Properties

Label 135.4.a
Level $135$
Weight $4$
Character orbit 135.a
Rep. character $\chi_{135}(1,\cdot)$
Character field $\Q$
Dimension $16$
Newform subspaces $8$
Sturm bound $72$
Trace bound $2$

Related objects

Downloads

Learn more

Defining parameters

Level: \( N \) \(=\) \( 135 = 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 135.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 8 \)
Sturm bound: \(72\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(135))\).

Total New Old
Modular forms 60 16 44
Cusp forms 48 16 32
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.
\(+\)\(+\)\(+\)\(4\)
\(+\)\(-\)\(-\)\(2\)
\(-\)\(+\)\(-\)\(4\)
\(-\)\(-\)\(+\)\(6\)
Plus space\(+\)\(10\)
Minus space\(-\)\(6\)

Trace form

\( 16 q + 58 q^{4} + 68 q^{7} + O(q^{10}) \) \( 16 q + 58 q^{4} + 68 q^{7} + 30 q^{10} - 100 q^{13} + 754 q^{16} - 64 q^{19} - 348 q^{22} + 400 q^{25} + 236 q^{28} + 296 q^{31} + 570 q^{34} - 412 q^{37} + 480 q^{40} - 736 q^{43} - 3090 q^{46} - 540 q^{49} - 772 q^{52} + 60 q^{55} - 3744 q^{58} + 668 q^{61} + 4600 q^{64} + 4244 q^{67} - 540 q^{70} + 788 q^{73} - 3814 q^{76} + 716 q^{79} - 2556 q^{82} + 120 q^{85} - 10212 q^{88} - 2084 q^{91} + 4116 q^{94} + 3212 q^{97} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(135))\) 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
135.4.a.a 135.a 1.a $1$ $7.965$ \(\Q\) None \(-2\) \(0\) \(5\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}-4q^{4}+5q^{5}+24q^{8}-10q^{10}+\cdots\)
135.4.a.b 135.a 1.a $1$ $7.965$ \(\Q\) None \(-1\) \(0\) \(-5\) \(-6\) $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-7q^{4}-5q^{5}-6q^{7}+15q^{8}+\cdots\)
135.4.a.c 135.a 1.a $1$ $7.965$ \(\Q\) None \(1\) \(0\) \(5\) \(-6\) $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-7q^{4}+5q^{5}-6q^{7}-15q^{8}+\cdots\)
135.4.a.d 135.a 1.a $1$ $7.965$ \(\Q\) None \(2\) \(0\) \(-5\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}-4q^{4}-5q^{5}-24q^{8}-10q^{10}+\cdots\)
135.4.a.e 135.a 1.a $3$ $7.965$ 3.3.1772.1 None \(-5\) \(0\) \(-15\) \(-4\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(-2+\beta _{1})q^{2}+(7-3\beta _{1}+\beta _{2})q^{4}+\cdots\)
135.4.a.f 135.a 1.a $3$ $7.965$ 3.3.5637.1 None \(-1\) \(0\) \(15\) \(44\) $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+(7+\beta _{1}+\beta _{2})q^{4}+5q^{5}+\cdots\)
135.4.a.g 135.a 1.a $3$ $7.965$ 3.3.5637.1 None \(1\) \(0\) \(-15\) \(44\) $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(7+\beta _{1}+\beta _{2})q^{4}-5q^{5}+\cdots\)
135.4.a.h 135.a 1.a $3$ $7.965$ 3.3.1772.1 None \(5\) \(0\) \(15\) \(-4\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(2-\beta _{1})q^{2}+(7-3\beta _{1}+\beta _{2})q^{4}+5q^{5}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(135))\) into lower level spaces

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