Properties

Label 1296.4.a.h
Level $1296$
Weight $4$
Character orbit 1296.a
Self dual yes
Analytic conductor $76.466$
Analytic rank $1$
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: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 162)
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)
 
\(f(q)\) \(=\) \( q + 21 q^{5} - 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + 21 q^{5} - 8 q^{7} - 36 q^{11} - 49 q^{13} + 21 q^{17} + 112 q^{19} - 180 q^{23} + 316 q^{25} - 135 q^{29} - 308 q^{31} - 168 q^{35} - q^{37} - 42 q^{41} - 20 q^{43} - 84 q^{47} - 279 q^{49} - 174 q^{53} - 756 q^{55} - 504 q^{59} - 385 q^{61} - 1029 q^{65} - 272 q^{67} + 888 q^{71} + 371 q^{73} + 288 q^{77} + 652 q^{79} - 84 q^{83} + 441 q^{85} + 21 q^{89} + 392 q^{91} + 2352 q^{95} - 1246 q^{97}+O(q^{100}) \) 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 21.0000 0 −8.00000 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.h 1
3.b odd 2 1 1296.4.a.a 1
4.b odd 2 1 162.4.a.b 1
12.b even 2 1 162.4.a.c yes 1
36.f odd 6 2 162.4.c.e 2
36.h even 6 2 162.4.c.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
162.4.a.b 1 4.b odd 2 1
162.4.a.c yes 1 12.b even 2 1
162.4.c.d 2 36.h even 6 2
162.4.c.e 2 36.f odd 6 2
1296.4.a.a 1 3.b odd 2 1
1296.4.a.h 1 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} - 21 \) 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
$5$ \( T - 21 \) Copy content Toggle raw display
$7$ \( T + 8 \) Copy content Toggle raw display
$11$ \( T + 36 \) Copy content Toggle raw display
$13$ \( T + 49 \) Copy content Toggle raw display
$17$ \( T - 21 \) Copy content Toggle raw display
$19$ \( T - 112 \) Copy content Toggle raw display
$23$ \( T + 180 \) Copy content Toggle raw display
$29$ \( T + 135 \) Copy content Toggle raw display
$31$ \( T + 308 \) Copy content Toggle raw display
$37$ \( T + 1 \) Copy content Toggle raw display
$41$ \( T + 42 \) Copy content Toggle raw display
$43$ \( T + 20 \) Copy content Toggle raw display
$47$ \( T + 84 \) Copy content Toggle raw display
$53$ \( T + 174 \) Copy content Toggle raw display
$59$ \( T + 504 \) Copy content Toggle raw display
$61$ \( T + 385 \) Copy content Toggle raw display
$67$ \( T + 272 \) Copy content Toggle raw display
$71$ \( T - 888 \) Copy content Toggle raw display
$73$ \( T - 371 \) Copy content Toggle raw display
$79$ \( T - 652 \) Copy content Toggle raw display
$83$ \( T + 84 \) Copy content Toggle raw display
$89$ \( T - 21 \) Copy content Toggle raw display
$97$ \( T + 1246 \) Copy content Toggle raw display
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