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Properties

Label 3360.2.j.e
Level $3360$
Weight $2$
Character orbit 3360.j
Analytic conductor $26.830$
Analytic rank $0$
Dimension $32$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [3360,2,Mod(1009,3360)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3360, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0, 1, 0])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3360.1009"); 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.j (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [32,0,-32,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(26.8297350792\)
Analytic rank: \(0\)
Dimension: \(32\)
Twist minimal: no (minimal twist has level 840)
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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 32 q^{3} + 32 q^{9} + 32 q^{13} - 16 q^{25} - 32 q^{27} - 32 q^{31} - 4 q^{35} - 8 q^{37} - 32 q^{39} + 24 q^{41} + 8 q^{43} - 32 q^{49} - 24 q^{53} - 24 q^{55} + 8 q^{65} - 24 q^{67} - 40 q^{71}+ \cdots + 48 q^{95}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1009.1 0 −1.00000 0 −2.20890 − 0.347530i 0 − 1.00000i 0 1.00000 0
1009.2 0 −1.00000 0 −2.20890 + 0.347530i 0 1.00000i 0 1.00000 0
1009.3 0 −1.00000 0 −2.15737 − 0.588016i 0 1.00000i 0 1.00000 0
1009.4 0 −1.00000 0 −2.15737 + 0.588016i 0 − 1.00000i 0 1.00000 0
1009.5 0 −1.00000 0 −2.03532 − 0.926004i 0 − 1.00000i 0 1.00000 0
1009.6 0 −1.00000 0 −2.03532 + 0.926004i 0 1.00000i 0 1.00000 0
1009.7 0 −1.00000 0 −1.48054 − 1.67571i 0 − 1.00000i 0 1.00000 0
1009.8 0 −1.00000 0 −1.48054 + 1.67571i 0 1.00000i 0 1.00000 0
1009.9 0 −1.00000 0 −0.980656 − 2.00956i 0 1.00000i 0 1.00000 0
1009.10 0 −1.00000 0 −0.980656 + 2.00956i 0 − 1.00000i 0 1.00000 0
1009.11 0 −1.00000 0 −0.836338 − 2.07377i 0 − 1.00000i 0 1.00000 0
1009.12 0 −1.00000 0 −0.836338 + 2.07377i 0 1.00000i 0 1.00000 0
1009.13 0 −1.00000 0 −0.725275 − 2.11518i 0 1.00000i 0 1.00000 0
1009.14 0 −1.00000 0 −0.725275 + 2.11518i 0 − 1.00000i 0 1.00000 0
1009.15 0 −1.00000 0 −0.241469 − 2.22299i 0 − 1.00000i 0 1.00000 0
1009.16 0 −1.00000 0 −0.241469 + 2.22299i 0 1.00000i 0 1.00000 0
1009.17 0 −1.00000 0 −0.123450 − 2.23266i 0 1.00000i 0 1.00000 0
1009.18 0 −1.00000 0 −0.123450 + 2.23266i 0 − 1.00000i 0 1.00000 0
1009.19 0 −1.00000 0 0.961089 − 2.01899i 0 1.00000i 0 1.00000 0
1009.20 0 −1.00000 0 0.961089 + 2.01899i 0 − 1.00000i 0 1.00000 0
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1009.32
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
40.f 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.j.e 32
4.b odd 2 1 840.2.j.f yes 32
5.b even 2 1 3360.2.j.f 32
8.b even 2 1 3360.2.j.f 32
8.d odd 2 1 840.2.j.e ✓ 32
20.d odd 2 1 840.2.j.e ✓ 32
40.e odd 2 1 840.2.j.f yes 32
40.f even 2 1 inner 3360.2.j.e 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
840.2.j.e ✓ 32 8.d odd 2 1
840.2.j.e ✓ 32 20.d odd 2 1
840.2.j.f yes 32 4.b odd 2 1
840.2.j.f yes 32 40.e odd 2 1
3360.2.j.e 32 1.a even 1 1 trivial
3360.2.j.e 32 40.f even 2 1 inner
3360.2.j.f 32 5.b even 2 1
3360.2.j.f 32 8.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3360, [\chi])\):

\( T_{11}^{32} + 200 T_{11}^{30} + 17624 T_{11}^{28} + 901760 T_{11}^{26} + 29739664 T_{11}^{24} + \cdots + 1677721600 \) Copy content Toggle raw display
\( T_{13}^{16} - 16 T_{13}^{15} + 24 T_{13}^{14} + 888 T_{13}^{13} - 5072 T_{13}^{12} - 6112 T_{13}^{11} + \cdots - 262144 \) Copy content Toggle raw display