Properties

Label 270.2.a
Level $270$
Weight $2$
Character orbit 270.a
Rep. character $\chi_{270}(1,\cdot)$
Character field $\Q$
Dimension $4$
Newform subspaces $4$
Sturm bound $108$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 270 = 2 \cdot 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 270.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 4 \)
Sturm bound: \(108\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(11\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(270))\).

Total New Old
Modular forms 66 4 62
Cusp forms 43 4 39
Eisenstein series 23 0 23

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

\(2\)\(3\)\(5\)FrickeDim.
\(+\)\(+\)\(-\)\(-\)\(1\)
\(+\)\(-\)\(+\)\(-\)\(1\)
\(-\)\(+\)\(+\)\(-\)\(1\)
\(-\)\(-\)\(-\)\(-\)\(1\)
Plus space\(+\)\(0\)
Minus space\(-\)\(4\)

Trace form

\( 4 q + 4 q^{4} + 8 q^{7} + O(q^{10}) \) \( 4 q + 4 q^{4} + 8 q^{7} + 8 q^{13} + 4 q^{16} + 8 q^{19} + 4 q^{25} + 8 q^{28} - 4 q^{31} - 12 q^{34} - 16 q^{37} - 16 q^{43} - 12 q^{46} - 12 q^{49} + 8 q^{52} - 12 q^{55} - 24 q^{58} - 16 q^{61} + 4 q^{64} - 16 q^{67} - 16 q^{73} + 8 q^{76} - 28 q^{79} - 24 q^{82} + 16 q^{91} + 12 q^{94} + 32 q^{97} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(270))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 3 5
270.2.a.a \(1\) \(2.156\) \(\Q\) None \(-1\) \(0\) \(-1\) \(2\) \(+\) \(-\) \(+\) \(q-q^{2}+q^{4}-q^{5}+2q^{7}-q^{8}+q^{10}+\cdots\)
270.2.a.b \(1\) \(2.156\) \(\Q\) None \(-1\) \(0\) \(1\) \(2\) \(+\) \(+\) \(-\) \(q-q^{2}+q^{4}+q^{5}+2q^{7}-q^{8}-q^{10}+\cdots\)
270.2.a.c \(1\) \(2.156\) \(\Q\) None \(1\) \(0\) \(-1\) \(2\) \(-\) \(+\) \(+\) \(q+q^{2}+q^{4}-q^{5}+2q^{7}+q^{8}-q^{10}+\cdots\)
270.2.a.d \(1\) \(2.156\) \(\Q\) None \(1\) \(0\) \(1\) \(2\) \(-\) \(-\) \(-\) \(q+q^{2}+q^{4}+q^{5}+2q^{7}+q^{8}+q^{10}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(270))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(270)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(27))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(54))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(90))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(135))\)\(^{\oplus 2}\)