Properties

Label 8800.2.a.t
Level $8800$
Weight $2$
Character orbit 8800.a
Self dual yes
Analytic conductor $70.268$
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] = [8800,2,Mod(1,8800)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8800, 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("8800.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8800 = 2^{5} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8800.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: \(70.2683537787\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 352)
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)
 
\(f(q)\) \(=\) \( q + q^{3} + 4 q^{7} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} + 4 q^{7} - 2 q^{9} - q^{11} + 2 q^{13} + 2 q^{19} + 4 q^{21} + 9 q^{23} - 5 q^{27} + 4 q^{29} - 5 q^{31} - q^{33} + 9 q^{37} + 2 q^{39} + 2 q^{41} - 6 q^{43} - 4 q^{47} + 9 q^{49} + 6 q^{53} + 2 q^{57} + 5 q^{59} - 8 q^{63} - 13 q^{67} + 9 q^{69} + q^{71} - 14 q^{73} - 4 q^{77} + 10 q^{79} + q^{81} + 14 q^{83} + 4 q^{87} - 13 q^{89} + 8 q^{91} - 5 q^{93} + 19 q^{97} + 2 q^{99}+O(q^{100}) \) 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 4.00000 0 −2.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(5\) \( +1 \)
\(11\) \( +1 \)

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 8800.2.a.t 1
4.b odd 2 1 8800.2.a.i 1
5.b even 2 1 352.2.a.c 1
15.d odd 2 1 3168.2.a.g 1
20.d odd 2 1 352.2.a.e yes 1
40.e odd 2 1 704.2.a.d 1
40.f even 2 1 704.2.a.g 1
55.d odd 2 1 3872.2.a.e 1
60.h even 2 1 3168.2.a.j 1
80.k odd 4 2 2816.2.c.a 2
80.q even 4 2 2816.2.c.m 2
120.i odd 2 1 6336.2.a.bq 1
120.m even 2 1 6336.2.a.bv 1
220.g even 2 1 3872.2.a.j 1
440.c even 2 1 7744.2.a.i 1
440.o odd 2 1 7744.2.a.y 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
352.2.a.c 1 5.b even 2 1
352.2.a.e yes 1 20.d odd 2 1
704.2.a.d 1 40.e odd 2 1
704.2.a.g 1 40.f even 2 1
2816.2.c.a 2 80.k odd 4 2
2816.2.c.m 2 80.q even 4 2
3168.2.a.g 1 15.d odd 2 1
3168.2.a.j 1 60.h even 2 1
3872.2.a.e 1 55.d odd 2 1
3872.2.a.j 1 220.g even 2 1
6336.2.a.bq 1 120.i odd 2 1
6336.2.a.bv 1 120.m even 2 1
7744.2.a.i 1 440.c even 2 1
7744.2.a.y 1 440.o odd 2 1
8800.2.a.i 1 4.b odd 2 1
8800.2.a.t 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8800))\):

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