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Properties

Label 912.2.bh.g
Level $912$
Weight $2$
Character orbit 912.bh
Analytic conductor $7.282$
Analytic rank $0$
Dimension $24$
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] = [912,2,Mod(239,912)] 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("912.239"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(912, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 0, 3, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 912 = 2^{4} \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 912.bh (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [24,0,6,0,0,0,0,0,-8,0,0,0,-6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.28235666434\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 6 q^{3} - 8 q^{9} - 6 q^{13} - 9 q^{15} + 6 q^{19} - 12 q^{21} + 26 q^{25} + 9 q^{33} - 32 q^{37} + 42 q^{43} - 10 q^{45} - 32 q^{49} + 51 q^{51} - 24 q^{55} - 9 q^{57} - 6 q^{61} + 12 q^{67} - 14 q^{69}+ \cdots - 33 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
239.1 0 −1.62428 + 0.601422i 0 −3.50986 − 2.02642i 0 0.196665i 0 2.27658 − 1.95376i 0
239.2 0 −1.53419 + 0.803908i 0 2.72078 + 1.57084i 0 − 4.27804i 0 1.70747 − 2.46669i 0
239.3 0 −0.579078 + 1.63238i 0 2.37836 + 1.37315i 0 4.29408i 0 −2.32934 − 1.89055i 0
239.4 0 −0.462645 + 1.66912i 0 0.299633 + 0.172993i 0 − 0.167877i 0 −2.57192 − 1.54442i 0
239.5 0 −0.291294 − 1.70738i 0 3.50986 + 2.02642i 0 0.196665i 0 −2.83030 + 0.994699i 0
239.6 0 −0.0708896 − 1.73060i 0 −2.72078 − 1.57084i 0 − 4.27804i 0 −2.98995 + 0.245363i 0
239.7 0 0.560953 + 1.63870i 0 −1.84530 − 1.06538i 0 − 2.59062i 0 −2.37066 + 1.83847i 0
239.8 0 1.12414 − 1.31769i 0 −2.37836 − 1.37315i 0 4.29408i 0 −0.472598 − 2.96254i 0
239.9 0 1.21418 − 1.23522i 0 −0.299633 − 0.172993i 0 − 0.167877i 0 −0.0515467 − 2.99956i 0
239.10 0 1.34700 + 1.08885i 0 1.83758 + 1.06093i 0 2.54579i 0 0.628815 + 2.93336i 0
239.11 0 1.61647 + 0.622111i 0 −1.83758 − 1.06093i 0 2.54579i 0 2.22596 + 2.01125i 0
239.12 0 1.69963 − 0.333549i 0 1.84530 + 1.06538i 0 − 2.59062i 0 2.77749 − 1.13382i 0
767.1 0 −1.62428 − 0.601422i 0 −3.50986 + 2.02642i 0 − 0.196665i 0 2.27658 + 1.95376i 0
767.2 0 −1.53419 − 0.803908i 0 2.72078 − 1.57084i 0 4.27804i 0 1.70747 + 2.46669i 0
767.3 0 −0.579078 − 1.63238i 0 2.37836 − 1.37315i 0 − 4.29408i 0 −2.32934 + 1.89055i 0
767.4 0 −0.462645 − 1.66912i 0 0.299633 − 0.172993i 0 0.167877i 0 −2.57192 + 1.54442i 0
767.5 0 −0.291294 + 1.70738i 0 3.50986 − 2.02642i 0 − 0.196665i 0 −2.83030 − 0.994699i 0
767.6 0 −0.0708896 + 1.73060i 0 −2.72078 + 1.57084i 0 4.27804i 0 −2.98995 − 0.245363i 0
767.7 0 0.560953 − 1.63870i 0 −1.84530 + 1.06538i 0 2.59062i 0 −2.37066 − 1.83847i 0
767.8 0 1.12414 + 1.31769i 0 −2.37836 + 1.37315i 0 − 4.29408i 0 −0.472598 + 2.96254i 0
See all 24 embeddings
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 239.12
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
76.g odd 6 1 inner
228.m even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 912.2.bh.g yes 24
3.b odd 2 1 inner 912.2.bh.g yes 24
4.b odd 2 1 912.2.bh.e ✓ 24
12.b even 2 1 912.2.bh.e ✓ 24
19.c even 3 1 912.2.bh.e ✓ 24
57.h odd 6 1 912.2.bh.e ✓ 24
76.g odd 6 1 inner 912.2.bh.g yes 24
228.m even 6 1 inner 912.2.bh.g yes 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
912.2.bh.e ✓ 24 4.b odd 2 1
912.2.bh.e ✓ 24 12.b even 2 1
912.2.bh.e ✓ 24 19.c even 3 1
912.2.bh.e ✓ 24 57.h odd 6 1
912.2.bh.g yes 24 1.a even 1 1 trivial
912.2.bh.g yes 24 3.b odd 2 1 inner
912.2.bh.g yes 24 76.g odd 6 1 inner
912.2.bh.g yes 24 228.m even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(912, [\chi])\):

\( T_{5}^{24} - 43 T_{5}^{22} + 1157 T_{5}^{20} - 19244 T_{5}^{18} + 233812 T_{5}^{16} - 2026568 T_{5}^{14} + \cdots + 8952064 \) Copy content Toggle raw display
\( T_{7}^{12} + 50T_{7}^{10} + 869T_{7}^{8} + 6108T_{7}^{6} + 15084T_{7}^{4} + 988T_{7}^{2} + 16 \) Copy content Toggle raw display
\( T_{43}^{12} - 21 T_{43}^{11} + 4 T_{43}^{10} + 3003 T_{43}^{9} - 5312 T_{43}^{8} - 381579 T_{43}^{7} + \cdots + 2834497600 \) Copy content Toggle raw display