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Properties

Label 2496.2.j.f
Level $2496$
Weight $2$
Character orbit 2496.j
Analytic conductor $19.931$
Analytic rank $0$
Dimension $32$
Inner twists $4$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2496,2,Mod(287,2496)] 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("2496.287"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2496, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2496 = 2^{6} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2496.j (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [32,0,0,0,0,0,0,0,-4,0,0,0,0,0,0,0,0,0,32] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(19.9306603445\)
Analytic rank: \(0\)
Dimension: \(32\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 4 q^{9} + 32 q^{19} + 56 q^{25} - 24 q^{27} + 32 q^{33} + 16 q^{43} + 8 q^{49} - 40 q^{51} - 8 q^{57} - 16 q^{67} - 16 q^{73} - 120 q^{75} - 44 q^{81} - 80 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
287.1 0 −1.68940 − 0.382025i 0 −2.10522 0 − 1.06700i 0 2.70811 + 1.29078i 0
287.2 0 −1.68940 − 0.382025i 0 2.10522 0 1.06700i 0 2.70811 + 1.29078i 0
287.3 0 −1.68940 + 0.382025i 0 −2.10522 0 1.06700i 0 2.70811 − 1.29078i 0
287.4 0 −1.68940 + 0.382025i 0 2.10522 0 − 1.06700i 0 2.70811 − 1.29078i 0
287.5 0 −1.31574 − 1.12642i 0 −4.10885 0 − 3.88378i 0 0.462352 + 2.96416i 0
287.6 0 −1.31574 − 1.12642i 0 4.10885 0 3.88378i 0 0.462352 + 2.96416i 0
287.7 0 −1.31574 + 1.12642i 0 −4.10885 0 3.88378i 0 0.462352 − 2.96416i 0
287.8 0 −1.31574 + 1.12642i 0 4.10885 0 − 3.88378i 0 0.462352 − 2.96416i 0
287.9 0 −1.22223 − 1.22726i 0 −3.17169 0 − 1.30638i 0 −0.0123113 + 2.99997i 0
287.10 0 −1.22223 − 1.22726i 0 3.17169 0 1.30638i 0 −0.0123113 + 2.99997i 0
287.11 0 −1.22223 + 1.22726i 0 −3.17169 0 1.30638i 0 −0.0123113 − 2.99997i 0
287.12 0 −1.22223 + 1.22726i 0 3.17169 0 − 1.30638i 0 −0.0123113 − 2.99997i 0
287.13 0 −0.0969806 − 1.72933i 0 −3.03737 0 2.31810i 0 −2.98119 + 0.335424i 0
287.14 0 −0.0969806 − 1.72933i 0 3.03737 0 − 2.31810i 0 −2.98119 + 0.335424i 0
287.15 0 −0.0969806 + 1.72933i 0 −3.03737 0 − 2.31810i 0 −2.98119 − 0.335424i 0
287.16 0 −0.0969806 + 1.72933i 0 3.03737 0 2.31810i 0 −2.98119 − 0.335424i 0
287.17 0 0.517777 − 1.65285i 0 −0.379917 0 − 2.38318i 0 −2.46381 − 1.71161i 0
287.18 0 0.517777 − 1.65285i 0 0.379917 0 2.38318i 0 −2.46381 − 1.71161i 0
287.19 0 0.517777 + 1.65285i 0 −0.379917 0 2.38318i 0 −2.46381 + 1.71161i 0
287.20 0 0.517777 + 1.65285i 0 0.379917 0 − 2.38318i 0 −2.46381 + 1.71161i 0
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 287.32
Significant digits:
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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 2496.2.j.f yes 32
3.b odd 2 1 inner 2496.2.j.f yes 32
4.b odd 2 1 2496.2.j.e ✓ 32
8.b even 2 1 2496.2.j.e ✓ 32
8.d odd 2 1 inner 2496.2.j.f yes 32
12.b even 2 1 2496.2.j.e ✓ 32
24.f even 2 1 inner 2496.2.j.f yes 32
24.h odd 2 1 2496.2.j.e ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2496.2.j.e ✓ 32 4.b odd 2 1
2496.2.j.e ✓ 32 8.b even 2 1
2496.2.j.e ✓ 32 12.b even 2 1
2496.2.j.e ✓ 32 24.h odd 2 1
2496.2.j.f yes 32 1.a even 1 1 trivial
2496.2.j.f yes 32 3.b odd 2 1 inner
2496.2.j.f yes 32 8.d odd 2 1 inner
2496.2.j.f yes 32 24.f even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2496, [\chi])\):

\( T_{5}^{16} - 54 T_{5}^{14} + 1149 T_{5}^{12} - 12248 T_{5}^{10} + 68228 T_{5}^{8} - 186336 T_{5}^{6} + \cdots + 2368 \) Copy content Toggle raw display
\( T_{19}^{8} - 8T_{19}^{7} - 68T_{19}^{6} + 588T_{19}^{5} + 708T_{19}^{4} - 10440T_{19}^{3} + 16920T_{19}^{2} - 1728T_{19} - 6912 \) Copy content Toggle raw display
\( T_{43}^{8} - 4T_{43}^{7} - 163T_{43}^{6} + 808T_{43}^{5} + 5772T_{43}^{4} - 33760T_{43}^{3} + 22816T_{43}^{2} + 17472T_{43} - 11456 \) Copy content Toggle raw display