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Properties

Label 3744.2.j.a
Level $3744$
Weight $2$
Character orbit 3744.j
Analytic conductor $29.896$
Analytic rank $0$
Dimension $48$
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [3744,2,Mod(2159,3744)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3744, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 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("3744.2159"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 3744 = 2^{5} \cdot 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3744.j (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(29.8959905168\)
Analytic rank: \(0\)
Dimension: \(48\)
Twist minimal: no (minimal twist has level 936)
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.

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 32 q^{19} + 48 q^{25} + 32 q^{43} - 48 q^{49} - 32 q^{67} - 32 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2159.1 0 0 0 −3.97430 0 0.281432i 0 0 0
2159.2 0 0 0 −3.97430 0 − 0.281432i 0 0 0
2159.3 0 0 0 −3.68104 0 5.03597i 0 0 0
2159.4 0 0 0 −3.68104 0 − 5.03597i 0 0 0
2159.5 0 0 0 −3.43430 0 1.48385i 0 0 0
2159.6 0 0 0 −3.43430 0 − 1.48385i 0 0 0
2159.7 0 0 0 −3.23777 0 − 4.33596i 0 0 0
2159.8 0 0 0 −3.23777 0 4.33596i 0 0 0
2159.9 0 0 0 −2.74521 0 0.0889956i 0 0 0
2159.10 0 0 0 −2.74521 0 − 0.0889956i 0 0 0
2159.11 0 0 0 −2.44507 0 − 2.55268i 0 0 0
2159.12 0 0 0 −2.44507 0 2.55268i 0 0 0
2159.13 0 0 0 −2.22260 0 − 0.534598i 0 0 0
2159.14 0 0 0 −2.22260 0 0.534598i 0 0 0
2159.15 0 0 0 −0.802827 0 1.93839i 0 0 0
2159.16 0 0 0 −0.802827 0 − 1.93839i 0 0 0
2159.17 0 0 0 −0.743520 0 1.72000i 0 0 0
2159.18 0 0 0 −0.743520 0 − 1.72000i 0 0 0
2159.19 0 0 0 −0.678099 0 4.56073i 0 0 0
2159.20 0 0 0 −0.678099 0 − 4.56073i 0 0 0
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2159.48
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
8.d odd 2 1 inner
24.f even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3744.2.j.a 48
3.b odd 2 1 inner 3744.2.j.a 48
4.b odd 2 1 936.2.j.a ✓ 48
8.b even 2 1 936.2.j.a ✓ 48
8.d odd 2 1 inner 3744.2.j.a 48
12.b even 2 1 936.2.j.a ✓ 48
24.f even 2 1 inner 3744.2.j.a 48
24.h odd 2 1 936.2.j.a ✓ 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
936.2.j.a ✓ 48 4.b odd 2 1
936.2.j.a ✓ 48 8.b even 2 1
936.2.j.a ✓ 48 12.b even 2 1
936.2.j.a ✓ 48 24.h odd 2 1
3744.2.j.a 48 1.a even 1 1 trivial
3744.2.j.a 48 3.b odd 2 1 inner
3744.2.j.a 48 8.d odd 2 1 inner
3744.2.j.a 48 24.f even 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(3744, [\chi])\).