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Properties

Label 3360.2.q.a
Level $3360$
Weight $2$
Character orbit 3360.q
Analytic conductor $26.830$
Analytic rank $0$
Dimension $48$
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] = [3360,2,Mod(2239,3360)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3360, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3360.2239");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3360 = 2^{5} \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3360.q (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: \(26.8297350792\)
Analytic rank: \(0\)
Dimension: \(48\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 8 q^{5} - 48 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q - 8 q^{5} - 48 q^{9} + 8 q^{25} - 16 q^{33} + 8 q^{45} - 16 q^{49} + 16 q^{73} + 48 q^{77} + 48 q^{81} - 40 q^{85} + 48 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2239.1 0 − 1.00000i 0 −2.14018 − 0.647801i 0 1.87696 + 1.86467i 0 −1.00000 0
2239.2 0 1.00000i 0 −2.14018 + 0.647801i 0 1.87696 − 1.86467i 0 −1.00000 0
2239.3 0 − 1.00000i 0 −1.64466 − 1.51495i 0 1.85401 − 1.88750i 0 −1.00000 0
2239.4 0 1.00000i 0 −1.64466 + 1.51495i 0 1.85401 + 1.88750i 0 −1.00000 0
2239.5 0 − 1.00000i 0 0.742720 + 2.10912i 0 1.31337 + 2.29675i 0 −1.00000 0
2239.6 0 1.00000i 0 0.742720 − 2.10912i 0 1.31337 − 2.29675i 0 −1.00000 0
2239.7 0 − 1.00000i 0 0.500413 + 2.17935i 0 −2.63707 − 0.214187i 0 −1.00000 0
2239.8 0 1.00000i 0 0.500413 − 2.17935i 0 −2.63707 + 0.214187i 0 −1.00000 0
2239.9 0 − 1.00000i 0 −2.21304 − 0.320091i 0 −1.58416 + 2.11906i 0 −1.00000 0
2239.10 0 1.00000i 0 −2.21304 + 0.320091i 0 −1.58416 − 2.11906i 0 −1.00000 0
2239.11 0 − 1.00000i 0 −1.64466 + 1.51495i 0 −1.85401 − 1.88750i 0 −1.00000 0
2239.12 0 1.00000i 0 −1.64466 − 1.51495i 0 −1.85401 + 1.88750i 0 −1.00000 0
2239.13 0 − 1.00000i 0 2.00218 + 0.995638i 0 −1.37846 − 2.25829i 0 −1.00000 0
2239.14 0 1.00000i 0 2.00218 − 0.995638i 0 −1.37846 + 2.25829i 0 −1.00000 0
2239.15 0 − 1.00000i 0 −1.90730 − 1.16714i 0 −0.473165 − 2.60310i 0 −1.00000 0
2239.16 0 1.00000i 0 −1.90730 + 1.16714i 0 −0.473165 + 2.60310i 0 −1.00000 0
2239.17 0 − 1.00000i 0 −0.590126 + 2.15679i 0 −2.22832 + 1.42640i 0 −1.00000 0
2239.18 0 1.00000i 0 −0.590126 − 2.15679i 0 −2.22832 − 1.42640i 0 −1.00000 0
2239.19 0 − 1.00000i 0 2.00218 − 0.995638i 0 1.37846 − 2.25829i 0 −1.00000 0
2239.20 0 1.00000i 0 2.00218 + 0.995638i 0 1.37846 + 2.25829i 0 −1.00000 0
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2239.48
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
35.c odd 2 1 inner
140.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3360.2.q.a ✓ 48
4.b odd 2 1 inner 3360.2.q.a ✓ 48
5.b even 2 1 3360.2.q.b yes 48
7.b odd 2 1 3360.2.q.b yes 48
20.d odd 2 1 3360.2.q.b yes 48
28.d even 2 1 3360.2.q.b yes 48
35.c odd 2 1 inner 3360.2.q.a ✓ 48
140.c even 2 1 inner 3360.2.q.a ✓ 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3360.2.q.a ✓ 48 1.a even 1 1 trivial
3360.2.q.a ✓ 48 4.b odd 2 1 inner
3360.2.q.a ✓ 48 35.c odd 2 1 inner
3360.2.q.a ✓ 48 140.c even 2 1 inner
3360.2.q.b yes 48 5.b even 2 1
3360.2.q.b yes 48 7.b odd 2 1
3360.2.q.b yes 48 20.d odd 2 1
3360.2.q.b yes 48 28.d even 2 1