Properties

Label 115.2.j.a
Level $115$
Weight $2$
Character orbit 115.j
Analytic conductor $0.918$
Analytic rank $0$
Dimension $100$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 115 = 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 115.j (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.918279623245\)
Analytic rank: \(0\)
Dimension: \(100\)
Relative dimension: \(10\) over \(\Q(\zeta_{22})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{22}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 100q - 14q^{4} - 9q^{5} - 18q^{6} - 12q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 100q - 14q^{4} - 9q^{5} - 18q^{6} - 12q^{9} - 13q^{10} - 26q^{11} - 26q^{14} - 10q^{15} - 18q^{16} - 14q^{19} + 49q^{20} - 22q^{21} - 68q^{24} + 21q^{25} - 42q^{26} - 24q^{29} + 19q^{30} - 12q^{31} + 8q^{34} - 37q^{35} - 10q^{36} + 14q^{39} - q^{40} + 8q^{41} + 166q^{44} - 42q^{45} - 18q^{46} + 32q^{49} - 23q^{50} - 22q^{51} + 116q^{54} + 27q^{55} - 116q^{56} + 50q^{59} + 123q^{60} - 38q^{61} + 10q^{64} + 76q^{65} - 28q^{66} + 80q^{69} + 102q^{70} - 110q^{71} + 22q^{74} + 6q^{75} + 4q^{76} + 42q^{79} + 18q^{80} + 204q^{81} + 56q^{84} - 121q^{85} + 132q^{86} - 66q^{89} - 198q^{90} + 76q^{91} - 70q^{94} - 74q^{95} + 236q^{96} - 60q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1 −2.04322 + 0.933106i 0.0755218 + 0.257204i 1.99433 2.30158i 0.256006 2.22136i −0.394306 0.455053i 0.783997 0.112722i −0.661572 + 2.25311i 2.46331 1.58307i 1.54969 + 4.77761i
4.2 −1.55935 + 0.712130i −0.478796 1.63063i 0.614711 0.709414i −1.26571 + 1.84336i 1.90783 + 2.20175i 4.84873 0.697142i 0.512574 1.74567i 0.0940571 0.0604468i 0.660971 3.77579i
4.3 −1.49755 + 0.683908i 0.900592 + 3.06714i 0.465201 0.536871i 2.15615 + 0.592474i −3.44632 3.97726i 0.632042 0.0908739i 0.598154 2.03713i −6.07249 + 3.90255i −3.63413 + 0.587347i
4.4 −0.477984 + 0.218288i −0.663307 2.25902i −1.12890 + 1.30282i −1.43085 1.71833i 0.810167 + 0.934983i −3.03882 + 0.436916i 0.551291 1.87752i −2.13942 + 1.37492i 1.05902 + 0.508999i
4.5 −0.175347 + 0.0800783i 0.167074 + 0.569000i −1.28539 + 1.48342i 0.481985 + 2.18350i −0.0748604 0.0863935i −3.28793 + 0.472733i 0.215217 0.732961i 2.22791 1.43179i −0.259366 0.344274i
4.6 0.175347 0.0800783i −0.167074 0.569000i −1.28539 + 1.48342i 2.18641 0.468631i −0.0748604 0.0863935i 3.28793 0.472733i −0.215217 + 0.732961i 2.22791 1.43179i 0.345853 0.257257i
4.7 0.477984 0.218288i 0.663307 + 2.25902i −1.12890 + 1.30282i −2.15745 0.587726i 0.810167 + 0.934983i 3.03882 0.436916i −0.551291 + 1.87752i −2.13942 + 1.37492i −1.15952 + 0.190022i
4.8 1.49755 0.683908i −0.900592 3.06714i 0.465201 0.536871i 1.43463 + 1.71518i −3.44632 3.97726i −0.632042 + 0.0908739i −0.598154 + 2.03713i −6.07249 + 3.90255i 3.32145 + 1.58741i
4.9 1.55935 0.712130i 0.478796 + 1.63063i 0.614711 0.709414i 1.15098 1.91709i 1.90783 + 2.20175i −4.84873 + 0.697142i −0.512574 + 1.74567i 0.0940571 0.0604468i 0.429565 3.80906i
4.10 2.04322 0.933106i −0.0755218 0.257204i 1.99433 2.30158i −1.91428 + 1.15566i −0.394306 0.455053i −0.783997 + 0.112722i 0.661572 2.25311i 2.46331 1.58307i −2.83293 + 4.14748i
9.1 −0.743795 2.53313i −1.34601 + 1.16633i −4.18102 + 2.68698i −2.23607 + 0.00123367i 3.95561 + 2.54212i 1.87277 + 0.855264i 5.92582 + 5.13475i 0.0244873 0.170313i 1.66630 + 5.66334i
9.2 −0.651256 2.21797i 0.917405 0.794936i −2.81277 + 1.80765i 1.79096 1.33883i −2.36061 1.51707i −1.70619 0.779189i 2.34716 + 2.03383i −0.217236 + 1.51091i −4.13586 3.10038i
9.3 −0.394732 1.34433i −1.97191 + 1.70867i 0.0310904 0.0199806i 2.16612 + 0.554912i 3.07540 + 1.97644i 2.68179 + 1.22473i −2.15687 1.86894i 0.541936 3.76925i −0.109049 3.13103i
9.4 −0.320885 1.09283i 1.24484 1.07866i 0.591189 0.379934i 0.429235 + 2.19448i −1.57825 1.01428i 1.40950 + 0.643697i −2.32646 2.01589i −0.0408222 + 0.283925i 2.26047 1.17326i
9.5 −0.176688 0.601743i 0.563711 0.488459i 1.35163 0.868641i −1.82300 1.29486i −0.393527 0.252905i 0.620531 + 0.283387i −1.70945 1.48124i −0.347766 + 2.41877i −0.457073 + 1.32576i
9.6 0.176688 + 0.601743i −0.563711 + 0.488459i 1.35163 0.868641i 1.38435 1.75601i −0.393527 0.252905i −0.620531 0.283387i 1.70945 + 1.48124i −0.347766 + 2.41877i 1.30126 + 0.522758i
9.7 0.320885 + 1.09283i −1.24484 + 1.07866i 0.591189 0.379934i 0.206409 + 2.22652i −1.57825 1.01428i −1.40950 0.643697i 2.32646 + 2.01589i −0.0408222 + 0.283925i −2.36698 + 0.940028i
9.8 0.394732 + 1.34433i 1.97191 1.70867i 0.0310904 0.0199806i −1.92204 + 1.14270i 3.07540 + 1.97644i −2.68179 1.22473i 2.15687 + 1.86894i 0.541936 3.76925i −2.29486 2.13280i
9.9 0.651256 + 2.21797i −0.917405 + 0.794936i −2.81277 + 1.80765i −2.09560 0.780027i −2.36061 1.51707i 1.70619 + 0.779189i −2.34716 2.03383i −0.217236 + 1.51091i 0.365306 5.15599i
9.10 0.743795 + 2.53313i 1.34601 1.16633i −4.18102 + 2.68698i 2.14584 0.628789i 3.95561 + 2.54212i −1.87277 0.855264i −5.92582 5.13475i 0.0244873 0.170313i 3.18887 + 4.96800i
See all 100 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 104.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
23.c even 11 1 inner
115.j even 22 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 115.2.j.a 100
5.b even 2 1 inner 115.2.j.a 100
5.c odd 4 2 575.2.k.g 100
23.c even 11 1 inner 115.2.j.a 100
115.j even 22 1 inner 115.2.j.a 100
115.k odd 44 2 575.2.k.g 100
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
115.2.j.a 100 1.a even 1 1 trivial
115.2.j.a 100 5.b even 2 1 inner
115.2.j.a 100 23.c even 11 1 inner
115.2.j.a 100 115.j even 22 1 inner
575.2.k.g 100 5.c odd 4 2
575.2.k.g 100 115.k odd 44 2

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(115, [\chi])\).