LMFDB - Embedded newform 1521.2.a.t.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1521,2,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, 2, names="a")

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

chi := DirichletCharacter("1521.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1521 = 3^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
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:	$12.1452461474$
Analytic rank:	$0$
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} $
Twist minimal:	no (minimal twist has level 117)
Fricke sign:	$-1$
Sato-Tate group:	$N(\mathrm{U}(1))$

Embedding invariants

Embedding label			1.1
Root			$-0.835000$ 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)

$f(q)$	$=$	$q-2.75782 q^{2} +5.60555 q^{4} -1.67000 q^{5} -9.94345 q^{8} +O(q^{10})$	$q-2.75782 q^{2} +5.60555 q^{4} -1.67000 q^{5} -9.94345 q^{8} +4.60555 q^{10} +3.84563 q^{11} +16.2111 q^{16} -9.36127 q^{20} -10.6056 q^{22} -2.21110 q^{25} -24.8203 q^{32} +16.6056 q^{40} +12.7013 q^{41} +4.00000 q^{43} +21.5569 q^{44} -0.505635 q^{47} -7.00000 q^{49} +6.09781 q^{50} -6.42221 q^{55} -10.5256 q^{59} +7.21110 q^{61} +36.0278 q^{64} +14.8769 q^{71} -14.4222 q^{79} -27.0725 q^{80} -35.0278 q^{82} -18.2169 q^{83} -11.0313 q^{86} -38.2389 q^{88} +16.0413 q^{89} +1.39445 q^{94} +19.3047 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 8 q^{4}+O(q^{10}) $ 4 * q + 8 * q^4	$ 4 q + 8 q^{4} + 4 q^{10} + 36 q^{16} - 28 q^{22} + 20 q^{25} + 52 q^{40} + 16 q^{43} - 28 q^{49} + 32 q^{55} + 72 q^{64} - 68 q^{82} - 52 q^{88} + 20 q^{94}+O(q^{100}) $ 4 * q + 8 * q^4 + 4 * q^10 + 36 * q^16 - 28 * q^22 + 20 * q^25 + 52 * q^40 + 16 * q^43 - 28 * q^49 + 32 * q^55 + 72 * q^64 - 68 * q^82 - 52 * q^88 + 20 * q^94

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 $

$2$	−2.75782	−1.95007	−0.975035	−	0.222050i	$-0.928725\pi$
$2$	−2.75782	−1.95007	−0.975035	+	0.222050i	$0.928725\pi$
$3$	0	0
$3$	0	0
$4$	5.60555	2.80278
$5$	−1.67000	−0.746846	−0.373423	−	0.927661i	$-0.621816\pi$
$5$	−1.67000	−0.746846	−0.373423	+	0.927661i	$0.621816\pi$
$6$	0	0
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	−9.94345	−3.51554
$9$	0	0
$10$	4.60555	1.45640
$11$	3.84563	1.15950	0.579751	−	0.814794i	$-0.303150\pi$
$11$	3.84563	1.15950	0.579751	+	0.814794i	$0.303150\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	0	0
$16$	16.2111	4.05278
$17$	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	$0.5\pi$
$20$	−9.36127	−2.09324
$21$	0	0
$22$	−10.6056	−2.26111
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	−2.21110	−0.442221
$26$	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	$0.5\pi$
$32$	−24.8203	−4.38766
$33$	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
$40$	16.6056	2.62557
$41$	12.7013	1.98360	0.991802	−	0.127783i	$-0.0407861\pi$
$41$	12.7013	1.98360	0.991802	+	0.127783i	$0.0407861\pi$
$42$	0	0
$43$	4.00000	0.609994	0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000	0.609994	0.304997	+	0.952353i	$0.401344\pi$
$44$	21.5569	3.24982
$45$	0	0
$46$	0	0
$47$	−0.505635	−0.0737545	−0.0368772	−	0.999320i	$-0.511741\pi$
$47$	−0.505635	−0.0737545	−0.0368772	+	0.999320i	$0.511741\pi$
$48$	0	0
$49$	−7.00000	−1.00000
$50$	6.09781	0.862361
$51$	0	0
$52$	0	0
$53$	0	0		−	1.00000i	$-0.5\pi$
$53$	0	0			1.00000i	$0.5\pi$
$54$	0	0
$55$	−6.42221	−0.865970
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−10.5256	−1.37032	−0.685160	−	0.728392i	$-0.740268\pi$
$59$	−10.5256	−1.37032	−0.685160	+	0.728392i	$0.740268\pi$
$60$	0	0
$61$	7.21110	0.923287	0.461644	−	0.887066i	$-0.347260\pi$
$61$	7.21110	0.923287	0.461644	+	0.887066i	$0.347260\pi$
$62$	0	0
$63$	0	0
$64$	36.0278	4.50347
$65$	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
$71$	14.8769	1.76556	0.882782	−	0.469784i	$-0.155668\pi$
$71$	14.8769	1.76556	0.882782	+	0.469784i	$0.155668\pi$
$72$	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
$79$	−14.4222	−1.62262	−0.811312	−	0.584613i	$-0.801246\pi$
$79$	−14.4222	−1.62262	−0.811312	+	0.584613i	$0.801246\pi$
$80$	−27.0725	−3.02680
$81$	0	0
$82$	−35.0278	−3.86817
$83$	−18.2169	−1.99956	−0.999782	−	0.0208726i	$-0.993356\pi$
$83$	−18.2169	−1.99956	−0.999782	+	0.0208726i	$0.993356\pi$
$84$	0	0
$85$	0	0
$86$	−11.0313	−1.18953
$87$	0	0
$88$	−38.2389	−4.07628
$89$	16.0413	1.70037	0.850185	−	0.526484i	$-0.176490\pi$
$89$	16.0413	1.70037	0.850185	+	0.526484i	$0.176490\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	1.39445	0.143826
$95$	0	0
$96$	0	0
$97$	0	0		−	1.00000i	$-0.5\pi$
$97$	0	0			1.00000i	$0.5\pi$
$98$	19.3047	1.95007
$99$	0	0
$100$	−12.3944	−1.23944
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	0	0
$103$	16.0000	1.57653	0.788263	−	0.615338i	$-0.210980\pi$
$103$	16.0000	1.57653	0.788263	+	0.615338i	$0.210980\pi$
$104$	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	$0.5\pi$
$110$	17.7113	1.68870
$111$	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
$118$	29.0278	2.67222
$119$	0	0
$120$	0	0
$121$	3.78890	0.344445
$122$	−19.8869	−1.80048
$123$	0	0
$124$	0	0
$125$	12.0425	1.07712
$126$	0	0
$127$	14.4222	1.27976	0.639882	−	0.768473i	$-0.278983\pi$
$127$	14.4222	1.27976	0.639882	+	0.768473i	$0.278983\pi$
$128$	−49.7172	−4.39442
$129$	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
$137$	−2.68127	−0.229076	−0.114538	−	0.993419i	$-0.536539\pi$
$137$	−2.68127	−0.229076	−0.114538	+	0.993419i	$0.536539\pi$
$138$	0	0
$139$	20.0000	1.69638	0.848189	−	0.529694i	$-0.177693\pi$
$139$	20.0000	1.69638	0.848189	+	0.529694i	$0.177693\pi$
$140$	0	0
$141$	0	0
$142$	−41.0278	−3.44297
$143$	0	0
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	17.0525	1.39700	0.698499	−	0.715611i	$-0.253852\pi$
$149$	17.0525	1.39700	0.698499	+	0.715611i	$0.253852\pi$
$150$	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
$157$	2.00000	0.159617	0.0798087	−	0.996810i	$-0.474569\pi$
$157$	2.00000	0.159617	0.0798087	+	0.996810i	$0.474569\pi$
$158$	39.7738	3.16423
$159$	0	0
$160$	41.4500	3.27691
$161$	0	0
$162$	0	0
$163$	0	0		−	1.00000i	$-0.5\pi$
$163$	0	0			1.00000i	$0.5\pi$
$164$	71.1976	5.55960
$165$	0	0
$166$	50.2389	3.89929
$167$	8.19690	0.634295	0.317148	−	0.948376i	$-0.397275\pi$
$167$	8.19690	0.634295	0.317148	+	0.948376i	$0.397275\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	0	0
$172$	22.4222	1.70968
$173$	0	0		−	1.00000i	$-0.5\pi$
$173$	0	0			1.00000i	$0.5\pi$
$174$	0	0
$175$	0	0
$176$	62.3420	4.69920
$177$	0	0
$178$	−44.2389	−3.31584
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	0	0
$181$	10.0000	0.743294	0.371647	−	0.928374i	$-0.378793\pi$
$181$	10.0000	0.743294	0.371647	+	0.928374i	$0.378793\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0	0
$188$	−2.83436	−0.206717
$189$	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
$196$	−39.2389	−2.80278
$197$	27.0725	1.92884	0.964419	−	0.264379i	$-0.0851669\pi$
$197$	27.0725	1.92884	0.964419	+	0.264379i	$0.0851669\pi$
$198$	0	0
$199$	14.4222	1.02236	0.511182	−	0.859473i	$-0.329208\pi$
$199$	14.4222	1.02236	0.511182	+	0.859473i	$0.329208\pi$
$200$	21.9860	1.55464
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−21.2111	−1.48145
$206$	−44.1251	−3.07434
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	28.8444	1.98573	0.992866	−	0.119239i	$-0.0380456\pi$
$211$	28.8444	1.98573	0.992866	+	0.119239i	$0.0380456\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−6.68000	−0.455572
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	−36.0000	−2.42712
$221$	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
$227$	24.8969	1.65246	0.826232	−	0.563329i	$-0.190480\pi$
$227$	24.8969	1.65246	0.826232	+	0.563329i	$0.190480\pi$
$228$	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
$235$	0.844410	0.0550832
$236$	−59.0020	−3.84070
$237$	0	0
$238$	0	0
$239$	6.17436	0.399386	0.199693	−	0.979858i	$-0.436005\pi$
$239$	6.17436	0.399386	0.199693	+	0.979858i	$0.436005\pi$
$240$	0	0
$241$	0	0		−	1.00000i	$-0.5\pi$
$241$	0	0			1.00000i	$0.5\pi$
$242$	−10.4491	−0.671693
$243$	0	0
$244$	40.4222	2.58777
$245$	11.6900	0.746846
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	−33.2111	−2.10045
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	0	0
$253$	0	0
$254$	−39.7738	−2.49563
$255$	0	0
$256$	65.0555	4.06597
$257$	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
$274$	7.39445	0.446715
$275$	−8.50309	−0.512756
$276$	0	0
$277$	22.0000	1.32185	0.660926	−	0.750451i	$-0.270164\pi$
$277$	22.0000	1.32185	0.660926	+	0.750451i	$0.270164\pi$
$278$	−55.1563	−3.30806
$279$	0	0
$280$	0	0
$281$	−31.4238	−1.87459	−0.937293	−	0.348542i	$-0.886677\pi$
$281$	−31.4238	−1.87459	−0.937293	+	0.348542i	$0.886677\pi$
$282$	0	0
$283$	−28.8444	−1.71462	−0.857311	−	0.514799i	$-0.827867\pi$
$283$	−28.8444	−1.71462	−0.857311	+	0.514799i	$0.827867\pi$
$284$	83.3932	4.94848
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−17.0000	−1.00000
$290$	0	0
$291$	0	0
$292$	0	0
$293$	30.4125	1.77672	0.888359	−	0.459149i	$-0.151846\pi$
$293$	30.4125	1.77672	0.888359	+	0.459149i	$0.151846\pi$
$294$	0	0
$295$	17.5778	1.02342
$296$	0	0
$297$	0	0
$298$	−47.0278	−2.72424
$299$	0	0
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−12.0425	−0.689554
$306$	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
$313$	7.21110	0.407596	0.203798	−	0.979013i	$-0.434671\pi$
$313$	7.21110	0.407596	0.203798	+	0.979013i	$0.434671\pi$
$314$	−5.51563	−0.311265
$315$	0	0
$316$	−80.8444	−4.54785
$317$	−10.3725	−0.582580	−0.291290	−	0.956635i	$-0.594084\pi$
$317$	−10.3725	−0.582580	−0.291290	+	0.956635i	$0.594084\pi$
$318$	0	0
$319$	0	0
$320$	−60.1663	−3.36340
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	−126.294	−6.97344
$329$	0	0
$330$	0	0
$331$	0	0		−	1.00000i	$-0.5\pi$
$331$	0	0			1.00000i	$0.5\pi$
$332$	−102.116	−5.60433
$333$	0	0
$334$	−22.6056	−1.23692
$335$	0	0
$336$	0	0
$337$	36.0555	1.96407	0.982034	−	0.188702i	$-0.0604279\pi$
$337$	36.0555	1.96407	0.982034	+	0.188702i	$0.0604279\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	0	0
$344$	−39.7738	−2.14446
$345$	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
$352$	−95.4500	−5.08750
$353$	−24.7438	−1.31698	−0.658490	−	0.752590i	$-0.728804\pi$
$353$	−24.7438	−1.31698	−0.658490	+	0.752590i	$0.728804\pi$
$354$	0	0
$355$	−24.8444	−1.31860
$356$	89.9201	4.76576
$357$	0	0
$358$	0	0
$359$	28.2369	1.49029	0.745143	−	0.666905i	$-0.232381\pi$
$359$	28.2369	1.49029	0.745143	+	0.666905i	$0.232381\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	−27.5782	−1.44948
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	8.00000	0.417597	0.208798	−	0.977959i	$-0.433045\pi$
$367$	8.00000	0.417597	0.208798	+	0.977959i	$0.433045\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−36.0555	−1.86688	−0.933442	−	0.358729i	$-0.883210\pi$
$373$	−36.0555	−1.86688	−0.933442	+	0.358729i	$0.883210\pi$
$374$	0	0
$375$	0	0
$376$	5.02776	0.259287
$377$	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
$383$	−36.9394	−1.88752	−0.943758	−	0.330636i	$-0.892737\pi$
$383$	−36.9394	−1.88752	−0.943758	+	0.330636i	$0.892737\pi$
$384$	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
$392$	69.6041	3.51554
$393$	0	0
$394$	−74.6611	−3.76137
$395$	24.0851	1.21185
$396$	0	0
$397$	0	0		−	1.00000i	$-0.5\pi$
$397$	0	0			1.00000i	$0.5\pi$
$398$	−39.7738	−1.99368
$399$	0	0
$400$	−35.8444	−1.79222
$401$	−3.99873	−0.199687	−0.0998435	−	0.995003i	$-0.531834\pi$
$401$	−3.99873	−0.199687	−0.0998435	+	0.995003i	$0.531834\pi$
$402$	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	$0.5\pi$
$410$	58.4963	2.88893
$411$	0	0
$412$	89.6888	4.41865
$413$	0	0
$414$	0	0
$415$	30.4222	1.49337
$416$	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	$0.5\pi$
$422$	−79.5476	−3.87232
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	18.4222	0.888398
$431$	20.5456	0.989648	0.494824	−	0.868993i	$-0.335233\pi$
$431$	20.5456	0.989648	0.494824	+	0.868993i	$0.335233\pi$
$432$	0	0
$433$	36.0555	1.73272	0.866359	−	0.499422i	$-0.166454\pi$
$433$	36.0555	1.73272	0.866359	+	0.499422i	$0.166454\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	40.0000	1.90910	0.954548	−	0.298057i	$-0.0963387\pi$
$439$	40.0000	1.90910	0.954548	+	0.298057i	$0.0963387\pi$
$440$	63.8589	3.04435
$441$	0	0
$442$	0	0
$443$	0	0		−	1.00000i	$-0.5\pi$
$443$	0	0			1.00000i	$0.5\pi$
$444$	0	0
$445$	−26.7889	−1.26992
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−21.4038	−1.01011	−0.505054	−	0.863088i	$-0.668527\pi$
$449$	−21.4038	−1.01011	−0.505054	+	0.863088i	$0.668527\pi$
$450$	0	0
$451$	48.8444	2.29999
$452$	0	0
$453$	0	0
$454$	−68.6611	−3.22242
$455$	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
$461$	7.03254	0.327538	0.163769	−	0.986499i	$-0.447635\pi$
$461$	7.03254	0.327538	0.163769	+	0.986499i	$0.447635\pi$
$462$	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
$470$	−2.32873	−0.107416
$471$	0	0
$472$	104.661	4.81742
$473$	15.3825	0.707290
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	−17.0278	−0.778831
$479$	−43.6194	−1.99302	−0.996511	−	0.0834585i	$-0.973403\pi$
$479$	−43.6194	−1.99302	−0.996511	+	0.0834585i	$0.973403\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	21.2389	0.965403
$485$	0	0
$486$	0	0
$487$	0	0		−	1.00000i	$-0.5\pi$
$487$	0	0			1.00000i	$0.5\pi$
$488$	−71.7032	−3.24585
$489$	0	0
$490$	−32.2389	−1.45640
$491$	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	$0.5\pi$
$500$	67.5051	3.01892
$501$	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
$508$	80.8444	3.58689
$509$	−39.1151	−1.73374	−0.866872	−	0.498530i	$-0.833873\pi$
$509$	−39.1151	−1.73374	−0.866872	+	0.498530i	$0.833873\pi$
$510$	0	0
$511$	0	0
$512$	−79.9767	−3.53450
$513$	0	0
$514$	0	0
$515$	−26.7200	−1.17742
$516$	0	0
$517$	−1.94449	−0.0855185
$518$	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
$523$	44.0000	1.92399	0.961993	−	0.273075i	$-0.0880406\pi$
$523$	44.0000	1.92399	0.961993	+	0.273075i	$0.0880406\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−23.0000	−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
$539$	−26.9194	−1.15950
$540$	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
$547$	−28.0000	−1.19719	−0.598597	−	0.801050i	$-0.704275\pi$
$547$	−28.0000	−1.19719	−0.598597	+	0.801050i	$0.704275\pi$
$548$	−15.0300	−0.642049
$549$	0	0
$550$	23.4500	0.999910
$551$	0	0
$552$	0	0
$553$	0	0
$554$	−60.6720	−2.57770
$555$	0	0
$556$	112.111	4.75457
$557$	−35.7751	−1.51584	−0.757919	−	0.652349i	$-0.773784\pi$
$557$	−35.7751	−1.51584	−0.757919	+	0.652349i	$0.773784\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	86.6611	3.65558
$563$	0	0		−	1.00000i	$-0.5\pi$
$563$	0	0			1.00000i	$0.5\pi$
$564$	0	0
$565$	0	0
$566$	79.5476	3.34363
$567$	0	0
$568$	−147.928	−6.20691
$569$	0	0		−	1.00000i	$-0.5\pi$
$569$	0	0			1.00000i	$0.5\pi$
$570$	0	0
$571$	−28.8444	−1.20710	−0.603550	−	0.797325i	$-0.706248\pi$
$571$	−28.8444	−1.20710	−0.603550	+	0.797325i	$0.706248\pi$
$572$	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	$0.5\pi$
$578$	46.8829	1.95007
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	−83.8722	−3.46473
$587$	−46.9594	−1.93822	−0.969111	−	0.246623i	$-0.920679\pi$
$587$	−46.9594	−1.93822	−0.969111	+	0.246623i	$0.920679\pi$
$588$	0	0
$589$	0	0
$590$	−48.4763	−1.99574
$591$	0	0
$592$	0	0
$593$	7.33873	0.301365	0.150683	−	0.988582i	$-0.451853\pi$
$593$	7.33873	0.301365	0.150683	+	0.988582i	$0.451853\pi$
$594$	0	0
$595$	0	0
$596$	95.5889	3.91547
$597$	0	0
$598$	0	0
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	0	0
$601$	7.21110	0.294147	0.147074	−	0.989126i	$-0.453015\pi$
$601$	7.21110	0.294147	0.147074	+	0.989126i	$0.453015\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−6.32746	−0.257248
$606$	0	0
$607$	32.0000	1.29884	0.649420	−	0.760430i	$-0.275012\pi$
$607$	32.0000	1.29884	0.649420	+	0.760430i	$0.275012\pi$
$608$	0	0
$609$	0	0
$610$	33.2111	1.34468
$611$	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
$617$	−44.7838	−1.80293	−0.901464	−	0.432855i	$-0.857506\pi$
$617$	−44.7838	−1.80293	−0.901464	+	0.432855i	$0.857506\pi$
$618$	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
$625$	−9.05551	−0.362221
$626$	−19.8869	−0.794840
$627$	0	0
$628$	11.2111	0.447372
$629$	0	0
$630$	0	0
$631$	0	0		−	1.00000i	$-0.5\pi$
$631$	0	0			1.00000i	$0.5\pi$
$632$	143.406	5.70440
$633$	0	0
$634$	28.6056	1.13607
$635$	−24.0851	−0.955787
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	83.0278	3.28196
$641$	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
$649$	−40.4777	−1.58889
$650$	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
$656$	205.902	8.03910
$657$	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
$664$	181.139	7.02955
$665$	0	0
$666$	0	0
$667$	0	0
$668$	45.9482	1.77779
$669$	0	0
$670$	0	0
$671$	27.7313	1.07055
$672$	0	0
$673$	−14.0000	−0.539660	−0.269830	−	0.962908i	$-0.586968\pi$
$673$	−14.0000	−0.539660	−0.269830	+	0.962908i	$0.586968\pi$
$674$	−99.4345	−3.83007
$675$	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
$683$	41.2907	1.57994	0.789972	−	0.613142i	$-0.210095\pi$
$683$	41.2907	1.57994	0.789972	+	0.613142i	$0.210095\pi$
$684$	0	0
$685$	4.47772	0.171085
$686$	0	0
$687$	0	0
$688$	64.8444	2.47217
$689$	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
$695$	−33.4000	−1.26693
$696$	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
$704$	138.550	5.22178
$705$	0	0
$706$	68.2389	2.56820
$707$	0	0
$708$	0	0
$709$	0	0		−	1.00000i	$-0.5\pi$
$709$	0	0			1.00000i	$0.5\pi$
$710$	68.5163	2.57137
$711$	0	0
$712$	−159.505	−5.97772
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	−77.8722	−2.90616
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	0	0
$722$	52.3985	1.95007
$723$	0	0
$724$	56.0555	2.08329
$725$	0	0
$726$	0	0
$727$	14.4222	0.534890	0.267445	−	0.963573i	$-0.413821\pi$
$727$	14.4222	0.534890	0.267445	+	0.963573i	$0.413821\pi$
$728$	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	$0.5\pi$
$734$	−22.0625	−0.814343
$735$	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
$743$	34.9169	1.28098	0.640488	−	0.767968i	$-0.278732\pi$
$743$	34.9169	1.28098	0.640488	+	0.767968i	$0.278732\pi$
$744$	0	0
$745$	−28.4777	−1.04334
$746$	99.4345	3.64056
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	40.0000	1.45962	0.729810	−	0.683650i	$-0.239608\pi$
$751$	40.0000	1.45962	0.729810	+	0.683650i	$0.239608\pi$
$752$	−8.19690	−0.298910
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−36.0555	−1.31046	−0.655230	−	0.755429i	$-0.727428\pi$
$757$	−36.0555	−1.31046	−0.655230	+	0.755429i	$0.727428\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	14.7238	0.533738	0.266869	−	0.963733i	$-0.414011\pi$
$761$	14.7238	0.533738	0.266869	+	0.963733i	$0.414011\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	101.872	3.68079
$767$	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
$773$	−29.0951	−1.04648	−0.523238	−	0.852186i	$-0.675276\pi$
$773$	−29.0951	−1.04648	−0.523238	+	0.852186i	$0.675276\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	57.2111	2.04717
$782$	0	0
$783$	0	0
$784$	−113.478	−4.05278
$785$	−3.34000	−0.119210
$786$	0	0
$787$	0	0		−	1.00000i	$-0.5\pi$
$787$	0	0			1.00000i	$0.5\pi$
$788$	151.756	5.40610
$789$	0	0
$790$	−66.4222	−2.36320
$791$	0	0
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	80.8444	2.86545
$797$	0	0		−	1.00000i	$-0.5\pi$
$797$	0	0			1.00000i	$0.5\pi$
$798$	0	0
$799$	0	0
$800$	54.8803	1.94031
$801$	0	0
$802$	11.0278	0.389404
$803$	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
$820$	−118.900	−4.15217
$821$	−40.4325	−1.41110	−0.705552	−	0.708658i	$-0.749301\pi$
$821$	−40.4325	−1.41110	−0.705552	+	0.708658i	$0.749301\pi$
$822$	0	0
$823$	−56.0000	−1.95204	−0.976019	−	0.217687i	$-0.930149\pi$
$823$	−56.0000	−1.95204	−0.976019	+	0.217687i	$0.930149\pi$
$824$	−159.095	−5.54234
$825$	0	0
$826$	0	0
$827$	−16.1944	−0.563133	−0.281566	−	0.959542i	$-0.590854\pi$
$827$	−16.1944	−0.563133	−0.281566	+	0.959542i	$0.590854\pi$
$828$	0	0
$829$	−50.4777	−1.75316	−0.876582	−	0.481253i	$-0.840182\pi$
$829$	−50.4777	−1.75316	−0.876582	+	0.481253i	$0.840182\pi$
$830$	−83.8989	−2.91217
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−13.6888	−0.473721
$836$	0	0
$837$	0	0
$838$	0	0
$839$	5.16309	0.178250	0.0891249	−	0.996020i	$-0.471593\pi$
$839$	5.16309	0.178250	0.0891249	+	0.996020i	$0.471593\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	0	0
$843$	0	0
$844$	161.689	5.56556
$845$	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
$859$	−57.6888	−1.96832	−0.984159	−	0.177290i	$-0.943267\pi$
$859$	−57.6888	−1.96832	−0.984159	+	0.177290i	$0.943267\pi$
$860$	−37.4451	−1.27687
$861$	0	0
$862$	−56.6611	−1.92988
$863$	57.9907	1.97403	0.987013	−	0.160640i	$-0.0513559\pi$
$863$	57.9907	1.97403	0.987013	+	0.160640i	$0.0513559\pi$
$864$	0	0
$865$	0	0
$866$	−99.4345	−3.37892
$867$	0	0
$868$	0	0
$869$	−55.4625	−1.88144
$870$	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	$0.5\pi$
$878$	−110.313	−3.72287
$879$	0	0
$880$	−104.111	−3.50958
$881$	0	0		−	1.00000i	$-0.5\pi$
$881$	0	0			1.00000i	$0.5\pi$
$882$	0	0
$883$	−28.8444	−0.970692	−0.485346	−	0.874322i	$-0.661306\pi$
$883$	−28.8444	−0.970692	−0.485346	+	0.874322i	$0.661306\pi$
$884$	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
$890$	73.8789	2.47643
$891$	0	0
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	59.0278	1.96978
$899$	0	0
$900$	0	0
$901$	0	0
$902$	−134.704	−4.48515
$903$	0	0
$904$	0	0
$905$	−16.7000	−0.555127
$906$	0	0
$907$	57.6888	1.91553	0.957763	−	0.287559i	$-0.0928437\pi$
$907$	57.6888	1.91553	0.957763	+	0.287559i	$0.0928437\pi$
$908$	139.561	4.63149
$909$	0	0
$910$	0	0
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	0	0
$913$	−70.0555	−2.31850
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−14.4222	−0.475745	−0.237872	−	0.971296i	$-0.576450\pi$
$919$	−14.4222	−0.475745	−0.237872	+	0.971296i	$0.576450\pi$
$920$	0	0
$921$	0	0
$922$	−19.3944	−0.638722
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	32.7413	1.07421	0.537103	−	0.843517i	$-0.319519\pi$
$929$	32.7413	1.07421	0.537103	+	0.843517i	$0.319519\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	50.4777	1.64904	0.824518	−	0.565836i	$-0.191447\pi$
$937$	50.4777	1.64904	0.824518	+	0.565836i	$0.191447\pi$
$938$	0	0
$939$	0	0
$940$	4.73338	0.154386
$941$	25.7551	0.839591	0.419796	−	0.907619i	$-0.362102\pi$
$941$	25.7551	0.839591	0.419796	+	0.907619i	$0.362102\pi$
$942$	0	0
$943$	0	0
$944$	−170.632	−5.55360
$945$	0	0
$946$	−42.4222	−1.37926
$947$	55.6620	1.80877	0.904385	−	0.426717i	$-0.140330\pi$
$947$	55.6620	1.80877	0.904385	+	0.426717i	$0.140330\pi$
$948$	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
$956$	34.6107	1.11939
$957$	0	0
$958$	120.294	3.88653
$959$	0	0
$960$	0	0
$961$	−31.0000	−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	$0.5\pi$
$968$	−37.6747	−1.21091
$969$	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
$976$	116.900	3.74188
$977$	11.3838	0.364200	0.182100	−	0.983280i	$-0.441711\pi$
$977$	11.3838	0.364200	0.182100	+	0.983280i	$0.441711\pi$
$978$	0	0
$979$	61.6888	1.97158
$980$	65.5289	2.09324
$981$	0	0
$982$	0	0
$983$	−52.3220	−1.66881	−0.834406	−	0.551151i	$-0.814189\pi$
$983$	−52.3220	−1.66881	−0.834406	+	0.551151i	$0.814189\pi$
$984$	0	0
$985$	−45.2111	−1.44055
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	8.00000	0.254128	0.127064	−	0.991894i	$-0.459445\pi$
$991$	8.00000	0.254128	0.127064	+	0.991894i	$0.459445\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−24.0851	−0.763548
$996$	0	0
$997$	−58.0000	−1.83688	−0.918439	−	0.395562i	$-0.870550\pi$
$997$	−58.0000	−1.83688	−0.918439	+	0.395562i	$0.870550\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1521.2.a.t.1.1		4
3.2	odd	2	inner	1521.2.a.t.1.4		4
13.5	odd	4		117.2.b.b.64.4	yes	4
13.8	odd	4		117.2.b.b.64.1	✓	4
13.12	even	2	inner	1521.2.a.t.1.4		4
39.5	even	4		117.2.b.b.64.1	✓	4
39.8	even	4		117.2.b.b.64.4	yes	4
39.38	odd	2	CM	1521.2.a.t.1.1		4
52.31	even	4		1872.2.c.k.1585.3		4
52.47	even	4		1872.2.c.k.1585.2		4
156.47	odd	4		1872.2.c.k.1585.3		4
156.83	odd	4		1872.2.c.k.1585.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
117.2.b.b.64.1	✓	4	13.8	odd	4
117.2.b.b.64.1	✓	4	39.5	even	4
117.2.b.b.64.4	yes	4	13.5	odd	4
117.2.b.b.64.4	yes	4	39.8	even	4
1521.2.a.t.1.1		4	1.1	even	1	trivial
1521.2.a.t.1.1		4	39.38	odd	2	CM
1521.2.a.t.1.4		4	3.2	odd	2	inner
1521.2.a.t.1.4		4	13.12	even	2	inner
1872.2.c.k.1585.2		4	52.47	even	4
1872.2.c.k.1585.2		4	156.83	odd	4
1872.2.c.k.1585.3		4	52.31	even	4
1872.2.c.k.1585.3		4	156.47	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-2.75782 q^{2} +5.60555 q^{4} -1.67000 q^{5} -9.94345 q^{8} +O(q^{10})\)	\(q-2.75782 q^{2} +5.60555 q^{4} -1.67000 q^{5} -9.94345 q^{8} +4.60555 q^{10} +3.84563 q^{11} +16.2111 q^{16} -9.36127 q^{20} -10.6056 q^{22} -2.21110 q^{25} -24.8203 q^{32} +16.6056 q^{40} +12.7013 q^{41} +4.00000 q^{43} +21.5569 q^{44} -0.505635 q^{47} -7.00000 q^{49} +6.09781 q^{50} -6.42221 q^{55} -10.5256 q^{59} +7.21110 q^{61} +36.0278 q^{64} +14.8769 q^{71} -14.4222 q^{79} -27.0725 q^{80} -35.0278 q^{82} -18.2169 q^{83} -11.0313 q^{86} -38.2389 q^{88} +16.0413 q^{89} +1.39445 q^{94} +19.3047 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8 q^{4}+O(q^{10}) \) 4 * q + 8 * q^4	\( 4 q + 8 q^{4} + 4 q^{10} + 36 q^{16} - 28 q^{22} + 20 q^{25} + 52 q^{40} + 16 q^{43} - 28 q^{49} + 32 q^{55} + 72 q^{64} - 68 q^{82} - 52 q^{88} + 20 q^{94}+O(q^{100}) \) 4 * q + 8 * q^4 + 4 * q^10 + 36 * q^16 - 28 * q^22 + 20 * q^25 + 52 * q^40 + 16 * q^43 - 28 * q^49 + 32 * q^55 + 72 * q^64 - 68 * q^82 - 52 * q^88 + 20 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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