Properties

Label 135.2.b
Level $135$
Weight $2$
Character orbit 135.b
Rep. character $\chi_{135}(109,\cdot)$
Character field $\Q$
Dimension $8$
Newform subspaces $2$
Sturm bound $36$
Trace bound $4$

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Defining parameters

Level: \( N \) \(=\) \( 135 = 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 135.b (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 5 \)
Character field: \(\Q\)
Newform subspaces: \( 2 \)
Sturm bound: \(36\)
Trace bound: \(4\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(135, [\chi])\).

Total New Old
Modular forms 24 8 16
Cusp forms 12 8 4
Eisenstein series 12 0 12

Trace form

\( 8 q - 6 q^{4} + O(q^{10}) \) \( 8 q - 6 q^{4} - 14 q^{10} + 10 q^{16} + 4 q^{19} - 4 q^{25} - 8 q^{31} - 10 q^{34} + 32 q^{40} + 38 q^{46} + 20 q^{49} - 36 q^{55} - 56 q^{61} - 40 q^{64} + 36 q^{70} - 102 q^{76} + 52 q^{79} + 28 q^{85} + 36 q^{91} + 56 q^{94} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(135, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
135.2.b.a 135.b 5.b $4$ $1.078$ \(\Q(i, \sqrt{5})\) \(\Q(\sqrt{-15}) \) 135.2.b.a \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q-\beta _{1}q^{2}+(-2+\beta _{3})q^{4}+(-\beta _{1}+\beta _{2}+\cdots)q^{5}+\cdots\)
135.2.b.b 135.b 5.b $4$ $1.078$ \(\Q(\zeta_{8})\) None 135.2.b.b \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\zeta_{8}^{2}q^{2}+\zeta_{8}^{3}q^{5}-\zeta_{8}q^{7}+2\zeta_{8}^{2}q^{8}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(135, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(135, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(45, [\chi])\)\(^{\oplus 2}\)