LMFDB - Embedded newform 1521.2.a.t.1.2

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			1.2
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)

$f(q)$	$=$	$q-0.628052 q^{2} -1.60555 q^{4} +4.14863 q^{5} +2.26447 q^{8} +O(q^{10})$	$q-0.628052 q^{2} -1.60555 q^{4} +4.14863 q^{5} +2.26447 q^{8} -2.60555 q^{10} +5.40473 q^{11} +1.78890 q^{16} -6.66083 q^{20} -3.39445 q^{22} +12.2111 q^{25} -5.65246 q^{32} +9.39445 q^{40} -1.63642 q^{41} +4.00000 q^{43} -8.67757 q^{44} -13.7020 q^{47} -7.00000 q^{49} -7.66920 q^{50} +22.4222 q^{55} +11.1898 q^{59} -7.21110 q^{61} -0.0277564 q^{64} +7.91694 q^{71} +14.4222 q^{79} +7.42147 q^{80} +1.02776 q^{82} +0.380317 q^{83} -2.51221 q^{86} +12.2389 q^{88} -9.93367 q^{89} +8.60555 q^{94} +4.39636 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−0.628052	−0.444099	−0.222050	−	0.975035i	$-0.571275\pi$
$2$	−0.628052	−0.444099	−0.222050	+	0.975035i	$0.571275\pi$
$3$	0	0
$3$	0	0
$4$	−1.60555	−0.802776
$5$	4.14863	1.85532	0.927661	−	0.373423i	$-0.121816\pi$
$5$	4.14863	1.85532	0.927661	+	0.373423i	$0.121816\pi$
$6$	0	0
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	2.26447	0.800612
$9$	0	0
$10$	−2.60555	−0.823948
$11$	5.40473	1.62959	0.814794	−	0.579751i	$-0.196850\pi$
$11$	5.40473	1.62959	0.814794	+	0.579751i	$0.196850\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	0	0
$16$	1.78890	0.447224
$17$	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	$0.5\pi$
$20$	−6.66083	−1.48941
$21$	0	0
$22$	−3.39445	−0.723699
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	12.2111	2.44222
$26$	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	$0.5\pi$
$32$	−5.65246	−0.999224
$33$	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
$40$	9.39445	1.48539
$41$	−1.63642	−0.255566	−0.127783	−	0.991802i	$-0.540786\pi$
$41$	−1.63642	−0.255566	−0.127783	+	0.991802i	$0.540786\pi$
$42$	0	0
$43$	4.00000	0.609994	0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000	0.609994	0.304997	+	0.952353i	$0.401344\pi$
$44$	−8.67757	−1.30819
$45$	0	0
$46$	0	0
$47$	−13.7020	−1.99864	−0.999320	−	0.0368772i	$-0.988259\pi$
$47$	−13.7020	−1.99864	−0.999320	+	0.0368772i	$0.988259\pi$
$48$	0	0
$49$	−7.00000	−1.00000
$50$	−7.66920	−1.08459
$51$	0	0
$52$	0	0
$53$	0	0		−	1.00000i	$-0.5\pi$
$53$	0	0			1.00000i	$0.5\pi$
$54$	0	0
$55$	22.4222	3.02341
$56$	0	0
$57$	0	0
$58$	0	0
$59$	11.1898	1.45678	0.728392	−	0.685160i	$-0.240268\pi$
$59$	11.1898	1.45678	0.728392	+	0.685160i	$0.240268\pi$
$60$	0	0
$61$	−7.21110	−0.923287	−0.461644	−	0.887066i	$-0.652740\pi$
$61$	−7.21110	−0.923287	−0.461644	+	0.887066i	$0.652740\pi$
$62$	0	0
$63$	0	0
$64$	−0.0277564	−0.00346955
$65$	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
$71$	7.91694	0.939567	0.469784	−	0.882782i	$-0.344332\pi$
$71$	7.91694	0.939567	0.469784	+	0.882782i	$0.344332\pi$
$72$	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	$-0.198754\pi$
$79$	14.4222	1.62262	0.811312	+	0.584613i	$0.198754\pi$
$80$	7.42147	0.829745
$81$	0	0
$82$	1.02776	0.113497
$83$	0.380317	0.0417453	0.0208726	−	0.999782i	$-0.493356\pi$
$83$	0.380317	0.0417453	0.0208726	+	0.999782i	$0.493356\pi$
$84$	0	0
$85$	0	0
$86$	−2.51221	−0.270898
$87$	0	0
$88$	12.2389	1.30467
$89$	−9.93367	−1.05297	−0.526484	−	0.850185i	$-0.676490\pi$
$89$	−9.93367	−1.05297	−0.526484	+	0.850185i	$0.676490\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	8.60555	0.887595
$95$	0	0
$96$	0	0
$97$	0	0		−	1.00000i	$-0.5\pi$
$97$	0	0			1.00000i	$0.5\pi$
$98$	4.39636	0.444099
$99$	0	0
$100$	−19.6056	−1.96056
$101$	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	$0.5\pi$
$110$	−14.0823	−1.34269
$111$	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
$118$	−7.02776	−0.646957
$119$	0	0
$120$	0	0
$121$	18.2111	1.65555
$122$	4.52894	0.410031
$123$	0	0
$124$	0	0
$125$	29.9162	2.67578
$126$	0	0
$127$	−14.4222	−1.27976	−0.639882	−	0.768473i	$-0.721017\pi$
$127$	−14.4222	−1.27976	−0.639882	+	0.768473i	$0.721017\pi$
$128$	11.3224	1.00076
$129$	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
$137$	−23.2553	−1.98684	−0.993419	−	0.114538i	$-0.963461\pi$
$137$	−23.2553	−1.98684	−0.993419	+	0.114538i	$0.963461\pi$
$138$	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
$142$	−4.97224	−0.417261
$143$	0	0
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	17.4703	1.43122	0.715611	−	0.698499i	$-0.246148\pi$
$149$	17.4703	1.43122	0.715611	+	0.698499i	$0.246148\pi$
$150$	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	$0.474569\pi$
$158$	−9.05789	−0.720607
$159$	0	0
$160$	−23.4500	−1.85388
$161$	0	0
$162$	0	0
$163$	0	0		−	1.00000i	$-0.5\pi$
$163$	0	0			1.00000i	$0.5\pi$
$164$	2.62736	0.205162
$165$	0	0
$166$	−0.238859	−0.0185390
$167$	24.5114	1.89675	0.948376	−	0.317148i	$-0.102725\pi$
$167$	24.5114	1.89675	0.948376	+	0.317148i	$0.102725\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	0	0
$172$	−6.42221	−0.489689
$173$	0	0		−	1.00000i	$-0.5\pi$
$173$	0	0			1.00000i	$0.5\pi$
$174$	0	0
$175$	0	0
$176$	9.66851	0.728791
$177$	0	0
$178$	6.23886	0.467622
$179$	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
$188$	21.9992	1.60446
$189$	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
$196$	11.2389	0.802776
$197$	−7.42147	−0.528758	−0.264379	−	0.964419i	$-0.585167\pi$
$197$	−7.42147	−0.528758	−0.264379	+	0.964419i	$0.585167\pi$
$198$	0	0
$199$	−14.4222	−1.02236	−0.511182	−	0.859473i	$-0.670792\pi$
$199$	−14.4222	−1.02236	−0.511182	+	0.859473i	$0.670792\pi$
$200$	27.6517	1.95527
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−6.78890	−0.474157
$206$	−10.0488	−0.700135
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−28.8444	−1.98573	−0.992866	−	0.119239i	$-0.961954\pi$
$211$	−28.8444	−1.98573	−0.992866	+	0.119239i	$0.961954\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	16.5945	1.13174
$216$	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
$227$	−16.9748	−1.12666	−0.563329	−	0.826232i	$-0.690480\pi$
$227$	−16.9748	−1.12666	−0.563329	+	0.826232i	$0.690480\pi$
$228$	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
$235$	−56.8444	−3.70812
$236$	−17.9658	−1.16947
$237$	0	0
$238$	0	0
$239$	−30.2965	−1.95972	−0.979858	−	0.199693i	$-0.936005\pi$
$239$	−30.2965	−1.95972	−0.979858	+	0.199693i	$0.936005\pi$
$240$	0	0
$241$	0	0		−	1.00000i	$-0.5\pi$
$241$	0	0			1.00000i	$0.5\pi$
$242$	−11.4375	−0.735231
$243$	0	0
$244$	11.5778	0.741192
$245$	−29.0404	−1.85532
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	−18.7889	−1.18831
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	0	0
$253$	0	0
$254$	9.05789	0.568342
$255$	0	0
$256$	−7.05551	−0.440970
$257$	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
$274$	14.6056	0.882354
$275$	65.9977	3.97981
$276$	0	0
$277$	22.0000	1.32185	0.660926	−	0.750451i	$-0.270164\pi$
$277$	22.0000	1.32185	0.660926	+	0.750451i	$0.270164\pi$
$278$	−12.5610	−0.753361
$279$	0	0
$280$	0	0
$281$	−11.6852	−0.697083	−0.348542	−	0.937293i	$-0.613323\pi$
$281$	−11.6852	−0.697083	−0.348542	+	0.937293i	$0.613323\pi$
$282$	0	0
$283$	28.8444	1.71462	0.857311	−	0.514799i	$-0.172133\pi$
$283$	28.8444	1.71462	0.857311	+	0.514799i	$0.172133\pi$
$284$	−12.7110	−0.754262
$285$	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
$293$	−15.7187	−0.918297	−0.459149	−	0.888359i	$-0.651846\pi$
$293$	−15.7187	−0.918297	−0.459149	+	0.888359i	$0.651846\pi$
$294$	0	0
$295$	46.4222	2.70281
$296$	0	0
$297$	0	0
$298$	−10.9722	−0.635605
$299$	0	0
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−29.9162	−1.71300
$306$	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	$-0.565329\pi$
$313$	−7.21110	−0.407596	−0.203798	+	0.979013i	$0.565329\pi$
$314$	−1.25610	−0.0708860
$315$	0	0
$316$	−23.1556	−1.30260
$317$	−34.0648	−1.91327	−0.956635	−	0.291290i	$-0.905916\pi$
$317$	−34.0648	−1.91327	−0.956635	+	0.291290i	$0.905916\pi$
$318$	0	0
$319$	0	0
$320$	−0.115151	−0.00643713
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	−3.70563	−0.204609
$329$	0	0
$330$	0	0
$331$	0	0		−	1.00000i	$-0.5\pi$
$331$	0	0			1.00000i	$0.5\pi$
$332$	−0.610619	−0.0335121
$333$	0	0
$334$	−15.3944	−0.842347
$335$	0	0
$336$	0	0
$337$	−36.0555	−1.96407	−0.982034	−	0.188702i	$-0.939572\pi$
$337$	−36.0555	−1.96407	−0.982034	+	0.188702i	$0.939572\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	0	0
$344$	9.05789	0.488369
$345$	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
$352$	−30.5500	−1.62832
$353$	−28.2798	−1.50518	−0.752590	−	0.658490i	$-0.771196\pi$
$353$	−28.2798	−1.50518	−0.752590	+	0.658490i	$0.771196\pi$
$354$	0	0
$355$	32.8444	1.74320
$356$	15.9490	0.845296
$357$	0	0
$358$	0	0
$359$	−25.2721	−1.33381	−0.666905	−	0.745143i	$-0.732381\pi$
$359$	−25.2721	−1.33381	−0.666905	+	0.745143i	$0.732381\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	−6.28052	−0.330097
$363$	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	$-0.116790\pi$
$373$	36.0555	1.86688	0.933442	+	0.358729i	$0.116790\pi$
$374$	0	0
$375$	0	0
$376$	−31.0278	−1.60013
$377$	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
$383$	−12.9413	−0.661272	−0.330636	−	0.943758i	$-0.607263\pi$
$383$	−12.9413	−0.661272	−0.330636	+	0.943758i	$0.607263\pi$
$384$	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
$392$	−15.8513	−0.800612
$393$	0	0
$394$	4.66106	0.234821
$395$	59.8323	3.01049
$396$	0	0
$397$	0	0		−	1.00000i	$-0.5\pi$
$397$	0	0			1.00000i	$0.5\pi$
$398$	9.05789	0.454031
$399$	0	0
$400$	21.8444	1.09222
$401$	39.8498	1.99001	0.995003	−	0.0998435i	$-0.0318342\pi$
$401$	39.8498	1.99001	0.995003	+	0.0998435i	$0.0318342\pi$
$402$	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	$0.5\pi$
$410$	4.26378	0.210573
$411$	0	0
$412$	−25.6888	−1.26560
$413$	0	0
$414$	0	0
$415$	1.57779	0.0774509
$416$	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	$0.5\pi$
$422$	18.1158	0.881862
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	−10.4222	−0.502603
$431$	−36.0815	−1.73799	−0.868993	−	0.494824i	$-0.835233\pi$
$431$	−36.0815	−1.73799	−0.868993	+	0.494824i	$0.835233\pi$
$432$	0	0
$433$	−36.0555	−1.73272	−0.866359	−	0.499422i	$-0.833546\pi$
$433$	−36.0555	−1.73272	−0.866359	+	0.499422i	$0.833546\pi$
$434$	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	$0.0963387\pi$
$440$	50.7745	2.42058
$441$	0	0
$442$	0	0
$443$	0	0		−	1.00000i	$-0.5\pi$
$443$	0	0			1.00000i	$0.5\pi$
$444$	0	0
$445$	−41.2111	−1.95359
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−36.5770	−1.72618	−0.863088	−	0.505054i	$-0.831473\pi$
$449$	−36.5770	−1.72618	−0.863088	+	0.505054i	$0.831473\pi$
$450$	0	0
$451$	−8.84441	−0.416467
$452$	0	0
$453$	0	0
$454$	10.6611	0.500349
$455$	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
$461$	42.3621	1.97300	0.986499	−	0.163769i	$-0.0523652\pi$
$461$	42.3621	1.97300	0.986499	+	0.163769i	$0.0523652\pi$
$462$	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
$470$	35.7012	1.64677
$471$	0	0
$472$	25.3389	1.16632
$473$	21.6189	0.994039
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	19.0278	0.870309
$479$	3.65316	0.166917	0.0834585	−	0.996511i	$-0.473403\pi$
$479$	3.65316	0.166917	0.0834585	+	0.996511i	$0.473403\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	−29.2389	−1.32904
$485$	0	0
$486$	0	0
$487$	0	0		−	1.00000i	$-0.5\pi$
$487$	0	0			1.00000i	$0.5\pi$
$488$	−16.3293	−0.739194
$489$	0	0
$490$	18.2389	0.823948
$491$	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	$0.5\pi$
$500$	−48.0319	−2.14805
$501$	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
$508$	23.1556	1.02736
$509$	−22.4947	−0.997060	−0.498530	−	0.866872i	$-0.666127\pi$
$509$	−22.4947	−0.997060	−0.498530	+	0.866872i	$0.666127\pi$
$510$	0	0
$511$	0	0
$512$	−18.2135	−0.804930
$513$	0	0
$514$	0	0
$515$	66.3780	2.92497
$516$	0	0
$517$	−74.0555	−3.25696
$518$	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
$539$	−37.8331	−1.62959
$540$	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	$0.704275\pi$
$548$	37.3376	1.59498
$549$	0	0
$550$	−41.4500	−1.76743
$551$	0	0
$552$	0	0
$553$	0	0
$554$	−13.8171	−0.587034
$555$	0	0
$556$	−32.1110	−1.36181
$557$	−30.7920	−1.30470	−0.652349	−	0.757919i	$-0.726216\pi$
$557$	−30.7920	−1.30470	−0.652349	+	0.757919i	$0.726216\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	7.33894	0.309574
$563$	0	0		−	1.00000i	$-0.5\pi$
$563$	0	0			1.00000i	$0.5\pi$
$564$	0	0
$565$	0	0
$566$	−18.1158	−0.761463
$567$	0	0
$568$	17.9277	0.752229
$569$	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	$-0.293752\pi$
$571$	28.8444	1.20710	0.603550	+	0.797325i	$0.293752\pi$
$572$	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	$0.5\pi$
$578$	10.6769	0.444099
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	9.87217	0.407815
$587$	11.9504	0.493246	0.246623	−	0.969111i	$-0.420679\pi$
$587$	11.9504	0.493246	0.246623	+	0.969111i	$0.420679\pi$
$588$	0	0
$589$	0	0
$590$	−29.1555	−1.20031
$591$	0	0
$592$	0	0
$593$	−48.1471	−1.97716	−0.988582	−	0.150683i	$-0.951853\pi$
$593$	−48.1471	−1.97716	−0.988582	+	0.150683i	$0.951853\pi$
$594$	0	0
$595$	0	0
$596$	−28.0494	−1.14895
$597$	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	$-0.546985\pi$
$601$	−7.21110	−0.294147	−0.147074	+	0.989126i	$0.546985\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	75.5511	3.07159
$606$	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
$610$	18.7889	0.760740
$611$	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
$617$	21.5038	0.865709	0.432855	−	0.901464i	$-0.357506\pi$
$617$	21.5038	0.865709	0.432855	+	0.901464i	$0.357506\pi$
$618$	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
$625$	63.0555	2.52222
$626$	4.52894	0.181013
$627$	0	0
$628$	−3.21110	−0.128137
$629$	0	0
$630$	0	0
$631$	0	0		−	1.00000i	$-0.5\pi$
$631$	0	0			1.00000i	$0.5\pi$
$632$	32.6587	1.29909
$633$	0	0
$634$	21.3944	0.849682
$635$	−59.8323	−2.37437
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	46.9722	1.85674
$641$	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
$649$	60.4777	2.37396
$650$	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
$656$	−2.92739	−0.114295
$657$	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
$664$	0.861218	0.0334217
$665$	0	0
$666$	0	0
$667$	0	0
$668$	−39.3544	−1.52267
$669$	0	0
$670$	0	0
$671$	−38.9741	−1.50458
$672$	0	0
$673$	−14.0000	−0.539660	−0.269830	−	0.962908i	$-0.586968\pi$
$673$	−14.0000	−0.539660	−0.269830	+	0.962908i	$0.586968\pi$
$674$	22.6447	0.872242
$675$	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
$683$	32.0481	1.22628	0.613142	−	0.789972i	$-0.289905\pi$
$683$	32.0481	1.22628	0.613142	+	0.789972i	$0.289905\pi$
$684$	0	0
$685$	−96.4777	−3.68622
$686$	0	0
$687$	0	0
$688$	7.15559	0.272804
$689$	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
$695$	82.9725	3.14733
$696$	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
$704$	−0.150016	−0.00565393
$705$	0	0
$706$	17.7611	0.668449
$707$	0	0
$708$	0	0
$709$	0	0		−	1.00000i	$-0.5\pi$
$709$	0	0			1.00000i	$0.5\pi$
$710$	−20.6280	−0.774154
$711$	0	0
$712$	−22.4945	−0.843018
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	15.8722	0.592344
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	0	0
$722$	11.9330	0.444099
$723$	0	0
$724$	−16.0555	−0.596698
$725$	0	0
$726$	0	0
$727$	−14.4222	−0.534890	−0.267445	−	0.963573i	$-0.586179\pi$
$727$	−14.4222	−0.534890	−0.267445	+	0.963573i	$0.586179\pi$
$728$	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	$0.5\pi$
$734$	−5.02441	−0.185454
$735$	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
$743$	−41.8666	−1.53594	−0.767968	−	0.640488i	$-0.778732\pi$
$743$	−41.8666	−1.53594	−0.767968	+	0.640488i	$0.778732\pi$
$744$	0	0
$745$	72.4777	2.65538
$746$	−22.6447	−0.829082
$747$	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	$0.239608\pi$
$752$	−24.5114	−0.893840
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	36.0555	1.31046	0.655230	−	0.755429i	$-0.272572\pi$
$757$	36.0555	1.31046	0.655230	+	0.755429i	$0.272572\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	53.1715	1.92747	0.963733	−	0.266869i	$-0.0859890\pi$
$761$	53.1715	1.92747	0.963733	+	0.266869i	$0.0859890\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	8.12783	0.293671
$767$	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
$773$	−47.3865	−1.70437	−0.852186	−	0.523238i	$-0.824724\pi$
$773$	−47.3865	−1.70437	−0.852186	+	0.523238i	$0.824724\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	42.7889	1.53111
$782$	0	0
$783$	0	0
$784$	−12.5223	−0.447224
$785$	8.29725	0.296142
$786$	0	0
$787$	0	0		−	1.00000i	$-0.5\pi$
$787$	0	0			1.00000i	$0.5\pi$
$788$	11.9155	0.424474
$789$	0	0
$790$	−37.5778	−1.33696
$791$	0	0
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	23.1556	0.820728
$797$	0	0		−	1.00000i	$-0.5\pi$
$797$	0	0			1.00000i	$0.5\pi$
$798$	0	0
$799$	0	0
$800$	−69.0228	−2.44033
$801$	0	0
$802$	−25.0278	−0.883761
$803$	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
$820$	10.8999	0.380642
$821$	40.6105	1.41732	0.708658	−	0.705552i	$-0.249301\pi$
$821$	40.6105	1.41732	0.708658	+	0.705552i	$0.249301\pi$
$822$	0	0
$823$	−56.0000	−1.95204	−0.976019	−	0.217687i	$-0.930149\pi$
$823$	−56.0000	−1.95204	−0.976019	+	0.217687i	$0.930149\pi$
$824$	36.2316	1.26219
$825$	0	0
$826$	0	0
$827$	55.1882	1.91908	0.959542	−	0.281566i	$-0.0908540\pi$
$827$	55.1882	1.91908	0.959542	+	0.281566i	$0.0908540\pi$
$828$	0	0
$829$	50.4777	1.75316	0.876582	−	0.481253i	$-0.159818\pi$
$829$	50.4777	1.75316	0.876582	+	0.481253i	$0.159818\pi$
$830$	−0.990937	−0.0343959
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	101.689	3.51909
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−57.7005	−1.99204	−0.996020	−	0.0891249i	$-0.971593\pi$
$839$	−57.7005	−1.99204	−0.996020	+	0.0891249i	$0.971593\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	0	0
$843$	0	0
$844$	46.3112	1.59410
$845$	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	$-0.0567332\pi$
$859$	57.6888	1.96832	0.984159	+	0.177290i	$0.0567332\pi$
$860$	−26.6433	−0.908530
$861$	0	0
$862$	22.6611	0.771839
$863$	−9.43821	−0.321280	−0.160640	−	0.987013i	$-0.551356\pi$
$863$	−9.43821	−0.321280	−0.160640	+	0.987013i	$0.551356\pi$
$864$	0	0
$865$	0	0
$866$	22.6447	0.769499
$867$	0	0
$868$	0	0
$869$	77.9481	2.64421
$870$	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	$0.5\pi$
$878$	−25.1221	−0.847829
$879$	0	0
$880$	40.1110	1.35214
$881$	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	$-0.338694\pi$
$883$	28.8444	0.970692	0.485346	+	0.874322i	$0.338694\pi$
$884$	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
$890$	25.8827	0.867590
$891$	0	0
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	22.9722	0.766594
$899$	0	0
$900$	0	0
$901$	0	0
$902$	5.55475	0.184953
$903$	0	0
$904$	0	0
$905$	41.4863	1.37905
$906$	0	0
$907$	−57.6888	−1.91553	−0.957763	−	0.287559i	$-0.907156\pi$
$907$	−57.6888	−1.91553	−0.957763	+	0.287559i	$0.907156\pi$
$908$	27.2539	0.904454
$909$	0	0
$910$	0	0
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	0	0
$913$	2.05551	0.0680275
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	14.4222	0.475745	0.237872	−	0.971296i	$-0.423550\pi$
$919$	14.4222	0.475745	0.237872	+	0.971296i	$0.423550\pi$
$920$	0	0
$921$	0	0
$922$	−26.6056	−0.876207
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−51.4199	−1.68703	−0.843517	−	0.537103i	$-0.819519\pi$
$929$	−51.4199	−1.68703	−0.843517	+	0.537103i	$0.819519\pi$
$930$	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	$-0.808553\pi$
$937$	−50.4777	−1.64904	−0.824518	+	0.565836i	$0.808553\pi$
$938$	0	0
$939$	0	0
$940$	91.2666	2.97679
$941$	55.6837	1.81524	0.907619	−	0.419796i	$-0.137898\pi$
$941$	55.6837	1.81524	0.907619	+	0.419796i	$0.137898\pi$
$942$	0	0
$943$	0	0
$944$	20.0174	0.651510
$945$	0	0
$946$	−13.5778	−0.441452
$947$	26.2630	0.853433	0.426717	−	0.904385i	$-0.359670\pi$
$947$	26.2630	0.853433	0.426717	+	0.904385i	$0.359670\pi$
$948$	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
$956$	48.6426	1.57321
$957$	0	0
$958$	−2.29437	−0.0741278
$959$	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	$0.5\pi$
$968$	41.2385	1.32546
$969$	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
$976$	−12.8999	−0.412916
$977$	61.4688	1.96656	0.983280	−	0.182100i	$-0.0582895\pi$
$977$	61.4688	1.96656	0.983280	+	0.182100i	$0.0582895\pi$
$978$	0	0
$979$	−53.6888	−1.71590
$980$	46.6258	1.48941
$981$	0	0
$982$	0	0
$983$	−34.5603	−1.10230	−0.551151	−	0.834406i	$-0.685811\pi$
$983$	−34.5603	−1.10230	−0.551151	+	0.834406i	$0.685811\pi$
$984$	0	0
$985$	−30.7889	−0.981016
$986$	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
$995$	−59.8323	−1.89681
$996$	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.1.2		4
3.2	odd	2	inner	1521.2.a.t.1.3		4
13.5	odd	4		117.2.b.b.64.3	yes	4
13.8	odd	4		117.2.b.b.64.2	✓	4
13.12	even	2	inner	1521.2.a.t.1.3		4
39.5	even	4		117.2.b.b.64.2	✓	4
39.8	even	4		117.2.b.b.64.3	yes	4
39.38	odd	2	CM	1521.2.a.t.1.2		4
52.31	even	4		1872.2.c.k.1585.1		4
52.47	even	4		1872.2.c.k.1585.4		4
156.47	odd	4		1872.2.c.k.1585.1		4
156.83	odd	4		1872.2.c.k.1585.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
117.2.b.b.64.2	✓	4	13.8	odd	4
117.2.b.b.64.2	✓	4	39.5	even	4
117.2.b.b.64.3	yes	4	13.5	odd	4
117.2.b.b.64.3	yes	4	39.8	even	4
1521.2.a.t.1.2		4	1.1	even	1	trivial
1521.2.a.t.1.2		4	39.38	odd	2	CM
1521.2.a.t.1.3		4	3.2	odd	2	inner
1521.2.a.t.1.3		4	13.12	even	2	inner
1872.2.c.k.1585.1		4	52.31	even	4
1872.2.c.k.1585.1		4	156.47	odd	4
1872.2.c.k.1585.4		4	52.47	even	4
1872.2.c.k.1585.4		4	156.83	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)

\(f(q)\)	\(=\)	\(q-0.628052 q^{2} -1.60555 q^{4} +4.14863 q^{5} +2.26447 q^{8} +O(q^{10})\)	\(q-0.628052 q^{2} -1.60555 q^{4} +4.14863 q^{5} +2.26447 q^{8} -2.60555 q^{10} +5.40473 q^{11} +1.78890 q^{16} -6.66083 q^{20} -3.39445 q^{22} +12.2111 q^{25} -5.65246 q^{32} +9.39445 q^{40} -1.63642 q^{41} +4.00000 q^{43} -8.67757 q^{44} -13.7020 q^{47} -7.00000 q^{49} -7.66920 q^{50} +22.4222 q^{55} +11.1898 q^{59} -7.21110 q^{61} -0.0277564 q^{64} +7.91694 q^{71} +14.4222 q^{79} +7.42147 q^{80} +1.02776 q^{82} +0.380317 q^{83} -2.51221 q^{86} +12.2389 q^{88} -9.93367 q^{89} +8.60555 q^{94} +4.39636 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8 q^{4}+O(q^{10}) \) 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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