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Properties

Label 966.2.f.c
Level $966$
Weight $2$
Character orbit 966.f
Analytic conductor $7.714$
Analytic rank $0$
Dimension $28$
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: \(28\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 28q - 28q^{4} + 4q^{7} - 4q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 28q - 28q^{4} + 4q^{7} - 4q^{9} + 16q^{15} + 28q^{16} - 16q^{18} + 4q^{21} + 80q^{25} - 4q^{28} + 12q^{30} + 4q^{36} + 20q^{37} - 20q^{39} + 28q^{42} - 28q^{43} - 28q^{46} - 28q^{49} + 16q^{51} - 8q^{57} - 36q^{58} - 16q^{60} + 36q^{63} - 28q^{64} - 8q^{67} - 60q^{70} + 16q^{72} + 16q^{78} - 76q^{81} - 4q^{84} - 24q^{85} + 36q^{91} + 48q^{93} + 72q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

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 −1.64221 − 0.550576i −1.00000 −1.99217 −0.550576 + 1.64221i 2.64482 − 0.0703415i 1.00000i 2.39373 + 1.80833i 1.99217i
461.2 − 1.00000i −1.55720 + 0.758370i −1.00000 −2.55377 0.758370 + 1.55720i 0.900399 − 2.48783i 1.00000i 1.84975 − 2.36187i 2.55377i
461.3 − 1.00000i −1.37202 + 1.05715i −1.00000 −2.38722 1.05715 + 1.37202i −2.59254 − 0.527932i 1.00000i 0.764872 − 2.90086i 2.38722i
461.4 − 1.00000i −1.18002 − 1.26789i −1.00000 3.87735 −1.26789 + 1.18002i 0.302862 − 2.62836i 1.00000i −0.215085 + 2.99228i − 3.87735i
461.5 − 1.00000i −0.952183 + 1.44684i −1.00000 3.39470 1.44684 + 0.952183i 1.73823 − 1.99463i 1.00000i −1.18670 − 2.75531i − 3.39470i
461.6 − 1.00000i −0.711304 + 1.57925i −1.00000 0.334212 1.57925 + 0.711304i −0.168570 + 2.64038i 1.00000i −1.98809 − 2.24666i − 0.334212i
461.7 − 1.00000i −0.436762 − 1.67608i −1.00000 −3.48442 −1.67608 + 0.436762i −1.82520 + 1.91537i 1.00000i −2.61848 + 1.46409i 3.48442i
461.8 − 1.00000i 0.436762 + 1.67608i −1.00000 3.48442 1.67608 − 0.436762i −1.82520 − 1.91537i 1.00000i −2.61848 + 1.46409i − 3.48442i
461.9 − 1.00000i 0.711304 − 1.57925i −1.00000 −0.334212 −1.57925 − 0.711304i −0.168570 − 2.64038i 1.00000i −1.98809 − 2.24666i 0.334212i
461.10 − 1.00000i 0.952183 − 1.44684i −1.00000 −3.39470 −1.44684 − 0.952183i 1.73823 + 1.99463i 1.00000i −1.18670 − 2.75531i 3.39470i
461.11 − 1.00000i 1.18002 + 1.26789i −1.00000 −3.87735 1.26789 − 1.18002i 0.302862 + 2.62836i 1.00000i −0.215085 + 2.99228i 3.87735i
461.12 − 1.00000i 1.37202 − 1.05715i −1.00000 2.38722 −1.05715 − 1.37202i −2.59254 + 0.527932i 1.00000i 0.764872 − 2.90086i − 2.38722i
461.13 − 1.00000i 1.55720 − 0.758370i −1.00000 2.55377 −0.758370 − 1.55720i 0.900399 + 2.48783i 1.00000i 1.84975 − 2.36187i − 2.55377i
461.14 − 1.00000i 1.64221 + 0.550576i −1.00000 1.99217 0.550576 − 1.64221i 2.64482 + 0.0703415i 1.00000i 2.39373 + 1.80833i − 1.99217i
461.15 1.00000i −1.64221 + 0.550576i −1.00000 −1.99217 −0.550576 − 1.64221i 2.64482 + 0.0703415i − 1.00000i 2.39373 − 1.80833i − 1.99217i
461.16 1.00000i −1.55720 − 0.758370i −1.00000 −2.55377 0.758370 − 1.55720i 0.900399 + 2.48783i − 1.00000i 1.84975 + 2.36187i − 2.55377i
461.17 1.00000i −1.37202 − 1.05715i −1.00000 −2.38722 1.05715 − 1.37202i −2.59254 + 0.527932i − 1.00000i 0.764872 + 2.90086i − 2.38722i
461.18 1.00000i −1.18002 + 1.26789i −1.00000 3.87735 −1.26789 − 1.18002i 0.302862 + 2.62836i − 1.00000i −0.215085 − 2.99228i 3.87735i
461.19 1.00000i −0.952183 − 1.44684i −1.00000 3.39470 1.44684 − 0.952183i 1.73823 + 1.99463i − 1.00000i −1.18670 + 2.75531i 3.39470i
461.20 1.00000i −0.711304 − 1.57925i −1.00000 0.334212 1.57925 − 0.711304i −0.168570 − 2.64038i − 1.00000i −1.98809 + 2.24666i 0.334212i
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 461.28
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.c ✓ 28
3.b odd 2 1 inner 966.2.f.c ✓ 28
7.b odd 2 1 inner 966.2.f.c ✓ 28
21.c even 2 1 inner 966.2.f.c ✓ 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
966.2.f.c ✓ 28 1.a even 1 1 trivial
966.2.f.c ✓ 28 3.b odd 2 1 inner
966.2.f.c ✓ 28 7.b odd 2 1 inner
966.2.f.c ✓ 28 21.c even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{14} - 55 T_{5}^{12} + 1214 T_{5}^{10} - 13726 T_{5}^{8} + 83744 T_{5}^{6} - 262496 T_{5}^{4} + 338560 T_{5}^{2} - 34656 \) acting on \(S_{2}^{\mathrm{new}}(966, [\chi])\).