Space of modular forms of level 475, weight 2, and trivial character

• Label: 475.2.a
• Level: $475$
• Weight: $2$
• Character orbit: 475.a
• Rep. character: $\chi_{475}(1,\cdot)$
• Character field: $\Q$
• Dimension: $28$
• Newform subspaces: $10$
• Sturm bound: $100$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 475 = 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	475.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 10 \)
Sturm bound:			\(100\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(475))\).

	Total	New	Old
Modular forms	56	28	28
Cusp forms	45	28	17
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.

\(5\)	\(19\)	Fricke		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(11\)	\(6\)	\(5\)		\(9\)	\(6\)	\(3\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(-\)		\(17\)	\(8\)	\(9\)		\(14\)	\(8\)	\(6\)		\(3\)	\(0\)	\(3\)
\(-\)	\(+\)	\(-\)		\(15\)	\(9\)	\(6\)		\(12\)	\(9\)	\(3\)		\(3\)	\(0\)	\(3\)
\(-\)	\(-\)	\(+\)		\(13\)	\(5\)	\(8\)		\(10\)	\(5\)	\(5\)		\(3\)	\(0\)	\(3\)
Plus space		\(+\)		\(24\)	\(11\)	\(13\)		\(19\)	\(11\)	\(8\)		\(5\)	\(0\)	\(5\)
Minus space		\(-\)		\(32\)	\(17\)	\(15\)		\(26\)	\(17\)	\(9\)		\(6\)	\(0\)	\(6\)

Trace form

28 * q + q^2 - 2 * q^3 + 29 * q^4 - 3 * q^7 + 9 * q^8 + 16 * q^9 - 3 * q^11 - 6 * q^13 + 23 * q^16 - 3 * q^17 + 29 * q^18 - 2 * q^19 - 6 * q^21 + 12 * q^22 + 12 * q^23 + 8 * q^24 + 6 * q^26 - 8 * q^27 - 6 * q^28 - 16 * q^31 + 9 * q^32 - 6 * q^33 - 10 * q^34 - 27 * q^36 - 16 * q^37 + 3 * q^38 - 28 * q^39 + 4 * q^41 - 16 * q^42 + q^43 - 2 * q^44 + 4 * q^46 + 15 * q^47 - 40 * q^48 - 13 * q^49 - 2 * q^51 - 30 * q^52 - 18 * q^53 - 52 * q^54 - 32 * q^56 + 2 * q^57 - 2 * q^58 + 2 * q^59 - 21 * q^61 - 48 * q^62 + 13 * q^63 + 79 * q^64 - 36 * q^66 + 20 * q^67 - 8 * q^68 + 18 * q^71 + 65 * q^72 - 23 * q^73 - 2 * q^74 - 5 * q^76 + 35 * q^77 - 24 * q^79 + 16 * q^81 + 2 * q^82 + 20 * q^83 - 56 * q^84 + 40 * q^86 + 76 * q^87 + 16 * q^88 + 2 * q^89 - 24 * q^91 + 20 * q^92 + 48 * q^93 + 24 * q^94 - 52 * q^96 - 58 * q^97 - 23 * q^98 + 9 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(475))\) 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	19
475.2.a.a	$475$	$2$	475.a	1.a	$1$	$1$	$1$	$3.793$	\(\Q\)	None	✓				95.2.b.a	$1$	$1$	\(-1\)	\(0\)	\(0\)	\(2\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{4}+2q^{7}+3q^{8}-3q^{9}-4q^{11}+\cdots\)
475.2.a.b	$475$	$2$	475.a	1.a	$1$	$1$	$1$	$3.793$	\(\Q\)	None	✓				19.2.a.a	$1$	$0$	\(0\)	\(2\)	\(0\)	\(1\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}-2q^{4}+q^{7}+q^{9}+3q^{11}+\cdots\)
475.2.a.c	$475$	$2$	475.a	1.a	$1$	$1$	$1$	$3.793$	\(\Q\)	None	✓				95.2.b.a	$1$	$1$	\(1\)	\(0\)	\(0\)	\(-2\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{4}-2q^{7}-3q^{8}-3q^{9}-4q^{11}+\cdots\)
475.2.a.d	$475$	$2$	475.a	1.a	$1$	$3$	$3$	$3.793$	\(\Q(\zeta_{14})^+\)	None	✓	✓			475.2.a.d	$1$	$1$	\(-4\)	\(-2\)	\(0\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{1})q^{2}+(-\beta _{1}+\beta _{2})q^{3}+(1+\cdots)q^{4}+\cdots\)
475.2.a.e	$475$	$2$	475.a	1.a	$1$	$3$	$3$	$3.793$	3.3.169.1	None	✓	✓	✓		475.2.a.e	$1$	$1$	\(-2\)	\(-2\)	\(0\)	\(-4\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+(-1-\beta _{2})q^{3}+(2+\cdots)q^{4}+\cdots\)
475.2.a.f	$475$	$2$	475.a	1.a	$1$	$3$	$3$	$3.793$	3.3.148.1	None	✓				95.2.a.a	$1$	$1$	\(-1\)	\(-2\)	\(0\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(-1+\beta _{1}+\beta _{2})q^{3}+(\beta _{1}+\cdots)q^{4}+\cdots\)
475.2.a.g	$475$	$2$	475.a	1.a	$1$	$3$	$3$	$3.793$	3.3.169.1	None	✓	✓			475.2.a.e	$1$	$0$	\(2\)	\(2\)	\(0\)	\(4\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+(1+\beta _{2})q^{3}+(2-\beta _{1}+\cdots)q^{4}+\cdots\)
475.2.a.h	$475$	$2$	475.a	1.a	$1$	$3$	$3$	$3.793$	\(\Q(\zeta_{14})^+\)	None	✓	✓			475.2.a.d	$1$	$0$	\(4\)	\(2\)	\(0\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta _{1})q^{2}+(\beta _{1}-\beta _{2})q^{3}+(1+2\beta _{1}+\cdots)q^{4}+\cdots\)
475.2.a.i	$475$	$2$	475.a	1.a	$1$	$4$	$4$	$3.793$	4.4.11344.1	None	✓		✓		95.2.a.b	$1$	$0$	\(2\)	\(-2\)	\(0\)	\(-4\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{2}q^{2}+\beta _{3}q^{3}+(1+\beta _{1}-\beta _{2})q^{4}+\cdots\)
475.2.a.j	$475$	$2$	475.a	1.a	$1$	$6$	$6$	$3.793$	6.6.66064384.1	None	✓		✓		95.2.b.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{4}q^{2}+(-\beta _{1}-\beta _{5})q^{3}+(1-\beta _{3}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(475))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(475)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(19))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(95))\)\(^{\oplus 2}\)