Properties

Label 539.3.c.a
Level $539$
Weight $3$
Character orbit 539.c
Self dual yes
Analytic conductor $14.687$
Analytic rank $0$
Dimension $1$
CM discriminant -11
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [539,3,Mod(197,539)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(539, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("539.197");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 539 = 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 539.c (of order \(2\), degree \(1\), 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: yes
Analytic conductor: \(14.6866862490\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 11)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q + 5 q^{3} + 4 q^{4} + q^{5} + 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 5 q^{3} + 4 q^{4} + q^{5} + 16 q^{9} - 11 q^{11} + 20 q^{12} + 5 q^{15} + 16 q^{16} + 4 q^{20} + 35 q^{23} - 24 q^{25} + 35 q^{27} + 37 q^{31} - 55 q^{33} + 64 q^{36} - 25 q^{37} - 44 q^{44} + 16 q^{45} - 50 q^{47} + 80 q^{48} - 70 q^{53} - 11 q^{55} - 107 q^{59} + 20 q^{60} + 64 q^{64} + 35 q^{67} + 175 q^{69} - 133 q^{71} - 120 q^{75} + 16 q^{80} + 31 q^{81} + 97 q^{89} + 140 q^{92} + 185 q^{93} - 95 q^{97} - 176 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(199\) \(442\)
\(\chi(n)\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
197.1
0
0 5.00000 4.00000 1.00000 0 0 0 16.0000 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
11.b odd 2 1 CM by \(\Q(\sqrt{-11}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 539.3.c.a 1
7.b odd 2 1 11.3.b.a 1
11.b odd 2 1 CM 539.3.c.a 1
21.c even 2 1 99.3.c.a 1
28.d even 2 1 176.3.h.a 1
35.c odd 2 1 275.3.c.a 1
35.f even 4 2 275.3.d.a 2
56.e even 2 1 704.3.h.a 1
56.h odd 2 1 704.3.h.b 1
77.b even 2 1 11.3.b.a 1
77.j odd 10 4 121.3.d.b 4
77.l even 10 4 121.3.d.b 4
84.h odd 2 1 1584.3.j.a 1
231.h odd 2 1 99.3.c.a 1
308.g odd 2 1 176.3.h.a 1
385.h even 2 1 275.3.c.a 1
385.l odd 4 2 275.3.d.a 2
616.g odd 2 1 704.3.h.a 1
616.o even 2 1 704.3.h.b 1
924.n even 2 1 1584.3.j.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
11.3.b.a 1 7.b odd 2 1
11.3.b.a 1 77.b even 2 1
99.3.c.a 1 21.c even 2 1
99.3.c.a 1 231.h odd 2 1
121.3.d.b 4 77.j odd 10 4
121.3.d.b 4 77.l even 10 4
176.3.h.a 1 28.d even 2 1
176.3.h.a 1 308.g odd 2 1
275.3.c.a 1 35.c odd 2 1
275.3.c.a 1 385.h even 2 1
275.3.d.a 2 35.f even 4 2
275.3.d.a 2 385.l odd 4 2
539.3.c.a 1 1.a even 1 1 trivial
539.3.c.a 1 11.b odd 2 1 CM
704.3.h.a 1 56.e even 2 1
704.3.h.a 1 616.g odd 2 1
704.3.h.b 1 56.h odd 2 1
704.3.h.b 1 616.o even 2 1
1584.3.j.a 1 84.h odd 2 1
1584.3.j.a 1 924.n even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(539, [\chi])\):

\( T_{2} \) Copy content Toggle raw display
\( T_{3} - 5 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T - 5 \) Copy content Toggle raw display
$5$ \( T - 1 \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T + 11 \) Copy content Toggle raw display
$13$ \( T \) Copy content Toggle raw display
$17$ \( T \) Copy content Toggle raw display
$19$ \( T \) Copy content Toggle raw display
$23$ \( T - 35 \) Copy content Toggle raw display
$29$ \( T \) Copy content Toggle raw display
$31$ \( T - 37 \) Copy content Toggle raw display
$37$ \( T + 25 \) Copy content Toggle raw display
$41$ \( T \) Copy content Toggle raw display
$43$ \( T \) Copy content Toggle raw display
$47$ \( T + 50 \) Copy content Toggle raw display
$53$ \( T + 70 \) Copy content Toggle raw display
$59$ \( T + 107 \) Copy content Toggle raw display
$61$ \( T \) Copy content Toggle raw display
$67$ \( T - 35 \) Copy content Toggle raw display
$71$ \( T + 133 \) Copy content Toggle raw display
$73$ \( T \) Copy content Toggle raw display
$79$ \( T \) Copy content Toggle raw display
$83$ \( T \) Copy content Toggle raw display
$89$ \( T - 97 \) Copy content Toggle raw display
$97$ \( T + 95 \) Copy content Toggle raw display
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