Properties

Label 7104.2.a.f
Level $7104$
Weight $2$
Character orbit 7104.a
Self dual yes
Analytic conductor $56.726$
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] = [7104,2,Mod(1,7104)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7104, 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("7104.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7104 = 2^{6} \cdot 3 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7104.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: \(56.7257255959\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 222)
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)
 
\(f(q)\) \(=\) \( q - q^{3} - 3 q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{3} - 3 q^{7} + q^{9} + q^{11} - q^{13} - 3 q^{17} + 3 q^{19} + 3 q^{21} + q^{23} - 5 q^{25} - q^{27} + 4 q^{29} + 6 q^{31} - q^{33} + q^{37} + q^{39} - 10 q^{41} + 12 q^{43} + 6 q^{47} + 2 q^{49} + 3 q^{51} + q^{53} - 3 q^{57} - 2 q^{61} - 3 q^{63} + 2 q^{67} - q^{69} - 3 q^{73} + 5 q^{75} - 3 q^{77} - 14 q^{79} + q^{81} + 9 q^{83} - 4 q^{87} - 3 q^{89} + 3 q^{91} - 6 q^{93} - 10 q^{97} + q^{99}+O(q^{100}) \) 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 0 0 −3.00000 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(3\) \( +1 \)
\(37\) \( -1 \)

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 7104.2.a.f 1
4.b odd 2 1 7104.2.a.v 1
8.b even 2 1 1776.2.a.h 1
8.d odd 2 1 222.2.a.d 1
24.f even 2 1 666.2.a.b 1
24.h odd 2 1 5328.2.a.h 1
40.e odd 2 1 5550.2.a.n 1
296.h odd 2 1 8214.2.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
222.2.a.d 1 8.d odd 2 1
666.2.a.b 1 24.f even 2 1
1776.2.a.h 1 8.b even 2 1
5328.2.a.h 1 24.h odd 2 1
5550.2.a.n 1 40.e odd 2 1
7104.2.a.f 1 1.a even 1 1 trivial
7104.2.a.v 1 4.b odd 2 1
8214.2.a.b 1 296.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7104))\):

\( T_{5} \) Copy content Toggle raw display
\( T_{7} + 3 \) Copy content Toggle raw display
\( T_{11} - 1 \) Copy content Toggle raw display
\( T_{19} - 3 \) Copy content Toggle raw display
\( T_{23} - 1 \) 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
$5$ \( T \) Copy content Toggle raw display
$7$ \( T + 3 \) Copy content Toggle raw display
$11$ \( T - 1 \) Copy content Toggle raw display
$13$ \( T + 1 \) Copy content Toggle raw display
$17$ \( T + 3 \) Copy content Toggle raw display
$19$ \( T - 3 \) Copy content Toggle raw display
$23$ \( T - 1 \) Copy content Toggle raw display
$29$ \( T - 4 \) Copy content Toggle raw display
$31$ \( T - 6 \) Copy content Toggle raw display
$37$ \( T - 1 \) Copy content Toggle raw display
$41$ \( T + 10 \) Copy content Toggle raw display
$43$ \( T - 12 \) Copy content Toggle raw display
$47$ \( T - 6 \) Copy content Toggle raw display
$53$ \( T - 1 \) Copy content Toggle raw display
$59$ \( T \) Copy content Toggle raw display
$61$ \( T + 2 \) Copy content Toggle raw display
$67$ \( T - 2 \) Copy content Toggle raw display
$71$ \( T \) Copy content Toggle raw display
$73$ \( T + 3 \) Copy content Toggle raw display
$79$ \( T + 14 \) Copy content Toggle raw display
$83$ \( T - 9 \) Copy content Toggle raw display
$89$ \( T + 3 \) Copy content Toggle raw display
$97$ \( T + 10 \) Copy content Toggle raw display
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