Properties

Label 1339.2.a
Level 1339
Weight 2
Character orbit a
Rep. character \(\chi_{1339}(1,\cdot)\)
Character field \(\Q\)
Dimension 103
Newforms 7
Sturm bound 242
Trace bound 3

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Defining parameters

Level: \( N \) = \( 1339 = 13 \cdot 103 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 1339.a (trivial)
Character field: \(\Q\)
Newforms: \( 7 \)
Sturm bound: \(242\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(2\), \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1339))\).

Total New Old
Modular forms 122 103 19
Cusp forms 119 103 16
Eisenstein series 3 0 3

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

\(13\)\(103\)FrickeDim.
\(+\)\(+\)\(+\)\(22\)
\(+\)\(-\)\(-\)\(30\)
\(-\)\(+\)\(-\)\(29\)
\(-\)\(-\)\(+\)\(22\)
Plus space\(+\)\(44\)
Minus space\(-\)\(59\)

Trace form

\(103q \) \(\mathstrut -\mathstrut 3q^{2} \) \(\mathstrut +\mathstrut 97q^{4} \) \(\mathstrut +\mathstrut 2q^{5} \) \(\mathstrut -\mathstrut 3q^{8} \) \(\mathstrut +\mathstrut 103q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(103q \) \(\mathstrut -\mathstrut 3q^{2} \) \(\mathstrut +\mathstrut 97q^{4} \) \(\mathstrut +\mathstrut 2q^{5} \) \(\mathstrut -\mathstrut 3q^{8} \) \(\mathstrut +\mathstrut 103q^{9} \) \(\mathstrut +\mathstrut 10q^{10} \) \(\mathstrut -\mathstrut 4q^{11} \) \(\mathstrut +\mathstrut 16q^{12} \) \(\mathstrut -\mathstrut q^{13} \) \(\mathstrut +\mathstrut 16q^{14} \) \(\mathstrut +\mathstrut 4q^{15} \) \(\mathstrut +\mathstrut 81q^{16} \) \(\mathstrut -\mathstrut 14q^{17} \) \(\mathstrut +\mathstrut 9q^{18} \) \(\mathstrut -\mathstrut 16q^{19} \) \(\mathstrut +\mathstrut 26q^{20} \) \(\mathstrut +\mathstrut 8q^{21} \) \(\mathstrut -\mathstrut 4q^{22} \) \(\mathstrut -\mathstrut 16q^{23} \) \(\mathstrut +\mathstrut 24q^{24} \) \(\mathstrut +\mathstrut 125q^{25} \) \(\mathstrut -\mathstrut 3q^{26} \) \(\mathstrut -\mathstrut 12q^{27} \) \(\mathstrut -\mathstrut 20q^{28} \) \(\mathstrut +\mathstrut 6q^{29} \) \(\mathstrut -\mathstrut 12q^{30} \) \(\mathstrut +\mathstrut 16q^{31} \) \(\mathstrut -\mathstrut 19q^{32} \) \(\mathstrut -\mathstrut 12q^{33} \) \(\mathstrut -\mathstrut 54q^{34} \) \(\mathstrut -\mathstrut 16q^{35} \) \(\mathstrut +\mathstrut 73q^{36} \) \(\mathstrut +\mathstrut 6q^{37} \) \(\mathstrut +\mathstrut 36q^{38} \) \(\mathstrut +\mathstrut 4q^{39} \) \(\mathstrut +\mathstrut 10q^{40} \) \(\mathstrut -\mathstrut 18q^{41} \) \(\mathstrut -\mathstrut 48q^{42} \) \(\mathstrut -\mathstrut 20q^{43} \) \(\mathstrut -\mathstrut 36q^{44} \) \(\mathstrut +\mathstrut 14q^{45} \) \(\mathstrut -\mathstrut 24q^{46} \) \(\mathstrut +\mathstrut 24q^{47} \) \(\mathstrut +\mathstrut 20q^{48} \) \(\mathstrut +\mathstrut 111q^{49} \) \(\mathstrut -\mathstrut 53q^{50} \) \(\mathstrut -\mathstrut 24q^{51} \) \(\mathstrut +\mathstrut q^{52} \) \(\mathstrut +\mathstrut 2q^{53} \) \(\mathstrut -\mathstrut 52q^{54} \) \(\mathstrut -\mathstrut 4q^{55} \) \(\mathstrut -\mathstrut 12q^{56} \) \(\mathstrut -\mathstrut 36q^{57} \) \(\mathstrut -\mathstrut 14q^{58} \) \(\mathstrut -\mathstrut 16q^{59} \) \(\mathstrut -\mathstrut 60q^{60} \) \(\mathstrut -\mathstrut 14q^{61} \) \(\mathstrut -\mathstrut 88q^{62} \) \(\mathstrut -\mathstrut 12q^{63} \) \(\mathstrut +\mathstrut 5q^{64} \) \(\mathstrut +\mathstrut 2q^{65} \) \(\mathstrut -\mathstrut 84q^{66} \) \(\mathstrut -\mathstrut 16q^{67} \) \(\mathstrut -\mathstrut 18q^{68} \) \(\mathstrut +\mathstrut 32q^{69} \) \(\mathstrut +\mathstrut 48q^{70} \) \(\mathstrut +\mathstrut 16q^{71} \) \(\mathstrut -\mathstrut 39q^{72} \) \(\mathstrut +\mathstrut 10q^{73} \) \(\mathstrut +\mathstrut 6q^{74} \) \(\mathstrut +\mathstrut 4q^{75} \) \(\mathstrut -\mathstrut 64q^{76} \) \(\mathstrut -\mathstrut 40q^{77} \) \(\mathstrut +\mathstrut 12q^{78} \) \(\mathstrut +\mathstrut 4q^{79} \) \(\mathstrut +\mathstrut 86q^{80} \) \(\mathstrut +\mathstrut 111q^{81} \) \(\mathstrut +\mathstrut 22q^{82} \) \(\mathstrut -\mathstrut 4q^{83} \) \(\mathstrut -\mathstrut 100q^{84} \) \(\mathstrut -\mathstrut 24q^{86} \) \(\mathstrut +\mathstrut 20q^{87} \) \(\mathstrut -\mathstrut 16q^{88} \) \(\mathstrut +\mathstrut 78q^{89} \) \(\mathstrut +\mathstrut 22q^{90} \) \(\mathstrut -\mathstrut 8q^{91} \) \(\mathstrut -\mathstrut 76q^{92} \) \(\mathstrut +\mathstrut 76q^{93} \) \(\mathstrut -\mathstrut 36q^{94} \) \(\mathstrut +\mathstrut 16q^{95} \) \(\mathstrut -\mathstrut 4q^{96} \) \(\mathstrut -\mathstrut 6q^{97} \) \(\mathstrut +\mathstrut 17q^{98} \) \(\mathstrut -\mathstrut 28q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1339))\) into irreducible Hecke orbits

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 13 103
1339.2.a.a \(1\) \(10.692\) \(\Q\) None \(1\) \(-1\) \(1\) \(4\) \(+\) \(+\) \(q+q^{2}-q^{3}-q^{4}+q^{5}-q^{6}+4q^{7}+\cdots\)
1339.2.a.b \(1\) \(10.692\) \(\Q\) None \(1\) \(0\) \(0\) \(-4\) \(-\) \(+\) \(q+q^{2}-q^{4}-4q^{7}-3q^{8}-3q^{9}+6q^{11}+\cdots\)
1339.2.a.c \(3\) \(10.692\) 3.3.148.1 None \(-1\) \(-1\) \(-7\) \(2\) \(-\) \(-\) \(q-\beta _{1}q^{2}+(-\beta _{1}-\beta _{2})q^{3}+(\beta _{1}+\beta _{2})q^{4}+\cdots\)
1339.2.a.d \(19\) \(10.692\) \(\mathbb{Q}[x]/(x^{19} - \cdots)\) None \(-9\) \(-2\) \(-18\) \(-8\) \(-\) \(-\) \(q-\beta _{1}q^{2}-\beta _{14}q^{3}+(\beta _{1}+\beta _{2})q^{4}+(-1+\cdots)q^{5}+\cdots\)
1339.2.a.e \(21\) \(10.692\) None \(-1\) \(-6\) \(-18\) \(-2\) \(+\) \(+\)
1339.2.a.f \(28\) \(10.692\) None \(6\) \(5\) \(27\) \(6\) \(-\) \(+\)
1339.2.a.g \(30\) \(10.692\) None \(0\) \(5\) \(17\) \(2\) \(+\) \(-\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1339))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1339)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(103))\)\(^{\oplus 2}\)