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Properties

Label 540.3.o
Level $540$
Weight $3$
Character orbit 540.o
Rep. character $\chi_{540}(341,\cdot)$
Character field $\Q(\zeta_{6})$
Dimension $16$
Newform subspaces $2$
Sturm bound $324$
Trace bound $1$

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Defining parameters

Level: \( N \) \(=\) \( 540 = 2^{2} \cdot 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 540.o (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 9 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 2 \)
Sturm bound: \(324\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(540, [\chi])\).

Total New Old
Modular forms 468 16 452
Cusp forms 396 16 380
Eisenstein series 72 0 72

Trace form

\( 16 q + 2 q^{7} + O(q^{10}) \) \( 16 q + 2 q^{7} - 54 q^{11} - 10 q^{13} - 4 q^{19} + 54 q^{23} + 40 q^{25} - 90 q^{29} - 16 q^{31} + 44 q^{37} + 92 q^{43} - 216 q^{47} - 6 q^{49} + 126 q^{59} - 34 q^{61} + 90 q^{65} - 154 q^{67} - 172 q^{73} - 126 q^{77} - 40 q^{79} + 198 q^{83} + 30 q^{85} + 520 q^{91} - 180 q^{95} + 98 q^{97} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{3}^{\mathrm{new}}(540, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
540.3.o.a 540.o 9.d $4$ $14.714$ \(\Q(\sqrt{-3}, \sqrt{-5})\) None \(0\) \(0\) \(0\) \(8\) $\mathrm{SU}(2)[C_{6}]$ \(q+\beta _{1}q^{5}+(-2\beta _{1}+4\beta _{2}-2\beta _{3})q^{7}+\cdots\)
540.3.o.b 540.o 9.d $12$ $14.714$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(0\) \(0\) \(0\) \(-6\) $\mathrm{SU}(2)[C_{6}]$ \(q+\beta _{4}q^{5}+(-1-\beta _{2}-\beta _{3}+2\beta _{4}+\beta _{9}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{3}^{\mathrm{old}}(540, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(540, [\chi]) \cong \) \(S_{3}^{\mathrm{new}}(9, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(18, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(27, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(36, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(45, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(54, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(90, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(108, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(135, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(180, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(270, [\chi])\)\(^{\oplus 2}\)