LMFDB - Embedded newform 240.8.a.i.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 60)
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)

$f(q)$

$=$

$q+27.0000 q^{3} -125.000 q^{5} -1028.00 q^{7} +729.000 q^{9} +O(q^{10})$

$q+27.0000 q^{3} -125.000 q^{5} -1028.00 q^{7} +729.000 q^{9} -3096.00 q^{11} -13030.0 q^{13} -3375.00 q^{15} +1878.00 q^{17} +31180.0 q^{19} -27756.0 q^{21} +33288.0 q^{23} +15625.0 q^{25} +19683.0 q^{27} -213054. q^{29} +172696. q^{31} -83592.0 q^{33} +128500. q^{35} +27434.0 q^{37} -351810. q^{39} +532650. q^{41} +911908. q^{43} -91125.0 q^{45} +732648. q^{47} +233241. q^{49} +50706.0 q^{51} +409074. q^{53} +387000. q^{55} +841860. q^{57} -1.50814e6 q^{59} -302578. q^{61} -749412. q^{63} +1.62875e6 q^{65} -1.25433e6 q^{67} +898776. q^{69} -4.78128e6 q^{71} -502414. q^{73} +421875. q^{75} +3.18269e6 q^{77} +1.99137e6 q^{79} +531441. q^{81} +8.09927e6 q^{83} -234750. q^{85} -5.75246e6 q^{87} +7.48797e6 q^{89} +1.33948e7 q^{91} +4.66279e6 q^{93} -3.89750e6 q^{95} -1.71726e7 q^{97} -2.25698e6 q^{99} +O(q^{100})$

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
$7$	−1028.00	−1.13279	−0.566396	−	0.824133i	$-0.691663\pi$
$7$	−1028.00	−1.13279	−0.566396	+	0.824133i	$0.691663\pi$
$8$	0	0
$9$	729.000	0.333333
$10$	0	0
$11$	−3096.00	−0.701337	−0.350668	−	0.936500i	$-0.614046\pi$
$11$	−3096.00	−0.701337	−0.350668	+	0.936500i	$0.614046\pi$
$12$	0	0
$13$	−13030.0	−1.64491	−0.822456	−	0.568829i	$-0.807397\pi$
$13$	−13030.0	−1.64491	−0.822456	+	0.568829i	$0.807397\pi$
$14$	0	0
$15$	−3375.00	−0.258199
$16$	0	0
$17$	1878.00	0.0927095	0.0463548	−	0.998925i	$-0.485240\pi$
$17$	1878.00	0.0927095	0.0463548	+	0.998925i	$0.485240\pi$
$18$	0	0
$19$	31180.0	1.04289	0.521445	−	0.853285i	$-0.325393\pi$
$19$	31180.0	1.04289	0.521445	+	0.853285i	$0.325393\pi$
$20$	0	0
$21$	−27756.0	−0.654017
$22$	0	0
$23$	33288.0	0.570480	0.285240	−	0.958456i	$-0.407927\pi$
$23$	33288.0	0.570480	0.285240	+	0.958456i	$0.407927\pi$
$24$	0	0
$25$	15625.0	0.200000
$26$	0	0
$27$	19683.0	0.192450
$28$	0	0
$29$	−213054.	−1.62217	−0.811086	−	0.584927i	$-0.801123\pi$
$29$	−213054.	−1.62217	−0.811086	+	0.584927i	$0.801123\pi$
$30$	0	0
$31$	172696.	1.04116	0.520579	−	0.853814i	$-0.325716\pi$
$31$	172696.	1.04116	0.520579	+	0.853814i	$0.325716\pi$
$32$	0	0
$33$	−83592.0	−0.404917
$34$	0	0
$35$	128500.	0.506600
$36$	0	0
$37$	27434.0	0.0890396	0.0445198	−	0.999009i	$-0.485824\pi$
$37$	27434.0	0.0890396	0.0445198	+	0.999009i	$0.485824\pi$
$38$	0	0
$39$	−351810.	−0.949690
$40$	0	0
$41$	532650.	1.20698	0.603488	−	0.797372i	$-0.293777\pi$
$41$	532650.	1.20698	0.603488	+	0.797372i	$0.293777\pi$
$42$	0	0
$43$	911908.	1.74909	0.874544	−	0.484947i	$-0.161161\pi$
$43$	911908.	1.74909	0.874544	+	0.484947i	$0.161161\pi$
$44$	0	0
$45$	−91125.0	−0.149071
$46$	0	0
$47$	732648.	1.02933	0.514663	−	0.857393i	$-0.327917\pi$
$47$	732648.	1.02933	0.514663	+	0.857393i	$0.327917\pi$
$48$	0	0
$49$	233241.	0.283217
$50$	0	0
$51$	50706.0	0.0535259
$52$	0	0
$53$	409074.	0.377430	0.188715	−	0.982032i	$-0.439568\pi$
$53$	409074.	0.377430	0.188715	+	0.982032i	$0.439568\pi$
$54$	0	0
$55$	387000.	0.313647
$56$	0	0
$57$	841860.	0.602113
$58$	0	0
$59$	−1.50814e6	−0.956001	−0.478001	−	0.878359i	$-0.658638\pi$
$59$	−1.50814e6	−0.956001	−0.478001	+	0.878359i	$0.658638\pi$
$60$	0	0
$61$	−302578.	−0.170680	−0.0853401	−	0.996352i	$-0.527198\pi$
$61$	−302578.	−0.170680	−0.0853401	+	0.996352i	$0.527198\pi$
$62$	0	0
$63$	−749412.	−0.377597
$64$	0	0
$65$	1.62875e6	0.735627
$66$	0	0
$67$	−1.25433e6	−0.509508	−0.254754	−	0.967006i	$-0.581994\pi$
$67$	−1.25433e6	−0.509508	−0.254754	+	0.967006i	$0.581994\pi$
$68$	0	0
$69$	898776.	0.329367
$70$	0	0
$71$	−4.78128e6	−1.58540	−0.792702	−	0.609609i	$-0.791326\pi$
$71$	−4.78128e6	−1.58540	−0.792702	+	0.609609i	$0.791326\pi$
$72$	0	0
$73$	−502414.	−0.151158	−0.0755791	−	0.997140i	$-0.524081\pi$
$73$	−502414.	−0.151158	−0.0755791	+	0.997140i	$0.524081\pi$
$74$	0	0
$75$	421875.	0.115470
$76$	0	0
$77$	3.18269e6	0.794468
$78$	0	0
$79$	1.99137e6	0.454419	0.227210	−	0.973846i	$-0.427040\pi$
$79$	1.99137e6	0.454419	0.227210	+	0.973846i	$0.427040\pi$
$80$	0	0
$81$	531441.	0.111111
$82$	0	0
$83$	8.09927e6	1.55479	0.777396	−	0.629011i	$-0.216540\pi$
$83$	8.09927e6	1.55479	0.777396	+	0.629011i	$0.216540\pi$
$84$	0	0
$85$	−234750.	−0.0414610
$86$	0	0
$87$	−5.75246e6	−0.936561
$88$	0	0
$89$	7.48797e6	1.12590	0.562949	−	0.826492i	$-0.309667\pi$
$89$	7.48797e6	1.12590	0.562949	+	0.826492i	$0.309667\pi$
$90$	0	0
$91$	1.33948e7	1.86334
$92$	0	0
$93$	4.66279e6	0.601112
$94$	0	0
$95$	−3.89750e6	−0.466395
$96$	0	0
$97$	−1.71726e7	−1.91044	−0.955222	−	0.295890i	$-0.904384\pi$
$97$	−1.71726e7	−1.91044	−0.955222	+	0.295890i	$0.904384\pi$
$98$	0	0
$99$	−2.25698e6	−0.233779
$100$	0	0
$101$	−9.74434e6	−0.941083	−0.470541	−	0.882378i	$-0.655941\pi$
$101$	−9.74434e6	−0.941083	−0.470541	+	0.882378i	$0.655941\pi$
$102$	0	0
$103$	1.69855e7	1.53161	0.765803	−	0.643076i	$-0.222342\pi$
$103$	1.69855e7	1.53161	0.765803	+	0.643076i	$0.222342\pi$
$104$	0	0
$105$	3.46950e6	0.292486
$106$	0	0
$107$	1.20386e7	0.950017	0.475008	−	0.879981i	$-0.342445\pi$
$107$	1.20386e7	0.950017	0.475008	+	0.879981i	$0.342445\pi$
$108$	0	0
$109$	1.18251e7	0.874605	0.437303	−	0.899314i	$-0.355934\pi$
$109$	1.18251e7	0.874605	0.437303	+	0.899314i	$0.355934\pi$
$110$	0	0
$111$	740718.	0.0514070
$112$	0	0
$113$	1.73862e7	1.13352	0.566761	−	0.823882i	$-0.308196\pi$
$113$	1.73862e7	1.13352	0.566761	+	0.823882i	$0.308196\pi$
$114$	0	0
$115$	−4.16100e6	−0.255126
$116$	0	0
$117$	−9.49887e6	−0.548304
$118$	0	0
$119$	−1.93058e6	−0.105021
$120$	0	0
$121$	−9.90196e6	−0.508127
$122$	0	0
$123$	1.43816e7	0.696848
$124$	0	0
$125$	−1.95312e6	−0.0894427
$126$	0	0
$127$	1.43900e7	0.623371	0.311686	−	0.950185i	$-0.399106\pi$
$127$	1.43900e7	0.623371	0.311686	+	0.950185i	$0.399106\pi$
$128$	0	0
$129$	2.46215e7	1.00984
$130$	0	0
$131$	1.80590e7	0.701849	0.350924	−	0.936404i	$-0.385867\pi$
$131$	1.80590e7	0.701849	0.350924	+	0.936404i	$0.385867\pi$
$132$	0	0
$133$	−3.20530e7	−1.18138
$134$	0	0
$135$	−2.46038e6	−0.0860663
$136$	0	0
$137$	4.51304e7	1.49950	0.749751	−	0.661720i	$-0.230173\pi$
$137$	4.51304e7	1.49950	0.749751	+	0.661720i	$0.230173\pi$
$138$	0	0
$139$	−1.34776e7	−0.425658	−0.212829	−	0.977089i	$-0.568268\pi$
$139$	−1.34776e7	−0.425658	−0.212829	+	0.977089i	$0.568268\pi$
$140$	0	0
$141$	1.97815e7	0.594282
$142$	0	0
$143$	4.03409e7	1.15364
$144$	0	0
$145$	2.66317e7	0.725457
$146$	0	0
$147$	6.29751e6	0.163515
$148$	0	0
$149$	7.31740e7	1.81219	0.906097	−	0.423069i	$-0.139047\pi$
$149$	7.31740e7	1.81219	0.906097	+	0.423069i	$0.139047\pi$
$150$	0	0
$151$	3.51556e6	0.0830951	0.0415475	−	0.999137i	$-0.486771\pi$
$151$	3.51556e6	0.0830951	0.0415475	+	0.999137i	$0.486771\pi$
$152$	0	0
$153$	1.36906e6	0.0309032
$154$	0	0
$155$	−2.15870e7	−0.465620
$156$	0	0
$157$	4.84023e7	0.998198	0.499099	−	0.866545i	$-0.333664\pi$
$157$	4.84023e7	0.998198	0.499099	+	0.866545i	$0.333664\pi$
$158$	0	0
$159$	1.10450e7	0.217909
$160$	0	0
$161$	−3.42201e7	−0.646235
$162$	0	0
$163$	−7.46558e6	−0.135023	−0.0675114	−	0.997719i	$-0.521506\pi$
$163$	−7.46558e6	−0.135023	−0.0675114	+	0.997719i	$0.521506\pi$
$164$	0	0
$165$	1.04490e7	0.181084
$166$	0	0
$167$	−1.12908e7	−0.187594	−0.0937968	−	0.995591i	$-0.529900\pi$
$167$	−1.12908e7	−0.187594	−0.0937968	+	0.995591i	$0.529900\pi$
$168$	0	0
$169$	1.07032e8	1.70574
$170$	0	0
$171$	2.27302e7	0.347630
$172$	0	0
$173$	−9.89209e7	−1.45254	−0.726268	−	0.687412i	$-0.758747\pi$
$173$	−9.89209e7	−1.45254	−0.726268	+	0.687412i	$0.758747\pi$
$174$	0	0
$175$	−1.60625e7	−0.226558
$176$	0	0
$177$	−4.07197e7	−0.551948
$178$	0	0
$179$	−7.70144e7	−1.00366	−0.501830	−	0.864966i	$-0.667340\pi$
$179$	−7.70144e7	−1.00366	−0.501830	+	0.864966i	$0.667340\pi$
$180$	0	0
$181$	1.06546e8	1.33555	0.667775	−	0.744363i	$-0.267247\pi$
$181$	1.06546e8	1.33555	0.667775	+	0.744363i	$0.267247\pi$
$182$	0	0
$183$	−8.16961e6	−0.0985422
$184$	0	0
$185$	−3.42925e6	−0.0398197
$186$	0	0
$187$	−5.81429e6	−0.0650206
$188$	0	0
$189$	−2.02341e7	−0.218006
$190$	0	0
$191$	2.98531e7	0.310008	0.155004	−	0.987914i	$-0.450461\pi$
$191$	2.98531e7	0.310008	0.155004	+	0.987914i	$0.450461\pi$
$192$	0	0
$193$	−7.96359e7	−0.797367	−0.398683	−	0.917089i	$-0.630533\pi$
$193$	−7.96359e7	−0.797367	−0.398683	+	0.917089i	$0.630533\pi$
$194$	0	0
$195$	4.39763e7	0.424714
$196$	0	0
$197$	−8.82207e7	−0.822127	−0.411063	−	0.911607i	$-0.634843\pi$
$197$	−8.82207e7	−0.822127	−0.411063	+	0.911607i	$0.634843\pi$
$198$	0	0
$199$	−1.13663e8	−1.02243	−0.511215	−	0.859453i	$-0.670804\pi$
$199$	−1.13663e8	−1.02243	−0.511215	+	0.859453i	$0.670804\pi$
$200$	0	0
$201$	−3.38670e7	−0.294164
$202$	0	0
$203$	2.19020e8	1.83758
$204$	0	0
$205$	−6.65813e7	−0.539776
$206$	0	0
$207$	2.42670e7	0.190160
$208$	0	0
$209$	−9.65333e7	−0.731417
$210$	0	0
$211$	7.54455e7	0.552898	0.276449	−	0.961029i	$-0.410842\pi$
$211$	7.54455e7	0.552898	0.276449	+	0.961029i	$0.410842\pi$
$212$	0	0
$213$	−1.29095e8	−0.915333
$214$	0	0
$215$	−1.13988e8	−0.782216
$216$	0	0
$217$	−1.77531e8	−1.17941
$218$	0	0
$219$	−1.35652e7	−0.0872712
$220$	0	0
$221$	−2.44703e7	−0.152499
$222$	0	0
$223$	−1.69318e8	−1.02244	−0.511218	−	0.859451i	$-0.670805\pi$
$223$	−1.69318e8	−1.02244	−0.511218	+	0.859451i	$0.670805\pi$
$224$	0	0
$225$	1.13906e7	0.0666667
$226$	0	0
$227$	−1.32506e7	−0.0751873	−0.0375936	−	0.999293i	$-0.511969\pi$
$227$	−1.32506e7	−0.0751873	−0.0375936	+	0.999293i	$0.511969\pi$
$228$	0	0
$229$	1.44714e8	0.796315	0.398158	−	0.917317i	$-0.369650\pi$
$229$	1.44714e8	0.796315	0.398158	+	0.917317i	$0.369650\pi$
$230$	0	0
$231$	8.59326e7	0.458686
$232$	0	0
$233$	2.13205e8	1.10421	0.552104	−	0.833775i	$-0.313825\pi$
$233$	2.13205e8	1.10421	0.552104	+	0.833775i	$0.313825\pi$
$234$	0	0
$235$	−9.15810e7	−0.460328
$236$	0	0
$237$	5.37669e7	0.262359
$238$	0	0
$239$	3.36644e8	1.59507	0.797533	−	0.603275i	$-0.206138\pi$
$239$	3.36644e8	1.59507	0.797533	+	0.603275i	$0.206138\pi$
$240$	0	0
$241$	−2.40329e8	−1.10598	−0.552989	−	0.833188i	$-0.686513\pi$
$241$	−2.40329e8	−1.10598	−0.552989	+	0.833188i	$0.686513\pi$
$242$	0	0
$243$	1.43489e7	0.0641500
$244$	0	0
$245$	−2.91551e7	−0.126658
$246$	0	0
$247$	−4.06275e8	−1.71546
$248$	0	0
$249$	2.18680e8	0.897660
$250$	0	0
$251$	−8.75235e7	−0.349355	−0.174677	−	0.984626i	$-0.555888\pi$
$251$	−8.75235e7	−0.349355	−0.174677	+	0.984626i	$0.555888\pi$
$252$	0	0
$253$	−1.03060e8	−0.400098
$254$	0	0
$255$	−6.33825e6	−0.0239375
$256$	0	0
$257$	1.05874e8	0.389068	0.194534	−	0.980896i	$-0.437681\pi$
$257$	1.05874e8	0.389068	0.194534	+	0.980896i	$0.437681\pi$
$258$	0	0
$259$	−2.82022e7	−0.100863
$260$	0	0
$261$	−1.55316e8	−0.540724
$262$	0	0
$263$	−2.81542e7	−0.0954330	−0.0477165	−	0.998861i	$-0.515194\pi$
$263$	−2.81542e7	−0.0954330	−0.0477165	+	0.998861i	$0.515194\pi$
$264$	0	0
$265$	−5.11343e7	−0.168792
$266$	0	0
$267$	2.02175e8	0.650038
$268$	0	0
$269$	−3.80408e8	−1.19156	−0.595781	−	0.803147i	$-0.703157\pi$
$269$	−3.80408e8	−1.19156	−0.595781	+	0.803147i	$0.703157\pi$
$270$	0	0
$271$	−6.13368e8	−1.87210	−0.936049	−	0.351870i	$-0.885546\pi$
$271$	−6.13368e8	−1.87210	−0.936049	+	0.351870i	$0.885546\pi$
$272$	0	0
$273$	3.61661e8	1.07580
$274$	0	0
$275$	−4.83750e7	−0.140267
$276$	0	0
$277$	−2.86407e8	−0.809662	−0.404831	−	0.914391i	$-0.632670\pi$
$277$	−2.86407e8	−0.809662	−0.404831	+	0.914391i	$0.632670\pi$
$278$	0	0
$279$	1.25895e8	0.347052
$280$	0	0
$281$	4.38705e8	1.17951	0.589753	−	0.807584i	$-0.299225\pi$
$281$	4.38705e8	1.17951	0.589753	+	0.807584i	$0.299225\pi$
$282$	0	0
$283$	−4.85747e8	−1.27397	−0.636983	−	0.770878i	$-0.719818\pi$
$283$	−4.85747e8	−1.27397	−0.636983	+	0.770878i	$0.719818\pi$
$284$	0	0
$285$	−1.05232e8	−0.269273
$286$	0	0
$287$	−5.47564e8	−1.36725
$288$	0	0
$289$	−4.06812e8	−0.991405
$290$	0	0
$291$	−4.63659e8	−1.10300
$292$	0	0
$293$	3.76441e8	0.874300	0.437150	−	0.899389i	$-0.355988\pi$
$293$	3.76441e8	0.874300	0.437150	+	0.899389i	$0.355988\pi$
$294$	0	0
$295$	1.88517e8	0.427537
$296$	0	0
$297$	−6.09386e7	−0.134972
$298$	0	0
$299$	−4.33743e8	−0.938389
$300$	0	0
$301$	−9.37441e8	−1.98135
$302$	0	0
$303$	−2.63097e8	−0.543334
$304$	0	0
$305$	3.78222e7	0.0763305
$306$	0	0
$307$	8.22474e8	1.62232	0.811162	−	0.584821i	$-0.198835\pi$
$307$	8.22474e8	1.62232	0.811162	+	0.584821i	$0.198835\pi$
$308$	0	0
$309$	4.58607e8	0.884273
$310$	0	0
$311$	4.61865e8	0.870671	0.435335	−	0.900268i	$-0.356630\pi$
$311$	4.61865e8	0.870671	0.435335	+	0.900268i	$0.356630\pi$
$312$	0	0
$313$	2.99153e8	0.551427	0.275714	−	0.961240i	$-0.411086\pi$
$313$	2.99153e8	0.551427	0.275714	+	0.961240i	$0.411086\pi$
$314$	0	0
$315$	9.36765e7	0.168867
$316$	0	0
$317$	6.25971e8	1.10369	0.551844	−	0.833947i	$-0.313924\pi$
$317$	6.25971e8	1.10369	0.551844	+	0.833947i	$0.313924\pi$
$318$	0	0
$319$	6.59615e8	1.13769
$320$	0	0
$321$	3.25041e8	0.548492
$322$	0	0
$323$	5.85560e7	0.0966858
$324$	0	0
$325$	−2.03594e8	−0.328982
$326$	0	0
$327$	3.19278e8	0.504954
$328$	0	0
$329$	−7.53162e8	−1.16601
$330$	0	0
$331$	−1.12189e9	−1.70041	−0.850203	−	0.526455i	$-0.823521\pi$
$331$	−1.12189e9	−1.70041	−0.850203	+	0.526455i	$0.823521\pi$
$332$	0	0
$333$	1.99994e7	0.0296799
$334$	0	0
$335$	1.56792e8	0.227859
$336$	0	0
$337$	−2.06060e8	−0.293285	−0.146642	−	0.989190i	$-0.546847\pi$
$337$	−2.06060e8	−0.293285	−0.146642	+	0.989190i	$0.546847\pi$
$338$	0	0
$339$	4.69428e8	0.654440
$340$	0	0
$341$	−5.34667e8	−0.730202
$342$	0	0
$343$	6.06830e8	0.811966
$344$	0	0
$345$	−1.12347e8	−0.147297
$346$	0	0
$347$	−5.20291e8	−0.668487	−0.334244	−	0.942487i	$-0.608481\pi$
$347$	−5.20291e8	−0.668487	−0.334244	+	0.942487i	$0.608481\pi$
$348$	0	0
$349$	−1.25363e9	−1.57863	−0.789313	−	0.613992i	$-0.789563\pi$
$349$	−1.25363e9	−1.57863	−0.789313	+	0.613992i	$0.789563\pi$
$350$	0	0
$351$	−2.56469e8	−0.316563
$352$	0	0
$353$	−3.54823e8	−0.429339	−0.214670	−	0.976687i	$-0.568868\pi$
$353$	−3.54823e8	−0.429339	−0.214670	+	0.976687i	$0.568868\pi$
$354$	0	0
$355$	5.97660e8	0.709014
$356$	0	0
$357$	−5.21258e7	−0.0606336
$358$	0	0
$359$	1.64905e9	1.88106	0.940530	−	0.339712i	$-0.110330\pi$
$359$	1.64905e9	1.88106	0.940530	+	0.339712i	$0.110330\pi$
$360$	0	0
$361$	7.83207e7	0.0876196
$362$	0	0
$363$	−2.67353e8	−0.293367
$364$	0	0
$365$	6.28017e7	0.0676000
$366$	0	0
$367$	3.96700e8	0.418920	0.209460	−	0.977817i	$-0.432829\pi$
$367$	3.96700e8	0.418920	0.209460	+	0.977817i	$0.432829\pi$
$368$	0	0
$369$	3.88302e8	0.402325
$370$	0	0
$371$	−4.20528e8	−0.427549
$372$	0	0
$373$	−2.20145e8	−0.219648	−0.109824	−	0.993951i	$-0.535029\pi$
$373$	−2.20145e8	−0.219648	−0.109824	+	0.993951i	$0.535029\pi$
$374$	0	0
$375$	−5.27344e7	−0.0516398
$376$	0	0
$377$	2.77609e9	2.66833
$378$	0	0
$379$	−6.05423e8	−0.571245	−0.285622	−	0.958342i	$-0.592200\pi$
$379$	−6.05423e8	−0.571245	−0.285622	+	0.958342i	$0.592200\pi$
$380$	0	0
$381$	3.88529e8	0.359904
$382$	0	0
$383$	1.18802e6	0.00108051	0.000540256	−	1.00000i	$-0.499828\pi$
$383$	1.18802e6	0.00108051	0.000540256		1.00000i	$0.499828\pi$
$384$	0	0
$385$	−3.97836e8	−0.355297
$386$	0	0
$387$	6.64781e8	0.583029
$388$	0	0
$389$	−2.16065e9	−1.86106	−0.930529	−	0.366217i	$-0.880653\pi$
$389$	−2.16065e9	−1.86106	−0.930529	+	0.366217i	$0.880653\pi$
$390$	0	0
$391$	6.25149e7	0.0528889
$392$	0	0
$393$	4.87592e8	0.405212
$394$	0	0
$395$	−2.48921e8	−0.203222
$396$	0	0
$397$	−1.30762e9	−1.04885	−0.524427	−	0.851455i	$-0.675720\pi$
$397$	−1.30762e9	−1.04885	−0.524427	+	0.851455i	$0.675720\pi$
$398$	0	0
$399$	−8.65432e8	−0.682068
$400$	0	0
$401$	1.23318e9	0.955041	0.477521	−	0.878621i	$-0.341536\pi$
$401$	1.23318e9	0.955041	0.477521	+	0.878621i	$0.341536\pi$
$402$	0	0
$403$	−2.25023e9	−1.71261
$404$	0	0
$405$	−6.64301e7	−0.0496904
$406$	0	0
$407$	−8.49357e7	−0.0624467
$408$	0	0
$409$	2.74339e9	1.98269	0.991346	−	0.131274i	$-0.0419067\pi$
$409$	2.74339e9	1.98269	0.991346	+	0.131274i	$0.0419067\pi$
$410$	0	0
$411$	1.21852e9	0.865738
$412$	0	0
$413$	1.55036e9	1.08295
$414$	0	0
$415$	−1.01241e9	−0.695324
$416$	0	0
$417$	−3.63895e8	−0.245754
$418$	0	0
$419$	1.57548e9	1.04632	0.523158	−	0.852236i	$-0.324754\pi$
$419$	1.57548e9	1.04632	0.523158	+	0.852236i	$0.324754\pi$
$420$	0	0
$421$	−5.15396e8	−0.336631	−0.168315	−	0.985733i	$-0.553833\pi$
$421$	−5.15396e8	−0.336631	−0.168315	+	0.985733i	$0.553833\pi$
$422$	0	0
$423$	5.34100e8	0.343109
$424$	0	0
$425$	2.93438e7	0.0185419
$426$	0	0
$427$	3.11050e8	0.193345
$428$	0	0
$429$	1.08920e9	0.666053
$430$	0	0
$431$	−9.20461e8	−0.553777	−0.276888	−	0.960902i	$-0.589303\pi$
$431$	−9.20461e8	−0.553777	−0.276888	+	0.960902i	$0.589303\pi$
$432$	0	0
$433$	9.01254e8	0.533507	0.266753	−	0.963765i	$-0.414049\pi$
$433$	9.01254e8	0.533507	0.266753	+	0.963765i	$0.414049\pi$
$434$	0	0
$435$	7.19057e8	0.418843
$436$	0	0
$437$	1.03792e9	0.594948
$438$	0	0
$439$	−7.57869e8	−0.427531	−0.213766	−	0.976885i	$-0.568573\pi$
$439$	−7.57869e8	−0.427531	−0.213766	+	0.976885i	$0.568573\pi$
$440$	0	0
$441$	1.70033e8	0.0944055
$442$	0	0
$443$	2.21270e9	1.20923	0.604615	−	0.796518i	$-0.293327\pi$
$443$	2.21270e9	1.20923	0.604615	+	0.796518i	$0.293327\pi$
$444$	0	0
$445$	−9.35996e8	−0.503517
$446$	0	0
$447$	1.97570e9	1.04627
$448$	0	0
$449$	2.65506e9	1.38424	0.692121	−	0.721781i	$-0.256676\pi$
$449$	2.65506e9	1.38424	0.692121	+	0.721781i	$0.256676\pi$
$450$	0	0
$451$	−1.64908e9	−0.846496
$452$	0	0
$453$	9.49201e7	0.0479750
$454$	0	0
$455$	−1.67436e9	−0.833312
$456$	0	0
$457$	3.38812e9	1.66055	0.830275	−	0.557354i	$-0.188183\pi$
$457$	3.38812e9	1.66055	0.830275	+	0.557354i	$0.188183\pi$
$458$	0	0
$459$	3.69647e7	0.0178420
$460$	0	0
$461$	−2.55176e9	−1.21307	−0.606536	−	0.795056i	$-0.707442\pi$
$461$	−2.55176e9	−1.21307	−0.606536	+	0.795056i	$0.707442\pi$
$462$	0	0
$463$	−9.40849e8	−0.440541	−0.220271	−	0.975439i	$-0.570694\pi$
$463$	−9.40849e8	−0.440541	−0.220271	+	0.975439i	$0.570694\pi$
$464$	0	0
$465$	−5.82849e8	−0.268826
$466$	0	0
$467$	2.81198e9	1.27762	0.638811	−	0.769364i	$-0.279427\pi$
$467$	2.81198e9	1.27762	0.638811	+	0.769364i	$0.279427\pi$
$468$	0	0
$469$	1.28945e9	0.577166
$470$	0	0
$471$	1.30686e9	0.576310
$472$	0	0
$473$	−2.82327e9	−1.22670
$474$	0	0
$475$	4.87188e8	0.208578
$476$	0	0
$477$	2.98215e8	0.125810
$478$	0	0
$479$	4.41140e9	1.83401	0.917006	−	0.398873i	$-0.130599\pi$
$479$	4.41140e9	1.83401	0.917006	+	0.398873i	$0.130599\pi$
$480$	0	0
$481$	−3.57465e8	−0.146462
$482$	0	0
$483$	−9.23942e8	−0.373104
$484$	0	0
$485$	2.14657e9	0.854377
$486$	0	0
$487$	1.58482e9	0.621768	0.310884	−	0.950448i	$-0.399375\pi$
$487$	1.58482e9	0.621768	0.310884	+	0.950448i	$0.399375\pi$
$488$	0	0
$489$	−2.01571e8	−0.0779554
$490$	0	0
$491$	2.25266e9	0.858836	0.429418	−	0.903106i	$-0.358719\pi$
$491$	2.25266e9	0.858836	0.429418	+	0.903106i	$0.358719\pi$
$492$	0	0
$493$	−4.00115e8	−0.150391
$494$	0	0
$495$	2.82123e8	0.104549
$496$	0	0
$497$	4.91516e9	1.79593
$498$	0	0
$499$	−3.21377e9	−1.15788	−0.578939	−	0.815371i	$-0.696533\pi$
$499$	−3.21377e9	−1.15788	−0.578939	+	0.815371i	$0.696533\pi$
$500$	0	0
$501$	−3.04852e8	−0.108307
$502$	0	0
$503$	1.46673e9	0.513879	0.256940	−	0.966427i	$-0.417286\pi$
$503$	1.46673e9	0.513879	0.256940	+	0.966427i	$0.417286\pi$
$504$	0	0
$505$	1.21804e9	0.420865
$506$	0	0
$507$	2.88987e9	0.984807
$508$	0	0
$509$	−6.22223e8	−0.209138	−0.104569	−	0.994518i	$-0.533346\pi$
$509$	−6.22223e8	−0.209138	−0.104569	+	0.994518i	$0.533346\pi$
$510$	0	0
$511$	5.16482e8	0.171231
$512$	0	0
$513$	6.13716e8	0.200704
$514$	0	0
$515$	−2.12318e9	−0.684955
$516$	0	0
$517$	−2.26828e9	−0.721904
$518$	0	0
$519$	−2.67086e9	−0.838622
$520$	0	0
$521$	−3.80030e9	−1.17730	−0.588648	−	0.808390i	$-0.700339\pi$
$521$	−3.80030e9	−1.17730	−0.588648	+	0.808390i	$0.700339\pi$
$522$	0	0
$523$	3.04889e9	0.931936	0.465968	−	0.884802i	$-0.345706\pi$
$523$	3.04889e9	0.931936	0.465968	+	0.884802i	$0.345706\pi$
$524$	0	0
$525$	−4.33688e8	−0.130803
$526$	0	0
$527$	3.24323e8	0.0965252
$528$	0	0
$529$	−2.29673e9	−0.674553
$530$	0	0
$531$	−1.09943e9	−0.318667
$532$	0	0
$533$	−6.94043e9	−1.98537
$534$	0	0
$535$	−1.50482e9	−0.424860
$536$	0	0
$537$	−2.07939e9	−0.579463
$538$	0	0
$539$	−7.22114e8	−0.198630
$540$	0	0
$541$	−4.90621e9	−1.33216	−0.666079	−	0.745882i	$-0.732028\pi$
$541$	−4.90621e9	−1.33216	−0.666079	+	0.745882i	$0.732028\pi$
$542$	0	0
$543$	2.87673e9	0.771081
$544$	0	0
$545$	−1.47814e9	−0.391135
$546$	0	0
$547$	4.24362e9	1.10862	0.554308	−	0.832312i	$-0.312983\pi$
$547$	4.24362e9	1.10862	0.554308	+	0.832312i	$0.312983\pi$
$548$	0	0
$549$	−2.20579e8	−0.0568934
$550$	0	0
$551$	−6.64302e9	−1.69175
$552$	0	0
$553$	−2.04713e9	−0.514762
$554$	0	0
$555$	−9.25898e7	−0.0229899
$556$	0	0
$557$	6.83397e9	1.67564	0.837818	−	0.545949i	$-0.183831\pi$
$557$	6.83397e9	1.67564	0.837818	+	0.545949i	$0.183831\pi$
$558$	0	0
$559$	−1.18822e10	−2.87709
$560$	0	0
$561$	−1.56986e8	−0.0375397
$562$	0	0
$563$	5.19575e9	1.22707	0.613534	−	0.789668i	$-0.289747\pi$
$563$	5.19575e9	1.22707	0.613534	+	0.789668i	$0.289747\pi$
$564$	0	0
$565$	−2.17328e9	−0.506927
$566$	0	0
$567$	−5.46321e8	−0.125866
$568$	0	0
$569$	4.99097e9	1.13577	0.567887	−	0.823107i	$-0.307761\pi$
$569$	4.99097e9	1.13577	0.567887	+	0.823107i	$0.307761\pi$
$570$	0	0
$571$	−5.63624e9	−1.26696	−0.633480	−	0.773759i	$-0.718374\pi$
$571$	−5.63624e9	−1.26696	−0.633480	+	0.773759i	$0.718374\pi$
$572$	0	0
$573$	8.06034e8	0.178983
$574$	0	0
$575$	5.20125e8	0.114096
$576$	0	0
$577$	2.34583e8	0.0508372	0.0254186	−	0.999677i	$-0.491908\pi$
$577$	2.34583e8	0.0508372	0.0254186	+	0.999677i	$0.491908\pi$
$578$	0	0
$579$	−2.15017e9	−0.460360
$580$	0	0
$581$	−8.32605e9	−1.76126
$582$	0	0
$583$	−1.26649e9	−0.264705
$584$	0	0
$585$	1.18736e9	0.245209
$586$	0	0
$587$	−6.81690e9	−1.39108	−0.695542	−	0.718485i	$-0.744836\pi$
$587$	−6.81690e9	−1.39108	−0.695542	+	0.718485i	$0.744836\pi$
$588$	0	0
$589$	5.38466e9	1.08581
$590$	0	0
$591$	−2.38196e9	−0.474655
$592$	0	0
$593$	7.44761e9	1.46665	0.733323	−	0.679880i	$-0.237968\pi$
$593$	7.44761e9	1.46665	0.733323	+	0.679880i	$0.237968\pi$
$594$	0	0
$595$	2.41323e8	0.0469666
$596$	0	0
$597$	−3.06890e9	−0.590300
$598$	0	0
$599$	4.83339e9	0.918878	0.459439	−	0.888209i	$-0.348050\pi$
$599$	4.83339e9	0.918878	0.459439	+	0.888209i	$0.348050\pi$
$600$	0	0
$601$	3.34645e9	0.628816	0.314408	−	0.949288i	$-0.398194\pi$
$601$	3.34645e9	0.628816	0.314408	+	0.949288i	$0.398194\pi$
$602$	0	0
$603$	−9.14408e8	−0.169836
$604$	0	0
$605$	1.23774e9	0.227241
$606$	0	0
$607$	8.46327e9	1.53595	0.767976	−	0.640479i	$-0.221264\pi$
$607$	8.46327e9	1.53595	0.767976	+	0.640479i	$0.221264\pi$
$608$	0	0
$609$	5.91353e9	1.06093
$610$	0	0
$611$	−9.54640e9	−1.69315
$612$	0	0
$613$	−6.10381e9	−1.07026	−0.535130	−	0.844769i	$-0.679738\pi$
$613$	−6.10381e9	−1.07026	−0.535130	+	0.844769i	$0.679738\pi$
$614$	0	0
$615$	−1.79769e9	−0.311640
$616$	0	0
$617$	1.77521e9	0.304265	0.152133	−	0.988360i	$-0.451386\pi$
$617$	1.77521e9	0.304265	0.152133	+	0.988360i	$0.451386\pi$
$618$	0	0
$619$	8.84220e9	1.49845	0.749226	−	0.662314i	$-0.230426\pi$
$619$	8.84220e9	1.49845	0.749226	+	0.662314i	$0.230426\pi$
$620$	0	0
$621$	6.55208e8	0.109789
$622$	0	0
$623$	−7.69763e9	−1.27541
$624$	0	0
$625$	2.44141e8	0.0400000
$626$	0	0
$627$	−2.60640e9	−0.422284
$628$	0	0
$629$	5.15211e7	0.00825482
$630$	0	0
$631$	5.56994e9	0.882567	0.441284	−	0.897368i	$-0.354523\pi$
$631$	5.56994e9	0.882567	0.441284	+	0.897368i	$0.354523\pi$
$632$	0	0
$633$	2.03703e9	0.319216
$634$	0	0
$635$	−1.79875e9	−0.278780
$636$	0	0
$637$	−3.03913e9	−0.465866
$638$	0	0
$639$	−3.48555e9	−0.528468
$640$	0	0
$641$	−1.17103e10	−1.75616	−0.878079	−	0.478516i	$-0.841175\pi$
$641$	−1.17103e10	−1.75616	−0.878079	+	0.478516i	$0.841175\pi$
$642$	0	0
$643$	6.05369e9	0.898011	0.449005	−	0.893529i	$-0.351778\pi$
$643$	6.05369e9	0.898011	0.449005	+	0.893529i	$0.351778\pi$
$644$	0	0
$645$	−3.07769e9	−0.451612
$646$	0	0
$647$	−1.16775e10	−1.69506	−0.847529	−	0.530749i	$-0.821911\pi$
$647$	−1.16775e10	−1.69506	−0.847529	+	0.530749i	$0.821911\pi$
$648$	0	0
$649$	4.66919e9	0.670479
$650$	0	0
$651$	−4.79335e9	−0.680935
$652$	0	0
$653$	−8.81248e9	−1.23852	−0.619259	−	0.785187i	$-0.712567\pi$
$653$	−8.81248e9	−1.23852	−0.619259	+	0.785187i	$0.712567\pi$
$654$	0	0
$655$	−2.25737e9	−0.313876
$656$	0	0
$657$	−3.66260e8	−0.0503861
$658$	0	0
$659$	−1.83941e9	−0.250368	−0.125184	−	0.992134i	$-0.539952\pi$
$659$	−1.83941e9	−0.250368	−0.125184	+	0.992134i	$0.539952\pi$
$660$	0	0
$661$	−5.81427e9	−0.783050	−0.391525	−	0.920167i	$-0.628052\pi$
$661$	−5.81427e9	−0.783050	−0.391525	+	0.920167i	$0.628052\pi$
$662$	0	0
$663$	−6.60699e8	−0.0880453
$664$	0	0
$665$	4.00663e9	0.528328
$666$	0	0
$667$	−7.09214e9	−0.925416
$668$	0	0
$669$	−4.57159e9	−0.590304
$670$	0	0
$671$	9.36781e8	0.119704
$672$	0	0
$673$	8.53129e9	1.07885	0.539426	−	0.842033i	$-0.318641\pi$
$673$	8.53129e9	1.07885	0.539426	+	0.842033i	$0.318641\pi$
$674$	0	0
$675$	3.07547e8	0.0384900
$676$	0	0
$677$	−1.03399e10	−1.28073	−0.640364	−	0.768071i	$-0.721217\pi$
$677$	−1.03399e10	−1.28073	−0.640364	+	0.768071i	$0.721217\pi$
$678$	0	0
$679$	1.76534e10	2.16413
$680$	0	0
$681$	−3.57766e8	−0.0434094
$682$	0	0
$683$	7.04112e9	0.845609	0.422804	−	0.906221i	$-0.361046\pi$
$683$	7.04112e9	0.845609	0.422804	+	0.906221i	$0.361046\pi$
$684$	0	0
$685$	−5.64130e9	−0.670598
$686$	0	0
$687$	3.90727e9	0.459753
$688$	0	0
$689$	−5.33023e9	−0.620839
$690$	0	0
$691$	−4.87229e9	−0.561772	−0.280886	−	0.959741i	$-0.590628\pi$
$691$	−4.87229e9	−0.561772	−0.280886	+	0.959741i	$0.590628\pi$
$692$	0	0
$693$	2.32018e9	0.264823
$694$	0	0
$695$	1.68470e9	0.190360
$696$	0	0
$697$	1.00032e9	0.111898
$698$	0	0
$699$	5.75653e9	0.637515
$700$	0	0
$701$	1.44547e10	1.58488	0.792439	−	0.609951i	$-0.208811\pi$
$701$	1.44547e10	1.58488	0.792439	+	0.609951i	$0.208811\pi$
$702$	0	0
$703$	8.55392e8	0.0928585
$704$	0	0
$705$	−2.47269e9	−0.265771
$706$	0	0
$707$	1.00172e10	1.06605
$708$	0	0
$709$	−4.76442e9	−0.502052	−0.251026	−	0.967980i	$-0.580768\pi$
$709$	−4.76442e9	−0.502052	−0.251026	+	0.967980i	$0.580768\pi$
$710$	0	0
$711$	1.45171e9	0.151473
$712$	0	0
$713$	5.74870e9	0.593959
$714$	0	0
$715$	−5.04261e9	−0.515922
$716$	0	0
$717$	9.08940e9	0.920912
$718$	0	0
$719$	−3.59296e9	−0.360497	−0.180248	−	0.983621i	$-0.557690\pi$
$719$	−3.59296e9	−0.360497	−0.180248	+	0.983621i	$0.557690\pi$
$720$	0	0
$721$	−1.74610e10	−1.73499
$722$	0	0
$723$	−6.48889e9	−0.638537
$724$	0	0
$725$	−3.32897e9	−0.324434
$726$	0	0
$727$	1.34517e10	1.29840	0.649198	−	0.760620i	$-0.275105\pi$
$727$	1.34517e10	1.29840	0.649198	+	0.760620i	$0.275105\pi$
$728$	0	0
$729$	3.87420e8	0.0370370
$730$	0	0
$731$	1.71256e9	0.162157
$732$	0	0
$733$	1.21392e10	1.13848	0.569242	−	0.822170i	$-0.307237\pi$
$733$	1.21392e10	1.13848	0.569242	+	0.822170i	$0.307237\pi$
$734$	0	0
$735$	−7.87188e8	−0.0731262
$736$	0	0
$737$	3.88341e9	0.357336
$738$	0	0
$739$	3.59124e9	0.327333	0.163666	−	0.986516i	$-0.447668\pi$
$739$	3.59124e9	0.327333	0.163666	+	0.986516i	$0.447668\pi$
$740$	0	0
$741$	−1.09694e10	−0.990423
$742$	0	0
$743$	−1.30847e10	−1.17031	−0.585156	−	0.810921i	$-0.698967\pi$
$743$	−1.30847e10	−1.17031	−0.585156	+	0.810921i	$0.698967\pi$
$744$	0	0
$745$	−9.14675e9	−0.810438
$746$	0	0
$747$	5.90437e9	0.518264
$748$	0	0
$749$	−1.23756e10	−1.07617
$750$	0	0
$751$	−8.09743e9	−0.697602	−0.348801	−	0.937197i	$-0.613411\pi$
$751$	−8.09743e9	−0.697602	−0.348801	+	0.937197i	$0.613411\pi$
$752$	0	0
$753$	−2.36313e9	−0.201700
$754$	0	0
$755$	−4.39445e8	−0.0371612
$756$	0	0
$757$	1.65837e10	1.38946	0.694729	−	0.719271i	$-0.255524\pi$
$757$	1.65837e10	1.38946	0.694729	+	0.719271i	$0.255524\pi$
$758$	0	0
$759$	−2.78261e9	−0.230997
$760$	0	0
$761$	−2.34197e9	−0.192635	−0.0963175	−	0.995351i	$-0.530706\pi$
$761$	−2.34197e9	−0.192635	−0.0963175	+	0.995351i	$0.530706\pi$
$762$	0	0
$763$	−1.21562e10	−0.990746
$764$	0	0
$765$	−1.71133e8	−0.0138203
$766$	0	0
$767$	1.96510e10	1.57254
$768$	0	0
$769$	1.38432e9	0.109773	0.0548863	−	0.998493i	$-0.482520\pi$
$769$	1.38432e9	0.109773	0.0548863	+	0.998493i	$0.482520\pi$
$770$	0	0
$771$	2.85861e9	0.224628
$772$	0	0
$773$	1.23896e10	0.964785	0.482392	−	0.875955i	$-0.339768\pi$
$773$	1.23896e10	0.964785	0.482392	+	0.875955i	$0.339768\pi$
$774$	0	0
$775$	2.69838e9	0.208231
$776$	0	0
$777$	−7.61458e8	−0.0582334
$778$	0	0
$779$	1.66080e10	1.25874
$780$	0	0
$781$	1.48028e10	1.11190
$782$	0	0
$783$	−4.19354e9	−0.312187
$784$	0	0
$785$	−6.05028e9	−0.446408
$786$	0	0
$787$	−3.44776e9	−0.252130	−0.126065	−	0.992022i	$-0.540235\pi$
$787$	−3.44776e9	−0.252130	−0.126065	+	0.992022i	$0.540235\pi$
$788$	0	0
$789$	−7.60164e8	−0.0550983
$790$	0	0
$791$	−1.78730e10	−1.28404
$792$	0	0
$793$	3.94259e9	0.280754
$794$	0	0
$795$	−1.38062e9	−0.0974520
$796$	0	0
$797$	−1.58177e9	−0.110673	−0.0553364	−	0.998468i	$-0.517623\pi$
$797$	−1.58177e9	−0.110673	−0.0553364	+	0.998468i	$0.517623\pi$
$798$	0	0
$799$	1.37591e9	0.0954283
$800$	0	0
$801$	5.45873e9	0.375299
$802$	0	0
$803$	1.55547e9	0.106013
$804$	0	0
$805$	4.27751e9	0.289005
$806$	0	0
$807$	−1.02710e10	−0.687948
$808$	0	0
$809$	6.06280e9	0.402581	0.201291	−	0.979532i	$-0.435486\pi$
$809$	6.06280e9	0.402581	0.201291	+	0.979532i	$0.435486\pi$
$810$	0	0
$811$	−1.09241e10	−0.719140	−0.359570	−	0.933118i	$-0.617077\pi$
$811$	−1.09241e10	−0.719140	−0.359570	+	0.933118i	$0.617077\pi$
$812$	0	0
$813$	−1.65609e10	−1.08086
$814$	0	0
$815$	9.33198e8	0.0603840
$816$	0	0
$817$	2.84333e10	1.82411
$818$	0	0
$819$	9.76484e9	0.621114
$820$	0	0
$821$	9.10830e9	0.574429	0.287215	−	0.957866i	$-0.407271\pi$
$821$	9.10830e9	0.574429	0.287215	+	0.957866i	$0.407271\pi$
$822$	0	0
$823$	2.13411e9	0.133449	0.0667247	−	0.997771i	$-0.478745\pi$
$823$	2.13411e9	0.133449	0.0667247	+	0.997771i	$0.478745\pi$
$824$	0	0
$825$	−1.30612e9	−0.0809834
$826$	0	0
$827$	−1.74657e10	−1.07379	−0.536893	−	0.843651i	$-0.680402\pi$
$827$	−1.74657e10	−1.07379	−0.536893	+	0.843651i	$0.680402\pi$
$828$	0	0
$829$	2.16797e10	1.32164	0.660818	−	0.750546i	$-0.270210\pi$
$829$	2.16797e10	1.32164	0.660818	+	0.750546i	$0.270210\pi$
$830$	0	0
$831$	−7.73298e9	−0.467459
$832$	0	0
$833$	4.38027e8	0.0262569
$834$	0	0
$835$	1.41135e9	0.0838944
$836$	0	0
$837$	3.39918e9	0.200371
$838$	0	0
$839$	1.51956e10	0.888285	0.444142	−	0.895956i	$-0.353508\pi$
$839$	1.51956e10	0.888285	0.444142	+	0.895956i	$0.353508\pi$
$840$	0	0
$841$	2.81421e10	1.63144
$842$	0	0
$843$	1.18450e10	0.680988
$844$	0	0
$845$	−1.33790e10	−0.762828
$846$	0	0
$847$	1.01792e10	0.575602
$848$	0	0
$849$	−1.31152e10	−0.735524
$850$	0	0
$851$	9.13223e8	0.0507953
$852$	0	0
$853$	1.88790e10	1.04150	0.520749	−	0.853710i	$-0.325653\pi$
$853$	1.88790e10	1.04150	0.520749	+	0.853710i	$0.325653\pi$
$854$	0	0
$855$	−2.84128e9	−0.155465
$856$	0	0
$857$	−2.49105e9	−0.135192	−0.0675958	−	0.997713i	$-0.521533\pi$
$857$	−2.49105e9	−0.135192	−0.0675958	+	0.997713i	$0.521533\pi$
$858$	0	0
$859$	1.61520e10	0.869460	0.434730	−	0.900561i	$-0.356844\pi$
$859$	1.61520e10	0.869460	0.434730	+	0.900561i	$0.356844\pi$
$860$	0	0
$861$	−1.47842e10	−0.789383
$862$	0	0
$863$	−1.91876e10	−1.01621	−0.508104	−	0.861296i	$-0.669654\pi$
$863$	−1.91876e10	−1.01621	−0.508104	+	0.861296i	$0.669654\pi$
$864$	0	0
$865$	1.23651e10	0.649594
$866$	0	0
$867$	−1.09839e10	−0.572388
$868$	0	0
$869$	−6.16528e9	−0.318701
$870$	0	0
$871$	1.63439e10	0.838095
$872$	0	0
$873$	−1.25188e10	−0.636815
$874$	0	0
$875$	2.00781e9	0.101320
$876$	0	0
$877$	2.04959e10	1.02605	0.513026	−	0.858373i	$-0.328524\pi$
$877$	2.04959e10	1.02605	0.513026	+	0.858373i	$0.328524\pi$
$878$	0	0
$879$	1.01639e10	0.504777
$880$	0	0
$881$	−2.07168e10	−1.02072	−0.510360	−	0.859961i	$-0.670488\pi$
$881$	−2.07168e10	−1.02072	−0.510360	+	0.859961i	$0.670488\pi$
$882$	0	0
$883$	5.28980e9	0.258569	0.129285	−	0.991608i	$-0.458732\pi$
$883$	5.28980e9	0.258569	0.129285	+	0.991608i	$0.458732\pi$
$884$	0	0
$885$	5.08996e9	0.246838
$886$	0	0
$887$	2.67479e10	1.28694	0.643468	−	0.765473i	$-0.277495\pi$
$887$	2.67479e10	1.28694	0.643468	+	0.765473i	$0.277495\pi$
$888$	0	0
$889$	−1.47929e10	−0.706150
$890$	0	0
$891$	−1.64534e9	−0.0779263
$892$	0	0
$893$	2.28440e10	1.07347
$894$	0	0
$895$	9.62680e9	0.448850
$896$	0	0
$897$	−1.17111e10	−0.541779
$898$	0	0
$899$	−3.67936e10	−1.68894
$900$	0	0
$901$	7.68241e8	0.0349913
$902$	0	0
$903$	−2.53109e10	−1.14393
$904$	0	0
$905$	−1.33182e10	−0.597277
$906$	0	0
$907$	−3.16032e8	−0.0140639	−0.00703194	−	0.999975i	$-0.502238\pi$
$907$	−3.16032e8	−0.0140639	−0.00703194	+	0.999975i	$0.502238\pi$
$908$	0	0
$909$	−7.10363e9	−0.313694
$910$	0	0
$911$	1.82747e10	0.800822	0.400411	−	0.916336i	$-0.368867\pi$
$911$	1.82747e10	0.800822	0.400411	+	0.916336i	$0.368867\pi$
$912$	0	0
$913$	−2.50753e10	−1.09043
$914$	0	0
$915$	1.02120e9	0.0440694
$916$	0	0
$917$	−1.85646e10	−0.795048
$918$	0	0
$919$	4.07018e10	1.72985	0.864927	−	0.501897i	$-0.167364\pi$
$919$	4.07018e10	1.72985	0.864927	+	0.501897i	$0.167364\pi$
$920$	0	0
$921$	2.22068e10	0.936650
$922$	0	0
$923$	6.23001e10	2.60785
$924$	0	0
$925$	4.28656e8	0.0178079
$926$	0	0
$927$	1.23824e10	0.510535
$928$	0	0
$929$	3.93743e10	1.61123	0.805616	−	0.592438i	$-0.201835\pi$
$929$	3.93743e10	1.61123	0.805616	+	0.592438i	$0.201835\pi$
$930$	0	0
$931$	7.27245e9	0.295364
$932$	0	0
$933$	1.24704e10	0.502682
$934$	0	0
$935$	7.26786e8	0.0290781
$936$	0	0
$937$	−2.40969e10	−0.956912	−0.478456	−	0.878112i	$-0.658803\pi$
$937$	−2.40969e10	−0.956912	−0.478456	+	0.878112i	$0.658803\pi$
$938$	0	0
$939$	8.07713e9	0.318367
$940$	0	0
$941$	−1.57229e10	−0.615131	−0.307566	−	0.951527i	$-0.599514\pi$
$941$	−1.57229e10	−0.615131	−0.307566	+	0.951527i	$0.599514\pi$
$942$	0	0
$943$	1.77309e10	0.688555
$944$	0	0
$945$	2.52927e9	0.0974952
$946$	0	0
$947$	1.11795e10	0.427758	0.213879	−	0.976860i	$-0.431390\pi$
$947$	1.11795e10	0.427758	0.213879	+	0.976860i	$0.431390\pi$
$948$	0	0
$949$	6.54645e9	0.248642
$950$	0	0
$951$	1.69012e10	0.637215
$952$	0	0
$953$	7.02371e9	0.262870	0.131435	−	0.991325i	$-0.458041\pi$
$953$	7.02371e9	0.262870	0.131435	+	0.991325i	$0.458041\pi$
$954$	0	0
$955$	−3.73164e9	−0.138640
$956$	0	0
$957$	1.78096e10	0.656845
$958$	0	0
$959$	−4.63940e10	−1.69862
$960$	0	0
$961$	2.31129e9	0.0840085
$962$	0	0
$963$	8.77611e9	0.316672
$964$	0	0
$965$	9.95448e9	0.356593
$966$	0	0
$967$	−8.09820e9	−0.288002	−0.144001	−	0.989578i	$-0.545997\pi$
$967$	−8.09820e9	−0.288002	−0.144001	+	0.989578i	$0.545997\pi$
$968$	0	0
$969$	1.58101e9	0.0558216
$970$	0	0
$971$	−3.51850e10	−1.23336	−0.616681	−	0.787213i	$-0.711523\pi$
$971$	−3.51850e10	−1.23336	−0.616681	+	0.787213i	$0.711523\pi$
$972$	0	0
$973$	1.38550e10	0.482182
$974$	0	0
$975$	−5.49703e9	−0.189938
$976$	0	0
$977$	−3.58518e10	−1.22993	−0.614964	−	0.788555i	$-0.710830\pi$
$977$	−3.58518e10	−1.22993	−0.614964	+	0.788555i	$0.710830\pi$
$978$	0	0
$979$	−2.31828e10	−0.789634
$980$	0	0
$981$	8.62050e9	0.291535
$982$	0	0
$983$	1.86814e10	0.627295	0.313648	−	0.949539i	$-0.398449\pi$
$983$	1.86814e10	0.627295	0.313648	+	0.949539i	$0.398449\pi$
$984$	0	0
$985$	1.10276e10	0.367666
$986$	0	0
$987$	−2.03354e10	−0.673197
$988$	0	0
$989$	3.03556e10	0.997819
$990$	0	0
$991$	6.86496e9	0.224068	0.112034	−	0.993704i	$-0.464263\pi$
$991$	6.86496e9	0.224068	0.112034	+	0.993704i	$0.464263\pi$
$992$	0	0
$993$	−3.02910e10	−0.981730
$994$	0	0
$995$	1.42079e10	0.457244
$996$	0	0
$997$	4.82086e9	0.154060	0.0770302	−	0.997029i	$-0.475456\pi$
$997$	4.82086e9	0.154060	0.0770302	+	0.997029i	$0.475456\pi$
$998$	0	0
$999$	5.39983e8	0.0171357

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	240.8.a.i.1.1		1
4.3	odd	2		60.8.a.a.1.1	✓	1
12.11	even	2		180.8.a.e.1.1		1
20.3	even	4		300.8.d.d.49.1		2
20.7	even	4		300.8.d.d.49.2		2
20.19	odd	2		300.8.a.e.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
60.8.a.a.1.1	✓	1	4.3	odd	2
180.8.a.e.1.1		1	12.11	even	2
240.8.a.i.1.1		1	1.1	even	1	trivial
300.8.a.e.1.1		1	20.19	odd	2
300.8.d.d.49.1		2	20.3	even	4
300.8.d.d.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:	\( N \)	\(=\)	\( 240 = 2^{4} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	240.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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