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Properties

Label 5520.2.be.d
Level $5520$
Weight $2$
Character orbit 5520.be
Analytic conductor $44.077$
Analytic rank $0$
Dimension $32$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [5520,2,Mod(1471,5520)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5520, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("5520.1471");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5520 = 2^{4} \cdot 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5520.be (of order \(2\), degree \(1\), minimal)

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: no
Analytic conductor: \(44.0774219157\)
Analytic rank: \(0\)
Dimension: \(32\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 8 q^{7} - 32 q^{9} - 8 q^{11} - 8 q^{13} + 32 q^{15} - 32 q^{25} + 4 q^{29} + 20 q^{41} + 52 q^{49} + 4 q^{51} - 8 q^{63} - 32 q^{67} - 40 q^{73} - 24 q^{77} - 32 q^{79} + 32 q^{81} - 4 q^{85} + 48 q^{91}+ \cdots + 8 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1471.1 0 − 1.00000i 0 1.00000i 0 −3.34851 0 −1.00000 0
1471.2 0 1.00000i 0 − 1.00000i 0 −3.34851 0 −1.00000 0
1471.3 0 − 1.00000i 0 1.00000i 0 1.52302 0 −1.00000 0
1471.4 0 1.00000i 0 − 1.00000i 0 1.52302 0 −1.00000 0
1471.5 0 − 1.00000i 0 1.00000i 0 −3.12483 0 −1.00000 0
1471.6 0 1.00000i 0 − 1.00000i 0 −3.12483 0 −1.00000 0
1471.7 0 − 1.00000i 0 1.00000i 0 −3.67125 0 −1.00000 0
1471.8 0 1.00000i 0 − 1.00000i 0 −3.67125 0 −1.00000 0
1471.9 0 − 1.00000i 0 1.00000i 0 1.53388 0 −1.00000 0
1471.10 0 1.00000i 0 − 1.00000i 0 1.53388 0 −1.00000 0
1471.11 0 − 1.00000i 0 1.00000i 0 2.13359 0 −1.00000 0
1471.12 0 1.00000i 0 − 1.00000i 0 2.13359 0 −1.00000 0
1471.13 0 − 1.00000i 0 1.00000i 0 3.74981 0 −1.00000 0
1471.14 0 1.00000i 0 − 1.00000i 0 3.74981 0 −1.00000 0
1471.15 0 − 1.00000i 0 1.00000i 0 −3.60511 0 −1.00000 0
1471.16 0 1.00000i 0 − 1.00000i 0 −3.60511 0 −1.00000 0
1471.17 0 − 1.00000i 0 1.00000i 0 0.143131 0 −1.00000 0
1471.18 0 1.00000i 0 − 1.00000i 0 0.143131 0 −1.00000 0
1471.19 0 − 1.00000i 0 1.00000i 0 2.01693 0 −1.00000 0
1471.20 0 1.00000i 0 − 1.00000i 0 2.01693 0 −1.00000 0
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1471.32
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
92.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 5520.2.be.d yes 32
4.b odd 2 1 5520.2.be.c ✓ 32
23.b odd 2 1 5520.2.be.c ✓ 32
92.b even 2 1 inner 5520.2.be.d yes 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5520.2.be.c ✓ 32 4.b odd 2 1
5520.2.be.c ✓ 32 23.b odd 2 1
5520.2.be.d yes 32 1.a even 1 1 trivial
5520.2.be.d yes 32 92.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{16} - 4 T_{7}^{15} - 61 T_{7}^{14} + 220 T_{7}^{13} + 1519 T_{7}^{12} - 4876 T_{7}^{11} + \cdots - 239616 \) acting on \(S_{2}^{\mathrm{new}}(5520, [\chi])\). Copy content Toggle raw display