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

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

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

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