Properties

Label 5239.2.a.a
Level 5239
Weight 2
Character orbit 5239.a
Self dual yes
Analytic conductor 41.834
Analytic rank 0
Dimension 1
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 5239 = 13^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5239.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(41.8336256189\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 403)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - q^{2} - q^{4} + 3q^{5} - 2q^{7} + 3q^{8} - 3q^{9} + O(q^{10}) \) \( q - q^{2} - q^{4} + 3q^{5} - 2q^{7} + 3q^{8} - 3q^{9} - 3q^{10} + 2q^{14} - q^{16} + 5q^{17} + 3q^{18} - 4q^{19} - 3q^{20} + 4q^{23} + 4q^{25} + 2q^{28} + 3q^{29} + q^{31} - 5q^{32} - 5q^{34} - 6q^{35} + 3q^{36} + 5q^{37} + 4q^{38} + 9q^{40} - 7q^{41} - 2q^{43} - 9q^{45} - 4q^{46} + 6q^{47} - 3q^{49} - 4q^{50} - 9q^{53} - 6q^{56} - 3q^{58} + 6q^{59} - q^{61} - q^{62} + 6q^{63} + 7q^{64} - 10q^{67} - 5q^{68} + 6q^{70} + 4q^{71} - 9q^{72} + 3q^{73} - 5q^{74} + 4q^{76} + 2q^{79} - 3q^{80} + 9q^{81} + 7q^{82} + 2q^{83} + 15q^{85} + 2q^{86} + 6q^{89} + 9q^{90} - 4q^{92} - 6q^{94} - 12q^{95} + 18q^{97} + 3q^{98} + O(q^{100}) \)

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
−1.00000 0 −1.00000 3.00000 0 −2.00000 3.00000 −3.00000 −3.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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 5239.2.a.a 1
13.b even 2 1 5239.2.a.d 1
13.e even 6 2 403.2.f.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
403.2.f.a 2 13.e even 6 2
5239.2.a.a 1 1.a even 1 1 trivial
5239.2.a.d 1 13.b even 2 1

Atkin-Lehner signs

\( p \) Sign
\(13\) \(1\)
\(31\) \(-1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5239))\):

\( T_{2} + 1 \)
\( T_{5} - 3 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T + 2 T^{2} \)
$3$ \( 1 + 3 T^{2} \)
$5$ \( 1 - 3 T + 5 T^{2} \)
$7$ \( 1 + 2 T + 7 T^{2} \)
$11$ \( 1 + 11 T^{2} \)
$13$ 1
$17$ \( 1 - 5 T + 17 T^{2} \)
$19$ \( 1 + 4 T + 19 T^{2} \)
$23$ \( 1 - 4 T + 23 T^{2} \)
$29$ \( 1 - 3 T + 29 T^{2} \)
$31$ \( 1 - T \)
$37$ \( 1 - 5 T + 37 T^{2} \)
$41$ \( 1 + 7 T + 41 T^{2} \)
$43$ \( 1 + 2 T + 43 T^{2} \)
$47$ \( 1 - 6 T + 47 T^{2} \)
$53$ \( 1 + 9 T + 53 T^{2} \)
$59$ \( 1 - 6 T + 59 T^{2} \)
$61$ \( 1 + T + 61 T^{2} \)
$67$ \( 1 + 10 T + 67 T^{2} \)
$71$ \( 1 - 4 T + 71 T^{2} \)
$73$ \( 1 - 3 T + 73 T^{2} \)
$79$ \( 1 - 2 T + 79 T^{2} \)
$83$ \( 1 - 2 T + 83 T^{2} \)
$89$ \( 1 - 6 T + 89 T^{2} \)
$97$ \( 1 - 18 T + 97 T^{2} \)
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