Properties

Label 503.2.a.c
Level $503$
Weight $2$
Character orbit 503.a
Self dual yes
Analytic conductor $4.016$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more about

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: \(0\)
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

\(f(q)\) \(=\) \( q + q^{2} + 3q^{3} - q^{4} - 2q^{5} + 3q^{6} + 3q^{7} - 3q^{8} + 6q^{9} + O(q^{10}) \) \( q + q^{2} + 3q^{3} - q^{4} - 2q^{5} + 3q^{6} + 3q^{7} - 3q^{8} + 6q^{9} - 2q^{10} + 3q^{11} - 3q^{12} + 5q^{13} + 3q^{14} - 6q^{15} - q^{16} - 8q^{17} + 6q^{18} + 4q^{19} + 2q^{20} + 9q^{21} + 3q^{22} - 5q^{23} - 9q^{24} - q^{25} + 5q^{26} + 9q^{27} - 3q^{28} - 6q^{30} - 2q^{31} + 5q^{32} + 9q^{33} - 8q^{34} - 6q^{35} - 6q^{36} + 4q^{37} + 4q^{38} + 15q^{39} + 6q^{40} - 10q^{41} + 9q^{42} - q^{43} - 3q^{44} - 12q^{45} - 5q^{46} - 3q^{47} - 3q^{48} + 2q^{49} - q^{50} - 24q^{51} - 5q^{52} - 12q^{53} + 9q^{54} - 6q^{55} - 9q^{56} + 12q^{57} + 12q^{59} + 6q^{60} - 11q^{61} - 2q^{62} + 18q^{63} + 7q^{64} - 10q^{65} + 9q^{66} + 7q^{67} + 8q^{68} - 15q^{69} - 6q^{70} - 8q^{71} - 18q^{72} - 6q^{73} + 4q^{74} - 3q^{75} - 4q^{76} + 9q^{77} + 15q^{78} - 4q^{79} + 2q^{80} + 9q^{81} - 10q^{82} + 15q^{83} - 9q^{84} + 16q^{85} - q^{86} - 9q^{88} - 12q^{90} + 15q^{91} + 5q^{92} - 6q^{93} - 3q^{94} - 8q^{95} + 15q^{96} - 6q^{97} + 2q^{98} + 18q^{99} + O(q^{100}) \)

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 3.00000 −1.00000 −2.00000 3.00000 3.00000 −3.00000 6.00000 −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.c 1
3.b odd 2 1 4527.2.a.d 1
4.b odd 2 1 8048.2.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
503.2.a.c 1 1.a even 1 1 trivial
4527.2.a.d 1 3.b odd 2 1
8048.2.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(503))\):

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

Hecke characteristic polynomials

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