Properties

Label 135.2.a
Level 135
Weight 2
Character orbit a
Rep. character \(\chi_{135}(1,\cdot)\)
Character field \(\Q\)
Dimension 6
Newforms 4
Sturm bound 36
Trace bound 2

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Defining parameters

Level: \( N \) = \( 135 = 3^{3} \cdot 5 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 135.a (trivial)
Character field: \(\Q\)
Newforms: \( 4 \)
Sturm bound: \(36\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(135))\).

Total New Old
Modular forms 24 6 18
Cusp forms 13 6 7
Eisenstein series 11 0 11

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

\(3\)\(5\)FrickeDim.
\(+\)\(+\)\(+\)\(1\)
\(+\)\(-\)\(-\)\(3\)
\(-\)\(+\)\(-\)\(2\)
Plus space\(+\)\(1\)
Minus space\(-\)\(5\)

Trace form

\(6q \) \(\mathstrut +\mathstrut 10q^{4} \) \(\mathstrut -\mathstrut 2q^{7} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(6q \) \(\mathstrut +\mathstrut 10q^{4} \) \(\mathstrut -\mathstrut 2q^{7} \) \(\mathstrut +\mathstrut 2q^{10} \) \(\mathstrut +\mathstrut 2q^{13} \) \(\mathstrut -\mathstrut 14q^{16} \) \(\mathstrut +\mathstrut 2q^{19} \) \(\mathstrut -\mathstrut 20q^{22} \) \(\mathstrut +\mathstrut 6q^{25} \) \(\mathstrut -\mathstrut 32q^{28} \) \(\mathstrut -\mathstrut 8q^{31} \) \(\mathstrut +\mathstrut 10q^{34} \) \(\mathstrut +\mathstrut 18q^{37} \) \(\mathstrut -\mathstrut 12q^{40} \) \(\mathstrut -\mathstrut 4q^{43} \) \(\mathstrut -\mathstrut 18q^{46} \) \(\mathstrut +\mathstrut 32q^{49} \) \(\mathstrut +\mathstrut 24q^{52} \) \(\mathstrut +\mathstrut 8q^{55} \) \(\mathstrut +\mathstrut 8q^{58} \) \(\mathstrut +\mathstrut 26q^{61} \) \(\mathstrut -\mathstrut 24q^{64} \) \(\mathstrut -\mathstrut 50q^{67} \) \(\mathstrut +\mathstrut 12q^{70} \) \(\mathstrut +\mathstrut 26q^{73} \) \(\mathstrut +\mathstrut 30q^{76} \) \(\mathstrut -\mathstrut 38q^{79} \) \(\mathstrut +\mathstrut 68q^{82} \) \(\mathstrut +\mathstrut 8q^{85} \) \(\mathstrut -\mathstrut 12q^{88} \) \(\mathstrut -\mathstrut 10q^{91} \) \(\mathstrut +\mathstrut 40q^{94} \) \(\mathstrut +\mathstrut 6q^{97} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(135))\) into irreducible Hecke orbits

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 3 5
135.2.a.a \(1\) \(1.078\) \(\Q\) None \(-2\) \(0\) \(-1\) \(-3\) \(+\) \(+\) \(q-2q^{2}+2q^{4}-q^{5}-3q^{7}+2q^{10}+\cdots\)
135.2.a.b \(1\) \(1.078\) \(\Q\) None \(2\) \(0\) \(1\) \(-3\) \(+\) \(-\) \(q+2q^{2}+2q^{4}+q^{5}-3q^{7}+2q^{10}+\cdots\)
135.2.a.c \(2\) \(1.078\) \(\Q(\sqrt{13}) \) None \(-1\) \(0\) \(2\) \(2\) \(+\) \(-\) \(q-\beta q^{2}+(1+\beta )q^{4}+q^{5}+(2-2\beta )q^{7}+\cdots\)
135.2.a.d \(2\) \(1.078\) \(\Q(\sqrt{13}) \) None \(1\) \(0\) \(-2\) \(2\) \(-\) \(+\) \(q+\beta q^{2}+(1+\beta )q^{4}-q^{5}+(2-2\beta )q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(135))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(135)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(27))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 2}\)