LMFDB - Newform orbit 108.13.c.a

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [108,13,Mod(53,108)] 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("108.53"); S:= CuspForms(chi, 13); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(108, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 13, names="a")

Level:	$ N $	$=$	$ 108 = 2^{2} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 13 $
Character orbit:	$[\chi]$	$=$	108.c (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [1,0,0,0,0,0,-77689] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$98.7113322658$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

$q$-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 $\left(\mathbb{Z}/108\mathbb{Z}\right)^\times$.

$n$	$29$	$55$
$\chi(n)$	$-1$	$1$

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} $

53.1

−77689.0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	CM by $\Q(\sqrt{-3}) $

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	108.13.c.a	✓	1
3.b	odd	2	1	CM	108.13.c.a	✓	1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
108.13.c.a	✓	1	1.a	even	1	1	trivial
108.13.c.a	✓	1	3.b	odd	2	1	CM

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{13}^{\mathrm{new}}(108, [\chi])$:

$ T_{5} $

T5

$ T_{7} + 77689 $

T7 + 77689

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T $ T
$3$	$ T $ T
$5$	$ T $ T
$7$	$ T + 77689 $ T + 77689
$11$	$ T $ T
$13$	$ T - 2786111 $ T - 2786111
$17$	$ T $ T
$19$	$ T + 88943401 $ T + 88943401
$23$	$ T $ T
$29$	$ T $ T
$31$	$ T + 530187838 $ T + 530187838
$37$	$ T - 2296056671 $ T - 2296056671
$41$	$ T $ T
$43$	$ T + 235885102 $ T + 235885102
$47$	$ T $ T
$53$	$ T $ T
$59$	$ T $ T
$61$	$ T - 25008001199 $ T - 25008001199
$67$	$ T + 10741538809 $ T + 10741538809
$71$	$ T $ T
$73$	$ T - 193782843071 $ T - 193782843071
$79$	$ T - 51215247719 $ T - 51215247719
$83$	$ T $ T
$89$	$ T $ T
$97$	$ T - 920565867311 $ T - 920565867311
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Level:	\( N \)	\(=\)	\( 108 = 2^{2} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 13 \)
Character orbit:	\([\chi]\)	\(=\)	108.c (of order \(2\), degree \(1\), minimal)

\(n\)	\(29\)	\(55\)
\(\chi(n)\)	\(-1\)	\(1\)

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