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Properties

Label 2340.2.fo.a
Level $2340$
Weight $2$
Character orbit 2340.fo
Analytic conductor $18.685$
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] = [2340,2,Mod(1241,2340)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2340, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([0, 6, 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("2340.1241");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2340 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2340.fo (of order \(12\), degree \(4\), 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: \(18.6849940730\)
Analytic rank: \(0\)
Dimension: \(40\)
Relative dimension: \(10\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 40 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q - 12 q^{13} - 20 q^{19} - 28 q^{31} + 12 q^{49} + 16 q^{55} + 48 q^{61} + 16 q^{67} - 20 q^{73} + 80 q^{79} + 20 q^{85} + 4 q^{91} - 40 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1241.1 0 0 0 −0.707107 − 0.707107i 0 −0.541475 + 2.02081i 0 0 0
1241.2 0 0 0 −0.707107 − 0.707107i 0 0.718874 − 2.68287i 0 0 0
1241.3 0 0 0 −0.707107 − 0.707107i 0 −0.420712 + 1.57012i 0 0 0
1241.4 0 0 0 −0.707107 − 0.707107i 0 −0.185315 + 0.691603i 0 0 0
1241.5 0 0 0 −0.707107 − 0.707107i 0 1.29465 − 4.83171i 0 0 0
1241.6 0 0 0 0.707107 + 0.707107i 0 1.29465 − 4.83171i 0 0 0
1241.7 0 0 0 0.707107 + 0.707107i 0 −0.420712 + 1.57012i 0 0 0
1241.8 0 0 0 0.707107 + 0.707107i 0 0.718874 − 2.68287i 0 0 0
1241.9 0 0 0 0.707107 + 0.707107i 0 −0.185315 + 0.691603i 0 0 0
1241.10 0 0 0 0.707107 + 0.707107i 0 −0.541475 + 2.02081i 0 0 0
1601.1 0 0 0 −0.707107 − 0.707107i 0 −4.01056 + 1.07463i 0 0 0
1601.2 0 0 0 −0.707107 − 0.707107i 0 −3.16623 + 0.848389i 0 0 0
1601.3 0 0 0 −0.707107 − 0.707107i 0 1.13373 − 0.303782i 0 0 0
1601.4 0 0 0 −0.707107 − 0.707107i 0 2.64656 − 0.709144i 0 0 0
1601.5 0 0 0 −0.707107 − 0.707107i 0 2.53048 − 0.678039i 0 0 0
1601.6 0 0 0 0.707107 + 0.707107i 0 1.13373 − 0.303782i 0 0 0
1601.7 0 0 0 0.707107 + 0.707107i 0 −3.16623 + 0.848389i 0 0 0
1601.8 0 0 0 0.707107 + 0.707107i 0 −4.01056 + 1.07463i 0 0 0
1601.9 0 0 0 0.707107 + 0.707107i 0 2.53048 − 0.678039i 0 0 0
1601.10 0 0 0 0.707107 + 0.707107i 0 2.64656 − 0.709144i 0 0 0
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1241.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
13.f odd 12 1 inner
39.k even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2340.2.fo.a ✓ 40
3.b odd 2 1 inner 2340.2.fo.a ✓ 40
13.f odd 12 1 inner 2340.2.fo.a ✓ 40
39.k even 12 1 inner 2340.2.fo.a ✓ 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2340.2.fo.a ✓ 40 1.a even 1 1 trivial
2340.2.fo.a ✓ 40 3.b odd 2 1 inner
2340.2.fo.a ✓ 40 13.f odd 12 1 inner
2340.2.fo.a ✓ 40 39.k even 12 1 inner