Properties

Label 2888.2.a.d
Level $2888$
Weight $2$
Character orbit 2888.a
Self dual yes
Analytic conductor $23.061$
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] = [2888,2,Mod(1,2888)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2888, 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("2888.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2888 = 2^{3} \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2888.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: \(23.0607961037\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 152)
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)
 
\(f(q)\) \(=\) \( q + q^{3} - 4 q^{5} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} - 4 q^{5} - 2 q^{9} + 3 q^{11} + 2 q^{13} - 4 q^{15} + 2 q^{17} + 6 q^{23} + 11 q^{25} - 5 q^{27} - 4 q^{29} - 10 q^{31} + 3 q^{33} + 2 q^{37} + 2 q^{39} + 9 q^{41} - 4 q^{43} + 8 q^{45} - 12 q^{47} - 7 q^{49} + 2 q^{51} - 2 q^{53} - 12 q^{55} - q^{59} - 8 q^{61} - 8 q^{65} + 9 q^{67} + 6 q^{69} - 6 q^{71} - 9 q^{73} + 11 q^{75} - 4 q^{79} + q^{81} - 5 q^{83} - 8 q^{85} - 4 q^{87} - 18 q^{89} - 10 q^{93} + q^{97} - 6 q^{99}+O(q^{100}) \) 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 −4.00000 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 2888.2.a.d 1
4.b odd 2 1 5776.2.a.e 1
19.b odd 2 1 2888.2.a.a 1
19.c even 3 2 152.2.i.b 2
57.h odd 6 2 1368.2.s.a 2
76.d even 2 1 5776.2.a.j 1
76.g odd 6 2 304.2.i.d 2
152.k odd 6 2 1216.2.i.b 2
152.p even 6 2 1216.2.i.f 2
228.m even 6 2 2736.2.s.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
152.2.i.b 2 19.c even 3 2
304.2.i.d 2 76.g odd 6 2
1216.2.i.b 2 152.k odd 6 2
1216.2.i.f 2 152.p even 6 2
1368.2.s.a 2 57.h odd 6 2
2736.2.s.a 2 228.m even 6 2
2888.2.a.a 1 19.b odd 2 1
2888.2.a.d 1 1.a even 1 1 trivial
5776.2.a.e 1 4.b odd 2 1
5776.2.a.j 1 76.d even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2888))\):

\( T_{3} - 1 \) Copy content Toggle raw display
\( T_{5} + 4 \) 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
$5$ \( T + 4 \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T - 3 \) Copy content Toggle raw display
$13$ \( T - 2 \) Copy content Toggle raw display
$17$ \( T - 2 \) Copy content Toggle raw display
$19$ \( T \) Copy content Toggle raw display
$23$ \( T - 6 \) Copy content Toggle raw display
$29$ \( T + 4 \) Copy content Toggle raw display
$31$ \( T + 10 \) Copy content Toggle raw display
$37$ \( T - 2 \) Copy content Toggle raw display
$41$ \( T - 9 \) Copy content Toggle raw display
$43$ \( T + 4 \) Copy content Toggle raw display
$47$ \( T + 12 \) Copy content Toggle raw display
$53$ \( T + 2 \) Copy content Toggle raw display
$59$ \( T + 1 \) Copy content Toggle raw display
$61$ \( T + 8 \) Copy content Toggle raw display
$67$ \( T - 9 \) Copy content Toggle raw display
$71$ \( T + 6 \) Copy content Toggle raw display
$73$ \( T + 9 \) Copy content Toggle raw display
$79$ \( T + 4 \) Copy content Toggle raw display
$83$ \( T + 5 \) Copy content Toggle raw display
$89$ \( T + 18 \) Copy content Toggle raw display
$97$ \( T - 1 \) Copy content Toggle raw display
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