Properties

Label 540.2.a
Level $540$
Weight $2$
Character orbit 540.a
Rep. character $\chi_{540}(1,\cdot)$
Character field $\Q$
Dimension $6$
Newform subspaces $6$
Sturm bound $216$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 540 = 2^{2} \cdot 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 540.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 6 \)
Sturm bound: \(216\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(7\), \(11\), \(17\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(540))\).

Total New Old
Modular forms 126 6 120
Cusp forms 91 6 85
Eisenstein series 35 0 35

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

Trace form

\( 6 q - 6 q^{7} + O(q^{10}) \) \( 6 q - 6 q^{7} - 6 q^{13} - 6 q^{19} + 6 q^{25} + 12 q^{31} - 6 q^{37} + 12 q^{43} - 6 q^{61} - 6 q^{67} - 6 q^{73} - 6 q^{79} + 42 q^{91} - 42 q^{97} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(540))\) 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
540.2.a.a \(1\) \(4.312\) \(\Q\) None \(0\) \(0\) \(-1\) \(-4\) \(-\) \(-\) \(+\) \(q-q^{5}-4q^{7}+6q^{11}-4q^{13}-3q^{17}+\cdots\)
540.2.a.b \(1\) \(4.312\) \(\Q\) None \(0\) \(0\) \(-1\) \(-1\) \(-\) \(-\) \(+\) \(q-q^{5}-q^{7}-6q^{11}-q^{13}-q^{19}+\cdots\)
540.2.a.c \(1\) \(4.312\) \(\Q\) None \(0\) \(0\) \(-1\) \(2\) \(-\) \(+\) \(+\) \(q-q^{5}+2q^{7}+2q^{13}+3q^{17}+5q^{19}+\cdots\)
540.2.a.d \(1\) \(4.312\) \(\Q\) None \(0\) \(0\) \(1\) \(-4\) \(-\) \(+\) \(-\) \(q+q^{5}-4q^{7}-6q^{11}-4q^{13}+3q^{17}+\cdots\)
540.2.a.e \(1\) \(4.312\) \(\Q\) None \(0\) \(0\) \(1\) \(-1\) \(-\) \(-\) \(-\) \(q+q^{5}-q^{7}+6q^{11}-q^{13}-q^{19}+\cdots\)
540.2.a.f \(1\) \(4.312\) \(\Q\) None \(0\) \(0\) \(1\) \(2\) \(-\) \(-\) \(-\) \(q+q^{5}+2q^{7}+2q^{13}-3q^{17}+5q^{19}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(540))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(540)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(27))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(54))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(90))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(108))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(135))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(180))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(270))\)\(^{\oplus 2}\)