Properties

Label 380.2.d.b
Level $380$
Weight $2$
Character orbit 380.d
Analytic conductor $3.034$
Analytic rank $0$
Dimension $40$
CM no
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 380 = 2^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 380.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.03431527681\)
Analytic rank: \(0\)
Dimension: \(40\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 40 q - 4 q^{5} + 8 q^{6} - 8 q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 40 q - 4 q^{5} + 8 q^{6} - 8 q^{9} - 8 q^{16} - 20 q^{20} - 40 q^{24} - 84 q^{25} - 24 q^{26} + 24 q^{30} + 24 q^{36} - 40 q^{44} - 12 q^{45} + 128 q^{49} - 120 q^{54} + 24 q^{61} + 72 q^{64} + 112 q^{66} + 32 q^{74} + 56 q^{76} + 96 q^{80} - 72 q^{81} + 44 q^{85} - 40 q^{96} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
379.1 −1.39672 0.221721i 2.04365i 1.90168 + 0.619368i 0.665544 2.13473i −0.453121 + 2.85442i 2.07881 −2.51879 1.28673i −1.17650 −1.40290 + 2.83406i
379.2 −1.39672 0.221721i 2.04365i 1.90168 + 0.619368i 0.665544 + 2.13473i −0.453121 + 2.85442i −2.07881 −2.51879 1.28673i −1.17650 −0.456267 3.12919i
379.3 −1.39672 + 0.221721i 2.04365i 1.90168 0.619368i 0.665544 2.13473i −0.453121 2.85442i −2.07881 −2.51879 + 1.28673i −1.17650 −0.456267 + 3.12919i
379.4 −1.39672 + 0.221721i 2.04365i 1.90168 0.619368i 0.665544 + 2.13473i −0.453121 2.85442i 2.07881 −2.51879 + 1.28673i −1.17650 −1.40290 2.83406i
379.5 −1.27059 0.620975i 0.701989i 1.22878 + 1.57800i −1.10449 1.94425i 0.435918 0.891938i 1.18642 −0.581371 2.76803i 2.50721 0.196019 + 3.15620i
379.6 −1.27059 0.620975i 0.701989i 1.22878 + 1.57800i −1.10449 + 1.94425i 0.435918 0.891938i −1.18642 −0.581371 2.76803i 2.50721 2.61068 1.78448i
379.7 −1.27059 + 0.620975i 0.701989i 1.22878 1.57800i −1.10449 1.94425i 0.435918 + 0.891938i −1.18642 −0.581371 + 2.76803i 2.50721 2.61068 + 1.78448i
379.8 −1.27059 + 0.620975i 0.701989i 1.22878 1.57800i −1.10449 + 1.94425i 0.435918 + 0.891938i 1.18642 −0.581371 + 2.76803i 2.50721 0.196019 3.15620i
379.9 −0.948256 1.04920i 1.76151i −0.201622 + 1.98981i −1.85854 1.24332i −1.84816 + 1.67036i −4.29252 2.27889 1.67531i −0.102903 0.457886 + 3.12895i
379.10 −0.948256 1.04920i 1.76151i −0.201622 + 1.98981i −1.85854 + 1.24332i −1.84816 + 1.67036i 4.29252 2.27889 1.67531i −0.102903 3.06685 + 0.770986i
379.11 −0.948256 + 1.04920i 1.76151i −0.201622 1.98981i −1.85854 1.24332i −1.84816 1.67036i 4.29252 2.27889 + 1.67531i −0.102903 3.06685 0.770986i
379.12 −0.948256 + 1.04920i 1.76151i −0.201622 1.98981i −1.85854 + 1.24332i −1.84816 1.67036i −4.29252 2.27889 + 1.67531i −0.102903 0.457886 3.12895i
379.13 −0.584780 1.28765i 2.81267i −1.31606 + 1.50598i 0.390055 2.20178i 3.62172 1.64479i −4.13073 2.70878 + 0.813956i −4.91111 −3.06322 + 0.785308i
379.14 −0.584780 1.28765i 2.81267i −1.31606 + 1.50598i 0.390055 + 2.20178i 3.62172 1.64479i 4.13073 2.70878 + 0.813956i −4.91111 2.60702 1.78981i
379.15 −0.584780 + 1.28765i 2.81267i −1.31606 1.50598i 0.390055 2.20178i 3.62172 + 1.64479i 4.13073 2.70878 0.813956i −4.91111 2.60702 + 1.78981i
379.16 −0.584780 + 1.28765i 2.81267i −1.31606 1.50598i 0.390055 + 2.20178i 3.62172 + 1.64479i −4.13073 2.70878 0.813956i −4.91111 −3.06322 0.785308i
379.17 −0.440015 1.34402i 0.562756i −1.61277 + 1.18278i 1.40743 1.73757i −0.756355 + 0.247621i 3.12767 2.29932 + 1.64716i 2.68331 −2.95462 1.12705i
379.18 −0.440015 1.34402i 0.562756i −1.61277 + 1.18278i 1.40743 + 1.73757i −0.756355 + 0.247621i −3.12767 2.29932 + 1.64716i 2.68331 1.71604 2.65617i
379.19 −0.440015 + 1.34402i 0.562756i −1.61277 1.18278i 1.40743 1.73757i −0.756355 0.247621i −3.12767 2.29932 1.64716i 2.68331 1.71604 + 2.65617i
379.20 −0.440015 + 1.34402i 0.562756i −1.61277 1.18278i 1.40743 + 1.73757i −0.756355 0.247621i 3.12767 2.29932 1.64716i 2.68331 −2.95462 + 1.12705i
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 379.40
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.b even 2 1 inner
19.b odd 2 1 inner
20.d odd 2 1 inner
76.d even 2 1 inner
95.d odd 2 1 inner
380.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 380.2.d.b 40
4.b odd 2 1 inner 380.2.d.b 40
5.b even 2 1 inner 380.2.d.b 40
19.b odd 2 1 inner 380.2.d.b 40
20.d odd 2 1 inner 380.2.d.b 40
76.d even 2 1 inner 380.2.d.b 40
95.d odd 2 1 inner 380.2.d.b 40
380.d even 2 1 inner 380.2.d.b 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
380.2.d.b 40 1.a even 1 1 trivial
380.2.d.b 40 4.b odd 2 1 inner
380.2.d.b 40 5.b even 2 1 inner
380.2.d.b 40 19.b odd 2 1 inner
380.2.d.b 40 20.d odd 2 1 inner
380.2.d.b 40 76.d even 2 1 inner
380.2.d.b 40 95.d odd 2 1 inner
380.2.d.b 40 380.d even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{10} + 16 T_{3}^{8} + 83 T_{3}^{6} + 162 T_{3}^{4} + 94 T_{3}^{2} + 16 \) acting on \(S_{2}^{\mathrm{new}}(380, [\chi])\).