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		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(14\)	\(1\)	\(13\)		\(11\)	\(1\)	\(10\)		\(3\)	\(0\)	\(3\)
\(+\)	\(+\)	\(-\)	\(-\)		\(20\)	\(2\)	\(18\)		\(16\)	\(2\)	\(14\)		\(4\)	\(0\)	\(4\)
\(+\)	\(-\)	\(+\)	\(-\)		\(18\)	\(2\)	\(16\)		\(14\)	\(2\)	\(12\)		\(4\)	\(0\)	\(4\)
\(+\)	\(-\)	\(-\)	\(+\)		\(20\)	\(1\)	\(19\)		\(16\)	\(1\)	\(15\)		\(4\)	\(0\)	\(4\)
\(-\)	\(+\)	\(+\)	\(-\)		\(22\)	\(2\)	\(20\)		\(18\)	\(2\)	\(16\)		\(4\)	\(0\)	\(4\)
\(-\)	\(+\)	\(-\)	\(+\)		\(16\)	\(1\)	\(15\)		\(12\)	\(1\)	\(11\)		\(4\)	\(0\)	\(4\)
\(-\)	\(-\)	\(+\)	\(+\)		\(18\)	\(1\)	\(17\)		\(14\)	\(1\)	\(13\)		\(4\)	\(0\)	\(4\)
\(-\)	\(-\)	\(-\)	\(-\)		\(16\)	\(2\)	\(14\)		\(12\)	\(2\)	\(10\)		\(4\)	\(0\)	\(4\)
Plus space			\(+\)		\(68\)	\(4\)	\(64\)		\(53\)	\(4\)	\(49\)		\(15\)	\(0\)	\(15\)
Minus space			\(-\)		\(76\)	\(8\)	\(68\)		\(60\)	\(8\)	\(52\)		\(16\)	\(0\)	\(16\)

Trace form

12 * q - 8 * q^5 + 12 * q^9 - 8 * q^13 + 24 * q^17 + 36 * q^25 - 8 * q^29 - 8 * q^37 + 24 * q^41 - 8 * q^45 + 12 * q^49 - 8 * q^53 - 8 * q^61 + 48 * q^65 + 24 * q^73 + 12 * q^81 - 32 * q^85 - 8 * q^89 - 16 * q^93 - 8 * q^97

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

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

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