Space of modular forms of level 150, weight 2, and trivial character

• Label: 150.2.a
• Level: $150$
• Weight: $2$
• Character orbit: 150.a
• Rep. character: $\chi_{150}(1,\cdot)$
• Character field: $\Q$
• Dimension: $3$
• Newform subspaces: $3$
• Sturm bound: $60$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 150 = 2 \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	150.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 3 \)
Sturm bound:			\(60\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(150))\).

	Total	New	Old
Modular forms	42	3	39
Cusp forms	19	3	16
Eisenstein series	23	0	23

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
\(+\)	\(+\)	\(-\)	\(-\)	\(1\)
\(-\)	\(+\)	\(+\)	\(-\)	\(1\)
\(-\)	\(-\)	\(-\)	\(-\)	\(1\)
Plus space			\(+\)	\(0\)
Minus space			\(-\)	\(3\)

Trace form

3 * q + q^2 - q^3 + 3 * q^4 + q^6 + 4 * q^7 + q^8 + 3 * q^9 + 4 * q^11 - q^12 - 2 * q^13 + 3 * q^16 - 6 * q^17 + q^18 - 4 * q^19 - 8 * q^21 + q^24 - 14 * q^26 - q^27 + 4 * q^28 - 6 * q^29 - 8 * q^31 + q^32 - 10 * q^34 + 3 * q^36 - 2 * q^37 - 4 * q^38 - 10 * q^39 - 2 * q^41 - 4 * q^42 + 4 * q^43 + 4 * q^44 + 8 * q^46 - q^48 + 3 * q^49 + 2 * q^51 - 2 * q^52 + 6 * q^53 + q^54 + 4 * q^57 - 6 * q^58 + 20 * q^59 - 6 * q^61 + 8 * q^62 + 4 * q^63 + 3 * q^64 + 4 * q^66 + 4 * q^67 - 6 * q^68 + 8 * q^69 + 24 * q^71 + q^72 - 2 * q^73 - 6 * q^74 - 4 * q^76 + 2 * q^78 + 8 * q^79 + 3 * q^81 - 6 * q^82 - 12 * q^83 - 8 * q^84 + 12 * q^86 + 6 * q^87 - 2 * q^89 + 16 * q^91 - 8 * q^93 + 16 * q^94 + q^96 - 2 * q^97 + 9 * q^98 + 4 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(150))\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Twist minimal	Largest	Maximal	Minimal twist	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
150.2.a.a	$150$	$2$	150.a	1.a	$1$	$1$	$1$	$1.198$	\(\Q\)	None	✓		✓	✓	30.2.c.a	$1$	$0$	\(-1\)	\(-1\)	\(0\)	\(2\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{3}+q^{4}+q^{6}+2q^{7}-q^{8}+\cdots\)
150.2.a.b	$150$	$2$	150.a	1.a	$1$	$1$	$1$	$1.198$	\(\Q\)	None	✓		✓	✓	30.2.a.a	$1$	$0$	\(1\)	\(-1\)	\(0\)	\(4\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{3}+q^{4}-q^{6}+4q^{7}+q^{8}+\cdots\)
150.2.a.c	$150$	$2$	150.a	1.a	$1$	$1$	$1$	$1.198$	\(\Q\)	None	✓		✓	✓	30.2.c.a	$1$	$0$	\(1\)	\(1\)	\(0\)	\(-2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{3}+q^{4}+q^{6}-2q^{7}+q^{8}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(150))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(150)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(75))\)\(^{\oplus 2}\)