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Properties

Label 135.6.a
Level $135$
Weight $6$
Character orbit 135.a
Rep. character $\chi_{135}(1,\cdot)$
Character field $\Q$
Dimension $26$
Newform subspaces $10$
Sturm bound $108$
Trace bound $4$

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Defining parameters

Level: \( N \) \(=\) \( 135 = 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 135.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 10 \)
Sturm bound: \(108\)
Trace bound: \(4\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(135))\).

Total New Old
Modular forms 96 26 70
Cusp forms 84 26 58
Eisenstein series 12 0 12

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(3\)\(5\)FrickeDim
\(+\)\(+\)$+$\(6\)
\(+\)\(-\)$-$\(8\)
\(-\)\(+\)$-$\(7\)
\(-\)\(-\)$+$\(5\)
Plus space\(+\)\(11\)
Minus space\(-\)\(15\)

Trace form

\( 26 q + 394 q^{4} - 138 q^{7} + O(q^{10}) \) \( 26 q + 394 q^{4} - 138 q^{7} - 50 q^{10} + 1598 q^{13} - 638 q^{16} + 1346 q^{19} + 22340 q^{22} + 16250 q^{25} + 14652 q^{28} + 10200 q^{31} - 22030 q^{34} - 10418 q^{37} + 10800 q^{40} + 54044 q^{43} + 71238 q^{46} + 32248 q^{49} + 69196 q^{52} - 15800 q^{55} - 165008 q^{58} + 29750 q^{61} - 13816 q^{64} - 100770 q^{67} + 3300 q^{70} - 110626 q^{73} + 146362 q^{76} + 215106 q^{79} + 116212 q^{82} - 66200 q^{85} + 89532 q^{88} - 160938 q^{91} - 84532 q^{94} - 588926 q^{97} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(135))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 3 5
135.6.a.a 135.a 1.a $1$ $21.652$ \(\Q\) None \(-10\) \(0\) \(25\) \(6\) $-$ $-$ $\mathrm{SU}(2)$ \(q-10q^{2}+68q^{4}+5^{2}q^{5}+6q^{7}-360q^{8}+\cdots\)
135.6.a.b 135.a 1.a $1$ $21.652$ \(\Q\) None \(10\) \(0\) \(-25\) \(6\) $-$ $+$ $\mathrm{SU}(2)$ \(q+10q^{2}+68q^{4}-5^{2}q^{5}+6q^{7}+360q^{8}+\cdots\)
135.6.a.c 135.a 1.a $2$ $21.652$ \(\Q(\sqrt{73}) \) None \(-5\) \(0\) \(-50\) \(-199\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(-2-\beta )q^{2}+(-10+5\beta )q^{4}-5^{2}q^{5}+\cdots\)
135.6.a.d 135.a 1.a $2$ $21.652$ \(\Q(\sqrt{5}) \) None \(-2\) \(0\) \(50\) \(-150\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(-1-\beta )q^{2}+(14+2\beta )q^{4}+5^{2}q^{5}+\cdots\)
135.6.a.e 135.a 1.a $2$ $21.652$ \(\Q(\sqrt{5}) \) None \(2\) \(0\) \(-50\) \(-150\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{2}+(14+2\beta )q^{4}-5^{2}q^{5}+\cdots\)
135.6.a.f 135.a 1.a $2$ $21.652$ \(\Q(\sqrt{73}) \) None \(5\) \(0\) \(50\) \(-199\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(3-\beta )q^{2}+(-5-5\beta )q^{4}+5^{2}q^{5}+\cdots\)
135.6.a.g 135.a 1.a $4$ $21.652$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(-3\) \(0\) \(-100\) \(128\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{2}+(8-2\beta _{1}+\beta _{2})q^{4}+\cdots\)
135.6.a.h 135.a 1.a $4$ $21.652$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(-3\) \(0\) \(-100\) \(146\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{2}+(22-2\beta _{1}+\beta _{3})q^{4}+\cdots\)
135.6.a.i 135.a 1.a $4$ $21.652$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(3\) \(0\) \(100\) \(128\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{2}+(8-2\beta _{1}+\beta _{2})q^{4}+5^{2}q^{5}+\cdots\)
135.6.a.j 135.a 1.a $4$ $21.652$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(3\) \(0\) \(100\) \(146\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{2}+(22-2\beta _{1}+\beta _{3})q^{4}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(135))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_0(135)) \cong \) \(S_{6}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(27))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 2}\)