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Properties

Label 936.2.be.a
Level $936$
Weight $2$
Character orbit 936.be
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(685,936)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(936, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 3, 0, 4])) 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("936.685"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 936 = 2^{3} \cdot 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 936.be (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [24,1,0,-1,0,0,-2,10,0,-3,0,0,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(14)] == 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 \(\Q(\zeta_{6})\)
Twist minimal: no (minimal twist has level 104)
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + q^{2} - q^{4} - 2 q^{7} + 10 q^{8} - 3 q^{10} - 8 q^{14} - q^{16} + 11 q^{20} - 2 q^{22} + 14 q^{23} - 12 q^{25} + 3 q^{26} - 4 q^{28} - 8 q^{31} + 21 q^{32} + 14 q^{34} - 12 q^{38} + 54 q^{40}+ \cdots + 17 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
685.1 −1.33456 + 0.467912i 0 1.56212 − 1.24892i 0.497079i 0 −0.845740 − 1.46486i −1.50036 + 2.39769i 0 −0.232589 − 0.663383i
685.2 −1.24938 − 0.662601i 0 1.12192 + 1.65569i 1.78237i 0 1.65832 + 2.87229i −0.304647 − 2.81197i 0 1.18100 − 2.22687i
685.3 −1.02392 + 0.975496i 0 0.0968157 − 1.99766i − 2.24007i 0 0.471952 + 0.817445i 1.84957 + 2.13988i 0 2.18518 + 2.29365i
685.4 −0.758596 − 1.19354i 0 −0.849063 + 1.81083i 0.556100i 0 −2.30251 − 3.98807i 2.80539 − 0.360297i 0 0.663726 − 0.421855i
685.5 −0.477121 − 1.33130i 0 −1.54471 + 1.27038i − 4.18204i 0 0.818571 + 1.41781i 2.42827 + 1.45035i 0 −5.56755 + 1.99534i
685.6 −0.332845 + 1.37449i 0 −1.77843 − 0.914983i 2.24007i 0 0.471952 + 0.817445i 1.84957 − 2.13988i 0 −3.07895 − 0.745597i
685.7 0.262058 + 1.38972i 0 −1.86265 + 0.728375i − 0.497079i 0 −0.845740 − 1.46486i −1.50036 − 2.39769i 0 0.690801 − 0.130263i
685.8 0.628723 − 1.26677i 0 −1.20941 − 1.59290i 2.59989i 0 −0.300588 − 0.520633i −2.77822 + 0.530559i 0 3.29346 + 1.63461i
685.9 0.782694 − 1.17788i 0 −0.774781 − 1.84383i − 2.59989i 0 −0.300588 − 0.520633i −2.77822 − 0.530559i 0 −3.06235 − 2.03492i
685.10 1.19852 + 0.750697i 0 0.872907 + 1.79945i − 1.78237i 0 1.65832 + 2.87229i −0.304647 + 2.81197i 0 1.33802 − 2.13621i
685.11 1.39150 − 0.252450i 0 1.87254 − 0.702569i 4.18204i 0 0.818571 + 1.41781i 2.42827 − 1.45035i 0 1.05576 + 5.81931i
685.12 1.41293 + 0.0601950i 0 1.99275 + 0.170103i − 0.556100i 0 −2.30251 − 3.98807i 2.80539 + 0.360297i 0 0.0334744 − 0.785731i
757.1 −1.33456 − 0.467912i 0 1.56212 + 1.24892i − 0.497079i 0 −0.845740 + 1.46486i −1.50036 − 2.39769i 0 −0.232589 + 0.663383i
757.2 −1.24938 + 0.662601i 0 1.12192 − 1.65569i − 1.78237i 0 1.65832 − 2.87229i −0.304647 + 2.81197i 0 1.18100 + 2.22687i
757.3 −1.02392 − 0.975496i 0 0.0968157 + 1.99766i 2.24007i 0 0.471952 − 0.817445i 1.84957 − 2.13988i 0 2.18518 − 2.29365i
757.4 −0.758596 + 1.19354i 0 −0.849063 − 1.81083i − 0.556100i 0 −2.30251 + 3.98807i 2.80539 + 0.360297i 0 0.663726 + 0.421855i
757.5 −0.477121 + 1.33130i 0 −1.54471 − 1.27038i 4.18204i 0 0.818571 − 1.41781i 2.42827 − 1.45035i 0 −5.56755 − 1.99534i
757.6 −0.332845 − 1.37449i 0 −1.77843 + 0.914983i − 2.24007i 0 0.471952 − 0.817445i 1.84957 + 2.13988i 0 −3.07895 + 0.745597i
757.7 0.262058 − 1.38972i 0 −1.86265 − 0.728375i 0.497079i 0 −0.845740 + 1.46486i −1.50036 + 2.39769i 0 0.690801 + 0.130263i
757.8 0.628723 + 1.26677i 0 −1.20941 + 1.59290i − 2.59989i 0 −0.300588 + 0.520633i −2.77822 − 0.530559i 0 3.29346 − 1.63461i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 685.12
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner
13.c even 3 1 inner
104.r even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 936.2.be.a 24
3.b odd 2 1 104.2.r.a ✓ 24
8.b even 2 1 inner 936.2.be.a 24
12.b even 2 1 416.2.z.a 24
13.c even 3 1 inner 936.2.be.a 24
24.f even 2 1 416.2.z.a 24
24.h odd 2 1 104.2.r.a ✓ 24
39.i odd 6 1 104.2.r.a ✓ 24
104.r even 6 1 inner 936.2.be.a 24
156.p even 6 1 416.2.z.a 24
312.bh odd 6 1 104.2.r.a ✓ 24
312.bn even 6 1 416.2.z.a 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
104.2.r.a ✓ 24 3.b odd 2 1
104.2.r.a ✓ 24 24.h odd 2 1
104.2.r.a ✓ 24 39.i odd 6 1
104.2.r.a ✓ 24 312.bh odd 6 1
416.2.z.a 24 12.b even 2 1
416.2.z.a 24 24.f even 2 1
416.2.z.a 24 156.p even 6 1
416.2.z.a 24 312.bn even 6 1
936.2.be.a 24 1.a even 1 1 trivial
936.2.be.a 24 8.b even 2 1 inner
936.2.be.a 24 13.c even 3 1 inner
936.2.be.a 24 104.r even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{12} + 33T_{5}^{10} + 351T_{5}^{8} + 1543T_{5}^{6} + 2664T_{5}^{4} + 1152T_{5}^{2} + 144 \) acting on \(S_{2}^{\mathrm{new}}(936, [\chi])\). Copy content Toggle raw display