Properties

Label 309.2.a
Level 309
Weight 2
Character orbit a
Rep. character \(\chi_{309}(1,\cdot)\)
Character field \(\Q\)
Dimension 17
Newforms 4
Sturm bound 69
Trace bound 1

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Defining parameters

Level: \( N \) = \( 309 = 3 \cdot 103 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 309.a (trivial)
Character field: \(\Q\)
Newforms: \( 4 \)
Sturm bound: \(69\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(309))\).

Total New Old
Modular forms 36 17 19
Cusp forms 33 17 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.

\(3\)\(103\)FrickeDim.
\(+\)\(+\)\(+\)\(5\)
\(+\)\(-\)\(-\)\(3\)
\(-\)\(+\)\(-\)\(8\)
\(-\)\(-\)\(+\)\(1\)
Plus space\(+\)\(6\)
Minus space\(-\)\(11\)

Trace form

\( 17q - 3q^{2} + q^{3} + 13q^{4} - 6q^{5} - q^{6} - 3q^{8} + 17q^{9} + O(q^{10}) \) \( 17q - 3q^{2} + q^{3} + 13q^{4} - 6q^{5} - q^{6} - 3q^{8} + 17q^{9} + 10q^{10} + 7q^{12} + 2q^{13} - 4q^{14} + 2q^{15} + q^{16} - 10q^{17} - 3q^{18} + 4q^{19} - 30q^{20} + 8q^{21} - 8q^{23} + 3q^{24} + 15q^{25} - 16q^{26} + q^{27} + 2q^{28} - 18q^{29} - 2q^{30} + 16q^{31} - 9q^{32} + 8q^{33} - 22q^{34} - 8q^{35} + 13q^{36} - 6q^{37} + 26q^{38} + 6q^{39} + 10q^{40} + 2q^{41} - 4q^{42} + 4q^{43} + 4q^{44} - 6q^{45} - 14q^{46} + 12q^{47} + 15q^{48} + 37q^{49} - 17q^{50} + 2q^{51} + 46q^{52} - 22q^{53} - q^{54} - 16q^{55} - 26q^{56} + 12q^{57} - 18q^{58} - 4q^{59} - 14q^{60} - 38q^{61} - 32q^{62} - 55q^{64} - 32q^{65} - 4q^{66} + 24q^{67} - 12q^{68} - 12q^{69} - 12q^{70} + 4q^{71} - 3q^{72} + 34q^{73} + 22q^{74} + 7q^{75} + 4q^{76} - 32q^{77} - 2q^{78} + 8q^{79} + 30q^{80} + 17q^{81} - 24q^{82} - 16q^{83} - 8q^{84} - 4q^{85} - 12q^{86} + 14q^{87} - 4q^{88} + 26q^{89} + 10q^{90} - 16q^{91} - 36q^{92} + 28q^{93} + 4q^{95} - 25q^{96} + 6q^{97} - 57q^{98} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(309))\) into irreducible Hecke orbits

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 3 103
309.2.a.a \(1\) \(2.467\) \(\Q\) None \(-1\) \(1\) \(-1\) \(-2\) \(-\) \(-\) \(q-q^{2}+q^{3}-q^{4}-q^{5}-q^{6}-2q^{7}+\cdots\)
309.2.a.b \(3\) \(2.467\) 3.3.148.1 None \(1\) \(-3\) \(1\) \(-2\) \(+\) \(-\) \(q+\beta _{1}q^{2}-q^{3}+(\beta _{1}+\beta _{2})q^{4}+\beta _{1}q^{5}+\cdots\)
309.2.a.c \(5\) \(2.467\) 5.5.81509.1 None \(-2\) \(-5\) \(-5\) \(-2\) \(+\) \(+\) \(q+\beta _{3}q^{2}-q^{3}+(-\beta _{3}-\beta _{4})q^{4}+(-1+\cdots)q^{5}+\cdots\)
309.2.a.d \(8\) \(2.467\) \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(-1\) \(8\) \(-1\) \(6\) \(-\) \(+\) \(q-\beta _{1}q^{2}+q^{3}+(1+\beta _{2})q^{4}+\beta _{7}q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(309))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(309)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(103))\)\(^{\oplus 2}\)