Properties

Label 88.2.a.a
Level $88$
Weight $2$
Character orbit 88.a
Self dual yes
Analytic conductor $0.703$
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] = [88,2,Mod(1,88)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(88, 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("88.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 88 = 2^{3} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 88.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: \(0.702683537787\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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)
 
\(f(q)\) \(=\) \( q - 3 q^{3} - 3 q^{5} - 2 q^{7} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 3 q^{3} - 3 q^{5} - 2 q^{7} + 6 q^{9} - q^{11} + 9 q^{15} - 6 q^{17} + 4 q^{19} + 6 q^{21} + q^{23} + 4 q^{25} - 9 q^{27} - 8 q^{29} - 7 q^{31} + 3 q^{33} + 6 q^{35} - q^{37} + 4 q^{41} + 6 q^{43} - 18 q^{45} - 8 q^{47} - 3 q^{49} + 18 q^{51} + 2 q^{53} + 3 q^{55} - 12 q^{57} - q^{59} + 4 q^{61} - 12 q^{63} - 5 q^{67} - 3 q^{69} + 3 q^{71} + 16 q^{73} - 12 q^{75} + 2 q^{77} + 2 q^{79} + 9 q^{81} - 2 q^{83} + 18 q^{85} + 24 q^{87} + 15 q^{89} + 21 q^{93} - 12 q^{95} - 7 q^{97} - 6 q^{99}+O(q^{100}) \) 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 −3.00000 0 −3.00000 0 −2.00000 0 6.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(11\) \( +1 \)

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 88.2.a.a 1
3.b odd 2 1 792.2.a.g 1
4.b odd 2 1 176.2.a.c 1
5.b even 2 1 2200.2.a.k 1
5.c odd 4 2 2200.2.b.a 2
7.b odd 2 1 4312.2.a.l 1
8.b even 2 1 704.2.a.l 1
8.d odd 2 1 704.2.a.b 1
11.b odd 2 1 968.2.a.a 1
11.c even 5 4 968.2.i.j 4
11.d odd 10 4 968.2.i.i 4
12.b even 2 1 1584.2.a.q 1
16.e even 4 2 2816.2.c.i 2
16.f odd 4 2 2816.2.c.d 2
20.d odd 2 1 4400.2.a.a 1
20.e even 4 2 4400.2.b.b 2
24.f even 2 1 6336.2.a.k 1
24.h odd 2 1 6336.2.a.h 1
28.d even 2 1 8624.2.a.c 1
33.d even 2 1 8712.2.a.x 1
44.c even 2 1 1936.2.a.l 1
88.b odd 2 1 7744.2.a.bk 1
88.g even 2 1 7744.2.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
88.2.a.a 1 1.a even 1 1 trivial
176.2.a.c 1 4.b odd 2 1
704.2.a.b 1 8.d odd 2 1
704.2.a.l 1 8.b even 2 1
792.2.a.g 1 3.b odd 2 1
968.2.a.a 1 11.b odd 2 1
968.2.i.i 4 11.d odd 10 4
968.2.i.j 4 11.c even 5 4
1584.2.a.q 1 12.b even 2 1
1936.2.a.l 1 44.c even 2 1
2200.2.a.k 1 5.b even 2 1
2200.2.b.a 2 5.c odd 4 2
2816.2.c.d 2 16.f odd 4 2
2816.2.c.i 2 16.e even 4 2
4312.2.a.l 1 7.b odd 2 1
4400.2.a.a 1 20.d odd 2 1
4400.2.b.b 2 20.e even 4 2
6336.2.a.h 1 24.h odd 2 1
6336.2.a.k 1 24.f even 2 1
7744.2.a.b 1 88.g even 2 1
7744.2.a.bk 1 88.b odd 2 1
8624.2.a.c 1 28.d even 2 1
8712.2.a.x 1 33.d even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} + 3 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(88))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T + 3 \) Copy content Toggle raw display
$5$ \( T + 3 \) Copy content Toggle raw display
$7$ \( T + 2 \) Copy content Toggle raw display
$11$ \( T + 1 \) Copy content Toggle raw display
$13$ \( T \) Copy content Toggle raw display
$17$ \( T + 6 \) Copy content Toggle raw display
$19$ \( T - 4 \) Copy content Toggle raw display
$23$ \( T - 1 \) Copy content Toggle raw display
$29$ \( T + 8 \) Copy content Toggle raw display
$31$ \( T + 7 \) Copy content Toggle raw display
$37$ \( T + 1 \) Copy content Toggle raw display
$41$ \( T - 4 \) Copy content Toggle raw display
$43$ \( T - 6 \) Copy content Toggle raw display
$47$ \( T + 8 \) Copy content Toggle raw display
$53$ \( T - 2 \) Copy content Toggle raw display
$59$ \( T + 1 \) Copy content Toggle raw display
$61$ \( T - 4 \) Copy content Toggle raw display
$67$ \( T + 5 \) Copy content Toggle raw display
$71$ \( T - 3 \) Copy content Toggle raw display
$73$ \( T - 16 \) Copy content Toggle raw display
$79$ \( T - 2 \) Copy content Toggle raw display
$83$ \( T + 2 \) Copy content Toggle raw display
$89$ \( T - 15 \) Copy content Toggle raw display
$97$ \( T + 7 \) Copy content Toggle raw display
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