Space of modular forms of level 150, weight 4, and trivial character

• Label: 150.4.a
• Level: $150$
• Weight: $4$
• Character orbit: 150.a
• Rep. character: $\chi_{150}(1,\cdot)$
• Character field: $\Q$
• Dimension: $9$
• Newform subspaces: $9$
• Sturm bound: $120$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 150 = 2 \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	150.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 9 \)
Sturm bound:			\(120\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(150))\).

	Total	New	Old
Modular forms	102	9	93
Cusp forms	78	9	69
Eisenstein series	24	0	24

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

\(2\)	\(3\)	\(5\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	$+$	\(1\)
\(+\)	\(+\)	\(-\)	$-$	\(1\)
\(+\)	\(-\)	\(+\)	$-$	\(1\)
\(+\)	\(-\)	\(-\)	$+$	\(1\)
\(-\)	\(+\)	\(+\)	$-$	\(1\)
\(-\)	\(+\)	\(-\)	$+$	\(2\)
\(-\)	\(-\)	\(+\)	$+$	\(2\)
Plus space			\(+\)	\(6\)
Minus space			\(-\)	\(3\)

Trace form

\( 9 q + 2 q^{2} - 3 q^{3} + 36 q^{4} - 6 q^{6} - 12 q^{7} + 8 q^{8} + 81 q^{9} + O(q^{10}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(150))\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Inner twists	Rank*	Traces					A-L signs			Fricke sign	Coefficient ring index	Sato-Tate	$q$-expansion
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		2	3	5
150.4.a.a	$150$	$4$	150.a	1.a	$1$	$1$	$1$	$8.850$	\(\Q\)	None	✓	$1$	$1$	\(-2\)	\(-3\)	\(0\)	\(-1\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}-3q^{3}+4q^{4}+6q^{6}-q^{7}+\cdots\)
150.4.a.b	$150$	$4$	150.a	1.a	$1$	$1$	$1$	$8.850$	\(\Q\)	None	✓	$1$	$0$	\(-2\)	\(-3\)	\(0\)	\(4\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}-3q^{3}+4q^{4}+6q^{6}+4q^{7}+\cdots\)
150.4.a.c	$150$	$4$	150.a	1.a	$1$	$1$	$1$	$8.850$	\(\Q\)	None	✓	$1$	$1$	\(-2\)	\(3\)	\(0\)	\(-23\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+3q^{3}+4q^{4}-6q^{6}-23q^{7}+\cdots\)
150.4.a.d	$150$	$4$	150.a	1.a	$1$	$1$	$1$	$8.850$	\(\Q\)	None	✓	$1$	$0$	\(-2\)	\(3\)	\(0\)	\(2\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+3q^{3}+4q^{4}-6q^{6}+2q^{7}+\cdots\)
150.4.a.e	$150$	$4$	150.a	1.a	$1$	$1$	$1$	$8.850$	\(\Q\)	None	✓	$1$	$1$	\(2\)	\(-3\)	\(0\)	\(-32\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}-3q^{3}+4q^{4}-6q^{6}-2^{5}q^{7}+\cdots\)
150.4.a.f	$150$	$4$	150.a	1.a	$1$	$1$	$1$	$8.850$	\(\Q\)	None	✓	$1$	$0$	\(2\)	\(-3\)	\(0\)	\(-2\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}-3q^{3}+4q^{4}-6q^{6}-2q^{7}+\cdots\)
150.4.a.g	$150$	$4$	150.a	1.a	$1$	$1$	$1$	$8.850$	\(\Q\)	None	✓	$1$	$0$	\(2\)	\(-3\)	\(0\)	\(23\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}-3q^{3}+4q^{4}-6q^{6}+23q^{7}+\cdots\)
150.4.a.h	$150$	$4$	150.a	1.a	$1$	$1$	$1$	$8.850$	\(\Q\)	None	✓	$1$	$0$	\(2\)	\(3\)	\(0\)	\(1\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+3q^{3}+4q^{4}+6q^{6}+q^{7}+\cdots\)
150.4.a.i	$150$	$4$	150.a	1.a	$1$	$1$	$1$	$8.850$	\(\Q\)	None	✓	$1$	$0$	\(2\)	\(3\)	\(0\)	\(16\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+3q^{3}+4q^{4}+6q^{6}+2^{4}q^{7}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(150))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(150)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(25))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(75))\)\(^{\oplus 2}\)