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Properties

Label 503.2.a
Level $503$
Weight $2$
Character orbit 503.a
Rep. character $\chi_{503}(1,\cdot)$
Character field $\Q$
Dimension $42$
Newform subspaces $6$
Sturm bound $84$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 503 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 503.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 6 \)
Sturm bound: \(84\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(2\), \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(503))\).

Total New Old
Modular forms 43 43 0
Cusp forms 42 42 0
Eisenstein series 1 1 0

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(503\)Dim
\(+\)\(11\)
\(-\)\(31\)

Trace form

\( 42 q + q^{2} + 2 q^{3} + 41 q^{4} - 2 q^{6} + 4 q^{7} + 3 q^{8} + 42 q^{9} + O(q^{10}) \) \( 42 q + q^{2} + 2 q^{3} + 41 q^{4} - 2 q^{6} + 4 q^{7} + 3 q^{8} + 42 q^{9} + 16 q^{12} + 8 q^{13} - 4 q^{14} + 4 q^{15} + 43 q^{16} + 10 q^{17} + 5 q^{18} - 2 q^{19} - 14 q^{20} - 4 q^{21} + 18 q^{22} + 14 q^{23} - 10 q^{24} + 60 q^{25} + 6 q^{26} + 26 q^{27} - 10 q^{28} - 4 q^{29} - 20 q^{30} - 3 q^{32} + 6 q^{33} - 30 q^{34} - 16 q^{35} + 43 q^{36} + 24 q^{37} - 30 q^{38} + 2 q^{39} - 26 q^{40} - 38 q^{42} + 14 q^{43} - 20 q^{44} - 6 q^{45} - 12 q^{46} - 8 q^{47} + 14 q^{48} + 40 q^{49} - 13 q^{50} - 2 q^{51} + 26 q^{52} + 14 q^{53} - 48 q^{54} - 14 q^{55} - 6 q^{56} + 6 q^{57} + 44 q^{58} - 2 q^{59} - 56 q^{60} + 4 q^{61} - 12 q^{62} + 12 q^{63} + 61 q^{64} - 12 q^{65} - 10 q^{66} + 18 q^{67} + 30 q^{68} - 42 q^{69} + 16 q^{70} - 26 q^{71} - 35 q^{72} + 36 q^{73} - 54 q^{74} + 34 q^{75} - 14 q^{76} - 16 q^{77} - 40 q^{78} + 20 q^{79} - 68 q^{80} + 58 q^{81} - 28 q^{82} - 6 q^{83} - 94 q^{84} + 16 q^{85} + 2 q^{86} - 10 q^{87} + 72 q^{88} - 6 q^{89} - 132 q^{90} - 2 q^{91} + 36 q^{92} - 8 q^{94} + 50 q^{95} - 66 q^{96} + 76 q^{97} - 17 q^{98} - 10 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(503))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 503
503.2.a.a 503.a 1.a $1$ $4.016$ \(\Q\) None \(-1\) \(1\) \(-4\) \(-3\) $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}-q^{4}-4q^{5}-q^{6}-3q^{7}+\cdots\)
503.2.a.b 503.a 1.a $1$ $4.016$ \(\Q\) None \(1\) \(1\) \(-2\) \(-3\) $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}-q^{4}-2q^{5}+q^{6}-3q^{7}+\cdots\)
503.2.a.c 503.a 1.a $1$ $4.016$ \(\Q\) None \(1\) \(3\) \(-2\) \(3\) $-$ $\mathrm{SU}(2)$ \(q+q^{2}+3q^{3}-q^{4}-2q^{5}+3q^{6}+3q^{7}+\cdots\)
503.2.a.d 503.a 1.a $3$ $4.016$ 3.3.257.1 None \(0\) \(1\) \(0\) \(1\) $-$ $\mathrm{SU}(2)$ \(q+\beta _{2}q^{2}+\beta _{1}q^{3}+(1+\beta _{1}-\beta _{2})q^{4}+\cdots\)
503.2.a.e 503.a 1.a $10$ $4.016$ \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None \(-4\) \(-8\) \(-1\) \(-5\) $+$ $\mathrm{SU}(2)$ \(q-\beta _{9}q^{2}+(-1+\beta _{1})q^{3}+\beta _{5}q^{4}-\beta _{2}q^{5}+\cdots\)
503.2.a.f 503.a 1.a $26$ $4.016$ None \(4\) \(4\) \(9\) \(11\) $-$ $\mathrm{SU}(2)$