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Properties

Label 6840.2.r.b
Level $6840$
Weight $2$
Character orbit 6840.r
Analytic conductor $54.618$
Analytic rank $0$
Dimension $40$
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] = [6840,2,Mod(3761,6840)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6840, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 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("6840.3761");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6840 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6840.r (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: \(54.6176749826\)
Analytic rank: \(0\)
Dimension: \(40\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 40 q - 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q - 8 q^{7} - 4 q^{19} - 40 q^{25} + 24 q^{29} - 8 q^{41} + 8 q^{43} + 64 q^{49} + 32 q^{53} + 16 q^{59} - 16 q^{61} + 16 q^{65} - 32 q^{71} - 24 q^{89} + 4 q^{95}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3761.1 0 0 0 − 1.00000i 0 2.72832 0 0 0
3761.2 0 0 0 1.00000i 0 2.72832 0 0 0
3761.3 0 0 0 − 1.00000i 0 5.20796 0 0 0
3761.4 0 0 0 1.00000i 0 5.20796 0 0 0
3761.5 0 0 0 − 1.00000i 0 −3.38570 0 0 0
3761.6 0 0 0 1.00000i 0 −3.38570 0 0 0
3761.7 0 0 0 − 1.00000i 0 −3.35691 0 0 0
3761.8 0 0 0 1.00000i 0 −3.35691 0 0 0
3761.9 0 0 0 − 1.00000i 0 1.73028 0 0 0
3761.10 0 0 0 1.00000i 0 1.73028 0 0 0
3761.11 0 0 0 − 1.00000i 0 2.51084 0 0 0
3761.12 0 0 0 1.00000i 0 2.51084 0 0 0
3761.13 0 0 0 − 1.00000i 0 −1.49028 0 0 0
3761.14 0 0 0 1.00000i 0 −1.49028 0 0 0
3761.15 0 0 0 − 1.00000i 0 1.29517 0 0 0
3761.16 0 0 0 1.00000i 0 1.29517 0 0 0
3761.17 0 0 0 − 1.00000i 0 −0.979785 0 0 0
3761.18 0 0 0 1.00000i 0 −0.979785 0 0 0
3761.19 0 0 0 − 1.00000i 0 −0.450010 0 0 0
3761.20 0 0 0 1.00000i 0 −0.450010 0 0 0
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3761.40
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
57.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 6840.2.r.b yes 40
3.b odd 2 1 6840.2.r.a ✓ 40
19.b odd 2 1 6840.2.r.a ✓ 40
57.d even 2 1 inner 6840.2.r.b yes 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6840.2.r.a ✓ 40 3.b odd 2 1
6840.2.r.a ✓ 40 19.b odd 2 1
6840.2.r.b yes 40 1.a even 1 1 trivial
6840.2.r.b yes 40 57.d even 2 1 inner