Space of modular forms of level 225, weight 2, and trivial character

• Label: 225.2.a
• Level: $225$
• Weight: $2$
• Character orbit: 225.a
• Rep. character: $\chi_{225}(1,\cdot)$
• Character field: $\Q$
• Dimension: $7$
• Newform subspaces: $6$
• Sturm bound: $60$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 225 = 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	225.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 6 \)
Sturm bound:			\(60\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(2\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(225))\).

	Total	New	Old
Modular forms	42	10	32
Cusp forms	19	7	12
Eisenstein series	23	3	20

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

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

Trace form

7 * q - q^2 + 5 * q^4 + 3 * q^8 + 2 * q^13 + 12 * q^14 - 3 * q^16 + 2 * q^17 - 4 * q^22 - 6 * q^26 - 18 * q^29 - 4 * q^31 - 5 * q^32 - 14 * q^34 + 10 * q^37 - 4 * q^38 + 6 * q^41 - 4 * q^43 - 12 * q^44 - 16 * q^46 + 8 * q^47 + 19 * q^49 - 2 * q^52 - 10 * q^53 - 2 * q^58 + 24 * q^59 - 10 * q^61 - 51 * q^64 - 12 * q^67 - 2 * q^68 + 24 * q^71 - 10 * q^73 - 18 * q^74 + 4 * q^76 + 24 * q^79 + 10 * q^82 + 12 * q^83 + 12 * q^88 + 6 * q^89 + 44 * q^91 + 40 * q^94 - 2 * q^97 + 7 * q^98

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(225))\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		3	5
225.2.a.a	$225$	$2$	225.a	1.a	$1$	$1$	$1$	$1.797$	\(\Q\)	None	✓		✓	✓	75.2.a.a	$1$	$1$	\(-2\)	\(0\)	\(0\)	\(-3\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+2q^{4}-3q^{7}-2q^{11}+q^{13}+\cdots\)
225.2.a.b	$225$	$2$	225.a	1.a	$1$	$1$	$1$	$1.797$	\(\Q\)	None	✓				15.2.a.a	$1$	$0$	\(-1\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{4}+3q^{8}+4q^{11}+2q^{13}+\cdots\)
225.2.a.c	$225$	$2$	225.a	1.a	$1$	$1$	$1$	$1.797$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓	✓	✓	✓	225.2.a.c	$2$	$1$	\(0\)	\(0\)	\(0\)	\(-5\)		$+$	$+$	$+$	$1$	$N(\mathrm{U}(1))$	\(q-2q^{4}-5q^{7}-5q^{13}+4q^{16}-q^{19}+\cdots\)
225.2.a.d	$225$	$2$	225.a	1.a	$1$	$1$	$1$	$1.797$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓	✓			225.2.a.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(5\)		$+$	$-$	$-$	$1$	$N(\mathrm{U}(1))$	\(q-2q^{4}+5q^{7}+5q^{13}+4q^{16}-q^{19}+\cdots\)
225.2.a.e	$225$	$2$	225.a	1.a	$1$	$1$	$1$	$1.797$	\(\Q\)	None	✓				75.2.a.a	$1$	$0$	\(2\)	\(0\)	\(0\)	\(3\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+2q^{4}+3q^{7}-2q^{11}-q^{13}+\cdots\)
225.2.a.f	$225$	$2$	225.a	1.a	$1$	$2$	$2$	$1.797$	\(\Q(\sqrt{5}) \)	\(\Q(\sqrt{-15}) \)	✓		✓		45.2.b.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$	$2$	$N(\mathrm{U}(1))$	\(q-\beta q^{2}+3q^{4}-\beta q^{8}-q^{16}+2\beta q^{17}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(225))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(225)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(75))\)\(^{\oplus 2}\)