Properties

Label 6840.2.r.a
Level $6840$
Weight $2$
Character orbit 6840.r
Analytic conductor $54.618$
Analytic rank $0$
Dimension $40$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [6840,2,Mod(3761,6840)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma:// 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);
 
Copy content sage: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")
 
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

Copy content comment:select newform
 
Copy content sage:traces = [40,0,0,0,0,0,-8,0,0,0,0,0,0,0,0,0,0,0,-4,0,0,0,0,0,-40,0,0,0, -24] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(29)] == traces)
 
Copy content 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 algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 40 q - 8 q^{7} - 4 q^{19} - 40 q^{25} - 24 q^{29} + 8 q^{41} + 8 q^{43} + 64 q^{49} - 32 q^{53} - 16 q^{59} - 16 q^{61} - 16 q^{65} + 32 q^{71} + 24 q^{89} - 4 q^{95}+O(q^{100}) \) Copy content Toggle raw display

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.

Copy content comment:embeddings in the coefficient field
 
Copy content 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 −3.52359 0 0 0
3761.2 0 0 0 1.00000i 0 −3.52359 0 0 0
3761.3 0 0 0 − 1.00000i 0 3.88206 0 0 0
3761.4 0 0 0 1.00000i 0 3.88206 0 0 0
3761.5 0 0 0 − 1.00000i 0 −4.28640 0 0 0
3761.6 0 0 0 1.00000i 0 −4.28640 0 0 0
3761.7 0 0 0 − 1.00000i 0 −4.62344 0 0 0
3761.8 0 0 0 1.00000i 0 −4.62344 0 0 0
3761.9 0 0 0 − 1.00000i 0 −4.17466 0 0 0
3761.10 0 0 0 1.00000i 0 −4.17466 0 0 0
3761.11 0 0 0 − 1.00000i 0 3.06856 0 0 0
3761.12 0 0 0 1.00000i 0 3.06856 0 0 0
3761.13 0 0 0 − 1.00000i 0 −0.245139 0 0 0
3761.14 0 0 0 1.00000i 0 −0.245139 0 0 0
3761.15 0 0 0 − 1.00000i 0 2.46536 0 0 0
3761.16 0 0 0 1.00000i 0 2.46536 0 0 0
3761.17 0 0 0 − 1.00000i 0 −0.0196792 0 0 0
3761.18 0 0 0 1.00000i 0 −0.0196792 0 0 0
3761.19 0 0 0 − 1.00000i 0 −0.352958 0 0 0
3761.20 0 0 0 1.00000i 0 −0.352958 0 0 0
See all 40 embeddings
\(n\): e.g. 2-40 or 80-90
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.a ✓ 40
3.b odd 2 1 6840.2.r.b yes 40
19.b odd 2 1 6840.2.r.b yes 40
57.d even 2 1 inner 6840.2.r.a ✓ 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6840.2.r.a ✓ 40 1.a even 1 1 trivial
6840.2.r.a ✓ 40 57.d even 2 1 inner
6840.2.r.b yes 40 3.b odd 2 1
6840.2.r.b yes 40 19.b odd 2 1