Space of modular forms of level 564, weight 2, and trivial character

• Label: 564.2.a
• Level: $564$
• Weight: $2$
• Character orbit: 564.a
• Rep. character: $\chi_{564}(1,\cdot)$
• Character field: $\Q$
• Dimension: $8$
• Newform subspaces: $4$
• Sturm bound: $192$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 564 = 2^{2} \cdot 3 \cdot 47 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	564.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(192\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(564))\).

	Total	New	Old
Modular forms	102	8	94
Cusp forms	91	8	83
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.

\(2\)	\(3\)	\(47\)	Fricke		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(7\)	\(0\)	\(7\)		\(6\)	\(0\)	\(6\)		\(1\)	\(0\)	\(1\)
\(+\)	\(+\)	\(-\)	\(-\)		\(19\)	\(0\)	\(19\)		\(17\)	\(0\)	\(17\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(+\)	\(-\)		\(9\)	\(0\)	\(9\)		\(7\)	\(0\)	\(7\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(-\)	\(+\)		\(17\)	\(0\)	\(17\)		\(15\)	\(0\)	\(15\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(+\)	\(-\)		\(11\)	\(3\)	\(8\)		\(10\)	\(3\)	\(7\)		\(1\)	\(0\)	\(1\)
\(-\)	\(+\)	\(-\)	\(+\)		\(14\)	\(1\)	\(13\)		\(13\)	\(1\)	\(12\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(+\)	\(+\)		\(9\)	\(1\)	\(8\)		\(8\)	\(1\)	\(7\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(-\)	\(-\)		\(16\)	\(3\)	\(13\)		\(15\)	\(3\)	\(12\)		\(1\)	\(0\)	\(1\)
Plus space			\(+\)		\(47\)	\(2\)	\(45\)		\(42\)	\(2\)	\(40\)		\(5\)	\(0\)	\(5\)
Minus space			\(-\)		\(55\)	\(6\)	\(49\)		\(49\)	\(6\)	\(43\)		\(6\)	\(0\)	\(6\)

Trace form

8 * q + 4 * q^5 + 4 * q^7 + 8 * q^9 + 4 * q^17 - 8 * q^19 + 20 * q^25 + 4 * q^29 - 8 * q^31 + 8 * q^35 + 12 * q^37 - 8 * q^39 + 12 * q^41 + 16 * q^43 + 4 * q^45 - 4 * q^49 + 4 * q^51 + 28 * q^53 + 4 * q^55 + 4 * q^57 + 4 * q^59 + 8 * q^61 + 4 * q^63 + 28 * q^65 - 8 * q^67 - 16 * q^69 - 40 * q^71 - 8 * q^73 - 8 * q^75 - 16 * q^77 - 28 * q^79 + 8 * q^81 - 20 * q^83 - 8 * q^85 + 8 * q^87 - 16 * q^89 - 16 * q^91 - 12 * q^93 - 24 * q^95 + 4 * q^97

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(564))\) 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	3	47
564.2.a.a	$564$	$2$	564.a	1.a	$1$	$1$	$1$	$4.504$	\(\Q\)	None	✓	✓	✓	✓	564.2.a.a	$1$	$1$	\(0\)	\(-1\)	\(-1\)	\(-1\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{5}-q^{7}+q^{9}+3q^{11}-2q^{13}+\cdots\)
564.2.a.b	$564$	$2$	564.a	1.a	$1$	$1$	$1$	$4.504$	\(\Q\)	None	✓	✓	✓	✓	564.2.a.b	$1$	$1$	\(0\)	\(1\)	\(-3\)	\(-1\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-3q^{5}-q^{7}+q^{9}-3q^{11}-4q^{13}+\cdots\)
564.2.a.c	$564$	$2$	564.a	1.a	$1$	$3$	$3$	$4.504$	3.3.2700.1	None	✓	✓	✓	✓	564.2.a.c	$1$	$0$	\(0\)	\(-3\)	\(3\)	\(3\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+(1-\beta _{1})q^{5}+(1+\beta _{2})q^{7}+q^{9}+\cdots\)
564.2.a.d	$564$	$2$	564.a	1.a	$1$	$3$	$3$	$4.504$	3.3.316.1	None	✓	✓	✓	✓	564.2.a.d	$1$	$0$	\(0\)	\(3\)	\(5\)	\(3\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+(2-\beta _{1})q^{5}+(1+\beta _{2})q^{7}+q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(564))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(564)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(47))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(94))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(141))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(188))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(282))\)\(^{\oplus 2}\)