Properties

Label 539.2.a.c
Level $539$
Weight $2$
Character orbit 539.a
Self dual yes
Analytic conductor $4.304$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 539 = 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 539.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(4.30393666895\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 77)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 3 q^{3} - 2 q^{4} + q^{5} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 3 q^{3} - 2 q^{4} + q^{5} + 6 q^{9} - q^{11} - 6 q^{12} + 4 q^{13} + 3 q^{15} + 4 q^{16} - 2 q^{17} + 6 q^{19} - 2 q^{20} - 5 q^{23} - 4 q^{25} + 9 q^{27} + 10 q^{29} - q^{31} - 3 q^{33} - 12 q^{36} - 5 q^{37} + 12 q^{39} + 2 q^{41} - 8 q^{43} + 2 q^{44} + 6 q^{45} - 8 q^{47} + 12 q^{48} - 6 q^{51} - 8 q^{52} - 6 q^{53} - q^{55} + 18 q^{57} - 3 q^{59} - 6 q^{60} + 2 q^{61} - 8 q^{64} + 4 q^{65} - 3 q^{67} + 4 q^{68} - 15 q^{69} + q^{71} - 10 q^{73} - 12 q^{75} - 12 q^{76} + 6 q^{79} + 4 q^{80} + 9 q^{81} - 12 q^{83} - 2 q^{85} + 30 q^{87} + 15 q^{89} + 10 q^{92} - 3 q^{93} + 6 q^{95} + 5 q^{97} - 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
0 3.00000 −2.00000 1.00000 0 0 0 6.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(7\) \(-1\)
\(11\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 539.2.a.c 1
3.b odd 2 1 4851.2.a.j 1
4.b odd 2 1 8624.2.a.a 1
7.b odd 2 1 77.2.a.a 1
7.c even 3 2 539.2.e.c 2
7.d odd 6 2 539.2.e.f 2
11.b odd 2 1 5929.2.a.f 1
21.c even 2 1 693.2.a.c 1
28.d even 2 1 1232.2.a.l 1
35.c odd 2 1 1925.2.a.h 1
35.f even 4 2 1925.2.b.e 2
56.e even 2 1 4928.2.a.a 1
56.h odd 2 1 4928.2.a.bj 1
77.b even 2 1 847.2.a.b 1
77.j odd 10 4 847.2.f.i 4
77.l even 10 4 847.2.f.h 4
231.h odd 2 1 7623.2.a.j 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.2.a.a 1 7.b odd 2 1
539.2.a.c 1 1.a even 1 1 trivial
539.2.e.c 2 7.c even 3 2
539.2.e.f 2 7.d odd 6 2
693.2.a.c 1 21.c even 2 1
847.2.a.b 1 77.b even 2 1
847.2.f.h 4 77.l even 10 4
847.2.f.i 4 77.j odd 10 4
1232.2.a.l 1 28.d even 2 1
1925.2.a.h 1 35.c odd 2 1
1925.2.b.e 2 35.f even 4 2
4851.2.a.j 1 3.b odd 2 1
4928.2.a.a 1 56.e even 2 1
4928.2.a.bj 1 56.h odd 2 1
5929.2.a.f 1 11.b odd 2 1
7623.2.a.j 1 231.h odd 2 1
8624.2.a.a 1 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(539))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T - 3 \) Copy content Toggle raw display
$5$ \( T - 1 \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T + 1 \) Copy content Toggle raw display
$13$ \( T - 4 \) Copy content Toggle raw display
$17$ \( T + 2 \) Copy content Toggle raw display
$19$ \( T - 6 \) Copy content Toggle raw display
$23$ \( T + 5 \) Copy content Toggle raw display
$29$ \( T - 10 \) Copy content Toggle raw display
$31$ \( T + 1 \) Copy content Toggle raw display
$37$ \( T + 5 \) Copy content Toggle raw display
$41$ \( T - 2 \) Copy content Toggle raw display
$43$ \( T + 8 \) Copy content Toggle raw display
$47$ \( T + 8 \) Copy content Toggle raw display
$53$ \( T + 6 \) Copy content Toggle raw display
$59$ \( T + 3 \) Copy content Toggle raw display
$61$ \( T - 2 \) Copy content Toggle raw display
$67$ \( T + 3 \) Copy content Toggle raw display
$71$ \( T - 1 \) Copy content Toggle raw display
$73$ \( T + 10 \) Copy content Toggle raw display
$79$ \( T - 6 \) Copy content Toggle raw display
$83$ \( T + 12 \) Copy content Toggle raw display
$89$ \( T - 15 \) Copy content Toggle raw display
$97$ \( T - 5 \) Copy content Toggle raw display
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