Space of modular forms of level 756, weight 2, and trivial character

• Label: 756.2.a
• Level: $756$
• Weight: $2$
• Character orbit: 756.a
• Rep. character: $\chi_{756}(1,\cdot)$
• Character field: $\Q$
• Dimension: $8$
• Newform subspaces: $7$
• Sturm bound: $288$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 756 = 2^{2} \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	756.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 7 \)
Sturm bound:			\(288\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(5\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(756))\).

	Total	New	Old
Modular forms	162	8	154
Cusp forms	127	8	119
Eisenstein series	35	0	35

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

\(2\)	\(3\)	\(7\)	Fricke	Dim
\(-\)	\(+\)	\(+\)	$-$	\(3\)
\(-\)	\(+\)	\(-\)	$+$	\(1\)
\(-\)	\(-\)	\(+\)	$+$	\(1\)
\(-\)	\(-\)	\(-\)	$-$	\(3\)
Plus space			\(+\)	\(2\)
Minus space			\(-\)	\(6\)

Trace form

\( 8 q + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(756))\) 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	7
756.2.a.a	$756$	$2$	756.a	1.a	$1$	$1$	$1$	$6.037$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(-3\)	\(1\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{5}+q^{7}-3q^{11}+2q^{13}+6q^{17}+\cdots\)
756.2.a.b	$756$	$2$	756.a	1.a	$1$	$1$	$1$	$6.037$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(-3\)	\(1\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{5}+q^{7}+2q^{13}-3q^{17}-4q^{19}+\cdots\)
756.2.a.c	$756$	$2$	756.a	1.a	$1$	$1$	$1$	$6.037$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(-1\)	\(-1\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}-q^{7}-2q^{11}-5q^{17}+2q^{19}+\cdots\)
756.2.a.d	$756$	$2$	756.a	1.a	$1$	$1$	$1$	$6.037$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(1\)	\(-1\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}-q^{7}+2q^{11}+5q^{17}+2q^{19}+\cdots\)
756.2.a.e	$756$	$2$	756.a	1.a	$1$	$1$	$1$	$6.037$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(3\)	\(1\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{5}+q^{7}+2q^{13}+3q^{17}-4q^{19}+\cdots\)
756.2.a.f	$756$	$2$	756.a	1.a	$1$	$1$	$1$	$6.037$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(3\)	\(1\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{5}+q^{7}+3q^{11}+2q^{13}-6q^{17}+\cdots\)
756.2.a.g	$756$	$2$	756.a	1.a	$1$	$2$	$2$	$6.037$	\(\Q(\sqrt{13}) \)	None	✓	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-2\)		$-$	$+$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q-\beta q^{5}-q^{7}+\beta q^{11}+6q^{13}-2\beta q^{17}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(756))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(756)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(27))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(28))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(42))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(54))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(63))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(84))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(108))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(126))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(189))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(252))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(378))\)\(^{\oplus 2}\)