Space of modular forms of level 476, weight 2, and trivial character

• Label: 476.2.a
• Level: $476$
• Weight: $2$
• Character orbit: 476.a
• Rep. character: $\chi_{476}(1,\cdot)$
• Character field: $\Q$
• Dimension: $8$
• Newform subspaces: $4$
• Sturm bound: $144$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 476 = 2^{2} \cdot 7 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	476.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(144\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(476))\).

	Total	New	Old
Modular forms	78	8	70
Cusp forms	67	8	59
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.

\(2\)	\(7\)	\(17\)	Fricke	Dim
\(-\)	\(+\)	\(+\)	\(-\)	\(2\)
\(-\)	\(+\)	\(-\)	\(+\)	\(2\)
\(-\)	\(-\)	\(+\)	\(+\)	\(2\)
\(-\)	\(-\)	\(-\)	\(-\)	\(2\)
Plus space			\(+\)	\(4\)
Minus space			\(-\)	\(4\)

Trace form

8 * q - 4 * q^3 - 4 * q^5 + 4 * q^9 + 4 * q^11 - 4 * q^13 + 8 * q^15 - 4 * q^19 + 8 * q^23 - 12 * q^25 - 16 * q^27 - 4 * q^29 - 4 * q^31 + 4 * q^33 - 4 * q^35 - 8 * q^37 - 4 * q^39 + 4 * q^41 - 12 * q^43 + 16 * q^45 + 16 * q^47 + 8 * q^49 + 4 * q^51 - 4 * q^53 - 4 * q^55 + 4 * q^57 - 16 * q^59 + 4 * q^61 + 8 * q^63 + 24 * q^65 - 4 * q^67 - 4 * q^69 - 8 * q^71 + 4 * q^73 + 24 * q^75 - 4 * q^79 + 8 * q^81 + 12 * q^83 - 12 * q^87 - 32 * q^89 + 4 * q^91 - 12 * q^95 - 28 * q^97 - 16 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(476))\) 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
476.2.a.a	$476$	$2$	476.a	1.a	$1$	$2$	$2$	$3.801$	\(\Q(\sqrt{13}) \)	None	✓	✓	✓	✓	476.2.a.a	$1$	$1$	\(0\)	\(-3\)	\(-3\)	\(2\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta )q^{3}+(-2+\beta )q^{5}+q^{7}+\cdots\)
476.2.a.b	$476$	$2$	476.a	1.a	$1$	$2$	$2$	$3.801$	\(\Q(\sqrt{5}) \)	None	✓	✓	✓	✓	476.2.a.b	$1$	$1$	\(0\)	\(-1\)	\(-1\)	\(-2\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{3}+(-1+\beta )q^{5}-q^{7}+(-2+\beta )q^{9}+\cdots\)
476.2.a.c	$476$	$2$	476.a	1.a	$1$	$2$	$2$	$3.801$	\(\Q(\sqrt{13}) \)	None	✓	✓	✓	✓	476.2.a.c	$1$	$0$	\(0\)	\(-1\)	\(1\)	\(-2\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{3}+(1-\beta )q^{5}-q^{7}+\beta q^{9}+4q^{11}+\cdots\)
476.2.a.d	$476$	$2$	476.a	1.a	$1$	$2$	$2$	$3.801$	\(\Q(\sqrt{13}) \)	None	✓	✓	✓	✓	476.2.a.d	$1$	$0$	\(0\)	\(1\)	\(-1\)	\(2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+(-1+\beta )q^{5}+q^{7}+\beta q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(476))\) into lower level spaces

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