Properties

Label 240.2.f
Level $240$
Weight $2$
Character orbit 240.f
Rep. character $\chi_{240}(49,\cdot)$
Character field $\Q$
Dimension $6$
Newform subspaces $3$
Sturm bound $96$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 240 = 2^{4} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 240.f (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 5 \)
Character field: \(\Q\)
Newform subspaces: \( 3 \)
Sturm bound: \(96\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(7\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(240, [\chi])\).

Total New Old
Modular forms 60 6 54
Cusp forms 36 6 30
Eisenstein series 24 0 24

Trace form

\( 6 q + 2 q^{5} - 6 q^{9} + O(q^{10}) \) \( 6 q + 2 q^{5} - 6 q^{9} + 16 q^{19} + 6 q^{25} - 4 q^{29} + 8 q^{31} - 24 q^{35} - 8 q^{39} - 12 q^{41} - 2 q^{45} - 6 q^{49} - 16 q^{51} + 8 q^{55} + 16 q^{59} + 12 q^{61} - 8 q^{65} - 8 q^{69} - 16 q^{71} + 8 q^{75} - 40 q^{79} + 6 q^{81} + 8 q^{85} + 28 q^{89} - 16 q^{91} + 32 q^{95} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(240, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
240.2.f.a 240.f 5.b $2$ $1.916$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(-4\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+iq^{3}+(-2+i)q^{5}+2iq^{7}-q^{9}+\cdots\)
240.2.f.b 240.f 5.b $2$ $1.916$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(2\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+iq^{3}+(1-2i)q^{5}-4iq^{7}-q^{9}+\cdots\)
240.2.f.c 240.f 5.b $2$ $1.916$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(4\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+iq^{3}+(2+i)q^{5}+2iq^{7}-q^{9}-2q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(240, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(240, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(30, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(40, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(60, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(80, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(120, [\chi])\)\(^{\oplus 2}\)