Space of modular forms of level 575, weight 2, and trivial character

• Label: 575.2.a
• Level: $575$
• Weight: $2$
• Character orbit: 575.a
• Rep. character: $\chi_{575}(1,\cdot)$
• Character field: $\Q$
• Dimension: $35$
• Newform subspaces: $12$
• Sturm bound: $120$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 575 = 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	575.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 12 \)
Sturm bound:			\(120\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(2\), \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(575))\).

	Total	New	Old
Modular forms	66	35	31
Cusp forms	55	35	20
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.

\(5\)	\(23\)	Fricke		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(12\)	\(4\)	\(8\)		\(10\)	\(4\)	\(6\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(-\)		\(21\)	\(13\)	\(8\)		\(18\)	\(13\)	\(5\)		\(3\)	\(0\)	\(3\)
\(-\)	\(+\)	\(-\)		\(21\)	\(12\)	\(9\)		\(18\)	\(12\)	\(6\)		\(3\)	\(0\)	\(3\)
\(-\)	\(-\)	\(+\)		\(12\)	\(6\)	\(6\)		\(9\)	\(6\)	\(3\)		\(3\)	\(0\)	\(3\)
Plus space		\(+\)		\(24\)	\(10\)	\(14\)		\(19\)	\(10\)	\(9\)		\(5\)	\(0\)	\(5\)
Minus space		\(-\)		\(42\)	\(25\)	\(17\)		\(36\)	\(25\)	\(11\)		\(6\)	\(0\)	\(6\)

Trace form

35 * q + 4 * q^3 + 36 * q^4 + 3 * q^6 + 2 * q^7 - 3 * q^8 + 37 * q^9 - 2 * q^11 + 17 * q^12 + 4 * q^13 + 22 * q^16 - 4 * q^17 - q^18 + 20 * q^19 + 6 * q^21 + 10 * q^22 + 3 * q^23 - 2 * q^24 - 13 * q^26 + 4 * q^27 + 14 * q^28 + 4 * q^29 + 12 * q^31 - 6 * q^32 - 10 * q^33 + 30 * q^34 + 3 * q^36 - 4 * q^37 - 30 * q^38 - 16 * q^39 + 8 * q^41 - 8 * q^42 + 12 * q^43 - 16 * q^44 - 2 * q^46 - 16 * q^47 + 49 * q^48 + 55 * q^49 + 10 * q^51 + 35 * q^52 - 10 * q^53 + 9 * q^54 - 18 * q^56 - 43 * q^58 + 4 * q^59 + 34 * q^61 + 19 * q^62 + 8 * q^63 - 23 * q^64 - 78 * q^66 + 2 * q^67 - 28 * q^68 - 4 * q^69 - 56 * q^71 - 51 * q^72 + 4 * q^73 - 44 * q^74 + 50 * q^76 + 4 * q^77 - 11 * q^78 + 20 * q^79 + 67 * q^81 - 5 * q^82 + 38 * q^83 - 82 * q^84 - 16 * q^86 + 8 * q^87 - 22 * q^88 + 10 * q^89 + 70 * q^91 + 6 * q^92 + 42 * q^93 - 49 * q^94 - 61 * q^96 - 28 * q^98 - 36 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(575))\) 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	23
575.2.a.a	$575$	$2$	575.a	1.a	$1$	$1$	$1$	$4.591$	\(\Q\)	None	✓				115.2.b.a	$1$	$1$	\(-2\)	\(-2\)	\(0\)	\(-1\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}-2q^{3}+2q^{4}+4q^{6}-q^{7}+\cdots\)
575.2.a.b	$575$	$2$	575.a	1.a	$1$	$1$	$1$	$4.591$	\(\Q\)	None	✓				115.2.a.a	$1$	$1$	\(-2\)	\(0\)	\(0\)	\(-1\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+2q^{4}-q^{7}-3q^{9}+2q^{11}+\cdots\)
575.2.a.c	$575$	$2$	575.a	1.a	$1$	$1$	$1$	$4.591$	\(\Q\)	None	✓	✓			575.2.a.c	$1$	$1$	\(-1\)	\(0\)	\(0\)	\(1\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{4}+q^{7}+3q^{8}-3q^{9}-q^{11}+\cdots\)
575.2.a.d	$575$	$2$	575.a	1.a	$1$	$1$	$1$	$4.591$	\(\Q\)	None	✓	✓			575.2.a.c	$1$	$1$	\(1\)	\(0\)	\(0\)	\(-1\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{4}-q^{7}-3q^{8}-3q^{9}-q^{11}+\cdots\)
575.2.a.e	$575$	$2$	575.a	1.a	$1$	$1$	$1$	$4.591$	\(\Q\)	None	✓				115.2.b.a	$1$	$0$	\(2\)	\(2\)	\(0\)	\(1\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+2q^{3}+2q^{4}+4q^{6}+q^{7}+\cdots\)
575.2.a.f	$575$	$2$	575.a	1.a	$1$	$2$	$2$	$4.591$	\(\Q(\sqrt{5}) \)	None	✓		✓		23.2.a.a	$1$	$1$	\(1\)	\(0\)	\(0\)	\(-2\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+(1-2\beta )q^{3}+(-1+\beta )q^{4}+\cdots\)
575.2.a.g	$575$	$2$	575.a	1.a	$1$	$2$	$2$	$4.591$	\(\Q(\sqrt{5}) \)	None	✓				115.2.a.b	$1$	$0$	\(3\)	\(2\)	\(0\)	\(2\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{2}+q^{3}+3\beta q^{4}+(1+\beta )q^{6}+\cdots\)
575.2.a.h	$575$	$2$	575.a	1.a	$1$	$4$	$4$	$4.591$	\(\Q(\sqrt{11 +2 \sqrt{17}})\)	None	✓				115.2.a.c	$1$	$0$	\(-2\)	\(2\)	\(0\)	\(3\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(1+\beta _{2})q^{3}+(1+\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
575.2.a.i	$575$	$2$	575.a	1.a	$1$	$4$	$4$	$4.591$	4.4.5744.1	None	✓		✓		115.2.b.b	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(-6\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(-1+\beta _{1}+\beta _{2})q^{3}+(\beta _{1}+\cdots)q^{4}+\cdots\)
575.2.a.j	$575$	$2$	575.a	1.a	$1$	$4$	$4$	$4.591$	4.4.5744.1	None	✓				115.2.b.b	$1$	$0$	\(0\)	\(2\)	\(0\)	\(6\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(1-\beta _{1}-\beta _{2})q^{3}+(\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
575.2.a.k	$575$	$2$	575.a	1.a	$1$	$7$	$7$	$4.591$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	✓	✓		575.2.a.k	$1$	$0$	\(-1\)	\(0\)	\(0\)	\(3\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+\beta _{5}q^{3}+(2+\beta _{2})q^{4}+(1-\beta _{1}+\cdots)q^{6}+\cdots\)
575.2.a.l	$575$	$2$	575.a	1.a	$1$	$7$	$7$	$4.591$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	✓	✓		575.2.a.k	$1$	$0$	\(1\)	\(0\)	\(0\)	\(-3\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}-\beta _{5}q^{3}+(2+\beta _{2})q^{4}+(1-\beta _{1}+\cdots)q^{6}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(575))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(575)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(23))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(115))\)\(^{\oplus 2}\)