Properties

Label 2904.2.a.g
Level $2904$
Weight $2$
Character orbit 2904.a
Self dual yes
Analytic conductor $23.189$
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] = [2904,2,Mod(1,2904)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2904, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2904.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2904 = 2^{3} \cdot 3 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2904.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: \(23.1885567470\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 264)
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)
 
\(f(q)\) \(=\) \( q - q^{3} + 2 q^{5} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{3} + 2 q^{5} + q^{9} - 2 q^{13} - 2 q^{15} - 6 q^{17} + 4 q^{23} - q^{25} - q^{27} - 2 q^{29} - 10 q^{37} + 2 q^{39} - 6 q^{41} + 8 q^{43} + 2 q^{45} - 4 q^{47} - 7 q^{49} + 6 q^{51} - 6 q^{53} - 12 q^{59} - 2 q^{61} - 4 q^{65} + 4 q^{67} - 4 q^{69} + 12 q^{71} + 14 q^{73} + q^{75} - 16 q^{79} + q^{81} + 12 q^{83} - 12 q^{85} + 2 q^{87} + 10 q^{89} - 14 q^{97}+O(q^{100}) \) 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 −1.00000 0 2.00000 0 0 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(11\) \(-1\)

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 2904.2.a.g 1
3.b odd 2 1 8712.2.a.f 1
4.b odd 2 1 5808.2.a.be 1
11.b odd 2 1 264.2.a.a 1
33.d even 2 1 792.2.a.b 1
44.c even 2 1 528.2.a.i 1
55.d odd 2 1 6600.2.a.y 1
55.e even 4 2 6600.2.d.s 2
88.b odd 2 1 2112.2.a.s 1
88.g even 2 1 2112.2.a.d 1
132.d odd 2 1 1584.2.a.d 1
264.m even 2 1 6336.2.a.cd 1
264.p odd 2 1 6336.2.a.cb 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
264.2.a.a 1 11.b odd 2 1
528.2.a.i 1 44.c even 2 1
792.2.a.b 1 33.d even 2 1
1584.2.a.d 1 132.d odd 2 1
2112.2.a.d 1 88.g even 2 1
2112.2.a.s 1 88.b odd 2 1
2904.2.a.g 1 1.a even 1 1 trivial
5808.2.a.be 1 4.b odd 2 1
6336.2.a.cb 1 264.p odd 2 1
6336.2.a.cd 1 264.m even 2 1
6600.2.a.y 1 55.d odd 2 1
6600.2.d.s 2 55.e even 4 2
8712.2.a.f 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2904))\):

\( T_{5} - 2 \) Copy content Toggle raw display
\( T_{7} \) Copy content Toggle raw display
\( T_{13} + 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T + 1 \) Copy content Toggle raw display
$5$ \( T - 2 \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T \) Copy content Toggle raw display
$13$ \( T + 2 \) Copy content Toggle raw display
$17$ \( T + 6 \) Copy content Toggle raw display
$19$ \( T \) Copy content Toggle raw display
$23$ \( T - 4 \) Copy content Toggle raw display
$29$ \( T + 2 \) Copy content Toggle raw display
$31$ \( T \) Copy content Toggle raw display
$37$ \( T + 10 \) Copy content Toggle raw display
$41$ \( T + 6 \) Copy content Toggle raw display
$43$ \( T - 8 \) Copy content Toggle raw display
$47$ \( T + 4 \) Copy content Toggle raw display
$53$ \( T + 6 \) Copy content Toggle raw display
$59$ \( T + 12 \) Copy content Toggle raw display
$61$ \( T + 2 \) Copy content Toggle raw display
$67$ \( T - 4 \) Copy content Toggle raw display
$71$ \( T - 12 \) Copy content Toggle raw display
$73$ \( T - 14 \) Copy content Toggle raw display
$79$ \( T + 16 \) Copy content Toggle raw display
$83$ \( T - 12 \) Copy content Toggle raw display
$89$ \( T - 10 \) Copy content Toggle raw display
$97$ \( T + 14 \) Copy content Toggle raw display
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