Space of modular forms of level 484, weight 6, and trivial character

• Label: 484.6.a
• Level: $484$
• Weight: $6$
• Character orbit: 484.a
• Rep. character: $\chi_{484}(1,\cdot)$
• Character field: $\Q$
• Dimension: $46$
• Newform subspaces: $10$
• Sturm bound: $396$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 484 = 2^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	484.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 10 \)
Sturm bound:			\(396\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(3\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(484))\).

	Total	New	Old
Modular forms	348	46	302
Cusp forms	312	46	266
Eisenstein series	36	0	36

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

\(2\)	\(11\)	Fricke	Dim
\(-\)	\(+\)	$-$	\(24\)
\(-\)	\(-\)	$+$	\(22\)
Plus space		\(+\)	\(22\)
Minus space		\(-\)	\(24\)

Trace form

\( 46 q + 12 q^{3} + 18 q^{5} - 130 q^{7} + 3850 q^{9} + O(q^{10}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(484))\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	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	11
484.6.a.a	$484$	$6$	484.a	1.a	$1$	$1$	$1$	$77.626$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-12\)	\(54\)	\(88\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-12q^{3}+54q^{5}+88q^{7}-99q^{9}+\cdots\)
484.6.a.b	$484$	$6$	484.a	1.a	$1$	$1$	$1$	$77.626$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(7\)	\(-79\)	\(50\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+7q^{3}-79q^{5}+50q^{7}-194q^{9}+\cdots\)
484.6.a.c	$484$	$6$	484.a	1.a	$1$	$2$	$2$	$77.626$	\(\Q(\sqrt{2766}) \)	None	✓	$2$	$0$	\(0\)	\(-30\)	\(18\)	\(0\)		$-$	$+$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q-15q^{3}+9q^{5}-\beta q^{7}-18q^{9}-\beta q^{13}+\cdots\)
484.6.a.d	$484$	$6$	484.a	1.a	$1$	$2$	$2$	$77.626$	\(\Q(\sqrt{31}) \)	None	✓	$1$	$1$	\(0\)	\(-6\)	\(-22\)	\(-268\)		$-$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(-3+\beta )q^{3}+(-11+2\beta )q^{5}+(-134+\cdots)q^{7}+\cdots\)
484.6.a.e	$484$	$6$	484.a	1.a	$1$	$2$	$2$	$77.626$	\(\Q(\sqrt{33}) \)	\(\Q(\sqrt{-11}) \)	✓	$2$	$0$	\(0\)	\(31\)	\(-57\)	\(0\)		$-$	$+$	$-$	$1$	$N(\mathrm{U}(1))$	\(q+(2^{4}-\beta )q^{3}+(-14-29\beta )q^{5}+(21+\cdots)q^{9}+\cdots\)
484.6.a.f	$484$	$6$	484.a	1.a	$1$	$4$	$4$	$77.626$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(-8\)	\(-68\)	\(-40\)		$-$	$-$	$+$	$2^{7}$	$\mathrm{SU}(2)$	\(q+(-2+\beta _{1})q^{3}+(-17-\beta _{2})q^{5}+(-10+\cdots)q^{7}+\cdots\)
484.6.a.g	$484$	$6$	484.a	1.a	$1$	$4$	$4$	$77.626$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(-8\)	\(-68\)	\(40\)		$-$	$-$	$+$	$2^{7}$	$\mathrm{SU}(2)$	\(q+(-2+\beta _{1})q^{3}+(-17-\beta _{2})q^{5}+(10+\cdots)q^{7}+\cdots\)
484.6.a.h	$484$	$6$	484.a	1.a	$1$	$10$	$10$	$77.626$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(9\)	\(69\)	\(-21\)		$-$	$+$	$-$	$2^{8}\cdot 11^{4}$	$\mathrm{SU}(2)$	\(q+(1+\beta _{2})q^{3}+(7-\beta _{2}+\beta _{4})q^{5}+(-1+\cdots)q^{7}+\cdots\)
484.6.a.i	$484$	$6$	484.a	1.a	$1$	$10$	$10$	$77.626$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(9\)	\(69\)	\(21\)		$-$	$-$	$+$	$2^{8}\cdot 11^{4}$	$\mathrm{SU}(2)$	\(q+(1+\beta _{2})q^{3}+(7-\beta _{2}+\beta _{4})q^{5}+(1+\cdots)q^{7}+\cdots\)
484.6.a.j	$484$	$6$	484.a	1.a	$1$	$10$	$10$	$77.626$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓	$2$	$0$	\(0\)	\(20\)	\(102\)	\(0\)		$-$	$+$	$-$	$2^{12}\cdot 3^{4}\cdot 11^{4}$	$\mathrm{SU}(2)$	\(q+(2+\beta _{2})q^{3}+(10+\beta _{2}-\beta _{3})q^{5}+\beta _{1}q^{7}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(484))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_0(484)) \cong \) \(S_{6}^{\mathrm{new}}(\Gamma_0(4))\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(22))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(44))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(121))\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(242))\)\(^{\oplus 2}\)