LMFDB - Embedded newform 2252.1.d.a.1125.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2252,1,Mod(1125,2252)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2252.1125"); S:= CuspForms(chi, 1); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2252, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]

Level:	$N$	$=$	$2252 = 2^{2} \cdot 563$
Weight:	$k$	$=$	$1$
Character orbit:	$[\chi]$	$=$	2252.d (of order $2$ , degree $1$ , minimal)

Newform invariants

comment:select newform

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

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

Self dual:	yes
Analytic conductor:	$1.12389440845$
Analytic rank:	$0$
Dimension:	$9$
Coefficient field:	$\Q(\zeta_{54})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{9} - 9x^{7} + 27x^{5} - 30x^{3} + 9x - 1$ x^9 - 9x^7 + 27x^5 - 30x^3 + 9x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$1$
Twist minimal:	yes
Projective image:	$D_{27}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{27} - \cdots)$

Embedding invariants

Embedding label			1125.4
Root			$0.573606$ of defining polynomial
Character	$\chi$	$=$	2252.1125

$q$ -expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-0.573606 q^{3} +0.792160 q^{7} -0.670976 q^{9} -1.87939 q^{11} +0.347296 q^{13} +1.78727 q^{17} -0.116290 q^{19} -0.454388 q^{21} +1.19432 q^{23} +1.00000 q^{25} +0.958482 q^{27} +1.07803 q^{33} -0.199211 q^{39} +1.94609 q^{47} -0.372483 q^{49} -1.02519 q^{51} +0.0667045 q^{57} -1.98648 q^{59} +1.94609 q^{61} -0.531520 q^{63} +1.78727 q^{67} -0.685068 q^{69} -1.37248 q^{71} -0.573606 q^{75} -1.48877 q^{77} +0.121184 q^{81} +0.275114 q^{91} +1.26102 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$9 q + 9 q^{9} + 9 q^{25} + 9 q^{49} + 9 q^{81}+O(q^{100})$ 9 * q + 9 * q^9 + 9 * q^25 + 9 * q^49 + 9 * q^81

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2252\mathbb{Z}\right)^\times$ .

$n$	$565$	$1127$
$\chi(n)$	$-1$	$1$

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
$2$	0	0
$3$	−0.573606	−0.573606	−0.286803	−	0.957990i	$-0.592593\pi$
$3$	−0.573606	−0.573606	−0.286803	+	0.957990i	$0.592593\pi$
$4$	0	0
$5$	0	0	1.00000			$0$
$5$	0	0	−1.00000			$\pi$
$6$	0	0
$7$	0.792160	0.792160	0.396080	−	0.918216i	$-0.370370\pi$
$7$	0.792160	0.792160	0.396080	+	0.918216i	$0.370370\pi$
$8$	0	0
$9$	−0.670976	−0.670976
$10$	0	0
$11$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$11$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$12$	0	0
$13$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$13$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1.78727	1.78727	0.893633	−	0.448799i	$-0.148148\pi$
$17$	1.78727	1.78727	0.893633	+	0.448799i	$0.148148\pi$
$18$	0	0
$19$	−0.116290	−0.116290	−0.0581448	−	0.998308i	$-0.518519\pi$
$19$	−0.116290	−0.116290	−0.0581448	+	0.998308i	$0.518519\pi$
$20$	0	0
$21$	−0.454388	−0.454388
$22$	0	0
$23$	1.19432	1.19432	0.597159	−	0.802123i	$-0.296296\pi$
$23$	1.19432	1.19432	0.597159	+	0.802123i	$0.296296\pi$
$24$	0	0
$25$	1.00000	1.00000
$26$	0	0
$27$	0.958482	0.958482
$28$	0	0
$29$	0	0	1.00000			$0$
$29$	0	0	−1.00000			$\pi$
$30$	0	0
$31$	0	0	1.00000			$0$
$31$	0	0	−1.00000			$\pi$
$32$	0	0
$33$	1.07803	1.07803
$34$	0	0
$35$	0	0
$36$	0	0
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	0	0
$39$	−0.199211	−0.199211
$40$	0	0
$41$	0	0	1.00000			$0$
$41$	0	0	−1.00000			$\pi$
$42$	0	0
$43$	0	0	1.00000			$0$
$43$	0	0	−1.00000			$\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	1.94609	1.94609	0.973045	−	0.230616i	$-0.0740741\pi$
$47$	1.94609	1.94609	0.973045	+	0.230616i	$0.0740741\pi$
$48$	0	0
$49$	−0.372483	−0.372483
$50$	0	0
$51$	−1.02519	−1.02519
$52$	0	0
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0.0667045	0.0667045
$58$	0	0
$59$	−1.98648	−1.98648	−0.993238	−	0.116093i	$-0.962963\pi$
$59$	−1.98648	−1.98648	−0.993238	+	0.116093i	$0.962963\pi$
$60$	0	0
$61$	1.94609	1.94609	0.973045	−	0.230616i	$-0.0740741\pi$
$61$	1.94609	1.94609	0.973045	+	0.230616i	$0.0740741\pi$
$62$	0	0
$63$	−0.531520	−0.531520
$64$	0	0
$65$	0	0
$66$	0	0
$67$	1.78727	1.78727	0.893633	−	0.448799i	$-0.148148\pi$
$67$	1.78727	1.78727	0.893633	+	0.448799i	$0.148148\pi$
$68$	0	0
$69$	−0.685068	−0.685068
$70$	0	0
$71$	−1.37248	−1.37248	−0.686242	−	0.727374i	$-0.740741\pi$
$71$	−1.37248	−1.37248	−0.686242	+	0.727374i	$0.740741\pi$
$72$	0	0
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	0	0
$75$	−0.573606	−0.573606
$76$	0	0
$77$	−1.48877	−1.48877
$78$	0	0
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	0	0
$81$	0.121184	0.121184
$82$	0	0
$83$	0	0	1.00000			$0$
$83$	0	0	−1.00000			$\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	0	0	1.00000			$0$
$89$	0	0	−1.00000			$\pi$
$90$	0	0
$91$	0.275114	0.275114
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	0	0
$99$	1.26102	1.26102
$100$	0	0
$101$	0.792160	0.792160	0.396080	−	0.918216i	$-0.370370\pi$
$101$	0.792160	0.792160	0.396080	+	0.918216i	$0.370370\pi$
$102$	0	0
$103$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$103$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−1.67098	−1.67098	−0.835488	−	0.549509i	$-0.814815\pi$
$107$	−1.67098	−1.67098	−0.835488	+	0.549509i	$0.814815\pi$
$108$	0	0
$109$	0	0	1.00000			$0$
$109$	0	0	−1.00000			$\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−1.98648	−1.98648	−0.993238	−	0.116093i	$-0.962963\pi$
$113$	−1.98648	−1.98648	−0.993238	+	0.116093i	$0.962963\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−0.233027	−0.233027
$118$	0	0
$119$	1.41580	1.41580
$120$	0	0
$121$	2.53209	2.53209
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−1.37248	−1.37248	−0.686242	−	0.727374i	$-0.740741\pi$
$127$	−1.37248	−1.37248	−0.686242	+	0.727374i	$0.740741\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	0	0	1.00000			$0$
$131$	0	0	−1.00000			$\pi$
$132$	0	0
$133$	−0.0921200	−0.0921200
$134$	0	0
$135$	0	0
$136$	0	0
$137$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$137$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$138$	0	0
$139$	0	0	1.00000			$0$
$139$	0	0	−1.00000			$\pi$
$140$	0	0
$141$	−1.11629	−1.11629
$142$	0	0
$143$	−0.652704	−0.652704
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0.213659	0.213659
$148$	0	0
$149$	1.94609	1.94609	0.973045	−	0.230616i	$-0.0740741\pi$
$149$	1.94609	1.94609	0.973045	+	0.230616i	$0.0740741\pi$
$150$	0	0
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	0	0
$153$	−1.19921	−1.19921
$154$	0	0
$155$	0	0
$156$	0	0
$157$	0	0	1.00000			$0$
$157$	0	0	−1.00000			$\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0.946090	0.946090
$162$	0	0
$163$	0	0	1.00000			$0$
$163$	0	0	−1.00000			$\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	0	0	1.00000			$0$
$167$	0	0	−1.00000			$\pi$
$168$	0	0
$169$	−0.879385	−0.879385
$170$	0	0
$171$	0.0780275	0.0780275
$172$	0	0
$173$	0	0	1.00000			$0$
$173$	0	0	−1.00000			$\pi$
$174$	0	0
$175$	0.792160	0.792160
$176$	0	0
$177$	1.13946	1.13946
$178$	0	0
$179$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$179$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$180$	0	0
$181$	1.19432	1.19432	0.597159	−	0.802123i	$-0.296296\pi$
$181$	1.19432	1.19432	0.597159	+	0.802123i	$0.296296\pi$
$182$	0	0
$183$	−1.11629	−1.11629
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−3.35896	−3.35896
$188$	0	0
$189$	0.759271	0.759271
$190$	0	0
$191$	−1.67098	−1.67098	−0.835488	−	0.549509i	$-0.814815\pi$
$191$	−1.67098	−1.67098	−0.835488	+	0.549509i	$0.814815\pi$
$192$	0	0
$193$	−0.573606	−0.573606	−0.286803	−	0.957990i	$-0.592593\pi$
$193$	−0.573606	−0.573606	−0.286803	+	0.957990i	$0.592593\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$197$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$198$	0	0
$199$	0	0	1.00000			$0$
$199$	0	0	−1.00000			$\pi$
$200$	0	0
$201$	−1.02519	−1.02519
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−0.801358	−0.801358
$208$	0	0
$209$	0.218553	0.218553
$210$	0	0
$211$	1.19432	1.19432	0.597159	−	0.802123i	$-0.296296\pi$
$211$	1.19432	1.19432	0.597159	+	0.802123i	$0.296296\pi$
$212$	0	0
$213$	0.787265	0.787265
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0.620711	0.620711
$222$	0	0
$223$	−1.67098	−1.67098	−0.835488	−	0.549509i	$-0.814815\pi$
$223$	−1.67098	−1.67098	−0.835488	+	0.549509i	$0.814815\pi$
$224$	0	0
$225$	−0.670976	−0.670976
$226$	0	0
$227$	0	0	1.00000			$0$
$227$	0	0	−1.00000			$\pi$
$228$	0	0
$229$	0	0	1.00000			$0$
$229$	0	0	−1.00000			$\pi$
$230$	0	0
$231$	0.853970	0.853970
$232$	0	0
$233$	0	0	1.00000			$0$
$233$	0	0	−1.00000			$\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	0	0	1.00000			$0$
$239$	0	0	−1.00000			$\pi$
$240$	0	0
$241$	−1.98648	−1.98648	−0.993238	−	0.116093i	$-0.962963\pi$
$241$	−1.98648	−1.98648	−0.993238	+	0.116093i	$0.962963\pi$
$242$	0	0
$243$	−1.02799	−1.02799
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−0.0403870	−0.0403870
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$251$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$252$	0	0
$253$	−2.24458	−2.24458
$254$	0	0
$255$	0	0
$256$	0	0
$257$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$257$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−1.67098	−1.67098	−0.835488	−	0.549509i	$-0.814815\pi$
$269$	−1.67098	−1.67098	−0.835488	+	0.549509i	$0.814815\pi$
$270$	0	0
$271$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$271$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$272$	0	0
$273$	−0.157807	−0.157807
$274$	0	0
$275$	−1.87939	−1.87939
$276$	0	0
$277$	−0.116290	−0.116290	−0.0581448	−	0.998308i	$-0.518519\pi$
$277$	−0.116290	−0.116290	−0.0581448	+	0.998308i	$0.518519\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−1.37248	−1.37248	−0.686242	−	0.727374i	$-0.740741\pi$
$281$	−1.37248	−1.37248	−0.686242	+	0.727374i	$0.740741\pi$
$282$	0	0
$283$	0	0	1.00000			$0$
$283$	0	0	−1.00000			$\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	2.19432	2.19432
$290$	0	0
$291$	0	0
$292$	0	0
$293$	0	0	1.00000			$0$
$293$	0	0	−1.00000			$\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−1.80136	−1.80136
$298$	0	0
$299$	0.414782	0.414782
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−0.454388	−0.454388
$304$	0	0
$305$	0	0
$306$	0	0
$307$	0	0	1.00000			$0$
$307$	0	0	−1.00000			$\pi$
$308$	0	0
$309$	−0.878816	−0.878816
$310$	0	0
$311$	0	0	1.00000			$0$
$311$	0	0	−1.00000			$\pi$
$312$	0	0
$313$	0	0	1.00000			$0$
$313$	0	0	−1.00000			$\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	0	0	1.00000			$0$
$317$	0	0	−1.00000			$\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0.958482	0.958482
$322$	0	0
$323$	−0.207840	−0.207840
$324$	0	0
$325$	0.347296	0.347296
$326$	0	0
$327$	0	0
$328$	0	0
$329$	1.54161	1.54161
$330$	0	0
$331$	0	0	1.00000			$0$
$331$	0	0	−1.00000			$\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	1.94609	1.94609	0.973045	−	0.230616i	$-0.0740741\pi$
$337$	1.94609	1.94609	0.973045	+	0.230616i	$0.0740741\pi$
$338$	0	0
$339$	1.13946	1.13946
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−1.08723	−1.08723
$344$	0	0
$345$	0	0
$346$	0	0
$347$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$347$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$348$	0	0
$349$	−0.573606	−0.573606	−0.286803	−	0.957990i	$-0.592593\pi$
$349$	−0.573606	−0.573606	−0.286803	+	0.957990i	$0.592593\pi$
$350$	0	0
$351$	0.332877	0.332877
$352$	0	0
$353$	0	0	1.00000			$0$
$353$	0	0	−1.00000			$\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−0.812112	−0.812112
$358$	0	0
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	0	0
$361$	−0.986477	−0.986477
$362$	0	0
$363$	−1.45242	−1.45242
$364$	0	0
$365$	0	0
$366$	0	0
$367$	0	0	1.00000			$0$
$367$	0	0	−1.00000			$\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$379$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$380$	0	0
$381$	0.787265	0.787265
$382$	0	0
$383$	1.78727	1.78727	0.893633	−	0.448799i	$-0.148148\pi$
$383$	1.78727	1.78727	0.893633	+	0.448799i	$0.148148\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	0	0	1.00000			$0$
$389$	0	0	−1.00000			$\pi$
$390$	0	0
$391$	2.13456	2.13456
$392$	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
$399$	0.0528406	0.0528406
$400$	0	0
$401$	−0.116290	−0.116290	−0.0581448	−	0.998308i	$-0.518519\pi$
$401$	−0.116290	−0.116290	−0.0581448	+	0.998308i	$0.518519\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−0.116290	−0.116290	−0.0581448	−	0.998308i	$-0.518519\pi$
$409$	−0.116290	−0.116290	−0.0581448	+	0.998308i	$0.518519\pi$
$410$	0	0
$411$	−0.878816	−0.878816
$412$	0	0
$413$	−1.57361	−1.57361
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	0	0	1.00000			$0$
$419$	0	0	−1.00000			$\pi$
$420$	0	0
$421$	−1.37248	−1.37248	−0.686242	−	0.727374i	$-0.740741\pi$
$421$	−1.37248	−1.37248	−0.686242	+	0.727374i	$0.740741\pi$
$422$	0	0
$423$	−1.30578	−1.30578
$424$	0	0
$425$	1.78727	1.78727
$426$	0	0
$427$	1.54161	1.54161
$428$	0	0
$429$	0.374395	0.374395
$430$	0	0
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	0	0
$433$	0	0	1.00000			$0$
$433$	0	0	−1.00000			$\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−0.138887	−0.138887
$438$	0	0
$439$	1.19432	1.19432	0.597159	−	0.802123i	$-0.296296\pi$
$439$	1.19432	1.19432	0.597159	+	0.802123i	$0.296296\pi$
$440$	0	0
$441$	0.249927	0.249927
$442$	0	0
$443$	0	0	1.00000			$0$
$443$	0	0	−1.00000			$\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−1.11629	−1.11629
$448$	0	0
$449$	−0.573606	−0.573606	−0.286803	−	0.957990i	$-0.592593\pi$
$449$	−0.573606	−0.573606	−0.286803	+	0.957990i	$0.592593\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	0	0	1.00000			$0$
$457$	0	0	−1.00000			$\pi$
$458$	0	0
$459$	1.71306	1.71306
$460$	0	0
$461$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$461$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$462$	0	0
$463$	0	0	1.00000			$0$
$463$	0	0	−1.00000			$\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	1.94609	1.94609	0.973045	−	0.230616i	$-0.0740741\pi$
$467$	1.94609	1.94609	0.973045	+	0.230616i	$0.0740741\pi$
$468$	0	0
$469$	1.41580	1.41580
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−0.116290	−0.116290
$476$	0	0
$477$	0	0
$478$	0	0
$479$	0	0	1.00000			$0$
$479$	0	0	−1.00000			$\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	−0.542683	−0.542683
$484$	0	0
$485$	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
$491$	1.94609	1.94609	0.973045	−	0.230616i	$-0.0740741\pi$
$491$	1.94609	1.94609	0.973045	+	0.230616i	$0.0740741\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−1.08723	−1.08723
$498$	0	0
$499$	0	0	1.00000			$0$
$499$	0	0	−1.00000			$\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−1.98648	−1.98648	−0.993238	−	0.116093i	$-0.962963\pi$
$503$	−1.98648	−1.98648	−0.993238	+	0.116093i	$0.962963\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0.504421	0.504421
$508$	0	0
$509$	−1.98648	−1.98648	−0.993238	−	0.116093i	$-0.962963\pi$
$509$	−1.98648	−1.98648	−0.993238	+	0.116093i	$0.962963\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−0.111462	−0.111462
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−3.65745	−3.65745
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$521$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$522$	0	0
$523$	0	0	1.00000			$0$
$523$	0	0	−1.00000			$\pi$
$524$	0	0
$525$	−0.454388	−0.454388
$526$	0	0
$527$	0	0
$528$	0	0
$529$	0.426394	0.426394
$530$	0	0
$531$	1.33288	1.33288
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−0.199211	−0.199211
$538$	0	0
$539$	0.700040	0.700040
$540$	0	0
$541$	−1.37248	−1.37248	−0.686242	−	0.727374i	$-0.740741\pi$
$541$	−1.37248	−1.37248	−0.686242	+	0.727374i	$0.740741\pi$
$542$	0	0
$543$	−0.685068	−0.685068
$544$	0	0
$545$	0	0
$546$	0	0
$547$	0	0	1.00000			$0$
$547$	0	0	−1.00000			$\pi$
$548$	0	0
$549$	−1.30578	−1.30578
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	1.19432	1.19432	0.597159	−	0.802123i	$-0.296296\pi$
$557$	1.19432	1.19432	0.597159	+	0.802123i	$0.296296\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	1.92672	1.92672
$562$	0	0
$563$	1.00000	1.00000
$563$	1.00000	1.00000
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0.0959970	0.0959970
$568$	0	0
$569$	0	0	1.00000			$0$
$569$	0	0	−1.00000			$\pi$
$570$	0	0
$571$	0	0	1.00000			$0$
$571$	0	0	−1.00000			$\pi$
$572$	0	0
$573$	0.958482	0.958482
$574$	0	0
$575$	1.19432	1.19432
$576$	0	0
$577$	0	0	1.00000			$0$
$577$	0	0	−1.00000			$\pi$
$578$	0	0
$579$	0.329024	0.329024
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	0	0	1.00000			$0$
$587$	0	0	−1.00000			$\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0.573606	0.573606
$592$	0	0
$593$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$593$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	2.00000	2.00000	1.00000			$0$
$599$	2.00000	2.00000	1.00000			$0$
$600$	0	0
$601$	0	0	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	0	0
$603$	−1.19921	−1.19921
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−1.67098	−1.67098	−0.835488	−	0.549509i	$-0.814815\pi$
$607$	−1.67098	−1.67098	−0.835488	+	0.549509i	$0.814815\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0.675870	0.675870
$612$	0	0
$613$	0	0	1.00000			$0$
$613$	0	0	−1.00000			$\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	0	0	1.00000			$0$
$617$	0	0	−1.00000			$\pi$
$618$	0	0
$619$	0	0	1.00000			$0$
$619$	0	0	−1.00000			$\pi$
$620$	0	0
$621$	1.14473	1.14473
$622$	0	0
$623$	0	0
$624$	0	0
$625$	1.00000	1.00000
$626$	0	0
$627$	−0.125363	−0.125363
$628$	0	0
$629$	0	0
$630$	0	0
$631$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$631$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$632$	0	0
$633$	−0.685068	−0.685068
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−0.129362	−0.129362
$638$	0	0
$639$	0.920903	0.920903
$640$	0	0
$641$	0	0	1.00000			$0$
$641$	0	0	−1.00000			$\pi$
$642$	0	0
$643$	0	0	1.00000			$0$
$643$	0	0	−1.00000			$\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$647$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$648$	0	0
$649$	3.73336	3.73336
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$653$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	0	0	1.00000			$0$
$659$	0	0	−1.00000			$\pi$
$660$	0	0
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	0	0
$663$	−0.356044	−0.356044
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	0.958482	0.958482
$670$	0	0
$671$	−3.65745	−3.65745
$672$	0	0
$673$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$673$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$674$	0	0
$675$	0.958482	0.958482
$676$	0	0
$677$	0	0	1.00000			$0$
$677$	0	0	−1.00000			$\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−0.116290	−0.116290	−0.0581448	−	0.998308i	$-0.518519\pi$
$683$	−0.116290	−0.116290	−0.0581448	+	0.998308i	$0.518519\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	0	0	1.00000			$0$
$691$	0	0	−1.00000			$\pi$
$692$	0	0
$693$	0.998930	0.998930
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	0	0
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0.627517	0.627517
$708$	0	0
$709$	1.78727	1.78727	0.893633	−	0.448799i	$-0.148148\pi$
$709$	1.78727	1.78727	0.893633	+	0.448799i	$0.148148\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$719$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$720$	0	0
$721$	1.21366	1.21366
$722$	0	0
$723$	1.13946	1.13946
$724$	0	0
$725$	0	0
$726$	0	0
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	0	0
$729$	0.468480	0.468480
$730$	0	0
$731$	0	0
$732$	0	0
$733$	0.792160	0.792160	0.396080	−	0.918216i	$-0.370370\pi$
$733$	0.792160	0.792160	0.396080	+	0.918216i	$0.370370\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−3.35896	−3.35896
$738$	0	0
$739$	−1.98648	−1.98648	−0.993238	−	0.116093i	$-0.962963\pi$
$739$	−1.98648	−1.98648	−0.993238	+	0.116093i	$0.962963\pi$
$740$	0	0
$741$	0.0231662	0.0231662
$742$	0	0
$743$	0	0	1.00000			$0$
$743$	0	0	−1.00000			$\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−1.32368	−1.32368
$750$	0	0
$751$	−0.573606	−0.573606	−0.286803	−	0.957990i	$-0.592593\pi$
$751$	−0.573606	−0.573606	−0.286803	+	0.957990i	$0.592593\pi$
$752$	0	0
$753$	0.573606	0.573606
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−1.98648	−1.98648	−0.993238	−	0.116093i	$-0.962963\pi$
$757$	−1.98648	−1.98648	−0.993238	+	0.116093i	$0.962963\pi$
$758$	0	0
$759$	1.28751	1.28751
$760$	0	0
$761$	0	0	1.00000			$0$
$761$	0	0	−1.00000			$\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−0.689896	−0.689896
$768$	0	0
$769$	0	0	1.00000			$0$
$769$	0	0	−1.00000			$\pi$
$770$	0	0
$771$	−0.199211	−0.199211
$772$	0	0
$773$	−1.98648	−1.98648	−0.993238	−	0.116093i	$-0.962963\pi$
$773$	−1.98648	−1.98648	−0.993238	+	0.116093i	$0.962963\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	2.57942	2.57942
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	0	0	1.00000			$0$
$787$	0	0	−1.00000			$\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−1.57361	−1.57361
$792$	0	0
$793$	0.675870	0.675870
$794$	0	0
$795$	0	0
$796$	0	0
$797$	0	0	1.00000			$0$
$797$	0	0	−1.00000			$\pi$
$798$	0	0
$799$	3.47818	3.47818
$800$	0	0
$801$	0	0
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0.958482	0.958482
$808$	0	0
$809$	−0.116290	−0.116290	−0.0581448	−	0.998308i	$-0.518519\pi$
$809$	−0.116290	−0.116290	−0.0581448	+	0.998308i	$0.518519\pi$
$810$	0	0
$811$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$811$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$812$	0	0
$813$	1.07803	1.07803
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	0	0
$819$	−0.184595	−0.184595
$820$	0	0
$821$	0.792160	0.792160	0.396080	−	0.918216i	$-0.370370\pi$
$821$	0.792160	0.792160	0.396080	+	0.918216i	$0.370370\pi$
$822$	0	0
$823$	0	0	1.00000			$0$
$823$	0	0	−1.00000			$\pi$
$824$	0	0
$825$	1.07803	1.07803
$826$	0	0
$827$	0	0	1.00000			$0$
$827$	0	0	−1.00000			$\pi$
$828$	0	0
$829$	0	0	1.00000			$0$
$829$	0	0	−1.00000			$\pi$
$830$	0	0
$831$	0.0667045	0.0667045
$832$	0	0
$833$	−0.665726	−0.665726
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−0.116290	−0.116290	−0.0581448	−	0.998308i	$-0.518519\pi$
$839$	−0.116290	−0.116290	−0.0581448	+	0.998308i	$0.518519\pi$
$840$	0	0
$841$	1.00000	1.00000
$842$	0	0
$843$	0.787265	0.787265
$844$	0	0
$845$	0	0
$846$	0	0
$847$	2.00582	2.00582
$848$	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
$857$	0	0	1.00000			$0$
$857$	0	0	−1.00000			$\pi$
$858$	0	0
$859$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$859$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	1.78727	1.78727	0.893633	−	0.448799i	$-0.148148\pi$
$863$	1.78727	1.78727	0.893633	+	0.448799i	$0.148148\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−1.25867	−1.25867
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0.620711	0.620711
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$877$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$881$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$882$	0	0
$883$	0	0	1.00000			$0$
$883$	0	0	−1.00000			$\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	2.00000	2.00000	1.00000			$0$
$887$	2.00000	2.00000	1.00000			$0$
$888$	0	0
$889$	−1.08723	−1.08723
$890$	0	0
$891$	−0.227751	−0.227751
$892$	0	0
$893$	−0.226310	−0.226310
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−0.237922	−0.237922
$898$	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
$907$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$907$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$908$	0	0
$909$	−0.531520	−0.531520
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−0.476658	−0.476658
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−1.02799	−1.02799
$928$	0	0
$929$	0	0	1.00000			$0$
$929$	0	0	−1.00000			$\pi$
$930$	0	0
$931$	0.0433160	0.0433160
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	0	0	1.00000			$0$
$937$	0	0	−1.00000			$\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	0	0	1.00000			$0$
$941$	0	0	−1.00000			$\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	0	0	1.00000			$0$
$947$	0	0	−1.00000			$\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$953$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	1.21366	1.21366
$960$	0	0
$961$	1.00000	1.00000
$962$	0	0
$963$	1.12118	1.12118
$964$	0	0
$965$	0	0
$966$	0	0
$967$	1.78727	1.78727	0.893633	−	0.448799i	$-0.148148\pi$
$967$	1.78727	1.78727	0.893633	+	0.448799i	$0.148148\pi$
$968$	0	0
$969$	0.119219	0.119219
$970$	0	0
$971$	0	0	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−0.199211	−0.199211
$976$	0	0
$977$	0	0	1.00000			$0$
$977$	0	0	−1.00000			$\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	0	0
$983$	0	0	1.00000			$0$
$983$	0	0	−1.00000			$\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−0.884279	−0.884279
$988$	0	0
$989$	0	0
$990$	0	0
$991$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$991$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$997$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$998$	0	0
$999$	0	0

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2252.1.d.a.1125.4	✓	9
563.562	odd	2	CM	2252.1.d.a.1125.4	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2252.1.d.a.1125.4	✓	9	1.1	even	1	trivial
2252.1.d.a.1125.4	✓	9	563.562	odd	2	CM

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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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Character values

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	2252.1.d.a.1125.4

Level	$2252$
Weight	$1$
Character	2252.1125
Self dual	yes
Analytic conductor	$1.124$
Analytic rank	$0$
Dimension	$9$
Projective image	$D_{27}$
CM discriminant	-563
Inner twists	$2$