Properties

Label 539.2.q.g
Level $539$
Weight $2$
Character orbit 539.q
Analytic conductor $4.304$
Analytic rank $0$
Dimension $32$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 539 = 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 539.q (of order \(15\), degree \(8\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.30393666895\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(4\) over \(\Q(\zeta_{15})\)
Twist minimal: no (minimal twist has level 77)
Sato-Tate group: $\mathrm{SU}(2)[C_{15}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 3 q^{2} + 2 q^{3} + 11 q^{4} + 5 q^{5} + 6 q^{6} - 10 q^{8} + 12 q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 32 q + 3 q^{2} + 2 q^{3} + 11 q^{4} + 5 q^{5} + 6 q^{6} - 10 q^{8} + 12 q^{9} - 12 q^{10} + 3 q^{11} - 18 q^{12} - 14 q^{13} - 36 q^{15} - 17 q^{16} + 5 q^{17} - 11 q^{18} - 19 q^{19} + 2 q^{20} - 66 q^{22} - 32 q^{23} + 35 q^{24} - 7 q^{25} + 27 q^{26} + 20 q^{27} + 6 q^{29} + 2 q^{30} + 7 q^{31} - 32 q^{32} + 26 q^{33} - 48 q^{34} + 104 q^{36} - 4 q^{37} + 5 q^{38} - 11 q^{39} + 10 q^{40} - 20 q^{41} - 16 q^{43} + 38 q^{44} - 70 q^{45} + 42 q^{46} + 23 q^{47} - 72 q^{48} + 104 q^{50} + 29 q^{51} - 33 q^{52} - 4 q^{53} - 60 q^{54} - 24 q^{55} - 22 q^{57} - 20 q^{58} - 17 q^{59} + 30 q^{60} + 7 q^{61} + 158 q^{62} + 14 q^{64} + 8 q^{65} - 8 q^{66} + 38 q^{67} + 2 q^{68} + 20 q^{69} - 28 q^{71} + 35 q^{73} + 29 q^{74} - 9 q^{75} + 104 q^{76} - 116 q^{78} - 15 q^{79} + 87 q^{80} + 14 q^{81} - 19 q^{82} + 10 q^{83} + 12 q^{85} + 52 q^{86} + 72 q^{87} - 55 q^{88} - 74 q^{89} - 28 q^{90} - 110 q^{92} - 32 q^{93} + 24 q^{94} - 32 q^{95} + 42 q^{96} + 40 q^{97} + 32 q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
214.1 −1.62629 + 1.80618i 1.31021 0.583344i −0.408403 3.88570i −1.23113 + 0.261684i −1.07716 + 3.31516i 0 3.74989 + 2.72445i −0.631025 + 0.700825i 1.52952 2.64921i
214.2 −0.942984 + 1.04729i −1.97635 + 0.879926i 0.00145969 + 0.0138880i 1.79137 0.380767i 0.942126 2.89957i 0 −2.29616 1.66826i 1.12429 1.24865i −1.29046 + 2.23514i
214.3 0.448146 0.497717i 2.86833 1.27706i 0.162170 + 1.54294i −2.09693 + 0.445717i 0.649815 1.99992i 0 1.92429 + 1.39808i 4.58902 5.09662i −0.717891 + 1.24342i
214.4 1.70758 1.89646i −0.375100 + 0.167005i −0.471675 4.48769i −3.35405 + 0.712924i −0.323795 + 0.996539i 0 −5.18703 3.76860i −1.89458 + 2.10415i −4.37528 + 7.57820i
312.1 −0.207438 + 1.97364i 1.86446 + 2.07069i −1.89594 0.402995i −0.0245966 0.0109511i −4.47357 + 3.25024i 0 −0.0378378 + 0.116453i −0.497972 + 4.73789i 0.0267159 0.0462732i
312.2 −0.116493 + 1.10836i −1.91292 2.12452i 0.741401 + 0.157590i 3.15728 + 1.40571i 2.57757 1.87272i 0 −0.949813 + 2.92322i −0.540709 + 5.14450i −1.92584 + 3.33566i
312.3 −0.0236455 + 0.224972i −0.146626 0.162845i 1.90624 + 0.405184i −2.27977 1.01502i 0.0401026 0.0291363i 0 −0.276036 + 0.849550i 0.308566 2.93581i 0.282258 0.488885i
312.4 0.178447 1.69781i 1.53335 + 1.70296i −0.894410 0.190113i 3.71481 + 1.65394i 3.16491 2.29944i 0 0.572703 1.76260i −0.235317 + 2.23889i 3.47097 6.01190i
324.1 −2.49618 0.530579i 0.0429192 0.408349i 4.12229 + 1.83536i 2.29443 + 2.54823i −0.323795 + 0.996539i 0 −5.18703 3.76860i 2.76954 + 0.588683i −4.37528 7.57820i
324.2 −0.655108 0.139248i −0.328196 + 3.12257i −1.41731 0.631029i 1.43447 + 1.59314i 0.649815 1.99992i 0 1.92429 + 1.39808i −6.70831 1.42589i −0.717891 1.24342i
324.3 1.37847 + 0.293003i 0.226135 2.15153i −0.0127572 0.00567988i −1.22544 1.36099i 0.942126 2.89957i 0 −2.29616 1.66826i −1.64350 0.349337i −1.29046 2.23514i
324.4 2.37734 + 0.505320i −0.149915 + 1.42635i 3.56931 + 1.58916i 0.842189 + 0.935345i −1.07716 + 3.31516i 0 3.74989 + 2.72445i 0.922445 + 0.196072i 1.52952 + 2.64921i
361.1 −2.49618 + 0.530579i 0.0429192 + 0.408349i 4.12229 1.83536i 2.29443 2.54823i −0.323795 0.996539i 0 −5.18703 + 3.76860i 2.76954 0.588683i −4.37528 + 7.57820i
361.2 −0.655108 + 0.139248i −0.328196 3.12257i −1.41731 + 0.631029i 1.43447 1.59314i 0.649815 + 1.99992i 0 1.92429 1.39808i −6.70831 + 1.42589i −0.717891 + 1.24342i
361.3 1.37847 0.293003i 0.226135 + 2.15153i −0.0127572 + 0.00567988i −1.22544 + 1.36099i 0.942126 + 2.89957i 0 −2.29616 + 1.66826i −1.64350 + 0.349337i −1.29046 + 2.23514i
361.4 2.37734 0.505320i −0.149915 1.42635i 3.56931 1.58916i 0.842189 0.935345i −1.07716 3.31516i 0 3.74989 2.72445i 0.922445 0.196072i 1.52952 2.64921i
410.1 −1.55957 0.694364i −2.24148 0.476441i 0.611847 + 0.679525i −0.425051 + 4.04409i 3.16491 + 2.29944i 0 0.572703 + 1.76260i 2.05659 + 0.915654i 3.47097 6.01190i
410.2 0.206654 + 0.0920084i 0.214341 + 0.0455596i −1.30402 1.44826i 0.260853 2.48185i 0.0401026 + 0.0291363i 0 −0.276036 0.849550i −2.69677 1.20068i 0.282258 0.488885i
410.3 1.01812 + 0.453294i 2.79635 + 0.594382i −0.507177 0.563277i −0.361259 + 3.43715i 2.57757 + 1.87272i 0 −0.949813 2.92322i 4.72563 + 2.10398i −1.92584 + 3.33566i
410.4 1.81294 + 0.807175i −2.72550 0.579324i 1.29698 + 1.44044i 0.00281436 0.0267768i −4.47357 3.25024i 0 −0.0378378 0.116453i 4.35212 + 1.93769i 0.0267159 0.0462732i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 520.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
11.c even 5 1 inner
77.m even 15 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 539.2.q.g 32
7.b odd 2 1 539.2.q.f 32
7.c even 3 1 77.2.f.b 16
7.c even 3 1 inner 539.2.q.g 32
7.d odd 6 1 539.2.f.e 16
7.d odd 6 1 539.2.q.f 32
11.c even 5 1 inner 539.2.q.g 32
21.h odd 6 1 693.2.m.i 16
77.h odd 6 1 847.2.f.x 16
77.j odd 10 1 539.2.q.f 32
77.m even 15 1 77.2.f.b 16
77.m even 15 1 inner 539.2.q.g 32
77.m even 15 1 847.2.a.p 8
77.m even 15 2 847.2.f.w 16
77.n even 30 1 5929.2.a.bs 8
77.o odd 30 1 847.2.a.o 8
77.o odd 30 2 847.2.f.v 16
77.o odd 30 1 847.2.f.x 16
77.p odd 30 1 539.2.f.e 16
77.p odd 30 1 539.2.q.f 32
77.p odd 30 1 5929.2.a.bt 8
231.z odd 30 1 693.2.m.i 16
231.z odd 30 1 7623.2.a.ct 8
231.be even 30 1 7623.2.a.cw 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.2.f.b 16 7.c even 3 1
77.2.f.b 16 77.m even 15 1
539.2.f.e 16 7.d odd 6 1
539.2.f.e 16 77.p odd 30 1
539.2.q.f 32 7.b odd 2 1
539.2.q.f 32 7.d odd 6 1
539.2.q.f 32 77.j odd 10 1
539.2.q.f 32 77.p odd 30 1
539.2.q.g 32 1.a even 1 1 trivial
539.2.q.g 32 7.c even 3 1 inner
539.2.q.g 32 11.c even 5 1 inner
539.2.q.g 32 77.m even 15 1 inner
693.2.m.i 16 21.h odd 6 1
693.2.m.i 16 231.z odd 30 1
847.2.a.o 8 77.o odd 30 1
847.2.a.p 8 77.m even 15 1
847.2.f.v 16 77.o odd 30 2
847.2.f.w 16 77.m even 15 2
847.2.f.x 16 77.h odd 6 1
847.2.f.x 16 77.o odd 30 1
5929.2.a.bs 8 77.n even 30 1
5929.2.a.bt 8 77.p odd 30 1
7623.2.a.ct 8 231.z odd 30 1
7623.2.a.cw 8 231.be even 30 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(539, [\chi])\):

\(T_{2}^{32} - \cdots\)
\(T_{3}^{32} - \cdots\)