LMFDB - Embedded newform 1521.4.a.y.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1521,4,Mod(1,1521)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1521, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1521.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1521 = 3^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1521.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:	$89.7419051187$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.8112.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 5x^{2} + 3 $ x^4 - 5*x^2 + 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2^{2}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 117)
Fricke sign:	$-1$
Sato-Tate group:	$N(\mathrm{U}(1))$

Embedding invariants

Embedding label			1.2
Root			$2.07431$ of defining polynomial
Character	$\chi$	$=$	1521.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.52058 q^{2} +4.39445 q^{4} -9.17304 q^{5} +12.6936 q^{8} +32.2944 q^{10} +20.4780 q^{11} -79.8444 q^{16} -40.3104 q^{20} -72.0942 q^{22} -40.8554 q^{25} +179.549 q^{32} -116.439 q^{40} -196.898 q^{41} -452.000 q^{43} +89.9894 q^{44} +640.490 q^{47} -343.000 q^{49} +143.834 q^{50} -187.845 q^{55} +579.506 q^{59} +944.654 q^{61} +6.63840 q^{64} +1190.09 q^{71} -418.244 q^{79} +732.416 q^{80} +693.193 q^{82} +94.6440 q^{83} +1591.30 q^{86} +259.939 q^{88} -1672.06 q^{89} -2254.89 q^{94} +1207.56 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 32 q^{4} - 116 q^{10} - 204 q^{16} + 476 q^{22} + 500 q^{25} - 884 q^{40} - 1808 q^{43} - 1372 q^{49} - 3232 q^{55} - 1632 q^{64} - 2924 q^{82} + 2756 q^{88} - 5140 q^{94}+O(q^{100}) $ 4 * q + 32 * q^4 - 116 * q^10 - 204 * q^16 + 476 * q^22 + 500 * q^25 - 884 * q^40 - 1808 * q^43 - 1372 * q^49 - 3232 * q^55 - 1632 * q^64 - 2924 * q^82 + 2756 * q^88 - 5140 * q^94

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 $

$2$	−3.52058	−1.24471	−0.622356	−	0.782735i	$-0.713824\pi$
$2$	−3.52058	−1.24471	−0.622356	+	0.782735i	$0.713824\pi$
$3$	0	0
$3$	0	0
$4$	4.39445	0.549306
$5$	−9.17304	−0.820462	−0.410231	−	0.911982i	$-0.634552\pi$
$5$	−9.17304	−0.820462	−0.410231	+	0.911982i	$0.634552\pi$
$6$	0	0
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	12.6936	0.560984
$9$	0	0
$10$	32.2944	1.02124
$11$	20.4780	0.561304	0.280652	−	0.959810i	$-0.409449\pi$
$11$	20.4780	0.561304	0.280652	+	0.959810i	$0.409449\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	0	0
$16$	−79.8444	−1.24757
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	−40.3104	−0.450685
$21$	0	0
$22$	−72.0942	−0.698661
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	−40.8554	−0.326843
$26$	0	0
$27$	0	0
$28$	0	0
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	179.549	0.991879
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	0	0		−	1.00000i	$-0.5\pi$
$37$	0	0			1.00000i	$0.5\pi$
$38$	0	0
$39$	0	0
$40$	−116.439	−0.460266
$41$	−196.898	−0.750006	−0.375003	−	0.927024i	$-0.622358\pi$
$41$	−196.898	−0.750006	−0.375003	+	0.927024i	$0.622358\pi$
$42$	0	0
$43$	−452.000	−1.60301	−0.801504	−	0.597989i	$-0.795967\pi$
$43$	−452.000	−1.60301	−0.801504	+	0.597989i	$0.795967\pi$
$44$	89.9894	0.308327
$45$	0	0
$46$	0	0
$47$	640.490	1.98777	0.993884	−	0.110431i	$-0.0352231\pi$
$47$	640.490	1.98777	0.993884	+	0.110431i	$0.0352231\pi$
$48$	0	0
$49$	−343.000	−1.00000
$50$	143.834	0.406825
$51$	0	0
$52$	0	0
$53$	0	0		−	1.00000i	$-0.5\pi$
$53$	0	0			1.00000i	$0.5\pi$
$54$	0	0
$55$	−187.845	−0.460528
$56$	0	0
$57$	0	0
$58$	0	0
$59$	579.506	1.27873	0.639367	−	0.768902i	$-0.279197\pi$
$59$	579.506	1.27873	0.639367	+	0.768902i	$0.279197\pi$
$60$	0	0
$61$	944.654	1.98280	0.991398	−	0.130879i	$-0.0417798\pi$
$61$	944.654	1.98280	0.991398	+	0.130879i	$0.0417798\pi$
$62$	0	0
$63$	0	0
$64$	6.63840	0.0129656
$65$	0	0
$66$	0	0
$67$	0	0		−	1.00000i	$-0.5\pi$
$67$	0	0			1.00000i	$0.5\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	1190.09	1.98926	0.994632	−	0.103474i	$-0.0329957\pi$
$71$	1190.09	1.98926	0.994632	+	0.103474i	$0.0329957\pi$
$72$	0	0
$73$	0	0		−	1.00000i	$-0.5\pi$
$73$	0	0			1.00000i	$0.5\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−418.244	−0.595647	−0.297824	−	0.954621i	$-0.596261\pi$
$79$	−418.244	−0.595647	−0.297824	+	0.954621i	$0.596261\pi$
$80$	732.416	1.02358
$81$	0	0
$82$	693.193	0.933541
$83$	94.6440	0.125163	0.0625815	−	0.998040i	$-0.480067\pi$
$83$	94.6440	0.125163	0.0625815	+	0.998040i	$0.480067\pi$
$84$	0	0
$85$	0	0
$86$	1591.30	1.99528
$87$	0	0
$88$	259.939	0.314882
$89$	−1672.06	−1.99143	−0.995717	−	0.0924493i	$-0.970530\pi$
$89$	−1672.06	−1.99143	−0.995717	+	0.0924493i	$0.970530\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	−2254.89	−2.47420
$95$	0	0
$96$	0	0
$97$	0	0		−	1.00000i	$-0.5\pi$
$97$	0	0			1.00000i	$0.5\pi$
$98$	1207.56	1.24471
$99$	0	0
$100$	−179.537	−0.179537
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	0	0
$103$	−848.000	−0.811223	−0.405611	−	0.914046i	$-0.632941\pi$
$103$	−848.000	−0.811223	−0.405611	+	0.914046i	$0.632941\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	0	0
$109$	0	0		−	1.00000i	$-0.5\pi$
$109$	0	0			1.00000i	$0.5\pi$
$110$	661.323	0.573224
$111$	0	0
$112$	0	0
$113$	0	0		−	1.00000i	$-0.5\pi$
$113$	0	0			1.00000i	$0.5\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	−2040.20	−1.59165
$119$	0	0
$120$	0	0
$121$	−911.653	−0.684938
$122$	−3325.73	−2.46801
$123$	0	0
$124$	0	0
$125$	1521.40	1.08862
$126$	0	0
$127$	2495.04	1.74330	0.871650	−	0.490129i	$-0.163050\pi$
$127$	2495.04	1.74330	0.871650	+	0.490129i	$0.163050\pi$
$128$	−1459.77	−1.00802
$129$	0	0
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	3018.79	1.88258	0.941288	−	0.337604i	$-0.109616\pi$
$137$	3018.79	1.88258	0.941288	+	0.337604i	$0.109616\pi$
$138$	0	0
$139$	−340.000	−0.207471	−0.103735	−	0.994605i	$-0.533079\pi$
$139$	−340.000	−0.207471	−0.103735	+	0.994605i	$0.533079\pi$
$140$	0	0
$141$	0	0
$142$	−4189.80	−2.47606
$143$	0	0
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	2477.09	1.36196	0.680978	−	0.732304i	$-0.261555\pi$
$149$	2477.09	1.36196	0.680978	+	0.732304i	$0.261555\pi$
$150$	0	0
$151$	0	0		−	1.00000i	$-0.5\pi$
$151$	0	0			1.00000i	$0.5\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−934.000	−0.474785	−0.237393	−	0.971414i	$-0.576293\pi$
$157$	−934.000	−0.474785	−0.237393	+	0.971414i	$0.576293\pi$
$158$	1472.46	0.741409
$159$	0	0
$160$	−1647.01	−0.813799
$161$	0	0
$162$	0	0
$163$	0	0		−	1.00000i	$-0.5\pi$
$163$	0	0			1.00000i	$0.5\pi$
$164$	−865.256	−0.411983
$165$	0	0
$166$	−333.201	−0.155792
$167$	−2446.51	−1.13363	−0.566815	−	0.823845i	$-0.691825\pi$
$167$	−2446.51	−1.13363	−0.566815	+	0.823845i	$0.691825\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	0	0
$172$	−1986.29	−0.880542
$173$	0	0		−	1.00000i	$-0.5\pi$
$173$	0	0			1.00000i	$0.5\pi$
$174$	0	0
$175$	0	0
$176$	−1635.05	−0.700265
$177$	0	0
$178$	5886.60	2.47876
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	0	0
$181$	−4430.00	−1.81922	−0.909611	−	0.415460i	$-0.863621\pi$
$181$	−4430.00	−1.81922	−0.909611	+	0.415460i	$0.863621\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0	0
$188$	2814.60	1.09189
$189$	0	0
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	0	0
$193$	0	0		−	1.00000i	$-0.5\pi$
$193$	0	0			1.00000i	$0.5\pi$
$194$	0	0
$195$	0	0
$196$	−1507.30	−0.549306
$197$	−3977.33	−1.43844	−0.719220	−	0.694782i	$-0.755501\pi$
$197$	−3977.33	−1.43844	−0.719220	+	0.694782i	$0.755501\pi$
$198$	0	0
$199$	5610.24	1.99849	0.999244	−	0.0388706i	$-0.0123760\pi$
$199$	5610.24	1.99849	0.999244	+	0.0388706i	$0.0123760\pi$
$200$	−518.602	−0.183354
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	1806.15	0.615351
$206$	2985.45	1.00974
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−5740.04	−1.87280	−0.936399	−	0.350937i	$-0.885863\pi$
$211$	−5740.04	−1.87280	−0.936399	+	0.350937i	$0.885863\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	4146.21	1.31521
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	−825.476	−0.252971
$221$	0	0
$222$	0	0
$223$	0	0		−	1.00000i	$-0.5\pi$
$223$	0	0			1.00000i	$0.5\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−6668.65	−1.94984	−0.974920	−	0.222554i	$-0.928561\pi$
$227$	−6668.65	−1.94984	−0.974920	+	0.222554i	$0.928561\pi$
$228$	0	0
$229$	0	0		−	1.00000i	$-0.5\pi$
$229$	0	0			1.00000i	$0.5\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	0	0		−	1.00000i	$-0.5\pi$
$233$	0	0			1.00000i	$0.5\pi$
$234$	0	0
$235$	−5875.24	−1.63089
$236$	2546.61	0.702416
$237$	0	0
$238$	0	0
$239$	6085.88	1.64712	0.823562	−	0.567226i	$-0.191984\pi$
$239$	6085.88	1.64712	0.823562	+	0.567226i	$0.191984\pi$
$240$	0	0
$241$	0	0		−	1.00000i	$-0.5\pi$
$241$	0	0			1.00000i	$0.5\pi$
$242$	3209.54	0.852550
$243$	0	0
$244$	4151.24	1.08916
$245$	3146.35	0.820462
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	−5356.19	−1.35502
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	0	0
$253$	0	0
$254$	−8783.98	−2.16991
$255$	0	0
$256$	5086.11	1.24173
$257$	0	0		−	1.00000i	$-0.5\pi$
$257$	0	0			1.00000i	$0.5\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	0	0		−	1.00000i	$-0.5\pi$
$269$	0	0			1.00000i	$0.5\pi$
$270$	0	0
$271$	0	0		−	1.00000i	$-0.5\pi$
$271$	0	0			1.00000i	$0.5\pi$
$272$	0	0
$273$	0	0
$274$	−10627.9	−2.34326
$275$	−836.635	−0.183458
$276$	0	0
$277$	−7634.00	−1.65589	−0.827947	−	0.560806i	$-0.810491\pi$
$277$	−7634.00	−1.65589	−0.827947	+	0.560806i	$0.810491\pi$
$278$	1197.00	0.258241
$279$	0	0
$280$	0	0
$281$	−8255.10	−1.75252	−0.876260	−	0.481839i	$-0.839969\pi$
$281$	−8255.10	−1.75252	−0.876260	+	0.481839i	$0.839969\pi$
$282$	0	0
$283$	−490.355	−0.102999	−0.0514993	−	0.998673i	$-0.516400\pi$
$283$	−490.355	−0.102999	−0.0514993	+	0.998673i	$0.516400\pi$
$284$	5229.79	1.09272
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−4913.00	−1.00000
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−9933.00	−1.98052	−0.990260	−	0.139232i	$-0.955537\pi$
$293$	−9933.00	−1.98052	−0.990260	+	0.139232i	$0.955537\pi$
$294$	0	0
$295$	−5315.83	−1.04915
$296$	0	0
$297$	0	0
$298$	−8720.79	−1.69524
$299$	0	0
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−8665.35	−1.62681
$306$	0	0
$307$	0	0		−	1.00000i	$-0.5\pi$
$307$	0	0			1.00000i	$0.5\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	6396.25	1.15507	0.577536	−	0.816365i	$-0.304014\pi$
$313$	6396.25	1.15507	0.577536	+	0.816365i	$0.304014\pi$
$314$	3288.22	0.590971
$315$	0	0
$316$	−1837.95	−0.327193
$317$	7133.53	1.26391	0.631954	−	0.775006i	$-0.282253\pi$
$317$	7133.53	1.26391	0.631954	+	0.775006i	$0.282253\pi$
$318$	0	0
$319$	0	0
$320$	−60.8943	−0.0106378
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	−2499.34	−0.420741
$329$	0	0
$330$	0	0
$331$	0	0		−	1.00000i	$-0.5\pi$
$331$	0	0			1.00000i	$0.5\pi$
$332$	415.908	0.0687528
$333$	0	0
$334$	8613.11	1.41104
$335$	0	0
$336$	0	0
$337$	−10420.0	−1.68432	−0.842160	−	0.539228i	$-0.818716\pi$
$337$	−10420.0	−1.68432	−0.842160	+	0.539228i	$0.818716\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	0	0
$344$	−5737.51	−0.899262
$345$	0	0
$346$	0	0
$347$	0	0		−	1.00000i	$-0.5\pi$
$347$	0	0			1.00000i	$0.5\pi$
$348$	0	0
$349$	0	0		−	1.00000i	$-0.5\pi$
$349$	0	0			1.00000i	$0.5\pi$
$350$	0	0
$351$	0	0
$352$	3676.81	0.556745
$353$	−7331.69	−1.10546	−0.552728	−	0.833361i	$-0.686413\pi$
$353$	−7331.69	−1.10546	−0.552728	+	0.833361i	$0.686413\pi$
$354$	0	0
$355$	−10916.7	−1.63211
$356$	−7347.77	−1.09391
$357$	0	0
$358$	0	0
$359$	−11077.3	−1.62852	−0.814259	−	0.580502i	$-0.802857\pi$
$359$	−11077.3	−1.62852	−0.814259	+	0.580502i	$0.802857\pi$
$360$	0	0
$361$	−6859.00	−1.00000
$362$	15596.1	2.26441
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−8296.00	−1.17997	−0.589983	−	0.807416i	$-0.700866\pi$
$367$	−8296.00	−1.17997	−0.589983	+	0.807416i	$0.700866\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	6526.05	0.905914	0.452957	−	0.891532i	$-0.350369\pi$
$373$	6526.05	0.905914	0.452957	+	0.891532i	$0.350369\pi$
$374$	0	0
$375$	0	0
$376$	8130.13	1.11511
$377$	0	0
$378$	0	0
$379$	0	0		−	1.00000i	$-0.5\pi$
$379$	0	0			1.00000i	$0.5\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−12702.2	−1.69465	−0.847327	−	0.531071i	$-0.821790\pi$
$383$	−12702.2	−1.69465	−0.847327	+	0.531071i	$0.821790\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	0	0		−	1.00000i	$-0.5\pi$
$389$	0	0			1.00000i	$0.5\pi$
$390$	0	0
$391$	0	0
$392$	−4353.91	−0.560984
$393$	0	0
$394$	14002.5	1.79044
$395$	3836.57	0.488706
$396$	0	0
$397$	0	0		−	1.00000i	$-0.5\pi$
$397$	0	0			1.00000i	$0.5\pi$
$398$	−19751.3	−2.48754
$399$	0	0
$400$	3262.07	0.407759
$401$	−15342.6	−1.91066	−0.955328	−	0.295549i	$-0.904497\pi$
$401$	−15342.6	−1.91066	−0.955328	+	0.295549i	$0.904497\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	0	0		−	1.00000i	$-0.5\pi$
$409$	0	0			1.00000i	$0.5\pi$
$410$	−6358.68	−0.765934
$411$	0	0
$412$	−3726.49	−0.445609
$413$	0	0
$414$	0	0
$415$	−868.173	−0.102691
$416$	0	0
$417$	0	0
$418$	0	0
$419$	0	0		−	1.00000i	$-0.5\pi$
$419$	0	0			1.00000i	$0.5\pi$
$420$	0	0
$421$	0	0		−	1.00000i	$-0.5\pi$
$421$	0	0			1.00000i	$0.5\pi$
$422$	20208.2	2.33109
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	−14597.1	−1.63705
$431$	320.305	0.0357971	0.0178985	−	0.999840i	$-0.494302\pi$
$431$	320.305	0.0357971	0.0178985	+	0.999840i	$0.494302\pi$
$432$	0	0
$433$	−36.0555	−0.00400166	−0.00200083	−	0.999998i	$-0.500637\pi$
$433$	−36.0555	−0.00400166	−0.00200083	+	0.999998i	$0.500637\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	11320.0	1.23069	0.615346	−	0.788257i	$-0.289016\pi$
$439$	11320.0	1.23069	0.615346	+	0.788257i	$0.289016\pi$
$440$	−2384.43	−0.258349
$441$	0	0
$442$	0	0
$443$	0	0		−	1.00000i	$-0.5\pi$
$443$	0	0			1.00000i	$0.5\pi$
$444$	0	0
$445$	15337.8	1.63390
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−333.683	−0.0350723	−0.0175361	−	0.999846i	$-0.505582\pi$
$449$	−333.683	−0.0350723	−0.0175361	+	0.999846i	$0.505582\pi$
$450$	0	0
$451$	−4032.06	−0.420981
$452$	0	0
$453$	0	0
$454$	23477.5	2.42699
$455$	0	0
$456$	0	0
$457$	0	0		−	1.00000i	$-0.5\pi$
$457$	0	0			1.00000i	$0.5\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−17433.8	−1.76133	−0.880666	−	0.473738i	$-0.842905\pi$
$461$	−17433.8	−1.76133	−0.880666	+	0.473738i	$0.842905\pi$
$462$	0	0
$463$	0	0		−	1.00000i	$-0.5\pi$
$463$	0	0			1.00000i	$0.5\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	0	0		−	1.00000i	$-0.5\pi$
$467$	0	0			1.00000i	$0.5\pi$
$468$	0	0
$469$	0	0
$470$	20684.2	2.02998
$471$	0	0
$472$	7356.03	0.717349
$473$	−9256.04	−0.899774
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	−21425.8	−2.05019
$479$	5200.84	0.496101	0.248050	−	0.968747i	$-0.420210\pi$
$479$	5200.84	0.496101	0.248050	+	0.968747i	$0.420210\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	−4006.21	−0.376241
$485$	0	0
$486$	0	0
$487$	0	0		−	1.00000i	$-0.5\pi$
$487$	0	0			1.00000i	$0.5\pi$
$488$	11991.1	1.11232
$489$	0	0
$490$	−11077.0	−1.02124
$491$	0	0		−	1.00000i	$-0.5\pi$
$491$	0	0			1.00000i	$0.5\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	0	0		−	1.00000i	$-0.5\pi$
$499$	0	0			1.00000i	$0.5\pi$
$500$	6685.70	0.597988
$501$	0	0
$502$	0	0
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	10964.3	0.957605
$509$	−22966.8	−1.99997	−0.999987	−	0.00508931i	$-0.998380\pi$
$509$	−22966.8	−1.99997	−0.999987	+	0.00508931i	$0.998380\pi$
$510$	0	0
$511$	0	0
$512$	−6227.90	−0.537572
$513$	0	0
$514$	0	0
$515$	7778.74	0.665577
$516$	0	0
$517$	13115.9	1.11574
$518$	0	0
$519$	0	0
$520$	0	0
$521$	0	0		−	1.00000i	$-0.5\pi$
$521$	0	0			1.00000i	$0.5\pi$
$522$	0	0
$523$	16148.0	1.35010	0.675050	−	0.737772i	$-0.264122\pi$
$523$	16148.0	1.35010	0.675050	+	0.737772i	$0.264122\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−12167.0	−1.00000
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−7023.94	−0.561304
$540$	0	0
$541$	0	0		−	1.00000i	$-0.5\pi$
$541$	0	0			1.00000i	$0.5\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	23996.0	1.87568	0.937838	−	0.347073i	$-0.112824\pi$
$547$	23996.0	1.87568	0.937838	+	0.347073i	$0.112824\pi$
$548$	13265.9	1.03411
$549$	0	0
$550$	2945.44	0.228352
$551$	0	0
$552$	0	0
$553$	0	0
$554$	26876.1	2.06111
$555$	0	0
$556$	−1494.11	−0.113965
$557$	−22258.1	−1.69319	−0.846595	−	0.532237i	$-0.821352\pi$
$557$	−22258.1	−1.69319	−0.846595	+	0.532237i	$0.821352\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	29062.7	2.18138
$563$	0	0		−	1.00000i	$-0.5\pi$
$563$	0	0			1.00000i	$0.5\pi$
$564$	0	0
$565$	0	0
$566$	1726.33	0.128203
$567$	0	0
$568$	15106.6	1.11595
$569$	0	0		−	1.00000i	$-0.5\pi$
$569$	0	0			1.00000i	$0.5\pi$
$570$	0	0
$571$	−25411.9	−1.86244	−0.931222	−	0.364451i	$-0.881257\pi$
$571$	−25411.9	−1.86244	−0.931222	+	0.364451i	$0.881257\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	0	0		−	1.00000i	$-0.5\pi$
$577$	0	0			1.00000i	$0.5\pi$
$578$	17296.6	1.24471
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	34969.9	2.46517
$587$	19338.0	1.35974	0.679868	−	0.733335i	$-0.262037\pi$
$587$	19338.0	1.35974	0.679868	+	0.733335i	$0.262037\pi$
$588$	0	0
$589$	0	0
$590$	18714.8	1.30589
$591$	0	0
$592$	0	0
$593$	25958.2	1.79760	0.898798	−	0.438363i	$-0.144442\pi$
$593$	25958.2	1.79760	0.898798	+	0.438363i	$0.144442\pi$
$594$	0	0
$595$	0	0
$596$	10885.5	0.748131
$597$	0	0
$598$	0	0
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	0	0
$601$	12626.6	0.856991	0.428495	−	0.903544i	$-0.359044\pi$
$601$	12626.6	0.856991	0.428495	+	0.903544i	$0.359044\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	8362.63	0.561966
$606$	0	0
$607$	−25504.0	−1.70540	−0.852698	−	0.522404i	$-0.825035\pi$
$607$	−25504.0	−1.70540	−0.852698	+	0.522404i	$0.825035\pi$
$608$	0	0
$609$	0	0
$610$	30507.0	2.02491
$611$	0	0
$612$	0	0
$613$	0	0		−	1.00000i	$-0.5\pi$
$613$	0	0			1.00000i	$0.5\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	29859.9	1.94832	0.974160	−	0.225860i	$-0.0725191\pi$
$617$	29859.9	1.94832	0.974160	+	0.225860i	$0.0725191\pi$
$618$	0	0
$619$	0	0		−	1.00000i	$-0.5\pi$
$619$	0	0			1.00000i	$0.5\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−8848.92	−0.566331
$626$	−22518.5	−1.43773
$627$	0	0
$628$	−4104.42	−0.260803
$629$	0	0
$630$	0	0
$631$	0	0		−	1.00000i	$-0.5\pi$
$631$	0	0			1.00000i	$0.5\pi$
$632$	−5309.03	−0.334148
$633$	0	0
$634$	−25114.1	−1.57320
$635$	−22887.1	−1.43031
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	13390.5	0.827040
$641$	0	0		−	1.00000i	$-0.5\pi$
$641$	0	0			1.00000i	$0.5\pi$
$642$	0	0
$643$	0	0		−	1.00000i	$-0.5\pi$
$643$	0	0			1.00000i	$0.5\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	0	0		−	1.00000i	$-0.5\pi$
$647$	0	0			1.00000i	$0.5\pi$
$648$	0	0
$649$	11867.1	0.717758
$650$	0	0
$651$	0	0
$652$	0	0
$653$	0	0		−	1.00000i	$-0.5\pi$
$653$	0	0			1.00000i	$0.5\pi$
$654$	0	0
$655$	0	0
$656$	15721.2	0.935684
$657$	0	0
$658$	0	0
$659$	0	0		−	1.00000i	$-0.5\pi$
$659$	0	0			1.00000i	$0.5\pi$
$660$	0	0
$661$	0	0		−	1.00000i	$-0.5\pi$
$661$	0	0			1.00000i	$0.5\pi$
$662$	0	0
$663$	0	0
$664$	1201.37	0.0702144
$665$	0	0
$666$	0	0
$667$	0	0
$668$	−10751.0	−0.622710
$669$	0	0
$670$	0	0
$671$	19344.6	1.11295
$672$	0	0
$673$	25522.0	1.46181	0.730907	−	0.682477i	$-0.239097\pi$
$673$	25522.0	1.46181	0.730907	+	0.682477i	$0.239097\pi$
$674$	36684.5	2.09649
$675$	0	0
$676$	0	0
$677$	0	0		−	1.00000i	$-0.5\pi$
$677$	0	0			1.00000i	$0.5\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	32750.6	1.83480	0.917399	−	0.397968i	$-0.130285\pi$
$683$	32750.6	1.83480	0.917399	+	0.397968i	$0.130285\pi$
$684$	0	0
$685$	−27691.5	−1.54458
$686$	0	0
$687$	0	0
$688$	36089.7	1.99986
$689$	0	0
$690$	0	0
$691$	0	0		−	1.00000i	$-0.5\pi$
$691$	0	0			1.00000i	$0.5\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	3118.83	0.170222
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	0	0
$701$	0	0		−	1.00000i	$-0.5\pi$
$701$	0	0			1.00000i	$0.5\pi$
$702$	0	0
$703$	0	0
$704$	135.941	0.00727765
$705$	0	0
$706$	25811.8	1.37597
$707$	0	0
$708$	0	0
$709$	0	0		−	1.00000i	$-0.5\pi$
$709$	0	0			1.00000i	$0.5\pi$
$710$	38433.2	2.03151
$711$	0	0
$712$	−21224.4	−1.11716
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	38998.5	2.02704
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	0	0
$722$	24147.6	1.24471
$723$	0	0
$724$	−19467.4	−0.999310
$725$	0	0
$726$	0	0
$727$	28455.0	1.45163	0.725817	−	0.687888i	$-0.241462\pi$
$727$	28455.0	1.45163	0.725817	+	0.687888i	$0.241462\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	0	0
$732$	0	0
$733$	0	0		−	1.00000i	$-0.5\pi$
$733$	0	0			1.00000i	$0.5\pi$
$734$	29206.7	1.46872
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	0	0		−	1.00000i	$-0.5\pi$
$739$	0	0			1.00000i	$0.5\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−19936.4	−0.984382	−0.492191	−	0.870487i	$-0.663804\pi$
$743$	−19936.4	−0.984382	−0.492191	+	0.870487i	$0.663804\pi$
$744$	0	0
$745$	−22722.5	−1.11743
$746$	−22975.4	−1.12760
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−26120.0	−1.26915	−0.634575	−	0.772861i	$-0.718825\pi$
$751$	−26120.0	−1.26915	−0.634575	+	0.772861i	$0.718825\pi$
$752$	−51139.6	−2.47988
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−35009.9	−1.68092	−0.840460	−	0.541873i	$-0.817715\pi$
$757$	−35009.9	−1.68092	−0.840460	+	0.541873i	$0.817715\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−28936.4	−1.37838	−0.689189	−	0.724582i	$-0.742033\pi$
$761$	−28936.4	−1.37838	−0.689189	+	0.724582i	$0.742033\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	44719.1	2.10936
$767$	0	0
$768$	0	0
$769$	0	0		−	1.00000i	$-0.5\pi$
$769$	0	0			1.00000i	$0.5\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−3484.00	−0.162110	−0.0810548	−	0.996710i	$-0.525829\pi$
$773$	−3484.00	−0.162110	−0.0810548	+	0.996710i	$0.525829\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	24370.6	1.11658
$782$	0	0
$783$	0	0
$784$	27386.6	1.24757
$785$	8567.62	0.389543
$786$	0	0
$787$	0	0		−	1.00000i	$-0.5\pi$
$787$	0	0			1.00000i	$0.5\pi$
$788$	−17478.2	−0.790144
$789$	0	0
$790$	−13506.9	−0.608297
$791$	0	0
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	24653.9	1.09778
$797$	0	0		−	1.00000i	$-0.5\pi$
$797$	0	0			1.00000i	$0.5\pi$
$798$	0	0
$799$	0	0
$800$	−7335.55	−0.324189
$801$	0	0
$802$	54014.8	2.37821
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	0	0		−	1.00000i	$-0.5\pi$
$809$	0	0			1.00000i	$0.5\pi$
$810$	0	0
$811$	0	0		−	1.00000i	$-0.5\pi$
$811$	0	0			1.00000i	$0.5\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0	0
$820$	7937.03	0.338016
$821$	33048.4	1.40487	0.702433	−	0.711749i	$-0.252097\pi$
$821$	33048.4	1.40487	0.702433	+	0.711749i	$0.252097\pi$
$822$	0	0
$823$	−37352.0	−1.58203	−0.791014	−	0.611798i	$-0.790446\pi$
$823$	−37352.0	−1.58203	−0.791014	+	0.611798i	$0.790446\pi$
$824$	−10764.2	−0.455083
$825$	0	0
$826$	0	0
$827$	−31167.2	−1.31051	−0.655253	−	0.755409i	$-0.727438\pi$
$827$	−31167.2	−1.31051	−0.655253	+	0.755409i	$0.727438\pi$
$828$	0	0
$829$	3079.14	0.129002	0.0645012	−	0.997918i	$-0.479454\pi$
$829$	3079.14	0.129002	0.0645012	+	0.997918i	$0.479454\pi$
$830$	3056.47	0.127821
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	22441.9	0.930101
$836$	0	0
$837$	0	0
$838$	0	0
$839$	46872.5	1.92875	0.964374	−	0.264543i	$-0.0852210\pi$
$839$	46872.5	1.92875	0.964374	+	0.264543i	$0.0852210\pi$
$840$	0	0
$841$	−24389.0	−1.00000
$842$	0	0
$843$	0	0
$844$	−25224.3	−1.02874
$845$	0	0
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	0	0		−	1.00000i	$-0.5\pi$
$853$	0	0			1.00000i	$0.5\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	0	0		−	1.00000i	$-0.5\pi$
$857$	0	0			1.00000i	$0.5\pi$
$858$	0	0
$859$	43324.3	1.72085	0.860423	−	0.509581i	$-0.170200\pi$
$859$	43324.3	1.72085	0.860423	+	0.509581i	$0.170200\pi$
$860$	18220.3	0.722451
$861$	0	0
$862$	−1127.66	−0.0445571
$863$	−23594.8	−0.930678	−0.465339	−	0.885133i	$-0.654068\pi$
$863$	−23594.8	−0.930678	−0.465339	+	0.885133i	$0.654068\pi$
$864$	0	0
$865$	0	0
$866$	126.936	0.00498091
$867$	0	0
$868$	0	0
$869$	−8564.79	−0.334339
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	0	0		−	1.00000i	$-0.5\pi$
$877$	0	0			1.00000i	$0.5\pi$
$878$	−39852.9	−1.53186
$879$	0	0
$880$	14998.4	0.574540
$881$	0	0		−	1.00000i	$-0.5\pi$
$881$	0	0			1.00000i	$0.5\pi$
$882$	0	0
$883$	−52410.3	−1.99745	−0.998724	−	0.0504988i	$-0.983919\pi$
$883$	−52410.3	−1.99745	−0.998724	+	0.0504988i	$0.983919\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	0	0		−	1.00000i	$-0.5\pi$
$887$	0	0			1.00000i	$0.5\pi$
$888$	0	0
$889$	0	0
$890$	−53998.0	−2.03373
$891$	0	0
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	1174.75	0.0436549
$899$	0	0
$900$	0	0
$901$	0	0
$902$	14195.2	0.524000
$903$	0	0
$904$	0	0
$905$	40636.6	1.49260
$906$	0	0
$907$	−35017.1	−1.28195	−0.640973	−	0.767564i	$-0.721469\pi$
$907$	−35017.1	−1.28195	−0.640973	+	0.767564i	$0.721469\pi$
$908$	−29305.0	−1.07106
$909$	0	0
$910$	0	0
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	0	0
$913$	1938.12	0.0702545
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−36762.2	−1.31956	−0.659779	−	0.751460i	$-0.729350\pi$
$919$	−36762.2	−1.31956	−0.659779	+	0.751460i	$0.729350\pi$
$920$	0	0
$921$	0	0
$922$	61377.1	2.19235
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−7352.53	−0.259665	−0.129832	−	0.991536i	$-0.541444\pi$
$929$	−7352.53	−0.259665	−0.129832	+	0.991536i	$0.541444\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	13275.6	0.462856	0.231428	−	0.972852i	$-0.425660\pi$
$937$	13275.6	0.462856	0.231428	+	0.972852i	$0.425660\pi$
$938$	0	0
$939$	0	0
$940$	−25818.4	−0.895856
$941$	−15462.1	−0.535652	−0.267826	−	0.963467i	$-0.586305\pi$
$941$	−15462.1	−0.535652	−0.267826	+	0.963467i	$0.586305\pi$
$942$	0	0
$943$	0	0
$944$	−46270.3	−1.59531
$945$	0	0
$946$	32586.6	1.11996
$947$	56498.4	1.93870	0.969352	−	0.245678i	$-0.0790105\pi$
$947$	56498.4	1.93870	0.969352	+	0.245678i	$0.0790105\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	0	0		−	1.00000i	$-0.5\pi$
$953$	0	0			1.00000i	$0.5\pi$
$954$	0	0
$955$	0	0
$956$	26744.1	0.904775
$957$	0	0
$958$	−18309.9	−0.617502
$959$	0	0
$960$	0	0
$961$	−29791.0	−1.00000
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	0	0		−	1.00000i	$-0.5\pi$
$967$	0	0			1.00000i	$0.5\pi$
$968$	−11572.2	−0.384239
$969$	0	0
$970$	0	0
$971$	0	0		−	1.00000i	$-0.5\pi$
$971$	0	0			1.00000i	$0.5\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	−75425.4	−2.47368
$977$	−52089.2	−1.70571	−0.852856	−	0.522146i	$-0.825132\pi$
$977$	−52089.2	−1.70571	−0.852856	+	0.522146i	$0.825132\pi$
$978$	0	0
$979$	−34240.3	−1.11780
$980$	13826.5	0.450685
$981$	0	0
$982$	0	0
$983$	−60639.0	−1.96753	−0.983766	−	0.179454i	$-0.942567\pi$
$983$	−60639.0	−1.96753	−0.983766	+	0.179454i	$0.942567\pi$
$984$	0	0
$985$	36484.2	1.18019
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	−23272.0	−0.745973	−0.372987	−	0.927837i	$-0.621666\pi$
$991$	−23272.0	−0.745973	−0.372987	+	0.927837i	$0.621666\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−51462.9	−1.63968
$996$	0	0
$997$	−21634.0	−0.687217	−0.343609	−	0.939113i	$-0.611649\pi$
$997$	−21634.0	−0.687217	−0.343609	+	0.939113i	$0.611649\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1521.4.a.y.1.2		4
3.2	odd	2	inner	1521.4.a.y.1.3		4
13.5	odd	4		117.4.b.c.64.3	yes	4
13.8	odd	4		117.4.b.c.64.2	✓	4
13.12	even	2	inner	1521.4.a.y.1.3		4
39.5	even	4		117.4.b.c.64.2	✓	4
39.8	even	4		117.4.b.c.64.3	yes	4
39.38	odd	2	CM	1521.4.a.y.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
117.4.b.c.64.2	✓	4	13.8	odd	4
117.4.b.c.64.2	✓	4	39.5	even	4
117.4.b.c.64.3	yes	4	13.5	odd	4
117.4.b.c.64.3	yes	4	39.8	even	4
1521.4.a.y.1.2		4	1.1	even	1	trivial
1521.4.a.y.1.2		4	39.38	odd	2	CM
1521.4.a.y.1.3		4	3.2	odd	2	inner
1521.4.a.y.1.3		4	13.12	even	2	inner

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:	\( N \)	\(=\)	\( 1521 = 3^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1521.a (trivial)

\(f(q)\)	\(=\)	\(q-3.52058 q^{2} +4.39445 q^{4} -9.17304 q^{5} +12.6936 q^{8} +32.2944 q^{10} +20.4780 q^{11} -79.8444 q^{16} -40.3104 q^{20} -72.0942 q^{22} -40.8554 q^{25} +179.549 q^{32} -116.439 q^{40} -196.898 q^{41} -452.000 q^{43} +89.9894 q^{44} +640.490 q^{47} -343.000 q^{49} +143.834 q^{50} -187.845 q^{55} +579.506 q^{59} +944.654 q^{61} +6.63840 q^{64} +1190.09 q^{71} -418.244 q^{79} +732.416 q^{80} +693.193 q^{82} +94.6440 q^{83} +1591.30 q^{86} +259.939 q^{88} -1672.06 q^{89} -2254.89 q^{94} +1207.56 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 32 q^{4} - 116 q^{10} - 204 q^{16} + 476 q^{22} + 500 q^{25} - 884 q^{40} - 1808 q^{43} - 1372 q^{49} - 3232 q^{55} - 1632 q^{64} - 2924 q^{82} + 2756 q^{88} - 5140 q^{94}+O(q^{100}) \) 4 * q + 32 * q^4 - 116 * q^10 - 204 * q^16 + 476 * q^22 + 500 * q^25 - 884 * q^40 - 1808 * q^43 - 1372 * q^49 - 3232 * q^55 - 1632 * q^64 - 2924 * q^82 + 2756 * q^88 - 5140 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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