Space of modular forms of level 672, weight 2, and trivial character

• Label: 672.2.a
• Level: $672$
• Weight: $2$
• Character orbit: 672.a
• Rep. character: $\chi_{672}(1,\cdot)$
• Character field: $\Q$
• Dimension: $12$
• Newform subspaces: $10$
• Sturm bound: $256$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 672 = 2^{5} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	672.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 10 \)
Sturm bound:			\(256\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(5\), \(11\), \(19\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(672))\).

	Total	New	Old
Modular forms	144	12	132
Cusp forms	113	12	101
Eisenstein series	31	0	31

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
\(+\)	\(+\)	\(+\)	$+$	\(1\)
\(+\)	\(+\)	\(-\)	$-$	\(2\)
\(+\)	\(-\)	\(+\)	$-$	\(2\)
\(+\)	\(-\)	\(-\)	$+$	\(1\)
\(-\)	\(+\)	\(+\)	$-$	\(2\)
\(-\)	\(+\)	\(-\)	$+$	\(1\)
\(-\)	\(-\)	\(+\)	$+$	\(1\)
\(-\)	\(-\)	\(-\)	$-$	\(2\)
Plus space			\(+\)	\(4\)
Minus space			\(-\)	\(8\)

Trace form

\( 12 q - 8 q^{5} + 12 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(672))\) 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
672.2.a.a	$672$	$2$	672.a	1.a	$1$	$1$	$1$	$5.366$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-1\)	\(-4\)	\(-1\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-4q^{5}-q^{7}+q^{9}+2q^{11}-2q^{13}+\cdots\)
672.2.a.b	$672$	$2$	672.a	1.a	$1$	$1$	$1$	$5.366$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-1\)	\(-2\)	\(1\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-2q^{5}+q^{7}+q^{9}+4q^{11}-6q^{13}+\cdots\)
672.2.a.c	$672$	$2$	672.a	1.a	$1$	$1$	$1$	$5.366$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(-1\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{7}+q^{9}-2q^{11}-2q^{13}+\cdots\)
672.2.a.d	$672$	$2$	672.a	1.a	$1$	$1$	$1$	$5.366$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-1\)	\(2\)	\(-1\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+2q^{5}-q^{7}+q^{9}+2q^{13}-2q^{15}+\cdots\)
672.2.a.e	$672$	$2$	672.a	1.a	$1$	$1$	$1$	$5.366$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(1\)	\(-4\)	\(1\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-4q^{5}+q^{7}+q^{9}-2q^{11}-2q^{13}+\cdots\)
672.2.a.f	$672$	$2$	672.a	1.a	$1$	$1$	$1$	$5.366$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(1\)	\(-2\)	\(-1\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-2q^{5}-q^{7}+q^{9}-4q^{11}-6q^{13}+\cdots\)
672.2.a.g	$672$	$2$	672.a	1.a	$1$	$1$	$1$	$5.366$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(1\)	\(0\)	\(1\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{7}+q^{9}+2q^{11}-2q^{13}+\cdots\)
672.2.a.h	$672$	$2$	672.a	1.a	$1$	$1$	$1$	$5.366$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(1\)	\(2\)	\(1\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+2q^{5}+q^{7}+q^{9}+2q^{13}+2q^{15}+\cdots\)
672.2.a.i	$672$	$2$	672.a	1.a	$1$	$2$	$2$	$5.366$	\(\Q(\sqrt{3}) \)	None	✓	$1$	$0$	\(0\)	\(-2\)	\(0\)	\(2\)		$+$	$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q-q^{3}+\beta q^{5}+q^{7}+q^{9}+(-2+\beta )q^{11}+\cdots\)
672.2.a.j	$672$	$2$	672.a	1.a	$1$	$2$	$2$	$5.366$	\(\Q(\sqrt{3}) \)	None	✓	$1$	$0$	\(0\)	\(2\)	\(0\)	\(-2\)		$+$	$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+q^{3}+\beta q^{5}-q^{7}+q^{9}+(2-\beta )q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(672))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(672)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(28))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(42))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(48))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(56))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(84))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(96))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(112))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(168))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(224))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(336))\)\(^{\oplus 2}\)