Space of modular forms of level 450, weight 2, and trivial character

• Label: 450.2.a
• Level: $450$
• Weight: $2$
• Character orbit: 450.a
• Rep. character: $\chi_{450}(1,\cdot)$
• Character field: $\Q$
• Dimension: $7$
• Newform subspaces: $7$
• Sturm bound: $180$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	450.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 7 \)
Sturm bound:			\(180\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(450))\).

	Total	New	Old
Modular forms	114	7	107
Cusp forms	67	7	60
Eisenstein series	47	0	47

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		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(12\)	\(1\)	\(11\)		\(7\)	\(1\)	\(6\)		\(5\)	\(0\)	\(5\)
\(+\)	\(+\)	\(-\)	\(-\)		\(16\)	\(0\)	\(16\)		\(10\)	\(0\)	\(10\)		\(6\)	\(0\)	\(6\)
\(+\)	\(-\)	\(+\)	\(-\)		\(15\)	\(2\)	\(13\)		\(9\)	\(2\)	\(7\)		\(6\)	\(0\)	\(6\)
\(+\)	\(-\)	\(-\)	\(+\)		\(14\)	\(1\)	\(13\)		\(8\)	\(1\)	\(7\)		\(6\)	\(0\)	\(6\)
\(-\)	\(+\)	\(+\)	\(-\)		\(15\)	\(1\)	\(14\)		\(9\)	\(1\)	\(8\)		\(6\)	\(0\)	\(6\)
\(-\)	\(+\)	\(-\)	\(+\)		\(13\)	\(0\)	\(13\)		\(7\)	\(0\)	\(7\)		\(6\)	\(0\)	\(6\)
\(-\)	\(-\)	\(+\)	\(+\)		\(15\)	\(0\)	\(15\)		\(9\)	\(0\)	\(9\)		\(6\)	\(0\)	\(6\)
\(-\)	\(-\)	\(-\)	\(-\)		\(14\)	\(2\)	\(12\)		\(8\)	\(2\)	\(6\)		\(6\)	\(0\)	\(6\)
Plus space			\(+\)		\(54\)	\(2\)	\(52\)		\(31\)	\(2\)	\(29\)		\(23\)	\(0\)	\(23\)
Minus space			\(-\)		\(60\)	\(5\)	\(55\)		\(36\)	\(5\)	\(31\)		\(24\)	\(0\)	\(24\)

Trace form

7 * q - q^2 + 7 * q^4 - q^8 + 2 * q^11 + 6 * q^13 + 4 * q^14 + 7 * q^16 + 6 * q^17 - 2 * q^19 + 12 * q^22 + 6 * q^26 + 6 * q^29 - 12 * q^31 - q^32 + 8 * q^34 - 18 * q^37 + 4 * q^38 + 8 * q^41 - 12 * q^43 + 2 * q^44 - 4 * q^46 - 9 * q^49 + 6 * q^52 - 6 * q^53 + 4 * q^56 - 18 * q^58 - 20 * q^59 + 2 * q^61 - 8 * q^62 + 7 * q^64 + 12 * q^67 + 6 * q^68 - 48 * q^71 + 18 * q^73 + 10 * q^74 - 2 * q^76 - 20 * q^79 - 6 * q^82 + 12 * q^83 - 20 * q^86 + 12 * q^88 - 28 * q^89 - 16 * q^91 - 8 * q^94 - 6 * q^97 - 9 * q^98

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(450))\) 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
450.2.a.a	$450$	$2$	450.a	1.a	$1$	$1$	$1$	$3.593$	\(\Q\)	None	✓		✓	✓	90.2.a.a	$1$	$1$	\(-1\)	\(0\)	\(0\)	\(-2\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}-2q^{7}-q^{8}-6q^{11}+4q^{13}+\cdots\)
450.2.a.b	$450$	$2$	450.a	1.a	$1$	$1$	$1$	$3.593$	\(\Q\)	None	✓		✓	✓	30.2.c.a	$1$	$1$	\(-1\)	\(0\)	\(0\)	\(-2\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}-2q^{7}-q^{8}-2q^{11}-6q^{13}+\cdots\)
450.2.a.c	$450$	$2$	450.a	1.a	$1$	$1$	$1$	$3.593$	\(\Q\)	None	✓				50.2.a.a	$1$	$0$	\(-1\)	\(0\)	\(0\)	\(-2\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}-2q^{7}-q^{8}+3q^{11}+4q^{13}+\cdots\)
450.2.a.d	$450$	$2$	450.a	1.a	$1$	$1$	$1$	$3.593$	\(\Q\)	None	✓				30.2.a.a	$1$	$0$	\(-1\)	\(0\)	\(0\)	\(4\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+4q^{7}-q^{8}-2q^{13}-4q^{14}+\cdots\)
450.2.a.e	$450$	$2$	450.a	1.a	$1$	$1$	$1$	$3.593$	\(\Q\)	None	✓		✓	✓	90.2.a.a	$1$	$0$	\(1\)	\(0\)	\(0\)	\(-2\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}-2q^{7}+q^{8}+6q^{11}+4q^{13}+\cdots\)
450.2.a.f	$450$	$2$	450.a	1.a	$1$	$1$	$1$	$3.593$	\(\Q\)	None	✓				30.2.c.a	$1$	$0$	\(1\)	\(0\)	\(0\)	\(2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+2q^{7}+q^{8}-2q^{11}+6q^{13}+\cdots\)
450.2.a.g	$450$	$2$	450.a	1.a	$1$	$1$	$1$	$3.593$	\(\Q\)	None	✓				50.2.a.a	$1$	$0$	\(1\)	\(0\)	\(0\)	\(2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+2q^{7}+q^{8}+3q^{11}-4q^{13}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(450))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(450)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(75))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(90))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(150))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(225))\)\(^{\oplus 2}\)