	0	0
$65$	1.05314e6	0.475652
$66$	0	0
$67$	−1.41624e6	−0.575273	−0.287636	−	0.957740i	$-0.592869\pi$
$67$	−1.41624e6	−0.575273	−0.287636	+	0.957740i	$0.592869\pi$
$68$	0	0
$69$	1.92046e6	0.703773
$70$	0	0
$71$	−3.51130e6	−1.16430	−0.582149	−	0.813082i	$-0.697788\pi$
$71$	−3.51130e6	−1.16430	−0.582149	+	0.813082i	$0.697788\pi$
$72$	0	0
$73$	4.73862e6	1.42568	0.712839	−	0.701327i	$-0.247409\pi$
$73$	4.73862e6	1.42568	0.712839	+	0.701327i	$0.247409\pi$
$74$	0	0
$75$	1.78268e6	0.487930
$76$	0	0
$77$	−1.92125e6	−0.479585
$78$	0	0
$79$	4.66149e6	1.06373	0.531863	−	0.846830i	$-0.321492\pi$
$79$	4.66149e6	1.06373	0.531863	+	0.846830i	$0.321492\pi$
$80$	0	0
$81$	531441.	0.111111
$82$	0	0
$83$	5.72925e6	1.09983	0.549914	−	0.835221i	$-0.314661\pi$
$83$	5.72925e6	1.09983	0.549914	+	0.835221i	$0.314661\pi$
$84$	0	0
$85$	−2.87078e6	−0.507030
$86$	0	0
$87$	2.00507e6	0.326447
$88$	0	0
$89$	1.19935e7	1.80336	0.901678	−	0.432408i	$-0.142336\pi$
$89$	1.19935e7	1.80336	0.901678	+	0.432408i	$0.142336\pi$
$90$	0	0
$91$	−4.82530e6	−0.671242
$92$	0	0
$93$	7.44336e6	0.959575
$94$	0	0
$95$	−4.21388e6	−0.504254
$96$	0	0
$97$	7.15075e6	0.795519	0.397760	−	0.917490i	$-0.369788\pi$
$97$	7.15075e6	0.795519	0.397760	+	0.917490i	$0.369788\pi$
$98$	0	0
$99$	−2.77895e6	−0.287844
$100$	0	0
$101$	8.78373e6	0.848309	0.424155	−	0.905590i	$-0.360571\pi$
$101$	8.78373e6	0.848309	0.424155	+	0.905590i	$0.360571\pi$
$102$	0	0
$103$	−8.01610e6	−0.722825	−0.361412	−	0.932406i	$-0.617705\pi$
$103$	−8.01610e6	−0.722825	−0.361412	+	0.932406i	$0.617705\pi$
$104$	0	0
$105$	1.49688e6	0.126190
$106$	0	0
$107$	5.15123e6	0.406507	0.203253	−	0.979126i	$-0.434849\pi$
$107$	5.15123e6	0.406507	0.203253	+	0.979126i	$0.434849\pi$
$108$	0	0
$109$	2.41280e7	1.78455	0.892274	−	0.451493i	$-0.149109\pi$
$109$	2.41280e7	1.78455	0.892274	+	0.451493i	$0.149109\pi$
$110$	0	0
$111$	−7.19847e6	−0.499585
$112$	0	0
$113$	2.04827e6	0.133541	0.0667703	−	0.997768i	$-0.478731\pi$
$113$	2.04827e6	0.133541	0.0667703	+	0.997768i	$0.478731\pi$
$114$	0	0
$115$	7.82408e6	0.479723
$116$	0	0
$117$	−6.97945e6	−0.402875
$118$	0	0
$119$	1.31534e7	0.715523
$120$	0	0
$121$	−4.95583e6	−0.254312
$122$	0	0
$123$	−1.84886e7	−0.895850
$124$	0	0
$125$	1.58565e7	0.726143
$126$	0	0
$127$	1.36634e6	0.0591895	0.0295947	−	0.999562i	$-0.490578\pi$
$127$	1.36634e6	0.0591895	0.0295947	+	0.999562i	$0.490578\pi$
$128$	0	0
$129$	6.64081e6	0.272369
$130$	0	0
$131$	−3.84645e7	−1.49489	−0.747447	−	0.664321i	$-0.768721\pi$
$131$	−3.84645e7	−1.49489	−0.747447	+	0.664321i	$0.768721\pi$
$132$	0	0
$133$	1.93072e7	0.711605
$134$	0	0
$135$	2.16513e6	0.0757383
$136$	0	0
$137$	7.62585e6	0.253376	0.126688	−	0.991943i	$-0.459565\pi$
$137$	7.62585e6	0.253376	0.126688	+	0.991943i	$0.459565\pi$
$138$	0	0
$139$	−5.32324e6	−0.168122	−0.0840609	−	0.996461i	$-0.526789\pi$
$139$	−5.32324e6	−0.168122	−0.0840609	+	0.996461i	$0.526789\pi$
$140$	0	0
$141$	−1.29276e7	−0.388375
$142$	0	0
$143$	3.64961e7	1.04369
$144$	0	0
$145$	8.16882e6	0.222521
$146$	0	0
$147$	1.53772e7	0.399271
$148$	0	0
$149$	−7.61366e6	−0.188557	−0.0942783	−	0.995546i	$-0.530054\pi$
$149$	−7.61366e6	−0.188557	−0.0942783	+	0.995546i	$0.530054\pi$
$150$	0	0
$151$	−2.50221e7	−0.591432	−0.295716	−	0.955276i	$-0.595558\pi$
$151$	−2.50221e7	−0.591432	−0.295716	+	0.955276i	$0.595558\pi$
$152$	0	0
$153$	1.90254e7	0.429452
$154$	0	0
$155$	3.03248e7	0.654089
$156$	0	0
$157$	−3.93145e7	−0.810782	−0.405391	−	0.914143i	$-0.632865\pi$
$157$	−3.93145e7	−0.810782	−0.405391	+	0.914143i	$0.632865\pi$
$158$	0	0
$159$	−1.53741e7	−0.303319
$160$	0	0
$161$	−3.58485e7	−0.676987
$162$	0	0
$163$	6.28387e7	1.13650	0.568252	−	0.822855i	$-0.307620\pi$
$163$	6.28387e7	1.13650	0.568252	+	0.822855i	$0.307620\pi$
$164$	0	0
$165$	−1.13216e7	−0.196207
$166$	0	0
$167$	1.04133e7	0.173014	0.0865072	−	0.996251i	$-0.472429\pi$
$167$	1.04133e7	0.173014	0.0865072	+	0.996251i	$0.472429\pi$
$168$	0	0
$169$	2.89130e7	0.460775
$170$	0	0
$171$	2.79265e7	0.427101
$172$	0	0
$173$	8.03551e7	1.17992	0.589959	−	0.807433i	$-0.299144\pi$
$173$	8.03551e7	1.17992	0.589959	+	0.807433i	$0.299144\pi$
$174$	0	0
$175$	−3.32766e7	−0.469360
$176$	0	0
$177$	−4.11837e7	−0.558238
$178$	0	0
$179$	8.40084e7	1.09481	0.547403	−	0.836869i	$-0.315617\pi$
$179$	8.40084e7	1.09481	0.547403	+	0.836869i	$0.315617\pi$
$180$	0	0
$181$	−1.15469e8	−1.44741	−0.723703	−	0.690112i	$-0.757561\pi$
$181$	−1.15469e8	−1.44741	−0.723703	+	0.690112i	$0.757561\pi$
$182$	0	0
$183$	−7.12924e7	−0.859932
$184$	0	0
$185$	−2.93271e7	−0.340540
$186$	0	0
$187$	−9.94856e7	−1.11254
$188$	0	0
$189$	−9.92023e6	−0.106882
$190$	0	0
$191$	9.97154e7	1.03549	0.517744	−	0.855535i	$-0.326772\pi$
$191$	9.97154e7	1.03549	0.517744	+	0.855535i	$0.326772\pi$
$192$	0	0
$193$	−1.86157e7	−0.186393	−0.0931965	−	0.995648i	$-0.529708\pi$
$193$	−1.86157e7	−0.186393	−0.0931965	+	0.995648i	$0.529708\pi$
$194$	0	0
$195$	−2.84348e7	−0.274618
$196$	0	0
$197$	−9.30384e7	−0.867022	−0.433511	−	0.901148i	$-0.642726\pi$
$197$	−9.30384e7	−0.867022	−0.433511	+	0.901148i	$0.642726\pi$
$198$	0	0
$199$	7.39686e7	0.665367	0.332684	−	0.943038i	$-0.392046\pi$
$199$	7.39686e7	0.665367	0.332684	+	0.943038i	$0.392046\pi$
$200$	0	0
$201$	3.82384e7	0.332134
$202$	0	0
$203$	−3.74280e7	−0.314023
$204$	0	0
$205$	−7.53238e7	−0.610652
$206$	0	0
$207$	−5.18523e7	−0.406323
$208$	0	0
$209$	−1.46030e8	−1.10645
$210$	0	0
$211$	−1.85163e8	−1.35695	−0.678476	−	0.734623i	$-0.737359\pi$
$211$	−1.85163e8	−1.35695	−0.678476	+	0.734623i	$0.737359\pi$
$212$	0	0
$213$	9.48052e7	0.672208
$214$	0	0
$215$	2.70552e7	0.185659
$216$	0	0
$217$	−1.38943e8	−0.923053
$218$	0	0
$219$	−1.27943e8	−0.823116
$220$	0	0
$221$	−2.49862e8	−1.55714
$222$	0	0
$223$	−1.20862e8	−0.729830	−0.364915	−	0.931041i	$-0.618902\pi$
$223$	−1.20862e8	−0.729830	−0.364915	+	0.931041i	$0.618902\pi$
$224$	0	0
$225$	−4.81322e7	−0.281707
$226$	0	0
$227$	2.82315e8	1.60193	0.800966	−	0.598710i	$-0.204320\pi$
$227$	2.82315e8	1.60193	0.800966	+	0.598710i	$0.204320\pi$
$228$	0	0
$229$	8.91913e7	0.490793	0.245397	−	0.969423i	$-0.421082\pi$
$229$	8.91913e7	0.490793	0.245397	+	0.969423i	$0.421082\pi$
$230$	0	0
$231$	5.18737e7	0.276889
$232$	0	0
$233$	−2.32240e8	−1.20279	−0.601396	−	0.798951i	$-0.705389\pi$
$233$	−2.32240e8	−1.20279	−0.601396	+	0.798951i	$0.705389\pi$
$234$	0	0
$235$	−5.26680e7	−0.264734
$236$	0	0
$237$	−1.25860e8	−0.614142
$238$	0	0
$239$	3.21986e8	1.52561	0.762807	−	0.646626i	$-0.223821\pi$
$239$	3.21986e8	1.52561	0.762807	+	0.646626i	$0.223821\pi$
$240$	0	0
$241$	−2.00366e8	−0.922072	−0.461036	−	0.887381i	$-0.652522\pi$
$241$	−2.00366e8	−0.922072	−0.461036	+	0.887381i	$0.652522\pi$
$242$	0	0
$243$	−1.43489e7	−0.0641500
$244$	0	0
$245$	6.26480e7	0.272161
$246$	0	0
$247$	−3.66761e8	−1.54862
$248$	0	0
$249$	−1.54690e8	−0.634986
$250$	0	0
$251$	8.70560e7	0.347489	0.173744	−	0.984791i	$-0.444413\pi$
$251$	8.70560e7	0.347489	0.173744	+	0.984791i	$0.444413\pi$
$252$	0	0
$253$	2.71140e8	1.05262
$254$	0	0
$255$	7.75111e7	0.292734
$256$	0	0
$257$	5.22879e8	1.92148	0.960738	−	0.277457i	$-0.0894915\pi$
$257$	5.22879e8	1.92148	0.960738	+	0.277457i	$0.0894915\pi$
$258$	0	0
$259$	1.34371e8	0.480571
$260$	0	0
$261$	−5.41370e7	−0.188474
$262$	0	0
$263$	−4.06215e8	−1.37693	−0.688464	−	0.725270i	$-0.741715\pi$
$263$	−4.06215e8	−1.37693	−0.688464	+	0.725270i	$0.741715\pi$
$264$	0	0
$265$	−6.26351e7	−0.206756
$266$	0	0
$267$	−3.23825e8	−1.04117
$268$	0	0
$269$	−3.82347e8	−1.19764	−0.598818	−	0.800885i	$-0.704363\pi$
$269$	−3.82347e8	−1.19764	−0.598818	+	0.800885i	$0.704363\pi$
$270$	0	0
$271$	2.84165e8	0.867317	0.433658	−	0.901077i	$-0.357222\pi$
$271$	2.84165e8	0.867317	0.433658	+	0.901077i	$0.357222\pi$
$272$	0	0
$273$	1.30283e8	0.387542
$274$	0	0
$275$	2.51687e8	0.729788
$276$	0	0
$277$	−2.93752e8	−0.830427	−0.415213	−	0.909724i	$-0.636293\pi$
$277$	−2.93752e8	−0.830427	−0.415213	+	0.909724i	$0.636293\pi$
$278$	0	0
$279$	−2.00971e8	−0.554011
$280$	0	0
$281$	4.15399e8	1.11685	0.558424	−	0.829556i	$-0.311406\pi$
$281$	4.15399e8	1.11685	0.558424	+	0.829556i	$0.311406\pi$
$282$	0	0
$283$	−5.06429e7	−0.132821	−0.0664104	−	0.997792i	$-0.521155\pi$
$283$	−5.06429e7	−0.132821	−0.0664104	+	0.997792i	$0.521155\pi$
$284$	0	0
$285$	1.13775e8	0.291131
$286$	0	0
$287$	3.45120e8	0.861754
$288$	0	0
$289$	2.70767e8	0.659862
$290$	0	0
$291$	−1.93070e8	−0.459293
$292$	0	0
$293$	−7.47714e7	−0.173660	−0.0868298	−	0.996223i	$-0.527674\pi$
$293$	−7.47714e7	−0.173660	−0.0868298	+	0.996223i	$0.527674\pi$
$294$	0	0
$295$	−1.67786e8	−0.380520
$296$	0	0
$297$	7.50316e7	0.166187
$298$	0	0
$299$	6.80979e8	1.47328
$300$	0	0
$301$	−1.23962e8	−0.262002
$302$	0	0
$303$	−2.37161e8	−0.489772
$304$	0	0
$305$	−2.90450e8	−0.586169
$306$	0	0
$307$	−8.52577e7	−0.168170	−0.0840851	−	0.996459i	$-0.526797\pi$
$307$	−8.52577e7	−0.168170	−0.0840851	+	0.996459i	$0.526797\pi$
$308$	0	0
$309$	2.16435e8	0.417323
$310$	0	0
$311$	9.39129e8	1.77037	0.885184	−	0.465240i	$-0.154032\pi$
$311$	9.39129e8	1.77037	0.885184	+	0.465240i	$0.154032\pi$
$312$	0	0
$313$	−3.43040e8	−0.632323	−0.316162	−	0.948705i	$-0.602394\pi$
$313$	−3.43040e8	−0.632323	−0.316162	+	0.948705i	$0.602394\pi$
$314$	0	0
$315$	−4.04158e7	−0.0728557
$316$	0	0
$317$	1.03960e9	1.83298	0.916492	−	0.400054i	$-0.131009\pi$
$317$	1.03960e9	1.83298	0.916492	+	0.400054i	$0.131009\pi$
$318$	0	0
$319$	2.83087e8	0.488261
$320$	0	0
$321$	−1.39083e8	−0.234697
$322$	0	0
$323$	9.99762e8	1.65077
$324$	0	0
$325$	6.32123e8	1.02143
$326$	0	0
$327$	−6.51456e8	−1.03031
$328$	0	0
$329$	2.41315e8	0.373593
$330$	0	0
$331$	−1.10022e9	−1.66756	−0.833779	−	0.552098i	$-0.813827\pi$
$331$	−1.10022e9	−1.66756	−0.833779	+	0.552098i	$0.813827\pi$
$332$	0	0
$333$	1.94359e8	0.288436
$334$	0	0
$335$	1.55786e8	0.226397
$336$	0	0
$337$	1.28272e9	1.82569	0.912847	−	0.408302i	$-0.133879\pi$
$337$	1.28272e9	1.82569	0.912847	+	0.408302i	$0.133879\pi$
$338$	0	0
$339$	−5.53034e7	−0.0770997
$340$	0	0
$341$	1.05089e9	1.43522
$342$	0	0
$343$	−7.02107e8	−0.939451
$344$	0	0
$345$	−2.11250e8	−0.276968
$346$	0	0
$347$	−1.25822e9	−1.61660	−0.808301	−	0.588770i	$-0.799612\pi$
$347$	−1.25822e9	−1.61660	−0.808301	+	0.588770i	$0.799612\pi$
$348$	0	0
$349$	−1.35371e8	−0.170465	−0.0852327	−	0.996361i	$-0.527163\pi$
$349$	−1.35371e8	−0.170465	−0.0852327	+	0.996361i	$0.527163\pi$
$350$	0	0
$351$	1.88445e8	0.232600
$352$	0	0
$353$	−8.49221e8	−1.02756	−0.513782	−	0.857921i	$-0.671756\pi$
$353$	−8.49221e8	−1.02756	−0.513782	+	0.857921i	$0.671756\pi$
$354$	0	0
$355$	3.86243e8	0.458207
$356$	0	0
$357$	−3.55142e8	−0.413107
$358$	0	0
$359$	2.20121e8	0.251091	0.125546	−	0.992088i	$-0.459932\pi$
$359$	2.20121e8	0.251091	0.125546	+	0.992088i	$0.459932\pi$
$360$	0	0
$361$	5.73631e8	0.641738
$362$	0	0
$363$	1.33807e8	0.146827
$364$	0	0
$365$	−5.21248e8	−0.561073
$366$	0	0
$367$	4.66505e8	0.492635	0.246317	−	0.969189i	$-0.420779\pi$
$367$	4.66505e8	0.492635	0.246317	+	0.969189i	$0.420779\pi$
$368$	0	0
$369$	4.99191e8	0.517220
$370$	0	0
$371$	2.86983e8	0.291774
$372$	0	0
$373$	2.98453e8	0.297780	0.148890	−	0.988854i	$-0.452430\pi$
$373$	2.98453e8	0.297780	0.148890	+	0.988854i	$0.452430\pi$
$374$	0	0
$375$	−4.28126e8	−0.419239
$376$	0	0
$377$	7.10984e8	0.683385
$378$	0	0
$379$	−1.46218e9	−1.37964	−0.689818	−	0.723983i	$-0.742309\pi$
$379$	−1.46218e9	−1.37964	−0.689818	+	0.723983i	$0.742309\pi$
$380$	0	0
$381$	−3.68911e7	−0.0341731
$382$	0	0
$383$	1.58702e9	1.44340	0.721700	−	0.692206i	$-0.243361\pi$
$383$	1.58702e9	1.44340	0.721700	+	0.692206i	$0.243361\pi$
$384$	0	0
$385$	2.11337e8	0.188740
$386$	0	0
$387$	−1.79302e8	−0.157252
$388$	0	0
$389$	3.14439e8	0.270840	0.135420	−	0.990788i	$-0.456762\pi$
$389$	3.14439e8	0.270840	0.135420	+	0.990788i	$0.456762\pi$
$390$	0	0
$391$	−1.85630e9	−1.57047
$392$	0	0
$393$	1.03854e9	0.863078
$394$	0	0
$395$	−5.12764e8	−0.418627
$396$	0	0
$397$	8.52757e8	0.684004	0.342002	−	0.939699i	$-0.388895\pi$
$397$	8.52757e8	0.684004	0.342002	+	0.939699i	$0.388895\pi$
$398$	0	0
$399$	−5.21295e8	−0.410846
$400$	0	0
$401$	6.92522e8	0.536325	0.268163	−	0.963374i	$-0.413584\pi$
$401$	6.92522e8	0.536325	0.268163	+	0.963374i	$0.413584\pi$
$402$	0	0
$403$	2.63936e9	2.00877
$404$	0	0
$405$	−5.84585e7	−0.0437276
$406$	0	0
$407$	−1.01632e9	−0.747221
$408$	0	0
$409$	6.17357e8	0.446174	0.223087	−	0.974799i	$-0.428387\pi$
$409$	6.17357e8	0.446174	0.223087	+	0.974799i	$0.428387\pi$
$410$	0	0
$411$	−2.05898e8	−0.146287
$412$	0	0
$413$	7.68763e8	0.536992
$414$	0	0
$415$	−6.30218e8	−0.432835
$416$	0	0
$417$	1.43727e8	0.0970651
$418$	0	0
$419$	1.65512e9	1.09921	0.549604	−	0.835425i	$-0.314779\pi$
$419$	1.65512e9	1.09921	0.549604	+	0.835425i	$0.314779\pi$
$420$	0	0
$421$	−7.01472e7	−0.0458166	−0.0229083	−	0.999738i	$-0.507293\pi$
$421$	−7.01472e7	−0.0458166	−0.0229083	+	0.999738i	$0.507293\pi$
$422$	0	0
$423$	3.49045e8	0.224228
$424$	0	0
$425$	−1.72312e9	−1.08882
$426$	0	0
$427$	1.33079e9	0.827203
$428$	0	0
$429$	−9.85394e8	−0.602573
$430$	0	0
$431$	1.81387e9	1.09128	0.545640	−	0.838020i	$-0.316287\pi$
$431$	1.81387e9	1.09128	0.545640	+	0.838020i	$0.316287\pi$
$432$	0	0
$433$	−2.59970e9	−1.53892	−0.769460	−	0.638695i	$-0.779475\pi$
$433$	−2.59970e9	−1.53892	−0.769460	+	0.638695i	$0.779475\pi$
$434$	0	0
$435$	−2.20558e8	−0.128473
$436$	0	0
$437$	−2.72477e9	−1.56187
$438$	0	0
$439$	−1.67431e9	−0.944517	−0.472258	−	0.881460i	$-0.656561\pi$
$439$	−1.67431e9	−0.944517	−0.472258	+	0.881460i	$0.656561\pi$
$440$	0	0
$441$	−4.15185e8	−0.230519
$442$	0	0
$443$	2.52711e8	0.138105	0.0690527	−	0.997613i	$-0.478002\pi$
$443$	2.52711e8	0.138105	0.0690527	+	0.997613i	$0.478002\pi$
$444$	0	0
$445$	−1.31929e9	−0.709707
$446$	0	0
$447$	2.05569e8	0.108863
$448$	0	0
$449$	7.55311e8	0.393789	0.196895	−	0.980425i	$-0.436914\pi$
$449$	7.55311e8	0.393789	0.196895	+	0.980425i	$0.436914\pi$
$450$	0	0
$451$	−2.61031e9	−1.33991
$452$	0	0
$453$	6.75597e8	0.341463
$454$	0	0
$455$	5.30783e8	0.264166
$456$	0	0
$457$	−1.51584e8	−0.0742928	−0.0371464	−	0.999310i	$-0.511827\pi$
$457$	−1.51584e8	−0.0742928	−0.0371464	+	0.999310i	$0.511827\pi$
$458$	0	0
$459$	−5.13687e8	−0.247944
$460$	0	0
$461$	7.78405e8	0.370043	0.185022	−	0.982734i	$-0.440764\pi$
$461$	7.78405e8	0.370043	0.185022	+	0.982734i	$0.440764\pi$
$462$	0	0
$463$	2.41052e9	1.12870	0.564349	−	0.825536i	$-0.309127\pi$
$463$	2.41052e9	1.12870	0.564349	+	0.825536i	$0.309127\pi$
$464$	0	0
$465$	−8.18770e8	−0.377639
$466$	0	0
$467$	1.76192e9	0.800527	0.400264	−	0.916400i	$-0.368919\pi$
$467$	1.76192e9	0.800527	0.400264	+	0.916400i	$0.368919\pi$
$468$	0	0
$469$	−7.13783e8	−0.319493
$470$	0	0
$471$	1.06149e9	0.468105
$472$	0	0
$473$	9.37584e8	0.407377
$474$	0	0
$475$	−2.52929e9	−1.08285
$476$	0	0
$477$	4.15100e8	0.175121
$478$	0	0
$479$	6.43811e8	0.267661	0.133830	−	0.991004i	$-0.457272\pi$
$479$	6.43811e8	0.267661	0.133830	+	0.991004i	$0.457272\pi$
$480$	0	0
$481$	−2.55252e9	−1.04583
$482$	0	0
$483$	9.67910e8	0.390859
$484$	0	0
$485$	−7.86583e8	−0.313075
$486$	0	0
$487$	3.16421e9	1.24141	0.620704	−	0.784045i	$-0.286847\pi$
$487$	3.16421e9	1.24141	0.620704	+	0.784045i	$0.286847\pi$
$488$	0	0
$489$	−1.69665e9	−0.656160
$490$	0	0
$491$	−3.62406e9	−1.38169	−0.690844	−	0.723004i	$-0.742761\pi$
$491$	−3.62406e9	−1.38169	−0.690844	+	0.723004i	$0.742761\pi$
$492$	0	0
$493$	−1.93809e9	−0.728467
$494$	0	0
$495$	3.05684e8	0.113280
$496$	0	0
$497$	−1.76970e9	−0.646624
$498$	0	0
$499$	1.35483e9	0.488128	0.244064	−	0.969759i	$-0.421519\pi$
$499$	1.35483e9	0.488128	0.244064	+	0.969759i	$0.421519\pi$
$500$	0	0
$501$	−2.81160e8	−0.0998899
$502$	0	0
$503$	−4.66389e9	−1.63403	−0.817015	−	0.576616i	$-0.804373\pi$
$503$	−4.66389e9	−1.63403	−0.817015	+	0.576616i	$0.804373\pi$
$504$	0	0
$505$	−9.66210e8	−0.333850
$506$	0	0
$507$	−7.80650e8	−0.266029
$508$	0	0
$509$	1.34292e9	0.451376	0.225688	−	0.974200i	$-0.427537\pi$
$509$	1.34292e9	0.451376	0.225688	+	0.974200i	$0.427537\pi$
$510$	0	0
$511$	2.38826e9	0.791788
$512$	0	0
$513$	−7.54016e8	−0.246587
$514$	0	0
$515$	8.81771e8	0.284466
$516$	0	0
$517$	−1.82519e9	−0.580885
$518$	0	0
$519$	−2.16959e9	−0.681226
$520$	0	0
$521$	−1.45400e9	−0.450435	−0.225217	−	0.974309i	$-0.572309\pi$
$521$	−1.45400e9	−0.450435	−0.225217	+	0.974309i	$0.572309\pi$
$522$	0	0
$523$	−4.90309e8	−0.149870	−0.0749349	−	0.997188i	$-0.523875\pi$
$523$	−4.90309e8	−0.149870	−0.0749349	+	0.997188i	$0.523875\pi$
$524$	0	0
$525$	8.98468e8	0.270985
$526$	0	0
$527$	−7.19470e9	−2.14129
$528$	0	0
$529$	1.65437e9	0.485889
$530$	0	0
$531$	1.11196e9	0.322299
$532$	0	0
$533$	−6.55591e9	−1.87537
$534$	0	0
$535$	−5.66635e8	−0.159980
$536$	0	0
$537$	−2.26823e9	−0.632086
$538$	0	0
$539$	2.17104e9	0.597182
$540$	0	0
$541$	4.82889e9	1.31116	0.655582	−	0.755124i	$-0.272423\pi$
$541$	4.82889e9	1.31116	0.655582	+	0.755124i	$0.272423\pi$
$542$	0	0
$543$	3.11766e9	0.835660
$544$	0	0
$545$	−2.65408e9	−0.702306
$546$	0	0
$547$	3.08793e9	0.806698	0.403349	−	0.915046i	$-0.367846\pi$
$547$	3.08793e9	0.806698	0.403349	+	0.915046i	$0.367846\pi$
$548$	0	0
$549$	1.92489e9	0.496482
$550$	0	0
$551$	−2.84483e9	−0.724479
$552$	0	0
$553$	2.34939e9	0.590768
$554$	0	0
$555$	7.91832e8	0.196611
$556$	0	0
$557$	−4.09889e9	−1.00502	−0.502508	−	0.864573i	$-0.667589\pi$
$557$	−4.09889e9	−1.00502	−0.502508	+	0.864573i	$0.667589\pi$
$558$	0	0
$559$	2.35478e9	0.570177
$560$	0	0
$561$	2.68611e9	0.642324
$562$	0	0
$563$	−4.97105e9	−1.17400	−0.587001	−	0.809587i	$-0.699691\pi$
$563$	−4.97105e9	−1.17400	−0.587001	+	0.809587i	$0.699691\pi$
$564$	0	0
$565$	−2.25310e8	−0.0525547
$566$	0	0
$567$	2.67846e8	0.0617085
$568$	0	0
$569$	3.71316e9	0.844988	0.422494	−	0.906366i	$-0.361155\pi$
$569$	3.71316e9	0.844988	0.422494	+	0.906366i	$0.361155\pi$
$570$	0	0
$571$	2.36205e9	0.530961	0.265481	−	0.964116i	$-0.414469\pi$
$571$	2.36205e9	0.530961	0.265481	+	0.964116i	$0.414469\pi$
$572$	0	0
$573$	−2.69232e9	−0.597840
$574$	0	0
$575$	4.69623e9	1.03018
$576$	0	0
$577$	−1.81146e8	−0.0392566	−0.0196283	−	0.999807i	$-0.506248\pi$
$577$	−1.81146e8	−0.0392566	−0.0196283	+	0.999807i	$0.506248\pi$
$578$	0	0
$579$	5.02625e8	0.107614
$580$	0	0
$581$	2.88754e9	0.610818
$582$	0	0
$583$	−2.17059e9	−0.453668
$584$	0	0
$585$	7.67739e8	0.158551
$586$	0	0
$587$	−2.31976e9	−0.473380	−0.236690	−	0.971585i	$-0.576063\pi$
$587$	−2.31976e9	−0.473380	−0.236690	+	0.971585i	$0.576063\pi$
$588$	0	0
$589$	−1.05607e10	−2.12957
$590$	0	0
$591$	2.51204e9	0.500576
$592$	0	0
$593$	2.27806e9	0.448615	0.224308	−	0.974518i	$-0.427988\pi$
$593$	2.27806e9	0.448615	0.224308	+	0.974518i	$0.427988\pi$
$594$	0	0
$595$	−1.44687e9	−0.281592
$596$	0	0
$597$	−1.99715e9	−0.384150
$598$	0	0
$599$	−4.88253e9	−0.928220	−0.464110	−	0.885778i	$-0.653626\pi$
$599$	−4.88253e9	−0.928220	−0.464110	+	0.885778i	$0.653626\pi$
$600$	0	0
$601$	−6.74758e9	−1.26791	−0.633954	−	0.773371i	$-0.718569\pi$
$601$	−6.74758e9	−1.26791	−0.633954	+	0.773371i	$0.718569\pi$
$602$	0	0
$603$	−1.03244e9	−0.191758
$604$	0	0
$605$	5.45141e8	0.100084
$606$	0	0
$607$	−9.05928e9	−1.64412	−0.822060	−	0.569401i	$-0.807175\pi$
$607$	−9.05928e9	−1.64412	−0.822060	+	0.569401i	$0.807175\pi$
$608$	0	0
$609$	1.01056e9	0.181301
$610$	0	0
$611$	−4.58403e9	−0.813024
$612$	0	0
$613$	8.48777e9	1.48827	0.744135	−	0.668029i	$-0.232862\pi$
$613$	8.48777e9	1.48827	0.744135	+	0.668029i	$0.232862\pi$
$614$	0	0
$615$	2.03374e9	0.352560
$616$	0	0
$617$	2.34407e9	0.401766	0.200883	−	0.979615i	$-0.435619\pi$
$617$	2.34407e9	0.401766	0.200883	+	0.979615i	$0.435619\pi$
$618$	0	0
$619$	1.01541e9	0.172077	0.0860384	−	0.996292i	$-0.472579\pi$
$619$	1.01541e9	0.172077	0.0860384	+	0.996292i	$0.472579\pi$
$620$	0	0
$621$	1.40001e9	0.234591
$622$	0	0
$623$	6.04473e9	1.00154
$624$	0	0
$625$	3.41399e9	0.559348
$626$	0	0
$627$	3.94281e9	0.638807
$628$	0	0
$629$	6.95799e9	1.11482
$630$	0	0
$631$	7.01911e9	1.11219	0.556095	−	0.831119i	$-0.312299\pi$
$631$	7.01911e9	1.11219	0.556095	+	0.831119i	$0.312299\pi$
$632$	0	0
$633$	4.99939e9	0.783437
$634$	0	0
$635$	−1.50297e8	−0.0232939
$636$	0	0
$637$	5.45265e9	0.835833
$638$	0	0
$639$	−2.55974e9	−0.388099
$640$	0	0
$641$	−4.52776e9	−0.679016	−0.339508	−	0.940603i	$-0.610261\pi$
$641$	−4.52776e9	−0.679016	−0.339508	+	0.940603i	$0.610261\pi$
$642$	0	0
$643$	8.63094e9	1.28032	0.640162	−	0.768240i	$-0.278867\pi$
$643$	8.63094e9	1.28032	0.640162	+	0.768240i	$0.278867\pi$
$644$	0	0
$645$	−7.30489e8	−0.107190
$646$	0	0
$647$	2.57401e9	0.373632	0.186816	−	0.982395i	$-0.440183\pi$
$647$	2.57401e9	0.373632	0.186816	+	0.982395i	$0.440183\pi$
$648$	0	0
$649$	−5.81454e9	−0.834946
$650$	0	0
$651$	3.75145e9	0.532925
$652$	0	0
$653$	9.31827e9	1.30960	0.654800	−	0.755802i	$-0.272753\pi$
$653$	9.31827e9	1.30960	0.654800	+	0.755802i	$0.272753\pi$
$654$	0	0
$655$	4.23109e9	0.588313
$656$	0	0
$657$	3.45445e9	0.475226
$658$	0	0
$659$	−1.04422e10	−1.42133	−0.710663	−	0.703532i	$-0.751605\pi$
$659$	−1.04422e10	−1.42133	−0.710663	+	0.703532i	$0.751605\pi$
$660$	0	0
$661$	1.04761e10	1.41090	0.705449	−	0.708761i	$-0.250746\pi$
$661$	1.04761e10	1.41090	0.705449	+	0.708761i	$0.250746\pi$
$662$	0	0
$663$	6.74628e9	0.899015
$664$	0	0
$665$	−2.12380e9	−0.280051
$666$	0	0
$667$	5.28211e9	0.689234
$668$	0	0
$669$	3.26327e9	0.421368
$670$	0	0
$671$	−1.00654e10	−1.28618
$672$	0	0
$673$	1.38891e10	1.75639	0.878197	−	0.478299i	$-0.158746\pi$
$673$	1.38891e10	1.75639	0.878197	+	0.478299i	$0.158746\pi$
$674$	0	0
$675$	1.29957e9	0.162643
$676$	0	0
$677$	7.48893e8	0.0927598	0.0463799	−	0.998924i	$-0.485232\pi$
$677$	7.48893e8	0.0927598	0.0463799	+	0.998924i	$0.485232\pi$
$678$	0	0
$679$	3.60398e9	0.441813
$680$	0	0
$681$	−7.62251e9	−0.924875
$682$	0	0
$683$	−1.15581e10	−1.38808	−0.694038	−	0.719938i	$-0.744170\pi$
$683$	−1.15581e10	−1.38808	−0.694038	+	0.719938i	$0.744170\pi$
$684$	0	0
$685$	−8.38844e8	−0.0997158
$686$	0	0
$687$	−2.40817e9	−0.283360
$688$	0	0
$689$	−5.45153e9	−0.634967
$690$	0	0
$691$	−3.34337e8	−0.0385489	−0.0192744	−	0.999814i	$-0.506136\pi$
$691$	−3.34337e8	−0.0385489	−0.0192744	+	0.999814i	$0.506136\pi$
$692$	0	0
$693$	−1.40059e9	−0.159862
$694$	0	0
$695$	5.85556e8	0.0661640
$696$	0	0
$697$	1.78709e10	1.99909
$698$	0	0
$699$	6.27047e9	0.694433
$700$	0	0
$701$	5.55383e9	0.608948	0.304474	−	0.952521i	$-0.401519\pi$
$701$	5.55383e9	0.608948	0.304474	+	0.952521i	$0.401519\pi$
$702$	0	0
$703$	1.02133e10	1.10872
$704$	0	0
$705$	1.42204e9	0.152844
$706$	0	0
$707$	4.42700e9	0.471131
$708$	0	0
$709$	−1.30817e10	−1.37849	−0.689243	−	0.724530i	$-0.742057\pi$
$709$	−1.30817e10	−1.37849	−0.689243	+	0.724530i	$0.742057\pi$
$710$	0	0
$711$	3.39822e9	0.354575
$712$	0	0
$713$	1.96086e10	2.02597
$714$	0	0
$715$	−4.01457e9	−0.410741
$716$	0	0
$717$	−8.69363e9	−0.880814
$718$	0	0
$719$	−1.10847e10	−1.11217	−0.556085	−	0.831125i	$-0.687697\pi$
$719$	−1.10847e10	−1.11217	−0.556085	+	0.831125i	$0.687697\pi$
$720$	0	0
$721$	−4.04012e9	−0.401440
$722$	0	0
$723$	5.40989e9	0.532358
$724$	0	0
$725$	4.90315e9	0.477850
$726$	0	0
$727$	−7.79416e9	−0.752314	−0.376157	−	0.926556i	$-0.622755\pi$
$727$	−7.79416e9	−0.752314	−0.376157	+	0.926556i	$0.622755\pi$
$728$	0	0
$729$	3.87420e8	0.0370370
$730$	0	0
$731$	−6.41896e9	−0.607790
$732$	0	0
$733$	−6.83552e9	−0.641073	−0.320537	−	0.947236i	$-0.603863\pi$
$733$	−6.83552e9	−0.641073	−0.320537	+	0.947236i	$0.603863\pi$
$734$	0	0
$735$	−1.69150e9	−0.157132
$736$	0	0
$737$	5.39869e9	0.496767
$738$	0	0
$739$	−1.73862e10	−1.58471	−0.792356	−	0.610059i	$-0.791146\pi$
$739$	−1.73862e10	−1.58471	−0.792356	+	0.610059i	$0.791146\pi$
$740$	0	0
$741$	9.90254e9	0.894094
$742$	0	0
$743$	2.25537e9	0.201724	0.100862	−	0.994900i	$-0.467840\pi$
$743$	2.25537e9	0.201724	0.100862	+	0.994900i	$0.467840\pi$
$744$	0	0
$745$	8.37503e8	0.0742061
$746$	0	0
$747$	4.17662e9	0.366609
$748$	0	0
$749$	2.59622e9	0.225764
$750$	0	0
$751$	−2.05027e10	−1.76632	−0.883162	−	0.469068i	$-0.844590\pi$
$751$	−2.05027e10	−1.76632	−0.883162	+	0.469068i	$0.844590\pi$
$752$	0	0
$753$	−2.35051e9	−0.200623
$754$	0	0
$755$	2.75243e9	0.232757
$756$	0	0
$757$	2.57872e8	0.0216057	0.0108029	−	0.999942i	$-0.496561\pi$
$757$	2.57872e8	0.0216057	0.0108029	+	0.999942i	$0.496561\pi$
$758$	0	0
$759$	−7.32078e9	−0.607731
$760$	0	0
$761$	−1.34452e10	−1.10591	−0.552957	−	0.833210i	$-0.686501\pi$
$761$	−1.34452e10	−1.10591	−0.552957	+	0.833210i	$0.686501\pi$
$762$	0	0
$763$	1.21605e10	0.991096
$764$	0	0
$765$	−2.09280e9	−0.169010
$766$	0	0
$767$	−1.46035e10	−1.16862
$768$	0	0
$769$	8.28541e9	0.657009	0.328505	−	0.944502i	$-0.393455\pi$
$769$	8.28541e9	0.657009	0.328505	+	0.944502i	$0.393455\pi$
$770$	0	0
$771$	−1.41177e10	−1.10936
$772$	0	0
$773$	−1.43430e10	−1.11689	−0.558447	−	0.829540i	$-0.688603\pi$
$773$	−1.43430e10	−1.11689	−0.558447	+	0.829540i	$0.688603\pi$
$774$	0	0
$775$	1.82018e10	1.40462
$776$	0	0
$777$	−3.62803e9	−0.277458
$778$	0	0
$779$	2.62319e10	1.98814
$780$	0	0
$781$	1.33851e10	1.00541
$782$	0	0
$783$	1.46170e9	0.108816
$784$	0	0
$785$	4.32460e9	0.319082
$786$	0	0
$787$	3.83137e9	0.280184	0.140092	−	0.990139i	$-0.455260\pi$
$787$	3.83137e9	0.280184	0.140092	+	0.990139i	$0.455260\pi$
$788$	0	0
$789$	1.09678e10	0.794970
$790$	0	0
$791$	1.03233e9	0.0741653
$792$	0	0
$793$	−2.52797e10	−1.80018
$794$	0	0
$795$	1.69115e9	0.119370
$796$	0	0
$797$	1.95859e9	0.137038	0.0685188	−	0.997650i	$-0.478173\pi$
$797$	1.95859e9	0.137038	0.0685188	+	0.997650i	$0.478173\pi$
$798$	0	0
$799$	1.24957e10	0.866658
$800$	0	0
$801$	8.74327e9	0.601119
$802$	0	0
$803$	−1.80636e10	−1.23112
$804$	0	0
$805$	3.94334e9	0.266427
$806$	0	0
$807$	1.03234e10	0.691455
$808$	0	0
$809$	2.07415e9	0.137727	0.0688637	−	0.997626i	$-0.478063\pi$
$809$	2.07415e9	0.137727	0.0688637	+	0.997626i	$0.478063\pi$
$810$	0	0
$811$	5.71508e9	0.376227	0.188113	−	0.982147i	$-0.439763\pi$
$811$	5.71508e9	0.376227	0.188113	+	0.982147i	$0.439763\pi$
$812$	0	0
$813$	−7.67245e9	−0.500746
$814$	0	0
$815$	−6.91226e9	−0.447268
$816$	0	0
$817$	−9.42208e9	−0.604463
$818$	0	0
$819$	−3.51764e9	−0.223747
$820$	0	0
$821$	−2.82748e10	−1.78319	−0.891596	−	0.452832i	$-0.850414\pi$
$821$	−2.82748e10	−1.78319	−0.891596	+	0.452832i	$0.850414\pi$
$822$	0	0
$823$	−2.09283e9	−0.130868	−0.0654342	−	0.997857i	$-0.520843\pi$
$823$	−2.09283e9	−0.130868	−0.0654342	+	0.997857i	$0.520843\pi$
$824$	0	0
$825$	−6.79556e9	−0.421343
$826$	0	0
$827$	3.71453e9	0.228368	0.114184	−	0.993460i	$-0.463575\pi$
$827$	3.71453e9	0.228368	0.114184	+	0.993460i	$0.463575\pi$
$828$	0	0
$829$	−3.37924e9	−0.206005	−0.103003	−	0.994681i	$-0.532845\pi$
$829$	−3.37924e9	−0.206005	−0.103003	+	0.994681i	$0.532845\pi$
$830$	0	0
$831$	7.93130e9	0.479447
$832$	0	0
$833$	−1.48635e10	−0.890972
$834$	0	0
$835$	−1.14547e9	−0.0680895
$836$	0	0
$837$	5.42621e9	0.319858
$838$	0	0
$839$	1.64907e10	0.963990	0.481995	−	0.876174i	$-0.339912\pi$
$839$	1.64907e10	0.963990	0.481995	+	0.876174i	$0.339912\pi$
$840$	0	0
$841$	−1.17350e10	−0.680297
$842$	0	0
$843$	−1.12158e10	−0.644812
$844$	0	0
$845$	−3.18043e9	−0.181337
$846$	0	0
$847$	−2.49774e9	−0.141239
$848$	0	0
$849$	1.36736e9	0.0766841
$850$	0	0
$851$	−1.89634e10	−1.05478
$852$	0	0
$853$	4.77028e9	0.263162	0.131581	−	0.991305i	$-0.457995\pi$
$853$	4.77028e9	0.263162	0.131581	+	0.991305i	$0.457995\pi$
$854$	0	0
$855$	−3.07192e9	−0.168085
$856$	0	0
$857$	1.61514e10	0.876554	0.438277	−	0.898840i	$-0.355589\pi$
$857$	1.61514e10	0.876554	0.438277	+	0.898840i	$0.355589\pi$
$858$	0	0
$859$	3.41593e8	0.0183879	0.00919397	−	0.999958i	$-0.497073\pi$
$859$	3.41593e8	0.0183879	0.00919397	+	0.999958i	$0.497073\pi$
$860$	0	0
$861$	−9.31824e9	−0.497534
$862$	0	0
$863$	6.07878e9	0.321943	0.160971	−	0.986959i	$-0.448537\pi$
$863$	6.07878e9	0.321943	0.160971	+	0.986959i	$0.448537\pi$
$864$	0	0
$865$	−8.83906e9	−0.464354
$866$	0	0
$867$	−7.31071e9	−0.380972
$868$	0	0
$869$	−1.77696e10	−0.918562
$870$	0	0
$871$	1.35590e10	0.695289
$872$	0	0
$873$	5.21290e9	0.265173
$874$	0	0
$875$	7.99168e9	0.403283
$876$	0	0
$877$	−1.23852e10	−0.620020	−0.310010	−	0.950733i	$-0.600332\pi$
$877$	−1.23852e10	−0.620020	−0.310010	+	0.950733i	$0.600332\pi$
$878$	0	0
$879$	2.01883e9	0.100262
$880$	0	0
$881$	−1.37801e10	−0.678949	−0.339475	−	0.940615i	$-0.610249\pi$
$881$	−1.37801e10	−0.678949	−0.339475	+	0.940615i	$0.610249\pi$
$882$	0	0
$883$	−1.89296e9	−0.0925292	−0.0462646	−	0.998929i	$-0.514732\pi$
$883$	−1.89296e9	−0.0925292	−0.0462646	+	0.998929i	$0.514732\pi$
$884$	0	0
$885$	4.53021e9	0.219693
$886$	0	0
$887$	4.23912e9	0.203959	0.101979	−	0.994787i	$-0.467482\pi$
$887$	4.23912e9	0.203959	0.101979	+	0.994787i	$0.467482\pi$
$888$	0	0
$889$	6.88633e8	0.0328724
$890$	0	0
$891$	−2.02585e9	−0.0959480
$892$	0	0
$893$	1.83419e10	0.861913
$894$	0	0
$895$	−9.24092e9	−0.430859
$896$	0	0
$897$	−1.83864e10	−0.850598
$898$	0	0
$899$	2.04725e10	0.939751
$900$	0	0
$901$	1.48605e10	0.676855
$902$	0	0
$903$	3.34697e9	0.151267
$904$	0	0
$905$	1.27016e10	0.569623
$906$	0	0
$907$	9.51367e9	0.423372	0.211686	−	0.977338i	$-0.432105\pi$
$907$	9.51367e9	0.423372	0.211686	+	0.977338i	$0.432105\pi$
$908$	0	0
$909$	6.40334e9	0.282770
$910$	0	0
$911$	−1.16235e10	−0.509359	−0.254680	−	0.967025i	$-0.581970\pi$
$911$	−1.16235e10	−0.509359	−0.254680	+	0.967025i	$0.581970\pi$
$912$	0	0
$913$	−2.18399e10	−0.949736
$914$	0	0
$915$	7.84216e9	0.338425
$916$	0	0
$917$	−1.93861e10	−0.830229
$918$	0	0
$919$	−9.22943e9	−0.392257	−0.196128	−	0.980578i	$-0.562837\pi$
$919$	−9.22943e9	−0.392257	−0.196128	+	0.980578i	$0.562837\pi$
$920$	0	0
$921$	2.30196e9	0.0970931
$922$	0	0
$923$	3.36172e10	1.40720
$924$	0	0
$925$	−1.76029e10	−0.731289
$926$	0	0
$927$	−5.84374e9	−0.240942
$928$	0	0
$929$	−4.09353e10	−1.67511	−0.837555	−	0.546353i	$-0.816016\pi$
$929$	−4.09353e10	−1.67511	−0.837555	+	0.546353i	$0.816016\pi$
$930$	0	0
$931$	−2.18174e10	−0.886094
$932$	0	0
$933$	−2.53565e10	−1.02212
$934$	0	0
$935$	1.09434e10	0.437837
$936$	0	0
$937$	−1.75085e9	−0.0695281	−0.0347641	−	0.999396i	$-0.511068\pi$
$937$	−1.75085e9	−0.0695281	−0.0347641	+	0.999396i	$0.511068\pi$
$938$	0	0
$939$	9.26207e9	0.365072
$940$	0	0
$941$	−5.91102e9	−0.231259	−0.115630	−	0.993292i	$-0.536889\pi$
$941$	−5.91102e9	−0.231259	−0.115630	+	0.993292i	$0.536889\pi$
$942$	0	0
$943$	−4.87058e10	−1.89143
$944$	0	0
$945$	1.09123e9	0.0420633
$946$	0	0
$947$	2.26089e10	0.865077	0.432538	−	0.901616i	$-0.357618\pi$
$947$	2.26089e10	0.865077	0.432538	+	0.901616i	$0.357618\pi$
$948$	0	0
$949$	−4.53675e10	−1.72311
$950$	0	0
$951$	−2.80692e10	−1.05827
$952$	0	0
$953$	1.11773e10	0.418322	0.209161	−	0.977881i	$-0.432927\pi$
$953$	1.11773e10	0.418322	0.209161	+	0.977881i	$0.432927\pi$
$954$	0	0
$955$	−1.09687e10	−0.407515
$956$	0	0
$957$	−7.64334e9	−0.281898
$958$	0	0
$959$	3.84343e9	0.140719
$960$	0	0
$961$	4.84868e10	1.76235
$962$	0	0
$963$	3.75525e9	0.135502
$964$	0	0
$965$	2.04773e9	0.0733546
$966$	0	0
$967$	1.55518e10	0.553078	0.276539	−	0.961003i	$-0.410812\pi$
$967$	1.55518e10	0.553078	0.276539	+	0.961003i	$0.410812\pi$
$968$	0	0
$969$	−2.69936e10	−0.953075
$970$	0	0
$971$	−8.34508e9	−0.292525	−0.146263	−	0.989246i	$-0.546724\pi$
$971$	−8.34508e9	−0.292525	−0.146263	+	0.989246i	$0.546724\pi$
$972$	0	0
$973$	−2.68291e9	−0.0933709
$974$	0	0
$975$	−1.70673e10	−0.589725
$976$	0	0
$977$	−9.85180e9	−0.337975	−0.168988	−	0.985618i	$-0.554050\pi$
$977$	−9.85180e9	−0.337975	−0.168988	+	0.985618i	$0.554050\pi$
$978$	0	0
$979$	−4.57193e10	−1.55726
$980$	0	0
$981$	1.75893e10	0.594850
$982$	0	0
$983$	3.70884e10	1.24538	0.622688	−	0.782470i	$-0.286041\pi$
$983$	3.70884e10	1.24538	0.622688	+	0.782470i	$0.286041\pi$
$984$	0	0
$985$	1.02342e10	0.341215
$986$	0	0
$987$	−6.51551e9	−0.215694
$988$	0	0
$989$	1.74944e10	0.575057
$990$	0	0
$991$	6.43526e9	0.210043	0.105022	−	0.994470i	$-0.466509\pi$
$991$	6.43526e9	0.210043	0.105022	+	0.994470i	$0.466509\pi$
$992$	0	0
$993$	2.97059e10	0.962765
$994$	0	0
$995$	−8.13654e9	−0.261854
$996$	0	0
$997$	1.23071e10	0.393299	0.196650	−	0.980474i	$-0.436994\pi$
$997$	1.23071e10	0.393299	0.196650	+	0.980474i	$0.436994\pi$
$998$	0	0
$999$	−5.24768e9	−0.166528

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	192.8.a.c.1.1		1
3.2	odd	2		576.8.a.t.1.1		1
4.3	odd	2		192.8.a.k.1.1		1
8.3	odd	2		48.8.a.c.1.1		1
8.5	even	2		24.8.a.c.1.1	✓	1
12.11	even	2		576.8.a.s.1.1		1
24.5	odd	2		72.8.a.b.1.1		1
24.11	even	2		144.8.a.d.1.1		1
40.13	odd	4		600.8.f.d.49.2		2
40.29	even	2		600.8.a.a.1.1		1
40.37	odd	4		600.8.f.d.49.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
24.8.a.c.1.1	✓	1	8.5	even	2
48.8.a.c.1.1		1	8.3	odd	2
72.8.a.b.1.1		1	24.5	odd	2
144.8.a.d.1.1		1	24.11	even	2
192.8.a.c.1.1		1	1.1	even	1	trivial
192.8.a.k.1.1		1	4.3	odd	2
576.8.a.s.1.1		1	12.11	even	2
576.8.a.t.1.1		1	3.2	odd	2
600.8.a.a.1.1		1	40.29	even	2
600.8.f.d.49.1		2	40.37	odd	4
600.8.f.d.49.2		2	40.13	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:	\( N \)	\(=\)	\( 192 = 2^{6} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	192.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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