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Properties

Label 728.2.bm
Level $728$
Weight $2$
Character orbit 728.bm
Rep. character $\chi_{728}(225,\cdot)$
Character field $\Q(\zeta_{6})$
Dimension $40$
Newform subspaces $3$
Sturm bound $224$
Trace bound $1$

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Defining parameters

Level: \( N \) \(=\) \( 728 = 2^{3} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 728.bm (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 13 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 3 \)
Sturm bound: \(224\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(728, [\chi])\).

Total New Old
Modular forms 240 40 200
Cusp forms 208 40 168
Eisenstein series 32 0 32

Trace form

\( 40 q - 16 q^{9} + O(q^{10}) \) \( 40 q - 16 q^{9} - 8 q^{13} + 4 q^{17} - 16 q^{25} + 20 q^{29} - 36 q^{33} - 12 q^{35} + 48 q^{37} + 16 q^{39} - 24 q^{41} + 8 q^{43} - 60 q^{45} + 20 q^{49} - 80 q^{51} - 16 q^{53} - 32 q^{55} - 36 q^{59} - 20 q^{61} + 12 q^{63} + 84 q^{67} + 28 q^{69} + 24 q^{71} + 20 q^{75} + 16 q^{77} + 24 q^{79} - 60 q^{85} + 16 q^{87} + 72 q^{89} + 8 q^{91} + 12 q^{93} - 8 q^{95} - 36 q^{97} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(728, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
728.2.bm.a 728.bm 13.e $4$ $5.813$ \(\Q(\zeta_{12})\) None \(0\) \(4\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+(2-2\zeta_{12}^{2})q^{3}+(1-2\zeta_{12}^{2})q^{5}+\cdots\)
728.2.bm.b 728.bm 13.e $12$ $5.813$ 12.0.\(\cdots\).1 None \(0\) \(-2\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+(\beta _{5}+\beta _{8}-\beta _{11})q^{3}+(\beta _{1}+\beta _{2}-\beta _{6}+\cdots)q^{5}+\cdots\)
728.2.bm.c 728.bm 13.e $24$ $5.813$ None \(0\) \(-2\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$

Decomposition of \(S_{2}^{\mathrm{old}}(728, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(728, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(13, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(26, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(52, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(91, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(104, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(182, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(364, [\chi])\)\(^{\oplus 2}\)