Properties

Label 2013.2.a
Level 2013
Weight 2
Character orbit a
Rep. character \(\chi_{2013}(1,\cdot)\)
Character field \(\Q\)
Dimension 99
Newform subspaces 8
Sturm bound 496
Trace bound 7

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Defining parameters

Level: \( N \) = \( 2013 = 3 \cdot 11 \cdot 61 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 2013.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 8 \)
Sturm bound: \(496\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(2013))\).

Total New Old
Modular forms 252 99 153
Cusp forms 245 99 146
Eisenstein series 7 0 7

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

\(3\)\(11\)\(61\)FrickeDim.
\(+\)\(+\)\(+\)\(+\)\(12\)
\(+\)\(+\)\(-\)\(-\)\(14\)
\(+\)\(-\)\(+\)\(-\)\(13\)
\(+\)\(-\)\(-\)\(+\)\(11\)
\(-\)\(+\)\(+\)\(-\)\(13\)
\(-\)\(+\)\(-\)\(+\)\(11\)
\(-\)\(-\)\(+\)\(+\)\(12\)
\(-\)\(-\)\(-\)\(-\)\(13\)
Plus space\(+\)\(46\)
Minus space\(-\)\(53\)

Trace form

\( 99q - 3q^{2} - q^{3} + 93q^{4} - 6q^{5} - 3q^{6} - 8q^{7} + 9q^{8} + 99q^{9} + O(q^{10}) \) \( 99q - 3q^{2} - q^{3} + 93q^{4} - 6q^{5} - 3q^{6} - 8q^{7} + 9q^{8} + 99q^{9} + 6q^{10} - q^{11} - 7q^{12} - 14q^{13} + 16q^{14} - 6q^{15} + 85q^{16} - 10q^{17} - 3q^{18} - 12q^{19} - 2q^{20} - 8q^{21} - 3q^{22} - 15q^{24} + 85q^{25} - 26q^{26} - q^{27} + 10q^{29} + 14q^{30} + 8q^{31} - 15q^{32} + 3q^{33} + 2q^{34} + 32q^{35} + 93q^{36} - 14q^{37} - 20q^{38} - 14q^{39} + 30q^{40} - 10q^{41} + 24q^{42} - 28q^{43} + 9q^{44} - 6q^{45} - 16q^{46} + 8q^{47} + q^{48} + 67q^{49} - 69q^{50} - 18q^{51} - 58q^{52} - 22q^{53} - 3q^{54} + 10q^{55} + 32q^{56} - 20q^{57} + 6q^{58} - 4q^{59} - 2q^{60} - q^{61} - 24q^{62} - 8q^{63} + 29q^{64} - 68q^{65} - 3q^{66} + 4q^{67} - 38q^{68} + 8q^{69} + 8q^{71} + 9q^{72} - 10q^{73} - 50q^{74} - 31q^{75} + 36q^{76} - 8q^{77} - 2q^{78} - 40q^{79} - 18q^{80} + 99q^{81} - 22q^{82} + 4q^{83} - 24q^{84} - 36q^{85} + 52q^{86} - 6q^{87} - 15q^{88} - 26q^{89} + 6q^{90} + 16q^{91} - 72q^{92} - 24q^{93} - 16q^{94} + 64q^{95} - 23q^{96} - 66q^{97} - 59q^{98} - q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(2013))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 3 11 61
2013.2.a.a \(11\) \(16.074\) \(\mathbb{Q}[x]/(x^{11} - \cdots)\) None \(-4\) \(11\) \(-13\) \(-5\) \(-\) \(+\) \(-\) \(q-\beta _{1}q^{2}+q^{3}+(1+\beta _{5}+\beta _{6})q^{4}+(-1+\cdots)q^{5}+\cdots\)
2013.2.a.b \(11\) \(16.074\) \(\mathbb{Q}[x]/(x^{11} - \cdots)\) None \(-2\) \(-11\) \(-1\) \(-11\) \(+\) \(-\) \(-\) \(q-\beta _{1}q^{2}-q^{3}+(1+\beta _{2})q^{4}+\beta _{8}q^{5}+\cdots\)
2013.2.a.c \(12\) \(16.074\) \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(-7\) \(12\) \(-7\) \(-15\) \(-\) \(-\) \(+\) \(q+(-1+\beta _{1})q^{2}+q^{3}+(2-\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
2013.2.a.d \(12\) \(16.074\) \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(1\) \(-12\) \(-3\) \(-9\) \(+\) \(+\) \(+\) \(q+\beta _{1}q^{2}-q^{3}+(1+\beta _{2})q^{4}-\beta _{3}q^{5}+\cdots\)
2013.2.a.e \(13\) \(16.074\) \(\mathbb{Q}[x]/(x^{13} - \cdots)\) None \(2\) \(-13\) \(3\) \(11\) \(+\) \(-\) \(+\) \(q+\beta _{1}q^{2}-q^{3}+(1+\beta _{2})q^{4}-\beta _{8}q^{5}+\cdots\)
2013.2.a.f \(13\) \(16.074\) \(\mathbb{Q}[x]/(x^{13} - \cdots)\) None \(4\) \(13\) \(7\) \(5\) \(-\) \(+\) \(+\) \(q+\beta _{1}q^{2}+q^{3}+(1+\beta _{1}+\beta _{2})q^{4}+(1+\cdots)q^{5}+\cdots\)
2013.2.a.g \(13\) \(16.074\) \(\mathbb{Q}[x]/(x^{13} - \cdots)\) None \(4\) \(13\) \(7\) \(7\) \(-\) \(-\) \(-\) \(q+\beta _{1}q^{2}+q^{3}+(1+\beta _{2})q^{4}+(1+\beta _{5}+\cdots)q^{5}+\cdots\)
2013.2.a.h \(14\) \(16.074\) \(\mathbb{Q}[x]/(x^{14} - \cdots)\) None \(-1\) \(-14\) \(1\) \(9\) \(+\) \(+\) \(-\) \(q-\beta _{1}q^{2}-q^{3}+(1+\beta _{2})q^{4}+\beta _{10}q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(2013))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(2013)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(33))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(61))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(183))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(671))\)\(^{\oplus 2}\)