LMFDB - Embedded newform 8.10.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8,10,Mod(1,8)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("8.1");

S:= CuspForms(chi, 10);

N := Newforms(S);

Level:	$ N $	$=$	$ 8 = 2^{3} $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	8.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:	$4.12028668931$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	8.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+68.0000 q^{3} +1510.00 q^{5} +10248.0 q^{7} -15059.0 q^{9} +O(q^{10})$

$q+68.0000 q^{3} +1510.00 q^{5} +10248.0 q^{7} -15059.0 q^{9} +3916.00 q^{11} -176594. q^{13} +102680. q^{15} +148370. q^{17} +499796. q^{19} +696864. q^{21} -1.88977e6 q^{23} +326975. q^{25} -2.36246e6 q^{27} -920898. q^{29} +1.37936e6 q^{31} +266288. q^{33} +1.54745e7 q^{35} +5.06497e6 q^{37} -1.20084e7 q^{39} -2.41008e7 q^{41} +2.57852e7 q^{43} -2.27391e7 q^{45} -6.07902e7 q^{47} +6.46679e7 q^{49} +1.00892e7 q^{51} +2.94962e7 q^{53} +5.91316e6 q^{55} +3.39861e7 q^{57} +5.18194e7 q^{59} +3.34269e7 q^{61} -1.54325e8 q^{63} -2.66657e8 q^{65} +1.44856e8 q^{67} -1.28504e8 q^{69} +6.83971e7 q^{71} +1.68216e8 q^{73} +2.22343e7 q^{75} +4.01312e7 q^{77} +2.35399e8 q^{79} +1.35759e8 q^{81} -6.46399e7 q^{83} +2.24039e8 q^{85} -6.26211e7 q^{87} -7.87827e7 q^{89} -1.80974e9 q^{91} +9.37965e7 q^{93} +7.54692e8 q^{95} -2.41136e7 q^{97} -5.89710e7 q^{99} +O(q^{100})$

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
$3$	68.0000	0.484689	0.242345	−	0.970190i	$-0.422084\pi$
$3$	68.0000	0.484689	0.242345	+	0.970190i	$0.422084\pi$
$4$	0	0
$5$	1510.00	1.08047	0.540234	−	0.841515i	$-0.318336\pi$
$5$	1510.00	1.08047	0.540234	+	0.841515i	$0.318336\pi$
$6$	0	0
$7$	10248.0	1.61324	0.806618	−	0.591073i	$-0.201295\pi$
$7$	10248.0	1.61324	0.806618	+	0.591073i	$0.201295\pi$
$8$	0	0
$9$	−15059.0	−0.765076
$10$	0	0
$11$	3916.00	0.0806447	0.0403223	−	0.999187i	$-0.487162\pi$
$11$	3916.00	0.0806447	0.0403223	+	0.999187i	$0.487162\pi$
$12$	0	0
$13$	−176594.	−1.71487	−0.857434	−	0.514593i	$-0.827943\pi$
$13$	−176594.	−1.71487	−0.857434	+	0.514593i	$0.827943\pi$
$14$	0	0
$15$	102680.	0.523691
$16$	0	0
$17$	148370.	0.430850	0.215425	−	0.976520i	$-0.430886\pi$
$17$	148370.	0.430850	0.215425	+	0.976520i	$0.430886\pi$
$18$	0	0
$19$	499796.	0.879836	0.439918	−	0.898038i	$-0.355008\pi$
$19$	499796.	0.879836	0.439918	+	0.898038i	$0.355008\pi$
$20$	0	0
$21$	696864.	0.781918
$22$	0	0
$23$	−1.88977e6	−1.40810	−0.704050	−	0.710151i	$-0.748627\pi$
$23$	−1.88977e6	−1.40810	−0.704050	+	0.710151i	$0.748627\pi$
$24$	0	0
$25$	326975.	0.167411
$26$	0	0
$27$	−2.36246e6	−0.855513
$28$	0	0
$29$	−920898.	−0.241780	−0.120890	−	0.992666i	$-0.538575\pi$
$29$	−920898.	−0.241780	−0.120890	+	0.992666i	$0.538575\pi$
$30$	0	0
$31$	1.37936e6	0.268256	0.134128	−	0.990964i	$-0.457177\pi$
$31$	1.37936e6	0.268256	0.134128	+	0.990964i	$0.457177\pi$
$32$	0	0
$33$	266288.	0.0390876
$34$	0	0
$35$	1.54745e7	1.74305
$36$	0	0
$37$	5.06497e6	0.444292	0.222146	−	0.975013i	$-0.428694\pi$
$37$	5.06497e6	0.444292	0.222146	+	0.975013i	$0.428694\pi$
$38$	0	0
$39$	−1.20084e7	−0.831178
$40$	0	0
$41$	−2.41008e7	−1.33200	−0.665999	−	0.745953i	$-0.731994\pi$
$41$	−2.41008e7	−1.33200	−0.665999	+	0.745953i	$0.731994\pi$
$42$	0	0
$43$	2.57852e7	1.15017	0.575085	−	0.818093i	$-0.304969\pi$
$43$	2.57852e7	1.15017	0.575085	+	0.818093i	$0.304969\pi$
$44$	0	0
$45$	−2.27391e7	−0.826641
$46$	0	0
$47$	−6.07902e7	−1.81716	−0.908580	−	0.417710i	$-0.862833\pi$
$47$	−6.07902e7	−1.81716	−0.908580	+	0.417710i	$0.862833\pi$
$48$	0	0
$49$	6.46679e7	1.60253
$50$	0	0
$51$	1.00892e7	0.208828
$52$	0	0
$53$	2.94962e7	0.513482	0.256741	−	0.966480i	$-0.417351\pi$
$53$	2.94962e7	0.513482	0.256741	+	0.966480i	$0.417351\pi$
$54$	0	0
$55$	5.91316e6	0.0871340
$56$	0	0
$57$	3.39861e7	0.426447
$58$	0	0
$59$	5.18194e7	0.556747	0.278374	−	0.960473i	$-0.410205\pi$
$59$	5.18194e7	0.556747	0.278374	+	0.960473i	$0.410205\pi$
$60$	0	0
$61$	3.34269e7	0.309109	0.154555	−	0.987984i	$-0.450606\pi$
$61$	3.34269e7	0.309109	0.154555	+	0.987984i	$0.450606\pi$
$62$	0	0
$63$	−1.54325e8	−1.23425
$64$	0	0
$65$	−2.66657e8	−1.85286
$66$	0	0
$67$	1.44856e8	0.878214	0.439107	−	0.898435i	$-0.355295\pi$
$67$	1.44856e8	0.878214	0.439107	+	0.898435i	$0.355295\pi$
$68$	0	0
$69$	−1.28504e8	−0.682490
$70$	0	0
$71$	6.83971e7	0.319430	0.159715	−	0.987163i	$-0.448943\pi$
$71$	6.83971e7	0.319430	0.159715	+	0.987163i	$0.448943\pi$
$72$	0	0
$73$	1.68216e8	0.693290	0.346645	−	0.937996i	$-0.387321\pi$
$73$	1.68216e8	0.693290	0.346645	+	0.937996i	$0.387321\pi$
$74$	0	0
$75$	2.22343e7	0.0811424
$76$	0	0
$77$	4.01312e7	0.130099
$78$	0	0
$79$	2.35399e8	0.679958	0.339979	−	0.940433i	$-0.389580\pi$
$79$	2.35399e8	0.679958	0.339979	+	0.940433i	$0.389580\pi$
$80$	0	0
$81$	1.35759e8	0.350418
$82$	0	0
$83$	−6.46399e7	−0.149503	−0.0747513	−	0.997202i	$-0.523816\pi$
$83$	−6.46399e7	−0.149503	−0.0747513	+	0.997202i	$0.523816\pi$
$84$	0	0
$85$	2.24039e8	0.465520
$86$	0	0
$87$	−6.26211e7	−0.117188
$88$	0	0
$89$	−7.87827e7	−0.133099	−0.0665497	−	0.997783i	$-0.521199\pi$
$89$	−7.87827e7	−0.133099	−0.0665497	+	0.997783i	$0.521199\pi$
$90$	0	0
$91$	−1.80974e9	−2.76649
$92$	0	0
$93$	9.37965e7	0.130021
$94$	0	0
$95$	7.54692e8	0.950634
$96$	0	0
$97$	−2.41136e7	−0.0276560	−0.0138280	−	0.999904i	$-0.504402\pi$
$97$	−2.41136e7	−0.0276560	−0.0138280	+	0.999904i	$0.504402\pi$
$98$	0	0
$99$	−5.89710e7	−0.0616993
$100$	0	0
$101$	−6.25963e8	−0.598553	−0.299277	−	0.954166i	$-0.596745\pi$
$101$	−6.25963e8	−0.598553	−0.299277	+	0.954166i	$0.596745\pi$
$102$	0	0
$103$	8.00618e8	0.700902	0.350451	−	0.936581i	$-0.386028\pi$
$103$	8.00618e8	0.700902	0.350451	+	0.936581i	$0.386028\pi$
$104$	0	0
$105$	1.05226e9	0.844837
$106$	0	0
$107$	2.45358e9	1.80956	0.904781	−	0.425877i	$-0.140034\pi$
$107$	2.45358e9	1.80956	0.904781	+	0.425877i	$0.140034\pi$
$108$	0	0
$109$	−9.29043e8	−0.630400	−0.315200	−	0.949025i	$-0.602072\pi$
$109$	−9.29043e8	−0.630400	−0.315200	+	0.949025i	$0.602072\pi$
$110$	0	0
$111$	3.44418e8	0.215344
$112$	0	0
$113$	−1.65129e9	−0.952731	−0.476366	−	0.879247i	$-0.658046\pi$
$113$	−1.65129e9	−0.952731	−0.476366	+	0.879247i	$0.658046\pi$
$114$	0	0
$115$	−2.85355e9	−1.52141
$116$	0	0
$117$	2.65933e9	1.31201
$118$	0	0
$119$	1.52050e9	0.695063
$120$	0	0
$121$	−2.34261e9	−0.993496
$122$	0	0
$123$	−1.63885e9	−0.645605
$124$	0	0
$125$	−2.45549e9	−0.899586
$126$	0	0
$127$	−1.83042e9	−0.624358	−0.312179	−	0.950023i	$-0.601059\pi$
$127$	−1.83042e9	−0.624358	−0.312179	+	0.950023i	$0.601059\pi$
$128$	0	0
$129$	1.75339e9	0.557475
$130$	0	0
$131$	5.60254e8	0.166213	0.0831064	−	0.996541i	$-0.473516\pi$
$131$	5.60254e8	0.166213	0.0831064	+	0.996541i	$0.473516\pi$
$132$	0	0
$133$	5.12191e9	1.41938
$134$	0	0
$135$	−3.56731e9	−0.924355
$136$	0	0
$137$	6.54232e9	1.58668	0.793340	−	0.608778i	$-0.208340\pi$
$137$	6.54232e9	1.58668	0.793340	+	0.608778i	$0.208340\pi$
$138$	0	0
$139$	5.52722e9	1.25586	0.627929	−	0.778271i	$-0.283903\pi$
$139$	5.52722e9	1.25586	0.627929	+	0.778271i	$0.283903\pi$
$140$	0	0
$141$	−4.13374e9	−0.880758
$142$	0	0
$143$	−6.91542e8	−0.138295
$144$	0	0
$145$	−1.39056e9	−0.261236
$146$	0	0
$147$	4.39742e9	0.776729
$148$	0	0
$149$	−3.25329e9	−0.540734	−0.270367	−	0.962757i	$-0.587145\pi$
$149$	−3.25329e9	−0.540734	−0.270367	+	0.962757i	$0.587145\pi$
$150$	0	0
$151$	−8.54419e9	−1.33744	−0.668721	−	0.743514i	$-0.733158\pi$
$151$	−8.54419e9	−1.33744	−0.668721	+	0.743514i	$0.733158\pi$
$152$	0	0
$153$	−2.23430e9	−0.329633
$154$	0	0
$155$	2.08283e9	0.289842
$156$	0	0
$157$	−2.69871e8	−0.0354493	−0.0177246	−	0.999843i	$-0.505642\pi$
$157$	−2.69871e8	−0.0354493	−0.0177246	+	0.999843i	$0.505642\pi$
$158$	0	0
$159$	2.00574e9	0.248879
$160$	0	0
$161$	−1.93663e10	−2.27160
$162$	0	0
$163$	1.02903e10	1.14178	0.570892	−	0.821025i	$-0.306597\pi$
$163$	1.02903e10	1.14178	0.570892	+	0.821025i	$0.306597\pi$
$164$	0	0
$165$	4.02095e8	0.0422329
$166$	0	0
$167$	1.23946e8	0.0123313	0.00616565	−	0.999981i	$-0.498037\pi$
$167$	1.23946e8	0.0123313	0.00616565	+	0.999981i	$0.498037\pi$
$168$	0	0
$169$	2.05809e10	1.94077
$170$	0	0
$171$	−7.52643e9	−0.673142
$172$	0	0
$173$	4.92770e9	0.418250	0.209125	−	0.977889i	$-0.432938\pi$
$173$	4.92770e9	0.418250	0.209125	+	0.977889i	$0.432938\pi$
$174$	0	0
$175$	3.35084e9	0.270074
$176$	0	0
$177$	3.52372e9	0.269849
$178$	0	0
$179$	−5.54564e9	−0.403751	−0.201875	−	0.979411i	$-0.564704\pi$
$179$	−5.54564e9	−0.403751	−0.201875	+	0.979411i	$0.564704\pi$
$180$	0	0
$181$	1.16150e10	0.804391	0.402196	−	0.915554i	$-0.368247\pi$
$181$	1.16150e10	0.804391	0.402196	+	0.915554i	$0.368247\pi$
$182$	0	0
$183$	2.27303e9	0.149822
$184$	0	0
$185$	7.64810e9	0.480044
$186$	0	0
$187$	5.81017e8	0.0347457
$188$	0	0
$189$	−2.42104e10	−1.38015
$190$	0	0
$191$	−2.72404e10	−1.48103	−0.740514	−	0.672041i	$-0.765418\pi$
$191$	−2.72404e10	−1.48103	−0.740514	+	0.672041i	$0.765418\pi$
$192$	0	0
$193$	−2.88743e10	−1.49797	−0.748987	−	0.662585i	$-0.769459\pi$
$193$	−2.88743e10	−1.49797	−0.748987	+	0.662585i	$0.769459\pi$
$194$	0	0
$195$	−1.81327e10	−0.898061
$196$	0	0
$197$	3.80169e10	1.79837	0.899184	−	0.437571i	$-0.144161\pi$
$197$	3.80169e10	1.79837	0.899184	+	0.437571i	$0.144161\pi$
$198$	0	0
$199$	2.57386e10	1.16344	0.581722	−	0.813387i	$-0.302379\pi$
$199$	2.57386e10	1.16344	0.581722	+	0.813387i	$0.302379\pi$
$200$	0	0
$201$	9.85022e9	0.425661
$202$	0	0
$203$	−9.43736e9	−0.390048
$204$	0	0
$205$	−3.63921e10	−1.43918
$206$	0	0
$207$	2.84580e10	1.07730
$208$	0	0
$209$	1.95720e9	0.0709540
$210$	0	0
$211$	5.50064e9	0.191048	0.0955239	−	0.995427i	$-0.469547\pi$
$211$	5.50064e9	0.191048	0.0955239	+	0.995427i	$0.469547\pi$
$212$	0	0
$213$	4.65100e9	0.154824
$214$	0	0
$215$	3.89356e10	1.24272
$216$	0	0
$217$	1.41357e10	0.432761
$218$	0	0
$219$	1.14387e10	0.336030
$220$	0	0
$221$	−2.62013e10	−0.738851
$222$	0	0
$223$	2.05983e10	0.557774	0.278887	−	0.960324i	$-0.410034\pi$
$223$	2.05983e10	0.557774	0.278887	+	0.960324i	$0.410034\pi$
$224$	0	0
$225$	−4.92392e9	−0.128082
$226$	0	0
$227$	−4.46842e10	−1.11696	−0.558480	−	0.829518i	$-0.688615\pi$
$227$	−4.46842e10	−1.11696	−0.558480	+	0.829518i	$0.688615\pi$
$228$	0	0
$229$	−5.69323e9	−0.136804	−0.0684020	−	0.997658i	$-0.521790\pi$
$229$	−5.69323e9	−0.136804	−0.0684020	+	0.997658i	$0.521790\pi$
$230$	0	0
$231$	2.72892e9	0.0630575
$232$	0	0
$233$	1.74032e10	0.386837	0.193419	−	0.981116i	$-0.438042\pi$
$233$	1.74032e10	0.386837	0.193419	+	0.981116i	$0.438042\pi$
$234$	0	0
$235$	−9.17932e10	−1.96338
$236$	0	0
$237$	1.60071e10	0.329568
$238$	0	0
$239$	−3.35988e10	−0.666090	−0.333045	−	0.942911i	$-0.608076\pi$
$239$	−3.35988e10	−0.666090	−0.333045	+	0.942911i	$0.608076\pi$
$240$	0	0
$241$	−6.08619e10	−1.16217	−0.581084	−	0.813844i	$-0.697371\pi$
$241$	−6.08619e10	−1.16217	−0.581084	+	0.813844i	$0.697371\pi$
$242$	0	0
$243$	5.57319e10	1.02536
$244$	0	0
$245$	9.76485e10	1.73148
$246$	0	0
$247$	−8.82610e10	−1.50880
$248$	0	0
$249$	−4.39551e9	−0.0724623
$250$	0	0
$251$	8.74389e10	1.39051	0.695253	−	0.718765i	$-0.255292\pi$
$251$	8.74389e10	1.39051	0.695253	+	0.718765i	$0.255292\pi$
$252$	0	0
$253$	−7.40033e9	−0.113556
$254$	0	0
$255$	1.52346e10	0.225632
$256$	0	0
$257$	7.11368e10	1.01717	0.508587	−	0.861011i	$-0.330168\pi$
$257$	7.11368e10	1.01717	0.508587	+	0.861011i	$0.330168\pi$
$258$	0	0
$259$	5.19058e10	0.716748
$260$	0	0
$261$	1.38678e10	0.184980
$262$	0	0
$263$	−5.24741e10	−0.676308	−0.338154	−	0.941091i	$-0.609802\pi$
$263$	−5.24741e10	−0.676308	−0.338154	+	0.941091i	$0.609802\pi$
$264$	0	0
$265$	4.45393e10	0.554800
$266$	0	0
$267$	−5.35722e9	−0.0645118
$268$	0	0
$269$	−1.37810e11	−1.60470	−0.802350	−	0.596853i	$-0.796417\pi$
$269$	−1.37810e11	−1.60470	−0.802350	+	0.596853i	$0.796417\pi$
$270$	0	0
$271$	1.18786e11	1.33784	0.668921	−	0.743333i	$-0.266756\pi$
$271$	1.18786e11	1.33784	0.668921	+	0.743333i	$0.266756\pi$
$272$	0	0
$273$	−1.23062e11	−1.34089
$274$	0	0
$275$	1.28043e9	0.0135008
$276$	0	0
$277$	−6.20716e10	−0.633481	−0.316741	−	0.948512i	$-0.602588\pi$
$277$	−6.20716e10	−0.633481	−0.316741	+	0.948512i	$0.602588\pi$
$278$	0	0
$279$	−2.07718e10	−0.205237
$280$	0	0
$281$	7.20927e10	0.689784	0.344892	−	0.938642i	$-0.387916\pi$
$281$	7.20927e10	0.689784	0.344892	+	0.938642i	$0.387916\pi$
$282$	0	0
$283$	3.90151e10	0.361571	0.180786	−	0.983523i	$-0.442136\pi$
$283$	3.90151e10	0.361571	0.180786	+	0.983523i	$0.442136\pi$
$284$	0	0
$285$	5.13191e10	0.460762
$286$	0	0
$287$	−2.46985e11	−2.14883
$288$	0	0
$289$	−9.65742e10	−0.814368
$290$	0	0
$291$	−1.63972e9	−0.0134045
$292$	0	0
$293$	−9.37214e10	−0.742907	−0.371454	−	0.928451i	$-0.621141\pi$
$293$	−9.37214e10	−0.742907	−0.371454	+	0.928451i	$0.621141\pi$
$294$	0	0
$295$	7.82473e10	0.601548
$296$	0	0
$297$	−9.25138e9	−0.0689926
$298$	0	0
$299$	3.33722e11	2.41471
$300$	0	0
$301$	2.64247e11	1.85550
$302$	0	0
$303$	−4.25655e10	−0.290112
$304$	0	0
$305$	5.04746e10	0.333983
$306$	0	0
$307$	−1.63239e11	−1.04882	−0.524411	−	0.851465i	$-0.675715\pi$
$307$	−1.63239e11	−1.04882	−0.524411	+	0.851465i	$0.675715\pi$
$308$	0	0
$309$	5.44420e10	0.339720
$310$	0	0
$311$	−5.54012e10	−0.335813	−0.167906	−	0.985803i	$-0.553701\pi$
$311$	−5.54012e10	−0.335813	−0.167906	+	0.985803i	$0.553701\pi$
$312$	0	0
$313$	2.35841e11	1.38889	0.694447	−	0.719544i	$-0.255649\pi$
$313$	2.35841e11	1.38889	0.694447	+	0.719544i	$0.255649\pi$
$314$	0	0
$315$	−2.33030e11	−1.33357
$316$	0	0
$317$	−7.74070e10	−0.430540	−0.215270	−	0.976555i	$-0.569063\pi$
$317$	−7.74070e10	−0.430540	−0.215270	+	0.976555i	$0.569063\pi$
$318$	0	0
$319$	−3.60624e9	−0.0194983
$320$	0	0
$321$	1.66844e11	0.877075
$322$	0	0
$323$	7.41547e10	0.379077
$324$	0	0
$325$	−5.77418e10	−0.287088
$326$	0	0
$327$	−6.31749e10	−0.305548
$328$	0	0
$329$	−6.22978e11	−2.93151
$330$	0	0
$331$	−7.90091e10	−0.361785	−0.180893	−	0.983503i	$-0.557899\pi$
$331$	−7.90091e10	−0.361785	−0.180893	+	0.983503i	$0.557899\pi$
$332$	0	0
$333$	−7.62733e10	−0.339918
$334$	0	0
$335$	2.18733e11	0.948882
$336$	0	0
$337$	2.15728e11	0.911114	0.455557	−	0.890207i	$-0.349440\pi$
$337$	2.15728e11	0.911114	0.455557	+	0.890207i	$0.349440\pi$
$338$	0	0
$339$	−1.12288e11	−0.461778
$340$	0	0
$341$	5.40157e9	0.0216334
$342$	0	0
$343$	2.49173e11	0.972024
$344$	0	0
$345$	−1.94041e11	−0.737409
$346$	0	0
$347$	−2.38976e11	−0.884854	−0.442427	−	0.896804i	$-0.645882\pi$
$347$	−2.38976e11	−0.884854	−0.442427	+	0.896804i	$0.645882\pi$
$348$	0	0
$349$	5.14491e9	0.0185637	0.00928183	−	0.999957i	$-0.497045\pi$
$349$	5.14491e9	0.0185637	0.00928183	+	0.999957i	$0.497045\pi$
$350$	0	0
$351$	4.17196e11	1.46709
$352$	0	0
$353$	5.71172e10	0.195786	0.0978928	−	0.995197i	$-0.468790\pi$
$353$	5.71172e10	0.195786	0.0978928	+	0.995197i	$0.468790\pi$
$354$	0	0
$355$	1.03280e11	0.345134
$356$	0	0
$357$	1.03394e11	0.336889
$358$	0	0
$359$	2.89030e11	0.918371	0.459185	−	0.888340i	$-0.348141\pi$
$359$	2.89030e11	0.918371	0.459185	+	0.888340i	$0.348141\pi$
$360$	0	0
$361$	−7.28917e10	−0.225889
$362$	0	0
$363$	−1.59298e11	−0.481537
$364$	0	0
$365$	2.54006e11	0.749078
$366$	0	0
$367$	−1.68812e11	−0.485742	−0.242871	−	0.970059i	$-0.578089\pi$
$367$	−1.68812e11	−0.485742	−0.242871	+	0.970059i	$0.578089\pi$
$368$	0	0
$369$	3.62933e11	1.01908
$370$	0	0
$371$	3.02277e11	0.828367
$372$	0	0
$373$	−6.06880e11	−1.62335	−0.811676	−	0.584108i	$-0.801445\pi$
$373$	−6.06880e11	−1.62335	−0.811676	+	0.584108i	$0.801445\pi$
$374$	0	0
$375$	−1.66973e11	−0.436019
$376$	0	0
$377$	1.62625e11	0.414621
$378$	0	0
$379$	−4.67636e11	−1.16421	−0.582105	−	0.813114i	$-0.697771\pi$
$379$	−4.67636e11	−1.16421	−0.582105	+	0.813114i	$0.697771\pi$
$380$	0	0
$381$	−1.24468e11	−0.302619
$382$	0	0
$383$	−3.90199e11	−0.926599	−0.463300	−	0.886202i	$-0.653335\pi$
$383$	−3.90199e11	−0.926599	−0.463300	+	0.886202i	$0.653335\pi$
$384$	0	0
$385$	6.05981e10	0.140568
$386$	0	0
$387$	−3.88299e11	−0.879969
$388$	0	0
$389$	−1.61508e11	−0.357620	−0.178810	−	0.983884i	$-0.557225\pi$
$389$	−1.61508e11	−0.357620	−0.178810	+	0.983884i	$0.557225\pi$
$390$	0	0
$391$	−2.80385e11	−0.606679
$392$	0	0
$393$	3.80973e10	0.0805615
$394$	0	0
$395$	3.55452e11	0.734673
$396$	0	0
$397$	8.18138e11	1.65299	0.826493	−	0.562947i	$-0.190332\pi$
$397$	8.18138e11	1.65299	0.826493	+	0.562947i	$0.190332\pi$
$398$	0	0
$399$	3.48290e11	0.687959
$400$	0	0
$401$	−1.08197e11	−0.208962	−0.104481	−	0.994527i	$-0.533318\pi$
$401$	−1.08197e11	−0.208962	−0.104481	+	0.994527i	$0.533318\pi$
$402$	0	0
$403$	−2.43587e11	−0.460024
$404$	0	0
$405$	2.04997e11	0.378616
$406$	0	0
$407$	1.98344e10	0.0358298
$408$	0	0
$409$	−7.56200e11	−1.33623	−0.668115	−	0.744058i	$-0.732899\pi$
$409$	−7.56200e11	−1.33623	−0.668115	+	0.744058i	$0.732899\pi$
$410$	0	0
$411$	4.44878e11	0.769047
$412$	0	0
$413$	5.31045e11	0.898165
$414$	0	0
$415$	−9.76062e10	−0.161533
$416$	0	0
$417$	3.75851e11	0.608701
$418$	0	0
$419$	6.64700e11	1.05357	0.526784	−	0.849999i	$-0.323398\pi$
$419$	6.64700e11	1.05357	0.526784	+	0.849999i	$0.323398\pi$
$420$	0	0
$421$	−6.96689e11	−1.08086	−0.540430	−	0.841389i	$-0.681739\pi$
$421$	−6.96689e11	−1.08086	−0.540430	+	0.841389i	$0.681739\pi$
$422$	0	0
$423$	9.15440e11	1.39027
$424$	0	0
$425$	4.85133e10	0.0721291
$426$	0	0
$427$	3.42559e11	0.498666
$428$	0	0
$429$	−4.70249e10	−0.0670301
$430$	0	0
$431$	−1.37728e12	−1.92254	−0.961268	−	0.275614i	$-0.911119\pi$
$431$	−1.37728e12	−1.92254	−0.961268	+	0.275614i	$0.911119\pi$
$432$	0	0
$433$	−2.26719e11	−0.309950	−0.154975	−	0.987918i	$-0.549530\pi$
$433$	−2.26719e11	−0.309950	−0.154975	+	0.987918i	$0.549530\pi$
$434$	0	0
$435$	−9.45578e10	−0.126618
$436$	0	0
$437$	−9.44498e11	−1.23890
$438$	0	0
$439$	4.03484e9	0.00518485	0.00259242	−	0.999997i	$-0.499175\pi$
$439$	4.03484e9	0.00518485	0.00259242	+	0.999997i	$0.499175\pi$
$440$	0	0
$441$	−9.73834e11	−1.22606
$442$	0	0
$443$	1.08930e12	1.34379	0.671893	−	0.740649i	$-0.265482\pi$
$443$	1.08930e12	1.34379	0.671893	+	0.740649i	$0.265482\pi$
$444$	0	0
$445$	−1.18962e11	−0.143810
$446$	0	0
$447$	−2.21223e11	−0.262088
$448$	0	0
$449$	1.06107e12	1.23207	0.616035	−	0.787719i	$-0.288738\pi$
$449$	1.06107e12	1.23207	0.616035	+	0.787719i	$0.288738\pi$
$450$	0	0
$451$	−9.43786e10	−0.107418
$452$	0	0
$453$	−5.81005e11	−0.648243
$454$	0	0
$455$	−2.73270e12	−2.98910
$456$	0	0
$457$	7.52127e11	0.806618	0.403309	−	0.915064i	$-0.367860\pi$
$457$	7.52127e11	0.806618	0.403309	+	0.915064i	$0.367860\pi$
$458$	0	0
$459$	−3.50518e11	−0.368598
$460$	0	0
$461$	1.22335e12	1.26152	0.630761	−	0.775977i	$-0.282743\pi$
$461$	1.22335e12	1.26152	0.630761	+	0.775977i	$0.282743\pi$
$462$	0	0
$463$	1.00710e12	1.01849	0.509246	−	0.860621i	$-0.329924\pi$
$463$	1.00710e12	1.01849	0.509246	+	0.860621i	$0.329924\pi$
$464$	0	0
$465$	1.41633e11	0.140483
$466$	0	0
$467$	1.11362e12	1.08345	0.541725	−	0.840556i	$-0.317771\pi$
$467$	1.11362e12	1.08345	0.541725	+	0.840556i	$0.317771\pi$
$468$	0	0
$469$	1.48449e12	1.41677
$470$	0	0
$471$	−1.83512e10	−0.0171819
$472$	0	0
$473$	1.00975e11	0.0927551
$474$	0	0
$475$	1.63421e11	0.147294
$476$	0	0
$477$	−4.44183e11	−0.392853
$478$	0	0
$479$	7.94324e11	0.689426	0.344713	−	0.938708i	$-0.387976\pi$
$479$	7.94324e11	0.689426	0.344713	+	0.938708i	$0.387976\pi$
$480$	0	0
$481$	−8.94443e11	−0.761903
$482$	0	0
$483$	−1.31691e12	−1.10102
$484$	0	0
$485$	−3.64115e10	−0.0298814
$486$	0	0
$487$	1.75226e12	1.41162	0.705810	−	0.708401i	$-0.250583\pi$
$487$	1.75226e12	1.41162	0.705810	+	0.708401i	$0.250583\pi$
$488$	0	0
$489$	6.99741e11	0.553411
$490$	0	0
$491$	−2.49979e12	−1.94105	−0.970526	−	0.240997i	$-0.922526\pi$
$491$	−2.49979e12	−1.94105	−0.970526	+	0.240997i	$0.922526\pi$
$492$	0	0
$493$	−1.36634e11	−0.104171
$494$	0	0
$495$	−8.90463e10	−0.0666642
$496$	0	0
$497$	7.00934e11	0.515316
$498$	0	0
$499$	−3.03714e11	−0.219286	−0.109643	−	0.993971i	$-0.534971\pi$
$499$	−3.03714e11	−0.219286	−0.109643	+	0.993971i	$0.534971\pi$
$500$	0	0
$501$	8.42834e9	0.00597685
$502$	0	0
$503$	−1.56882e12	−1.09274	−0.546372	−	0.837543i	$-0.683991\pi$
$503$	−1.56882e12	−1.09274	−0.546372	+	0.837543i	$0.683991\pi$
$504$	0	0
$505$	−9.45204e11	−0.646717
$506$	0	0
$507$	1.39950e12	0.940672
$508$	0	0
$509$	−2.83572e12	−1.87255	−0.936274	−	0.351272i	$-0.885749\pi$
$509$	−2.83572e12	−1.87255	−0.936274	+	0.351272i	$0.885749\pi$
$510$	0	0
$511$	1.72388e12	1.11844
$512$	0	0
$513$	−1.18075e12	−0.752711
$514$	0	0
$515$	1.20893e12	0.757303
$516$	0	0
$517$	−2.38055e11	−0.146544
$518$	0	0
$519$	3.35083e11	0.202721
$520$	0	0
$521$	−1.43643e12	−0.854111	−0.427056	−	0.904225i	$-0.640449\pi$
$521$	−1.43643e12	−0.854111	−0.427056	+	0.904225i	$0.640449\pi$
$522$	0	0
$523$	−9.78444e11	−0.571845	−0.285923	−	0.958253i	$-0.592300\pi$
$523$	−9.78444e11	−0.571845	−0.285923	+	0.958253i	$0.592300\pi$
$524$	0	0
$525$	2.27857e11	0.130902
$526$	0	0
$527$	2.04656e11	0.115578
$528$	0	0
$529$	1.77007e12	0.982743
$530$	0	0
$531$	−7.80348e11	−0.425954
$532$	0	0
$533$	4.25605e12	2.28420
$534$	0	0
$535$	3.70491e12	1.95517
$536$	0	0
$537$	−3.77104e11	−0.195694
$538$	0	0
$539$	2.53239e11	0.129236
$540$	0	0
$541$	−2.85542e11	−0.143312	−0.0716560	−	0.997429i	$-0.522828\pi$
$541$	−2.85542e11	−0.143312	−0.0716560	+	0.997429i	$0.522828\pi$
$542$	0	0
$543$	7.89823e11	0.389880
$544$	0	0
$545$	−1.40285e12	−0.681127
$546$	0	0
$547$	−2.87296e12	−1.37210	−0.686052	−	0.727552i	$-0.740658\pi$
$547$	−2.87296e12	−1.37210	−0.686052	+	0.727552i	$0.740658\pi$
$548$	0	0
$549$	−5.03376e11	−0.236492
$550$	0	0
$551$	−4.60261e11	−0.212727
$552$	0	0
$553$	2.41237e12	1.09693
$554$	0	0
$555$	5.20071e11	0.232672
$556$	0	0
$557$	1.40595e12	0.618899	0.309450	−	0.950916i	$-0.399855\pi$
$557$	1.40595e12	0.618899	0.309450	+	0.950916i	$0.399855\pi$
$558$	0	0
$559$	−4.55351e12	−1.97239
$560$	0	0
$561$	3.95092e10	0.0168409
$562$	0	0
$563$	−3.50500e11	−0.147028	−0.0735139	−	0.997294i	$-0.523421\pi$
$563$	−3.50500e11	−0.147028	−0.0735139	+	0.997294i	$0.523421\pi$
$564$	0	0
$565$	−2.49345e12	−1.02940
$566$	0	0
$567$	1.39126e12	0.565308
$568$	0	0
$569$	3.17729e12	1.27073	0.635363	−	0.772214i	$-0.280851\pi$
$569$	3.17729e12	1.27073	0.635363	+	0.772214i	$0.280851\pi$
$570$	0	0
$571$	−3.02125e12	−1.18939	−0.594695	−	0.803951i	$-0.702727\pi$
$571$	−3.02125e12	−1.18939	−0.594695	+	0.803951i	$0.702727\pi$
$572$	0	0
$573$	−1.85235e12	−0.717838
$574$	0	0
$575$	−6.17907e11	−0.235732
$576$	0	0
$577$	−3.01537e12	−1.13253	−0.566264	−	0.824224i	$-0.691612\pi$
$577$	−3.01537e12	−1.13253	−0.566264	+	0.824224i	$0.691612\pi$
$578$	0	0
$579$	−1.96345e12	−0.726051
$580$	0	0
$581$	−6.62429e11	−0.241183
$582$	0	0
$583$	1.15507e11	0.0414095
$584$	0	0
$585$	4.01559e12	1.41758
$586$	0	0
$587$	3.76756e12	1.30975	0.654876	−	0.755736i	$-0.272721\pi$
$587$	3.76756e12	1.30975	0.654876	+	0.755736i	$0.272721\pi$
$588$	0	0
$589$	6.89399e11	0.236022
$590$	0	0
$591$	2.58515e12	0.871649
$592$	0	0
$593$	−4.38478e12	−1.45614	−0.728068	−	0.685505i	$-0.759581\pi$
$593$	−4.38478e12	−1.45614	−0.728068	+	0.685505i	$0.759581\pi$
$594$	0	0
$595$	2.29595e12	0.750993
$596$	0	0
$597$	1.75022e12	0.563909
$598$	0	0
$599$	−1.67903e11	−0.0532890	−0.0266445	−	0.999645i	$-0.508482\pi$
$599$	−1.67903e11	−0.0532890	−0.0266445	+	0.999645i	$0.508482\pi$
$600$	0	0
$601$	2.30729e12	0.721384	0.360692	−	0.932685i	$-0.382541\pi$
$601$	2.30729e12	0.721384	0.360692	+	0.932685i	$0.382541\pi$
$602$	0	0
$603$	−2.18139e12	−0.671901
$604$	0	0
$605$	−3.53735e12	−1.07344
$606$	0	0
$607$	−1.50248e12	−0.449220	−0.224610	−	0.974449i	$-0.572111\pi$
$607$	−1.50248e12	−0.449220	−0.224610	+	0.974449i	$0.572111\pi$
$608$	0	0
$609$	−6.41741e11	−0.189052
$610$	0	0
$611$	1.07352e13	3.11619
$612$	0	0
$613$	3.94077e12	1.12722	0.563610	−	0.826041i	$-0.309412\pi$
$613$	3.94077e12	1.12722	0.563610	+	0.826041i	$0.309412\pi$
$614$	0	0
$615$	−2.47467e12	−0.697555
$616$	0	0
$617$	−3.75098e12	−1.04198	−0.520992	−	0.853561i	$-0.674438\pi$
$617$	−3.75098e12	−1.04198	−0.520992	+	0.853561i	$0.674438\pi$
$618$	0	0
$619$	3.29947e12	0.903308	0.451654	−	0.892193i	$-0.350834\pi$
$619$	3.29947e12	0.903308	0.451654	+	0.892193i	$0.350834\pi$
$620$	0	0
$621$	4.46449e12	1.20465
$622$	0	0
$623$	−8.07365e11	−0.214721
$624$	0	0
$625$	−4.34641e12	−1.13938
$626$	0	0
$627$	1.33090e11	0.0343907
$628$	0	0
$629$	7.51489e11	0.191423
$630$	0	0
$631$	−7.22548e12	−1.81441	−0.907203	−	0.420693i	$-0.861787\pi$
$631$	−7.22548e12	−1.81441	−0.907203	+	0.420693i	$0.861787\pi$
$632$	0	0
$633$	3.74043e11	0.0925988
$634$	0	0
$635$	−2.76393e12	−0.674599
$636$	0	0
$637$	−1.14200e13	−2.74813
$638$	0	0
$639$	−1.02999e12	−0.244388
$640$	0	0
$641$	4.43842e12	1.03841	0.519203	−	0.854651i	$-0.326229\pi$
$641$	4.43842e12	1.03841	0.519203	+	0.854651i	$0.326229\pi$
$642$	0	0
$643$	−2.95078e11	−0.0680751	−0.0340375	−	0.999421i	$-0.510837\pi$
$643$	−2.95078e11	−0.0680751	−0.0340375	+	0.999421i	$0.510837\pi$
$644$	0	0
$645$	2.64762e12	0.602334
$646$	0	0
$647$	2.07880e12	0.466383	0.233192	−	0.972431i	$-0.425083\pi$
$647$	2.07880e12	0.466383	0.233192	+	0.972431i	$0.425083\pi$
$648$	0	0
$649$	2.02925e11	0.0448987
$650$	0	0
$651$	9.61226e11	0.209754
$652$	0	0
$653$	−1.80860e12	−0.389254	−0.194627	−	0.980877i	$-0.562350\pi$
$653$	−1.80860e12	−0.389254	−0.194627	+	0.980877i	$0.562350\pi$
$654$	0	0
$655$	8.45984e11	0.179588
$656$	0	0
$657$	−2.53317e12	−0.530420
$658$	0	0
$659$	−6.58150e12	−1.35938	−0.679690	−	0.733500i	$-0.737886\pi$
$659$	−6.58150e12	−1.35938	−0.679690	+	0.733500i	$0.737886\pi$
$660$	0	0
$661$	7.79747e12	1.58872	0.794360	−	0.607448i	$-0.207807\pi$
$661$	7.79747e12	1.58872	0.794360	+	0.607448i	$0.207807\pi$
$662$	0	0
$663$	−1.78169e12	−0.358113
$664$	0	0
$665$	7.73408e12	1.53360
$666$	0	0
$667$	1.74028e12	0.340450
$668$	0	0
$669$	1.40068e12	0.270347
$670$	0	0
$671$	1.30900e11	0.0249280
$672$	0	0
$673$	7.86129e12	1.47716	0.738578	−	0.674168i	$-0.235498\pi$
$673$	7.86129e12	1.47716	0.738578	+	0.674168i	$0.235498\pi$
$674$	0	0
$675$	−7.72464e11	−0.143223
$676$	0	0
$677$	−6.11388e12	−1.11858	−0.559291	−	0.828971i	$-0.688927\pi$
$677$	−6.11388e12	−1.11858	−0.559291	+	0.828971i	$0.688927\pi$
$678$	0	0
$679$	−2.47116e11	−0.0446156
$680$	0	0
$681$	−3.03852e12	−0.541378
$682$	0	0
$683$	3.45133e11	0.0606867	0.0303434	−	0.999540i	$-0.490340\pi$
$683$	3.45133e11	0.0606867	0.0303434	+	0.999540i	$0.490340\pi$
$684$	0	0
$685$	9.87891e12	1.71436
$686$	0	0
$687$	−3.87140e11	−0.0663074
$688$	0	0
$689$	−5.20885e12	−0.880553
$690$	0	0
$691$	5.18228e12	0.864708	0.432354	−	0.901704i	$-0.357683\pi$
$691$	5.18228e12	0.864708	0.432354	+	0.901704i	$0.357683\pi$
$692$	0	0
$693$	−6.04335e11	−0.0995356
$694$	0	0
$695$	8.34611e12	1.35691
$696$	0	0
$697$	−3.57583e12	−0.573891
$698$	0	0
$699$	1.18342e12	0.187496
$700$	0	0
$701$	−9.33484e12	−1.46008	−0.730038	−	0.683406i	$-0.760498\pi$
$701$	−9.33484e12	−1.46008	−0.730038	+	0.683406i	$0.760498\pi$
$702$	0	0
$703$	2.53145e12	0.390904
$704$	0	0
$705$	−6.24194e12	−0.951631
$706$	0	0
$707$	−6.41487e12	−0.965607
$708$	0	0
$709$	−7.63912e11	−0.113536	−0.0567682	−	0.998387i	$-0.518080\pi$
$709$	−7.63912e11	−0.113536	−0.0567682	+	0.998387i	$0.518080\pi$
$710$	0	0
$711$	−3.54487e12	−0.520220
$712$	0	0
$713$	−2.60667e12	−0.377732
$714$	0	0
$715$	−1.04423e12	−0.149423
$716$	0	0
$717$	−2.28472e12	−0.322846
$718$	0	0
$719$	−3.65176e12	−0.509591	−0.254796	−	0.966995i	$-0.582008\pi$
$719$	−3.65176e12	−0.509591	−0.254796	+	0.966995i	$0.582008\pi$
$720$	0	0
$721$	8.20473e12	1.13072
$722$	0	0
$723$	−4.13861e12	−0.563290
$724$	0	0
$725$	−3.01111e11	−0.0404767
$726$	0	0
$727$	4.06692e12	0.539959	0.269979	−	0.962866i	$-0.412983\pi$
$727$	4.06692e12	0.539959	0.269979	+	0.962866i	$0.412983\pi$
$728$	0	0
$729$	1.11762e12	0.146561
$730$	0	0
$731$	3.82575e12	0.495551
$732$	0	0
$733$	−1.00125e13	−1.28107	−0.640537	−	0.767927i	$-0.721288\pi$
$733$	−1.00125e13	−1.28107	−0.640537	+	0.767927i	$0.721288\pi$
$734$	0	0
$735$	6.64010e12	0.839231
$736$	0	0
$737$	5.67257e11	0.0708233
$738$	0	0
$739$	−5.95622e12	−0.734634	−0.367317	−	0.930096i	$-0.619724\pi$
$739$	−5.95622e12	−0.734634	−0.367317	+	0.930096i	$0.619724\pi$
$740$	0	0
$741$	−6.00175e12	−0.731300
$742$	0	0
$743$	8.85106e12	1.06548	0.532740	−	0.846279i	$-0.321162\pi$
$743$	8.85106e12	1.06548	0.532740	+	0.846279i	$0.321162\pi$
$744$	0	0
$745$	−4.91246e12	−0.584246
$746$	0	0
$747$	9.73412e11	0.114381
$748$	0	0
$749$	2.51443e13	2.91925
$750$	0	0
$751$	1.31039e13	1.50321	0.751606	−	0.659613i	$-0.229280\pi$
$751$	1.31039e13	1.50321	0.751606	+	0.659613i	$0.229280\pi$
$752$	0	0
$753$	5.94584e12	0.673963
$754$	0	0
$755$	−1.29017e13	−1.44506
$756$	0	0
$757$	1.31072e13	1.45071	0.725353	−	0.688378i	$-0.241677\pi$
$757$	1.31072e13	1.45071	0.725353	+	0.688378i	$0.241677\pi$
$758$	0	0
$759$	−5.03223e11	−0.0550392
$760$	0	0
$761$	4.34902e12	0.470068	0.235034	−	0.971987i	$-0.424480\pi$
$761$	4.34902e12	0.470068	0.235034	+	0.971987i	$0.424480\pi$
$762$	0	0
$763$	−9.52083e12	−1.01698
$764$	0	0
$765$	−3.37380e12	−0.356158
$766$	0	0
$767$	−9.15099e12	−0.954749
$768$	0	0
$769$	−9.23095e12	−0.951871	−0.475935	−	0.879480i	$-0.657890\pi$
$769$	−9.23095e12	−0.951871	−0.475935	+	0.879480i	$0.657890\pi$
$770$	0	0
$771$	4.83730e12	0.493013
$772$	0	0
$773$	−4.89415e12	−0.493025	−0.246513	−	0.969140i	$-0.579285\pi$
$773$	−4.89415e12	−0.493025	−0.246513	+	0.969140i	$0.579285\pi$
$774$	0	0
$775$	4.51016e11	0.0449091
$776$	0	0
$777$	3.52959e12	0.347400
$778$	0	0
$779$	−1.20455e13	−1.17194
$780$	0	0
$781$	2.67843e11	0.0257603
$782$	0	0
$783$	2.17558e12	0.206846
$784$	0	0
$785$	−4.07505e11	−0.0383018
$786$	0	0
$787$	1.57772e13	1.46604	0.733018	−	0.680209i	$-0.238111\pi$
$787$	1.57772e13	1.46604	0.733018	+	0.680209i	$0.238111\pi$
$788$	0	0
$789$	−3.56824e12	−0.327799
$790$	0	0
$791$	−1.69224e13	−1.53698
$792$	0	0
$793$	−5.90299e12	−0.530082
$794$	0	0
$795$	3.02867e12	0.268906
$796$	0	0
$797$	1.39090e13	1.22105	0.610524	−	0.791998i	$-0.290959\pi$
$797$	1.39090e13	1.22105	0.610524	+	0.791998i	$0.290959\pi$
$798$	0	0
$799$	−9.01945e12	−0.782924
$800$	0	0
$801$	1.18639e12	0.101831
$802$	0	0
$803$	6.58735e11	0.0559101
$804$	0	0
$805$	−2.92432e13	−2.45439
$806$	0	0
$807$	−9.37105e12	−0.777781
$808$	0	0
$809$	3.81208e11	0.0312891	0.0156446	−	0.999878i	$-0.495020\pi$
$809$	3.81208e11	0.0312891	0.0156446	+	0.999878i	$0.495020\pi$
$810$	0	0
$811$	−9.34915e12	−0.758889	−0.379445	−	0.925214i	$-0.623885\pi$
$811$	−9.34915e12	−0.758889	−0.379445	+	0.925214i	$0.623885\pi$
$812$	0	0
$813$	8.07748e12	0.648438
$814$	0	0
$815$	1.55384e13	1.23366
$816$	0	0
$817$	1.28873e13	1.01196
$818$	0	0
$819$	2.72528e13	2.11657
$820$	0	0
$821$	−1.17330e13	−0.901290	−0.450645	−	0.892703i	$-0.648806\pi$
$821$	−1.17330e13	−0.901290	−0.450645	+	0.892703i	$0.648806\pi$
$822$	0	0
$823$	6.28533e12	0.477561	0.238781	−	0.971074i	$-0.423252\pi$
$823$	6.28533e12	0.477561	0.238781	+	0.971074i	$0.423252\pi$
$824$	0	0
$825$	8.70695e10	0.00654370
$826$	0	0
$827$	1.80738e13	1.34361	0.671806	−	0.740727i	$-0.265519\pi$
$827$	1.80738e13	1.34361	0.671806	+	0.740727i	$0.265519\pi$
$828$	0	0
$829$	−1.34766e13	−0.991028	−0.495514	−	0.868600i	$-0.665020\pi$
$829$	−1.34766e13	−0.991028	−0.495514	+	0.868600i	$0.665020\pi$
$830$	0	0
$831$	−4.22087e12	−0.307041
$832$	0	0
$833$	9.59478e12	0.690450
$834$	0	0
$835$	1.87159e11	0.0133236
$836$	0	0
$837$	−3.25868e12	−0.229497
$838$	0	0
$839$	2.14122e13	1.49188	0.745939	−	0.666014i	$-0.232001\pi$
$839$	2.14122e13	1.49188	0.745939	+	0.666014i	$0.232001\pi$
$840$	0	0
$841$	−1.36591e13	−0.941542
$842$	0	0
$843$	4.90231e12	0.334331
$844$	0	0
$845$	3.10772e13	2.09694
$846$	0	0
$847$	−2.40071e13	−1.60274
$848$	0	0
$849$	2.65303e12	0.175250
$850$	0	0
$851$	−9.57161e12	−0.625608
$852$	0	0
$853$	1.20087e13	0.776649	0.388324	−	0.921523i	$-0.373054\pi$
$853$	1.20087e13	0.776649	0.388324	+	0.921523i	$0.373054\pi$
$854$	0	0
$855$	−1.13649e13	−0.727308
$856$	0	0
$857$	−2.89490e13	−1.83324	−0.916620	−	0.399759i	$-0.869094\pi$
$857$	−2.89490e13	−1.83324	−0.916620	+	0.399759i	$0.869094\pi$
$858$	0	0
$859$	2.31413e13	1.45017	0.725083	−	0.688662i	$-0.241801\pi$
$859$	2.31413e13	1.45017	0.725083	+	0.688662i	$0.241801\pi$
$860$	0	0
$861$	−1.67950e13	−1.04151
$862$	0	0
$863$	−6.84504e12	−0.420075	−0.210038	−	0.977693i	$-0.567359\pi$
$863$	−6.84504e12	−0.420075	−0.210038	+	0.977693i	$0.567359\pi$
$864$	0	0
$865$	7.44082e12	0.451906
$866$	0	0
$867$	−6.56705e12	−0.394716
$868$	0	0
$869$	9.21821e11	0.0548350
$870$	0	0
$871$	−2.55807e13	−1.50602
$872$	0	0
$873$	3.63126e11	0.0211589
$874$	0	0
$875$	−2.51638e13	−1.45124
$876$	0	0
$877$	2.24289e13	1.28029	0.640147	−	0.768253i	$-0.278874\pi$
$877$	2.24289e13	1.28029	0.640147	+	0.768253i	$0.278874\pi$
$878$	0	0
$879$	−6.37306e12	−0.360079
$880$	0	0
$881$	1.24212e13	0.694658	0.347329	−	0.937743i	$-0.387089\pi$
$881$	1.24212e13	0.694658	0.347329	+	0.937743i	$0.387089\pi$
$882$	0	0
$883$	−5.52878e12	−0.306060	−0.153030	−	0.988222i	$-0.548903\pi$
$883$	−5.52878e12	−0.306060	−0.153030	+	0.988222i	$0.548903\pi$
$884$	0	0
$885$	5.32081e12	0.291564
$886$	0	0
$887$	1.84924e13	1.00308	0.501542	−	0.865133i	$-0.332766\pi$
$887$	1.84924e13	1.00308	0.501542	+	0.865133i	$0.332766\pi$
$888$	0	0
$889$	−1.87581e13	−1.00724
$890$	0	0
$891$	5.31633e11	0.0282594
$892$	0	0
$893$	−3.03827e13	−1.59880
$894$	0	0
$895$	−8.37392e12	−0.436240
$896$	0	0
$897$	2.26931e13	1.17038
$898$	0	0
$899$	−1.27025e12	−0.0648590
$900$	0	0
$901$	4.37635e12	0.221233
$902$	0	0
$903$	1.79688e13	0.899339
$904$	0	0
$905$	1.75387e13	0.869119
$906$	0	0
$907$	2.72494e13	1.33698	0.668489	−	0.743722i	$-0.266941\pi$
$907$	2.72494e13	1.33698	0.668489	+	0.743722i	$0.266941\pi$
$908$	0	0
$909$	9.42638e12	0.457939
$910$	0	0
$911$	−7.31194e12	−0.351722	−0.175861	−	0.984415i	$-0.556271\pi$
$911$	−7.31194e12	−0.351722	−0.175861	+	0.984415i	$0.556271\pi$
$912$	0	0
$913$	−2.53130e11	−0.0120566
$914$	0	0
$915$	3.43228e12	0.161878
$916$	0	0
$917$	5.74148e12	0.268140
$918$	0	0
$919$	−6.17500e12	−0.285573	−0.142786	−	0.989754i	$-0.545606\pi$
$919$	−6.17500e12	−0.285573	−0.142786	+	0.989754i	$0.545606\pi$
$920$	0	0
$921$	−1.11003e13	−0.508353
$922$	0	0
$923$	−1.20785e13	−0.547780
$924$	0	0
$925$	1.65612e12	0.0743795
$926$	0	0
$927$	−1.20565e13	−0.536244
$928$	0	0
$929$	4.90710e12	0.216149	0.108075	−	0.994143i	$-0.465531\pi$
$929$	4.90710e12	0.216149	0.108075	+	0.994143i	$0.465531\pi$
$930$	0	0
$931$	3.23208e13	1.40996
$932$	0	0
$933$	−3.76728e12	−0.162765
$934$	0	0
$935$	8.77336e11	0.0375417
$936$	0	0
$937$	2.66248e13	1.12839	0.564195	−	0.825642i	$-0.309187\pi$
$937$	2.66248e13	1.12839	0.564195	+	0.825642i	$0.309187\pi$
$938$	0	0
$939$	1.60372e13	0.673182
$940$	0	0
$941$	−3.78799e11	−0.0157491	−0.00787454	−	0.999969i	$-0.502507\pi$
$941$	−3.78799e11	−0.0157491	−0.00787454	+	0.999969i	$0.502507\pi$
$942$	0	0
$943$	4.55448e13	1.87558
$944$	0	0
$945$	−3.65578e13	−1.49120
$946$	0	0
$947$	−3.93752e13	−1.59092	−0.795460	−	0.606006i	$-0.792771\pi$
$947$	−3.93752e13	−1.59092	−0.795460	+	0.606006i	$0.792771\pi$
$948$	0	0
$949$	−2.97060e13	−1.18890
$950$	0	0
$951$	−5.26368e12	−0.208678
$952$	0	0
$953$	−3.77122e13	−1.48103	−0.740514	−	0.672041i	$-0.765418\pi$
$953$	−3.77122e13	−1.48103	−0.740514	+	0.672041i	$0.765418\pi$
$954$	0	0
$955$	−4.11330e13	−1.60020
$956$	0	0
$957$	−2.45224e11	−0.00945060
$958$	0	0
$959$	6.70457e13	2.55969
$960$	0	0
$961$	−2.45370e13	−0.928039
$962$	0	0
$963$	−3.69485e13	−1.38445
$964$	0	0
$965$	−4.36002e13	−1.61851
$966$	0	0
$967$	−3.28952e12	−0.120980	−0.0604900	−	0.998169i	$-0.519266\pi$
$967$	−3.28952e12	−0.120980	−0.0604900	+	0.998169i	$0.519266\pi$
$968$	0	0
$969$	5.04252e12	0.183735
$970$	0	0
$971$	2.68793e11	0.00970357	0.00485179	−	0.999988i	$-0.498456\pi$
$971$	2.68793e11	0.00970357	0.00485179	+	0.999988i	$0.498456\pi$
$972$	0	0
$973$	5.66430e13	2.02599
$974$	0	0
$975$	−3.92644e12	−0.139149
$976$	0	0
$977$	2.44634e13	0.858998	0.429499	−	0.903067i	$-0.358690\pi$
$977$	2.44634e13	0.858998	0.429499	+	0.903067i	$0.358690\pi$
$978$	0	0
$979$	−3.08513e11	−0.0107337
$980$	0	0
$981$	1.39905e13	0.482304
$982$	0	0
$983$	1.57199e13	0.536983	0.268491	−	0.963282i	$-0.413475\pi$
$983$	1.57199e13	0.536983	0.268491	+	0.963282i	$0.413475\pi$
$984$	0	0
$985$	5.74055e13	1.94308
$986$	0	0
$987$	−4.23625e13	−1.42087
$988$	0	0
$989$	−4.87280e13	−1.61955
$990$	0	0
$991$	−1.11195e13	−0.366229	−0.183114	−	0.983092i	$-0.558618\pi$
$991$	−1.11195e13	−0.366229	−0.183114	+	0.983092i	$0.558618\pi$
$992$	0	0
$993$	−5.37262e12	−0.175353
$994$	0	0
$995$	3.88652e13	1.25706
$996$	0	0
$997$	2.50821e13	0.803962	0.401981	−	0.915648i	$-0.368322\pi$
$997$	2.50821e13	0.803962	0.401981	+	0.915648i	$0.368322\pi$
$998$	0	0
$999$	−1.19658e13	−0.380098

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8.10.a.b.1.1	✓	1
3.2	odd	2		72.10.a.a.1.1		1
4.3	odd	2		16.10.a.b.1.1		1
5.2	odd	4		200.10.c.a.49.1		2
5.3	odd	4		200.10.c.a.49.2		2
5.4	even	2		200.10.a.a.1.1		1
7.6	odd	2		392.10.a.a.1.1		1
8.3	odd	2		64.10.a.g.1.1		1
8.5	even	2		64.10.a.c.1.1		1
12.11	even	2		144.10.a.b.1.1		1
16.3	odd	4		256.10.b.k.129.1		2
16.5	even	4		256.10.b.a.129.1		2
16.11	odd	4		256.10.b.k.129.2		2
16.13	even	4		256.10.b.a.129.2		2
20.3	even	4		400.10.c.f.49.1		2
20.7	even	4		400.10.c.f.49.2		2
20.19	odd	2		400.10.a.i.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8.10.a.b.1.1	✓	1	1.1	even	1	trivial
16.10.a.b.1.1		1	4.3	odd	2
64.10.a.c.1.1		1	8.5	even	2
64.10.a.g.1.1		1	8.3	odd	2
72.10.a.a.1.1		1	3.2	odd	2
144.10.a.b.1.1		1	12.11	even	2
200.10.a.a.1.1		1	5.4	even	2
200.10.c.a.49.1		2	5.2	odd	4
200.10.c.a.49.2		2	5.3	odd	4
256.10.b.a.129.1		2	16.5	even	4
256.10.b.a.129.2		2	16.13	even	4
256.10.b.k.129.1		2	16.3	odd	4
256.10.b.k.129.2		2	16.11	odd	4
392.10.a.a.1.1		1	7.6	odd	2
400.10.a.i.1.1		1	20.19	odd	2
400.10.c.f.49.1		2	20.3	even	4
400.10.c.f.49.2		2	20.7	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 8 = 2^{3} \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	8.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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