↑

Properties

Label 126.12.k.a
Level $126$
Weight $12$
Character orbit 126.k
Analytic conductor $96.811$
Analytic rank $0$
Dimension $56$
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [126,12,Mod(17,126)] 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("126.17"); S:= CuspForms(chi, 12); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(126, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 1])) N = Newforms(chi, 12, names="a")
 
Level: \( N \) \(=\) \( 126 = 2 \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 126.k (of order \(6\), degree \(2\), 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: \(96.8112407505\)
Analytic rank: \(0\)
Dimension: \(56\)
Relative dimension: \(28\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 56 q + 28672 q^{4} + 54388 q^{7} - 490368 q^{10} - 29360128 q^{16} - 3299448 q^{19} - 41411328 q^{22} - 187110112 q^{25} - 125980672 q^{28} + 94360452 q^{31} + 174871424 q^{37} - 502136832 q^{40} - 2133688480 q^{43}+ \cdots + 507889989888 q^{94}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1 −27.7128 + 16.0000i 0 512.000 − 886.810i −5242.35 − 9080.02i 0 4958.60 − 44189.8i 32768.0i 0 290561. + 167755.i
17.2 −27.7128 + 16.0000i 0 512.000 − 886.810i −4764.94 − 8253.11i 0 37556.4 + 23808.4i 32768.0i 0 264100. + 152478.i
17.3 −27.7128 + 16.0000i 0 512.000 − 886.810i −5312.73 − 9201.92i 0 −34960.4 − 27479.0i 32768.0i 0 294462. + 170007.i
17.4 −27.7128 + 16.0000i 0 512.000 − 886.810i −3290.58 − 5699.45i 0 30244.4 + 32597.6i 32768.0i 0 182383. + 105299.i
17.5 −27.7128 + 16.0000i 0 512.000 − 886.810i −2257.08 − 3909.39i 0 −44086.8 + 5803.28i 32768.0i 0 125100. + 72226.7i
17.6 −27.7128 + 16.0000i 0 512.000 − 886.810i −419.765 − 727.054i 0 34844.1 − 27626.3i 32768.0i 0 23265.7 + 13432.5i
17.7 −27.7128 + 16.0000i 0 512.000 − 886.810i 406.152 + 703.476i 0 3805.90 + 44304.0i 32768.0i 0 −22511.2 − 12996.9i
17.8 −27.7128 + 16.0000i 0 512.000 − 886.810i −110.383 − 191.189i 0 −39295.0 + 20814.3i 32768.0i 0 6118.05 + 3532.25i
17.9 −27.7128 + 16.0000i 0 512.000 − 886.810i 2396.90 + 4151.56i 0 −32574.3 − 30269.5i 32768.0i 0 −132850. − 76700.9i
17.10 −27.7128 + 16.0000i 0 512.000 − 886.810i 2477.66 + 4291.43i 0 4993.79 − 44185.8i 32768.0i 0 −137326. − 79285.0i
17.11 −27.7128 + 16.0000i 0 512.000 − 886.810i 3654.85 + 6330.38i 0 8564.03 − 43634.7i 32768.0i 0 −202572. − 116955.i
17.12 −27.7128 + 16.0000i 0 512.000 − 886.810i 3812.56 + 6603.54i 0 44444.3 + 1426.32i 32768.0i 0 −211313. − 122002.i
17.13 −27.7128 + 16.0000i 0 512.000 − 886.810i 4784.47 + 8286.95i 0 −40425.2 + 18523.7i 32768.0i 0 −265182. − 153103.i
17.14 −27.7128 + 16.0000i 0 512.000 − 886.810i 6077.08 + 10525.8i 0 35527.2 + 26742.2i 32768.0i 0 −336826. − 194466.i
17.15 27.7128 − 16.0000i 0 512.000 − 886.810i −6077.08 − 10525.8i 0 35527.2 + 26742.2i − 32768.0i 0 −336826. − 194466.i
17.16 27.7128 − 16.0000i 0 512.000 − 886.810i −4784.47 − 8286.95i 0 −40425.2 + 18523.7i − 32768.0i 0 −265182. − 153103.i
17.17 27.7128 − 16.0000i 0 512.000 − 886.810i −3812.56 − 6603.54i 0 44444.3 + 1426.32i − 32768.0i 0 −211313. − 122002.i
17.18 27.7128 − 16.0000i 0 512.000 − 886.810i −3654.85 − 6330.38i 0 8564.03 − 43634.7i − 32768.0i 0 −202572. − 116955.i
17.19 27.7128 − 16.0000i 0 512.000 − 886.810i −2477.66 − 4291.43i 0 4993.79 − 44185.8i − 32768.0i 0 −137326. − 79285.0i
17.20 27.7128 − 16.0000i 0 512.000 − 886.810i −2396.90 − 4151.56i 0 −32574.3 − 30269.5i − 32768.0i 0 −132850. − 76700.9i
See all 56 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.28
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.d odd 6 1 inner
21.g even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 126.12.k.a ✓ 56
3.b odd 2 1 inner 126.12.k.a ✓ 56
7.d odd 6 1 inner 126.12.k.a ✓ 56
21.g even 6 1 inner 126.12.k.a ✓ 56
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.12.k.a ✓ 56 1.a even 1 1 trivial
126.12.k.a ✓ 56 3.b odd 2 1 inner
126.12.k.a ✓ 56 7.d odd 6 1 inner
126.12.k.a ✓ 56 21.g even 6 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{12}^{\mathrm{new}}(126, [\chi])\).