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Properties

Label 936.2.w.j
Level $936$
Weight $2$
Character orbit 936.w
Analytic conductor $7.474$
Analytic rank $0$
Dimension $24$
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [936,2,Mod(307,936)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(936, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 2, 0, 1])) 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("936.307"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 936 = 2^{3} \cdot 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 936.w (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [24,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.47399762919\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(i)\)
Twist minimal: no (minimal twist has level 312)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 6 q^{8} - 8 q^{11} - 36 q^{14} + 28 q^{16} + 20 q^{19} + 20 q^{20} + 20 q^{22} - 12 q^{26} - 16 q^{28} + 30 q^{32} + 16 q^{34} - 16 q^{35} + 36 q^{40} + 12 q^{41} + 32 q^{44} - 44 q^{46} + 36 q^{50}+ \cdots - 68 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
307.1 −1.41280 − 0.0632999i 0 1.99199 + 0.178860i 1.61967 − 1.61967i 0 2.08158 + 2.08158i −2.80295 − 0.378785i 0 −2.39080 + 2.18574i
307.2 −1.30513 + 0.544643i 0 1.40673 − 1.42166i 0.220561 − 0.220561i 0 −0.834354 − 0.834354i −1.06167 + 2.62161i 0 −0.167734 + 0.407987i
307.3 −1.13060 − 0.849559i 0 0.556500 + 1.92102i −1.67211 + 1.67211i 0 1.16012 + 1.16012i 1.00284 − 2.64468i 0 3.31105 − 0.469928i
307.4 −0.849559 − 1.13060i 0 −0.556500 + 1.92102i 1.67211 − 1.67211i 0 −1.16012 − 1.16012i 2.64468 − 1.00284i 0 −3.31105 − 0.469928i
307.5 −0.437240 + 1.34492i 0 −1.61764 − 1.17611i 1.98058 − 1.98058i 0 3.05943 + 3.05943i 2.28908 − 1.66136i 0 1.79774 + 3.52971i
307.6 −0.0632999 − 1.41280i 0 −1.99199 + 0.178860i −1.61967 + 1.61967i 0 −2.08158 − 2.08158i 0.378785 + 2.80295i 0 2.39080 + 2.18574i
307.7 0.140104 + 1.40726i 0 −1.96074 + 0.394325i −3.08779 + 3.08779i 0 2.85816 + 2.85816i −0.829624 − 2.70402i 0 −4.77792 − 3.91270i
307.8 0.407446 + 1.35425i 0 −1.66797 + 1.10357i 0.273741 − 0.273741i 0 −1.75949 − 1.75949i −2.17411 − 1.80921i 0 0.482247 + 0.259178i
307.9 0.544643 − 1.30513i 0 −1.40673 − 1.42166i −0.220561 + 0.220561i 0 0.834354 + 0.834354i −2.62161 + 1.06167i 0 0.167734 + 0.407987i
307.10 1.34492 − 0.437240i 0 1.61764 − 1.17611i −1.98058 + 1.98058i 0 −3.05943 − 3.05943i 1.66136 − 2.28908i 0 −1.79774 + 3.52971i
307.11 1.35425 + 0.407446i 0 1.66797 + 1.10357i −0.273741 + 0.273741i 0 1.75949 + 1.75949i 1.80921 + 2.17411i 0 −0.482247 + 0.259178i
307.12 1.40726 + 0.140104i 0 1.96074 + 0.394325i 3.08779 − 3.08779i 0 −2.85816 − 2.85816i 2.70402 + 0.829624i 0 4.77792 − 3.91270i
811.1 −1.41280 + 0.0632999i 0 1.99199 − 0.178860i 1.61967 + 1.61967i 0 2.08158 − 2.08158i −2.80295 + 0.378785i 0 −2.39080 − 2.18574i
811.2 −1.30513 − 0.544643i 0 1.40673 + 1.42166i 0.220561 + 0.220561i 0 −0.834354 + 0.834354i −1.06167 − 2.62161i 0 −0.167734 − 0.407987i
811.3 −1.13060 + 0.849559i 0 0.556500 − 1.92102i −1.67211 − 1.67211i 0 1.16012 − 1.16012i 1.00284 + 2.64468i 0 3.31105 + 0.469928i
811.4 −0.849559 + 1.13060i 0 −0.556500 − 1.92102i 1.67211 + 1.67211i 0 −1.16012 + 1.16012i 2.64468 + 1.00284i 0 −3.31105 + 0.469928i
811.5 −0.437240 − 1.34492i 0 −1.61764 + 1.17611i 1.98058 + 1.98058i 0 3.05943 − 3.05943i 2.28908 + 1.66136i 0 1.79774 − 3.52971i
811.6 −0.0632999 + 1.41280i 0 −1.99199 − 0.178860i −1.61967 − 1.61967i 0 −2.08158 + 2.08158i 0.378785 − 2.80295i 0 2.39080 − 2.18574i
811.7 0.140104 − 1.40726i 0 −1.96074 − 0.394325i −3.08779 − 3.08779i 0 2.85816 − 2.85816i −0.829624 + 2.70402i 0 −4.77792 + 3.91270i
811.8 0.407446 − 1.35425i 0 −1.66797 − 1.10357i 0.273741 + 0.273741i 0 −1.75949 + 1.75949i −2.17411 + 1.80921i 0 0.482247 − 0.259178i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 307.12
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 inner
13.d odd 4 1 inner
104.m even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 936.2.w.j 24
3.b odd 2 1 312.2.t.e ✓ 24
8.d odd 2 1 inner 936.2.w.j 24
12.b even 2 1 1248.2.bb.f 24
13.d odd 4 1 inner 936.2.w.j 24
24.f even 2 1 312.2.t.e ✓ 24
24.h odd 2 1 1248.2.bb.f 24
39.f even 4 1 312.2.t.e ✓ 24
104.m even 4 1 inner 936.2.w.j 24
156.l odd 4 1 1248.2.bb.f 24
312.w odd 4 1 312.2.t.e ✓ 24
312.y even 4 1 1248.2.bb.f 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
312.2.t.e ✓ 24 3.b odd 2 1
312.2.t.e ✓ 24 24.f even 2 1
312.2.t.e ✓ 24 39.f even 4 1
312.2.t.e ✓ 24 312.w odd 4 1
936.2.w.j 24 1.a even 1 1 trivial
936.2.w.j 24 8.d odd 2 1 inner
936.2.w.j 24 13.d odd 4 1 inner
936.2.w.j 24 104.m even 4 1 inner
1248.2.bb.f 24 12.b even 2 1
1248.2.bb.f 24 24.h odd 2 1
1248.2.bb.f 24 156.l odd 4 1
1248.2.bb.f 24 312.y even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(936, [\chi])\):

\( T_{5}^{24} + 484T_{5}^{20} + 48256T_{5}^{16} + 1683456T_{5}^{12} + 19318784T_{5}^{8} + 615424T_{5}^{4} + 4096 \) Copy content Toggle raw display
\( T_{7}^{24} + 740T_{7}^{20} + 173184T_{7}^{16} + 13927936T_{7}^{12} + 385439744T_{7}^{8} + 2647475200T_{7}^{4} + 3782742016 \) Copy content Toggle raw display