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Properties

Label 966.2.k.a
Level $966$
Weight $2$
Character orbit 966.k
Analytic conductor $7.714$
Analytic rank $0$
Dimension $32$
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] = [966,2,Mod(229,966)] 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("966.229"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(966, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 5, 3])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 966 = 2 \cdot 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 966.k (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [32,-16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.71354883526\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(16\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 16 q^{2} - 16 q^{4} + 32 q^{8} + 16 q^{9} - 16 q^{16} + 16 q^{18} + 8 q^{23} - 24 q^{25} - 12 q^{26} - 8 q^{29} + 48 q^{31} - 16 q^{32} - 20 q^{35} - 32 q^{36} - 8 q^{39} + 8 q^{46} + 12 q^{47} + 24 q^{49}+ \cdots - 24 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
229.1 −0.500000 + 0.866025i −0.866025 + 0.500000i −0.500000 − 0.866025i −1.81715 + 3.14740i − 1.00000i −1.58446 + 2.11884i 1.00000 0.500000 − 0.866025i −1.81715 − 3.14740i
229.2 −0.500000 + 0.866025i −0.866025 + 0.500000i −0.500000 − 0.866025i −1.52184 + 2.63590i − 1.00000i −1.49399 − 2.18358i 1.00000 0.500000 − 0.866025i −1.52184 − 2.63590i
229.3 −0.500000 + 0.866025i −0.866025 + 0.500000i −0.500000 − 0.866025i −1.44868 + 2.50918i − 1.00000i 2.53486 − 0.757941i 1.00000 0.500000 − 0.866025i −1.44868 − 2.50918i
229.4 −0.500000 + 0.866025i −0.866025 + 0.500000i −0.500000 − 0.866025i −0.286936 + 0.496987i − 1.00000i 1.61243 + 2.09764i 1.00000 0.500000 − 0.866025i −0.286936 − 0.496987i
229.5 −0.500000 + 0.866025i −0.866025 + 0.500000i −0.500000 − 0.866025i 0.286936 − 0.496987i − 1.00000i −1.61243 − 2.09764i 1.00000 0.500000 − 0.866025i 0.286936 + 0.496987i
229.6 −0.500000 + 0.866025i −0.866025 + 0.500000i −0.500000 − 0.866025i 1.44868 − 2.50918i − 1.00000i −2.53486 + 0.757941i 1.00000 0.500000 − 0.866025i 1.44868 + 2.50918i
229.7 −0.500000 + 0.866025i −0.866025 + 0.500000i −0.500000 − 0.866025i 1.52184 − 2.63590i − 1.00000i 1.49399 + 2.18358i 1.00000 0.500000 − 0.866025i 1.52184 + 2.63590i
229.8 −0.500000 + 0.866025i −0.866025 + 0.500000i −0.500000 − 0.866025i 1.81715 − 3.14740i − 1.00000i 1.58446 − 2.11884i 1.00000 0.500000 − 0.866025i 1.81715 + 3.14740i
229.9 −0.500000 + 0.866025i 0.866025 − 0.500000i −0.500000 − 0.866025i −2.07087 + 3.58686i 1.00000i −1.78719 + 1.95088i 1.00000 0.500000 − 0.866025i −2.07087 − 3.58686i
229.10 −0.500000 + 0.866025i 0.866025 − 0.500000i −0.500000 − 0.866025i −0.856094 + 1.48280i 1.00000i 0.871223 + 2.49819i 1.00000 0.500000 − 0.866025i −0.856094 − 1.48280i
229.11 −0.500000 + 0.866025i 0.866025 − 0.500000i −0.500000 − 0.866025i −0.347206 + 0.601379i 1.00000i −2.64100 + 0.158433i 1.00000 0.500000 − 0.866025i −0.347206 − 0.601379i
229.12 −0.500000 + 0.866025i 0.866025 − 0.500000i −0.500000 − 0.866025i −0.242904 + 0.420722i 1.00000i −2.51079 − 0.834227i 1.00000 0.500000 − 0.866025i −0.242904 − 0.420722i
229.13 −0.500000 + 0.866025i 0.866025 − 0.500000i −0.500000 − 0.866025i 0.242904 − 0.420722i 1.00000i 2.51079 + 0.834227i 1.00000 0.500000 − 0.866025i 0.242904 + 0.420722i
229.14 −0.500000 + 0.866025i 0.866025 − 0.500000i −0.500000 − 0.866025i 0.347206 − 0.601379i 1.00000i 2.64100 − 0.158433i 1.00000 0.500000 − 0.866025i 0.347206 + 0.601379i
229.15 −0.500000 + 0.866025i 0.866025 − 0.500000i −0.500000 − 0.866025i 0.856094 − 1.48280i 1.00000i −0.871223 − 2.49819i 1.00000 0.500000 − 0.866025i 0.856094 + 1.48280i
229.16 −0.500000 + 0.866025i 0.866025 − 0.500000i −0.500000 − 0.866025i 2.07087 − 3.58686i 1.00000i 1.78719 − 1.95088i 1.00000 0.500000 − 0.866025i 2.07087 + 3.58686i
367.1 −0.500000 − 0.866025i −0.866025 − 0.500000i −0.500000 + 0.866025i −1.81715 − 3.14740i 1.00000i −1.58446 − 2.11884i 1.00000 0.500000 + 0.866025i −1.81715 + 3.14740i
367.2 −0.500000 − 0.866025i −0.866025 − 0.500000i −0.500000 + 0.866025i −1.52184 − 2.63590i 1.00000i −1.49399 + 2.18358i 1.00000 0.500000 + 0.866025i −1.52184 + 2.63590i
367.3 −0.500000 − 0.866025i −0.866025 − 0.500000i −0.500000 + 0.866025i −1.44868 − 2.50918i 1.00000i 2.53486 + 0.757941i 1.00000 0.500000 + 0.866025i −1.44868 + 2.50918i
367.4 −0.500000 − 0.866025i −0.866025 − 0.500000i −0.500000 + 0.866025i −0.286936 − 0.496987i 1.00000i 1.61243 − 2.09764i 1.00000 0.500000 + 0.866025i −0.286936 + 0.496987i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 229.16
Significant digits:
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Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 966.2.k.a ✓ 32
7.d odd 6 1 inner 966.2.k.a ✓ 32
23.b odd 2 1 inner 966.2.k.a ✓ 32
161.g even 6 1 inner 966.2.k.a ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
966.2.k.a ✓ 32 1.a even 1 1 trivial
966.2.k.a ✓ 32 7.d odd 6 1 inner
966.2.k.a ✓ 32 23.b odd 2 1 inner
966.2.k.a ✓ 32 161.g even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{32} + 52 T_{5}^{30} + 1669 T_{5}^{28} + 34076 T_{5}^{26} + 512018 T_{5}^{24} + 5540548 T_{5}^{22} + \cdots + 3748096 \) acting on \(S_{2}^{\mathrm{new}}(966, [\chi])\). Copy content Toggle raw display