Properties

Label 5776.2.a.k
Level $5776$
Weight $2$
Character orbit 5776.a
Self dual yes
Analytic conductor $46.122$
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] = [5776,2,Mod(1,5776)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5776, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("5776.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5776 = 2^{4} \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5776.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: \(46.1215922075\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 76)
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)
 
\(f(q)\) \(=\) \( q + q^{3} - q^{5} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} - q^{5} - 2 q^{9} + 4 q^{11} - q^{13} - q^{15} + 3 q^{17} - 5 q^{23} - 4 q^{25} - 5 q^{27} + 7 q^{29} - 4 q^{31} + 4 q^{33} + 10 q^{37} - q^{39} - 5 q^{41} + 5 q^{43} + 2 q^{45} + 7 q^{47} - 7 q^{49} + 3 q^{51} + 11 q^{53} - 4 q^{55} - 3 q^{59} + 11 q^{61} + q^{65} + 3 q^{67} - 5 q^{69} - 11 q^{71} + 15 q^{73} - 4 q^{75} + 13 q^{79} + q^{81} - 3 q^{85} + 7 q^{87} + 3 q^{89} - 4 q^{93} - 5 q^{97} - 8 q^{99}+O(q^{100}) \) 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 −1.00000 0 0 0 −2.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(19\) \(1\)

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 5776.2.a.k 1
4.b odd 2 1 1444.2.a.b 1
19.b odd 2 1 5776.2.a.f 1
19.c even 3 2 304.2.i.a 2
57.h odd 6 2 2736.2.s.g 2
76.d even 2 1 1444.2.a.c 1
76.f even 6 2 1444.2.e.b 2
76.g odd 6 2 76.2.e.a 2
152.k odd 6 2 1216.2.i.c 2
152.p even 6 2 1216.2.i.g 2
228.m even 6 2 684.2.k.b 2
380.p odd 6 2 1900.2.i.a 2
380.v even 12 4 1900.2.s.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.2.e.a 2 76.g odd 6 2
304.2.i.a 2 19.c even 3 2
684.2.k.b 2 228.m even 6 2
1216.2.i.c 2 152.k odd 6 2
1216.2.i.g 2 152.p even 6 2
1444.2.a.b 1 4.b odd 2 1
1444.2.a.c 1 76.d even 2 1
1444.2.e.b 2 76.f even 6 2
1900.2.i.a 2 380.p odd 6 2
1900.2.s.a 4 380.v even 12 4
2736.2.s.g 2 57.h odd 6 2
5776.2.a.f 1 19.b odd 2 1
5776.2.a.k 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5776))\):

\( T_{3} - 1 \) Copy content Toggle raw display
\( T_{5} + 1 \) Copy content Toggle raw display
\( T_{7} \) Copy content Toggle raw display
\( T_{11} - 4 \) Copy content Toggle raw display
\( T_{13} + 1 \) 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
$5$ \( T + 1 \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T - 4 \) Copy content Toggle raw display
$13$ \( T + 1 \) Copy content Toggle raw display
$17$ \( T - 3 \) Copy content Toggle raw display
$19$ \( T \) Copy content Toggle raw display
$23$ \( T + 5 \) Copy content Toggle raw display
$29$ \( T - 7 \) Copy content Toggle raw display
$31$ \( T + 4 \) Copy content Toggle raw display
$37$ \( T - 10 \) Copy content Toggle raw display
$41$ \( T + 5 \) Copy content Toggle raw display
$43$ \( T - 5 \) Copy content Toggle raw display
$47$ \( T - 7 \) Copy content Toggle raw display
$53$ \( T - 11 \) Copy content Toggle raw display
$59$ \( T + 3 \) Copy content Toggle raw display
$61$ \( T - 11 \) Copy content Toggle raw display
$67$ \( T - 3 \) Copy content Toggle raw display
$71$ \( T + 11 \) Copy content Toggle raw display
$73$ \( T - 15 \) Copy content Toggle raw display
$79$ \( T - 13 \) Copy content Toggle raw display
$83$ \( T \) Copy content Toggle raw display
$89$ \( T - 3 \) Copy content Toggle raw display
$97$ \( T + 5 \) Copy content Toggle raw display
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