Properties

Label 503.2.a.b
Level $503$
Weight $2$
Character orbit 503.a
Self dual yes
Analytic conductor $4.016$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 503 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 503.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(4.01647522167\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + q^{2} + q^{3} - q^{4} - 2q^{5} + q^{6} - 3q^{7} - 3q^{8} - 2q^{9} + O(q^{10}) \) \( q + q^{2} + q^{3} - q^{4} - 2q^{5} + q^{6} - 3q^{7} - 3q^{8} - 2q^{9} - 2q^{10} + q^{11} - q^{12} + q^{13} - 3q^{14} - 2q^{15} - q^{16} - 2q^{18} - 4q^{19} + 2q^{20} - 3q^{21} + q^{22} - 3q^{23} - 3q^{24} - q^{25} + q^{26} - 5q^{27} + 3q^{28} - 2q^{30} + 10q^{31} + 5q^{32} + q^{33} + 6q^{35} + 2q^{36} - 4q^{37} - 4q^{38} + q^{39} + 6q^{40} - 2q^{41} - 3q^{42} + 5q^{43} - q^{44} + 4q^{45} - 3q^{46} - 5q^{47} - q^{48} + 2q^{49} - q^{50} - q^{52} + 12q^{53} - 5q^{54} - 2q^{55} + 9q^{56} - 4q^{57} - 4q^{59} + 2q^{60} - 7q^{61} + 10q^{62} + 6q^{63} + 7q^{64} - 2q^{65} + q^{66} - 11q^{67} - 3q^{69} + 6q^{70} + 6q^{72} - 6q^{73} - 4q^{74} - q^{75} + 4q^{76} - 3q^{77} + q^{78} + 4q^{79} + 2q^{80} + q^{81} - 2q^{82} - 3q^{83} + 3q^{84} + 5q^{86} - 3q^{88} + 4q^{90} - 3q^{91} + 3q^{92} + 10q^{93} - 5q^{94} + 8q^{95} + 5q^{96} + 10q^{97} + 2q^{98} - 2q^{99} + O(q^{100}) \)

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

Atkin-Lehner signs

\( p \) Sign
\(503\) \(1\)

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 503.2.a.b 1
3.b odd 2 1 4527.2.a.c 1
4.b odd 2 1 8048.2.a.e 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
503.2.a.b 1 1.a even 1 1 trivial
4527.2.a.c 1 3.b odd 2 1
8048.2.a.e 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(503))\):

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

Hecke characteristic polynomials

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