Space of modular forms of level 473, weight 2, and trivial character

• Label: 473.2.a
• Level: $473$
• Weight: $2$
• Character orbit: 473.a
• Rep. character: $\chi_{473}(1,\cdot)$
• Character field: $\Q$
• Dimension: $35$
• Newform subspaces: $7$
• Sturm bound: $88$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 473 = 11 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	473.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 7 \)
Sturm bound:			\(88\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(2\), \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(473))\).

	Total	New	Old
Modular forms	46	35	11
Cusp forms	43	35	8
Eisenstein series	3	0	3

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

\(11\)	\(43\)	Fricke		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(9\)	\(8\)	\(1\)		\(9\)	\(8\)	\(1\)		\(0\)	\(0\)	\(0\)
\(+\)	\(-\)	\(-\)		\(14\)	\(11\)	\(3\)		\(13\)	\(11\)	\(2\)		\(1\)	\(0\)	\(1\)
\(-\)	\(+\)	\(-\)		\(12\)	\(9\)	\(3\)		\(11\)	\(9\)	\(2\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(+\)		\(11\)	\(7\)	\(4\)		\(10\)	\(7\)	\(3\)		\(1\)	\(0\)	\(1\)
Plus space		\(+\)		\(20\)	\(15\)	\(5\)		\(19\)	\(15\)	\(4\)		\(1\)	\(0\)	\(1\)
Minus space		\(-\)		\(26\)	\(20\)	\(6\)		\(24\)	\(20\)	\(4\)		\(2\)	\(0\)	\(2\)

Trace form

35 * q + q^2 + 2 * q^3 + 31 * q^4 - 4 * q^5 - 8 * q^6 + 8 * q^7 - 15 * q^8 + 41 * q^9 - 14 * q^10 - 3 * q^11 - 12 * q^12 - 2 * q^13 - 16 * q^14 - 22 * q^15 + 35 * q^16 - 14 * q^17 - 19 * q^18 + 2 * q^20 + 20 * q^21 + 3 * q^22 - 10 * q^23 - 8 * q^24 + 31 * q^25 - 2 * q^26 - 10 * q^27 - 8 * q^28 + 10 * q^29 + 24 * q^30 - 2 * q^31 - 39 * q^32 + 2 * q^33 - 38 * q^34 + 12 * q^35 + 39 * q^36 - 4 * q^37 - 4 * q^38 - 8 * q^39 - 50 * q^40 + 18 * q^41 + 32 * q^42 + q^43 - 3 * q^44 - 2 * q^45 + 12 * q^46 - 36 * q^47 - 24 * q^48 + 79 * q^49 + 35 * q^50 + 18 * q^52 - 22 * q^53 - 16 * q^54 - 8 * q^56 - 20 * q^57 - 34 * q^58 - 6 * q^59 - 4 * q^60 - 10 * q^61 + 52 * q^62 - 68 * q^63 - 5 * q^64 - 56 * q^65 - 16 * q^66 - 6 * q^67 - 14 * q^68 - 26 * q^69 + 12 * q^70 + 18 * q^71 - 43 * q^72 + 22 * q^73 + 54 * q^74 + 12 * q^75 - 8 * q^76 + 4 * q^77 - 12 * q^78 + 40 * q^79 + 10 * q^80 + 59 * q^81 - 58 * q^82 - 8 * q^83 + 48 * q^84 - 36 * q^85 - 7 * q^86 - 8 * q^87 - 9 * q^88 - 8 * q^89 + 2 * q^90 + 56 * q^91 + 16 * q^92 + 14 * q^93 - 8 * q^94 - 16 * q^95 + 56 * q^96 - 8 * q^97 - 7 * q^98 - 9 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(473))\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		11	43
473.2.a.a	$473$	$2$	473.a	1.a	$1$	$1$	$1$	$3.777$	\(\Q\)	None	✓	✓			473.2.a.a	$1$	$1$	\(-2\)	\(1\)	\(-1\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+q^{3}+2q^{4}-q^{5}-2q^{6}-2q^{9}+\cdots\)
473.2.a.b	$473$	$2$	473.a	1.a	$1$	$2$	$2$	$3.777$	\(\Q(\sqrt{5}) \)	None	✓	✓			473.2.a.b	$1$	$1$	\(1\)	\(-4\)	\(2\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}-2q^{3}+(-1+\beta )q^{4}+(2-2\beta )q^{5}+\cdots\)
473.2.a.c	$473$	$2$	473.a	1.a	$1$	$2$	$2$	$3.777$	\(\Q(\sqrt{5}) \)	None	✓	✓			473.2.a.c	$1$	$1$	\(1\)	\(-2\)	\(2\)	\(-4\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}-2\beta q^{3}+(-1+\beta )q^{4}+2\beta q^{5}+\cdots\)
473.2.a.d	$473$	$2$	473.a	1.a	$1$	$5$	$5$	$3.777$	5.5.173513.1	None	✓	✓	✓		473.2.a.d	$1$	$1$	\(-3\)	\(-1\)	\(-4\)	\(-9\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{3})q^{2}+(-1+\beta _{1}-\beta _{4})q^{3}+\cdots\)
473.2.a.e	$473$	$2$	473.a	1.a	$1$	$5$	$5$	$3.777$	5.5.38569.1	None	✓	✓	✓		473.2.a.e	$1$	$1$	\(1\)	\(-3\)	\(-6\)	\(-15\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(\beta _{1}+\beta _{3})q^{2}+(-1-\beta _{2}-\beta _{4})q^{3}+\cdots\)
473.2.a.f	$473$	$2$	473.a	1.a	$1$	$9$	$9$	$3.777$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓	✓	✓	✓	473.2.a.f	$1$	$0$	\(4\)	\(5\)	\(0\)	\(19\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(1+\beta _{4})q^{3}+(1+\beta _{2}-\beta _{3}+\cdots)q^{4}+\cdots\)
473.2.a.g	$473$	$2$	473.a	1.a	$1$	$11$	$11$	$3.777$	\(\mathbb{Q}[x]/(x^{11} - \cdots)\)	None	✓	✓	✓	✓	473.2.a.g	$1$	$0$	\(-1\)	\(6\)	\(3\)	\(17\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(1-\beta _{8})q^{3}+(1+\beta _{2})q^{4}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(473))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(473)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(43))\)\(^{\oplus 2}\)