Space of modular forms of level 552, weight 6, and trivial character

• Label: 552.6.a
• Level: $552$
• Weight: $6$
• Character orbit: 552.a
• Rep. character: $\chi_{552}(1,\cdot)$
• Character field: $\Q$
• Dimension: $54$
• Newform subspaces: $9$
• Sturm bound: $576$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 552 = 2^{3} \cdot 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	552.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 9 \)
Sturm bound:			\(576\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(552))\).

	Total	New	Old
Modular forms	488	54	434
Cusp forms	472	54	418
Eisenstein series	16	0	16

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

\(2\)	\(3\)	\(23\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	$+$	\(6\)
\(+\)	\(+\)	\(-\)	$-$	\(7\)
\(+\)	\(-\)	\(+\)	$-$	\(8\)
\(+\)	\(-\)	\(-\)	$+$	\(5\)
\(-\)	\(+\)	\(+\)	$-$	\(6\)
\(-\)	\(+\)	\(-\)	$+$	\(7\)
\(-\)	\(-\)	\(+\)	$+$	\(7\)
\(-\)	\(-\)	\(-\)	$-$	\(8\)
Plus space			\(+\)	\(25\)
Minus space			\(-\)	\(29\)

Trace form

\( 54 q + 18 q^{3} - 196 q^{5} - 48 q^{7} + 4374 q^{9} + O(q^{10}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(552))\) 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	3	23
552.6.a.a	$552$	$6$	552.a	1.a	$1$	$1$	$1$	$88.532$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-9\)	\(-82\)	\(-64\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-9q^{3}-82q^{5}-2^{6}q^{7}+3^{4}q^{9}-412q^{11}+\cdots\)
552.6.a.b	$552$	$6$	552.a	1.a	$1$	$5$	$5$	$88.532$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(45\)	\(16\)	\(-134\)		$+$	$-$	$-$	$+$	$2^{4}\cdot 3$	$\mathrm{SU}(2)$	\(q+9q^{3}+(3-\beta _{4})q^{5}+(-3^{3}-2\beta _{1}+\cdots)q^{7}+\cdots\)
552.6.a.c	$552$	$6$	552.a	1.a	$1$	$6$	$6$	$88.532$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(-54\)	\(-90\)	\(50\)		$-$	$+$	$+$	$-$	$2^{6}$	$\mathrm{SU}(2)$	\(q-9q^{3}+(-15+\beta _{5})q^{5}+(8-\beta _{1}+\beta _{3}+\cdots)q^{7}+\cdots\)
552.6.a.d	$552$	$6$	552.a	1.a	$1$	$6$	$6$	$88.532$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(-54\)	\(-70\)	\(144\)		$+$	$+$	$+$	$+$	$2^{7}\cdot 3^{3}\cdot 5$	$\mathrm{SU}(2)$	\(q-9q^{3}+(-12+\beta _{1})q^{5}+(24+\beta _{2}+\cdots)q^{7}+\cdots\)
552.6.a.e	$552$	$6$	552.a	1.a	$1$	$6$	$6$	$88.532$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(-54\)	\(42\)	\(-82\)		$-$	$+$	$-$	$+$	$2^{11}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q-9q^{3}+(7+\beta _{2})q^{5}+(-14+\beta _{1}+\beta _{2}+\cdots)q^{7}+\cdots\)
552.6.a.f	$552$	$6$	552.a	1.a	$1$	$7$	$7$	$88.532$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(-63\)	\(80\)	\(46\)		$+$	$+$	$-$	$-$	$2^{14}\cdot 3$	$\mathrm{SU}(2)$	\(q-9q^{3}+(11+\beta _{2})q^{5}+(7+\beta _{1}-\beta _{2}+\cdots)q^{7}+\cdots\)
552.6.a.g	$552$	$6$	552.a	1.a	$1$	$7$	$7$	$88.532$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(63\)	\(-104\)	\(-182\)		$-$	$-$	$+$	$+$	$2^{13}$	$\mathrm{SU}(2)$	\(q+9q^{3}+(-15-\beta _{1})q^{5}+(-26+\beta _{2}+\cdots)q^{7}+\cdots\)
552.6.a.h	$552$	$6$	552.a	1.a	$1$	$8$	$8$	$88.532$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(72\)	\(-4\)	\(210\)		$-$	$-$	$-$	$-$	$2^{13}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+9q^{3}+(-1+\beta _{1})q^{5}+(26+\beta _{3})q^{7}+\cdots\)
552.6.a.i	$552$	$6$	552.a	1.a	$1$	$8$	$8$	$88.532$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(72\)	\(16\)	\(-36\)		$+$	$-$	$+$	$-$	$2^{18}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+9q^{3}+(2+\beta _{1})q^{5}+(-5-\beta _{2})q^{7}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(552))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_0(552)) \cong \) \(S_{6}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 8}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(4))\)\(^{\oplus 8}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(23))\)\(^{\oplus 8}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(46))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(69))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(92))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(138))\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(184))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(276))\)\(^{\oplus 2}\)