# Properties

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

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [539,2,Mod(67,539)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(539, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([4, 0]))

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

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

chi := DirichletCharacter("539.67");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$539 = 7^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 539.e (of order $$3$$, degree $$2$$, 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: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{5})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} + 2x^{2} + x + 1$$ x^4 - x^3 + 2*x^2 + x + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 77) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_{2} q^{2} + ( - \beta_{3} - \beta_{2} + \beta_1) q^{3} + 3 \beta_1 q^{4} + (2 \beta_1 + 2) q^{5} + ( - \beta_{3} - 5) q^{6} - \beta_{3} q^{8} + (2 \beta_{2} - 3 \beta_1 - 3) q^{9}+O(q^{10})$$ q - b2 * q^2 + (-b3 - b2 + b1) * q^3 + 3*b1 * q^4 + (2*b1 + 2) * q^5 + (-b3 - 5) * q^6 - b3 * q^8 + (2*b2 - 3*b1 - 3) * q^9 $$q - \beta_{2} q^{2} + ( - \beta_{3} - \beta_{2} + \beta_1) q^{3} + 3 \beta_1 q^{4} + (2 \beta_1 + 2) q^{5} + ( - \beta_{3} - 5) q^{6} - \beta_{3} q^{8} + (2 \beta_{2} - 3 \beta_1 - 3) q^{9} + ( - 2 \beta_{3} - 2 \beta_{2}) q^{10} - \beta_1 q^{11} + (3 \beta_{2} - 3 \beta_1 - 3) q^{12} + ( - \beta_{3} + 1) q^{13} + ( - 2 \beta_{3} - 2) q^{15} + (\beta_1 + 1) q^{16} + ( - \beta_{3} - \beta_{2} - \beta_1) q^{17} + (3 \beta_{3} + 3 \beta_{2} - 10 \beta_1) q^{18} + ( - 2 \beta_{2} - 2 \beta_1 - 2) q^{19} - 6 q^{20} + \beta_{3} q^{22} + ( - 2 \beta_{2} + 2 \beta_1 + 2) q^{23} + (\beta_{3} + \beta_{2} - 5 \beta_1) q^{24} - \beta_1 q^{25} + ( - \beta_{2} - 5 \beta_1 - 5) q^{26} + (2 \beta_{3} + 10) q^{27} + ( - 2 \beta_{3} + 4) q^{29} + (2 \beta_{2} - 10 \beta_1 - 10) q^{30} + (\beta_{3} + \beta_{2} - 5 \beta_1) q^{31} + ( - 3 \beta_{3} - 3 \beta_{2}) q^{32} + ( - \beta_{2} + \beta_1 + 1) q^{33} + (\beta_{3} - 5) q^{34} + (6 \beta_{3} + 9) q^{36} + (2 \beta_{2} + 4 \beta_1 + 4) q^{37} + (2 \beta_{3} + 2 \beta_{2} + 10 \beta_1) q^{38} - 4 \beta_1 q^{39} + 2 \beta_{2} q^{40} + (\beta_{3} - 9) q^{41} + 8 q^{43} + (3 \beta_1 + 3) q^{44} + (4 \beta_{3} + 4 \beta_{2} - 6 \beta_1) q^{45} + ( - 2 \beta_{3} - 2 \beta_{2} + 10 \beta_1) q^{46} + (\beta_{2} - 5 \beta_1 - 5) q^{47} + ( - \beta_{3} - 1) q^{48} + \beta_{3} q^{50} + ( - 4 \beta_1 - 4) q^{51} + (3 \beta_{3} + 3 \beta_{2} + 3 \beta_1) q^{52} + ( - 2 \beta_{3} - 2 \beta_{2} + 4 \beta_1) q^{53} + ( - 10 \beta_{2} + 10 \beta_1 + 10) q^{54} + 2 q^{55} - 8 q^{57} + ( - 4 \beta_{2} - 10 \beta_1 - 10) q^{58} + ( - \beta_{3} - \beta_{2} + \beta_1) q^{59} + (6 \beta_{3} + 6 \beta_{2} - 6 \beta_1) q^{60} + (\beta_{2} + 5 \beta_1 + 5) q^{61} + (5 \beta_{3} + 5) q^{62} - 13 q^{64} + (2 \beta_{2} + 2 \beta_1 + 2) q^{65} + ( - \beta_{3} - \beta_{2} + 5 \beta_1) q^{66} + (2 \beta_{3} + 2 \beta_{2} + 10 \beta_1) q^{67} + (3 \beta_{2} + 3 \beta_1 + 3) q^{68} + ( - 4 \beta_{3} - 12) q^{69} + (2 \beta_{3} - 6) q^{71} + ( - 3 \beta_{2} + 10 \beta_1 + 10) q^{72} + (\beta_{3} + \beta_{2} - 3 \beta_1) q^{73} + ( - 4 \beta_{3} - 4 \beta_{2} - 10 \beta_1) q^{74} + ( - \beta_{2} + \beta_1 + 1) q^{75} + ( - 6 \beta_{3} + 6) q^{76} + 4 \beta_{3} q^{78} + 4 \beta_{2} q^{79} + 2 \beta_1 q^{80} + ( - 6 \beta_{3} - 6 \beta_{2} + 11 \beta_1) q^{81} + (9 \beta_{2} + 5 \beta_1 + 5) q^{82} + (6 \beta_{3} + 2) q^{83} + ( - 2 \beta_{3} + 2) q^{85} - 8 \beta_{2} q^{86} + ( - 2 \beta_{3} - 2 \beta_{2} - 6 \beta_1) q^{87} + ( - \beta_{3} - \beta_{2}) q^{88} + ( - 2 \beta_1 - 2) q^{89} + (6 \beta_{3} + 20) q^{90} + ( - 6 \beta_{3} - 6) q^{92} + ( - 6 \beta_{2} + 10 \beta_1 + 10) q^{93} + (5 \beta_{3} + 5 \beta_{2} - 5 \beta_1) q^{94} + ( - 4 \beta_{3} - 4 \beta_{2} - 4 \beta_1) q^{95} + (3 \beta_{2} - 15 \beta_1 - 15) q^{96} + ( - 6 \beta_{3} + 4) q^{97} + ( - 2 \beta_{3} - 3) q^{99}+O(q^{100})$$ q - b2 * q^2 + (-b3 - b2 + b1) * q^3 + 3*b1 * q^4 + (2*b1 + 2) * q^5 + (-b3 - 5) * q^6 - b3 * q^8 + (2*b2 - 3*b1 - 3) * q^9 + (-2*b3 - 2*b2) * q^10 - b1 * q^11 + (3*b2 - 3*b1 - 3) * q^12 + (-b3 + 1) * q^13 + (-2*b3 - 2) * q^15 + (b1 + 1) * q^16 + (-b3 - b2 - b1) * q^17 + (3*b3 + 3*b2 - 10*b1) * q^18 + (-2*b2 - 2*b1 - 2) * q^19 - 6 * q^20 + b3 * q^22 + (-2*b2 + 2*b1 + 2) * q^23 + (b3 + b2 - 5*b1) * q^24 - b1 * q^25 + (-b2 - 5*b1 - 5) * q^26 + (2*b3 + 10) * q^27 + (-2*b3 + 4) * q^29 + (2*b2 - 10*b1 - 10) * q^30 + (b3 + b2 - 5*b1) * q^31 + (-3*b3 - 3*b2) * q^32 + (-b2 + b1 + 1) * q^33 + (b3 - 5) * q^34 + (6*b3 + 9) * q^36 + (2*b2 + 4*b1 + 4) * q^37 + (2*b3 + 2*b2 + 10*b1) * q^38 - 4*b1 * q^39 + 2*b2 * q^40 + (b3 - 9) * q^41 + 8 * q^43 + (3*b1 + 3) * q^44 + (4*b3 + 4*b2 - 6*b1) * q^45 + (-2*b3 - 2*b2 + 10*b1) * q^46 + (b2 - 5*b1 - 5) * q^47 + (-b3 - 1) * q^48 + b3 * q^50 + (-4*b1 - 4) * q^51 + (3*b3 + 3*b2 + 3*b1) * q^52 + (-2*b3 - 2*b2 + 4*b1) * q^53 + (-10*b2 + 10*b1 + 10) * q^54 + 2 * q^55 - 8 * q^57 + (-4*b2 - 10*b1 - 10) * q^58 + (-b3 - b2 + b1) * q^59 + (6*b3 + 6*b2 - 6*b1) * q^60 + (b2 + 5*b1 + 5) * q^61 + (5*b3 + 5) * q^62 - 13 * q^64 + (2*b2 + 2*b1 + 2) * q^65 + (-b3 - b2 + 5*b1) * q^66 + (2*b3 + 2*b2 + 10*b1) * q^67 + (3*b2 + 3*b1 + 3) * q^68 + (-4*b3 - 12) * q^69 + (2*b3 - 6) * q^71 + (-3*b2 + 10*b1 + 10) * q^72 + (b3 + b2 - 3*b1) * q^73 + (-4*b3 - 4*b2 - 10*b1) * q^74 + (-b2 + b1 + 1) * q^75 + (-6*b3 + 6) * q^76 + 4*b3 * q^78 + 4*b2 * q^79 + 2*b1 * q^80 + (-6*b3 - 6*b2 + 11*b1) * q^81 + (9*b2 + 5*b1 + 5) * q^82 + (6*b3 + 2) * q^83 + (-2*b3 + 2) * q^85 - 8*b2 * q^86 + (-2*b3 - 2*b2 - 6*b1) * q^87 + (-b3 - b2) * q^88 + (-2*b1 - 2) * q^89 + (6*b3 + 20) * q^90 + (-6*b3 - 6) * q^92 + (-6*b2 + 10*b1 + 10) * q^93 + (5*b3 + 5*b2 - 5*b1) * q^94 + (-4*b3 - 4*b2 - 4*b1) * q^95 + (3*b2 - 15*b1 - 15) * q^96 + (-6*b3 + 4) * q^97 + (-2*b3 - 3) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 2 q^{3} - 6 q^{4} + 4 q^{5} - 20 q^{6} - 6 q^{9}+O(q^{10})$$ 4 * q - 2 * q^3 - 6 * q^4 + 4 * q^5 - 20 * q^6 - 6 * q^9 $$4 q - 2 q^{3} - 6 q^{4} + 4 q^{5} - 20 q^{6} - 6 q^{9} + 2 q^{11} - 6 q^{12} + 4 q^{13} - 8 q^{15} + 2 q^{16} + 2 q^{17} + 20 q^{18} - 4 q^{19} - 24 q^{20} + 4 q^{23} + 10 q^{24} + 2 q^{25} - 10 q^{26} + 40 q^{27} + 16 q^{29} - 20 q^{30} + 10 q^{31} + 2 q^{33} - 20 q^{34} + 36 q^{36} + 8 q^{37} - 20 q^{38} + 8 q^{39} - 36 q^{41} + 32 q^{43} + 6 q^{44} + 12 q^{45} - 20 q^{46} - 10 q^{47} - 4 q^{48} - 8 q^{51} - 6 q^{52} - 8 q^{53} + 20 q^{54} + 8 q^{55} - 32 q^{57} - 20 q^{58} - 2 q^{59} + 12 q^{60} + 10 q^{61} + 20 q^{62} - 52 q^{64} + 4 q^{65} - 10 q^{66} - 20 q^{67} + 6 q^{68} - 48 q^{69} - 24 q^{71} + 20 q^{72} + 6 q^{73} + 20 q^{74} + 2 q^{75} + 24 q^{76} - 4 q^{80} - 22 q^{81} + 10 q^{82} + 8 q^{83} + 8 q^{85} + 12 q^{87} - 4 q^{89} + 80 q^{90} - 24 q^{92} + 20 q^{93} + 10 q^{94} + 8 q^{95} - 30 q^{96} + 16 q^{97} - 12 q^{99}+O(q^{100})$$ 4 * q - 2 * q^3 - 6 * q^4 + 4 * q^5 - 20 * q^6 - 6 * q^9 + 2 * q^11 - 6 * q^12 + 4 * q^13 - 8 * q^15 + 2 * q^16 + 2 * q^17 + 20 * q^18 - 4 * q^19 - 24 * q^20 + 4 * q^23 + 10 * q^24 + 2 * q^25 - 10 * q^26 + 40 * q^27 + 16 * q^29 - 20 * q^30 + 10 * q^31 + 2 * q^33 - 20 * q^34 + 36 * q^36 + 8 * q^37 - 20 * q^38 + 8 * q^39 - 36 * q^41 + 32 * q^43 + 6 * q^44 + 12 * q^45 - 20 * q^46 - 10 * q^47 - 4 * q^48 - 8 * q^51 - 6 * q^52 - 8 * q^53 + 20 * q^54 + 8 * q^55 - 32 * q^57 - 20 * q^58 - 2 * q^59 + 12 * q^60 + 10 * q^61 + 20 * q^62 - 52 * q^64 + 4 * q^65 - 10 * q^66 - 20 * q^67 + 6 * q^68 - 48 * q^69 - 24 * q^71 + 20 * q^72 + 6 * q^73 + 20 * q^74 + 2 * q^75 + 24 * q^76 - 4 * q^80 - 22 * q^81 + 10 * q^82 + 8 * q^83 + 8 * q^85 + 12 * q^87 - 4 * q^89 + 80 * q^90 - 24 * q^92 + 20 * q^93 + 10 * q^94 + 8 * q^95 - 30 * q^96 + 16 * q^97 - 12 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} + 2x^{2} + x + 1$$ :

 $$\beta_{1}$$ $$=$$ $$( -\nu^{3} + 2\nu^{2} - 2\nu - 1 ) / 2$$ (-v^3 + 2*v^2 - 2*v - 1) / 2 $$\beta_{2}$$ $$=$$ $$( \nu^{3} - 2\nu^{2} + 6\nu - 1 ) / 2$$ (v^3 - 2*v^2 + 6*v - 1) / 2 $$\beta_{3}$$ $$=$$ $$\nu^{3} + 2$$ v^3 + 2
 $$\nu$$ $$=$$ $$( \beta_{2} + \beta _1 + 1 ) / 2$$ (b2 + b1 + 1) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{3} + \beta_{2} + 3\beta_1 ) / 2$$ (b3 + b2 + 3*b1) / 2 $$\nu^{3}$$ $$=$$ $$\beta_{3} - 2$$ b3 - 2

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/539\mathbb{Z}\right)^\times$$.

 $$n$$ $$199$$ $$442$$ $$\chi(n)$$ $$-1 - \beta_{1}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
67.1
 0.809017 + 1.40126i −0.309017 − 0.535233i 0.809017 − 1.40126i −0.309017 + 0.535233i
−1.11803 1.93649i 0.618034 1.07047i −1.50000 + 2.59808i 1.00000 + 1.73205i −2.76393 0 2.23607 0.736068 + 1.27491i 2.23607 3.87298i
67.2 1.11803 + 1.93649i −1.61803 + 2.80252i −1.50000 + 2.59808i 1.00000 + 1.73205i −7.23607 0 −2.23607 −3.73607 6.47106i −2.23607 + 3.87298i
177.1 −1.11803 + 1.93649i 0.618034 + 1.07047i −1.50000 2.59808i 1.00000 1.73205i −2.76393 0 2.23607 0.736068 1.27491i 2.23607 + 3.87298i
177.2 1.11803 1.93649i −1.61803 2.80252i −1.50000 2.59808i 1.00000 1.73205i −7.23607 0 −2.23607 −3.73607 + 6.47106i −2.23607 3.87298i
 $$n$$: e.g. 2-40 or 990-1000 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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 539.2.e.i 4
7.b odd 2 1 539.2.e.j 4
7.c even 3 1 77.2.a.d 2
7.c even 3 1 inner 539.2.e.i 4
7.d odd 6 1 539.2.a.f 2
7.d odd 6 1 539.2.e.j 4
21.g even 6 1 4851.2.a.y 2
21.h odd 6 1 693.2.a.h 2
28.f even 6 1 8624.2.a.ce 2
28.g odd 6 1 1232.2.a.m 2
35.j even 6 1 1925.2.a.r 2
35.l odd 12 2 1925.2.b.h 4
56.k odd 6 1 4928.2.a.bv 2
56.p even 6 1 4928.2.a.bm 2
77.h odd 6 1 847.2.a.f 2
77.i even 6 1 5929.2.a.m 2
77.m even 15 2 847.2.f.a 4
77.m even 15 2 847.2.f.n 4
77.o odd 30 2 847.2.f.b 4
77.o odd 30 2 847.2.f.m 4
231.l even 6 1 7623.2.a.bl 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.2.a.d 2 7.c even 3 1
539.2.a.f 2 7.d odd 6 1
539.2.e.i 4 1.a even 1 1 trivial
539.2.e.i 4 7.c even 3 1 inner
539.2.e.j 4 7.b odd 2 1
539.2.e.j 4 7.d odd 6 1
693.2.a.h 2 21.h odd 6 1
847.2.a.f 2 77.h odd 6 1
847.2.f.a 4 77.m even 15 2
847.2.f.b 4 77.o odd 30 2
847.2.f.m 4 77.o odd 30 2
847.2.f.n 4 77.m even 15 2
1232.2.a.m 2 28.g odd 6 1
1925.2.a.r 2 35.j even 6 1
1925.2.b.h 4 35.l odd 12 2
4851.2.a.y 2 21.g even 6 1
4928.2.a.bm 2 56.p even 6 1
4928.2.a.bv 2 56.k odd 6 1
5929.2.a.m 2 77.i even 6 1
7623.2.a.bl 2 231.l even 6 1
8624.2.a.ce 2 28.f even 6 1

## Hecke kernels

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

 $$T_{2}^{4} + 5T_{2}^{2} + 25$$ T2^4 + 5*T2^2 + 25 $$T_{3}^{4} + 2T_{3}^{3} + 8T_{3}^{2} - 8T_{3} + 16$$ T3^4 + 2*T3^3 + 8*T3^2 - 8*T3 + 16

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 5T^{2} + 25$$
$3$ $$T^{4} + 2 T^{3} + 8 T^{2} - 8 T + 16$$
$5$ $$(T^{2} - 2 T + 4)^{2}$$
$7$ $$T^{4}$$
$11$ $$(T^{2} - T + 1)^{2}$$
$13$ $$(T^{2} - 2 T - 4)^{2}$$
$17$ $$T^{4} - 2 T^{3} + 8 T^{2} + 8 T + 16$$
$19$ $$T^{4} + 4 T^{3} + 32 T^{2} - 64 T + 256$$
$23$ $$T^{4} - 4 T^{3} + 32 T^{2} + 64 T + 256$$
$29$ $$(T^{2} - 8 T - 4)^{2}$$
$31$ $$T^{4} - 10 T^{3} + 80 T^{2} + \cdots + 400$$
$37$ $$T^{4} - 8 T^{3} + 68 T^{2} + 32 T + 16$$
$41$ $$(T^{2} + 18 T + 76)^{2}$$
$43$ $$(T - 8)^{4}$$
$47$ $$T^{4} + 10 T^{3} + 80 T^{2} + \cdots + 400$$
$53$ $$T^{4} + 8 T^{3} + 68 T^{2} - 32 T + 16$$
$59$ $$T^{4} + 2 T^{3} + 8 T^{2} - 8 T + 16$$
$61$ $$T^{4} - 10 T^{3} + 80 T^{2} + \cdots + 400$$
$67$ $$T^{4} + 20 T^{3} + 320 T^{2} + \cdots + 6400$$
$71$ $$(T^{2} + 12 T + 16)^{2}$$
$73$ $$T^{4} - 6 T^{3} + 32 T^{2} - 24 T + 16$$
$79$ $$T^{4} + 80T^{2} + 6400$$
$83$ $$(T^{2} - 4 T - 176)^{2}$$
$89$ $$(T^{2} + 2 T + 4)^{2}$$
$97$ $$(T^{2} - 8 T - 164)^{2}$$