Properties

Label 966.2.k.b
Level $966$
Weight $2$
Character orbit 966.k
Analytic conductor $7.714$
Analytic rank $0$
Dimension $32$
CM no
Inner twists $4$

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Newspace parameters

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

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 dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 32q + 16q^{2} - 16q^{4} - 32q^{8} + 16q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 32q + 16q^{2} - 16q^{4} - 32q^{8} + 16q^{9} - 16q^{16} - 16q^{18} + 4q^{23} - 40q^{25} - 12q^{26} + 24q^{29} - 24q^{31} + 16q^{32} + 12q^{35} - 32q^{36} + 8q^{39} - 4q^{46} + 12q^{47} + 24q^{49} - 80q^{50} - 12q^{52} + 12q^{58} - 12q^{59} + 32q^{64} + 24q^{70} + 16q^{71} - 16q^{72} - 84q^{73} + 12q^{75} - 40q^{77} + 16q^{78} - 16q^{81} - 36q^{82} - 112q^{85} - 48q^{87} - 8q^{92} - 8q^{93} + 12q^{94} + 24q^{98} + O(q^{100}) \)

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.

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 −2.02969 + 3.51552i 1.00000i 2.03303 + 1.69316i −1.00000 0.500000 0.866025i 2.02969 + 3.51552i
229.2 0.500000 0.866025i −0.866025 + 0.500000i −0.500000 0.866025i −1.92123 + 3.32767i 1.00000i 0.661545 2.56171i −1.00000 0.500000 0.866025i 1.92123 + 3.32767i
229.3 0.500000 0.866025i −0.866025 + 0.500000i −0.500000 0.866025i −0.320775 + 0.555599i 1.00000i −2.04200 + 1.68233i −1.00000 0.500000 0.866025i 0.320775 + 0.555599i
229.4 0.500000 0.866025i −0.866025 + 0.500000i −0.500000 0.866025i −0.139134 + 0.240988i 1.00000i 2.42763 + 1.05197i −1.00000 0.500000 0.866025i 0.139134 + 0.240988i
229.5 0.500000 0.866025i −0.866025 + 0.500000i −0.500000 0.866025i 0.139134 0.240988i 1.00000i −2.42763 1.05197i −1.00000 0.500000 0.866025i −0.139134 0.240988i
229.6 0.500000 0.866025i −0.866025 + 0.500000i −0.500000 0.866025i 0.320775 0.555599i 1.00000i 2.04200 1.68233i −1.00000 0.500000 0.866025i −0.320775 0.555599i
229.7 0.500000 0.866025i −0.866025 + 0.500000i −0.500000 0.866025i 1.92123 3.32767i 1.00000i −0.661545 + 2.56171i −1.00000 0.500000 0.866025i −1.92123 3.32767i
229.8 0.500000 0.866025i −0.866025 + 0.500000i −0.500000 0.866025i 2.02969 3.51552i 1.00000i −2.03303 1.69316i −1.00000 0.500000 0.866025i −2.02969 3.51552i
229.9 0.500000 0.866025i 0.866025 0.500000i −0.500000 0.866025i −1.70402 + 2.95145i 1.00000i −2.64298 0.121148i −1.00000 0.500000 0.866025i 1.70402 + 2.95145i
229.10 0.500000 0.866025i 0.866025 0.500000i −0.500000 0.866025i −1.36115 + 2.35758i 1.00000i 2.49823 0.871108i −1.00000 0.500000 0.866025i 1.36115 + 2.35758i
229.11 0.500000 0.866025i 0.866025 0.500000i −0.500000 0.866025i −1.28362 + 2.22329i 1.00000i −1.74878 1.98539i −1.00000 0.500000 0.866025i 1.28362 + 2.22329i
229.12 0.500000 0.866025i 0.866025 0.500000i −0.500000 0.866025i −0.814185 + 1.41021i 1.00000i −0.285113 + 2.63034i −1.00000 0.500000 0.866025i 0.814185 + 1.41021i
229.13 0.500000 0.866025i 0.866025 0.500000i −0.500000 0.866025i 0.814185 1.41021i 1.00000i 0.285113 2.63034i −1.00000 0.500000 0.866025i −0.814185 1.41021i
229.14 0.500000 0.866025i 0.866025 0.500000i −0.500000 0.866025i 1.28362 2.22329i 1.00000i 1.74878 + 1.98539i −1.00000 0.500000 0.866025i −1.28362 2.22329i
229.15 0.500000 0.866025i 0.866025 0.500000i −0.500000 0.866025i 1.36115 2.35758i 1.00000i −2.49823 + 0.871108i −1.00000 0.500000 0.866025i −1.36115 2.35758i
229.16 0.500000 0.866025i 0.866025 0.500000i −0.500000 0.866025i 1.70402 2.95145i 1.00000i 2.64298 + 0.121148i −1.00000 0.500000 0.866025i −1.70402 2.95145i
367.1 0.500000 + 0.866025i −0.866025 0.500000i −0.500000 + 0.866025i −2.02969 3.51552i 1.00000i 2.03303 1.69316i −1.00000 0.500000 + 0.866025i 2.02969 3.51552i
367.2 0.500000 + 0.866025i −0.866025 0.500000i −0.500000 + 0.866025i −1.92123 3.32767i 1.00000i 0.661545 + 2.56171i −1.00000 0.500000 + 0.866025i 1.92123 3.32767i
367.3 0.500000 + 0.866025i −0.866025 0.500000i −0.500000 + 0.866025i −0.320775 0.555599i 1.00000i −2.04200 1.68233i −1.00000 0.500000 + 0.866025i 0.320775 0.555599i
367.4 0.500000 + 0.866025i −0.866025 0.500000i −0.500000 + 0.866025i −0.139134 0.240988i 1.00000i 2.42763 1.05197i −1.00000 0.500000 + 0.866025i 0.139134 0.240988i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 367.16
Significant digits:
Format:

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{5}^{32} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(966, [\chi])\).