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Properties

Label 240.2.s
Level $240$
Weight $2$
Character orbit 240.s
Rep. character $\chi_{240}(61,\cdot)$
Character field $\Q(\zeta_{4})$
Dimension $32$
Newform subspaces $3$
Sturm bound $96$
Trace bound $1$

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Defining parameters

Level: \( N \) \(=\) \( 240 = 2^{4} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 240.s (of order \(4\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 16 \)
Character field: \(\Q(i)\)
Newform subspaces: \( 3 \)
Sturm bound: \(96\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(7\)

Dimensions

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

Total New Old
Modular forms 104 32 72
Cusp forms 88 32 56
Eisenstein series 16 0 16

Trace form

\( 32 q + 8 q^{4} + O(q^{10}) \) \( 32 q + 8 q^{4} - 4 q^{10} + 16 q^{11} - 8 q^{14} + 8 q^{15} - 4 q^{16} - 8 q^{18} + 8 q^{19} - 16 q^{20} + 16 q^{22} - 4 q^{24} - 16 q^{28} + 32 q^{29} - 40 q^{32} - 36 q^{34} + 4 q^{36} + 32 q^{37} - 40 q^{38} - 16 q^{43} + 8 q^{44} + 12 q^{46} - 32 q^{49} + 8 q^{50} - 8 q^{51} - 56 q^{52} - 32 q^{53} - 4 q^{54} + 48 q^{56} + 64 q^{58} - 32 q^{59} - 16 q^{61} - 48 q^{62} - 16 q^{63} - 16 q^{64} - 24 q^{66} - 16 q^{67} - 8 q^{68} - 16 q^{69} + 16 q^{70} + 8 q^{72} + 32 q^{74} - 4 q^{76} - 32 q^{77} - 24 q^{78} - 16 q^{79} - 32 q^{81} + 40 q^{82} + 48 q^{84} + 16 q^{85} + 48 q^{86} + 8 q^{88} - 16 q^{91} + 104 q^{92} + 52 q^{94} + 40 q^{96} + 16 q^{98} + 16 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

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.s.a 240.s 16.e $4$ $1.916$ \(\Q(\zeta_{8})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+(-\zeta_{8}+\zeta_{8}^{3})q^{2}+\zeta_{8}^{3}q^{3}+2q^{4}+\cdots\)
240.2.s.b 240.s 16.e $8$ $1.916$ 8.0.18939904.2 None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+(-\beta _{4}+\beta _{5})q^{2}-\beta _{6}q^{3}+(-1+\beta _{2}+\cdots)q^{4}+\cdots\)
240.2.s.c 240.s 16.e $20$ $1.916$ \(\mathbb{Q}[x]/(x^{20} - \cdots)\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+\beta _{1}q^{2}-\beta _{4}q^{3}+\beta _{2}q^{4}+\beta _{5}q^{5}+\cdots\)

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

\( S_{2}^{\mathrm{old}}(240, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(16, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(48, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(80, [\chi])\)\(^{\oplus 2}\)