LMFDB - Embedded newform 240.8.a.k.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [240,8,Mod(1,240)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(240, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("240.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 240 = 2^{4} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	240.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:	$74.9724061162$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 30)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	240.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+27.0000 q^{3} -125.000 q^{5} +1084.00 q^{7} +729.000 q^{9} +O(q^{10})$

$q+27.0000 q^{3} -125.000 q^{5} +1084.00 q^{7} +729.000 q^{9} -6552.00 q^{11} +2522.00 q^{13} -3375.00 q^{15} +30486.0 q^{17} -12020.0 q^{19} +29268.0 q^{21} +3528.00 q^{23} +15625.0 q^{25} +19683.0 q^{27} +84930.0 q^{29} +94168.0 q^{31} -176904. q^{33} -135500. q^{35} -509974. q^{37} +68094.0 q^{39} +841002. q^{41} -889052. q^{43} -91125.0 q^{45} +852264. q^{47} +351513. q^{49} +823122. q^{51} +1.59448e6 q^{53} +819000. q^{55} -324540. q^{57} -752040. q^{59} +1.53870e6 q^{61} +790236. q^{63} -315250. q^{65} +947524. q^{67} +95256.0 q^{69} +3.82493e6 q^{71} -913678. q^{73} +421875. q^{75} -7.10237e6 q^{77} -1.62188e6 q^{79} +531441. q^{81} -1.47001e6 q^{83} -3.81075e6 q^{85} +2.29311e6 q^{87} +2.37501e6 q^{89} +2.73385e6 q^{91} +2.54254e6 q^{93} +1.50250e6 q^{95} +1.41388e7 q^{97} -4.77641e6 q^{99} +O(q^{100})$

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
$5$	−125.000	−0.447214
$5$	−125.000	−0.447214
$6$	0	0
$7$	1084.00	1.19450	0.597250	−	0.802055i	$-0.296260\pi$
$7$	1084.00	1.19450	0.597250	+	0.802055i	$0.296260\pi$
$8$	0	0
$9$	729.000	0.333333
$10$	0	0
$11$	−6552.00	−1.48422	−0.742112	−	0.670276i	$-0.766176\pi$
$11$	−6552.00	−1.48422	−0.742112	+	0.670276i	$0.766176\pi$
$12$	0	0
$13$	2522.00	0.318378	0.159189	−	0.987248i	$-0.449112\pi$
$13$	2522.00	0.318378	0.159189	+	0.987248i	$0.449112\pi$
$14$	0	0
$15$	−3375.00	−0.258199
$16$	0	0
$17$	30486.0	1.50497	0.752487	−	0.658607i	$-0.228854\pi$
$17$	30486.0	1.50497	0.752487	+	0.658607i	$0.228854\pi$
$18$	0	0
$19$	−12020.0	−0.402038	−0.201019	−	0.979587i	$-0.564425\pi$
$19$	−12020.0	−0.402038	−0.201019	+	0.979587i	$0.564425\pi$
$20$	0	0
$21$	29268.0	0.689645
$22$	0	0
$23$	3528.00	0.0604618	0.0302309	−	0.999543i	$-0.490376\pi$
$23$	3528.00	0.0604618	0.0302309	+	0.999543i	$0.490376\pi$
$24$	0	0
$25$	15625.0	0.200000
$26$	0	0
$27$	19683.0	0.192450
$28$	0	0
$29$	84930.0	0.646648	0.323324	−	0.946288i	$-0.395200\pi$
$29$	84930.0	0.646648	0.323324	+	0.946288i	$0.395200\pi$
$30$	0	0
$31$	94168.0	0.567724	0.283862	−	0.958865i	$-0.408384\pi$
$31$	94168.0	0.567724	0.283862	+	0.958865i	$0.408384\pi$
$32$	0	0
$33$	−176904.	−0.856917
$34$	0	0
$35$	−135500.	−0.534197
$36$	0	0
$37$	−509974.	−1.65517	−0.827584	−	0.561342i	$-0.810285\pi$
$37$	−509974.	−1.65517	−0.827584	+	0.561342i	$0.810285\pi$
$38$	0	0
$39$	68094.0	0.183816
$40$	0	0
$41$	841002.	1.90570	0.952848	−	0.303449i	$-0.0981381\pi$
$41$	841002.	1.90570	0.952848	+	0.303449i	$0.0981381\pi$
$42$	0	0
$43$	−889052.	−1.70525	−0.852624	−	0.522525i	$-0.824990\pi$
$43$	−889052.	−1.70525	−0.852624	+	0.522525i	$0.824990\pi$
$44$	0	0
$45$	−91125.0	−0.149071
$46$	0	0
$47$	852264.	1.19738	0.598690	−	0.800981i	$-0.295688\pi$
$47$	852264.	1.19738	0.598690	+	0.800981i	$0.295688\pi$
$48$	0	0
$49$	351513.	0.426830
$50$	0	0
$51$	823122.	0.868898
$52$	0	0
$53$	1.59448e6	1.47114	0.735570	−	0.677449i	$-0.236914\pi$
$53$	1.59448e6	1.47114	0.735570	+	0.677449i	$0.236914\pi$
$54$	0	0
$55$	819000.	0.663765
$56$	0	0
$57$	−324540.	−0.232117
$58$	0	0
$59$	−752040.	−0.476715	−0.238358	−	0.971177i	$-0.576609\pi$
$59$	−752040.	−0.476715	−0.238358	+	0.971177i	$0.576609\pi$
$60$	0	0
$61$	1.53870e6	0.867961	0.433980	−	0.900922i	$-0.357109\pi$
$61$	1.53870e6	0.867961	0.433980	+	0.900922i	$0.357109\pi$
$62$	0	0
$63$	790236.	0.398167
$64$	0	0
$65$	−315250.	−0.142383
$66$	0	0
$67$	947524.	0.384883	0.192441	−	0.981308i	$-0.438359\pi$
$67$	947524.	0.384883	0.192441	+	0.981308i	$0.438359\pi$
$68$	0	0
$69$	95256.0	0.0349076
$70$	0	0
$71$	3.82493e6	1.26829	0.634146	−	0.773214i	$-0.281352\pi$
$71$	3.82493e6	1.26829	0.634146	+	0.773214i	$0.281352\pi$
$72$	0	0
$73$	−913678.	−0.274893	−0.137446	−	0.990509i	$-0.543889\pi$
$73$	−913678.	−0.274893	−0.137446	+	0.990509i	$0.543889\pi$
$74$	0	0
$75$	421875.	0.115470
$76$	0	0
$77$	−7.10237e6	−1.77291
$78$	0	0
$79$	−1.62188e6	−0.370104	−0.185052	−	0.982729i	$-0.559245\pi$
$79$	−1.62188e6	−0.370104	−0.185052	+	0.982729i	$0.559245\pi$
$80$	0	0
$81$	531441.	0.111111
$82$	0	0
$83$	−1.47001e6	−0.282194	−0.141097	−	0.989996i	$-0.545063\pi$
$83$	−1.47001e6	−0.282194	−0.141097	+	0.989996i	$0.545063\pi$
$84$	0	0
$85$	−3.81075e6	−0.673045
$86$	0	0
$87$	2.29311e6	0.373343
$88$	0	0
$89$	2.37501e6	0.357109	0.178554	−	0.983930i	$-0.442858\pi$
$89$	2.37501e6	0.357109	0.178554	+	0.983930i	$0.442858\pi$
$90$	0	0
$91$	2.73385e6	0.380303
$92$	0	0
$93$	2.54254e6	0.327776
$94$	0	0
$95$	1.50250e6	0.179797
$96$	0	0
$97$	1.41388e7	1.57294	0.786468	−	0.617631i	$-0.211907\pi$
$97$	1.41388e7	1.57294	0.786468	+	0.617631i	$0.211907\pi$
$98$	0	0
$99$	−4.77641e6	−0.494741
$100$	0	0
$101$	2.11588e6	0.204346	0.102173	−	0.994767i	$-0.467420\pi$
$101$	2.11588e6	0.204346	0.102173	+	0.994767i	$0.467420\pi$
$102$	0	0
$103$	−1.71017e7	−1.54208	−0.771042	−	0.636785i	$-0.780264\pi$
$103$	−1.71017e7	−1.54208	−0.771042	+	0.636785i	$0.780264\pi$
$104$	0	0
$105$	−3.65850e6	−0.308419
$106$	0	0
$107$	1.73673e7	1.37053	0.685267	−	0.728292i	$-0.259686\pi$
$107$	1.73673e7	1.37053	0.685267	+	0.728292i	$0.259686\pi$
$108$	0	0
$109$	2.47014e7	1.82696	0.913479	−	0.406885i	$-0.133385\pi$
$109$	2.47014e7	1.82696	0.913479	+	0.406885i	$0.133385\pi$
$110$	0	0
$111$	−1.37693e7	−0.955612
$112$	0	0
$113$	1.48658e7	0.969202	0.484601	−	0.874735i	$-0.338965\pi$
$113$	1.48658e7	0.969202	0.484601	+	0.874735i	$0.338965\pi$
$114$	0	0
$115$	−441000.	−0.0270393
$116$	0	0
$117$	1.83854e6	0.106126
$118$	0	0
$119$	3.30468e7	1.79769
$120$	0	0
$121$	2.34415e7	1.20292
$122$	0	0
$123$	2.27071e7	1.10025
$124$	0	0
$125$	−1.95312e6	−0.0894427
$126$	0	0
$127$	3.04104e6	0.131738	0.0658688	−	0.997828i	$-0.479018\pi$
$127$	3.04104e6	0.131738	0.0658688	+	0.997828i	$0.479018\pi$
$128$	0	0
$129$	−2.40044e7	−0.984525
$130$	0	0
$131$	3.09449e6	0.120265	0.0601325	−	0.998190i	$-0.480848\pi$
$131$	3.09449e6	0.120265	0.0601325	+	0.998190i	$0.480848\pi$
$132$	0	0
$133$	−1.30297e7	−0.480234
$134$	0	0
$135$	−2.46038e6	−0.0860663
$136$	0	0
$137$	2.05296e7	0.682117	0.341058	−	0.940042i	$-0.389215\pi$
$137$	2.05296e7	0.682117	0.341058	+	0.940042i	$0.389215\pi$
$138$	0	0
$139$	6.15746e7	1.94469	0.972344	−	0.233553i	$-0.0750353\pi$
$139$	6.15746e7	1.94469	0.972344	+	0.233553i	$0.0750353\pi$
$140$	0	0
$141$	2.30111e7	0.691307
$142$	0	0
$143$	−1.65241e7	−0.472545
$144$	0	0
$145$	−1.06162e7	−0.289190
$146$	0	0
$147$	9.49085e6	0.246431
$148$	0	0
$149$	−1.47881e7	−0.366236	−0.183118	−	0.983091i	$-0.558619\pi$
$149$	−1.47881e7	−0.366236	−0.183118	+	0.983091i	$0.558619\pi$
$150$	0	0
$151$	2.95314e7	0.698014	0.349007	−	0.937120i	$-0.386519\pi$
$151$	2.95314e7	0.698014	0.349007	+	0.937120i	$0.386519\pi$
$152$	0	0
$153$	2.22243e7	0.501658
$154$	0	0
$155$	−1.17710e7	−0.253894
$156$	0	0
$157$	3.06106e7	0.631281	0.315641	−	0.948879i	$-0.397781\pi$
$157$	3.06106e7	0.631281	0.315641	+	0.948879i	$0.397781\pi$
$158$	0	0
$159$	4.30510e7	0.849363
$160$	0	0
$161$	3.82435e6	0.0722216
$162$	0	0
$163$	3.95944e7	0.716106	0.358053	−	0.933701i	$-0.383441\pi$
$163$	3.95944e7	0.716106	0.358053	+	0.933701i	$0.383441\pi$
$164$	0	0
$165$	2.21130e7	0.383225
$166$	0	0
$167$	−7.49947e7	−1.24601	−0.623007	−	0.782216i	$-0.714089\pi$
$167$	−7.49947e7	−1.24601	−0.623007	+	0.782216i	$0.714089\pi$
$168$	0	0
$169$	−5.63880e7	−0.898635
$170$	0	0
$171$	−8.76258e6	−0.134013
$172$	0	0
$173$	9.74475e7	1.43090	0.715450	−	0.698664i	$-0.246222\pi$
$173$	9.74475e7	1.43090	0.715450	+	0.698664i	$0.246222\pi$
$174$	0	0
$175$	1.69375e7	0.238900
$176$	0	0
$177$	−2.03051e7	−0.275232
$178$	0	0
$179$	6.84578e7	0.892149	0.446075	−	0.894996i	$-0.352822\pi$
$179$	6.84578e7	0.892149	0.446075	+	0.894996i	$0.352822\pi$
$180$	0	0
$181$	−1.32622e8	−1.66242	−0.831209	−	0.555960i	$-0.812351\pi$
$181$	−1.32622e8	−1.66242	−0.831209	+	0.555960i	$0.812351\pi$
$182$	0	0
$183$	4.15450e7	0.501117
$184$	0	0
$185$	6.37468e7	0.740213
$186$	0	0
$187$	−1.99744e8	−2.23372
$188$	0	0
$189$	2.13364e7	0.229882
$190$	0	0
$191$	−1.11671e8	−1.15964	−0.579820	−	0.814745i	$-0.696877\pi$
$191$	−1.11671e8	−1.15964	−0.579820	+	0.814745i	$0.696877\pi$
$192$	0	0
$193$	4.76490e7	0.477093	0.238547	−	0.971131i	$-0.423329\pi$
$193$	4.76490e7	0.477093	0.238547	+	0.971131i	$0.423329\pi$
$194$	0	0
$195$	−8.51175e6	−0.0822049
$196$	0	0
$197$	−1.19728e8	−1.11574	−0.557869	−	0.829929i	$-0.688381\pi$
$197$	−1.19728e8	−1.11574	−0.557869	+	0.829929i	$0.688381\pi$
$198$	0	0
$199$	−5.01434e7	−0.451053	−0.225527	−	0.974237i	$-0.572410\pi$
$199$	−5.01434e7	−0.451053	−0.225527	+	0.974237i	$0.572410\pi$
$200$	0	0
$201$	2.55831e7	0.222212
$202$	0	0
$203$	9.20641e7	0.772421
$204$	0	0
$205$	−1.05125e8	−0.852253
$206$	0	0
$207$	2.57191e6	0.0201539
$208$	0	0
$209$	7.87550e7	0.596714
$210$	0	0
$211$	2.16429e8	1.58609	0.793043	−	0.609165i	$-0.208495\pi$
$211$	2.16429e8	1.58609	0.793043	+	0.609165i	$0.208495\pi$
$212$	0	0
$213$	1.03273e8	0.732248
$214$	0	0
$215$	1.11132e8	0.762610
$216$	0	0
$217$	1.02078e8	0.678147
$218$	0	0
$219$	−2.46693e7	−0.158709
$220$	0	0
$221$	7.68857e7	0.479151
$222$	0	0
$223$	−2.59157e8	−1.56493	−0.782466	−	0.622693i	$-0.786039\pi$
$223$	−2.59157e8	−1.56493	−0.782466	+	0.622693i	$0.786039\pi$
$224$	0	0
$225$	1.13906e7	0.0666667
$226$	0	0
$227$	7.28829e7	0.413557	0.206778	−	0.978388i	$-0.433702\pi$
$227$	7.28829e7	0.413557	0.206778	+	0.978388i	$0.433702\pi$
$228$	0	0
$229$	−2.30378e8	−1.26770	−0.633851	−	0.773455i	$-0.718527\pi$
$229$	−2.30378e8	−1.26770	−0.633851	+	0.773455i	$0.718527\pi$
$230$	0	0
$231$	−1.91764e8	−1.02359
$232$	0	0
$233$	−2.12672e8	−1.10145	−0.550725	−	0.834687i	$-0.685649\pi$
$233$	−2.12672e8	−1.10145	−0.550725	+	0.834687i	$0.685649\pi$
$234$	0	0
$235$	−1.06533e8	−0.535484
$236$	0	0
$237$	−4.37908e7	−0.213680
$238$	0	0
$239$	1.88471e8	0.893003	0.446501	−	0.894783i	$-0.352670\pi$
$239$	1.88471e8	0.893003	0.446501	+	0.894783i	$0.352670\pi$
$240$	0	0
$241$	−9.80050e7	−0.451012	−0.225506	−	0.974242i	$-0.572404\pi$
$241$	−9.80050e7	−0.451012	−0.225506	+	0.974242i	$0.572404\pi$
$242$	0	0
$243$	1.43489e7	0.0641500
$244$	0	0
$245$	−4.39391e7	−0.190884
$246$	0	0
$247$	−3.03144e7	−0.128000
$248$	0	0
$249$	−3.96903e7	−0.162925
$250$	0	0
$251$	−4.49880e8	−1.79572	−0.897860	−	0.440282i	$-0.854878\pi$
$251$	−4.49880e8	−1.79572	−0.897860	+	0.440282i	$0.854878\pi$
$252$	0	0
$253$	−2.31155e7	−0.0897389
$254$	0	0
$255$	−1.02890e8	−0.388583
$256$	0	0
$257$	−3.65086e8	−1.34162	−0.670809	−	0.741630i	$-0.734053\pi$
$257$	−3.65086e8	−1.34162	−0.670809	+	0.741630i	$0.734053\pi$
$258$	0	0
$259$	−5.52812e8	−1.97710
$260$	0	0
$261$	6.19140e7	0.215549
$262$	0	0
$263$	−2.22137e8	−0.752965	−0.376483	−	0.926424i	$-0.622867\pi$
$263$	−2.22137e8	−0.752965	−0.376483	+	0.926424i	$0.622867\pi$
$264$	0	0
$265$	−1.99310e8	−0.657914
$266$	0	0
$267$	6.41253e7	0.206177
$268$	0	0
$269$	5.06542e8	1.58665	0.793327	−	0.608796i	$-0.208347\pi$
$269$	5.06542e8	1.58665	0.793327	+	0.608796i	$0.208347\pi$
$270$	0	0
$271$	1.07886e8	0.329285	0.164643	−	0.986353i	$-0.447353\pi$
$271$	1.07886e8	0.329285	0.164643	+	0.986353i	$0.447353\pi$
$272$	0	0
$273$	7.38139e7	0.219568
$274$	0	0
$275$	−1.02375e8	−0.296845
$276$	0	0
$277$	−2.50937e8	−0.709392	−0.354696	−	0.934982i	$-0.615416\pi$
$277$	−2.50937e8	−0.709392	−0.354696	+	0.934982i	$0.615416\pi$
$278$	0	0
$279$	6.86485e7	0.189241
$280$	0	0
$281$	8.57738e7	0.230612	0.115306	−	0.993330i	$-0.463215\pi$
$281$	8.57738e7	0.230612	0.115306	+	0.993330i	$0.463215\pi$
$282$	0	0
$283$	4.08257e8	1.07073	0.535367	−	0.844620i	$-0.320173\pi$
$283$	4.08257e8	1.07073	0.535367	+	0.844620i	$0.320173\pi$
$284$	0	0
$285$	4.05675e7	0.103806
$286$	0	0
$287$	9.11646e8	2.27635
$288$	0	0
$289$	5.19058e8	1.26495
$290$	0	0
$291$	3.81747e8	0.908135
$292$	0	0
$293$	−5.94603e8	−1.38099	−0.690495	−	0.723337i	$-0.742607\pi$
$293$	−5.94603e8	−1.38099	−0.690495	+	0.723337i	$0.742607\pi$
$294$	0	0
$295$	9.40050e7	0.213193
$296$	0	0
$297$	−1.28963e8	−0.285639
$298$	0	0
$299$	8.89762e6	0.0192497
$300$	0	0
$301$	−9.63732e8	−2.03692
$302$	0	0
$303$	5.71288e7	0.117979
$304$	0	0
$305$	−1.92338e8	−0.388164
$306$	0	0
$307$	−3.32730e8	−0.656308	−0.328154	−	0.944624i	$-0.606426\pi$
$307$	−3.32730e8	−0.656308	−0.328154	+	0.944624i	$0.606426\pi$
$308$	0	0
$309$	−4.61745e8	−0.890322
$310$	0	0
$311$	−1.83181e8	−0.345317	−0.172659	−	0.984982i	$-0.555236\pi$
$311$	−1.83181e8	−0.345317	−0.172659	+	0.984982i	$0.555236\pi$
$312$	0	0
$313$	1.85033e7	0.0341070	0.0170535	−	0.999855i	$-0.494571\pi$
$313$	1.85033e7	0.0341070	0.0170535	+	0.999855i	$0.494571\pi$
$314$	0	0
$315$	−9.87795e7	−0.178066
$316$	0	0
$317$	4.34331e8	0.765796	0.382898	−	0.923791i	$-0.374926\pi$
$317$	4.34331e8	0.765796	0.382898	+	0.923791i	$0.374926\pi$
$318$	0	0
$319$	−5.56461e8	−0.959771
$320$	0	0
$321$	4.68918e8	0.791278
$322$	0	0
$323$	−3.66442e8	−0.605057
$324$	0	0
$325$	3.94062e7	0.0636756
$326$	0	0
$327$	6.66938e8	1.05479
$328$	0	0
$329$	9.23854e8	1.43027
$330$	0	0
$331$	−1.08232e9	−1.64043	−0.820217	−	0.572053i	$-0.806147\pi$
$331$	−1.08232e9	−1.64043	−0.820217	+	0.572053i	$0.806147\pi$
$332$	0	0
$333$	−3.71771e8	−0.551723
$334$	0	0
$335$	−1.18440e8	−0.172125
$336$	0	0
$337$	5.13069e8	0.730249	0.365125	−	0.930959i	$-0.381026\pi$
$337$	5.13069e8	0.730249	0.365125	+	0.930959i	$0.381026\pi$
$338$	0	0
$339$	4.01377e8	0.559569
$340$	0	0
$341$	−6.16989e8	−0.842630
$342$	0	0
$343$	−5.11681e8	−0.684651
$344$	0	0
$345$	−1.19070e7	−0.0156112
$346$	0	0
$347$	−7.20149e8	−0.925271	−0.462636	−	0.886548i	$-0.653096\pi$
$347$	−7.20149e8	−0.925271	−0.462636	+	0.886548i	$0.653096\pi$
$348$	0	0
$349$	−5.53299e8	−0.696740	−0.348370	−	0.937357i	$-0.613265\pi$
$349$	−5.53299e8	−0.696740	−0.348370	+	0.937357i	$0.613265\pi$
$350$	0	0
$351$	4.96405e7	0.0612719
$352$	0	0
$353$	9.29308e8	1.12447	0.562235	−	0.826977i	$-0.309942\pi$
$353$	9.29308e8	1.12447	0.562235	+	0.826977i	$0.309942\pi$
$354$	0	0
$355$	−4.78116e8	−0.567197
$356$	0	0
$357$	8.92264e8	1.03790
$358$	0	0
$359$	−4.63305e8	−0.528489	−0.264245	−	0.964456i	$-0.585123\pi$
$359$	−4.63305e8	−0.528489	−0.264245	+	0.964456i	$0.585123\pi$
$360$	0	0
$361$	−7.49391e8	−0.838366
$362$	0	0
$363$	6.32921e8	0.694507
$364$	0	0
$365$	1.14210e8	0.122936
$366$	0	0
$367$	−1.50498e9	−1.58928	−0.794638	−	0.607084i	$-0.792339\pi$
$367$	−1.50498e9	−1.58928	−0.794638	+	0.607084i	$0.792339\pi$
$368$	0	0
$369$	6.13090e8	0.635232
$370$	0	0
$371$	1.72842e9	1.75728
$372$	0	0
$373$	−1.02471e9	−1.02240	−0.511199	−	0.859462i	$-0.670799\pi$
$373$	−1.02471e9	−1.02240	−0.511199	+	0.859462i	$0.670799\pi$
$374$	0	0
$375$	−5.27344e7	−0.0516398
$376$	0	0
$377$	2.14193e8	0.205879
$378$	0	0
$379$	1.17436e9	1.10807	0.554033	−	0.832495i	$-0.313088\pi$
$379$	1.17436e9	1.10807	0.554033	+	0.832495i	$0.313088\pi$
$380$	0	0
$381$	8.21082e7	0.0760587
$382$	0	0
$383$	5.43056e8	0.493911	0.246955	−	0.969027i	$-0.420570\pi$
$383$	5.43056e8	0.493911	0.246955	+	0.969027i	$0.420570\pi$
$384$	0	0
$385$	8.87796e8	0.792868
$386$	0	0
$387$	−6.48119e8	−0.568416
$388$	0	0
$389$	−2.12790e8	−0.183285	−0.0916425	−	0.995792i	$-0.529212\pi$
$389$	−2.12790e8	−0.183285	−0.0916425	+	0.995792i	$0.529212\pi$
$390$	0	0
$391$	1.07555e8	0.0909935
$392$	0	0
$393$	8.35512e7	0.0694350
$394$	0	0
$395$	2.02735e8	0.165516
$396$	0	0
$397$	3.11989e7	0.0250249	0.0125125	−	0.999922i	$-0.496017\pi$
$397$	3.11989e7	0.0250249	0.0125125	+	0.999922i	$0.496017\pi$
$398$	0	0
$399$	−3.51801e8	−0.277263
$400$	0	0
$401$	6.33676e8	0.490752	0.245376	−	0.969428i	$-0.421089\pi$
$401$	6.33676e8	0.490752	0.245376	+	0.969428i	$0.421089\pi$
$402$	0	0
$403$	2.37492e8	0.180751
$404$	0	0
$405$	−6.64301e7	−0.0496904
$406$	0	0
$407$	3.34135e9	2.45664
$408$	0	0
$409$	−1.55578e9	−1.12439	−0.562196	−	0.827004i	$-0.690043\pi$
$409$	−1.55578e9	−1.12439	−0.562196	+	0.827004i	$0.690043\pi$
$410$	0	0
$411$	5.54299e8	0.393820
$412$	0	0
$413$	−8.15211e8	−0.569436
$414$	0	0
$415$	1.83752e8	0.126201
$416$	0	0
$417$	1.66251e9	1.12277
$418$	0	0
$419$	1.24899e9	0.829489	0.414744	−	0.909938i	$-0.363871\pi$
$419$	1.24899e9	0.829489	0.414744	+	0.909938i	$0.363871\pi$
$420$	0	0
$421$	6.18173e8	0.403760	0.201880	−	0.979410i	$-0.435295\pi$
$421$	6.18173e8	0.403760	0.201880	+	0.979410i	$0.435295\pi$
$422$	0	0
$423$	6.21300e8	0.399126
$424$	0	0
$425$	4.76344e8	0.300995
$426$	0	0
$427$	1.66795e9	1.03678
$428$	0	0
$429$	−4.46152e8	−0.272824
$430$	0	0
$431$	−5.79114e8	−0.348412	−0.174206	−	0.984709i	$-0.555736\pi$
$431$	−5.79114e8	−0.348412	−0.174206	+	0.984709i	$0.555736\pi$
$432$	0	0
$433$	−9.19873e8	−0.544528	−0.272264	−	0.962223i	$-0.587772\pi$
$433$	−9.19873e8	−0.544528	−0.272264	+	0.962223i	$0.587772\pi$
$434$	0	0
$435$	−2.86639e8	−0.166964
$436$	0	0
$437$	−4.24066e7	−0.0243079
$438$	0	0
$439$	2.59160e7	0.0146198	0.00730990	−	0.999973i	$-0.497673\pi$
$439$	2.59160e7	0.0146198	0.00730990	+	0.999973i	$0.497673\pi$
$440$	0	0
$441$	2.56253e8	0.142277
$442$	0	0
$443$	6.95227e8	0.379939	0.189969	−	0.981790i	$-0.439161\pi$
$443$	6.95227e8	0.379939	0.189969	+	0.981790i	$0.439161\pi$
$444$	0	0
$445$	−2.96876e8	−0.159704
$446$	0	0
$447$	−3.99279e8	−0.211446
$448$	0	0
$449$	1.26402e9	0.659010	0.329505	−	0.944154i	$-0.393118\pi$
$449$	1.26402e9	0.659010	0.329505	+	0.944154i	$0.393118\pi$
$450$	0	0
$451$	−5.51025e9	−2.82848
$452$	0	0
$453$	7.97347e8	0.402999
$454$	0	0
$455$	−3.41731e8	−0.170077
$456$	0	0
$457$	2.68433e9	1.31561	0.657807	−	0.753186i	$-0.271484\pi$
$457$	2.68433e9	1.31561	0.657807	+	0.753186i	$0.271484\pi$
$458$	0	0
$459$	6.00056e8	0.289633
$460$	0	0
$461$	−3.18118e9	−1.51229	−0.756144	−	0.654405i	$-0.772919\pi$
$461$	−3.18118e9	−1.51229	−0.756144	+	0.654405i	$0.772919\pi$
$462$	0	0
$463$	3.22092e9	1.50816	0.754078	−	0.656785i	$-0.228084\pi$
$463$	3.22092e9	1.50816	0.754078	+	0.656785i	$0.228084\pi$
$464$	0	0
$465$	−3.17817e8	−0.146586
$466$	0	0
$467$	1.00598e9	0.457067	0.228534	−	0.973536i	$-0.426607\pi$
$467$	1.00598e9	0.457067	0.228534	+	0.973536i	$0.426607\pi$
$468$	0	0
$469$	1.02712e9	0.459742
$470$	0	0
$471$	8.26486e8	0.364470
$472$	0	0
$473$	5.82507e9	2.53097
$474$	0	0
$475$	−1.87812e8	−0.0804076
$476$	0	0
$477$	1.16238e9	0.490380
$478$	0	0
$479$	−2.35124e9	−0.977515	−0.488758	−	0.872420i	$-0.662550\pi$
$479$	−2.35124e9	−0.977515	−0.488758	+	0.872420i	$0.662550\pi$
$480$	0	0
$481$	−1.28615e9	−0.526969
$482$	0	0
$483$	1.03258e8	0.0416972
$484$	0	0
$485$	−1.76735e9	−0.703438
$486$	0	0
$487$	4.58792e9	1.79997	0.899983	−	0.435924i	$-0.143579\pi$
$487$	4.58792e9	1.79997	0.899983	+	0.435924i	$0.143579\pi$
$488$	0	0
$489$	1.06905e9	0.413444
$490$	0	0
$491$	−3.41881e9	−1.30343	−0.651717	−	0.758462i	$-0.725951\pi$
$491$	−3.41881e9	−1.30343	−0.651717	+	0.758462i	$0.725951\pi$
$492$	0	0
$493$	2.58918e9	0.973189
$494$	0	0
$495$	5.97051e8	0.221255
$496$	0	0
$497$	4.14622e9	1.51497
$498$	0	0
$499$	3.43084e9	1.23608	0.618042	−	0.786145i	$-0.287926\pi$
$499$	3.43084e9	1.23608	0.618042	+	0.786145i	$0.287926\pi$
$500$	0	0
$501$	−2.02486e9	−0.719386
$502$	0	0
$503$	−1.18094e9	−0.413754	−0.206877	−	0.978367i	$-0.566330\pi$
$503$	−1.18094e9	−0.413754	−0.206877	+	0.978367i	$0.566330\pi$
$504$	0	0
$505$	−2.64485e8	−0.0913864
$506$	0	0
$507$	−1.52248e9	−0.518827
$508$	0	0
$509$	8.27757e8	0.278221	0.139111	−	0.990277i	$-0.455576\pi$
$509$	8.27757e8	0.278221	0.139111	+	0.990277i	$0.455576\pi$
$510$	0	0
$511$	−9.90427e8	−0.328359
$512$	0	0
$513$	−2.36590e8	−0.0773722
$514$	0	0
$515$	2.13771e9	0.689641
$516$	0	0
$517$	−5.58403e9	−1.77718
$518$	0	0
$519$	2.63108e9	0.826130
$520$	0	0
$521$	7.71735e8	0.239076	0.119538	−	0.992830i	$-0.461859\pi$
$521$	7.71735e8	0.239076	0.119538	+	0.992830i	$0.461859\pi$
$522$	0	0
$523$	−2.42234e9	−0.740423	−0.370211	−	0.928948i	$-0.620715\pi$
$523$	−2.42234e9	−0.740423	−0.370211	+	0.928948i	$0.620715\pi$
$524$	0	0
$525$	4.57313e8	0.137929
$526$	0	0
$527$	2.87081e9	0.854411
$528$	0	0
$529$	−3.39238e9	−0.996344
$530$	0	0
$531$	−5.48237e8	−0.158905
$532$	0	0
$533$	2.12101e9	0.606732
$534$	0	0
$535$	−2.17092e9	−0.612921
$536$	0	0
$537$	1.84836e9	0.515083
$538$	0	0
$539$	−2.30311e9	−0.633512
$540$	0	0
$541$	−2.73881e9	−0.743656	−0.371828	−	0.928302i	$-0.621269\pi$
$541$	−2.73881e9	−0.743656	−0.371828	+	0.928302i	$0.621269\pi$
$542$	0	0
$543$	−3.58079e9	−0.959797
$544$	0	0
$545$	−3.08767e9	−0.817041
$546$	0	0
$547$	2.45569e9	0.641532	0.320766	−	0.947158i	$-0.396060\pi$
$547$	2.45569e9	0.641532	0.320766	+	0.947158i	$0.396060\pi$
$548$	0	0
$549$	1.12171e9	0.289320
$550$	0	0
$551$	−1.02086e9	−0.259977
$552$	0	0
$553$	−1.75812e9	−0.442089
$554$	0	0
$555$	1.72116e9	0.427362
$556$	0	0
$557$	2.75036e8	0.0674368	0.0337184	−	0.999431i	$-0.489265\pi$
$557$	2.75036e8	0.0674368	0.0337184	+	0.999431i	$0.489265\pi$
$558$	0	0
$559$	−2.24219e9	−0.542914
$560$	0	0
$561$	−5.39310e9	−1.28964
$562$	0	0
$563$	−1.02388e8	−0.0241807	−0.0120904	−	0.999927i	$-0.503849\pi$
$563$	−1.02388e8	−0.0241807	−0.0120904	+	0.999927i	$0.503849\pi$
$564$	0	0
$565$	−1.85823e9	−0.433440
$566$	0	0
$567$	5.76082e8	0.132722
$568$	0	0
$569$	5.73901e9	1.30600	0.653002	−	0.757356i	$-0.273509\pi$
$569$	5.73901e9	1.30600	0.653002	+	0.757356i	$0.273509\pi$
$570$	0	0
$571$	−2.71039e9	−0.609265	−0.304632	−	0.952470i	$-0.598534\pi$
$571$	−2.71039e9	−0.609265	−0.304632	+	0.952470i	$0.598534\pi$
$572$	0	0
$573$	−3.01511e9	−0.669518
$574$	0	0
$575$	5.51250e7	0.0120924
$576$	0	0
$577$	4.30877e9	0.933766	0.466883	−	0.884319i	$-0.345377\pi$
$577$	4.30877e9	0.933766	0.466883	+	0.884319i	$0.345377\pi$
$578$	0	0
$579$	1.28652e9	0.275450
$580$	0	0
$581$	−1.59349e9	−0.337081
$582$	0	0
$583$	−1.04470e10	−2.18350
$584$	0	0
$585$	−2.29817e8	−0.0474610
$586$	0	0
$587$	3.75153e9	0.765553	0.382777	−	0.923841i	$-0.374968\pi$
$587$	3.75153e9	0.765553	0.382777	+	0.923841i	$0.374968\pi$
$588$	0	0
$589$	−1.13190e9	−0.228247
$590$	0	0
$591$	−3.23264e9	−0.644172
$592$	0	0
$593$	−3.59825e9	−0.708598	−0.354299	−	0.935132i	$-0.615280\pi$
$593$	−3.59825e9	−0.708598	−0.354299	+	0.935132i	$0.615280\pi$
$594$	0	0
$595$	−4.13085e9	−0.803952
$596$	0	0
$597$	−1.35387e9	−0.260416
$598$	0	0
$599$	−3.07822e9	−0.585203	−0.292601	−	0.956235i	$-0.594521\pi$
$599$	−3.07822e9	−0.585203	−0.292601	+	0.956235i	$0.594521\pi$
$600$	0	0
$601$	6.79679e9	1.27715	0.638577	−	0.769558i	$-0.279523\pi$
$601$	6.79679e9	1.27715	0.638577	+	0.769558i	$0.279523\pi$
$602$	0	0
$603$	6.90745e8	0.128294
$604$	0	0
$605$	−2.93019e9	−0.537963
$606$	0	0
$607$	−1.23960e9	−0.224969	−0.112484	−	0.993654i	$-0.535881\pi$
$607$	−1.23960e9	−0.224969	−0.112484	+	0.993654i	$0.535881\pi$
$608$	0	0
$609$	2.48573e9	0.445958
$610$	0	0
$611$	2.14941e9	0.381219
$612$	0	0
$613$	−8.46150e9	−1.48367	−0.741833	−	0.670585i	$-0.766043\pi$
$613$	−8.46150e9	−1.48367	−0.741833	+	0.670585i	$0.766043\pi$
$614$	0	0
$615$	−2.83838e9	−0.492048
$616$	0	0
$617$	−7.96609e9	−1.36536	−0.682680	−	0.730717i	$-0.739186\pi$
$617$	−7.96609e9	−1.36536	−0.682680	+	0.730717i	$0.739186\pi$
$618$	0	0
$619$	4.87438e9	0.826042	0.413021	−	0.910722i	$-0.364474\pi$
$619$	4.87438e9	0.826042	0.413021	+	0.910722i	$0.364474\pi$
$620$	0	0
$621$	6.94416e7	0.0116359
$622$	0	0
$623$	2.57451e9	0.426566
$624$	0	0
$625$	2.44141e8	0.0400000
$626$	0	0
$627$	2.12639e9	0.344513
$628$	0	0
$629$	−1.55471e10	−2.49099
$630$	0	0
$631$	−1.11752e9	−0.177073	−0.0885364	−	0.996073i	$-0.528219\pi$
$631$	−1.11752e9	−0.177073	−0.0885364	+	0.996073i	$0.528219\pi$
$632$	0	0
$633$	5.84358e9	0.915728
$634$	0	0
$635$	−3.80130e8	−0.0589148
$636$	0	0
$637$	8.86516e8	0.135893
$638$	0	0
$639$	2.78837e9	0.422764
$640$	0	0
$641$	−6.56989e9	−0.985270	−0.492635	−	0.870236i	$-0.663966\pi$
$641$	−6.56989e9	−0.985270	−0.492635	+	0.870236i	$0.663966\pi$
$642$	0	0
$643$	−3.37408e9	−0.500514	−0.250257	−	0.968179i	$-0.580515\pi$
$643$	−3.37408e9	−0.500514	−0.250257	+	0.968179i	$0.580515\pi$
$644$	0	0
$645$	3.00055e9	0.440293
$646$	0	0
$647$	−7.23965e9	−1.05088	−0.525440	−	0.850831i	$-0.676099\pi$
$647$	−7.23965e9	−1.05088	−0.525440	+	0.850831i	$0.676099\pi$
$648$	0	0
$649$	4.92737e9	0.707552
$650$	0	0
$651$	2.75611e9	0.391528
$652$	0	0
$653$	2.12956e6	0.000299291 0	0.000149646	−	1.00000i	$-0.499952\pi$
$653$	2.12956e6	0.000299291 0	0.000149646		1.00000i	$0.499952\pi$
$654$	0	0
$655$	−3.86811e8	−0.0537841
$656$	0	0
$657$	−6.66071e8	−0.0916309
$658$	0	0
$659$	3.66306e9	0.498592	0.249296	−	0.968427i	$-0.419801\pi$
$659$	3.66306e9	0.498592	0.249296	+	0.968427i	$0.419801\pi$
$660$	0	0
$661$	3.16420e9	0.426146	0.213073	−	0.977036i	$-0.431653\pi$
$661$	3.16420e9	0.426146	0.213073	+	0.977036i	$0.431653\pi$
$662$	0	0
$663$	2.07591e9	0.276638
$664$	0	0
$665$	1.62871e9	0.214767
$666$	0	0
$667$	2.99633e8	0.0390975
$668$	0	0
$669$	−6.99724e9	−0.903514
$670$	0	0
$671$	−1.00816e10	−1.28825
$672$	0	0
$673$	2.69190e9	0.340413	0.170206	−	0.985408i	$-0.445557\pi$
$673$	2.69190e9	0.340413	0.170206	+	0.985408i	$0.445557\pi$
$674$	0	0
$675$	3.07547e8	0.0384900
$676$	0	0
$677$	8.61904e9	1.06758	0.533788	−	0.845618i	$-0.320768\pi$
$677$	8.61904e9	1.06758	0.533788	+	0.845618i	$0.320768\pi$
$678$	0	0
$679$	1.53264e10	1.87887
$680$	0	0
$681$	1.96784e9	0.238767
$682$	0	0
$683$	9.36016e9	1.12411	0.562057	−	0.827098i	$-0.310010\pi$
$683$	9.36016e9	1.12411	0.562057	+	0.827098i	$0.310010\pi$
$684$	0	0
$685$	−2.56620e9	−0.305052
$686$	0	0
$687$	−6.22021e9	−0.731908
$688$	0	0
$689$	4.02128e9	0.468379
$690$	0	0
$691$	−1.56831e9	−0.180825	−0.0904127	−	0.995904i	$-0.528819\pi$
$691$	−1.56831e9	−0.180825	−0.0904127	+	0.995904i	$0.528819\pi$
$692$	0	0
$693$	−5.17763e9	−0.590969
$694$	0	0
$695$	−7.69683e9	−0.869691
$696$	0	0
$697$	2.56388e10	2.86802
$698$	0	0
$699$	−5.74215e9	−0.635923
$700$	0	0
$701$	−1.28923e10	−1.41357	−0.706784	−	0.707429i	$-0.749855\pi$
$701$	−1.28923e10	−1.41357	−0.706784	+	0.707429i	$0.749855\pi$
$702$	0	0
$703$	6.12989e9	0.665440
$704$	0	0
$705$	−2.87639e9	−0.309162
$706$	0	0
$707$	2.29362e9	0.244092
$708$	0	0
$709$	7.16715e9	0.755239	0.377620	−	0.925961i	$-0.376743\pi$
$709$	7.16715e9	0.755239	0.377620	+	0.925961i	$0.376743\pi$
$710$	0	0
$711$	−1.18235e9	−0.123368
$712$	0	0
$713$	3.32225e8	0.0343256
$714$	0	0
$715$	2.06552e9	0.211328
$716$	0	0
$717$	5.08873e9	0.515575
$718$	0	0
$719$	7.05640e9	0.707998	0.353999	−	0.935246i	$-0.384822\pi$
$719$	7.05640e9	0.707998	0.353999	+	0.935246i	$0.384822\pi$
$720$	0	0
$721$	−1.85382e10	−1.84202
$722$	0	0
$723$	−2.64613e9	−0.260392
$724$	0	0
$725$	1.32703e9	0.129330
$726$	0	0
$727$	2.04175e9	0.197075	0.0985377	−	0.995133i	$-0.468583\pi$
$727$	2.04175e9	0.197075	0.0985377	+	0.995133i	$0.468583\pi$
$728$	0	0
$729$	3.87420e8	0.0370370
$730$	0	0
$731$	−2.71036e10	−2.56636
$732$	0	0
$733$	8.27298e9	0.775886	0.387943	−	0.921683i	$-0.373186\pi$
$733$	8.27298e9	0.775886	0.387943	+	0.921683i	$0.373186\pi$
$734$	0	0
$735$	−1.18636e9	−0.110207
$736$	0	0
$737$	−6.20818e9	−0.571252
$738$	0	0
$739$	−1.97116e10	−1.79666	−0.898331	−	0.439318i	$-0.855220\pi$
$739$	−1.97116e10	−1.79666	−0.898331	+	0.439318i	$0.855220\pi$
$740$	0	0
$741$	−8.18490e8	−0.0739009
$742$	0	0
$743$	−3.14462e9	−0.281259	−0.140630	−	0.990062i	$-0.544913\pi$
$743$	−3.14462e9	−0.281259	−0.140630	+	0.990062i	$0.544913\pi$
$744$	0	0
$745$	1.84851e9	0.163786
$746$	0	0
$747$	−1.07164e9	−0.0940646
$748$	0	0
$749$	1.88262e10	1.63710
$750$	0	0
$751$	7.37057e9	0.634982	0.317491	−	0.948261i	$-0.397160\pi$
$751$	7.37057e9	0.634982	0.317491	+	0.948261i	$0.397160\pi$
$752$	0	0
$753$	−1.21468e10	−1.03676
$754$	0	0
$755$	−3.69142e9	−0.312161
$756$	0	0
$757$	−9.28224e9	−0.777709	−0.388854	−	0.921299i	$-0.627129\pi$
$757$	−9.28224e9	−0.777709	−0.388854	+	0.921299i	$0.627129\pi$
$758$	0	0
$759$	−6.24117e8	−0.0518108
$760$	0	0
$761$	5.83380e9	0.479850	0.239925	−	0.970791i	$-0.422877\pi$
$761$	5.83380e9	0.479850	0.239925	+	0.970791i	$0.422877\pi$
$762$	0	0
$763$	2.67763e10	2.18230
$764$	0	0
$765$	−2.77804e9	−0.224348
$766$	0	0
$767$	−1.89664e9	−0.151776
$768$	0	0
$769$	1.72683e10	1.36933	0.684664	−	0.728859i	$-0.259949\pi$
$769$	1.72683e10	1.36933	0.684664	+	0.728859i	$0.259949\pi$
$770$	0	0
$771$	−9.85732e9	−0.774584
$772$	0	0
$773$	8.60200e9	0.669840	0.334920	−	0.942247i	$-0.391291\pi$
$773$	8.60200e9	0.669840	0.334920	+	0.942247i	$0.391291\pi$
$774$	0	0
$775$	1.47138e9	0.113545
$776$	0	0
$777$	−1.49259e10	−1.14148
$778$	0	0
$779$	−1.01088e10	−0.766162
$780$	0	0
$781$	−2.50609e10	−1.88243
$782$	0	0
$783$	1.67168e9	0.124448
$784$	0	0
$785$	−3.82632e9	−0.282318
$786$	0	0
$787$	−1.05678e10	−0.772814	−0.386407	−	0.922328i	$-0.626284\pi$
$787$	−1.05678e10	−0.772814	−0.386407	+	0.922328i	$0.626284\pi$
$788$	0	0
$789$	−5.99769e9	−0.434725
$790$	0	0
$791$	1.61146e10	1.15771
$792$	0	0
$793$	3.88061e9	0.276340
$794$	0	0
$795$	−5.38138e9	−0.379847
$796$	0	0
$797$	1.32243e10	0.925267	0.462634	−	0.886550i	$-0.346905\pi$
$797$	1.32243e10	0.925267	0.462634	+	0.886550i	$0.346905\pi$
$798$	0	0
$799$	2.59821e10	1.80203
$800$	0	0
$801$	1.73138e9	0.119036
$802$	0	0
$803$	5.98642e9	0.408002
$804$	0	0
$805$	−4.78044e8	−0.0322985
$806$	0	0
$807$	1.36766e10	0.916055
$808$	0	0
$809$	−1.36636e10	−0.907288	−0.453644	−	0.891183i	$-0.649876\pi$
$809$	−1.36636e10	−0.907288	−0.453644	+	0.891183i	$0.649876\pi$
$810$	0	0
$811$	1.05566e10	0.694947	0.347473	−	0.937690i	$-0.387040\pi$
$811$	1.05566e10	0.694947	0.347473	+	0.937690i	$0.387040\pi$
$812$	0	0
$813$	2.91292e9	0.190113
$814$	0	0
$815$	−4.94930e9	−0.320252
$816$	0	0
$817$	1.06864e10	0.685574
$818$	0	0
$819$	1.99298e9	0.126768
$820$	0	0
$821$	−1.52906e10	−0.964325	−0.482162	−	0.876082i	$-0.660149\pi$
$821$	−1.52906e10	−0.964325	−0.482162	+	0.876082i	$0.660149\pi$
$822$	0	0
$823$	1.26246e10	0.789441	0.394720	−	0.918801i	$-0.370841\pi$
$823$	1.26246e10	0.789441	0.394720	+	0.918801i	$0.370841\pi$
$824$	0	0
$825$	−2.76412e9	−0.171383
$826$	0	0
$827$	3.06814e10	1.88628	0.943140	−	0.332395i	$-0.107857\pi$
$827$	3.06814e10	1.88628	0.943140	+	0.332395i	$0.107857\pi$
$828$	0	0
$829$	2.11073e10	1.28675	0.643373	−	0.765553i	$-0.277534\pi$
$829$	2.11073e10	1.28675	0.643373	+	0.765553i	$0.277534\pi$
$830$	0	0
$831$	−6.77531e9	−0.409568
$832$	0	0
$833$	1.07162e10	0.642369
$834$	0	0
$835$	9.37434e9	0.557234
$836$	0	0
$837$	1.85351e9	0.109259
$838$	0	0
$839$	8.83619e9	0.516533	0.258266	−	0.966074i	$-0.416849\pi$
$839$	8.83619e9	0.516533	0.258266	+	0.966074i	$0.416849\pi$
$840$	0	0
$841$	−1.00368e10	−0.581846
$842$	0	0
$843$	2.31589e9	0.133144
$844$	0	0
$845$	7.04850e9	0.401882
$846$	0	0
$847$	2.54106e10	1.43689
$848$	0	0
$849$	1.10229e10	0.618188
$850$	0	0
$851$	−1.79919e9	−0.100074
$852$	0	0
$853$	2.46181e9	0.135810	0.0679052	−	0.997692i	$-0.478368\pi$
$853$	2.46181e9	0.135810	0.0679052	+	0.997692i	$0.478368\pi$
$854$	0	0
$855$	1.09532e9	0.0599323
$856$	0	0
$857$	−2.25771e10	−1.22528	−0.612639	−	0.790362i	$-0.709892\pi$
$857$	−2.25771e10	−1.22528	−0.612639	+	0.790362i	$0.709892\pi$
$858$	0	0
$859$	−2.02335e10	−1.08917	−0.544585	−	0.838706i	$-0.683313\pi$
$859$	−2.02335e10	−1.08917	−0.544585	+	0.838706i	$0.683313\pi$
$860$	0	0
$861$	2.46144e10	1.31425
$862$	0	0
$863$	−1.79249e10	−0.949335	−0.474668	−	0.880165i	$-0.657432\pi$
$863$	−1.79249e10	−0.949335	−0.474668	+	0.880165i	$0.657432\pi$
$864$	0	0
$865$	−1.21809e10	−0.639918
$866$	0	0
$867$	1.40146e10	0.730319
$868$	0	0
$869$	1.06266e10	0.549317
$870$	0	0
$871$	2.38966e9	0.122538
$872$	0	0
$873$	1.03072e10	0.524312
$874$	0	0
$875$	−2.11719e9	−0.106839
$876$	0	0
$877$	9.29307e9	0.465222	0.232611	−	0.972570i	$-0.425273\pi$
$877$	9.29307e9	0.465222	0.232611	+	0.972570i	$0.425273\pi$
$878$	0	0
$879$	−1.60543e10	−0.797315
$880$	0	0
$881$	1.86664e10	0.919697	0.459848	−	0.887997i	$-0.347904\pi$
$881$	1.86664e10	0.919697	0.459848	+	0.887997i	$0.347904\pi$
$882$	0	0
$883$	−6.30323e9	−0.308106	−0.154053	−	0.988063i	$-0.549233\pi$
$883$	−6.30323e9	−0.308106	−0.154053	+	0.988063i	$0.549233\pi$
$884$	0	0
$885$	2.53814e9	0.123087
$886$	0	0
$887$	−4.23957e9	−0.203981	−0.101990	−	0.994785i	$-0.532521\pi$
$887$	−4.23957e9	−0.203981	−0.101990	+	0.994785i	$0.532521\pi$
$888$	0	0
$889$	3.29649e9	0.157361
$890$	0	0
$891$	−3.48200e9	−0.164914
$892$	0	0
$893$	−1.02442e10	−0.481392
$894$	0	0
$895$	−8.55723e9	−0.398981
$896$	0	0
$897$	2.40236e8	0.0111138
$898$	0	0
$899$	7.99769e9	0.367118
$900$	0	0
$901$	4.86094e10	2.21403
$902$	0	0
$903$	−2.60208e10	−1.17602
$904$	0	0
$905$	1.65777e10	0.743456
$906$	0	0
$907$	1.49498e10	0.665288	0.332644	−	0.943052i	$-0.392059\pi$
$907$	1.49498e10	0.665288	0.332644	+	0.943052i	$0.392059\pi$
$908$	0	0
$909$	1.54248e9	0.0681154
$910$	0	0
$911$	−1.77320e10	−0.777039	−0.388520	−	0.921440i	$-0.627013\pi$
$911$	−1.77320e10	−0.777039	−0.388520	+	0.921440i	$0.627013\pi$
$912$	0	0
$913$	9.63152e9	0.418839
$914$	0	0
$915$	−5.19312e9	−0.224107
$916$	0	0
$917$	3.35442e9	0.143657
$918$	0	0
$919$	−7.50434e9	−0.318939	−0.159470	−	0.987203i	$-0.550978\pi$
$919$	−7.50434e9	−0.318939	−0.159470	+	0.987203i	$0.550978\pi$
$920$	0	0
$921$	−8.98371e9	−0.378919
$922$	0	0
$923$	9.64647e9	0.403796
$924$	0	0
$925$	−7.96834e9	−0.331034
$926$	0	0
$927$	−1.24671e10	−0.514028
$928$	0	0
$929$	5.09197e9	0.208368	0.104184	−	0.994558i	$-0.466777\pi$
$929$	5.09197e9	0.208368	0.104184	+	0.994558i	$0.466777\pi$
$930$	0	0
$931$	−4.22519e9	−0.171602
$932$	0	0
$933$	−4.94588e9	−0.199369
$934$	0	0
$935$	2.49680e10	0.998950
$936$	0	0
$937$	−1.94963e10	−0.774219	−0.387109	−	0.922034i	$-0.626526\pi$
$937$	−1.94963e10	−0.774219	−0.387109	+	0.922034i	$0.626526\pi$
$938$	0	0
$939$	4.99589e8	0.0196917
$940$	0	0
$941$	−2.27025e10	−0.888198	−0.444099	−	0.895978i	$-0.646476\pi$
$941$	−2.27025e10	−0.888198	−0.444099	+	0.895978i	$0.646476\pi$
$942$	0	0
$943$	2.96706e9	0.115222
$944$	0	0
$945$	−2.66705e9	−0.102806
$946$	0	0
$947$	−2.59286e10	−0.992099	−0.496049	−	0.868294i	$-0.665216\pi$
$947$	−2.59286e10	−0.992099	−0.496049	+	0.868294i	$0.665216\pi$
$948$	0	0
$949$	−2.30430e9	−0.0875198
$950$	0	0
$951$	1.17269e10	0.442132
$952$	0	0
$953$	−3.88558e10	−1.45422	−0.727112	−	0.686519i	$-0.759138\pi$
$953$	−3.88558e10	−1.45422	−0.727112	+	0.686519i	$0.759138\pi$
$954$	0	0
$955$	1.39589e10	0.518607
$956$	0	0
$957$	−1.50245e10	−0.554124
$958$	0	0
$959$	2.22541e10	0.814788
$960$	0	0
$961$	−1.86450e10	−0.677689
$962$	0	0
$963$	1.26608e10	0.456845
$964$	0	0
$965$	−5.95613e9	−0.213363
$966$	0	0
$967$	−4.04683e9	−0.143920	−0.0719601	−	0.997408i	$-0.522925\pi$
$967$	−4.04683e9	−0.143920	−0.0719601	+	0.997408i	$0.522925\pi$
$968$	0	0
$969$	−9.89393e9	−0.349330
$970$	0	0
$971$	6.94765e9	0.243540	0.121770	−	0.992558i	$-0.461143\pi$
$971$	6.94765e9	0.243540	0.121770	+	0.992558i	$0.461143\pi$
$972$	0	0
$973$	6.67469e10	2.32293
$974$	0	0
$975$	1.06397e9	0.0367632
$976$	0	0
$977$	2.79014e10	0.957182	0.478591	−	0.878038i	$-0.341148\pi$
$977$	2.79014e10	0.957182	0.478591	+	0.878038i	$0.341148\pi$
$978$	0	0
$979$	−1.55611e10	−0.530029
$980$	0	0
$981$	1.80073e10	0.608986
$982$	0	0
$983$	7.05702e9	0.236965	0.118483	−	0.992956i	$-0.462197\pi$
$983$	7.05702e9	0.236965	0.118483	+	0.992956i	$0.462197\pi$
$984$	0	0
$985$	1.49659e10	0.498973
$986$	0	0
$987$	2.49441e10	0.825766
$988$	0	0
$989$	−3.13658e9	−0.103102
$990$	0	0
$991$	4.35670e10	1.42200	0.711001	−	0.703191i	$-0.248242\pi$
$991$	4.35670e10	1.42200	0.711001	+	0.703191i	$0.248242\pi$
$992$	0	0
$993$	−2.92227e10	−0.947105
$994$	0	0
$995$	6.26792e9	0.201717
$996$	0	0
$997$	1.79367e10	0.573204	0.286602	−	0.958050i	$-0.407474\pi$
$997$	1.79367e10	0.573204	0.286602	+	0.958050i	$0.407474\pi$
$998$	0	0
$999$	−1.00378e10	−0.318537

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	240.8.a.k.1.1		1
4.3	odd	2		30.8.a.a.1.1	✓	1
12.11	even	2		90.8.a.i.1.1		1
20.3	even	4		150.8.c.j.49.2		2
20.7	even	4		150.8.c.j.49.1		2
20.19	odd	2		150.8.a.q.1.1		1
60.23	odd	4		450.8.c.b.199.1		2
60.47	odd	4		450.8.c.b.199.2		2
60.59	even	2		450.8.a.k.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
30.8.a.a.1.1	✓	1	4.3	odd	2
90.8.a.i.1.1		1	12.11	even	2
150.8.a.q.1.1		1	20.19	odd	2
150.8.c.j.49.1		2	20.7	even	4
150.8.c.j.49.2		2	20.3	even	4
240.8.a.k.1.1		1	1.1	even	1	trivial
450.8.a.k.1.1		1	60.59	even	2
450.8.c.b.199.1		2	60.23	odd	4
450.8.c.b.199.2		2	60.47	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:	\( N \)	\(=\)	\( 240 = 2^{4} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	240.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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