Properties

Label 690.2.j.a
Level $690$
Weight $2$
Character orbit 690.j
Analytic conductor $5.510$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 690 = 2 \cdot 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 690.j (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.50967773947\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24q - 24q^{6} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 24q - 24q^{6} - 24q^{16} - 8q^{23} - 16q^{25} - 16q^{26} + 16q^{31} - 16q^{35} + 24q^{36} - 8q^{46} - 8q^{47} + 24q^{50} + 24q^{55} + 16q^{58} - 56q^{62} - 32q^{70} - 16q^{71} - 48q^{73} - 24q^{81} + 24q^{82} + 16q^{87} - 8q^{92} + 56q^{93} + 24q^{95} + 24q^{96} - 32q^{98} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
367.1 −0.707107 0.707107i 0.707107 0.707107i 1.00000i −2.13084 + 0.677875i −1.00000 −1.05927 + 1.05927i 0.707107 0.707107i 1.00000i 1.98606 + 1.02740i
367.2 −0.707107 0.707107i 0.707107 0.707107i 1.00000i −0.899538 2.04715i −1.00000 −1.80902 + 1.80902i 0.707107 0.707107i 1.00000i −0.811485 + 2.08362i
367.3 −0.707107 0.707107i 0.707107 0.707107i 1.00000i −0.299475 2.21592i −1.00000 3.52616 3.52616i 0.707107 0.707107i 1.00000i −1.35513 + 1.77865i
367.4 −0.707107 0.707107i 0.707107 0.707107i 1.00000i 0.299475 + 2.21592i −1.00000 −3.52616 + 3.52616i 0.707107 0.707107i 1.00000i 1.35513 1.77865i
367.5 −0.707107 0.707107i 0.707107 0.707107i 1.00000i 0.899538 + 2.04715i −1.00000 1.80902 1.80902i 0.707107 0.707107i 1.00000i 0.811485 2.08362i
367.6 −0.707107 0.707107i 0.707107 0.707107i 1.00000i 2.13084 0.677875i −1.00000 1.05927 1.05927i 0.707107 0.707107i 1.00000i −1.98606 1.02740i
367.7 0.707107 + 0.707107i −0.707107 + 0.707107i 1.00000i −2.22154 0.254460i −1.00000 2.54671 2.54671i −0.707107 + 0.707107i 1.00000i −1.39094 1.75080i
367.8 0.707107 + 0.707107i −0.707107 + 0.707107i 1.00000i −1.57090 + 1.59131i −1.00000 −1.37926 + 1.37926i −0.707107 + 0.707107i 1.00000i −2.23602 + 0.0144369i
367.9 0.707107 + 0.707107i −0.707107 + 0.707107i 1.00000i −0.397104 2.20052i −1.00000 −1.66838 + 1.66838i −0.707107 + 0.707107i 1.00000i 1.27521 1.83680i
367.10 0.707107 + 0.707107i −0.707107 + 0.707107i 1.00000i 0.397104 + 2.20052i −1.00000 1.66838 1.66838i −0.707107 + 0.707107i 1.00000i −1.27521 + 1.83680i
367.11 0.707107 + 0.707107i −0.707107 + 0.707107i 1.00000i 1.57090 1.59131i −1.00000 1.37926 1.37926i −0.707107 + 0.707107i 1.00000i 2.23602 0.0144369i
367.12 0.707107 + 0.707107i −0.707107 + 0.707107i 1.00000i 2.22154 + 0.254460i −1.00000 −2.54671 + 2.54671i −0.707107 + 0.707107i 1.00000i 1.39094 + 1.75080i
643.1 −0.707107 + 0.707107i 0.707107 + 0.707107i 1.00000i −2.13084 0.677875i −1.00000 −1.05927 1.05927i 0.707107 + 0.707107i 1.00000i 1.98606 1.02740i
643.2 −0.707107 + 0.707107i 0.707107 + 0.707107i 1.00000i −0.899538 + 2.04715i −1.00000 −1.80902 1.80902i 0.707107 + 0.707107i 1.00000i −0.811485 2.08362i
643.3 −0.707107 + 0.707107i 0.707107 + 0.707107i 1.00000i −0.299475 + 2.21592i −1.00000 3.52616 + 3.52616i 0.707107 + 0.707107i 1.00000i −1.35513 1.77865i
643.4 −0.707107 + 0.707107i 0.707107 + 0.707107i 1.00000i 0.299475 2.21592i −1.00000 −3.52616 3.52616i 0.707107 + 0.707107i 1.00000i 1.35513 + 1.77865i
643.5 −0.707107 + 0.707107i 0.707107 + 0.707107i 1.00000i 0.899538 2.04715i −1.00000 1.80902 + 1.80902i 0.707107 + 0.707107i 1.00000i 0.811485 + 2.08362i
643.6 −0.707107 + 0.707107i 0.707107 + 0.707107i 1.00000i 2.13084 + 0.677875i −1.00000 1.05927 + 1.05927i 0.707107 + 0.707107i 1.00000i −1.98606 + 1.02740i
643.7 0.707107 0.707107i −0.707107 0.707107i 1.00000i −2.22154 + 0.254460i −1.00000 2.54671 + 2.54671i −0.707107 0.707107i 1.00000i −1.39094 + 1.75080i
643.8 0.707107 0.707107i −0.707107 0.707107i 1.00000i −1.57090 1.59131i −1.00000 −1.37926 1.37926i −0.707107 0.707107i 1.00000i −2.23602 0.0144369i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 643.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
23.b odd 2 1 inner
115.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 690.2.j.a 24
5.c odd 4 1 inner 690.2.j.a 24
23.b odd 2 1 inner 690.2.j.a 24
115.e even 4 1 inner 690.2.j.a 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
690.2.j.a 24 1.a even 1 1 trivial
690.2.j.a 24 5.c odd 4 1 inner
690.2.j.a 24 23.b odd 2 1 inner
690.2.j.a 24 115.e even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{24} + 880 T_{7}^{20} + 180320 T_{7}^{16} + 11978496 T_{7}^{12} + 320323840 T_{7}^{8} + 3331522560 T_{7}^{4} + 10070523904 \) acting on \(S_{2}^{\mathrm{new}}(690, [\chi])\).