Properties

Label 966.2.f.b
Level $966$
Weight $2$
Character orbit 966.f
Analytic conductor $7.714$
Analytic rank $0$
Dimension $24$
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.f (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.71354883526\)
Analytic rank: \(0\)
Dimension: \(24\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24q - 24q^{4} - 4q^{7} + 8q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 24q - 24q^{4} - 4q^{7} + 8q^{9} + 24q^{16} + 16q^{18} - 28q^{21} + 8q^{22} - 24q^{25} + 4q^{28} + 24q^{30} - 8q^{36} + 40q^{37} + 72q^{39} + 64q^{43} + 24q^{46} - 24q^{51} + 16q^{58} + 12q^{63} - 24q^{64} - 64q^{67} + 16q^{70} - 16q^{72} - 32q^{78} + 88q^{79} + 48q^{81} + 28q^{84} + 64q^{85} - 8q^{88} - 56q^{91} + 8q^{93} - 8q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
461.1 1.00000i −1.72384 0.168468i −1.00000 1.25574 −0.168468 + 1.72384i −1.98144 1.75325i 1.00000i 2.94324 + 0.580822i 1.25574i
461.2 1.00000i −1.66919 0.462387i −1.00000 −1.95027 −0.462387 + 1.66919i −0.602835 + 2.57616i 1.00000i 2.57240 + 1.54363i 1.95027i
461.3 1.00000i −1.62776 + 0.591954i −1.00000 2.12506 0.591954 + 1.62776i 2.32692 + 1.25915i 1.00000i 2.29918 1.92711i 2.12506i
461.4 1.00000i −0.950180 1.44816i −1.00000 0.575987 −1.44816 + 0.950180i 2.10044 1.60877i 1.00000i −1.19432 + 2.75202i 0.575987i
461.5 1.00000i −0.740879 1.56560i −1.00000 −3.67536 −1.56560 + 0.740879i −0.229372 2.63579i 1.00000i −1.90220 + 2.31984i 3.67536i
461.6 1.00000i −0.375300 + 1.69090i −1.00000 −0.513446 1.69090 + 0.375300i −2.61371 + 0.410499i 1.00000i −2.71830 1.26919i 0.513446i
461.7 1.00000i 0.375300 1.69090i −1.00000 0.513446 −1.69090 0.375300i −2.61371 0.410499i 1.00000i −2.71830 1.26919i 0.513446i
461.8 1.00000i 0.740879 + 1.56560i −1.00000 3.67536 1.56560 0.740879i −0.229372 + 2.63579i 1.00000i −1.90220 + 2.31984i 3.67536i
461.9 1.00000i 0.950180 + 1.44816i −1.00000 −0.575987 1.44816 0.950180i 2.10044 + 1.60877i 1.00000i −1.19432 + 2.75202i 0.575987i
461.10 1.00000i 1.62776 0.591954i −1.00000 −2.12506 −0.591954 1.62776i 2.32692 1.25915i 1.00000i 2.29918 1.92711i 2.12506i
461.11 1.00000i 1.66919 + 0.462387i −1.00000 1.95027 0.462387 1.66919i −0.602835 2.57616i 1.00000i 2.57240 + 1.54363i 1.95027i
461.12 1.00000i 1.72384 + 0.168468i −1.00000 −1.25574 0.168468 1.72384i −1.98144 + 1.75325i 1.00000i 2.94324 + 0.580822i 1.25574i
461.13 1.00000i −1.72384 + 0.168468i −1.00000 1.25574 −0.168468 1.72384i −1.98144 + 1.75325i 1.00000i 2.94324 0.580822i 1.25574i
461.14 1.00000i −1.66919 + 0.462387i −1.00000 −1.95027 −0.462387 1.66919i −0.602835 2.57616i 1.00000i 2.57240 1.54363i 1.95027i
461.15 1.00000i −1.62776 0.591954i −1.00000 2.12506 0.591954 1.62776i 2.32692 1.25915i 1.00000i 2.29918 + 1.92711i 2.12506i
461.16 1.00000i −0.950180 + 1.44816i −1.00000 0.575987 −1.44816 0.950180i 2.10044 + 1.60877i 1.00000i −1.19432 2.75202i 0.575987i
461.17 1.00000i −0.740879 + 1.56560i −1.00000 −3.67536 −1.56560 0.740879i −0.229372 + 2.63579i 1.00000i −1.90220 2.31984i 3.67536i
461.18 1.00000i −0.375300 1.69090i −1.00000 −0.513446 1.69090 0.375300i −2.61371 0.410499i 1.00000i −2.71830 + 1.26919i 0.513446i
461.19 1.00000i 0.375300 + 1.69090i −1.00000 0.513446 −1.69090 + 0.375300i −2.61371 + 0.410499i 1.00000i −2.71830 + 1.26919i 0.513446i
461.20 1.00000i 0.740879 1.56560i −1.00000 3.67536 1.56560 + 0.740879i −0.229372 2.63579i 1.00000i −1.90220 2.31984i 3.67536i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 461.24
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 966.2.f.b 24
3.b odd 2 1 inner 966.2.f.b 24
7.b odd 2 1 inner 966.2.f.b 24
21.c even 2 1 inner 966.2.f.b 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
966.2.f.b 24 1.a even 1 1 trivial
966.2.f.b 24 3.b odd 2 1 inner
966.2.f.b 24 7.b odd 2 1 inner
966.2.f.b 24 21.c even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{12} - 24 T_{5}^{10} + 178 T_{5}^{8} - 536 T_{5}^{6} + 640 T_{5}^{4} - 256 T_{5}^{2} + 32 \) acting on \(S_{2}^{\mathrm{new}}(966, [\chi])\).