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Properties

Label 2790.2.x.b
Level $2790$
Weight $2$
Character orbit 2790.x
Analytic conductor $22.278$
Analytic rank $0$
Dimension $40$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2790,2,Mod(161,2790)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2790, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 0, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2790.161");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2790 = 2 \cdot 3^{2} \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2790.x (of order \(6\), degree \(2\), 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: \(22.2782621639\)
Analytic rank: \(0\)
Dimension: \(40\)
Relative dimension: \(20\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 40 q - 40 q^{4}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q - 40 q^{4} + 20 q^{10} + 24 q^{13} + 40 q^{16} + 12 q^{22} + 20 q^{25} - 16 q^{31} - 12 q^{34} - 12 q^{37} - 20 q^{40} - 12 q^{43} - 28 q^{49} - 24 q^{52} - 12 q^{55} - 40 q^{64} - 44 q^{67} + 60 q^{79} + 16 q^{82} - 12 q^{88} + 8 q^{94} + 80 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
161.1 1.00000i 0 −1.00000 −0.866025 − 0.500000i 0 2.50916 + 4.34599i − 1.00000i 0 0.500000 − 0.866025i
161.2 1.00000i 0 −1.00000 −0.866025 − 0.500000i 0 −1.85599 − 3.21468i − 1.00000i 0 0.500000 − 0.866025i
161.3 1.00000i 0 −1.00000 −0.866025 − 0.500000i 0 −1.62830 − 2.82029i − 1.00000i 0 0.500000 − 0.866025i
161.4 1.00000i 0 −1.00000 −0.866025 − 0.500000i 0 −2.03719 − 3.52852i − 1.00000i 0 0.500000 − 0.866025i
161.5 1.00000i 0 −1.00000 −0.866025 − 0.500000i 0 −0.557748 − 0.966048i − 1.00000i 0 0.500000 − 0.866025i
161.6 − 1.00000i 0 −1.00000 0.866025 + 0.500000i 0 −0.176993 − 0.306561i 1.00000i 0 0.500000 − 0.866025i
161.7 1.00000i 0 −1.00000 −0.866025 − 0.500000i 0 0.445017 + 0.770793i − 1.00000i 0 0.500000 − 0.866025i
161.8 − 1.00000i 0 −1.00000 0.866025 + 0.500000i 0 −0.557748 − 0.966048i 1.00000i 0 0.500000 − 0.866025i
161.9 − 1.00000i 0 −1.00000 0.866025 + 0.500000i 0 0.445017 + 0.770793i 1.00000i 0 0.500000 − 0.866025i
161.10 1.00000i 0 −1.00000 −0.866025 − 0.500000i 0 −0.176993 − 0.306561i − 1.00000i 0 0.500000 − 0.866025i
161.11 − 1.00000i 0 −1.00000 0.866025 + 0.500000i 0 −1.85599 − 3.21468i 1.00000i 0 0.500000 − 0.866025i
161.12 1.00000i 0 −1.00000 −0.866025 − 0.500000i 0 1.50236 + 2.60216i − 1.00000i 0 0.500000 − 0.866025i
161.13 − 1.00000i 0 −1.00000 0.866025 + 0.500000i 0 1.04292 + 1.80639i 1.00000i 0 0.500000 − 0.866025i
161.14 − 1.00000i 0 −1.00000 0.866025 + 0.500000i 0 −2.03719 − 3.52852i 1.00000i 0 0.500000 − 0.866025i
161.15 − 1.00000i 0 −1.00000 0.866025 + 0.500000i 0 2.50916 + 4.34599i 1.00000i 0 0.500000 − 0.866025i
161.16 1.00000i 0 −1.00000 −0.866025 − 0.500000i 0 0.756767 + 1.31076i − 1.00000i 0 0.500000 − 0.866025i
161.17 − 1.00000i 0 −1.00000 0.866025 + 0.500000i 0 1.50236 + 2.60216i 1.00000i 0 0.500000 − 0.866025i
161.18 − 1.00000i 0 −1.00000 0.866025 + 0.500000i 0 0.756767 + 1.31076i 1.00000i 0 0.500000 − 0.866025i
161.19 1.00000i 0 −1.00000 −0.866025 − 0.500000i 0 1.04292 + 1.80639i − 1.00000i 0 0.500000 − 0.866025i
161.20 − 1.00000i 0 −1.00000 0.866025 + 0.500000i 0 −1.62830 − 2.82029i 1.00000i 0 0.500000 − 0.866025i
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 161.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
31.e odd 6 1 inner
93.g even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2790.2.x.b ✓ 40
3.b odd 2 1 inner 2790.2.x.b ✓ 40
31.e odd 6 1 inner 2790.2.x.b ✓ 40
93.g even 6 1 inner 2790.2.x.b ✓ 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2790.2.x.b ✓ 40 1.a even 1 1 trivial
2790.2.x.b ✓ 40 3.b odd 2 1 inner
2790.2.x.b ✓ 40 31.e odd 6 1 inner
2790.2.x.b ✓ 40 93.g even 6 1 inner