Space of modular forms of level 476, weight 4, and trivial character

• Label: 476.4.a
• Level: $476$
• Weight: $4$
• Character orbit: 476.a
• Rep. character: $\chi_{476}(1,\cdot)$
• Character field: $\Q$
• Dimension: $24$
• Newform subspaces: $4$
• Sturm bound: $288$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 476 = 2^{2} \cdot 7 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	476.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(288\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(476))\).

	Total	New	Old
Modular forms	222	24	198
Cusp forms	210	24	186
Eisenstein series	12	0	12

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

\(2\)	\(7\)	\(17\)	Fricke	Dim
\(-\)	\(+\)	\(+\)	\(-\)	\(6\)
\(-\)	\(+\)	\(-\)	\(+\)	\(6\)
\(-\)	\(-\)	\(+\)	\(+\)	\(6\)
\(-\)	\(-\)	\(-\)	\(-\)	\(6\)
Plus space			\(+\)	\(12\)
Minus space			\(-\)	\(12\)

Trace form

\( 24 q + 8 q^{3} - 16 q^{5} + 152 q^{9} + O(q^{10}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(476))\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		2	7	17
476.4.a.a	$476$	$4$	476.a	1.a	$1$	$6$	$6$	$28.085$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	✓	✓	✓	476.4.a.a	$1$	$1$	\(0\)	\(-4\)	\(-28\)	\(42\)		$-$	$-$	$-$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{2})q^{3}+(-5+\beta _{3})q^{5}+7q^{7}+\cdots\)
476.4.a.b	$476$	$4$	476.a	1.a	$1$	$6$	$6$	$28.085$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	✓	✓	✓	476.4.a.b	$1$	$0$	\(0\)	\(2\)	\(-14\)	\(-42\)		$-$	$+$	$-$	$+$	$2^{4}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(-2-\beta _{4})q^{5}-7q^{7}+(-4+\cdots)q^{9}+\cdots\)
476.4.a.c	$476$	$4$	476.a	1.a	$1$	$6$	$6$	$28.085$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	✓	✓	✓	476.4.a.c	$1$	$1$	\(0\)	\(2\)	\(10\)	\(-42\)		$-$	$+$	$+$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(2-\beta _{1}+\beta _{2})q^{5}-7q^{7}+\cdots\)
476.4.a.d	$476$	$4$	476.a	1.a	$1$	$6$	$6$	$28.085$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	✓	✓	✓	476.4.a.d	$1$	$0$	\(0\)	\(8\)	\(16\)	\(42\)		$-$	$-$	$+$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q+(1+\beta _{1})q^{3}+(3+\beta _{3})q^{5}+7q^{7}+(15+\cdots)q^{9}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(476))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(476)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(17))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(28))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(34))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(68))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(119))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(238))\)\(^{\oplus 2}\)