Properties

Label 432.6.a.b
Level $432$
Weight $6$
Character orbit 432.a
Self dual yes
Analytic conductor $69.286$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [432,6,Mod(1,432)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(432, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("432.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 432 = 2^{4} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 432.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(69.2858101592\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 54)
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)
 
\(f(q)\) \(=\) \( q - 33 q^{5} - 59 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - 33 q^{5} - 59 q^{7} + 147 q^{11} + 836 q^{13} + 1080 q^{17} - 2882 q^{19} - 4386 q^{23} - 2036 q^{25} - 1866 q^{29} + 3295 q^{31} + 1947 q^{35} - 3958 q^{37} + 20586 q^{41} + 8770 q^{43} + 12666 q^{47} - 13326 q^{49} + 9621 q^{53} - 4851 q^{55} - 21468 q^{59} + 36248 q^{61} - 27588 q^{65} - 5174 q^{67} + 63720 q^{71} + 57953 q^{73} - 8673 q^{77} - 16448 q^{79} + 69267 q^{83} - 35640 q^{85} + 54198 q^{89} - 49324 q^{91} + 95106 q^{95} - 132961 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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
0
0 0 0 −33.0000 0 −59.0000 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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 432.6.a.b 1
3.b odd 2 1 432.6.a.i 1
4.b odd 2 1 54.6.a.a 1
12.b even 2 1 54.6.a.f yes 1
36.f odd 6 2 162.6.c.k 2
36.h even 6 2 162.6.c.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
54.6.a.a 1 4.b odd 2 1
54.6.a.f yes 1 12.b even 2 1
162.6.c.b 2 36.h even 6 2
162.6.c.k 2 36.f odd 6 2
432.6.a.b 1 1.a even 1 1 trivial
432.6.a.i 1 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(432))\):

\( T_{5} + 33 \) Copy content Toggle raw display
\( T_{7} + 59 \) Copy content Toggle raw display
\( T_{11} - 147 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T + 33 \) Copy content Toggle raw display
$7$ \( T + 59 \) Copy content Toggle raw display
$11$ \( T - 147 \) Copy content Toggle raw display
$13$ \( T - 836 \) Copy content Toggle raw display
$17$ \( T - 1080 \) Copy content Toggle raw display
$19$ \( T + 2882 \) Copy content Toggle raw display
$23$ \( T + 4386 \) Copy content Toggle raw display
$29$ \( T + 1866 \) Copy content Toggle raw display
$31$ \( T - 3295 \) Copy content Toggle raw display
$37$ \( T + 3958 \) Copy content Toggle raw display
$41$ \( T - 20586 \) Copy content Toggle raw display
$43$ \( T - 8770 \) Copy content Toggle raw display
$47$ \( T - 12666 \) Copy content Toggle raw display
$53$ \( T - 9621 \) Copy content Toggle raw display
$59$ \( T + 21468 \) Copy content Toggle raw display
$61$ \( T - 36248 \) Copy content Toggle raw display
$67$ \( T + 5174 \) Copy content Toggle raw display
$71$ \( T - 63720 \) Copy content Toggle raw display
$73$ \( T - 57953 \) Copy content Toggle raw display
$79$ \( T + 16448 \) Copy content Toggle raw display
$83$ \( T - 69267 \) Copy content Toggle raw display
$89$ \( T - 54198 \) Copy content Toggle raw display
$97$ \( T + 132961 \) Copy content Toggle raw display
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