Properties

Label 1296.4.a.g
Level $1296$
Weight $4$
Character orbit 1296.a
Self dual yes
Analytic conductor $76.466$
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] = [1296,4,Mod(1,1296)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1296, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1296.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1296 = 2^{4} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1296.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: \(76.4664753674\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 18)
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)
 
\(f(q)\) \(=\) \( q + 9 q^{5} + 31 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + 9 q^{5} + 31 q^{7} - 15 q^{11} - 37 q^{13} + 42 q^{17} + 28 q^{19} + 195 q^{23} - 44 q^{25} - 111 q^{29} + 205 q^{31} + 279 q^{35} - 166 q^{37} + 261 q^{41} + 43 q^{43} + 177 q^{47} + 618 q^{49} - 114 q^{53} - 135 q^{55} + 159 q^{59} + 191 q^{61} - 333 q^{65} + 421 q^{67} + 156 q^{71} + 182 q^{73} - 465 q^{77} - 1133 q^{79} - 1083 q^{83} + 378 q^{85} + 1050 q^{89} - 1147 q^{91} + 252 q^{95} - 901 q^{97}+O(q^{100}) \) 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 9.00000 0 31.0000 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 1296.4.a.g 1
3.b odd 2 1 1296.4.a.b 1
4.b odd 2 1 162.4.a.a 1
9.c even 3 2 432.4.i.a 2
9.d odd 6 2 144.4.i.a 2
12.b even 2 1 162.4.a.d 1
36.f odd 6 2 54.4.c.a 2
36.h even 6 2 18.4.c.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.4.c.a 2 36.h even 6 2
54.4.c.a 2 36.f odd 6 2
144.4.i.a 2 9.d odd 6 2
162.4.a.a 1 4.b odd 2 1
162.4.a.d 1 12.b even 2 1
432.4.i.a 2 9.c even 3 2
1296.4.a.b 1 3.b odd 2 1
1296.4.a.g 1 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} - 9 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1296))\). 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
$5$ \( T - 9 \) Copy content Toggle raw display
$7$ \( T - 31 \) Copy content Toggle raw display
$11$ \( T + 15 \) Copy content Toggle raw display
$13$ \( T + 37 \) Copy content Toggle raw display
$17$ \( T - 42 \) Copy content Toggle raw display
$19$ \( T - 28 \) Copy content Toggle raw display
$23$ \( T - 195 \) Copy content Toggle raw display
$29$ \( T + 111 \) Copy content Toggle raw display
$31$ \( T - 205 \) Copy content Toggle raw display
$37$ \( T + 166 \) Copy content Toggle raw display
$41$ \( T - 261 \) Copy content Toggle raw display
$43$ \( T - 43 \) Copy content Toggle raw display
$47$ \( T - 177 \) Copy content Toggle raw display
$53$ \( T + 114 \) Copy content Toggle raw display
$59$ \( T - 159 \) Copy content Toggle raw display
$61$ \( T - 191 \) Copy content Toggle raw display
$67$ \( T - 421 \) Copy content Toggle raw display
$71$ \( T - 156 \) Copy content Toggle raw display
$73$ \( T - 182 \) Copy content Toggle raw display
$79$ \( T + 1133 \) Copy content Toggle raw display
$83$ \( T + 1083 \) Copy content Toggle raw display
$89$ \( T - 1050 \) Copy content Toggle raw display
$97$ \( T + 901 \) Copy content Toggle raw display
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