LMFDB - Embedded newform 153.2.d.b.118.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [153,2,Mod(118,153)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(153, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("153.118"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 153 = 3^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	153.d (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

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

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

Self dual:	no
Analytic conductor:	$1.22171115093$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-17}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 17 $ x^2 + 17
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			118.1
Root			$-4.12311i$ of defining polynomial
Character	$\chi$	$=$	153.118
Dual form			153.2.d.b.118.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000 q^{4} -4.12311i q^{5} -4.12311i q^{11} -1.00000 q^{13} +4.00000 q^{16} +4.12311i q^{17} +5.00000 q^{19} +8.24621i q^{20} -4.12311i q^{23} -12.0000 q^{25} +8.24621i q^{29} -4.12311i q^{41} +11.0000 q^{43} +8.24621i q^{44} +7.00000 q^{49} +2.00000 q^{52} -17.0000 q^{55} -8.00000 q^{64} +4.12311i q^{65} +8.00000 q^{67} -8.24621i q^{68} -16.4924i q^{71} -10.0000 q^{76} -16.4924i q^{80} +17.0000 q^{85} +8.24621i q^{92} -20.6155i q^{95} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{4} - 2 q^{13} + 8 q^{16} + 10 q^{19} - 24 q^{25} + 22 q^{43} + 14 q^{49} + 4 q^{52} - 34 q^{55} - 16 q^{64} + 16 q^{67} - 20 q^{76} + 34 q^{85}+O(q^{100}) $ 2 * q - 4 * q^4 - 2 * q^13 + 8 * q^16 + 10 * q^19 - 24 * q^25 + 22 * q^43 + 14 * q^49 + 4 * q^52 - 34 * q^55 - 16 * q^64 + 16 * q^67 - 20 * q^76 + 34 * q^85

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/153\mathbb{Z}\right)^\times$.

$n$	$37$	$137$
$\chi(n)$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0		−	1.00000i	$-0.5\pi$
$2$	0	0			1.00000i	$0.5\pi$
$3$	0	0
$3$	0	0
$4$	−2.00000	−1.00000
$5$	− 4.12311i	− 1.84391i	−0.387298	−	0.921954i	$-0.626592\pi$
$5$	− 4.12311i	− 1.84391i	0.387298	−	0.921954i	$-0.373408\pi$
$6$	0	0
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 4.12311i	− 1.24316i	−0.783349	−	0.621582i	$-0.786490\pi$
$11$	− 4.12311i	− 1.24316i	0.783349	−	0.621582i	$-0.213510\pi$
$12$	0	0
$13$	−1.00000	−0.277350	−0.138675	−	0.990338i	$-0.544284\pi$
$13$	−1.00000	−0.277350	−0.138675	+	0.990338i	$0.544284\pi$
$14$	0	0
$15$	0	0
$16$	4.00000	1.00000
$17$	4.12311i	1.00000i
$17$	4.12311i	1.00000i
$18$	0	0
$19$	5.00000	1.14708	0.573539	−	0.819178i	$-0.305570\pi$
$19$	5.00000	1.14708	0.573539	+	0.819178i	$0.305570\pi$
$20$	8.24621i	1.84391i
$21$	0	0
$22$	0	0
$23$	− 4.12311i	− 0.859727i	−0.902894	−	0.429863i	$-0.858562\pi$
$23$	− 4.12311i	− 0.859727i	0.902894	−	0.429863i	$-0.141438\pi$
$24$	0	0
$25$	−12.0000	−2.40000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	8.24621i	1.53128i	0.643268	+	0.765641i	$0.277578\pi$
$29$	8.24621i	1.53128i	−0.643268	+	0.765641i	$0.722422\pi$
$30$	0	0
$31$	0	0	1.00000			$0$
$31$	0	0	−1.00000			$\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 4.12311i	− 0.643921i	−0.946753	−	0.321960i	$-0.895658\pi$
$41$	− 4.12311i	− 0.643921i	0.946753	−	0.321960i	$-0.104342\pi$
$42$	0	0
$43$	11.0000	1.67748	0.838742	−	0.544529i	$-0.183292\pi$
$43$	11.0000	1.67748	0.838742	+	0.544529i	$0.183292\pi$
$44$	8.24621i	1.24316i
$45$	0	0
$46$	0	0
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	7.00000	1.00000
$50$	0	0
$51$	0	0
$52$	2.00000	0.277350
$53$	0	0		−	1.00000i	$-0.5\pi$
$53$	0	0			1.00000i	$0.5\pi$
$54$	0	0
$55$	−17.0000	−2.29228
$56$	0	0
$57$	0	0
$58$	0	0
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	0	0
$63$	0	0
$64$	−8.00000	−1.00000
$65$	4.12311i	0.511408i
$66$	0	0
$67$	8.00000	0.977356	0.488678	−	0.872464i	$-0.337479\pi$
$67$	8.00000	0.977356	0.488678	+	0.872464i	$0.337479\pi$
$68$	− 8.24621i	− 1.00000i
$69$	0	0
$70$	0	0
$71$	− 16.4924i	− 1.95729i	−0.205557	−	0.978645i	$-0.565900\pi$
$71$	− 16.4924i	− 1.95729i	0.205557	−	0.978645i	$-0.434100\pi$
$72$	0	0
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	0	0
$75$	0	0
$76$	−10.0000	−1.14708
$77$	0	0
$78$	0	0
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	− 16.4924i	− 1.84391i
$81$	0	0
$82$	0	0
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	0	0
$85$	17.0000	1.84391
$86$	0	0
$87$	0	0
$88$	0	0
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	0	0
$91$	0	0
$92$	8.24621i	0.859727i
$93$	0	0
$94$	0	0
$95$	− 20.6155i	− 2.11511i
$96$	0	0
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	0	0
$99$	0	0
$100$	24.0000	2.40000
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	0	0
$103$	−19.0000	−1.87213	−0.936063	−	0.351833i	$-0.885559\pi$
$103$	−19.0000	−1.87213	−0.936063	+	0.351833i	$0.885559\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	20.6155i	1.99298i	0.0837218	+	0.996489i	$0.473319\pi$
$107$	20.6155i	1.99298i	−0.0837218	+	0.996489i	$0.526681\pi$
$108$	0	0
$109$	0	0	1.00000			$0$
$109$	0	0	−1.00000			$\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	20.6155i	1.93935i	0.244406	+	0.969673i	$0.421407\pi$
$113$	20.6155i	1.93935i	−0.244406	+	0.969673i	$0.578593\pi$
$114$	0	0
$115$	−17.0000	−1.58526
$116$	− 16.4924i	− 1.53128i
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−6.00000	−0.545455
$122$	0	0
$123$	0	0
$124$	0	0
$125$	28.8617i	2.58147i
$126$	0	0
$127$	−7.00000	−0.621150	−0.310575	−	0.950549i	$-0.600522\pi$
$127$	−7.00000	−0.621150	−0.310575	+	0.950549i	$0.600522\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 4.12311i	− 0.360237i	−0.983645	−	0.180119i	$-0.942352\pi$
$131$	− 4.12311i	− 0.360237i	0.983645	−	0.180119i	$-0.0576482\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	0	0		−	1.00000i	$-0.5\pi$
$137$	0	0			1.00000i	$0.5\pi$
$138$	0	0
$139$	0	0	1.00000			$0$
$139$	0	0	−1.00000			$\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	4.12311i	0.344791i
$144$	0	0
$145$	34.0000	2.82355
$146$	0	0
$147$	0	0
$148$	0	0
$149$	0	0		−	1.00000i	$-0.5\pi$
$149$	0	0			1.00000i	$0.5\pi$
$150$	0	0
$151$	20.0000	1.62758	0.813788	−	0.581161i	$-0.197401\pi$
$151$	20.0000	1.62758	0.813788	+	0.581161i	$0.197401\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−13.0000	−1.03751	−0.518756	−	0.854922i	$-0.673605\pi$
$157$	−13.0000	−1.03751	−0.518756	+	0.854922i	$0.673605\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	0	0	1.00000			$0$
$163$	0	0	−1.00000			$\pi$
$164$	8.24621i	0.643921i
$165$	0	0
$166$	0	0
$167$	20.6155i	1.59528i	0.603136	+	0.797639i	$0.293918\pi$
$167$	20.6155i	1.59528i	−0.603136	+	0.797639i	$0.706082\pi$
$168$	0	0
$169$	−12.0000	−0.923077
$170$	0	0
$171$	0	0
$172$	−22.0000	−1.67748
$173$	− 4.12311i	− 0.313474i	−0.987640	−	0.156737i	$-0.949903\pi$
$173$	− 4.12311i	− 0.313474i	0.987640	−	0.156737i	$-0.0500975\pi$
$174$	0	0
$175$	0	0
$176$	− 16.4924i	− 1.24316i
$177$	0	0
$178$	0	0
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	0	0
$181$	0	0	1.00000			$0$
$181$	0	0	−1.00000			$\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	17.0000	1.24316
$188$	0	0
$189$	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	1.00000			$0$
$193$	0	0	−1.00000			$\pi$
$194$	0	0
$195$	0	0
$196$	−14.0000	−1.00000
$197$	20.6155i	1.46880i	0.678719	+	0.734398i	$0.262535\pi$
$197$	20.6155i	1.46880i	−0.678719	+	0.734398i	$0.737465\pi$
$198$	0	0
$199$	0	0	1.00000			$0$
$199$	0	0	−1.00000			$\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−17.0000	−1.18733
$206$	0	0
$207$	0	0
$208$	−4.00000	−0.277350
$209$	− 20.6155i	− 1.42601i
$210$	0	0
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	− 45.3542i	− 3.09313i
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	34.0000	2.29228
$221$	− 4.12311i	− 0.277350i
$222$	0	0
$223$	29.0000	1.94198	0.970992	−	0.239113i	$-0.0768565\pi$
$223$	29.0000	1.94198	0.970992	+	0.239113i	$0.0768565\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 28.8617i	− 1.91562i	−0.287401	−	0.957810i	$-0.592791\pi$
$227$	− 28.8617i	− 1.91562i	0.287401	−	0.957810i	$-0.407209\pi$
$228$	0	0
$229$	−10.0000	−0.660819	−0.330409	−	0.943838i	$-0.607187\pi$
$229$	−10.0000	−0.660819	−0.330409	+	0.943838i	$0.607187\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	20.6155i	1.35057i	0.737558	+	0.675284i	$0.235979\pi$
$233$	20.6155i	1.35057i	−0.737558	+	0.675284i	$0.764021\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 28.8617i	− 1.84391i
$246$	0	0
$247$	−5.00000	−0.318142
$248$	0	0
$249$	0	0
$250$	0	0
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	0	0
$253$	−17.0000	−1.06878
$254$	0	0
$255$	0	0
$256$	16.0000	1.00000
$257$	0	0		−	1.00000i	$-0.5\pi$
$257$	0	0			1.00000i	$0.5\pi$
$258$	0	0
$259$	0	0
$260$	− 8.24621i	− 0.511408i
$261$	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
$268$	−16.0000	−0.977356
$269$	− 28.8617i	− 1.75973i	−0.475223	−	0.879866i	$-0.657632\pi$
$269$	− 28.8617i	− 1.75973i	0.475223	−	0.879866i	$-0.342368\pi$
$270$	0	0
$271$	−25.0000	−1.51864	−0.759321	−	0.650716i	$-0.774469\pi$
$271$	−25.0000	−1.51864	−0.759321	+	0.650716i	$0.774469\pi$
$272$	16.4924i	1.00000i
$273$	0	0
$274$	0	0
$275$	49.4773i	2.98359i
$276$	0	0
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	0	0		−	1.00000i	$-0.5\pi$
$281$	0	0			1.00000i	$0.5\pi$
$282$	0	0
$283$	0	0	1.00000			$0$
$283$	0	0	−1.00000			$\pi$
$284$	32.9848i	1.95729i
$285$	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
$293$	0	0		−	1.00000i	$-0.5\pi$
$293$	0	0			1.00000i	$0.5\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	4.12311i	0.238445i
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	20.0000	1.14708
$305$	0	0
$306$	0	0
$307$	32.0000	1.82634	0.913168	−	0.407583i	$-0.133628\pi$
$307$	32.0000	1.82634	0.913168	+	0.407583i	$0.133628\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	− 16.4924i	− 0.935199i	−0.883940	−	0.467600i	$-0.845119\pi$
$311$	− 16.4924i	− 0.935199i	0.883940	−	0.467600i	$-0.154881\pi$
$312$	0	0
$313$	0	0	1.00000			$0$
$313$	0	0	−1.00000			$\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	8.24621i	0.463153i	0.972817	+	0.231577i	$0.0743884\pi$
$317$	8.24621i	0.463153i	−0.972817	+	0.231577i	$0.925612\pi$
$318$	0	0
$319$	34.0000	1.90363
$320$	32.9848i	1.84391i
$321$	0	0
$322$	0	0
$323$	20.6155i	1.14708i
$324$	0	0
$325$	12.0000	0.665640
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−7.00000	−0.384755	−0.192377	−	0.981321i	$-0.561620\pi$
$331$	−7.00000	−0.384755	−0.192377	+	0.981321i	$0.561620\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	− 32.9848i	− 1.80215i
$336$	0	0
$337$	0	0	1.00000			$0$
$337$	0	0	−1.00000			$\pi$
$338$	0	0
$339$	0	0
$340$	−34.0000	−1.84391
$341$	0	0
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	32.9848i	1.77072i	0.464907	+	0.885360i	$0.346088\pi$
$347$	32.9848i	1.77072i	−0.464907	+	0.885360i	$0.653912\pi$
$348$	0	0
$349$	11.0000	0.588817	0.294408	−	0.955680i	$-0.404877\pi$
$349$	11.0000	0.588817	0.294408	+	0.955680i	$0.404877\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	0	0		−	1.00000i	$-0.5\pi$
$353$	0	0			1.00000i	$0.5\pi$
$354$	0	0
$355$	−68.0000	−3.60907
$356$	0	0
$357$	0	0
$358$	0	0
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	0	0
$361$	6.00000	0.315789
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	0	0	1.00000			$0$
$367$	0	0	−1.00000			$\pi$
$368$	− 16.4924i	− 0.859727i
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	26.0000	1.34623	0.673114	−	0.739538i	$-0.264956\pi$
$373$	26.0000	1.34623	0.673114	+	0.739538i	$0.264956\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 8.24621i	− 0.424701i
$378$	0	0
$379$	0	0	1.00000			$0$
$379$	0	0	−1.00000			$\pi$
$380$	41.2311i	2.11511i
$381$	0	0
$382$	0	0
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	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
$391$	17.0000	0.859727
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	0	0	1.00000			$0$
$397$	0	0	−1.00000			$\pi$
$398$	0	0
$399$	0	0
$400$	−48.0000	−2.40000
$401$	− 4.12311i	− 0.205898i	−0.994687	−	0.102949i	$-0.967172\pi$
$401$	− 4.12311i	− 0.205898i	0.994687	−	0.102949i	$-0.0328279\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−19.0000	−0.939490	−0.469745	−	0.882802i	$-0.655654\pi$
$409$	−19.0000	−0.939490	−0.469745	+	0.882802i	$0.655654\pi$
$410$	0	0
$411$	0	0
$412$	38.0000	1.87213
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	32.9848i	1.61142i	0.592314	+	0.805708i	$0.298215\pi$
$419$	32.9848i	1.61142i	−0.592314	+	0.805708i	$0.701785\pi$
$420$	0	0
$421$	35.0000	1.70580	0.852898	−	0.522078i	$-0.174843\pi$
$421$	35.0000	1.70580	0.852898	+	0.522078i	$0.174843\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 49.4773i	− 2.40000i
$426$	0	0
$427$	0	0
$428$	− 41.2311i	− 1.99298i
$429$	0	0
$430$	0	0
$431$	− 16.4924i	− 0.794412i	−0.917729	−	0.397206i	$-0.869980\pi$
$431$	− 16.4924i	− 0.794412i	0.917729	−	0.397206i	$-0.130020\pi$
$432$	0	0
$433$	41.0000	1.97033	0.985167	−	0.171598i	$-0.0548929\pi$
$433$	41.0000	1.97033	0.985167	+	0.171598i	$0.0548929\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 20.6155i	− 0.986174i
$438$	0	0
$439$	0	0	1.00000			$0$
$439$	0	0	−1.00000			$\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	0	0		−	1.00000i	$-0.5\pi$
$443$	0	0			1.00000i	$0.5\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	8.24621i	0.389163i	0.980886	+	0.194581i	$0.0623348\pi$
$449$	8.24621i	0.389163i	−0.980886	+	0.194581i	$0.937665\pi$
$450$	0	0
$451$	−17.0000	−0.800499
$452$	− 41.2311i	− 1.93935i
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−37.0000	−1.73079	−0.865393	−	0.501093i	$-0.832931\pi$
$457$	−37.0000	−1.73079	−0.865393	+	0.501093i	$0.832931\pi$
$458$	0	0
$459$	0	0
$460$	34.0000	1.58526
$461$	0	0		−	1.00000i	$-0.5\pi$
$461$	0	0			1.00000i	$0.5\pi$
$462$	0	0
$463$	−4.00000	−0.185896	−0.0929479	−	0.995671i	$-0.529629\pi$
$463$	−4.00000	−0.185896	−0.0929479	+	0.995671i	$0.529629\pi$
$464$	32.9848i	1.53128i
$465$	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
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 45.3542i	− 2.08539i
$474$	0	0
$475$	−60.0000	−2.75299
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 28.8617i	− 1.31873i	−0.751825	−	0.659363i	$-0.770826\pi$
$479$	− 28.8617i	− 1.31873i	0.751825	−	0.659363i	$-0.229174\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	12.0000	0.545455
$485$	0	0
$486$	0	0
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	0	0		−	1.00000i	$-0.5\pi$
$491$	0	0			1.00000i	$0.5\pi$
$492$	0	0
$493$	−34.0000	−1.53128
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	0	0	1.00000			$0$
$499$	0	0	−1.00000			$\pi$
$500$	− 57.7235i	− 2.58147i
$501$	0	0
$502$	0	0
$503$	20.6155i	0.919201i	0.888126	+	0.459600i	$0.152007\pi$
$503$	20.6155i	0.919201i	−0.888126	+	0.459600i	$0.847993\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	14.0000	0.621150
$509$	0	0		−	1.00000i	$-0.5\pi$
$509$	0	0			1.00000i	$0.5\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0	0
$514$	0	0
$515$	78.3390i	3.45203i
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	45.3542i	1.98700i	0.113824	+	0.993501i	$0.463690\pi$
$521$	45.3542i	1.98700i	−0.113824	+	0.993501i	$0.536310\pi$
$522$	0	0
$523$	−16.0000	−0.699631	−0.349816	−	0.936819i	$-0.613756\pi$
$523$	−16.0000	−0.699631	−0.349816	+	0.936819i	$0.613756\pi$
$524$	8.24621i	0.360237i
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	6.00000	0.260870
$530$	0	0
$531$	0	0
$532$	0	0
$533$	4.12311i	0.178592i
$534$	0	0
$535$	85.0000	3.67487
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 28.8617i	− 1.24316i
$540$	0	0
$541$	0	0	1.00000			$0$
$541$	0	0	−1.00000			$\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	0	0	1.00000			$0$
$547$	0	0	−1.00000			$\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	41.2311i	1.75650i
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	0	0		−	1.00000i	$-0.5\pi$
$557$	0	0			1.00000i	$0.5\pi$
$558$	0	0
$559$	−11.0000	−0.465250
$560$	0	0
$561$	0	0
$562$	0	0
$563$	0	0		−	1.00000i	$-0.5\pi$
$563$	0	0			1.00000i	$0.5\pi$
$564$	0	0
$565$	85.0000	3.57598
$566$	0	0
$567$	0	0
$568$	0	0
$569$	0	0		−	1.00000i	$-0.5\pi$
$569$	0	0			1.00000i	$0.5\pi$
$570$	0	0
$571$	0	0	1.00000			$0$
$571$	0	0	−1.00000			$\pi$
$572$	− 8.24621i	− 0.344791i
$573$	0	0
$574$	0	0
$575$	49.4773i	2.06334i
$576$	0	0
$577$	−43.0000	−1.79011	−0.895057	−	0.445952i	$-0.852865\pi$
$577$	−43.0000	−1.79011	−0.895057	+	0.445952i	$0.852865\pi$
$578$	0	0
$579$	0	0
$580$	−68.0000	−2.82355
$581$	0	0
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	0	0		−	1.00000i	$-0.5\pi$
$587$	0	0			1.00000i	$0.5\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	0	0		−	1.00000i	$-0.5\pi$
$593$	0	0			1.00000i	$0.5\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	0	0
$601$	0	0	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	0	0
$603$	0	0
$604$	−40.0000	−1.62758
$605$	24.7386i	1.00577i
$606$	0	0
$607$	0	0	1.00000			$0$
$607$	0	0	−1.00000			$\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−49.0000	−1.97909	−0.989546	−	0.144220i	$-0.953933\pi$
$613$	−49.0000	−1.97909	−0.989546	+	0.144220i	$0.953933\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 41.2311i	− 1.65990i	−0.557838	−	0.829950i	$-0.688369\pi$
$617$	− 41.2311i	− 1.65990i	0.557838	−	0.829950i	$-0.311631\pi$
$618$	0	0
$619$	0	0	1.00000			$0$
$619$	0	0	−1.00000			$\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	59.0000	2.36000
$626$	0	0
$627$	0	0
$628$	26.0000	1.03751
$629$	0	0
$630$	0	0
$631$	5.00000	0.199047	0.0995234	−	0.995035i	$-0.468268\pi$
$631$	5.00000	0.199047	0.0995234	+	0.995035i	$0.468268\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	28.8617i	1.14534i
$636$	0	0
$637$	−7.00000	−0.277350
$638$	0	0
$639$	0	0
$640$	0	0
$641$	45.3542i	1.79138i	0.444677	+	0.895691i	$0.353318\pi$
$641$	45.3542i	1.79138i	−0.444677	+	0.895691i	$0.646682\pi$
$642$	0	0
$643$	0	0	1.00000			$0$
$643$	0	0	−1.00000			$\pi$
$644$	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
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	20.6155i	0.806748i	0.915035	+	0.403374i	$0.132163\pi$
$653$	20.6155i	0.806748i	−0.915035	+	0.403374i	$0.867837\pi$
$654$	0	0
$655$	−17.0000	−0.664245
$656$	− 16.4924i	− 0.643921i
$657$	0	0
$658$	0	0
$659$	0	0		−	1.00000i	$-0.5\pi$
$659$	0	0			1.00000i	$0.5\pi$
$660$	0	0
$661$	−37.0000	−1.43913	−0.719567	−	0.694423i	$-0.755660\pi$
$661$	−37.0000	−1.43913	−0.719567	+	0.694423i	$0.755660\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	34.0000	1.31649
$668$	− 41.2311i	− 1.59528i
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	0	0	1.00000			$0$
$673$	0	0	−1.00000			$\pi$
$674$	0	0
$675$	0	0
$676$	24.0000	0.923077
$677$	− 28.8617i	− 1.10925i	−0.832102	−	0.554623i	$-0.812862\pi$
$677$	− 28.8617i	− 1.10925i	0.832102	−	0.554623i	$-0.187138\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	45.3542i	1.73543i	0.497063	+	0.867714i	$0.334412\pi$
$683$	45.3542i	1.73543i	−0.497063	+	0.867714i	$0.665588\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	44.0000	1.67748
$689$	0	0
$690$	0	0
$691$	0	0	1.00000			$0$
$691$	0	0	−1.00000			$\pi$
$692$	8.24621i	0.313474i
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	17.0000	0.643921
$698$	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
$704$	32.9848i	1.24316i
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	0	0	1.00000			$0$
$709$	0	0	−1.00000			$\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	0	0
$714$	0	0
$715$	17.0000	0.635764
$716$	0	0
$717$	0	0
$718$	0	0
$719$	− 53.6004i	− 1.99896i	−0.0322973	−	0.999478i	$-0.510282\pi$
$719$	− 53.6004i	− 1.99896i	0.0322973	−	0.999478i	$-0.489718\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 98.9545i	− 3.67508i
$726$	0	0
$727$	−52.0000	−1.92857	−0.964287	−	0.264861i	$-0.914674\pi$
$727$	−52.0000	−1.92857	−0.964287	+	0.264861i	$0.914674\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	45.3542i	1.67748i
$732$	0	0
$733$	−46.0000	−1.69905	−0.849524	−	0.527549i	$-0.823111\pi$
$733$	−46.0000	−1.69905	−0.849524	+	0.527549i	$0.823111\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 32.9848i	− 1.21501i
$738$	0	0
$739$	41.0000	1.50821	0.754105	−	0.656754i	$-0.228071\pi$
$739$	41.0000	1.50821	0.754105	+	0.656754i	$0.228071\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 16.4924i	− 0.605048i	−0.953142	−	0.302524i	$-0.902171\pi$
$743$	− 16.4924i	− 0.605048i	0.953142	−	0.302524i	$-0.0978293\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	−34.0000	−1.24316
$749$	0	0
$750$	0	0
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 82.4621i	− 3.00110i
$756$	0	0
$757$	23.0000	0.835949	0.417975	−	0.908459i	$-0.362740\pi$
$757$	23.0000	0.835949	0.417975	+	0.908459i	$0.362740\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	0	0		−	1.00000i	$-0.5\pi$
$761$	0	0			1.00000i	$0.5\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−55.0000	−1.98335	−0.991675	−	0.128763i	$-0.958899\pi$
$769$	−55.0000	−1.98335	−0.991675	+	0.128763i	$0.958899\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	0	0		−	1.00000i	$-0.5\pi$
$773$	0	0			1.00000i	$0.5\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 20.6155i	− 0.738628i
$780$	0	0
$781$	−68.0000	−2.43323
$782$	0	0
$783$	0	0
$784$	28.0000	1.00000
$785$	53.6004i	1.91308i
$786$	0	0
$787$	0	0	1.00000			$0$
$787$	0	0	−1.00000			$\pi$
$788$	− 41.2311i	− 1.46880i
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	0	0		−	1.00000i	$-0.5\pi$
$797$	0	0			1.00000i	$0.5\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 53.6004i	− 1.88449i	−0.334926	−	0.942244i	$-0.608711\pi$
$809$	− 53.6004i	− 1.88449i	0.334926	−	0.942244i	$-0.391289\pi$
$810$	0	0
$811$	0	0	1.00000			$0$
$811$	0	0	−1.00000			$\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	55.0000	1.92421
$818$	0	0
$819$	0	0
$820$	34.0000	1.18733
$821$	− 4.12311i	− 0.143897i	−0.997408	−	0.0719487i	$-0.977078\pi$
$821$	− 4.12311i	− 0.143897i	0.997408	−	0.0719487i	$-0.0229218\pi$
$822$	0	0
$823$	0	0	1.00000			$0$
$823$	0	0	−1.00000			$\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	20.6155i	0.716872i	0.933554	+	0.358436i	$0.116690\pi$
$827$	20.6155i	0.716872i	−0.933554	+	0.358436i	$0.883310\pi$
$828$	0	0
$829$	50.0000	1.73657	0.868286	−	0.496064i	$-0.165222\pi$
$829$	50.0000	1.73657	0.868286	+	0.496064i	$0.165222\pi$
$830$	0	0
$831$	0	0
$832$	8.00000	0.277350
$833$	28.8617i	1.00000i
$834$	0	0
$835$	85.0000	2.94155
$836$	41.2311i	1.42601i
$837$	0	0
$838$	0	0
$839$	− 28.8617i	− 0.996418i	−0.867057	−	0.498209i	$-0.833991\pi$
$839$	− 28.8617i	− 0.996418i	0.867057	−	0.498209i	$-0.166009\pi$
$840$	0	0
$841$	−39.0000	−1.34483
$842$	0	0
$843$	0	0
$844$	0	0
$845$	49.4773i	1.70207i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	0	0	1.00000			$0$
$853$	0	0	−1.00000			$\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 41.2311i	− 1.40843i	−0.709989	−	0.704213i	$-0.751300\pi$
$857$	− 41.2311i	− 1.40843i	0.709989	−	0.704213i	$-0.248700\pi$
$858$	0	0
$859$	−40.0000	−1.36478	−0.682391	−	0.730987i	$-0.739060\pi$
$859$	−40.0000	−1.36478	−0.682391	+	0.730987i	$0.739060\pi$
$860$	90.7083i	3.09313i
$861$	0	0
$862$	0	0
$863$	0	0		−	1.00000i	$-0.5\pi$
$863$	0	0			1.00000i	$0.5\pi$
$864$	0	0
$865$	−17.0000	−0.578017
$866$	0	0
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−8.00000	−0.271070
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	0	0	1.00000			$0$
$877$	0	0	−1.00000			$\pi$
$878$	0	0
$879$	0	0
$880$	−68.0000	−2.29228
$881$	57.7235i	1.94475i	0.233417	+	0.972377i	$0.425009\pi$
$881$	57.7235i	1.94475i	−0.233417	+	0.972377i	$0.574991\pi$
$882$	0	0
$883$	59.0000	1.98551	0.992754	−	0.120164i	$-0.0383421\pi$
$883$	59.0000	1.98551	0.992754	+	0.120164i	$0.0383421\pi$
$884$	8.24621i	0.277350i
$885$	0	0
$886$	0	0
$887$	− 53.6004i	− 1.79972i	−0.436174	−	0.899862i	$-0.643667\pi$
$887$	− 53.6004i	− 1.79972i	0.436174	−	0.899862i	$-0.356333\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	−58.0000	−1.94198
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0	0
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	0	0	1.00000			$0$
$907$	0	0	−1.00000			$\pi$
$908$	57.7235i	1.91562i
$909$	0	0
$910$	0	0
$911$	45.3542i	1.50265i	0.659932	+	0.751325i	$0.270585\pi$
$911$	45.3542i	1.50265i	−0.659932	+	0.751325i	$0.729415\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	20.0000	0.660819
$917$	0	0
$918$	0	0
$919$	−49.0000	−1.61636	−0.808180	−	0.588935i	$-0.799547\pi$
$919$	−49.0000	−1.61636	−0.808180	+	0.588935i	$0.799547\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	16.4924i	0.542855i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 28.8617i	− 0.946923i	−0.880815	−	0.473461i	$-0.843004\pi$
$929$	− 28.8617i	− 0.946923i	0.880815	−	0.473461i	$-0.156996\pi$
$930$	0	0
$931$	35.0000	1.14708
$932$	− 41.2311i	− 1.35057i
$933$	0	0
$934$	0	0
$935$	− 70.0928i	− 2.29228i
$936$	0	0
$937$	−22.0000	−0.718709	−0.359354	−	0.933201i	$-0.617003\pi$
$937$	−22.0000	−0.718709	−0.359354	+	0.933201i	$0.617003\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	57.7235i	1.88173i	0.338779	+	0.940866i	$0.389986\pi$
$941$	57.7235i	1.88173i	−0.338779	+	0.940866i	$0.610014\pi$
$942$	0	0
$943$	−17.0000	−0.553596
$944$	0	0
$945$	0	0
$946$	0	0
$947$	32.9848i	1.07186i	0.844261	+	0.535932i	$0.180040\pi$
$947$	32.9848i	1.07186i	−0.844261	+	0.535932i	$0.819960\pi$
$948$	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
$956$	0	0
$957$	0	0
$958$	0	0
$959$	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
$967$	−37.0000	−1.18984	−0.594920	−	0.803785i	$-0.702816\pi$
$967$	−37.0000	−1.18984	−0.594920	+	0.803785i	$0.702816\pi$
$968$	0	0
$969$	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
$976$	0	0
$977$	0	0		−	1.00000i	$-0.5\pi$
$977$	0	0			1.00000i	$0.5\pi$
$978$	0	0
$979$	0	0
$980$	57.7235i	1.84391i
$981$	0	0
$982$	0	0
$983$	45.3542i	1.44657i	0.690548	+	0.723287i	$0.257369\pi$
$983$	45.3542i	1.44657i	−0.690548	+	0.723287i	$0.742631\pi$
$984$	0	0
$985$	85.0000	2.70833
$986$	0	0
$987$	0	0
$988$	10.0000	0.318142
$989$	− 45.3542i	− 1.44218i
$990$	0	0
$991$	0	0	1.00000			$0$
$991$	0	0	−1.00000			$\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	0	0	1.00000			$0$
$997$	0	0	−1.00000			$\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	153.2.d.b.118.1	✓	2
3.2	odd	2	inner	153.2.d.b.118.2	yes	2
4.3	odd	2		2448.2.c.g.577.1		2
12.11	even	2		2448.2.c.g.577.2		2
17.4	even	4		2601.2.a.o.1.1		2
17.13	even	4		2601.2.a.o.1.2		2
17.16	even	2	inner	153.2.d.b.118.2	yes	2
51.38	odd	4		2601.2.a.o.1.2		2
51.47	odd	4		2601.2.a.o.1.1		2
51.50	odd	2	CM	153.2.d.b.118.1	✓	2
68.67	odd	2		2448.2.c.g.577.2		2
204.203	even	2		2448.2.c.g.577.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
153.2.d.b.118.1	✓	2	1.1	even	1	trivial
153.2.d.b.118.1	✓	2	51.50	odd	2	CM
153.2.d.b.118.2	yes	2	3.2	odd	2	inner
153.2.d.b.118.2	yes	2	17.16	even	2	inner
2448.2.c.g.577.1		2	4.3	odd	2
2448.2.c.g.577.1		2	204.203	even	2
2448.2.c.g.577.2		2	12.11	even	2
2448.2.c.g.577.2		2	68.67	odd	2
2601.2.a.o.1.1		2	17.4	even	4
2601.2.a.o.1.1		2	51.47	odd	4
2601.2.a.o.1.2		2	17.13	even	4
2601.2.a.o.1.2		2	51.38	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 153 = 3^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	153.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.00000 q^{4} -4.12311i q^{5} -4.12311i q^{11} -1.00000 q^{13} +4.00000 q^{16} +4.12311i q^{17} +5.00000 q^{19} +8.24621i q^{20} -4.12311i q^{23} -12.0000 q^{25} +8.24621i q^{29} -4.12311i q^{41} +11.0000 q^{43} +8.24621i q^{44} +7.00000 q^{49} +2.00000 q^{52} -17.0000 q^{55} -8.00000 q^{64} +4.12311i q^{65} +8.00000 q^{67} -8.24621i q^{68} -16.4924i q^{71} -10.0000 q^{76} -16.4924i q^{80} +17.0000 q^{85} +8.24621i q^{92} -20.6155i q^{95} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{4} - 2 q^{13} + 8 q^{16} + 10 q^{19} - 24 q^{25} + 22 q^{43} + 14 q^{49} + 4 q^{52} - 34 q^{55} - 16 q^{64} + 16 q^{67} - 20 q^{76} + 34 q^{85}+O(q^{100}) \) 2 * q - 4 * q^4 - 2 * q^13 + 8 * q^16 + 10 * q^19 - 24 * q^25 + 22 * q^43 + 14 * q^49 + 4 * q^52 - 34 * q^55 - 16 * q^64 + 16 * q^67 - 20 * q^76 + 34 * q^85

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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