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Properties

Label 2760.2.k.f
Level $2760$
Weight $2$
Character orbit 2760.k
Analytic conductor $22.039$
Analytic rank $0$
Dimension $22$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2760,2,Mod(2209,2760)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2760, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2760.2209");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2760 = 2^{3} \cdot 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2760.k (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: \(22.0387109579\)
Analytic rank: \(0\)
Dimension: \(22\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 22 q - 2 q^{5} - 22 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 22 q - 2 q^{5} - 22 q^{9} - 16 q^{19} + 6 q^{21} + 2 q^{25} - 22 q^{29} + 6 q^{31} + 14 q^{35} - 12 q^{39} + 6 q^{41} + 2 q^{45} - 68 q^{49} + 10 q^{51} + 12 q^{55} - 18 q^{59} + 16 q^{61} + 44 q^{65} - 22 q^{69} - 30 q^{71} + 4 q^{75} + 36 q^{79} + 22 q^{81} + 34 q^{85} - 16 q^{89} + 28 q^{91} + 20 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2209.1 0 − 1.00000i 0 −2.22799 + 0.189879i 0 4.23369i 0 −1.00000 0
2209.2 0 − 1.00000i 0 −2.06524 + 0.857190i 0 − 3.22845i 0 −1.00000 0
2209.3 0 − 1.00000i 0 −1.61965 − 1.54167i 0 − 4.12396i 0 −1.00000 0
2209.4 0 − 1.00000i 0 −0.887264 + 2.05250i 0 1.58269i 0 −1.00000 0
2209.5 0 − 1.00000i 0 −0.885149 − 2.05341i 0 1.25349i 0 −1.00000 0
2209.6 0 − 1.00000i 0 −0.488181 − 2.18213i 0 4.05083i 0 −1.00000 0
2209.7 0 − 1.00000i 0 −0.299974 + 2.21586i 0 − 3.84917i 0 −1.00000 0
2209.8 0 − 1.00000i 0 1.46815 + 1.68658i 0 4.80325i 0 −1.00000 0
2209.9 0 − 1.00000i 0 1.78427 − 1.34773i 0 0.198322i 0 −1.00000 0
2209.10 0 − 1.00000i 0 2.08908 + 0.797325i 0 − 2.60191i 0 −1.00000 0
2209.11 0 − 1.00000i 0 2.13195 − 0.674383i 0 0.681214i 0 −1.00000 0
2209.12 0 1.00000i 0 −2.22799 − 0.189879i 0 − 4.23369i 0 −1.00000 0
2209.13 0 1.00000i 0 −2.06524 − 0.857190i 0 3.22845i 0 −1.00000 0
2209.14 0 1.00000i 0 −1.61965 + 1.54167i 0 4.12396i 0 −1.00000 0
2209.15 0 1.00000i 0 −0.887264 − 2.05250i 0 − 1.58269i 0 −1.00000 0
2209.16 0 1.00000i 0 −0.885149 + 2.05341i 0 − 1.25349i 0 −1.00000 0
2209.17 0 1.00000i 0 −0.488181 + 2.18213i 0 − 4.05083i 0 −1.00000 0
2209.18 0 1.00000i 0 −0.299974 − 2.21586i 0 3.84917i 0 −1.00000 0
2209.19 0 1.00000i 0 1.46815 − 1.68658i 0 − 4.80325i 0 −1.00000 0
2209.20 0 1.00000i 0 1.78427 + 1.34773i 0 − 0.198322i 0 −1.00000 0
See all 22 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2209.22
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2760.2.k.f ✓ 22
5.b even 2 1 inner 2760.2.k.f ✓ 22
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2760.2.k.f ✓ 22 1.a even 1 1 trivial
2760.2.k.f ✓ 22 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{22} + 111 T_{7}^{20} + 5263 T_{7}^{18} + 139013 T_{7}^{16} + 2235156 T_{7}^{14} + 22436528 T_{7}^{12} + \cdots + 8667136 \) acting on \(S_{2}^{\mathrm{new}}(2760, [\chi])\). Copy content Toggle raw display