LMFDB - Newform orbit 1296.2.a.d

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1296,2,Mod(1,1296)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1296, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1296.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 1296 = 2^{4} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1296.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,0,0,0,-1,0,0,0,0,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$10.3486121020$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 648)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

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} $

1.1

−1.00000

Atkin-Lehner signs

$ p $	Sign
$2$	$ +1 $
$3$	$ +1 $

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1296.2.a.d		1
3.b	odd	2	1		1296.2.a.h		1
4.b	odd	2	1		648.2.a.b	✓	1
8.b	even	2	1		5184.2.a.u		1
8.d	odd	2	1		5184.2.a.v		1
9.c	even	3	2		1296.2.i.k		2
9.d	odd	6	2		1296.2.i.f		2
12.b	even	2	1		648.2.a.d	yes	1
24.f	even	2	1		5184.2.a.k		1
24.h	odd	2	1		5184.2.a.l		1
36.f	odd	6	2		648.2.i.f		2
36.h	even	6	2		648.2.i.d		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
648.2.a.b	✓	1	4.b	odd	2	1
648.2.a.d	yes	1	12.b	even	2	1
648.2.i.d		2	36.h	even	6	2
648.2.i.f		2	36.f	odd	6	2
1296.2.a.d		1	1.a	even	1	1	trivial
1296.2.a.h		1	3.b	odd	2	1
1296.2.i.f		2	9.d	odd	6	2
1296.2.i.k		2	9.c	even	3	2
5184.2.a.k		1	24.f	even	2	1
5184.2.a.l		1	24.h	odd	2	1
5184.2.a.u		1	8.b	even	2	1
5184.2.a.v		1	8.d	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(\Gamma_0(1296))$:

$ T_{5} + 1 $

T5 + 1

$ T_{7} $

T7

$ T_{11} - 4 $

T11 - 4

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T $ T
$3$	$ T $ T
$5$	$ T + 1 $ T + 1
$7$	$ T $ T
$11$	$ T - 4 $ T - 4
$13$	$ T + 5 $ T + 5
$17$	$ T + 5 $ T + 5
$19$	$ T + 8 $ T + 8
$23$	$ T - 4 $ T - 4
$29$	$ T - 3 $ T - 3
$31$	$ T - 4 $ T - 4
$37$	$ T - 3 $ T - 3
$41$	$ T + 6 $ T + 6
$43$	$ T + 4 $ T + 4
$47$	$ T - 12 $ T - 12
$53$	$ T + 10 $ T + 10
$59$	$ T + 8 $ T + 8
$61$	$ T + 5 $ T + 5
$67$	$ T + 8 $ T + 8
$71$	$ T + 16 $ T + 16
$73$	$ T + 5 $ T + 5
$79$	$ T + 4 $ T + 4
$83$	$ T + 4 $ T + 4
$89$	$ T - 3 $ T - 3
$97$	$ T - 2 $ T - 2
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Level:	\( N \)	\(=\)	\( 1296 = 2^{4} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1296.a (trivial)

\(n\):		e.g. 2-40 or 990-1000
Significant digits:
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