LMFDB - Embedded newform 48.8.a.f.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [48,8,Mod(1,48)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(48, 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("48.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 48 = 2^{4} \cdot 3 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	48.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:	$14.9944812232$
Analytic rank:	$1$
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	$\chi$	$=$	48.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+27.0000 q^{3} -26.0000 q^{5} -1056.00 q^{7} +729.000 q^{9} +O(q^{10})$

$q+27.0000 q^{3} -26.0000 q^{5} -1056.00 q^{7} +729.000 q^{9} -6412.00 q^{11} +5206.00 q^{13} -702.000 q^{15} -6238.00 q^{17} -41492.0 q^{19} -28512.0 q^{21} +29432.0 q^{23} -77449.0 q^{25} +19683.0 q^{27} -210498. q^{29} -185240. q^{31} -173124. q^{33} +27456.0 q^{35} +507630. q^{37} +140562. q^{39} +360042. q^{41} -620044. q^{43} -18954.0 q^{45} +847680. q^{47} +291593. q^{49} -168426. q^{51} +1.42375e6 q^{53} +166712. q^{55} -1.12028e6 q^{57} +2.54872e6 q^{59} -706058. q^{61} -769824. q^{63} -135356. q^{65} +2.41880e6 q^{67} +794664. q^{69} -265976. q^{71} -5.79124e6 q^{73} -2.09112e6 q^{75} +6.77107e6 q^{77} -2.95569e6 q^{79} +531441. q^{81} -3.46293e6 q^{83} +162188. q^{85} -5.68345e6 q^{87} -2.21113e6 q^{89} -5.49754e6 q^{91} -5.00148e6 q^{93} +1.07879e6 q^{95} -1.55948e7 q^{97} -4.67435e6 q^{99} +O(q^{100})$

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
$5$	−26.0000	−0.0930204	−0.0465102	−	0.998918i	$-0.514810\pi$
$5$	−26.0000	−0.0930204	−0.0465102	+	0.998918i	$0.514810\pi$
$6$	0	0
$7$	−1056.00	−1.16365	−0.581823	−	0.813316i	$-0.697660\pi$
$7$	−1056.00	−1.16365	−0.581823	+	0.813316i	$0.697660\pi$
$8$	0	0
$9$	729.000	0.333333
$10$	0	0
$11$	−6412.00	−1.45251	−0.726255	−	0.687425i	$-0.758741\pi$
$11$	−6412.00	−1.45251	−0.726255	+	0.687425i	$0.758741\pi$
$12$	0	0
$13$	5206.00	0.657207	0.328604	−	0.944468i	$-0.393422\pi$
$13$	5206.00	0.657207	0.328604	+	0.944468i	$0.393422\pi$
$14$	0	0
$15$	−702.000	−0.0537054
$16$	0	0
$17$	−6238.00	−0.307946	−0.153973	−	0.988075i	$-0.549207\pi$
$17$	−6238.00	−0.307946	−0.153973	+	0.988075i	$0.549207\pi$
$18$	0	0
$19$	−41492.0	−1.38780	−0.693900	−	0.720072i	$-0.744109\pi$
$19$	−41492.0	−1.38780	−0.693900	+	0.720072i	$0.744109\pi$
$20$	0	0
$21$	−28512.0	−0.671831
$22$	0	0
$23$	29432.0	0.504397	0.252198	−	0.967676i	$-0.418846\pi$
$23$	29432.0	0.504397	0.252198	+	0.967676i	$0.418846\pi$
$24$	0	0
$25$	−77449.0	−0.991347
$26$	0	0
$27$	19683.0	0.192450
$28$	0	0
$29$	−210498.	−1.60271	−0.801355	−	0.598189i	$-0.795887\pi$
$29$	−210498.	−1.60271	−0.801355	+	0.598189i	$0.795887\pi$
$30$	0	0
$31$	−185240.	−1.11678	−0.558392	−	0.829578i	$-0.688581\pi$
$31$	−185240.	−1.11678	−0.558392	+	0.829578i	$0.688581\pi$
$32$	0	0
$33$	−173124.	−0.838607
$34$	0	0
$35$	27456.0	0.108243
$36$	0	0
$37$	507630.	1.64756	0.823780	−	0.566910i	$-0.191861\pi$
$37$	507630.	1.64756	0.823780	+	0.566910i	$0.191861\pi$
$38$	0	0
$39$	140562.	0.379439
$40$	0	0
$41$	360042.	0.815849	0.407924	−	0.913016i	$-0.366253\pi$
$41$	360042.	0.815849	0.407924	+	0.913016i	$0.366253\pi$
$42$	0	0
$43$	−620044.	−1.18928	−0.594638	−	0.803993i	$-0.702705\pi$
$43$	−620044.	−1.18928	−0.594638	+	0.803993i	$0.702705\pi$
$44$	0	0
$45$	−18954.0	−0.0310068
$46$	0	0
$47$	847680.	1.19094	0.595469	−	0.803378i	$-0.296966\pi$
$47$	847680.	1.19094	0.595469	+	0.803378i	$0.296966\pi$
$48$	0	0
$49$	291593.	0.354071
$50$	0	0
$51$	−168426.	−0.177793
$52$	0	0
$53$	1.42375e6	1.31362	0.656808	−	0.754058i	$-0.271906\pi$
$53$	1.42375e6	1.31362	0.656808	+	0.754058i	$0.271906\pi$
$54$	0	0
$55$	166712.	0.135113
$56$	0	0
$57$	−1.12028e6	−0.801246
$58$	0	0
$59$	2.54872e6	1.61563	0.807813	−	0.589439i	$-0.200651\pi$
$59$	2.54872e6	1.61563	0.807813	+	0.589439i	$0.200651\pi$
$60$	0	0
$61$	−706058.	−0.398278	−0.199139	−	0.979971i	$-0.563814\pi$
$61$	−706058.	−0.398278	−0.199139	+	0.979971i	$0.563814\pi$
$62$	0	0
$63$	−769824.	−0.387882
$64$	0	0
$65$	−135356.	−0.0611337
$66$	0	0
$67$	2.41880e6	0.982511	0.491256	−	0.871016i	$-0.336538\pi$
$67$	2.41880e6	0.982511	0.491256	+	0.871016i	$0.336538\pi$
$68$	0	0
$69$	794664.	0.291214
$70$	0	0
$71$	−265976.	−0.0881938	−0.0440969	−	0.999027i	$-0.514041\pi$
$71$	−265976.	−0.0881938	−0.0440969	+	0.999027i	$0.514041\pi$
$72$	0	0
$73$	−5.79124e6	−1.74237	−0.871187	−	0.490951i	$-0.836649\pi$
$73$	−5.79124e6	−1.74237	−0.871187	+	0.490951i	$0.836649\pi$
$74$	0	0
$75$	−2.09112e6	−0.572355
$76$	0	0
$77$	6.77107e6	1.69021
$78$	0	0
$79$	−2.95569e6	−0.674472	−0.337236	−	0.941420i	$-0.609492\pi$
$79$	−2.95569e6	−0.674472	−0.337236	+	0.941420i	$0.609492\pi$
$80$	0	0
$81$	531441.	0.111111
$82$	0	0
$83$	−3.46293e6	−0.664769	−0.332384	−	0.943144i	$-0.607853\pi$
$83$	−3.46293e6	−0.664769	−0.332384	+	0.943144i	$0.607853\pi$
$84$	0	0
$85$	162188.	0.0286452
$86$	0	0
$87$	−5.68345e6	−0.925325
$88$	0	0
$89$	−2.21113e6	−0.332467	−0.166233	−	0.986086i	$-0.553161\pi$
$89$	−2.21113e6	−0.332467	−0.166233	+	0.986086i	$0.553161\pi$
$90$	0	0
$91$	−5.49754e6	−0.764757
$92$	0	0
$93$	−5.00148e6	−0.644775
$94$	0	0
$95$	1.07879e6	0.129094
$96$	0	0
$97$	−1.55948e7	−1.73492	−0.867459	−	0.497508i	$-0.834248\pi$
$97$	−1.55948e7	−1.73492	−0.867459	+	0.497508i	$0.834248\pi$
$98$	0	0
$99$	−4.67435e6	−0.484170
$100$	0	0
$101$	1.29563e7	1.25129	0.625643	−	0.780110i	$-0.284837\pi$
$101$	1.29563e7	1.25129	0.625643	+	0.780110i	$0.284837\pi$
$102$	0	0
$103$	6.82002e6	0.614972	0.307486	−	0.951553i	$-0.400512\pi$
$103$	6.82002e6	0.614972	0.307486	+	0.951553i	$0.400512\pi$
$104$	0	0
$105$	741312.	0.0624940
$106$	0	0
$107$	1.37785e7	1.08733	0.543663	−	0.839304i	$-0.317037\pi$
$107$	1.37785e7	1.08733	0.543663	+	0.839304i	$0.317037\pi$
$108$	0	0
$109$	9.67721e6	0.715743	0.357872	−	0.933771i	$-0.383503\pi$
$109$	9.67721e6	0.715743	0.357872	+	0.933771i	$0.383503\pi$
$110$	0	0
$111$	1.37060e7	0.951219
$112$	0	0
$113$	−2.19184e7	−1.42901	−0.714504	−	0.699632i	$-0.753348\pi$
$113$	−2.19184e7	−1.42901	−0.714504	+	0.699632i	$0.753348\pi$
$114$	0	0
$115$	−765232.	−0.0469192
$116$	0	0
$117$	3.79517e6	0.219069
$118$	0	0
$119$	6.58733e6	0.358340
$120$	0	0
$121$	2.16266e7	1.10979
$122$	0	0
$123$	9.72113e6	0.471031
$124$	0	0
$125$	4.04492e6	0.185236
$126$	0	0
$127$	661016.	0.0286351	0.0143176	−	0.999897i	$-0.495442\pi$
$127$	661016.	0.0286351	0.0143176	+	0.999897i	$0.495442\pi$
$128$	0	0
$129$	−1.67412e7	−0.686629
$130$	0	0
$131$	−1.20309e6	−0.0467573	−0.0233786	−	0.999727i	$-0.507442\pi$
$131$	−1.20309e6	−0.0467573	−0.0233786	+	0.999727i	$0.507442\pi$
$132$	0	0
$133$	4.38156e7	1.61491
$134$	0	0
$135$	−511758.	−0.0179018
$136$	0	0
$137$	1.20046e7	0.398866	0.199433	−	0.979911i	$-0.436090\pi$
$137$	1.20046e7	0.398866	0.199433	+	0.979911i	$0.436090\pi$
$138$	0	0
$139$	−2.22914e7	−0.704019	−0.352010	−	0.935996i	$-0.614502\pi$
$139$	−2.22914e7	−0.704019	−0.352010	+	0.935996i	$0.614502\pi$
$140$	0	0
$141$	2.28874e7	0.687589
$142$	0	0
$143$	−3.33809e7	−0.954600
$144$	0	0
$145$	5.47295e6	0.149085
$146$	0	0
$147$	7.87301e6	0.204423
$148$	0	0
$149$	−3.92355e7	−0.971689	−0.485844	−	0.874045i	$-0.661488\pi$
$149$	−3.92355e7	−0.971689	−0.485844	+	0.874045i	$0.661488\pi$
$150$	0	0
$151$	−1.13013e7	−0.267122	−0.133561	−	0.991041i	$-0.542641\pi$
$151$	−1.13013e7	−0.267122	−0.133561	+	0.991041i	$0.542641\pi$
$152$	0	0
$153$	−4.54750e6	−0.102649
$154$	0	0
$155$	4.81624e6	0.103884
$156$	0	0
$157$	−2.22302e7	−0.458453	−0.229227	−	0.973373i	$-0.573620\pi$
$157$	−2.22302e7	−0.458453	−0.229227	+	0.973373i	$0.573620\pi$
$158$	0	0
$159$	3.84412e7	0.758416
$160$	0	0
$161$	−3.10802e7	−0.586939
$162$	0	0
$163$	3.23811e7	0.585645	0.292822	−	0.956167i	$-0.405406\pi$
$163$	3.23811e7	0.585645	0.292822	+	0.956167i	$0.405406\pi$
$164$	0	0
$165$	4.50122e6	0.0780076
$166$	0	0
$167$	−5.15095e7	−0.855815	−0.427908	−	0.903822i	$-0.640749\pi$
$167$	−5.15095e7	−0.855815	−0.427908	+	0.903822i	$0.640749\pi$
$168$	0	0
$169$	−3.56461e7	−0.568078
$170$	0	0
$171$	−3.02477e7	−0.462600
$172$	0	0
$173$	−1.00284e8	−1.47255	−0.736274	−	0.676683i	$-0.763417\pi$
$173$	−1.00284e8	−1.47255	−0.736274	+	0.676683i	$0.763417\pi$
$174$	0	0
$175$	8.17861e7	1.15358
$176$	0	0
$177$	6.88155e7	0.932782
$178$	0	0
$179$	−6.93179e7	−0.903358	−0.451679	−	0.892181i	$-0.649175\pi$
$179$	−6.93179e7	−0.903358	−0.451679	+	0.892181i	$0.649175\pi$
$180$	0	0
$181$	−7.46718e7	−0.936012	−0.468006	−	0.883725i	$-0.655028\pi$
$181$	−7.46718e7	−0.936012	−0.468006	+	0.883725i	$0.655028\pi$
$182$	0	0
$183$	−1.90636e7	−0.229946
$184$	0	0
$185$	−1.31984e7	−0.153257
$186$	0	0
$187$	3.99981e7	0.447294
$188$	0	0
$189$	−2.07852e7	−0.223944
$190$	0	0
$191$	−1.70910e8	−1.77480	−0.887402	−	0.460996i	$-0.847492\pi$
$191$	−1.70910e8	−1.77480	−0.887402	+	0.460996i	$0.847492\pi$
$192$	0	0
$193$	1.15066e7	0.115211	0.0576056	−	0.998339i	$-0.481653\pi$
$193$	1.15066e7	0.115211	0.0576056	+	0.998339i	$0.481653\pi$
$194$	0	0
$195$	−3.65461e6	−0.0352956
$196$	0	0
$197$	7.93036e7	0.739029	0.369514	−	0.929225i	$-0.379524\pi$
$197$	7.93036e7	0.739029	0.369514	+	0.929225i	$0.379524\pi$
$198$	0	0
$199$	−2.38620e7	−0.214645	−0.107322	−	0.994224i	$-0.534228\pi$
$199$	−2.38620e7	−0.214645	−0.107322	+	0.994224i	$0.534228\pi$
$200$	0	0
$201$	6.53075e7	0.567253
$202$	0	0
$203$	2.22286e8	1.86499
$204$	0	0
$205$	−9.36109e6	−0.0758906
$206$	0	0
$207$	2.14559e7	0.168132
$208$	0	0
$209$	2.66047e8	2.01579
$210$	0	0
$211$	−1.09739e8	−0.804216	−0.402108	−	0.915592i	$-0.631722\pi$
$211$	−1.09739e8	−0.804216	−0.402108	+	0.915592i	$0.631722\pi$
$212$	0	0
$213$	−7.18135e6	−0.0509187
$214$	0	0
$215$	1.61211e7	0.110627
$216$	0	0
$217$	1.95613e8	1.29954
$218$	0	0
$219$	−1.56363e8	−1.00596
$220$	0	0
$221$	−3.24750e7	−0.202384
$222$	0	0
$223$	2.71085e8	1.63696	0.818481	−	0.574534i	$-0.194817\pi$
$223$	2.71085e8	1.63696	0.818481	+	0.574534i	$0.194817\pi$
$224$	0	0
$225$	−5.64603e7	−0.330449
$226$	0	0
$227$	2.21029e7	0.125418	0.0627089	−	0.998032i	$-0.480026\pi$
$227$	2.21029e7	0.125418	0.0627089	+	0.998032i	$0.480026\pi$
$228$	0	0
$229$	−1.51778e8	−0.835190	−0.417595	−	0.908633i	$-0.637127\pi$
$229$	−1.51778e8	−0.835190	−0.417595	+	0.908633i	$0.637127\pi$
$230$	0	0
$231$	1.82819e8	0.975841
$232$	0	0
$233$	−8.94412e7	−0.463225	−0.231613	−	0.972808i	$-0.574400\pi$
$233$	−8.94412e7	−0.463225	−0.231613	+	0.972808i	$0.574400\pi$
$234$	0	0
$235$	−2.20397e7	−0.110782
$236$	0	0
$237$	−7.98036e7	−0.389406
$238$	0	0
$239$	2.69285e8	1.27591	0.637954	−	0.770075i	$-0.279781\pi$
$239$	2.69285e8	1.27591	0.637954	+	0.770075i	$0.279781\pi$
$240$	0	0
$241$	1.98807e8	0.914897	0.457449	−	0.889236i	$-0.348763\pi$
$241$	1.98807e8	0.914897	0.457449	+	0.889236i	$0.348763\pi$
$242$	0	0
$243$	1.43489e7	0.0641500
$244$	0	0
$245$	−7.58142e6	−0.0329359
$246$	0	0
$247$	−2.16007e8	−0.912072
$248$	0	0
$249$	−9.34992e7	−0.383804
$250$	0	0
$251$	3.38049e8	1.34934	0.674671	−	0.738118i	$-0.264285\pi$
$251$	3.38049e8	1.34934	0.674671	+	0.738118i	$0.264285\pi$
$252$	0	0
$253$	−1.88718e8	−0.732641
$254$	0	0
$255$	4.37908e6	0.0165383
$256$	0	0
$257$	−1.64489e8	−0.604464	−0.302232	−	0.953234i	$-0.597732\pi$
$257$	−1.64489e8	−0.604464	−0.302232	+	0.953234i	$0.597732\pi$
$258$	0	0
$259$	−5.36057e8	−1.91718
$260$	0	0
$261$	−1.53453e8	−0.534237
$262$	0	0
$263$	−4.21664e8	−1.42929	−0.714646	−	0.699486i	$-0.753412\pi$
$263$	−4.21664e8	−1.42929	−0.714646	+	0.699486i	$0.753412\pi$
$264$	0	0
$265$	−3.70175e7	−0.122193
$266$	0	0
$267$	−5.97004e7	−0.191950
$268$	0	0
$269$	2.13468e8	0.668651	0.334325	−	0.942458i	$-0.391492\pi$
$269$	2.13468e8	0.668651	0.334325	+	0.942458i	$0.391492\pi$
$270$	0	0
$271$	−2.23317e8	−0.681601	−0.340800	−	0.940136i	$-0.610698\pi$
$271$	−2.23317e8	−0.681601	−0.340800	+	0.940136i	$0.610698\pi$
$272$	0	0
$273$	−1.48433e8	−0.441532
$274$	0	0
$275$	4.96603e8	1.43994
$276$	0	0
$277$	1.72179e8	0.486743	0.243372	−	0.969933i	$-0.421747\pi$
$277$	1.72179e8	0.486743	0.243372	+	0.969933i	$0.421747\pi$
$278$	0	0
$279$	−1.35040e8	−0.372261
$280$	0	0
$281$	6.84836e8	1.84126	0.920628	−	0.390441i	$-0.127677\pi$
$281$	6.84836e8	1.84126	0.920628	+	0.390441i	$0.127677\pi$
$282$	0	0
$283$	7.80003e7	0.204571	0.102285	−	0.994755i	$-0.467384\pi$
$283$	7.80003e7	0.204571	0.102285	+	0.994755i	$0.467384\pi$
$284$	0	0
$285$	2.91274e7	0.0745323
$286$	0	0
$287$	−3.80204e8	−0.949359
$288$	0	0
$289$	−3.71426e8	−0.905169
$290$	0	0
$291$	−4.21060e8	−1.00166
$292$	0	0
$293$	−6.50814e8	−1.51154	−0.755771	−	0.654836i	$-0.772738\pi$
$293$	−6.50814e8	−1.51154	−0.755771	+	0.654836i	$0.772738\pi$
$294$	0	0
$295$	−6.62668e7	−0.150286
$296$	0	0
$297$	−1.26207e8	−0.279536
$298$	0	0
$299$	1.53223e8	0.331493
$300$	0	0
$301$	6.54766e8	1.38390
$302$	0	0
$303$	3.49820e8	0.722430
$304$	0	0
$305$	1.83575e7	0.0370480
$306$	0	0
$307$	−3.67913e8	−0.725707	−0.362853	−	0.931846i	$-0.618197\pi$
$307$	−3.67913e8	−0.725707	−0.362853	+	0.931846i	$0.618197\pi$
$308$	0	0
$309$	1.84140e8	0.355054
$310$	0	0
$311$	−4.42351e8	−0.833883	−0.416942	−	0.908933i	$-0.636898\pi$
$311$	−4.42351e8	−0.833883	−0.416942	+	0.908933i	$0.636898\pi$
$312$	0	0
$313$	−1.06172e9	−1.95705	−0.978527	−	0.206118i	$-0.933917\pi$
$313$	−1.06172e9	−1.95705	−0.978527	+	0.206118i	$0.933917\pi$
$314$	0	0
$315$	2.00154e7	0.0360809
$316$	0	0
$317$	4.30645e8	0.759298	0.379649	−	0.925131i	$-0.376045\pi$
$317$	4.30645e8	0.759298	0.379649	+	0.925131i	$0.376045\pi$
$318$	0	0
$319$	1.34971e9	2.32795
$320$	0	0
$321$	3.72020e8	0.627768
$322$	0	0
$323$	2.58827e8	0.427367
$324$	0	0
$325$	−4.03199e8	−0.651521
$326$	0	0
$327$	2.61285e8	0.413235
$328$	0	0
$329$	−8.95150e8	−1.38583
$330$	0	0
$331$	7.77479e8	1.17839	0.589197	−	0.807989i	$-0.299444\pi$
$331$	7.77479e8	1.17839	0.589197	+	0.807989i	$0.299444\pi$
$332$	0	0
$333$	3.70062e8	0.549187
$334$	0	0
$335$	−6.28887e7	−0.0913936
$336$	0	0
$337$	2.38365e8	0.339265	0.169632	−	0.985507i	$-0.445742\pi$
$337$	2.38365e8	0.339265	0.169632	+	0.985507i	$0.445742\pi$
$338$	0	0
$339$	−5.91797e8	−0.825038
$340$	0	0
$341$	1.18776e9	1.62214
$342$	0	0
$343$	5.61739e8	0.751632
$344$	0	0
$345$	−2.06613e7	−0.0270888
$346$	0	0
$347$	9.17421e8	1.17873	0.589367	−	0.807866i	$-0.299377\pi$
$347$	9.17421e8	1.17873	0.589367	+	0.807866i	$0.299377\pi$
$348$	0	0
$349$	3.96693e8	0.499535	0.249768	−	0.968306i	$-0.419646\pi$
$349$	3.96693e8	0.499535	0.249768	+	0.968306i	$0.419646\pi$
$350$	0	0
$351$	1.02470e8	0.126480
$352$	0	0
$353$	−2.27149e8	−0.274852	−0.137426	−	0.990512i	$-0.543883\pi$
$353$	−2.27149e8	−0.274852	−0.137426	+	0.990512i	$0.543883\pi$
$354$	0	0
$355$	6.91538e6	0.00820383
$356$	0	0
$357$	1.77858e8	0.206888
$358$	0	0
$359$	−1.59175e9	−1.81570	−0.907848	−	0.419300i	$-0.862276\pi$
$359$	−1.59175e9	−1.81570	−0.907848	+	0.419300i	$0.862276\pi$
$360$	0	0
$361$	8.27714e8	0.925988
$362$	0	0
$363$	5.83917e8	0.640735
$364$	0	0
$365$	1.50572e8	0.162076
$366$	0	0
$367$	−2.90259e8	−0.306517	−0.153258	−	0.988186i	$-0.548977\pi$
$367$	−2.90259e8	−0.306517	−0.153258	+	0.988186i	$0.548977\pi$
$368$	0	0
$369$	2.62471e8	0.271950
$370$	0	0
$371$	−1.50348e9	−1.52858
$372$	0	0
$373$	6.51931e8	0.650460	0.325230	−	0.945635i	$-0.394558\pi$
$373$	6.51931e8	0.650460	0.325230	+	0.945635i	$0.394558\pi$
$374$	0	0
$375$	1.09213e8	0.106946
$376$	0	0
$377$	−1.09585e9	−1.05331
$378$	0	0
$379$	−8.58738e8	−0.810258	−0.405129	−	0.914259i	$-0.632773\pi$
$379$	−8.58738e8	−0.810258	−0.405129	+	0.914259i	$0.632773\pi$
$380$	0	0
$381$	1.78474e7	0.0165325
$382$	0	0
$383$	1.24441e9	1.13179	0.565897	−	0.824476i	$-0.308530\pi$
$383$	1.24441e9	1.13179	0.565897	+	0.824476i	$0.308530\pi$
$384$	0	0
$385$	−1.76048e8	−0.157224
$386$	0	0
$387$	−4.52012e8	−0.396426
$388$	0	0
$389$	4.35597e8	0.375198	0.187599	−	0.982246i	$-0.439929\pi$
$389$	4.35597e8	0.375198	0.187599	+	0.982246i	$0.439929\pi$
$390$	0	0
$391$	−1.83597e8	−0.155327
$392$	0	0
$393$	−3.24835e7	−0.0269953
$394$	0	0
$395$	7.68479e7	0.0627397
$396$	0	0
$397$	−1.07940e9	−0.865796	−0.432898	−	0.901443i	$-0.642509\pi$
$397$	−1.07940e9	−0.865796	−0.432898	+	0.901443i	$0.642509\pi$
$398$	0	0
$399$	1.18302e9	0.932367
$400$	0	0
$401$	−2.22249e9	−1.72122	−0.860608	−	0.509269i	$-0.829916\pi$
$401$	−2.22249e9	−1.72122	−0.860608	+	0.509269i	$0.829916\pi$
$402$	0	0
$403$	−9.64359e8	−0.733958
$404$	0	0
$405$	−1.38175e7	−0.0103356
$406$	0	0
$407$	−3.25492e9	−2.39310
$408$	0	0
$409$	3.70641e8	0.267869	0.133934	−	0.990990i	$-0.457239\pi$
$409$	3.70641e8	0.267869	0.133934	+	0.990990i	$0.457239\pi$
$410$	0	0
$411$	3.24126e8	0.230286
$412$	0	0
$413$	−2.69145e9	−1.88002
$414$	0	0
$415$	9.00362e7	0.0618371
$416$	0	0
$417$	−6.01867e8	−0.406466
$418$	0	0
$419$	1.05917e9	0.703422	0.351711	−	0.936109i	$-0.385600\pi$
$419$	1.05917e9	0.703422	0.351711	+	0.936109i	$0.385600\pi$
$420$	0	0
$421$	2.15633e9	1.40840	0.704202	−	0.709999i	$-0.251305\pi$
$421$	2.15633e9	1.40840	0.704202	+	0.709999i	$0.251305\pi$
$422$	0	0
$423$	6.17959e8	0.396980
$424$	0	0
$425$	4.83127e8	0.305281
$426$	0	0
$427$	7.45597e8	0.463454
$428$	0	0
$429$	−9.01284e8	−0.551139
$430$	0	0
$431$	1.94009e9	1.16722	0.583608	−	0.812036i	$-0.301641\pi$
$431$	1.94009e9	1.16722	0.583608	+	0.812036i	$0.301641\pi$
$432$	0	0
$433$	−4.84393e8	−0.286741	−0.143371	−	0.989669i	$-0.545794\pi$
$433$	−4.84393e8	−0.286741	−0.143371	+	0.989669i	$0.545794\pi$
$434$	0	0
$435$	1.47770e8	0.0860741
$436$	0	0
$437$	−1.22119e9	−0.700002
$438$	0	0
$439$	−3.68469e8	−0.207862	−0.103931	−	0.994585i	$-0.533142\pi$
$439$	−3.68469e8	−0.207862	−0.103931	+	0.994585i	$0.533142\pi$
$440$	0	0
$441$	2.12571e8	0.118024
$442$	0	0
$443$	1.75727e9	0.960340	0.480170	−	0.877175i	$-0.340575\pi$
$443$	1.75727e9	0.960340	0.480170	+	0.877175i	$0.340575\pi$
$444$	0	0
$445$	5.74893e7	0.0309262
$446$	0	0
$447$	−1.05936e9	−0.561005
$448$	0	0
$449$	2.93704e9	1.53126	0.765628	−	0.643284i	$-0.222429\pi$
$449$	2.93704e9	1.53126	0.765628	+	0.643284i	$0.222429\pi$
$450$	0	0
$451$	−2.30859e9	−1.18503
$452$	0	0
$453$	−3.05135e8	−0.154223
$454$	0	0
$455$	1.42936e8	0.0711380
$456$	0	0
$457$	−4.11305e8	−0.201585	−0.100792	−	0.994907i	$-0.532138\pi$
$457$	−4.11305e8	−0.201585	−0.100792	+	0.994907i	$0.532138\pi$
$458$	0	0
$459$	−1.22783e8	−0.0592642
$460$	0	0
$461$	1.92967e9	0.917341	0.458670	−	0.888606i	$-0.348326\pi$
$461$	1.92967e9	0.917341	0.458670	+	0.888606i	$0.348326\pi$
$462$	0	0
$463$	−3.07870e8	−0.144157	−0.0720783	−	0.997399i	$-0.522963\pi$
$463$	−3.07870e8	−0.144157	−0.0720783	+	0.997399i	$0.522963\pi$
$464$	0	0
$465$	1.30038e8	0.0599773
$466$	0	0
$467$	−1.50298e9	−0.682878	−0.341439	−	0.939904i	$-0.610914\pi$
$467$	−1.50298e9	−0.682878	−0.341439	+	0.939904i	$0.610914\pi$
$468$	0	0
$469$	−2.55425e9	−1.14329
$470$	0	0
$471$	−6.00216e8	−0.264688
$472$	0	0
$473$	3.97572e9	1.72744
$474$	0	0
$475$	3.21351e9	1.37579
$476$	0	0
$477$	1.03791e9	0.437872
$478$	0	0
$479$	3.92209e7	0.0163058	0.00815292	−	0.999967i	$-0.497405\pi$
$479$	3.92209e7	0.0163058	0.00815292	+	0.999967i	$0.497405\pi$
$480$	0	0
$481$	2.64272e9	1.08279
$482$	0	0
$483$	−8.39165e8	−0.338870
$484$	0	0
$485$	4.05465e8	0.161383
$486$	0	0
$487$	−2.80262e9	−1.09955	−0.549773	−	0.835314i	$-0.685286\pi$
$487$	−2.80262e9	−1.09955	−0.549773	+	0.835314i	$0.685286\pi$
$488$	0	0
$489$	8.74288e8	0.338122
$490$	0	0
$491$	1.71459e9	0.653694	0.326847	−	0.945077i	$-0.394014\pi$
$491$	1.71459e9	0.653694	0.326847	+	0.945077i	$0.394014\pi$
$492$	0	0
$493$	1.31309e9	0.493548
$494$	0	0
$495$	1.21533e8	0.0450377
$496$	0	0
$497$	2.80871e8	0.102626
$498$	0	0
$499$	2.46873e9	0.889449	0.444724	−	0.895667i	$-0.353302\pi$
$499$	2.46873e9	0.889449	0.444724	+	0.895667i	$0.353302\pi$
$500$	0	0
$501$	−1.39076e9	−0.494105
$502$	0	0
$503$	1.68007e9	0.588626	0.294313	−	0.955709i	$-0.404909\pi$
$503$	1.68007e9	0.588626	0.294313	+	0.955709i	$0.404909\pi$
$504$	0	0
$505$	−3.36864e8	−0.116395
$506$	0	0
$507$	−9.62444e8	−0.327980
$508$	0	0
$509$	−2.51153e9	−0.844162	−0.422081	−	0.906558i	$-0.638700\pi$
$509$	−2.51153e9	−0.844162	−0.422081	+	0.906558i	$0.638700\pi$
$510$	0	0
$511$	6.11555e9	2.02751
$512$	0	0
$513$	−8.16687e8	−0.267082
$514$	0	0
$515$	−1.77320e8	−0.0572049
$516$	0	0
$517$	−5.43532e9	−1.72985
$518$	0	0
$519$	−2.70766e9	−0.850177
$520$	0	0
$521$	4.17452e9	1.29323	0.646614	−	0.762818i	$-0.276185\pi$
$521$	4.17452e9	1.29323	0.646614	+	0.762818i	$0.276185\pi$
$522$	0	0
$523$	−2.83172e8	−0.0865554	−0.0432777	−	0.999063i	$-0.513780\pi$
$523$	−2.83172e8	−0.0865554	−0.0432777	+	0.999063i	$0.513780\pi$
$524$	0	0
$525$	2.20823e9	0.666018
$526$	0	0
$527$	1.15553e9	0.343909
$528$	0	0
$529$	−2.53858e9	−0.745584
$530$	0	0
$531$	1.85802e9	0.538542
$532$	0	0
$533$	1.87438e9	0.536182
$534$	0	0
$535$	−3.58242e8	−0.101144
$536$	0	0
$537$	−1.87158e9	−0.521554
$538$	0	0
$539$	−1.86969e9	−0.514292
$540$	0	0
$541$	−6.66923e9	−1.81086	−0.905431	−	0.424493i	$-0.860452\pi$
$541$	−6.66923e9	−1.81086	−0.905431	+	0.424493i	$0.860452\pi$
$542$	0	0
$543$	−2.01614e9	−0.540407
$544$	0	0
$545$	−2.51607e8	−0.0665787
$546$	0	0
$547$	−5.12201e9	−1.33809	−0.669045	−	0.743222i	$-0.733297\pi$
$547$	−5.12201e9	−1.33809	−0.669045	+	0.743222i	$0.733297\pi$
$548$	0	0
$549$	−5.14716e8	−0.132759
$550$	0	0
$551$	8.73398e9	2.22424
$552$	0	0
$553$	3.12121e9	0.784846
$554$	0	0
$555$	−3.56356e8	−0.0884828
$556$	0	0
$557$	−1.45491e9	−0.356732	−0.178366	−	0.983964i	$-0.557081\pi$
$557$	−1.45491e9	−0.356732	−0.178366	+	0.983964i	$0.557081\pi$
$558$	0	0
$559$	−3.22795e9	−0.781601
$560$	0	0
$561$	1.07995e9	0.258245
$562$	0	0
$563$	−7.50227e9	−1.77179	−0.885897	−	0.463882i	$-0.846456\pi$
$563$	−7.50227e9	−1.77179	−0.885897	+	0.463882i	$0.846456\pi$
$564$	0	0
$565$	5.69879e8	0.132927
$566$	0	0
$567$	−5.61202e8	−0.129294
$568$	0	0
$569$	−4.23661e9	−0.964108	−0.482054	−	0.876142i	$-0.660109\pi$
$569$	−4.23661e9	−0.964108	−0.482054	+	0.876142i	$0.660109\pi$
$570$	0	0
$571$	1.48301e9	0.333364	0.166682	−	0.986011i	$-0.446695\pi$
$571$	1.48301e9	0.333364	0.166682	+	0.986011i	$0.446695\pi$
$572$	0	0
$573$	−4.61457e9	−1.02468
$574$	0	0
$575$	−2.27948e9	−0.500032
$576$	0	0
$577$	−2.36381e8	−0.0512268	−0.0256134	−	0.999672i	$-0.508154\pi$
$577$	−2.36381e8	−0.0512268	−0.0256134	+	0.999672i	$0.508154\pi$
$578$	0	0
$579$	3.10677e8	0.0665173
$580$	0	0
$581$	3.65686e9	0.773555
$582$	0	0
$583$	−9.12908e9	−1.90804
$584$	0	0
$585$	−9.86745e7	−0.0203779
$586$	0	0
$587$	−1.92918e9	−0.393675	−0.196838	−	0.980436i	$-0.563067\pi$
$587$	−1.92918e9	−0.393675	−0.196838	+	0.980436i	$0.563067\pi$
$588$	0	0
$589$	7.68598e9	1.54987
$590$	0	0
$591$	2.14120e9	0.426678
$592$	0	0
$593$	−4.58212e9	−0.902350	−0.451175	−	0.892436i	$-0.648995\pi$
$593$	−4.58212e9	−0.902350	−0.451175	+	0.892436i	$0.648995\pi$
$594$	0	0
$595$	−1.71271e8	−0.0333329
$596$	0	0
$597$	−6.44273e8	−0.123925
$598$	0	0
$599$	−4.35564e9	−0.828053	−0.414027	−	0.910265i	$-0.635878\pi$
$599$	−4.35564e9	−0.828053	−0.414027	+	0.910265i	$0.635878\pi$
$600$	0	0
$601$	1.60073e9	0.300786	0.150393	−	0.988626i	$-0.451946\pi$
$601$	1.60073e9	0.300786	0.150393	+	0.988626i	$0.451946\pi$
$602$	0	0
$603$	1.76330e9	0.327504
$604$	0	0
$605$	−5.62291e8	−0.103233
$606$	0	0
$607$	3.16856e9	0.575044	0.287522	−	0.957774i	$-0.407169\pi$
$607$	3.16856e9	0.575044	0.287522	+	0.957774i	$0.407169\pi$
$608$	0	0
$609$	6.00172e9	1.07675
$610$	0	0
$611$	4.41302e9	0.782694
$612$	0	0
$613$	5.29300e9	0.928090	0.464045	−	0.885811i	$-0.346398\pi$
$613$	5.29300e9	0.928090	0.464045	+	0.885811i	$0.346398\pi$
$614$	0	0
$615$	−2.52749e8	−0.0438155
$616$	0	0
$617$	−2.19316e8	−0.0375899	−0.0187950	−	0.999823i	$-0.505983\pi$
$617$	−2.19316e8	−0.0375899	−0.0187950	+	0.999823i	$0.505983\pi$
$618$	0	0
$619$	4.66552e9	0.790648	0.395324	−	0.918542i	$-0.370632\pi$
$619$	4.66552e9	0.790648	0.395324	+	0.918542i	$0.370632\pi$
$620$	0	0
$621$	5.79310e8	0.0970712
$622$	0	0
$623$	2.33495e9	0.386874
$624$	0	0
$625$	5.94554e9	0.974116
$626$	0	0
$627$	7.18326e9	1.16382
$628$	0	0
$629$	−3.16660e9	−0.507359
$630$	0	0
$631$	−3.93963e9	−0.624241	−0.312121	−	0.950042i	$-0.601039\pi$
$631$	−3.93963e9	−0.624241	−0.312121	+	0.950042i	$0.601039\pi$
$632$	0	0
$633$	−2.96295e9	−0.464314
$634$	0	0
$635$	−1.71864e7	−0.00266365
$636$	0	0
$637$	1.51803e9	0.232698
$638$	0	0
$639$	−1.93897e8	−0.0293979
$640$	0	0
$641$	−6.74548e9	−1.01160	−0.505801	−	0.862650i	$-0.668803\pi$
$641$	−6.74548e9	−1.01160	−0.505801	+	0.862650i	$0.668803\pi$
$642$	0	0
$643$	7.70175e9	1.14249	0.571243	−	0.820781i	$-0.306461\pi$
$643$	7.70175e9	1.14249	0.571243	+	0.820781i	$0.306461\pi$
$644$	0	0
$645$	4.35271e8	0.0638705
$646$	0	0
$647$	9.65642e9	1.40169	0.700844	−	0.713315i	$-0.252807\pi$
$647$	9.65642e9	1.40169	0.700844	+	0.713315i	$0.252807\pi$
$648$	0	0
$649$	−1.63424e10	−2.34671
$650$	0	0
$651$	5.28156e9	0.750290
$652$	0	0
$653$	2.95427e9	0.415197	0.207599	−	0.978214i	$-0.433435\pi$
$653$	2.95427e9	0.415197	0.207599	+	0.978214i	$0.433435\pi$
$654$	0	0
$655$	3.12804e7	0.00434938
$656$	0	0
$657$	−4.22181e9	−0.580791
$658$	0	0
$659$	−8.90170e9	−1.21164	−0.605821	−	0.795601i	$-0.707155\pi$
$659$	−8.90170e9	−1.21164	−0.605821	+	0.795601i	$0.707155\pi$
$660$	0	0
$661$	1.26579e10	1.70474	0.852369	−	0.522941i	$-0.175165\pi$
$661$	1.26579e10	1.70474	0.852369	+	0.522941i	$0.175165\pi$
$662$	0	0
$663$	−8.76826e8	−0.116847
$664$	0	0
$665$	−1.13920e9	−0.150219
$666$	0	0
$667$	−6.19538e9	−0.808402
$668$	0	0
$669$	7.31929e9	0.945100
$670$	0	0
$671$	4.52724e9	0.578502
$672$	0	0
$673$	7.30112e9	0.923287	0.461644	−	0.887065i	$-0.347260\pi$
$673$	7.30112e9	0.923287	0.461644	+	0.887065i	$0.347260\pi$
$674$	0	0
$675$	−1.52443e9	−0.190785
$676$	0	0
$677$	−6.15717e9	−0.762642	−0.381321	−	0.924443i	$-0.624531\pi$
$677$	−6.15717e9	−0.762642	−0.381321	+	0.924443i	$0.624531\pi$
$678$	0	0
$679$	1.64681e10	2.01883
$680$	0	0
$681$	5.96779e8	0.0724100
$682$	0	0
$683$	−1.07209e10	−1.28753	−0.643766	−	0.765222i	$-0.722629\pi$
$683$	−1.07209e10	−1.28753	−0.643766	+	0.765222i	$0.722629\pi$
$684$	0	0
$685$	−3.12121e8	−0.0371027
$686$	0	0
$687$	−4.09801e9	−0.482197
$688$	0	0
$689$	7.41204e9	0.863318
$690$	0	0
$691$	1.40404e10	1.61885	0.809425	−	0.587223i	$-0.199779\pi$
$691$	1.40404e10	1.61885	0.809425	+	0.587223i	$0.199779\pi$
$692$	0	0
$693$	4.93611e9	0.563402
$694$	0	0
$695$	5.79575e8	0.0654882
$696$	0	0
$697$	−2.24594e9	−0.251237
$698$	0	0
$699$	−2.41491e9	−0.267443
$700$	0	0
$701$	3.68495e9	0.404035	0.202018	−	0.979382i	$-0.435250\pi$
$701$	3.68495e9	0.404035	0.202018	+	0.979382i	$0.435250\pi$
$702$	0	0
$703$	−2.10626e10	−2.28648
$704$	0	0
$705$	−5.95071e8	−0.0639598
$706$	0	0
$707$	−1.36819e10	−1.45605
$708$	0	0
$709$	−1.55532e10	−1.63892	−0.819461	−	0.573135i	$-0.805727\pi$
$709$	−1.55532e10	−1.63892	−0.819461	+	0.573135i	$0.805727\pi$
$710$	0	0
$711$	−2.15470e9	−0.224824
$712$	0	0
$713$	−5.45198e9	−0.563302
$714$	0	0
$715$	8.67903e8	0.0887973
$716$	0	0
$717$	7.27069e9	0.736645
$718$	0	0
$719$	−7.47108e9	−0.749605	−0.374802	−	0.927105i	$-0.622289\pi$
$719$	−7.47108e9	−0.749605	−0.374802	+	0.927105i	$0.622289\pi$
$720$	0	0
$721$	−7.20194e9	−0.715609
$722$	0	0
$723$	5.36779e9	0.528216
$724$	0	0
$725$	1.63029e10	1.58884
$726$	0	0
$727$	1.77786e9	0.171604	0.0858020	−	0.996312i	$-0.472655\pi$
$727$	1.77786e9	0.171604	0.0858020	+	0.996312i	$0.472655\pi$
$728$	0	0
$729$	3.87420e8	0.0370370
$730$	0	0
$731$	3.86783e9	0.366233
$732$	0	0
$733$	1.33140e10	1.24866	0.624331	−	0.781160i	$-0.285372\pi$
$733$	1.33140e10	1.24866	0.624331	+	0.781160i	$0.285372\pi$
$734$	0	0
$735$	−2.04698e8	−0.0190155
$736$	0	0
$737$	−1.55093e10	−1.42711
$738$	0	0
$739$	3.06064e9	0.278970	0.139485	−	0.990224i	$-0.455455\pi$
$739$	3.06064e9	0.278970	0.139485	+	0.990224i	$0.455455\pi$
$740$	0	0
$741$	−5.83220e9	−0.526585
$742$	0	0
$743$	−1.03506e10	−0.925774	−0.462887	−	0.886417i	$-0.653186\pi$
$743$	−1.03506e10	−0.925774	−0.462887	+	0.886417i	$0.653186\pi$
$744$	0	0
$745$	1.02012e9	0.0903869
$746$	0	0
$747$	−2.52448e9	−0.221590
$748$	0	0
$749$	−1.45501e10	−1.26526
$750$	0	0
$751$	1.79720e9	0.154831	0.0774153	−	0.996999i	$-0.475333\pi$
$751$	1.79720e9	0.154831	0.0774153	+	0.996999i	$0.475333\pi$
$752$	0	0
$753$	9.12733e9	0.779043
$754$	0	0
$755$	2.93834e8	0.0248478
$756$	0	0
$757$	−1.61244e10	−1.35098	−0.675488	−	0.737371i	$-0.736067\pi$
$757$	−1.61244e10	−1.35098	−0.675488	+	0.737371i	$0.736067\pi$
$758$	0	0
$759$	−5.09539e9	−0.422991
$760$	0	0
$761$	7.65172e9	0.629380	0.314690	−	0.949195i	$-0.398099\pi$
$761$	7.65172e9	0.629380	0.314690	+	0.949195i	$0.398099\pi$
$762$	0	0
$763$	−1.02191e10	−0.832872
$764$	0	0
$765$	1.18235e8	0.00954841
$766$	0	0
$767$	1.32687e10	1.06180
$768$	0	0
$769$	−1.90645e10	−1.51176	−0.755882	−	0.654708i	$-0.772792\pi$
$769$	−1.90645e10	−1.51176	−0.755882	+	0.654708i	$0.772792\pi$
$770$	0	0
$771$	−4.44119e9	−0.348987
$772$	0	0
$773$	6.37260e9	0.496236	0.248118	−	0.968730i	$-0.420188\pi$
$773$	6.37260e9	0.496236	0.248118	+	0.968730i	$0.420188\pi$
$774$	0	0
$775$	1.43467e10	1.10712
$776$	0	0
$777$	−1.44735e10	−1.10688
$778$	0	0
$779$	−1.49389e10	−1.13223
$780$	0	0
$781$	1.70544e9	0.128102
$782$	0	0
$783$	−4.14323e9	−0.308442
$784$	0	0
$785$	5.77986e8	0.0426455
$786$	0	0
$787$	2.68873e9	0.196623	0.0983116	−	0.995156i	$-0.468656\pi$
$787$	2.68873e9	0.196623	0.0983116	+	0.995156i	$0.468656\pi$
$788$	0	0
$789$	−1.13849e10	−0.825203
$790$	0	0
$791$	2.31458e10	1.66286
$792$	0	0
$793$	−3.67574e9	−0.261751
$794$	0	0
$795$	−9.99472e8	−0.0705482
$796$	0	0
$797$	−5.10359e9	−0.357085	−0.178542	−	0.983932i	$-0.557138\pi$
$797$	−5.10359e9	−0.357085	−0.178542	+	0.983932i	$0.557138\pi$
$798$	0	0
$799$	−5.28783e9	−0.366744
$800$	0	0
$801$	−1.61191e9	−0.110822
$802$	0	0
$803$	3.71334e10	2.53082
$804$	0	0
$805$	8.08085e8	0.0545973
$806$	0	0
$807$	5.76363e9	0.386046
$808$	0	0
$809$	2.37206e10	1.57509	0.787547	−	0.616254i	$-0.211351\pi$
$809$	2.37206e10	1.57509	0.787547	+	0.616254i	$0.211351\pi$
$810$	0	0
$811$	−1.48151e10	−0.975287	−0.487644	−	0.873043i	$-0.662143\pi$
$811$	−1.48151e10	−0.975287	−0.487644	+	0.873043i	$0.662143\pi$
$812$	0	0
$813$	−6.02957e9	−0.393522
$814$	0	0
$815$	−8.41907e8	−0.0544769
$816$	0	0
$817$	2.57269e10	1.65048
$818$	0	0
$819$	−4.00770e9	−0.254919
$820$	0	0
$821$	7.38885e9	0.465989	0.232994	−	0.972478i	$-0.425148\pi$
$821$	7.38885e9	0.465989	0.232994	+	0.972478i	$0.425148\pi$
$822$	0	0
$823$	−1.23070e10	−0.769580	−0.384790	−	0.923004i	$-0.625726\pi$
$823$	−1.23070e10	−0.769580	−0.384790	+	0.923004i	$0.625726\pi$
$824$	0	0
$825$	1.34083e10	0.831351
$826$	0	0
$827$	−6.19040e9	−0.380583	−0.190292	−	0.981728i	$-0.560943\pi$
$827$	−6.19040e9	−0.380583	−0.190292	+	0.981728i	$0.560943\pi$
$828$	0	0
$829$	6.54741e9	0.399143	0.199572	−	0.979883i	$-0.436045\pi$
$829$	6.54741e9	0.399143	0.199572	+	0.979883i	$0.436045\pi$
$830$	0	0
$831$	4.64882e9	0.281021
$832$	0	0
$833$	−1.81896e9	−0.109035
$834$	0	0
$835$	1.33925e9	0.0796083
$836$	0	0
$837$	−3.64608e9	−0.214925
$838$	0	0
$839$	7.98560e9	0.466810	0.233405	−	0.972380i	$-0.425013\pi$
$839$	7.98560e9	0.466810	0.233405	+	0.972380i	$0.425013\pi$
$840$	0	0
$841$	2.70595e10	1.56868
$842$	0	0
$843$	1.84906e10	1.06305
$844$	0	0
$845$	9.26798e8	0.0528429
$846$	0	0
$847$	−2.28377e10	−1.29140
$848$	0	0
$849$	2.10601e9	0.118109
$850$	0	0
$851$	1.49406e10	0.831024
$852$	0	0
$853$	4.32266e9	0.238467	0.119234	−	0.992866i	$-0.461956\pi$
$853$	4.32266e9	0.238467	0.119234	+	0.992866i	$0.461956\pi$
$854$	0	0
$855$	7.86439e8	0.0430312
$856$	0	0
$857$	2.25925e10	1.22612	0.613058	−	0.790038i	$-0.289939\pi$
$857$	2.25925e10	1.22612	0.613058	+	0.790038i	$0.289939\pi$
$858$	0	0
$859$	4.20339e9	0.226268	0.113134	−	0.993580i	$-0.463911\pi$
$859$	4.20339e9	0.226268	0.113134	+	0.993580i	$0.463911\pi$
$860$	0	0
$861$	−1.02655e10	−0.548113
$862$	0	0
$863$	−1.09766e10	−0.581338	−0.290669	−	0.956824i	$-0.593878\pi$
$863$	−1.09766e10	−0.581338	−0.290669	+	0.956824i	$0.593878\pi$
$864$	0	0
$865$	2.60738e9	0.136977
$866$	0	0
$867$	−1.00285e10	−0.522600
$868$	0	0
$869$	1.89519e10	0.979677
$870$	0	0
$871$	1.25923e10	0.645713
$872$	0	0
$873$	−1.13686e10	−0.578306
$874$	0	0
$875$	−4.27144e9	−0.215549
$876$	0	0
$877$	−1.43474e10	−0.718246	−0.359123	−	0.933290i	$-0.616924\pi$
$877$	−1.43474e10	−0.718246	−0.359123	+	0.933290i	$0.616924\pi$
$878$	0	0
$879$	−1.75720e10	−0.872689
$880$	0	0
$881$	−1.65119e10	−0.813546	−0.406773	−	0.913529i	$-0.633346\pi$
$881$	−1.65119e10	−0.813546	−0.406773	+	0.913529i	$0.633346\pi$
$882$	0	0
$883$	−8.58509e9	−0.419645	−0.209823	−	0.977739i	$-0.567289\pi$
$883$	−8.58509e9	−0.419645	−0.209823	+	0.977739i	$0.567289\pi$
$884$	0	0
$885$	−1.78920e9	−0.0867678
$886$	0	0
$887$	−3.55541e9	−0.171063	−0.0855316	−	0.996335i	$-0.527259\pi$
$887$	−3.55541e9	−0.171063	−0.0855316	+	0.996335i	$0.527259\pi$
$888$	0	0
$889$	−6.98033e8	−0.0333211
$890$	0	0
$891$	−3.40760e9	−0.161390
$892$	0	0
$893$	−3.51719e10	−1.65278
$894$	0	0
$895$	1.80227e9	0.0840307
$896$	0	0
$897$	4.13702e9	0.191388
$898$	0	0
$899$	3.89926e10	1.78988
$900$	0	0
$901$	−8.88135e9	−0.404522
$902$	0	0
$903$	1.76787e10	0.798993
$904$	0	0
$905$	1.94147e9	0.0870683
$906$	0	0
$907$	2.50681e9	0.111557	0.0557783	−	0.998443i	$-0.482236\pi$
$907$	2.50681e9	0.111557	0.0557783	+	0.998443i	$0.482236\pi$
$908$	0	0
$909$	9.44515e9	0.417095
$910$	0	0
$911$	−1.82057e10	−0.797799	−0.398899	−	0.916995i	$-0.630608\pi$
$911$	−1.82057e10	−0.797799	−0.398899	+	0.916995i	$0.630608\pi$
$912$	0	0
$913$	2.22043e10	0.965583
$914$	0	0
$915$	4.95653e8	0.0213896
$916$	0	0
$917$	1.27047e9	0.0544089
$918$	0	0
$919$	−2.41571e9	−0.102669	−0.0513347	−	0.998682i	$-0.516348\pi$
$919$	−2.41571e9	−0.102669	−0.0513347	+	0.998682i	$0.516348\pi$
$920$	0	0
$921$	−9.93366e9	−0.418987
$922$	0	0
$923$	−1.38467e9	−0.0579616
$924$	0	0
$925$	−3.93154e10	−1.63330
$926$	0	0
$927$	4.97179e9	0.204991
$928$	0	0
$929$	1.61139e10	0.659396	0.329698	−	0.944087i	$-0.393053\pi$
$929$	1.61139e10	0.659396	0.329698	+	0.944087i	$0.393053\pi$
$930$	0	0
$931$	−1.20988e10	−0.491380
$932$	0	0
$933$	−1.19435e10	−0.481443
$934$	0	0
$935$	−1.03995e9	−0.0416075
$936$	0	0
$937$	−3.64699e10	−1.44826	−0.724129	−	0.689664i	$-0.757758\pi$
$937$	−3.64699e10	−1.44826	−0.724129	+	0.689664i	$0.757758\pi$
$938$	0	0
$939$	−2.86663e10	−1.12991
$940$	0	0
$941$	−2.19999e10	−0.860711	−0.430356	−	0.902659i	$-0.641612\pi$
$941$	−2.19999e10	−0.860711	−0.430356	+	0.902659i	$0.641612\pi$
$942$	0	0
$943$	1.05968e10	0.411512
$944$	0	0
$945$	5.40416e8	0.0208313
$946$	0	0
$947$	−1.52734e10	−0.584402	−0.292201	−	0.956357i	$-0.594388\pi$
$947$	−1.52734e10	−0.584402	−0.292201	+	0.956357i	$0.594388\pi$
$948$	0	0
$949$	−3.01492e10	−1.14510
$950$	0	0
$951$	1.16274e10	0.438381
$952$	0	0
$953$	−8.79256e9	−0.329071	−0.164536	−	0.986371i	$-0.552613\pi$
$953$	−8.79256e9	−0.329071	−0.164536	+	0.986371i	$0.552613\pi$
$954$	0	0
$955$	4.44366e9	0.165093
$956$	0	0
$957$	3.64423e10	1.34404
$958$	0	0
$959$	−1.26769e10	−0.464139
$960$	0	0
$961$	6.80124e9	0.247205
$962$	0	0
$963$	1.00445e10	0.362442
$964$	0	0
$965$	−2.99171e8	−0.0107170
$966$	0	0
$967$	4.31656e10	1.53513	0.767566	−	0.640970i	$-0.221468\pi$
$967$	4.31656e10	1.53513	0.767566	+	0.640970i	$0.221468\pi$
$968$	0	0
$969$	6.98833e9	0.246740
$970$	0	0
$971$	−2.84776e10	−0.998241	−0.499121	−	0.866533i	$-0.666344\pi$
$971$	−2.84776e10	−0.998241	−0.499121	+	0.866533i	$0.666344\pi$
$972$	0	0
$973$	2.35397e10	0.819229
$974$	0	0
$975$	−1.08864e10	−0.376156
$976$	0	0
$977$	−2.49908e9	−0.0857334	−0.0428667	−	0.999081i	$-0.513649\pi$
$977$	−2.49908e9	−0.0857334	−0.0428667	+	0.999081i	$0.513649\pi$
$978$	0	0
$979$	1.41777e10	0.482912
$980$	0	0
$981$	7.05468e9	0.238581
$982$	0	0
$983$	−4.08680e10	−1.37229	−0.686145	−	0.727465i	$-0.740698\pi$
$983$	−4.08680e10	−1.37229	−0.686145	+	0.727465i	$0.740698\pi$
$984$	0	0
$985$	−2.06189e9	−0.0687448
$986$	0	0
$987$	−2.41691e10	−0.800110
$988$	0	0
$989$	−1.82491e10	−0.599867
$990$	0	0
$991$	8.31493e9	0.271394	0.135697	−	0.990750i	$-0.456673\pi$
$991$	8.31493e9	0.271394	0.135697	+	0.990750i	$0.456673\pi$
$992$	0	0
$993$	2.09919e10	0.680346
$994$	0	0
$995$	6.20411e8	0.0199664
$996$	0	0
$997$	4.11316e10	1.31445	0.657223	−	0.753696i	$-0.271731\pi$
$997$	4.11316e10	1.31445	0.657223	+	0.753696i	$0.271731\pi$
$998$	0	0
$999$	9.99168e9	0.317073

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	48.8.a.f.1.1		1
3.2	odd	2		144.8.a.f.1.1		1
4.3	odd	2		24.8.a.a.1.1	✓	1
8.3	odd	2		192.8.a.l.1.1		1
8.5	even	2		192.8.a.d.1.1		1
12.11	even	2		72.8.a.c.1.1		1
20.3	even	4		600.8.f.e.49.1		2
20.7	even	4		600.8.f.e.49.2		2
20.19	odd	2		600.8.a.e.1.1		1
24.5	odd	2		576.8.a.o.1.1		1
24.11	even	2		576.8.a.p.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
24.8.a.a.1.1	✓	1	4.3	odd	2
48.8.a.f.1.1		1	1.1	even	1	trivial
72.8.a.c.1.1		1	12.11	even	2
144.8.a.f.1.1		1	3.2	odd	2
192.8.a.d.1.1		1	8.5	even	2
192.8.a.l.1.1		1	8.3	odd	2
576.8.a.o.1.1		1	24.5	odd	2
576.8.a.p.1.1		1	24.11	even	2
600.8.a.e.1.1		1	20.19	odd	2
600.8.f.e.49.1		2	20.3	even	4
600.8.f.e.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:	\( N \)	\(=\)	\( 48 = 2^{4} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	48.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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