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Properties

Label 12.51.d.a
Level $12$
Weight $51$
Character orbit 12.d
Analytic conductor $190.003$
Analytic rank $0$
Dimension $50$
Inner twists $2$

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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] = [12,51,Mod(7,12)] 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("12.7"); S:= CuspForms(chi, 51); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(12, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 51, names="a")
 
Level: \( N \) \(=\) \( 12 = 2^{2} \cdot 3 \)
Weight: \( k \) \(=\) \( 51 \)
Character orbit: \([\chi]\) \(=\) 12.d (of order \(2\), degree \(1\), 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: \(190.002544227\)
Analytic rank: \(0\)
Dimension: \(50\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 50 q + 35720278 q^{2} + 17\!\cdots\!04 q^{4} + 27\!\cdots\!16 q^{5} + 29\!\cdots\!50 q^{6} + 91\!\cdots\!76 q^{8} - 11\!\cdots\!50 q^{9} - 31\!\cdots\!96 q^{10} + 47\!\cdots\!68 q^{12} - 14\!\cdots\!36 q^{13}+ \cdots - 22\!\cdots\!02 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1 −3.35347e7 − 1.15025e6i − 4.89182e11i 1.12325e15 + 7.71463e13i −4.00199e16 −5.62680e17 + 1.64046e19i 2.26268e21i −3.75793e22 − 3.87910e21i −2.39299e23 1.34206e24 + 4.60328e22i
7.2 −3.35347e7 + 1.15025e6i 4.89182e11i 1.12325e15 − 7.71463e13i −4.00199e16 −5.62680e17 − 1.64046e19i − 2.26268e21i −3.75793e22 + 3.87910e21i −2.39299e23 1.34206e24 − 4.60328e22i
7.3 −3.34564e7 − 2.56244e6i − 4.89182e11i 1.11277e15 + 1.71460e14i 3.82081e17 −1.25350e18 + 1.63663e19i − 8.06297e20i −3.67899e22 − 8.58786e21i −2.39299e23 −1.27831e25 − 9.79060e23i
7.4 −3.34564e7 + 2.56244e6i 4.89182e11i 1.11277e15 − 1.71460e14i 3.82081e17 −1.25350e18 − 1.63663e19i 8.06297e20i −3.67899e22 + 8.58786e21i −2.39299e23 −1.27831e25 + 9.79060e23i
7.5 −3.20635e7 − 9.89084e6i − 4.89182e11i 9.30242e14 + 6.34271e14i −4.12563e17 −4.83842e18 + 1.56849e19i − 1.04951e21i −2.35534e22 − 2.95379e22i −2.39299e23 1.32282e25 + 4.08059e24i
7.6 −3.20635e7 + 9.89084e6i 4.89182e11i 9.30242e14 − 6.34271e14i −4.12563e17 −4.83842e18 − 1.56849e19i 1.04951e21i −2.35534e22 + 2.95379e22i −2.39299e23 1.32282e25 − 4.08059e24i
7.7 −3.14766e7 − 1.16241e7i 4.89182e11i 8.55659e14 + 7.31777e14i −2.34107e17 5.68632e18 − 1.53978e19i − 4.13824e19i −1.84270e22 − 3.29802e22i −2.39299e23 7.36889e24 + 2.72129e24i
7.8 −3.14766e7 + 1.16241e7i − 4.89182e11i 8.55659e14 − 7.31777e14i −2.34107e17 5.68632e18 + 1.53978e19i 4.13824e19i −1.84270e22 + 3.29802e22i −2.39299e23 7.36889e24 − 2.72129e24i
7.9 −2.73078e7 − 1.94982e7i 4.89182e11i 3.65537e14 + 1.06491e15i −4.24332e16 9.53820e18 − 1.33585e19i 1.31057e21i 1.07819e22 − 3.62077e22i −2.39299e23 1.15876e24 + 8.27373e23i
7.10 −2.73078e7 + 1.94982e7i − 4.89182e11i 3.65537e14 − 1.06491e15i −4.24332e16 9.53820e18 + 1.33585e19i − 1.31057e21i 1.07819e22 + 3.62077e22i −2.39299e23 1.15876e24 − 8.27373e23i
7.11 −2.48810e7 − 2.25130e7i − 4.89182e11i 1.12231e14 + 1.12029e15i 3.21108e17 −1.10130e19 + 1.21714e19i 1.53177e21i 2.24287e22 − 3.04007e22i −2.39299e23 −7.98949e24 − 7.22909e24i
7.12 −2.48810e7 + 2.25130e7i 4.89182e11i 1.12231e14 − 1.12029e15i 3.21108e17 −1.10130e19 − 1.21714e19i − 1.53177e21i 2.24287e22 + 3.04007e22i −2.39299e23 −7.98949e24 + 7.22909e24i
7.13 −2.14639e7 − 2.57915e7i − 4.89182e11i −2.04500e14 + 1.10717e15i −3.28882e17 −1.26167e19 + 1.04998e19i 1.23060e20i 3.29450e22 − 1.84899e22i −2.39299e23 7.05909e24 + 8.48234e24i
7.14 −2.14639e7 + 2.57915e7i 4.89182e11i −2.04500e14 − 1.10717e15i −3.28882e17 −1.26167e19 − 1.04998e19i − 1.23060e20i 3.29450e22 + 1.84899e22i −2.39299e23 7.05909e24 − 8.48234e24i
7.15 −1.95374e7 − 2.72798e7i 4.89182e11i −3.62479e14 + 1.06595e15i 5.46965e17 1.33448e19 − 9.55736e18i 5.19091e20i 3.61610e22 − 1.09376e22i −2.39299e23 −1.06863e25 − 1.49211e25i
7.16 −1.95374e7 + 2.72798e7i − 4.89182e11i −3.62479e14 − 1.06595e15i 5.46965e17 1.33448e19 + 9.55736e18i − 5.19091e20i 3.61610e22 + 1.09376e22i −2.39299e23 −1.06863e25 + 1.49211e25i
7.17 −1.94987e7 − 2.73075e7i 4.89182e11i −3.65499e14 + 1.06492e15i 8.85553e16 1.33583e19 − 9.53843e18i − 2.26140e21i 3.62071e22 − 1.07838e22i −2.39299e23 −1.72672e24 − 2.41822e24i
7.18 −1.94987e7 + 2.73075e7i − 4.89182e11i −3.65499e14 − 1.06492e15i 8.85553e16 1.33583e19 + 9.53843e18i 2.26140e21i 3.62071e22 + 1.07838e22i −2.39299e23 −1.72672e24 + 2.41822e24i
7.19 −8.16335e6 − 3.25463e7i − 4.89182e11i −9.92619e14 + 5.31373e14i −1.79762e16 −1.59211e19 + 3.99337e18i − 8.09641e20i 2.53973e22 + 2.79683e22i −2.39299e23 1.46746e23 + 5.85058e23i
7.20 −8.16335e6 + 3.25463e7i 4.89182e11i −9.92619e14 − 5.31373e14i −1.79762e16 −1.59211e19 − 3.99337e18i 8.09641e20i 2.53973e22 − 2.79683e22i −2.39299e23 1.46746e23 − 5.85058e23i
See all 50 embeddings
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 7.50
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 12.51.d.a ✓ 50
4.b odd 2 1 inner 12.51.d.a ✓ 50
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
12.51.d.a ✓ 50 1.a even 1 1 trivial
12.51.d.a ✓ 50 4.b odd 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{51}^{\mathrm{new}}(12, [\chi])\).