LMFDB - Embedded newform 2790.2.a.y.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2790,2,Mod(1,2790)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2790.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2790, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2790 = 2 \cdot 3^{2} \cdot 5 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2790.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,1,0,1,1,0,-3,1,0,1,-3,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$22.2782621639$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 930)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	2790.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -3.00000 q^{7} +1.00000 q^{8} +1.00000 q^{10} -3.00000 q^{11} -2.00000 q^{13} -3.00000 q^{14} +1.00000 q^{16} +4.00000 q^{17} -3.00000 q^{19} +1.00000 q^{20} -3.00000 q^{22} -5.00000 q^{23} +1.00000 q^{25} -2.00000 q^{26} -3.00000 q^{28} -4.00000 q^{29} +1.00000 q^{31} +1.00000 q^{32} +4.00000 q^{34} -3.00000 q^{35} -3.00000 q^{38} +1.00000 q^{40} -4.00000 q^{41} +1.00000 q^{43} -3.00000 q^{44} -5.00000 q^{46} -10.0000 q^{47} +2.00000 q^{49} +1.00000 q^{50} -2.00000 q^{52} -3.00000 q^{53} -3.00000 q^{55} -3.00000 q^{56} -4.00000 q^{58} -6.00000 q^{59} -2.00000 q^{61} +1.00000 q^{62} +1.00000 q^{64} -2.00000 q^{65} +2.00000 q^{67} +4.00000 q^{68} -3.00000 q^{70} -7.00000 q^{71} +5.00000 q^{73} -3.00000 q^{76} +9.00000 q^{77} -1.00000 q^{79} +1.00000 q^{80} -4.00000 q^{82} -12.0000 q^{83} +4.00000 q^{85} +1.00000 q^{86} -3.00000 q^{88} -1.00000 q^{89} +6.00000 q^{91} -5.00000 q^{92} -10.0000 q^{94} -3.00000 q^{95} -10.0000 q^{97} +2.00000 q^{98} +O(q^{100})$

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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−3.00000	−1.13389	−0.566947	−	0.823754i	$-0.691875\pi$
$7$	−3.00000	−1.13389	−0.566947	+	0.823754i	$0.691875\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	1.00000	0.316228
$11$	−3.00000	−0.904534	−0.452267	−	0.891883i	$-0.649385\pi$
$11$	−3.00000	−0.904534	−0.452267	+	0.891883i	$0.649385\pi$
$12$	0	0
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	−3.00000	−0.801784
$15$	0	0
$16$	1.00000	0.250000
$17$	4.00000	0.970143	0.485071	−	0.874475i	$-0.338794\pi$
$17$	4.00000	0.970143	0.485071	+	0.874475i	$0.338794\pi$
$18$	0	0
$19$	−3.00000	−0.688247	−0.344124	−	0.938924i	$-0.611824\pi$
$19$	−3.00000	−0.688247	−0.344124	+	0.938924i	$0.611824\pi$
$20$	1.00000	0.223607
$21$	0	0
$22$	−3.00000	−0.639602
$23$	−5.00000	−1.04257	−0.521286	−	0.853382i	$-0.674548\pi$
$23$	−5.00000	−1.04257	−0.521286	+	0.853382i	$0.674548\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−2.00000	−0.392232
$27$	0	0
$28$	−3.00000	−0.566947
$29$	−4.00000	−0.742781	−0.371391	−	0.928477i	$-0.621119\pi$
$29$	−4.00000	−0.742781	−0.371391	+	0.928477i	$0.621119\pi$
$30$	0	0
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	1.00000	0.176777
$33$	0	0
$34$	4.00000	0.685994
$35$	−3.00000	−0.507093
$36$	0	0
$37$	0	0		−	1.00000i	$-0.5\pi$
$37$	0	0			1.00000i	$0.5\pi$
$38$	−3.00000	−0.486664
$39$	0	0
$40$	1.00000	0.158114
$41$	−4.00000	−0.624695	−0.312348	−	0.949968i	$-0.601115\pi$
$41$	−4.00000	−0.624695	−0.312348	+	0.949968i	$0.601115\pi$
$42$	0	0
$43$	1.00000	0.152499	0.0762493	−	0.997089i	$-0.475706\pi$
$43$	1.00000	0.152499	0.0762493	+	0.997089i	$0.475706\pi$
$44$	−3.00000	−0.452267
$45$	0	0
$46$	−5.00000	−0.737210
$47$	−10.0000	−1.45865	−0.729325	−	0.684167i	$-0.760166\pi$
$47$	−10.0000	−1.45865	−0.729325	+	0.684167i	$0.760166\pi$
$48$	0	0
$49$	2.00000	0.285714
$50$	1.00000	0.141421
$51$	0	0
$52$	−2.00000	−0.277350
$53$	−3.00000	−0.412082	−0.206041	−	0.978543i	$-0.566058\pi$
$53$	−3.00000	−0.412082	−0.206041	+	0.978543i	$0.566058\pi$
$54$	0	0
$55$	−3.00000	−0.404520
$56$	−3.00000	−0.400892
$57$	0	0
$58$	−4.00000	−0.525226
$59$	−6.00000	−0.781133	−0.390567	−	0.920575i	$-0.627721\pi$
$59$	−6.00000	−0.781133	−0.390567	+	0.920575i	$0.627721\pi$
$60$	0	0
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	1.00000	0.127000
$63$	0	0
$64$	1.00000	0.125000
$65$	−2.00000	−0.248069
$66$	0	0
$67$	2.00000	0.244339	0.122169	−	0.992509i	$-0.461015\pi$
$67$	2.00000	0.244339	0.122169	+	0.992509i	$0.461015\pi$
$68$	4.00000	0.485071
$69$	0	0
$70$	−3.00000	−0.358569
$71$	−7.00000	−0.830747	−0.415374	−	0.909651i	$-0.636349\pi$
$71$	−7.00000	−0.830747	−0.415374	+	0.909651i	$0.636349\pi$
$72$	0	0
$73$	5.00000	0.585206	0.292603	−	0.956234i	$-0.405479\pi$
$73$	5.00000	0.585206	0.292603	+	0.956234i	$0.405479\pi$
$74$	0	0
$75$	0	0
$76$	−3.00000	−0.344124
$77$	9.00000	1.02565
$78$	0	0
$79$	−1.00000	−0.112509	−0.0562544	−	0.998416i	$-0.517916\pi$
$79$	−1.00000	−0.112509	−0.0562544	+	0.998416i	$0.517916\pi$
$80$	1.00000	0.111803
$81$	0	0
$82$	−4.00000	−0.441726
$83$	−12.0000	−1.31717	−0.658586	−	0.752506i	$-0.728845\pi$
$83$	−12.0000	−1.31717	−0.658586	+	0.752506i	$0.728845\pi$
$84$	0	0
$85$	4.00000	0.433861
$86$	1.00000	0.107833
$87$	0	0
$88$	−3.00000	−0.319801
$89$	−1.00000	−0.106000	−0.0529999	−	0.998595i	$-0.516878\pi$
$89$	−1.00000	−0.106000	−0.0529999	+	0.998595i	$0.516878\pi$
$90$	0	0
$91$	6.00000	0.628971
$92$	−5.00000	−0.521286
$93$	0	0
$94$	−10.0000	−1.03142
$95$	−3.00000	−0.307794
$96$	0	0
$97$	−10.0000	−1.01535	−0.507673	−	0.861550i	$-0.669494\pi$
$97$	−10.0000	−1.01535	−0.507673	+	0.861550i	$0.669494\pi$
$98$	2.00000	0.202031
$99$	0	0
$100$	1.00000	0.100000
$101$	1.00000	0.0995037	0.0497519	−	0.998762i	$-0.484157\pi$
$101$	1.00000	0.0995037	0.0497519	+	0.998762i	$0.484157\pi$
$102$	0	0
$103$	16.0000	1.57653	0.788263	−	0.615338i	$-0.210980\pi$
$103$	16.0000	1.57653	0.788263	+	0.615338i	$0.210980\pi$
$104$	−2.00000	−0.196116
$105$	0	0
$106$	−3.00000	−0.291386
$107$	−9.00000	−0.870063	−0.435031	−	0.900415i	$-0.643263\pi$
$107$	−9.00000	−0.870063	−0.435031	+	0.900415i	$0.643263\pi$
$108$	0	0
$109$	20.0000	1.91565	0.957826	−	0.287348i	$-0.0927736\pi$
$109$	20.0000	1.91565	0.957826	+	0.287348i	$0.0927736\pi$
$110$	−3.00000	−0.286039
$111$	0	0
$112$	−3.00000	−0.283473
$113$	−9.00000	−0.846649	−0.423324	−	0.905978i	$-0.639137\pi$
$113$	−9.00000	−0.846649	−0.423324	+	0.905978i	$0.639137\pi$
$114$	0	0
$115$	−5.00000	−0.466252
$116$	−4.00000	−0.371391
$117$	0	0
$118$	−6.00000	−0.552345
$119$	−12.0000	−1.10004
$120$	0	0
$121$	−2.00000	−0.181818
$122$	−2.00000	−0.181071
$123$	0	0
$124$	1.00000	0.0898027
$125$	1.00000	0.0894427
$126$	0	0
$127$	8.00000	0.709885	0.354943	−	0.934888i	$-0.384500\pi$
$127$	8.00000	0.709885	0.354943	+	0.934888i	$0.384500\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	−2.00000	−0.175412
$131$	−6.00000	−0.524222	−0.262111	−	0.965038i	$-0.584419\pi$
$131$	−6.00000	−0.524222	−0.262111	+	0.965038i	$0.584419\pi$
$132$	0	0
$133$	9.00000	0.780399
$134$	2.00000	0.172774
$135$	0	0
$136$	4.00000	0.342997
$137$	10.0000	0.854358	0.427179	−	0.904167i	$-0.359507\pi$
$137$	10.0000	0.854358	0.427179	+	0.904167i	$0.359507\pi$
$138$	0	0
$139$	−14.0000	−1.18746	−0.593732	−	0.804663i	$-0.702346\pi$
$139$	−14.0000	−1.18746	−0.593732	+	0.804663i	$0.702346\pi$
$140$	−3.00000	−0.253546
$141$	0	0
$142$	−7.00000	−0.587427
$143$	6.00000	0.501745
$144$	0	0
$145$	−4.00000	−0.332182
$146$	5.00000	0.413803
$147$	0	0
$148$	0	0
$149$	11.0000	0.901155	0.450578	−	0.892737i	$-0.351218\pi$
$149$	11.0000	0.901155	0.450578	+	0.892737i	$0.351218\pi$
$150$	0	0
$151$	0	0		−	1.00000i	$-0.5\pi$
$151$	0	0			1.00000i	$0.5\pi$
$152$	−3.00000	−0.243332
$153$	0	0
$154$	9.00000	0.725241
$155$	1.00000	0.0803219
$156$	0	0
$157$	−5.00000	−0.399043	−0.199522	−	0.979893i	$-0.563939\pi$
$157$	−5.00000	−0.399043	−0.199522	+	0.979893i	$0.563939\pi$
$158$	−1.00000	−0.0795557
$159$	0	0
$160$	1.00000	0.0790569
$161$	15.0000	1.18217
$162$	0	0
$163$	14.0000	1.09656	0.548282	−	0.836293i	$-0.315282\pi$
$163$	14.0000	1.09656	0.548282	+	0.836293i	$0.315282\pi$
$164$	−4.00000	−0.312348
$165$	0	0
$166$	−12.0000	−0.931381
$167$	19.0000	1.47026	0.735132	−	0.677924i	$-0.237120\pi$
$167$	19.0000	1.47026	0.735132	+	0.677924i	$0.237120\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	4.00000	0.306786
$171$	0	0
$172$	1.00000	0.0762493
$173$	22.0000	1.67263	0.836315	−	0.548250i	$-0.184706\pi$
$173$	22.0000	1.67263	0.836315	+	0.548250i	$0.184706\pi$
$174$	0	0
$175$	−3.00000	−0.226779
$176$	−3.00000	−0.226134
$177$	0	0
$178$	−1.00000	−0.0749532
$179$	−4.00000	−0.298974	−0.149487	−	0.988764i	$-0.547762\pi$
$179$	−4.00000	−0.298974	−0.149487	+	0.988764i	$0.547762\pi$
$180$	0	0
$181$	5.00000	0.371647	0.185824	−	0.982583i	$-0.440505\pi$
$181$	5.00000	0.371647	0.185824	+	0.982583i	$0.440505\pi$
$182$	6.00000	0.444750
$183$	0	0
$184$	−5.00000	−0.368605
$185$	0	0
$186$	0	0
$187$	−12.0000	−0.877527
$188$	−10.0000	−0.729325
$189$	0	0
$190$	−3.00000	−0.217643
$191$	−16.0000	−1.15772	−0.578860	−	0.815427i	$-0.696502\pi$
$191$	−16.0000	−1.15772	−0.578860	+	0.815427i	$0.696502\pi$
$192$	0	0
$193$	−6.00000	−0.431889	−0.215945	−	0.976406i	$-0.569283\pi$
$193$	−6.00000	−0.431889	−0.215945	+	0.976406i	$0.569283\pi$
$194$	−10.0000	−0.717958
$195$	0	0
$196$	2.00000	0.142857
$197$	6.00000	0.427482	0.213741	−	0.976890i	$-0.431435\pi$
$197$	6.00000	0.427482	0.213741	+	0.976890i	$0.431435\pi$
$198$	0	0
$199$	7.00000	0.496217	0.248108	−	0.968732i	$-0.420191\pi$
$199$	7.00000	0.496217	0.248108	+	0.968732i	$0.420191\pi$
$200$	1.00000	0.0707107
$201$	0	0
$202$	1.00000	0.0703598
$203$	12.0000	0.842235
$204$	0	0
$205$	−4.00000	−0.279372
$206$	16.0000	1.11477
$207$	0	0
$208$	−2.00000	−0.138675
$209$	9.00000	0.622543
$210$	0	0
$211$	−19.0000	−1.30801	−0.654007	−	0.756489i	$-0.726913\pi$
$211$	−19.0000	−1.30801	−0.654007	+	0.756489i	$0.726913\pi$
$212$	−3.00000	−0.206041
$213$	0	0
$214$	−9.00000	−0.615227
$215$	1.00000	0.0681994
$216$	0	0
$217$	−3.00000	−0.203653
$218$	20.0000	1.35457
$219$	0	0
$220$	−3.00000	−0.202260
$221$	−8.00000	−0.538138
$222$	0	0
$223$	−6.00000	−0.401790	−0.200895	−	0.979613i	$-0.564385\pi$
$223$	−6.00000	−0.401790	−0.200895	+	0.979613i	$0.564385\pi$
$224$	−3.00000	−0.200446
$225$	0	0
$226$	−9.00000	−0.598671
$227$	27.0000	1.79205	0.896026	−	0.444001i	$-0.146441\pi$
$227$	27.0000	1.79205	0.896026	+	0.444001i	$0.146441\pi$
$228$	0	0
$229$	13.0000	0.859064	0.429532	−	0.903052i	$-0.358679\pi$
$229$	13.0000	0.859064	0.429532	+	0.903052i	$0.358679\pi$
$230$	−5.00000	−0.329690
$231$	0	0
$232$	−4.00000	−0.262613
$233$	−15.0000	−0.982683	−0.491341	−	0.870967i	$-0.663493\pi$
$233$	−15.0000	−0.982683	−0.491341	+	0.870967i	$0.663493\pi$
$234$	0	0
$235$	−10.0000	−0.652328
$236$	−6.00000	−0.390567
$237$	0	0
$238$	−12.0000	−0.777844
$239$	−12.0000	−0.776215	−0.388108	−	0.921614i	$-0.626871\pi$
$239$	−12.0000	−0.776215	−0.388108	+	0.921614i	$0.626871\pi$
$240$	0	0
$241$	−18.0000	−1.15948	−0.579741	−	0.814801i	$-0.696846\pi$
$241$	−18.0000	−1.15948	−0.579741	+	0.814801i	$0.696846\pi$
$242$	−2.00000	−0.128565
$243$	0	0
$244$	−2.00000	−0.128037
$245$	2.00000	0.127775
$246$	0	0
$247$	6.00000	0.381771
$248$	1.00000	0.0635001
$249$	0	0
$250$	1.00000	0.0632456
$251$	12.0000	0.757433	0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000	0.757433	0.378717	+	0.925513i	$0.376365\pi$
$252$	0	0
$253$	15.0000	0.943042
$254$	8.00000	0.501965
$255$	0	0
$256$	1.00000	0.0625000
$257$	23.0000	1.43470	0.717350	−	0.696713i	$-0.245355\pi$
$257$	23.0000	1.43470	0.717350	+	0.696713i	$0.245355\pi$
$258$	0	0
$259$	0	0
$260$	−2.00000	−0.124035
$261$	0	0
$262$	−6.00000	−0.370681
$263$	−24.0000	−1.47990	−0.739952	−	0.672660i	$-0.765152\pi$
$263$	−24.0000	−1.47990	−0.739952	+	0.672660i	$0.765152\pi$
$264$	0	0
$265$	−3.00000	−0.184289
$266$	9.00000	0.551825
$267$	0	0
$268$	2.00000	0.122169
$269$	−2.00000	−0.121942	−0.0609711	−	0.998140i	$-0.519420\pi$
$269$	−2.00000	−0.121942	−0.0609711	+	0.998140i	$0.519420\pi$
$270$	0	0
$271$	13.0000	0.789694	0.394847	−	0.918747i	$-0.370798\pi$
$271$	13.0000	0.789694	0.394847	+	0.918747i	$0.370798\pi$
$272$	4.00000	0.242536
$273$	0	0
$274$	10.0000	0.604122
$275$	−3.00000	−0.180907
$276$	0	0
$277$	12.0000	0.721010	0.360505	−	0.932757i	$-0.382604\pi$
$277$	12.0000	0.721010	0.360505	+	0.932757i	$0.382604\pi$
$278$	−14.0000	−0.839664
$279$	0	0
$280$	−3.00000	−0.179284
$281$	−16.0000	−0.954480	−0.477240	−	0.878773i	$-0.658363\pi$
$281$	−16.0000	−0.954480	−0.477240	+	0.878773i	$0.658363\pi$
$282$	0	0
$283$	−24.0000	−1.42665	−0.713326	−	0.700832i	$-0.752812\pi$
$283$	−24.0000	−1.42665	−0.713326	+	0.700832i	$0.752812\pi$
$284$	−7.00000	−0.415374
$285$	0	0
$286$	6.00000	0.354787
$287$	12.0000	0.708338
$288$	0	0
$289$	−1.00000	−0.0588235
$290$	−4.00000	−0.234888
$291$	0	0
$292$	5.00000	0.292603
$293$	−30.0000	−1.75262	−0.876309	−	0.481749i	$-0.840002\pi$
$293$	−30.0000	−1.75262	−0.876309	+	0.481749i	$0.840002\pi$
$294$	0	0
$295$	−6.00000	−0.349334
$296$	0	0
$297$	0	0
$298$	11.0000	0.637213
$299$	10.0000	0.578315
$300$	0	0
$301$	−3.00000	−0.172917
$302$	0	0
$303$	0	0
$304$	−3.00000	−0.172062
$305$	−2.00000	−0.114520
$306$	0	0
$307$	16.0000	0.913168	0.456584	−	0.889680i	$-0.349073\pi$
$307$	16.0000	0.913168	0.456584	+	0.889680i	$0.349073\pi$
$308$	9.00000	0.512823
$309$	0	0
$310$	1.00000	0.0567962
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	30.0000	1.69570	0.847850	−	0.530236i	$-0.177897\pi$
$313$	30.0000	1.69570	0.847850	+	0.530236i	$0.177897\pi$
$314$	−5.00000	−0.282166
$315$	0	0
$316$	−1.00000	−0.0562544
$317$	8.00000	0.449325	0.224662	−	0.974437i	$-0.427872\pi$
$317$	8.00000	0.449325	0.224662	+	0.974437i	$0.427872\pi$
$318$	0	0
$319$	12.0000	0.671871
$320$	1.00000	0.0559017
$321$	0	0
$322$	15.0000	0.835917
$323$	−12.0000	−0.667698
$324$	0	0
$325$	−2.00000	−0.110940
$326$	14.0000	0.775388
$327$	0	0
$328$	−4.00000	−0.220863
$329$	30.0000	1.65395
$330$	0	0
$331$	12.0000	0.659580	0.329790	−	0.944054i	$-0.393022\pi$
$331$	12.0000	0.659580	0.329790	+	0.944054i	$0.393022\pi$
$332$	−12.0000	−0.658586
$333$	0	0
$334$	19.0000	1.03963
$335$	2.00000	0.109272
$336$	0	0
$337$	−2.00000	−0.108947	−0.0544735	−	0.998515i	$-0.517348\pi$
$337$	−2.00000	−0.108947	−0.0544735	+	0.998515i	$0.517348\pi$
$338$	−9.00000	−0.489535
$339$	0	0
$340$	4.00000	0.216930
$341$	−3.00000	−0.162459
$342$	0	0
$343$	15.0000	0.809924
$344$	1.00000	0.0539164
$345$	0	0
$346$	22.0000	1.18273
$347$	32.0000	1.71785	0.858925	−	0.512101i	$-0.171133\pi$
$347$	32.0000	1.71785	0.858925	+	0.512101i	$0.171133\pi$
$348$	0	0
$349$	−24.0000	−1.28469	−0.642345	−	0.766415i	$-0.722038\pi$
$349$	−24.0000	−1.28469	−0.642345	+	0.766415i	$0.722038\pi$
$350$	−3.00000	−0.160357
$351$	0	0
$352$	−3.00000	−0.159901
$353$	−18.0000	−0.958043	−0.479022	−	0.877803i	$-0.659008\pi$
$353$	−18.0000	−0.958043	−0.479022	+	0.877803i	$0.659008\pi$
$354$	0	0
$355$	−7.00000	−0.371521
$356$	−1.00000	−0.0529999
$357$	0	0
$358$	−4.00000	−0.211407
$359$	−15.0000	−0.791670	−0.395835	−	0.918322i	$-0.629545\pi$
$359$	−15.0000	−0.791670	−0.395835	+	0.918322i	$0.629545\pi$
$360$	0	0
$361$	−10.0000	−0.526316
$362$	5.00000	0.262794
$363$	0	0
$364$	6.00000	0.314485
$365$	5.00000	0.261712
$366$	0	0
$367$	2.00000	0.104399	0.0521996	−	0.998637i	$-0.483377\pi$
$367$	2.00000	0.104399	0.0521996	+	0.998637i	$0.483377\pi$
$368$	−5.00000	−0.260643
$369$	0	0
$370$	0	0
$371$	9.00000	0.467257
$372$	0	0
$373$	−9.00000	−0.466002	−0.233001	−	0.972476i	$-0.574855\pi$
$373$	−9.00000	−0.466002	−0.233001	+	0.972476i	$0.574855\pi$
$374$	−12.0000	−0.620505
$375$	0	0
$376$	−10.0000	−0.515711
$377$	8.00000	0.412021
$378$	0	0
$379$	15.0000	0.770498	0.385249	−	0.922813i	$-0.374116\pi$
$379$	15.0000	0.770498	0.385249	+	0.922813i	$0.374116\pi$
$380$	−3.00000	−0.153897
$381$	0	0
$382$	−16.0000	−0.818631
$383$	−12.0000	−0.613171	−0.306586	−	0.951843i	$-0.599187\pi$
$383$	−12.0000	−0.613171	−0.306586	+	0.951843i	$0.599187\pi$
$384$	0	0
$385$	9.00000	0.458682
$386$	−6.00000	−0.305392
$387$	0	0
$388$	−10.0000	−0.507673
$389$	26.0000	1.31825	0.659126	−	0.752032i	$-0.270926\pi$
$389$	26.0000	1.31825	0.659126	+	0.752032i	$0.270926\pi$
$390$	0	0
$391$	−20.0000	−1.01144
$392$	2.00000	0.101015
$393$	0	0
$394$	6.00000	0.302276
$395$	−1.00000	−0.0503155
$396$	0	0
$397$	35.0000	1.75660	0.878300	−	0.478110i	$-0.158678\pi$
$397$	35.0000	1.75660	0.878300	+	0.478110i	$0.158678\pi$
$398$	7.00000	0.350878
$399$	0	0
$400$	1.00000	0.0500000
$401$	−25.0000	−1.24844	−0.624220	−	0.781248i	$-0.714583\pi$
$401$	−25.0000	−1.24844	−0.624220	+	0.781248i	$0.714583\pi$
$402$	0	0
$403$	−2.00000	−0.0996271
$404$	1.00000	0.0497519
$405$	0	0
$406$	12.0000	0.595550
$407$	0	0
$408$	0	0
$409$	8.00000	0.395575	0.197787	−	0.980245i	$-0.436624\pi$
$409$	8.00000	0.395575	0.197787	+	0.980245i	$0.436624\pi$
$410$	−4.00000	−0.197546
$411$	0	0
$412$	16.0000	0.788263
$413$	18.0000	0.885722
$414$	0	0
$415$	−12.0000	−0.589057
$416$	−2.00000	−0.0980581
$417$	0	0
$418$	9.00000	0.440204
$419$	−12.0000	−0.586238	−0.293119	−	0.956076i	$-0.594693\pi$
$419$	−12.0000	−0.586238	−0.293119	+	0.956076i	$0.594693\pi$
$420$	0	0
$421$	−22.0000	−1.07221	−0.536107	−	0.844150i	$-0.680106\pi$
$421$	−22.0000	−1.07221	−0.536107	+	0.844150i	$0.680106\pi$
$422$	−19.0000	−0.924906
$423$	0	0
$424$	−3.00000	−0.145693
$425$	4.00000	0.194029
$426$	0	0
$427$	6.00000	0.290360
$428$	−9.00000	−0.435031
$429$	0	0
$430$	1.00000	0.0482243
$431$	−16.0000	−0.770693	−0.385346	−	0.922772i	$-0.625918\pi$
$431$	−16.0000	−0.770693	−0.385346	+	0.922772i	$0.625918\pi$
$432$	0	0
$433$	−1.00000	−0.0480569	−0.0240285	−	0.999711i	$-0.507649\pi$
$433$	−1.00000	−0.0480569	−0.0240285	+	0.999711i	$0.507649\pi$
$434$	−3.00000	−0.144005
$435$	0	0
$436$	20.0000	0.957826
$437$	15.0000	0.717547
$438$	0	0
$439$	18.0000	0.859093	0.429547	−	0.903045i	$-0.358673\pi$
$439$	18.0000	0.859093	0.429547	+	0.903045i	$0.358673\pi$
$440$	−3.00000	−0.143019
$441$	0	0
$442$	−8.00000	−0.380521
$443$	−15.0000	−0.712672	−0.356336	−	0.934358i	$-0.615974\pi$
$443$	−15.0000	−0.712672	−0.356336	+	0.934358i	$0.615974\pi$
$444$	0	0
$445$	−1.00000	−0.0474045
$446$	−6.00000	−0.284108
$447$	0	0
$448$	−3.00000	−0.141737
$449$	−2.00000	−0.0943858	−0.0471929	−	0.998886i	$-0.515028\pi$
$449$	−2.00000	−0.0943858	−0.0471929	+	0.998886i	$0.515028\pi$
$450$	0	0
$451$	12.0000	0.565058
$452$	−9.00000	−0.423324
$453$	0	0
$454$	27.0000	1.26717
$455$	6.00000	0.281284
$456$	0	0
$457$	10.0000	0.467780	0.233890	−	0.972263i	$-0.424854\pi$
$457$	10.0000	0.467780	0.233890	+	0.972263i	$0.424854\pi$
$458$	13.0000	0.607450
$459$	0	0
$460$	−5.00000	−0.233126
$461$	0	0		−	1.00000i	$-0.5\pi$
$461$	0	0			1.00000i	$0.5\pi$
$462$	0	0
$463$	4.00000	0.185896	0.0929479	−	0.995671i	$-0.470371\pi$
$463$	4.00000	0.185896	0.0929479	+	0.995671i	$0.470371\pi$
$464$	−4.00000	−0.185695
$465$	0	0
$466$	−15.0000	−0.694862
$467$	−24.0000	−1.11059	−0.555294	−	0.831654i	$-0.687394\pi$
$467$	−24.0000	−1.11059	−0.555294	+	0.831654i	$0.687394\pi$
$468$	0	0
$469$	−6.00000	−0.277054
$470$	−10.0000	−0.461266
$471$	0	0
$472$	−6.00000	−0.276172
$473$	−3.00000	−0.137940
$474$	0	0
$475$	−3.00000	−0.137649
$476$	−12.0000	−0.550019
$477$	0	0
$478$	−12.0000	−0.548867
$479$	−3.00000	−0.137073	−0.0685367	−	0.997649i	$-0.521833\pi$
$479$	−3.00000	−0.137073	−0.0685367	+	0.997649i	$0.521833\pi$
$480$	0	0
$481$	0	0
$482$	−18.0000	−0.819878
$483$	0	0
$484$	−2.00000	−0.0909091
$485$	−10.0000	−0.454077
$486$	0	0
$487$	32.0000	1.45006	0.725029	−	0.688718i	$-0.241826\pi$
$487$	32.0000	1.45006	0.725029	+	0.688718i	$0.241826\pi$
$488$	−2.00000	−0.0905357
$489$	0	0
$490$	2.00000	0.0903508
$491$	37.0000	1.66979	0.834893	−	0.550412i	$-0.185529\pi$
$491$	37.0000	1.66979	0.834893	+	0.550412i	$0.185529\pi$
$492$	0	0
$493$	−16.0000	−0.720604
$494$	6.00000	0.269953
$495$	0	0
$496$	1.00000	0.0449013
$497$	21.0000	0.941979
$498$	0	0
$499$	4.00000	0.179065	0.0895323	−	0.995984i	$-0.471463\pi$
$499$	4.00000	0.179065	0.0895323	+	0.995984i	$0.471463\pi$
$500$	1.00000	0.0447214
$501$	0	0
$502$	12.0000	0.535586
$503$	−20.0000	−0.891756	−0.445878	−	0.895094i	$-0.647108\pi$
$503$	−20.0000	−0.891756	−0.445878	+	0.895094i	$0.647108\pi$
$504$	0	0
$505$	1.00000	0.0444994
$506$	15.0000	0.666831
$507$	0	0
$508$	8.00000	0.354943
$509$	0	0		−	1.00000i	$-0.5\pi$
$509$	0	0			1.00000i	$0.5\pi$
$510$	0	0
$511$	−15.0000	−0.663561
$512$	1.00000	0.0441942
$513$	0	0
$514$	23.0000	1.01449
$515$	16.0000	0.705044
$516$	0	0
$517$	30.0000	1.31940
$518$	0	0
$519$	0	0
$520$	−2.00000	−0.0877058
$521$	18.0000	0.788594	0.394297	−	0.918983i	$-0.370988\pi$
$521$	18.0000	0.788594	0.394297	+	0.918983i	$0.370988\pi$
$522$	0	0
$523$	−43.0000	−1.88026	−0.940129	−	0.340818i	$-0.889296\pi$
$523$	−43.0000	−1.88026	−0.940129	+	0.340818i	$0.889296\pi$
$524$	−6.00000	−0.262111
$525$	0	0
$526$	−24.0000	−1.04645
$527$	4.00000	0.174243
$528$	0	0
$529$	2.00000	0.0869565
$530$	−3.00000	−0.130312
$531$	0	0
$532$	9.00000	0.390199
$533$	8.00000	0.346518
$534$	0	0
$535$	−9.00000	−0.389104
$536$	2.00000	0.0863868
$537$	0	0
$538$	−2.00000	−0.0862261
$539$	−6.00000	−0.258438
$540$	0	0
$541$	22.0000	0.945854	0.472927	−	0.881102i	$-0.343197\pi$
$541$	22.0000	0.945854	0.472927	+	0.881102i	$0.343197\pi$
$542$	13.0000	0.558398
$543$	0	0
$544$	4.00000	0.171499
$545$	20.0000	0.856706
$546$	0	0
$547$	22.0000	0.940652	0.470326	−	0.882493i	$-0.344136\pi$
$547$	22.0000	0.940652	0.470326	+	0.882493i	$0.344136\pi$
$548$	10.0000	0.427179
$549$	0	0
$550$	−3.00000	−0.127920
$551$	12.0000	0.511217
$552$	0	0
$553$	3.00000	0.127573
$554$	12.0000	0.509831
$555$	0	0
$556$	−14.0000	−0.593732
$557$	1.00000	0.0423714	0.0211857	−	0.999776i	$-0.493256\pi$
$557$	1.00000	0.0423714	0.0211857	+	0.999776i	$0.493256\pi$
$558$	0	0
$559$	−2.00000	−0.0845910
$560$	−3.00000	−0.126773
$561$	0	0
$562$	−16.0000	−0.674919
$563$	4.00000	0.168580	0.0842900	−	0.996441i	$-0.473138\pi$
$563$	4.00000	0.168580	0.0842900	+	0.996441i	$0.473138\pi$
$564$	0	0
$565$	−9.00000	−0.378633
$566$	−24.0000	−1.00880
$567$	0	0
$568$	−7.00000	−0.293713
$569$	23.0000	0.964210	0.482105	−	0.876113i	$-0.339872\pi$
$569$	23.0000	0.964210	0.482105	+	0.876113i	$0.339872\pi$
$570$	0	0
$571$	22.0000	0.920671	0.460336	−	0.887745i	$-0.347729\pi$
$571$	22.0000	0.920671	0.460336	+	0.887745i	$0.347729\pi$
$572$	6.00000	0.250873
$573$	0	0
$574$	12.0000	0.500870
$575$	−5.00000	−0.208514
$576$	0	0
$577$	4.00000	0.166522	0.0832611	−	0.996528i	$-0.473466\pi$
$577$	4.00000	0.166522	0.0832611	+	0.996528i	$0.473466\pi$
$578$	−1.00000	−0.0415945
$579$	0	0
$580$	−4.00000	−0.166091
$581$	36.0000	1.49353
$582$	0	0
$583$	9.00000	0.372742
$584$	5.00000	0.206901
$585$	0	0
$586$	−30.0000	−1.23929
$587$	−44.0000	−1.81607	−0.908037	−	0.418890i	$-0.862419\pi$
$587$	−44.0000	−1.81607	−0.908037	+	0.418890i	$0.862419\pi$
$588$	0	0
$589$	−3.00000	−0.123613
$590$	−6.00000	−0.247016
$591$	0	0
$592$	0	0
$593$	−22.0000	−0.903432	−0.451716	−	0.892162i	$-0.649188\pi$
$593$	−22.0000	−0.903432	−0.451716	+	0.892162i	$0.649188\pi$
$594$	0	0
$595$	−12.0000	−0.491952
$596$	11.0000	0.450578
$597$	0	0
$598$	10.0000	0.408930
$599$	27.0000	1.10319	0.551595	−	0.834112i	$-0.314019\pi$
$599$	27.0000	1.10319	0.551595	+	0.834112i	$0.314019\pi$
$600$	0	0
$601$	42.0000	1.71322	0.856608	−	0.515968i	$-0.172568\pi$
$601$	42.0000	1.71322	0.856608	+	0.515968i	$0.172568\pi$
$602$	−3.00000	−0.122271
$603$	0	0
$604$	0	0
$605$	−2.00000	−0.0813116
$606$	0	0
$607$	29.0000	1.17707	0.588537	−	0.808470i	$-0.299704\pi$
$607$	29.0000	1.17707	0.588537	+	0.808470i	$0.299704\pi$
$608$	−3.00000	−0.121666
$609$	0	0
$610$	−2.00000	−0.0809776
$611$	20.0000	0.809113
$612$	0	0
$613$	−22.0000	−0.888572	−0.444286	−	0.895885i	$-0.646543\pi$
$613$	−22.0000	−0.888572	−0.444286	+	0.895885i	$0.646543\pi$
$614$	16.0000	0.645707
$615$	0	0
$616$	9.00000	0.362620
$617$	27.0000	1.08698	0.543490	−	0.839416i	$-0.317103\pi$
$617$	27.0000	1.08698	0.543490	+	0.839416i	$0.317103\pi$
$618$	0	0
$619$	−38.0000	−1.52735	−0.763674	−	0.645601i	$-0.776607\pi$
$619$	−38.0000	−1.52735	−0.763674	+	0.645601i	$0.776607\pi$
$620$	1.00000	0.0401610
$621$	0	0
$622$	0	0
$623$	3.00000	0.120192
$624$	0	0
$625$	1.00000	0.0400000
$626$	30.0000	1.19904
$627$	0	0
$628$	−5.00000	−0.199522
$629$	0	0
$630$	0	0
$631$	17.0000	0.676759	0.338380	−	0.941010i	$-0.390121\pi$
$631$	17.0000	0.676759	0.338380	+	0.941010i	$0.390121\pi$
$632$	−1.00000	−0.0397779
$633$	0	0
$634$	8.00000	0.317721
$635$	8.00000	0.317470
$636$	0	0
$637$	−4.00000	−0.158486
$638$	12.0000	0.475085
$639$	0	0
$640$	1.00000	0.0395285
$641$	−26.0000	−1.02694	−0.513469	−	0.858108i	$-0.671640\pi$
$641$	−26.0000	−1.02694	−0.513469	+	0.858108i	$0.671640\pi$
$642$	0	0
$643$	19.0000	0.749287	0.374643	−	0.927169i	$-0.377765\pi$
$643$	19.0000	0.749287	0.374643	+	0.927169i	$0.377765\pi$
$644$	15.0000	0.591083
$645$	0	0
$646$	−12.0000	−0.472134
$647$	−3.00000	−0.117942	−0.0589711	−	0.998260i	$-0.518782\pi$
$647$	−3.00000	−0.117942	−0.0589711	+	0.998260i	$0.518782\pi$
$648$	0	0
$649$	18.0000	0.706562
$650$	−2.00000	−0.0784465
$651$	0	0
$652$	14.0000	0.548282
$653$	26.0000	1.01746	0.508729	−	0.860927i	$-0.330115\pi$
$653$	26.0000	1.01746	0.508729	+	0.860927i	$0.330115\pi$
$654$	0	0
$655$	−6.00000	−0.234439
$656$	−4.00000	−0.156174
$657$	0	0
$658$	30.0000	1.16952
$659$	4.00000	0.155818	0.0779089	−	0.996960i	$-0.475176\pi$
$659$	4.00000	0.155818	0.0779089	+	0.996960i	$0.475176\pi$
$660$	0	0
$661$	−14.0000	−0.544537	−0.272268	−	0.962221i	$-0.587774\pi$
$661$	−14.0000	−0.544537	−0.272268	+	0.962221i	$0.587774\pi$
$662$	12.0000	0.466393
$663$	0	0
$664$	−12.0000	−0.465690
$665$	9.00000	0.349005
$666$	0	0
$667$	20.0000	0.774403
$668$	19.0000	0.735132
$669$	0	0
$670$	2.00000	0.0772667
$671$	6.00000	0.231627
$672$	0	0
$673$	26.0000	1.00223	0.501113	−	0.865382i	$-0.332924\pi$
$673$	26.0000	1.00223	0.501113	+	0.865382i	$0.332924\pi$
$674$	−2.00000	−0.0770371
$675$	0	0
$676$	−9.00000	−0.346154
$677$	−27.0000	−1.03769	−0.518847	−	0.854867i	$-0.673639\pi$
$677$	−27.0000	−1.03769	−0.518847	+	0.854867i	$0.673639\pi$
$678$	0	0
$679$	30.0000	1.15129
$680$	4.00000	0.153393
$681$	0	0
$682$	−3.00000	−0.114876
$683$	19.0000	0.727015	0.363507	−	0.931591i	$-0.381579\pi$
$683$	19.0000	0.727015	0.363507	+	0.931591i	$0.381579\pi$
$684$	0	0
$685$	10.0000	0.382080
$686$	15.0000	0.572703
$687$	0	0
$688$	1.00000	0.0381246
$689$	6.00000	0.228582
$690$	0	0
$691$	−31.0000	−1.17930	−0.589648	−	0.807661i	$-0.700733\pi$
$691$	−31.0000	−1.17930	−0.589648	+	0.807661i	$0.700733\pi$
$692$	22.0000	0.836315
$693$	0	0
$694$	32.0000	1.21470
$695$	−14.0000	−0.531050
$696$	0	0
$697$	−16.0000	−0.606043
$698$	−24.0000	−0.908413
$699$	0	0
$700$	−3.00000	−0.113389
$701$	21.0000	0.793159	0.396580	−	0.918000i	$-0.370197\pi$
$701$	21.0000	0.793159	0.396580	+	0.918000i	$0.370197\pi$
$702$	0	0
$703$	0	0
$704$	−3.00000	−0.113067
$705$	0	0
$706$	−18.0000	−0.677439
$707$	−3.00000	−0.112827
$708$	0	0
$709$	−25.0000	−0.938895	−0.469447	−	0.882960i	$-0.655547\pi$
$709$	−25.0000	−0.938895	−0.469447	+	0.882960i	$0.655547\pi$
$710$	−7.00000	−0.262705
$711$	0	0
$712$	−1.00000	−0.0374766
$713$	−5.00000	−0.187251
$714$	0	0
$715$	6.00000	0.224387
$716$	−4.00000	−0.149487
$717$	0	0
$718$	−15.0000	−0.559795
$719$	−26.0000	−0.969636	−0.484818	−	0.874615i	$-0.661114\pi$
$719$	−26.0000	−0.969636	−0.484818	+	0.874615i	$0.661114\pi$
$720$	0	0
$721$	−48.0000	−1.78761
$722$	−10.0000	−0.372161
$723$	0	0
$724$	5.00000	0.185824
$725$	−4.00000	−0.148556
$726$	0	0
$727$	45.0000	1.66896	0.834479	−	0.551040i	$-0.185769\pi$
$727$	45.0000	1.66896	0.834479	+	0.551040i	$0.185769\pi$
$728$	6.00000	0.222375
$729$	0	0
$730$	5.00000	0.185058
$731$	4.00000	0.147945
$732$	0	0
$733$	−18.0000	−0.664845	−0.332423	−	0.943131i	$-0.607866\pi$
$733$	−18.0000	−0.664845	−0.332423	+	0.943131i	$0.607866\pi$
$734$	2.00000	0.0738213
$735$	0	0
$736$	−5.00000	−0.184302
$737$	−6.00000	−0.221013
$738$	0	0
$739$	−20.0000	−0.735712	−0.367856	−	0.929883i	$-0.619908\pi$
$739$	−20.0000	−0.735712	−0.367856	+	0.929883i	$0.619908\pi$
$740$	0	0
$741$	0	0
$742$	9.00000	0.330400
$743$	37.0000	1.35740	0.678699	−	0.734416i	$-0.262544\pi$
$743$	37.0000	1.35740	0.678699	+	0.734416i	$0.262544\pi$
$744$	0	0
$745$	11.0000	0.403009
$746$	−9.00000	−0.329513
$747$	0	0
$748$	−12.0000	−0.438763
$749$	27.0000	0.986559
$750$	0	0
$751$	−50.0000	−1.82453	−0.912263	−	0.409605i	$-0.865667\pi$
$751$	−50.0000	−1.82453	−0.912263	+	0.409605i	$0.865667\pi$
$752$	−10.0000	−0.364662
$753$	0	0
$754$	8.00000	0.291343
$755$	0	0
$756$	0	0
$757$	50.0000	1.81728	0.908640	−	0.417579i	$-0.137121\pi$
$757$	50.0000	1.81728	0.908640	+	0.417579i	$0.137121\pi$
$758$	15.0000	0.544825
$759$	0	0
$760$	−3.00000	−0.108821
$761$	−45.0000	−1.63125	−0.815624	−	0.578582i	$-0.803606\pi$
$761$	−45.0000	−1.63125	−0.815624	+	0.578582i	$0.803606\pi$
$762$	0	0
$763$	−60.0000	−2.17215
$764$	−16.0000	−0.578860
$765$	0	0
$766$	−12.0000	−0.433578
$767$	12.0000	0.433295
$768$	0	0
$769$	9.00000	0.324548	0.162274	−	0.986746i	$-0.448117\pi$
$769$	9.00000	0.324548	0.162274	+	0.986746i	$0.448117\pi$
$770$	9.00000	0.324337
$771$	0	0
$772$	−6.00000	−0.215945
$773$	31.0000	1.11499	0.557496	−	0.830179i	$-0.311762\pi$
$773$	31.0000	1.11499	0.557496	+	0.830179i	$0.311762\pi$
$774$	0	0
$775$	1.00000	0.0359211
$776$	−10.0000	−0.358979
$777$	0	0
$778$	26.0000	0.932145
$779$	12.0000	0.429945
$780$	0	0
$781$	21.0000	0.751439
$782$	−20.0000	−0.715199
$783$	0	0
$784$	2.00000	0.0714286
$785$	−5.00000	−0.178458
$786$	0	0
$787$	−35.0000	−1.24762	−0.623808	−	0.781578i	$-0.714415\pi$
$787$	−35.0000	−1.24762	−0.623808	+	0.781578i	$0.714415\pi$
$788$	6.00000	0.213741
$789$	0	0
$790$	−1.00000	−0.0355784
$791$	27.0000	0.960009
$792$	0	0
$793$	4.00000	0.142044
$794$	35.0000	1.24210
$795$	0	0
$796$	7.00000	0.248108
$797$	−2.00000	−0.0708436	−0.0354218	−	0.999372i	$-0.511277\pi$
$797$	−2.00000	−0.0708436	−0.0354218	+	0.999372i	$0.511277\pi$
$798$	0	0
$799$	−40.0000	−1.41510
$800$	1.00000	0.0353553
$801$	0	0
$802$	−25.0000	−0.882781
$803$	−15.0000	−0.529339
$804$	0	0
$805$	15.0000	0.528681
$806$	−2.00000	−0.0704470
$807$	0	0
$808$	1.00000	0.0351799
$809$	−17.0000	−0.597688	−0.298844	−	0.954302i	$-0.596601\pi$
$809$	−17.0000	−0.597688	−0.298844	+	0.954302i	$0.596601\pi$
$810$	0	0
$811$	−27.0000	−0.948098	−0.474049	−	0.880498i	$-0.657208\pi$
$811$	−27.0000	−0.948098	−0.474049	+	0.880498i	$0.657208\pi$
$812$	12.0000	0.421117
$813$	0	0
$814$	0	0
$815$	14.0000	0.490399
$816$	0	0
$817$	−3.00000	−0.104957
$818$	8.00000	0.279713
$819$	0	0
$820$	−4.00000	−0.139686
$821$	−40.0000	−1.39601	−0.698005	−	0.716093i	$-0.745929\pi$
$821$	−40.0000	−1.39601	−0.698005	+	0.716093i	$0.745929\pi$
$822$	0	0
$823$	−18.0000	−0.627441	−0.313720	−	0.949515i	$-0.601575\pi$
$823$	−18.0000	−0.627441	−0.313720	+	0.949515i	$0.601575\pi$
$824$	16.0000	0.557386
$825$	0	0
$826$	18.0000	0.626300
$827$	−20.0000	−0.695468	−0.347734	−	0.937593i	$-0.613049\pi$
$827$	−20.0000	−0.695468	−0.347734	+	0.937593i	$0.613049\pi$
$828$	0	0
$829$	−35.0000	−1.21560	−0.607800	−	0.794090i	$-0.707948\pi$
$829$	−35.0000	−1.21560	−0.607800	+	0.794090i	$0.707948\pi$
$830$	−12.0000	−0.416526
$831$	0	0
$832$	−2.00000	−0.0693375
$833$	8.00000	0.277184
$834$	0	0
$835$	19.0000	0.657522
$836$	9.00000	0.311272
$837$	0	0
$838$	−12.0000	−0.414533
$839$	−51.0000	−1.76072	−0.880358	−	0.474310i	$-0.842698\pi$
$839$	−51.0000	−1.76072	−0.880358	+	0.474310i	$0.842698\pi$
$840$	0	0
$841$	−13.0000	−0.448276
$842$	−22.0000	−0.758170
$843$	0	0
$844$	−19.0000	−0.654007
$845$	−9.00000	−0.309609
$846$	0	0
$847$	6.00000	0.206162
$848$	−3.00000	−0.103020
$849$	0	0
$850$	4.00000	0.137199
$851$	0	0
$852$	0	0
$853$	−41.0000	−1.40381	−0.701907	−	0.712269i	$-0.747668\pi$
$853$	−41.0000	−1.40381	−0.701907	+	0.712269i	$0.747668\pi$
$854$	6.00000	0.205316
$855$	0	0
$856$	−9.00000	−0.307614
$857$	−42.0000	−1.43469	−0.717346	−	0.696717i	$-0.754643\pi$
$857$	−42.0000	−1.43469	−0.717346	+	0.696717i	$0.754643\pi$
$858$	0	0
$859$	32.0000	1.09183	0.545913	−	0.837842i	$-0.316183\pi$
$859$	32.0000	1.09183	0.545913	+	0.837842i	$0.316183\pi$
$860$	1.00000	0.0340997
$861$	0	0
$862$	−16.0000	−0.544962
$863$	−43.0000	−1.46374	−0.731869	−	0.681446i	$-0.761351\pi$
$863$	−43.0000	−1.46374	−0.731869	+	0.681446i	$0.761351\pi$
$864$	0	0
$865$	22.0000	0.748022
$866$	−1.00000	−0.0339814
$867$	0	0
$868$	−3.00000	−0.101827
$869$	3.00000	0.101768
$870$	0	0
$871$	−4.00000	−0.135535
$872$	20.0000	0.677285
$873$	0	0
$874$	15.0000	0.507383
$875$	−3.00000	−0.101419
$876$	0	0
$877$	46.0000	1.55331	0.776655	−	0.629926i	$-0.216915\pi$
$877$	46.0000	1.55331	0.776655	+	0.629926i	$0.216915\pi$
$878$	18.0000	0.607471
$879$	0	0
$880$	−3.00000	−0.101130
$881$	−26.0000	−0.875962	−0.437981	−	0.898984i	$-0.644306\pi$
$881$	−26.0000	−0.875962	−0.437981	+	0.898984i	$0.644306\pi$
$882$	0	0
$883$	11.0000	0.370179	0.185090	−	0.982722i	$-0.440742\pi$
$883$	11.0000	0.370179	0.185090	+	0.982722i	$0.440742\pi$
$884$	−8.00000	−0.269069
$885$	0	0
$886$	−15.0000	−0.503935
$887$	46.0000	1.54453	0.772264	−	0.635301i	$-0.219124\pi$
$887$	46.0000	1.54453	0.772264	+	0.635301i	$0.219124\pi$
$888$	0	0
$889$	−24.0000	−0.804934
$890$	−1.00000	−0.0335201
$891$	0	0
$892$	−6.00000	−0.200895
$893$	30.0000	1.00391
$894$	0	0
$895$	−4.00000	−0.133705
$896$	−3.00000	−0.100223
$897$	0	0
$898$	−2.00000	−0.0667409
$899$	−4.00000	−0.133407
$900$	0	0
$901$	−12.0000	−0.399778
$902$	12.0000	0.399556
$903$	0	0
$904$	−9.00000	−0.299336
$905$	5.00000	0.166206
$906$	0	0
$907$	−52.0000	−1.72663	−0.863316	−	0.504664i	$-0.831616\pi$
$907$	−52.0000	−1.72663	−0.863316	+	0.504664i	$0.831616\pi$
$908$	27.0000	0.896026
$909$	0	0
$910$	6.00000	0.198898
$911$	−20.0000	−0.662630	−0.331315	−	0.943520i	$-0.607492\pi$
$911$	−20.0000	−0.662630	−0.331315	+	0.943520i	$0.607492\pi$
$912$	0	0
$913$	36.0000	1.19143
$914$	10.0000	0.330771
$915$	0	0
$916$	13.0000	0.429532
$917$	18.0000	0.594412
$918$	0	0
$919$	−38.0000	−1.25350	−0.626752	−	0.779219i	$-0.715616\pi$
$919$	−38.0000	−1.25350	−0.626752	+	0.779219i	$0.715616\pi$
$920$	−5.00000	−0.164845
$921$	0	0
$922$	0	0
$923$	14.0000	0.460816
$924$	0	0
$925$	0	0
$926$	4.00000	0.131448
$927$	0	0
$928$	−4.00000	−0.131306
$929$	41.0000	1.34517	0.672583	−	0.740022i	$-0.265185\pi$
$929$	41.0000	1.34517	0.672583	+	0.740022i	$0.265185\pi$
$930$	0	0
$931$	−6.00000	−0.196642
$932$	−15.0000	−0.491341
$933$	0	0
$934$	−24.0000	−0.785304
$935$	−12.0000	−0.392442
$936$	0	0
$937$	−52.0000	−1.69877	−0.849383	−	0.527777i	$-0.823026\pi$
$937$	−52.0000	−1.69877	−0.849383	+	0.527777i	$0.823026\pi$
$938$	−6.00000	−0.195907
$939$	0	0
$940$	−10.0000	−0.326164
$941$	−40.0000	−1.30396	−0.651981	−	0.758235i	$-0.726062\pi$
$941$	−40.0000	−1.30396	−0.651981	+	0.758235i	$0.726062\pi$
$942$	0	0
$943$	20.0000	0.651290
$944$	−6.00000	−0.195283
$945$	0	0
$946$	−3.00000	−0.0975384
$947$	−30.0000	−0.974869	−0.487435	−	0.873160i	$-0.662067\pi$
$947$	−30.0000	−0.974869	−0.487435	+	0.873160i	$0.662067\pi$
$948$	0	0
$949$	−10.0000	−0.324614
$950$	−3.00000	−0.0973329
$951$	0	0
$952$	−12.0000	−0.388922
$953$	52.0000	1.68445	0.842223	−	0.539130i	$-0.181247\pi$
$953$	52.0000	1.68445	0.842223	+	0.539130i	$0.181247\pi$
$954$	0	0
$955$	−16.0000	−0.517748
$956$	−12.0000	−0.388108
$957$	0	0
$958$	−3.00000	−0.0969256
$959$	−30.0000	−0.968751
$960$	0	0
$961$	1.00000	0.0322581
$962$	0	0
$963$	0	0
$964$	−18.0000	−0.579741
$965$	−6.00000	−0.193147
$966$	0	0
$967$	0	0		−	1.00000i	$-0.5\pi$
$967$	0	0			1.00000i	$0.5\pi$
$968$	−2.00000	−0.0642824
$969$	0	0
$970$	−10.0000	−0.321081
$971$	54.0000	1.73294	0.866471	−	0.499227i	$-0.166383\pi$
$971$	54.0000	1.73294	0.866471	+	0.499227i	$0.166383\pi$
$972$	0	0
$973$	42.0000	1.34646
$974$	32.0000	1.02535
$975$	0	0
$976$	−2.00000	−0.0640184
$977$	−38.0000	−1.21573	−0.607864	−	0.794041i	$-0.707973\pi$
$977$	−38.0000	−1.21573	−0.607864	+	0.794041i	$0.707973\pi$
$978$	0	0
$979$	3.00000	0.0958804
$980$	2.00000	0.0638877
$981$	0	0
$982$	37.0000	1.18072
$983$	56.0000	1.78612	0.893061	−	0.449935i	$-0.148553\pi$
$983$	56.0000	1.78612	0.893061	+	0.449935i	$0.148553\pi$
$984$	0	0
$985$	6.00000	0.191176
$986$	−16.0000	−0.509544
$987$	0	0
$988$	6.00000	0.190885
$989$	−5.00000	−0.158991
$990$	0	0
$991$	−25.0000	−0.794151	−0.397076	−	0.917786i	$-0.629975\pi$
$991$	−25.0000	−0.794151	−0.397076	+	0.917786i	$0.629975\pi$
$992$	1.00000	0.0317500
$993$	0	0
$994$	21.0000	0.666080
$995$	7.00000	0.221915
$996$	0	0
$997$	−42.0000	−1.33015	−0.665077	−	0.746775i	$-0.731601\pi$
$997$	−42.0000	−1.33015	−0.665077	+	0.746775i	$0.731601\pi$
$998$	4.00000	0.126618
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2790.2.a.y.1.1		1
3.2	odd	2		930.2.a.a.1.1	✓	1
12.11	even	2		7440.2.a.v.1.1		1
15.2	even	4		4650.2.d.y.3349.1		2
15.8	even	4		4650.2.d.y.3349.2		2
15.14	odd	2		4650.2.a.bv.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
930.2.a.a.1.1	✓	1	3.2	odd	2
2790.2.a.y.1.1		1	1.1	even	1	trivial
4650.2.a.bv.1.1		1	15.14	odd	2
4650.2.d.y.3349.1		2	15.2	even	4
4650.2.d.y.3349.2		2	15.8	even	4
7440.2.a.v.1.1		1	12.11	even	2

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2790 = 2 \cdot 3^{2} \cdot 5 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2790.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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