Properties

Label 380.2.y.b
Level $380$
Weight $2$
Character orbit 380.y
Analytic conductor $3.034$
Analytic rank $0$
Dimension $36$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 380 = 2^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 380.y (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.03431527681\)
Analytic rank: \(0\)
Dimension: \(36\)
Relative dimension: \(9\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 36q - 6q^{3} - 4q^{5} + 4q^{7} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 36q - 6q^{3} - 4q^{5} + 4q^{7} - 12q^{11} - 6q^{13} - 12q^{15} + 6q^{17} + 48q^{21} + 16q^{23} + 4q^{25} - 54q^{33} + 16q^{35} - 2q^{43} + 100q^{45} + 24q^{47} - 108q^{51} + 14q^{55} - 30q^{57} + 34q^{61} - 26q^{63} - 78q^{67} - 42q^{71} + 16q^{73} - 20q^{77} + 14q^{81} - 28q^{83} + 10q^{85} - 124q^{87} + 96q^{91} - 26q^{93} - 32q^{95} + 54q^{97} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
217.1 0 −2.89008 + 0.774393i 0 2.13589 + 0.661803i 0 −2.81505 2.81505i 0 5.15478 2.97611i 0
217.2 0 −2.08775 + 0.559410i 0 1.02821 1.98565i 0 1.41342 + 1.41342i 0 1.44767 0.835811i 0
217.3 0 −1.83970 + 0.492946i 0 −2.13067 + 0.678412i 0 −0.607167 0.607167i 0 0.543427 0.313748i 0
217.4 0 −1.24341 + 0.333170i 0 −1.30148 1.81828i 0 0.474734 + 0.474734i 0 −1.16302 + 0.671470i 0
217.5 0 −0.660873 + 0.177080i 0 1.44130 + 1.70958i 0 1.38808 + 1.38808i 0 −2.19268 + 1.26594i 0
217.6 0 0.500957 0.134231i 0 −1.65073 + 1.50834i 0 −3.08317 3.08317i 0 −2.36514 + 1.36551i 0
217.7 0 1.33262 0.357073i 0 2.06883 0.848495i 0 0.519503 + 0.519503i 0 −0.949712 + 0.548316i 0
217.8 0 1.42686 0.382326i 0 −2.03585 + 0.924834i 0 3.57243 + 3.57243i 0 −0.708320 + 0.408949i 0
217.9 0 3.09534 0.829395i 0 1.17655 + 1.90150i 0 0.137224 + 0.137224i 0 6.29517 3.63452i 0
293.1 0 −0.774393 2.89008i 0 −1.64108 1.51883i 0 −2.81505 + 2.81505i 0 −5.15478 + 2.97611i 0
293.2 0 −0.559410 2.08775i 0 1.20552 1.88328i 0 1.41342 1.41342i 0 −1.44767 + 0.835811i 0
293.3 0 −0.492946 1.83970i 0 0.477813 + 2.18442i 0 −0.607167 + 0.607167i 0 −0.543427 + 0.313748i 0
293.4 0 −0.333170 1.24341i 0 2.22542 + 0.217974i 0 0.474734 0.474734i 0 1.16302 0.671470i 0
293.5 0 −0.177080 0.660873i 0 −2.20119 0.393414i 0 1.38808 1.38808i 0 2.19268 1.26594i 0
293.6 0 0.134231 + 0.500957i 0 −0.480898 + 2.18374i 0 −3.08317 + 3.08317i 0 2.36514 1.36551i 0
293.7 0 0.357073 + 1.33262i 0 −0.299597 2.21591i 0 0.519503 0.519503i 0 0.949712 0.548316i 0
293.8 0 0.382326 + 1.42686i 0 0.216995 + 2.22551i 0 3.57243 3.57243i 0 0.708320 0.408949i 0
293.9 0 0.829395 + 3.09534i 0 −2.23503 0.0681732i 0 0.137224 0.137224i 0 −6.29517 + 3.63452i 0
297.1 0 −0.774393 + 2.89008i 0 −1.64108 + 1.51883i 0 −2.81505 2.81505i 0 −5.15478 2.97611i 0
297.2 0 −0.559410 + 2.08775i 0 1.20552 + 1.88328i 0 1.41342 + 1.41342i 0 −1.44767 0.835811i 0
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 373.9
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
19.d odd 6 1 inner
95.l even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 380.2.y.b 36
5.c odd 4 1 inner 380.2.y.b 36
19.d odd 6 1 inner 380.2.y.b 36
95.l even 12 1 inner 380.2.y.b 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
380.2.y.b 36 1.a even 1 1 trivial
380.2.y.b 36 5.c odd 4 1 inner
380.2.y.b 36 19.d odd 6 1 inner
380.2.y.b 36 95.l even 12 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{3}^{36} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(380, [\chi])\).