Properties

Label 380.3.h.a
Level $380$
Weight $3$
Character orbit 380.h
Analytic conductor $10.354$
Analytic rank $0$
Dimension $108$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 380 = 2^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 380.h (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(10.3542500457\)
Analytic rank: \(0\)
Dimension: \(108\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 108 q + 4 q^{5} + 324 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 108 q + 4 q^{5} + 324 q^{9} - 8 q^{10} + 8 q^{14} - 104 q^{16} - 16 q^{21} - 8 q^{24} - 76 q^{25} + 80 q^{26} - 88 q^{29} - 140 q^{30} - 88 q^{34} - 256 q^{36} + 44 q^{40} - 200 q^{41} - 8 q^{44} + 108 q^{45} + 272 q^{46} + 916 q^{49} - 276 q^{50} - 320 q^{54} - 328 q^{56} + 172 q^{60} + 200 q^{61} - 216 q^{64} - 192 q^{65} + 152 q^{66} - 592 q^{69} + 200 q^{70} - 232 q^{74} + 340 q^{80} + 1052 q^{81} + 208 q^{84} + 248 q^{85} - 1048 q^{86} + 760 q^{89} + 268 q^{90} - 320 q^{94} + 720 q^{96}+O(q^{100}) \) 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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
39.1 −1.99911 0.0595768i −2.57497 3.99290 + 0.238201i 2.22773 4.47629i 5.14765 + 0.153408i −1.63196 −7.96807 0.714075i −2.36954 −4.72017 + 8.81589i
39.2 −1.99911 + 0.0595768i −2.57497 3.99290 0.238201i 2.22773 + 4.47629i 5.14765 0.153408i −1.63196 −7.96807 + 0.714075i −2.36954 −4.72017 8.81589i
39.3 −1.99455 0.147607i −0.578687 3.95642 + 0.588817i −3.52665 3.54440i 1.15422 + 0.0854181i 11.0384 −7.80436 1.75842i −8.66512 6.51089 + 7.59002i
39.4 −1.99455 + 0.147607i −0.578687 3.95642 0.588817i −3.52665 + 3.54440i 1.15422 0.0854181i 11.0384 −7.80436 + 1.75842i −8.66512 6.51089 7.59002i
39.5 −1.94809 0.452727i 2.05284 3.59008 + 1.76390i 4.88304 1.07515i −3.99911 0.929375i −5.89695 −6.19521 5.06155i −4.78585 −9.99932 0.116190i
39.6 −1.94809 + 0.452727i 2.05284 3.59008 1.76390i 4.88304 + 1.07515i −3.99911 + 0.929375i −5.89695 −6.19521 + 5.06155i −4.78585 −9.99932 + 0.116190i
39.7 −1.94008 0.485899i 4.64902 3.52780 + 1.88537i 2.66799 + 4.22869i −9.01947 2.25896i 9.90681 −5.92811 5.37191i 12.6134 −3.12138 9.50037i
39.8 −1.94008 + 0.485899i 4.64902 3.52780 1.88537i 2.66799 4.22869i −9.01947 + 2.25896i 9.90681 −5.92811 + 5.37191i 12.6134 −3.12138 + 9.50037i
39.9 −1.91201 0.586697i −5.34961 3.31157 + 2.24354i −3.58581 + 3.48454i 10.2285 + 3.13860i 0.878631 −5.01548 6.23257i 19.6184 8.90047 4.55869i
39.10 −1.91201 + 0.586697i −5.34961 3.31157 2.24354i −3.58581 3.48454i 10.2285 3.13860i 0.878631 −5.01548 + 6.23257i 19.6184 8.90047 + 4.55869i
39.11 −1.86011 0.734843i 2.97071 2.92001 + 2.73378i −0.632199 + 4.95987i −5.52585 2.18301i −5.57969 −3.42265 7.23087i −0.174871 4.82068 8.76134i
39.12 −1.86011 + 0.734843i 2.97071 2.92001 2.73378i −0.632199 4.95987i −5.52585 + 2.18301i −5.57969 −3.42265 + 7.23087i −0.174871 4.82068 + 8.76134i
39.13 −1.85933 0.736815i −2.04408 2.91421 + 2.73996i −4.55181 2.06907i 3.80062 + 1.50611i −11.4458 −3.39962 7.24172i −4.82174 6.93879 + 7.20092i
39.14 −1.85933 + 0.736815i −2.04408 2.91421 2.73996i −4.55181 + 2.06907i 3.80062 1.50611i −11.4458 −3.39962 + 7.24172i −4.82174 6.93879 7.20092i
39.15 −1.75001 0.968228i −4.24708 2.12507 + 3.38882i 4.77157 + 1.49403i 7.43242 + 4.11214i 7.86306 −0.437742 7.98801i 9.03765 −6.90374 7.23453i
39.16 −1.75001 + 0.968228i −4.24708 2.12507 3.38882i 4.77157 1.49403i 7.43242 4.11214i 7.86306 −0.437742 + 7.98801i 9.03765 −6.90374 + 7.23453i
39.17 −1.70952 1.03804i 1.20444 1.84495 + 3.54911i 3.17819 3.85994i −2.05903 1.25026i 6.44137 0.530130 7.98242i −7.54932 −9.43996 + 3.29958i
39.18 −1.70952 + 1.03804i 1.20444 1.84495 3.54911i 3.17819 + 3.85994i −2.05903 + 1.25026i 6.44137 0.530130 + 7.98242i −7.54932 −9.43996 3.29958i
39.19 −1.68898 1.07114i 5.94620 1.70531 + 3.61828i 1.71987 4.69490i −10.0430 6.36923i −9.59523 0.995461 7.93782i 26.3573 −7.93373 + 6.08736i
39.20 −1.68898 + 1.07114i 5.94620 1.70531 3.61828i 1.71987 + 4.69490i −10.0430 + 6.36923i −9.59523 0.995461 + 7.93782i 26.3573 −7.93373 6.08736i
See next 80 embeddings (of 108 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 39.108
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
20.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 380.3.h.a 108
4.b odd 2 1 inner 380.3.h.a 108
5.b even 2 1 inner 380.3.h.a 108
20.d odd 2 1 inner 380.3.h.a 108
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
380.3.h.a 108 1.a even 1 1 trivial
380.3.h.a 108 4.b odd 2 1 inner
380.3.h.a 108 5.b even 2 1 inner
380.3.h.a 108 20.d odd 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(380, [\chi])\).