Properties

Label 380.2.n.a
Level $380$
Weight $2$
Character orbit 380.n
Analytic conductor $3.034$
Analytic rank $0$
Dimension $40$
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.n (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.03431527681\)
Analytic rank: \(0\)
Dimension: \(40\)
Relative dimension: \(20\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 40 q - 3 q^{2} + q^{4} + 20 q^{5} + 3 q^{6} - 22 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q - 3 q^{2} + q^{4} + 20 q^{5} + 3 q^{6} - 22 q^{9} - 3 q^{10} + 12 q^{13} + 18 q^{14} - 7 q^{16} + 4 q^{17} + 2 q^{20} + 12 q^{21} - 8 q^{24} - 20 q^{25} + 2 q^{26} + 8 q^{28} + 6 q^{30} - 18 q^{32} - 6 q^{33} - 27 q^{34} - 14 q^{36} + 38 q^{38} + 36 q^{41} - 21 q^{42} - 8 q^{44} - 44 q^{45} - 18 q^{48} - 60 q^{49} - 33 q^{52} + 42 q^{53} + 9 q^{54} + 12 q^{57} - 62 q^{58} + 3 q^{60} + 12 q^{61} - 23 q^{62} + 64 q^{64} + 2 q^{66} + 72 q^{68} + 18 q^{70} + 42 q^{72} - 18 q^{73} + 6 q^{74} - 62 q^{76} - 28 q^{77} - 24 q^{78} + 7 q^{80} - 48 q^{81} - q^{82} - 4 q^{85} + 78 q^{86} - 18 q^{89} + 39 q^{90} + 16 q^{92} + 8 q^{96} + 30 q^{97} - 12 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1 −1.41224 0.0746014i −1.09575 1.89790i 1.98887 + 0.210711i 0.500000 + 0.866025i 1.40588 + 2.76204i 2.65350i −2.79305 0.445948i −0.901338 + 1.56116i −0.641516 1.26034i
31.2 −1.39096 + 0.255414i 1.57351 + 2.72539i 1.86953 0.710539i 0.500000 + 0.866025i −2.88478 3.38901i 3.91622i −2.41895 + 1.46583i −3.45184 + 5.97877i −0.916674 1.07690i
31.3 −1.38644 + 0.278901i −0.418744 0.725286i 1.84443 0.773359i 0.500000 + 0.866025i 0.782847 + 0.888777i 3.40884i −2.34150 + 1.58663i 1.14931 1.99066i −0.934755 1.06124i
31.4 −1.27347 0.615033i 0.481867 + 0.834618i 1.24347 + 1.56646i 0.500000 + 0.866025i −0.100327 1.35923i 1.00869i −0.620104 2.75961i 1.03561 1.79373i −0.104103 1.41038i
31.5 −0.934755 + 1.06124i 0.418744 + 0.725286i −0.252465 1.98400i 0.500000 + 0.866025i −1.16113 0.233577i 3.40884i 2.34150 + 1.58663i 1.14931 1.99066i −1.38644 0.278901i
31.6 −0.916674 + 1.07690i −1.57351 2.72539i −0.319419 1.97433i 0.500000 + 0.866025i 4.37736 + 0.803789i 3.91622i 2.41895 + 1.46583i −3.45184 + 5.97877i −1.39096 0.255414i
31.7 −0.840559 1.13730i 1.52787 + 2.64635i −0.586922 + 1.91194i 0.500000 + 0.866025i 1.72544 3.96206i 2.58057i 2.66780 0.939591i −3.16877 + 5.48848i 0.564655 1.29660i
31.8 −0.776263 1.18212i −1.45074 2.51275i −0.794830 + 1.83528i 0.500000 + 0.866025i −1.84423 + 3.66550i 1.19935i 2.78652 0.485072i −2.70927 + 4.69260i 0.635617 1.26333i
31.9 −0.641516 + 1.26034i 1.09575 + 1.89790i −1.17692 1.61706i 0.500000 + 0.866025i −3.09494 + 0.163489i 2.65350i 2.79305 0.445948i −0.901338 + 1.56116i −1.41224 + 0.0746014i
31.10 −0.293307 1.38346i −0.256977 0.445098i −1.82794 + 0.811560i 0.500000 + 0.866025i −0.540403 + 0.486069i 1.09058i 1.65891 + 2.29085i 1.36793 2.36932i 1.05146 0.945743i
31.11 −0.104103 + 1.41038i −0.481867 0.834618i −1.97833 0.293648i 0.500000 + 0.866025i 1.22729 0.592728i 1.00869i 0.620104 2.75961i 1.03561 1.79373i −1.27347 + 0.615033i
31.12 0.270280 1.38815i 0.435984 + 0.755146i −1.85390 0.750377i 0.500000 + 0.866025i 1.16609 0.401108i 4.89965i −1.54270 + 2.37067i 1.11984 1.93961i 1.33731 0.460003i
31.13 0.564655 + 1.29660i −1.52787 2.64635i −1.36233 + 1.46426i 0.500000 + 0.866025i 2.56853 3.47531i 2.58057i −2.66780 0.939591i −3.16877 + 5.48848i −0.840559 + 1.13730i
31.14 0.635617 + 1.26333i 1.45074 + 2.51275i −1.19198 + 1.60598i 0.500000 + 0.866025i −2.25231 + 3.42990i 1.19935i −2.78652 0.485072i −2.70927 + 4.69260i −0.776263 + 1.18212i
31.15 0.759006 1.19328i 0.600896 + 1.04078i −0.847818 1.81141i 0.500000 + 0.866025i 1.69802 + 0.0729253i 4.07590i −2.80501 0.363190i 0.777849 1.34727i 1.41291 + 0.0606805i
31.16 1.05146 + 0.945743i 0.256977 + 0.445098i 0.211139 + 1.98882i 0.500000 + 0.866025i −0.150747 + 0.711038i 1.09058i −1.65891 + 2.29085i 1.36793 2.36932i −0.293307 + 1.38346i
31.17 1.15432 0.817038i −1.05340 1.82454i 0.664897 1.88624i 0.500000 + 0.866025i −2.70668 1.24543i 0.279006i −0.773631 2.72057i −0.719298 + 1.24586i 1.28473 + 0.591149i
31.18 1.28473 0.591149i 1.05340 + 1.82454i 1.30109 1.51894i 0.500000 + 0.866025i 2.43191 + 1.72133i 0.279006i 0.773631 2.72057i −0.719298 + 1.24586i 1.15432 + 0.817038i
31.19 1.33731 + 0.460003i −0.435984 0.755146i 1.57679 + 1.23033i 0.500000 + 0.866025i −0.235676 1.21042i 4.89965i 1.54270 + 2.37067i 1.11984 1.93961i 0.270280 + 1.38815i
31.20 1.41291 0.0606805i −0.600896 1.04078i 1.99264 0.171472i 0.500000 + 0.866025i −0.912167 1.43407i 4.07590i 2.80501 0.363190i 0.777849 1.34727i 0.759006 + 1.19328i
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 331.20
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
19.d odd 6 1 inner
76.f even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 380.2.n.a 40
4.b odd 2 1 inner 380.2.n.a 40
19.d odd 6 1 inner 380.2.n.a 40
76.f even 6 1 inner 380.2.n.a 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
380.2.n.a 40 1.a even 1 1 trivial
380.2.n.a 40 4.b odd 2 1 inner
380.2.n.a 40 19.d odd 6 1 inner
380.2.n.a 40 76.f even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{40} + 41 T_{3}^{38} + 992 T_{3}^{36} + 15975 T_{3}^{34} + 191688 T_{3}^{32} + 1746767 T_{3}^{30} + 12475873 T_{3}^{28} + 69587636 T_{3}^{26} + 306343209 T_{3}^{24} + 1047905415 T_{3}^{22} + \cdots + 9834496 \) acting on \(S_{2}^{\mathrm{new}}(380, [\chi])\). Copy content Toggle raw display