Properties

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

Related objects

Downloads

Learn more about

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 \(\Q(\zeta_{15})\)
Coefficient ring index: multiple of None
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.

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

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.f 32
7.b odd 2 1 539.2.q.g 32
7.c even 3 1 539.2.f.e 16
7.c even 3 1 inner 539.2.q.f 32
7.d odd 6 1 77.2.f.b 16
7.d odd 6 1 539.2.q.g 32
11.c even 5 1 inner 539.2.q.f 32
21.g even 6 1 693.2.m.i 16
77.i even 6 1 847.2.f.x 16
77.j odd 10 1 539.2.q.g 32
77.m even 15 1 539.2.f.e 16
77.m even 15 1 inner 539.2.q.f 32
77.m even 15 1 5929.2.a.bt 8
77.n even 30 1 847.2.a.o 8
77.n even 30 2 847.2.f.v 16
77.n even 30 1 847.2.f.x 16
77.o odd 30 1 5929.2.a.bs 8
77.p odd 30 1 77.2.f.b 16
77.p odd 30 1 539.2.q.g 32
77.p odd 30 1 847.2.a.p 8
77.p odd 30 2 847.2.f.w 16
231.bc even 30 1 693.2.m.i 16
231.bc even 30 1 7623.2.a.ct 8
231.bf odd 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.d odd 6 1
77.2.f.b 16 77.p odd 30 1
539.2.f.e 16 7.c even 3 1
539.2.f.e 16 77.m even 15 1
539.2.q.f 32 1.a even 1 1 trivial
539.2.q.f 32 7.c even 3 1 inner
539.2.q.f 32 11.c even 5 1 inner
539.2.q.f 32 77.m even 15 1 inner
539.2.q.g 32 7.b odd 2 1
539.2.q.g 32 7.d odd 6 1
539.2.q.g 32 77.j odd 10 1
539.2.q.g 32 77.p odd 30 1
693.2.m.i 16 21.g even 6 1
693.2.m.i 16 231.bc even 30 1
847.2.a.o 8 77.n even 30 1
847.2.a.p 8 77.p odd 30 1
847.2.f.v 16 77.n even 30 2
847.2.f.w 16 77.p odd 30 2
847.2.f.x 16 77.i even 6 1
847.2.f.x 16 77.n even 30 1
5929.2.a.bs 8 77.o odd 30 1
5929.2.a.bt 8 77.m even 15 1
7623.2.a.ct 8 231.bc even 30 1
7623.2.a.cw 8 231.bf odd 30 1

Hecke kernels

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

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

Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database