LMFDB - Newform orbit 264.1.p.a

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [264,1,Mod(131,264)] 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("264.131"); S:= CuspForms(chi, 1); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(264, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 1, 1])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]

Level:	$ N $	$=$	$ 264 = 2^{3} \cdot 3 \cdot 11 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	264.p (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [1,-1] 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:	yes
Analytic conductor:	$0.131753163335$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{2}$
Projective field:	Galois closure of $\Q(\sqrt{-2}, \sqrt{33})$
Artin image:	$D_4$
Artin field:	Galois closure of $\Q(\sqrt{-5 +2 \sqrt{-2}})$
Stark unit:	Root of $x^{4} - 3x^{3} - 4x^{2} - 3x + 1$

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/264\mathbb{Z}\right)^\times$.

$n$	$89$	$133$	$145$	$199$
$\chi(n)$	$-1$	$-1$	$-1$	$-1$

Embeddings

For each embedding $\iota_m$ of the coefficient field, the values $\iota_m(a_n)$ are shown below.

For more information on an embedded modular form you can click on its label.

comment:embeddings in the coefficient field

gp:mfembed(f)

Label

$\iota_m(\nu)$

$ a_{2} $

$ a_{3} $

$ a_{4} $

$ a_{5} $

$ a_{6} $

$ a_{7} $

$ a_{8} $

$ a_{9} $

$ a_{10} $

131.1

−1.00000

1.00000

−1.00000

1.00000

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
8.d	odd	2	1	CM by $\Q(\sqrt{-2}) $
33.d	even	2	1	RM by $\Q(\sqrt{33}) $
264.p	odd	2	1	CM by $\Q(\sqrt{-66}) $

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	264.1.p.a	✓	1
3.b	odd	2	1		264.1.p.b	yes	1
4.b	odd	2	1		1056.1.p.a		1
8.b	even	2	1		1056.1.p.a		1
8.d	odd	2	1	CM	264.1.p.a	✓	1
11.b	odd	2	1		264.1.p.b	yes	1
11.c	even	5	4		2904.1.r.f		4
11.d	odd	10	4		2904.1.r.b		4
12.b	even	2	1		1056.1.p.b		1
24.f	even	2	1		264.1.p.b	yes	1
24.h	odd	2	1		1056.1.p.b		1
33.d	even	2	1	RM	264.1.p.a	✓	1
33.f	even	10	4		2904.1.r.f		4
33.h	odd	10	4		2904.1.r.b		4
44.c	even	2	1		1056.1.p.b		1
88.b	odd	2	1		1056.1.p.b		1
88.g	even	2	1		264.1.p.b	yes	1
88.k	even	10	4		2904.1.r.b		4
88.l	odd	10	4		2904.1.r.f		4
132.d	odd	2	1		1056.1.p.a		1
264.m	even	2	1		1056.1.p.a		1
264.p	odd	2	1	CM	264.1.p.a	✓	1
264.r	odd	10	4		2904.1.r.f		4
264.w	even	10	4		2904.1.r.b		4

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
264.1.p.a	✓	1	1.a	even	1	1	trivial
264.1.p.a	✓	1	8.d	odd	2	1	CM
264.1.p.a	✓	1	33.d	even	2	1	RM
264.1.p.a	✓	1	264.p	odd	2	1	CM
264.1.p.b	yes	1	3.b	odd	2	1
264.1.p.b	yes	1	11.b	odd	2	1
264.1.p.b	yes	1	24.f	even	2	1
264.1.p.b	yes	1	88.g	even	2	1
1056.1.p.a		1	4.b	odd	2	1
1056.1.p.a		1	8.b	even	2	1
1056.1.p.a		1	132.d	odd	2	1
1056.1.p.a		1	264.m	even	2	1
1056.1.p.b		1	12.b	even	2	1
1056.1.p.b		1	24.h	odd	2	1
1056.1.p.b		1	44.c	even	2	1
1056.1.p.b		1	88.b	odd	2	1
2904.1.r.b		4	11.d	odd	10	4
2904.1.r.b		4	33.h	odd	10	4
2904.1.r.b		4	88.k	even	10	4
2904.1.r.b		4	264.w	even	10	4
2904.1.r.f		4	11.c	even	5	4
2904.1.r.f		4	33.f	even	10	4
2904.1.r.f		4	88.l	odd	10	4
2904.1.r.f		4	264.r	odd	10	4

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{17} - 2 $ T17 - 2 acting on $S_{1}^{\mathrm{new}}(264, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T + 1 $ T + 1
$3$	$ T + 1 $ T + 1
$5$	$ T $ T
$7$	$ T $ T
$11$	$ T - 1 $ T - 1
$13$	$ T $ T
$17$	$ T - 2 $ T - 2
$19$	$ T $ T
$23$	$ T $ T
$29$	$ T $ T
$31$	$ T $ T
$37$	$ T $ T
$41$	$ T + 2 $ T + 2
$43$	$ T $ T
$47$	$ T $ T
$53$	$ T $ T
$59$	$ T $ T
$61$	$ T $ T
$67$	$ T - 2 $ T - 2
$71$	$ T $ T
$73$	$ T $ T
$79$	$ T $ T
$83$	$ T + 2 $ T + 2
$89$	$ T $ T
$97$	$ T + 2 $ T + 2
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Level:	\( N \)	\(=\)	\( 264 = 2^{3} \cdot 3 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	264.p (of order \(2\), degree \(1\), minimal)

\(n\)	\(89\)	\(133\)	\(145\)	\(199\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(-1\)	\(-1\)

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