# 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 $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [539,2,Mod(214,539)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(539, base_ring=CyclotomicField(30))

chi = DirichletCharacter(H, H._module([20, 12]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("539.214");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 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.

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

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 $$S_{2}^{\mathrm{new}}(539, [\chi])$$:

 $$T_{2}^{32} - 3 T_{2}^{31} - 5 T_{2}^{30} + 22 T_{2}^{29} + 14 T_{2}^{28} - 77 T_{2}^{27} - 125 T_{2}^{26} + 328 T_{2}^{25} + 156 T_{2}^{24} + 210 T_{2}^{23} + 4375 T_{2}^{22} - 19805 T_{2}^{21} + 11570 T_{2}^{20} + 14130 T_{2}^{19} + 5420 T_{2}^{18} + \cdots + 625$$ T2^32 - 3*T2^31 - 5*T2^30 + 22*T2^29 + 14*T2^28 - 77*T2^27 - 125*T2^26 + 328*T2^25 + 156*T2^24 + 210*T2^23 + 4375*T2^22 - 19805*T2^21 + 11570*T2^20 + 14130*T2^19 + 5420*T2^18 + 15325*T2^17 + 95925*T2^16 - 403550*T2^15 + 175350*T2^14 - 59500*T2^13 + 928775*T2^12 - 1232125*T2^11 + 1094250*T2^10 - 1243500*T2^9 + 254500*T2^8 + 1052375*T2^7 - 792625*T2^6 + 35000*T2^5 + 258750*T2^4 - 93750*T2^3 + 22500*T2^2 - 5000*T2 + 625 $$T_{3}^{32} - 2 T_{3}^{31} - 10 T_{3}^{30} + 24 T_{3}^{29} + 20 T_{3}^{28} + 32 T_{3}^{27} + 234 T_{3}^{26} - 1740 T_{3}^{25} - 2647 T_{3}^{24} + 22174 T_{3}^{23} + 30888 T_{3}^{22} - 222742 T_{3}^{21} - 9459 T_{3}^{20} + 1042822 T_{3}^{19} + \cdots + 65536$$ T3^32 - 2*T3^31 - 10*T3^30 + 24*T3^29 + 20*T3^28 + 32*T3^27 + 234*T3^26 - 1740*T3^25 - 2647*T3^24 + 22174*T3^23 + 30888*T3^22 - 222742*T3^21 - 9459*T3^20 + 1042822*T3^19 - 1899947*T3^18 + 724842*T3^17 + 12609261*T3^16 - 23755248*T3^15 + 1967068*T3^14 + 135013072*T3^13 - 34861104*T3^12 - 25613632*T3^11 + 78207168*T3^10 - 579908736*T3^9 + 683738368*T3^8 + 438679040*T3^7 + 123142144*T3^6 + 46272512*T3^5 + 7659520*T3^4 - 6823936*T3^3 - 983040*T3^2 + 32768*T3 + 65536