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		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(6\)	\(3\)	\(3\)		\(4\)	\(3\)	\(1\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(-\)		\(12\)	\(5\)	\(7\)		\(9\)	\(5\)	\(4\)		\(3\)	\(0\)	\(3\)
\(-\)	\(+\)	\(-\)		\(10\)	\(6\)	\(4\)		\(7\)	\(6\)	\(1\)		\(3\)	\(0\)	\(3\)
\(-\)	\(-\)	\(+\)		\(8\)	\(2\)	\(6\)		\(5\)	\(2\)	\(3\)		\(3\)	\(0\)	\(3\)
Plus space		\(+\)		\(14\)	\(5\)	\(9\)		\(9\)	\(5\)	\(4\)		\(5\)	\(0\)	\(5\)
Minus space		\(-\)		\(22\)	\(11\)	\(11\)		\(16\)	\(11\)	\(5\)		\(6\)	\(0\)	\(6\)

Trace form

16 * q - q^2 + q^3 + 13 * q^4 - 2 * q^6 + 6 * q^7 - 3 * q^8 + 11 * q^9 - 2 * q^11 + 18 * q^12 + 2 * q^13 - 8 * q^14 + 3 * q^16 - 12 * q^17 - 11 * q^18 + 8 * q^19 - 6 * q^21 + q^22 - 3 * q^23 - 16 * q^24 + 22 * q^26 - 5 * q^27 + 8 * q^28 - 6 * q^29 + q^31 - 7 * q^32 + q^33 - 30 * q^34 - 17 * q^36 + 3 * q^37 + 4 * q^38 + 24 * q^39 - 2 * q^41 - 12 * q^42 + 14 * q^43 - 9 * q^44 - 30 * q^46 + 4 * q^47 - 4 * q^48 + 32 * q^49 - 2 * q^51 - 14 * q^52 - 4 * q^53 - 6 * q^54 - 12 * q^56 + 6 * q^58 - 25 * q^59 - 6 * q^61 + 22 * q^62 + 16 * q^63 - 13 * q^64 + 6 * q^66 + 15 * q^67 - 14 * q^68 - 37 * q^69 - 9 * q^71 - 39 * q^72 - 10 * q^73 + 24 * q^74 - 8 * q^76 + 6 * q^77 - 24 * q^78 + 22 * q^79 - 20 * q^81 - 30 * q^82 + 22 * q^83 - 64 * q^84 - 72 * q^86 - 32 * q^87 - 9 * q^88 - 21 * q^89 + 36 * q^91 + 22 * q^92 + 7 * q^93 + 64 * q^94 + 4 * q^96 + q^97 + 7 * q^98 + 9 * q^99

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	Twist minimal	Largest	Maximal	Minimal twist	Inner twists	Rank*	Traces					A-L signs		Fricke sign	Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		5	11
275.2.a.a	$275$	$2$	275.a	1.a	$1$	$1$	$1$	$2.196$	\(\Q\)	None	✓				55.2.a.a	$1$	$1$	\(-1\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{4}+3q^{8}-3q^{9}-q^{11}-2q^{13}+\cdots\)
275.2.a.b	$275$	$2$	275.a	1.a	$1$	$1$	$1$	$2.196$	\(\Q\)	None	✓				11.2.a.a	$1$	$0$	\(2\)	\(1\)	\(0\)	\(2\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+q^{3}+2q^{4}+2q^{6}+2q^{7}+\cdots\)
275.2.a.c	$275$	$2$	275.a	1.a	$1$	$2$	$2$	$2.196$	\(\Q(\sqrt{2}) \)	None	✓				55.2.a.b	$1$	$0$	\(-2\)	\(0\)	\(0\)	\(4\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{2}-2\beta q^{3}+(1-2\beta )q^{4}+\cdots\)
275.2.a.d	$275$	$2$	275.a	1.a	$1$	$2$	$2$	$2.196$	\(\Q(\sqrt{5}) \)	None	✓	✓	✓	✓	275.2.a.d	$1$	$1$	\(-1\)	\(-3\)	\(0\)	\(-1\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{2}+(-1-\beta )q^{3}+(-1+\beta )q^{4}+\cdots\)
275.2.a.e	$275$	$2$	275.a	1.a	$1$	$2$	$2$	$2.196$	\(\Q(\sqrt{13}) \)	None	✓	✓	✓		275.2.a.e	$1$	$1$	\(-1\)	\(-1\)	\(0\)	\(-5\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{2}+(-1+\beta )q^{3}+(1+\beta )q^{4}-3q^{6}+\cdots\)
275.2.a.f	$275$	$2$	275.a	1.a	$1$	$2$	$2$	$2.196$	\(\Q(\sqrt{13}) \)	None	✓	✓			275.2.a.e	$1$	$0$	\(1\)	\(1\)	\(0\)	\(5\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+(1-\beta )q^{3}+(1+\beta )q^{4}-3q^{6}+\cdots\)
275.2.a.g	$275$	$2$	275.a	1.a	$1$	$2$	$2$	$2.196$	\(\Q(\sqrt{5}) \)	None	✓	✓			275.2.a.d	$1$	$0$	\(1\)	\(3\)	\(0\)	\(1\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+(1+\beta )q^{3}+(-1+\beta )q^{4}+(1+\cdots)q^{6}+\cdots\)
275.2.a.h	$275$	$2$	275.a	1.a	$1$	$4$	$4$	$2.196$	\(\Q(\sqrt{3}, \sqrt{11})\)	None	✓		✓		55.2.b.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(\beta _{1}-\beta _{2})q^{3}+(2+\beta _{3})q^{4}+\cdots\)

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

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