Properties

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

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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: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \zeta_{6} + 1) q^{3} + ( - 2 \zeta_{6} + 2) q^{4} + 3 \zeta_{6} q^{5} + 2 \zeta_{6} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{6} + 1) q^{3} + ( - 2 \zeta_{6} + 2) q^{4} + 3 \zeta_{6} q^{5} + 2 \zeta_{6} q^{9} + ( - \zeta_{6} + 1) q^{11} - 2 \zeta_{6} q^{12} + 4 q^{13} + 3 q^{15} - 4 \zeta_{6} q^{16} + (6 \zeta_{6} - 6) q^{17} + 2 \zeta_{6} q^{19} + 6 q^{20} - 3 \zeta_{6} q^{23} + (4 \zeta_{6} - 4) q^{25} + 5 q^{27} - 6 q^{29} + ( - 5 \zeta_{6} + 5) q^{31} - \zeta_{6} q^{33} + 4 q^{36} - 11 \zeta_{6} q^{37} + ( - 4 \zeta_{6} + 4) q^{39} - 6 q^{41} + 8 q^{43} - 2 \zeta_{6} q^{44} + (6 \zeta_{6} - 6) q^{45} - 4 q^{48} + 6 \zeta_{6} q^{51} + ( - 8 \zeta_{6} + 8) q^{52} + ( - 6 \zeta_{6} + 6) q^{53} + 3 q^{55} + 2 q^{57} + (9 \zeta_{6} - 9) q^{59} + ( - 6 \zeta_{6} + 6) q^{60} - 10 \zeta_{6} q^{61} - 8 q^{64} + 12 \zeta_{6} q^{65} + (5 \zeta_{6} - 5) q^{67} + 12 \zeta_{6} q^{68} - 3 q^{69} + 9 q^{71} + ( - 2 \zeta_{6} + 2) q^{73} + 4 \zeta_{6} q^{75} + 4 q^{76} + 10 \zeta_{6} q^{79} + ( - 12 \zeta_{6} + 12) q^{80} + (\zeta_{6} - 1) q^{81} - 12 q^{83} - 18 q^{85} + (6 \zeta_{6} - 6) q^{87} - 3 \zeta_{6} q^{89} - 6 q^{92} - 5 \zeta_{6} q^{93} + (6 \zeta_{6} - 6) q^{95} + q^{97} + 2 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{3} + 2 q^{4} + 3 q^{5} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{3} + 2 q^{4} + 3 q^{5} + 2 q^{9} + q^{11} - 2 q^{12} + 8 q^{13} + 6 q^{15} - 4 q^{16} - 6 q^{17} + 2 q^{19} + 12 q^{20} - 3 q^{23} - 4 q^{25} + 10 q^{27} - 12 q^{29} + 5 q^{31} - q^{33} + 8 q^{36} - 11 q^{37} + 4 q^{39} - 12 q^{41} + 16 q^{43} - 2 q^{44} - 6 q^{45} - 8 q^{48} + 6 q^{51} + 8 q^{52} + 6 q^{53} + 6 q^{55} + 4 q^{57} - 9 q^{59} + 6 q^{60} - 10 q^{61} - 16 q^{64} + 12 q^{65} - 5 q^{67} + 12 q^{68} - 6 q^{69} + 18 q^{71} + 2 q^{73} + 4 q^{75} + 8 q^{76} + 10 q^{79} + 12 q^{80} - q^{81} - 24 q^{83} - 36 q^{85} - 6 q^{87} - 3 q^{89} - 12 q^{92} - 5 q^{93} - 6 q^{95} + 2 q^{97} + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(199\) \(442\)
\(\chi(n)\) \(-\zeta_{6}\) \(1\)

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
0.500000 + 0.866025i
0.500000 0.866025i
0 0.500000 0.866025i 1.00000 1.73205i 1.50000 + 2.59808i 0 0 0 1.00000 + 1.73205i 0
177.1 0 0.500000 + 0.866025i 1.00000 + 1.73205i 1.50000 2.59808i 0 0 0 1.00000 1.73205i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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

Hecke kernels

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

\( T_{2} \) Copy content Toggle raw display
\( T_{3}^{2} - T_{3} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$5$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$13$ \( (T - 4)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$19$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$23$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$29$ \( (T + 6)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 5T + 25 \) Copy content Toggle raw display
$37$ \( T^{2} + 11T + 121 \) Copy content Toggle raw display
$41$ \( (T + 6)^{2} \) Copy content Toggle raw display
$43$ \( (T - 8)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$59$ \( T^{2} + 9T + 81 \) Copy content Toggle raw display
$61$ \( T^{2} + 10T + 100 \) Copy content Toggle raw display
$67$ \( T^{2} + 5T + 25 \) Copy content Toggle raw display
$71$ \( (T - 9)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$79$ \( T^{2} - 10T + 100 \) Copy content Toggle raw display
$83$ \( (T + 12)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$97$ \( (T - 1)^{2} \) Copy content Toggle raw display
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