Properties

Label 432.8.a.g
Level $432$
Weight $8$
Character orbit 432.a
Self dual yes
Analytic conductor $134.950$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [432,8,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, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("432.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 432 = 2^{4} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 8 \)
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: \(134.950331009\)
Analytic rank: \(1\)
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)
 
\(f(q)\) \(=\) \( q + 120 q^{5} - 377 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + 120 q^{5} - 377 q^{7} - 600 q^{11} + 5369 q^{13} + 12168 q^{17} - 16211 q^{19} - 106392 q^{23} - 63725 q^{25} + 177216 q^{29} + 268060 q^{31} - 45240 q^{35} + 114959 q^{37} - 112128 q^{41} + 115048 q^{43} - 561336 q^{47} - 681414 q^{49} - 1787760 q^{53} - 72000 q^{55} + 1786344 q^{59} - 1306837 q^{61} + 644280 q^{65} + 2013817 q^{67} + 4060944 q^{71} - 3850639 q^{73} + 226200 q^{77} - 1037231 q^{79} - 9203568 q^{83} + 1460160 q^{85} - 1289304 q^{89} - 2024113 q^{91} - 1945320 q^{95} + 8555885 q^{97}+O(q^{100}) \) 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 0 120.000 0 −377.000 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 432.8.a.g 1
3.b odd 2 1 432.8.a.b 1
4.b odd 2 1 54.8.a.f yes 1
12.b even 2 1 54.8.a.a 1
36.f odd 6 2 162.8.c.b 2
36.h even 6 2 162.8.c.k 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
54.8.a.a 1 12.b even 2 1
54.8.a.f yes 1 4.b odd 2 1
162.8.c.b 2 36.f odd 6 2
162.8.c.k 2 36.h even 6 2
432.8.a.b 1 3.b odd 2 1
432.8.a.g 1 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(432))\):

\( T_{5} - 120 \) Copy content Toggle raw display
\( T_{7} + 377 \) 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
$5$ \( T - 120 \) Copy content Toggle raw display
$7$ \( T + 377 \) Copy content Toggle raw display
$11$ \( T + 600 \) Copy content Toggle raw display
$13$ \( T - 5369 \) Copy content Toggle raw display
$17$ \( T - 12168 \) Copy content Toggle raw display
$19$ \( T + 16211 \) Copy content Toggle raw display
$23$ \( T + 106392 \) Copy content Toggle raw display
$29$ \( T - 177216 \) Copy content Toggle raw display
$31$ \( T - 268060 \) Copy content Toggle raw display
$37$ \( T - 114959 \) Copy content Toggle raw display
$41$ \( T + 112128 \) Copy content Toggle raw display
$43$ \( T - 115048 \) Copy content Toggle raw display
$47$ \( T + 561336 \) Copy content Toggle raw display
$53$ \( T + 1787760 \) Copy content Toggle raw display
$59$ \( T - 1786344 \) Copy content Toggle raw display
$61$ \( T + 1306837 \) Copy content Toggle raw display
$67$ \( T - 2013817 \) Copy content Toggle raw display
$71$ \( T - 4060944 \) Copy content Toggle raw display
$73$ \( T + 3850639 \) Copy content Toggle raw display
$79$ \( T + 1037231 \) Copy content Toggle raw display
$83$ \( T + 9203568 \) Copy content Toggle raw display
$89$ \( T + 1289304 \) Copy content Toggle raw display
$97$ \( T - 8555885 \) Copy content Toggle raw display
show more
show less