Space of modular forms of level 275, weight 2, and trivial character

• Label: 275.2.a
• Level: $275$
• Weight: $2$
• Character orbit: 275.a
• Rep. character: $\chi_{275}(1,\cdot)$
• Character field: $\Q$
• Dimension: $16$
• Newform subspaces: $8$
• Sturm bound: $60$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 275 = 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	275.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 8 \)
Sturm bound:			\(60\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(275))\).

	Total	New	Old
Modular forms	36	16	20
Cusp forms	25	16	9
Eisenstein series	11	0	11

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

\(5\)	\(11\)	Fricke	Dim
\(+\)	\(+\)	$+$	\(3\)
\(+\)	\(-\)	$-$	\(5\)
\(-\)	\(+\)	$-$	\(6\)
\(-\)	\(-\)	$+$	\(2\)
Plus space		\(+\)	\(5\)
Minus space		\(-\)	\(11\)

Trace form

\( 16 q - q^{2} + q^{3} + 13 q^{4} - 2 q^{6} + 6 q^{7} - 3 q^{8} + 11 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(275))\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		5	11
275.2.a.a	$275$	$2$	275.a	1.a	$1$	$1$	$1$	$2.196$	\(\Q\)	None	✓	$1$	$1$	\(-1\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{4}+3q^{8}-3q^{9}-q^{11}-2q^{13}+\cdots\)
275.2.a.b	$275$	$2$	275.a	1.a	$1$	$1$	$1$	$2.196$	\(\Q\)	None	✓	$1$	$0$	\(2\)	\(1\)	\(0\)	\(2\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+q^{3}+2q^{4}+2q^{6}+2q^{7}+\cdots\)
275.2.a.c	$275$	$2$	275.a	1.a	$1$	$2$	$2$	$2.196$	\(\Q(\sqrt{2}) \)	None	✓	$1$	$0$	\(-2\)	\(0\)	\(0\)	\(4\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{2}-2\beta q^{3}+(1-2\beta )q^{4}+\cdots\)
275.2.a.d	$275$	$2$	275.a	1.a	$1$	$2$	$2$	$2.196$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$1$	\(-1\)	\(-3\)	\(0\)	\(-1\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{2}+(-1-\beta )q^{3}+(-1+\beta )q^{4}+\cdots\)
275.2.a.e	$275$	$2$	275.a	1.a	$1$	$2$	$2$	$2.196$	\(\Q(\sqrt{13}) \)	None	✓	$1$	$1$	\(-1\)	\(-1\)	\(0\)	\(-5\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{2}+(-1+\beta )q^{3}+(1+\beta )q^{4}-3q^{6}+\cdots\)
275.2.a.f	$275$	$2$	275.a	1.a	$1$	$2$	$2$	$2.196$	\(\Q(\sqrt{13}) \)	None	✓	$1$	$0$	\(1\)	\(1\)	\(0\)	\(5\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+(1-\beta )q^{3}+(1+\beta )q^{4}-3q^{6}+\cdots\)
275.2.a.g	$275$	$2$	275.a	1.a	$1$	$2$	$2$	$2.196$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$0$	\(1\)	\(3\)	\(0\)	\(1\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+(1+\beta )q^{3}+(-1+\beta )q^{4}+(1+\cdots)q^{6}+\cdots\)
275.2.a.h	$275$	$2$	275.a	1.a	$1$	$4$	$4$	$2.196$	\(\Q(\sqrt{3}, \sqrt{11})\)	None	✓	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(\beta _{1}-\beta _{2})q^{3}+(2+\beta _{3})q^{4}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(275))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(275)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(55))\)\(^{\oplus 2}\)