Space of modular forms of level 312, weight 2, and trivial character

• Label: 312.2.a
• Level: $312$
• Weight: $2$
• Character orbit: 312.a
• Rep. character: $\chi_{312}(1,\cdot)$
• Character field: $\Q$
• Dimension: $6$
• Newform subspaces: $6$
• Sturm bound: $112$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 312 = 2^{3} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	312.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 6 \)
Sturm bound:			\(112\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(312))\).

	Total	New	Old
Modular forms	64	6	58
Cusp forms	49	6	43
Eisenstein series	15	0	15

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

\(2\)	\(3\)	\(13\)	Fricke		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(4\)	\(1\)	\(3\)		\(3\)	\(1\)	\(2\)		\(1\)	\(0\)	\(1\)
\(+\)	\(+\)	\(-\)	\(-\)		\(12\)	\(1\)	\(11\)		\(10\)	\(1\)	\(9\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(+\)	\(-\)		\(9\)	\(1\)	\(8\)		\(7\)	\(1\)	\(6\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(-\)	\(+\)		\(7\)	\(0\)	\(7\)		\(5\)	\(0\)	\(5\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(+\)	\(-\)		\(8\)	\(1\)	\(7\)		\(6\)	\(1\)	\(5\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(-\)	\(+\)		\(8\)	\(0\)	\(8\)		\(6\)	\(0\)	\(6\)		\(2\)	\(0\)	\(2\)
\(-\)	\(-\)	\(+\)	\(+\)		\(11\)	\(1\)	\(10\)		\(9\)	\(1\)	\(8\)		\(2\)	\(0\)	\(2\)
\(-\)	\(-\)	\(-\)	\(-\)		\(5\)	\(1\)	\(4\)		\(3\)	\(1\)	\(2\)		\(2\)	\(0\)	\(2\)
Plus space			\(+\)		\(30\)	\(2\)	\(28\)		\(23\)	\(2\)	\(21\)		\(7\)	\(0\)	\(7\)
Minus space			\(-\)		\(34\)	\(4\)	\(30\)		\(26\)	\(4\)	\(22\)		\(8\)	\(0\)	\(8\)

Trace form

6 * q - 4 * q^7 + 6 * q^9 - 2 * q^13 - 4 * q^15 - 4 * q^17 + 12 * q^19 - 4 * q^21 + 24 * q^23 + 10 * q^25 - 4 * q^29 - 4 * q^31 + 8 * q^33 + 8 * q^35 - 20 * q^37 - 8 * q^41 - 16 * q^43 - 16 * q^47 + 6 * q^49 + 4 * q^53 - 12 * q^57 - 24 * q^59 + 4 * q^61 - 4 * q^63 + 4 * q^67 - 8 * q^69 - 8 * q^71 - 12 * q^73 + 16 * q^77 - 24 * q^79 + 6 * q^81 + 24 * q^83 + 32 * q^85 - 8 * q^87 - 32 * q^89 + 12 * q^91 + 12 * q^93 - 8 * q^95 - 12 * q^97

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(312))\) 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	13
312.2.a.a	$312$	$2$	312.a	1.a	$1$	$1$	$1$	$2.491$	\(\Q\)	None	✓	✓	✓	✓	312.2.a.a	$1$	$0$	\(0\)	\(-1\)	\(-2\)	\(4\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-2q^{5}+4q^{7}+q^{9}+q^{13}+2q^{15}+\cdots\)
312.2.a.b	$312$	$2$	312.a	1.a	$1$	$1$	$1$	$2.491$	\(\Q\)	None	✓	✓	✓	✓	312.2.a.b	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(-4\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-4q^{7}+q^{9}-2q^{11}-q^{13}+\cdots\)
312.2.a.c	$312$	$2$	312.a	1.a	$1$	$1$	$1$	$2.491$	\(\Q\)	None	✓	✓	✓	✓	312.2.a.c	$1$	$0$	\(0\)	\(-1\)	\(4\)	\(0\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+4q^{5}+q^{9}-2q^{11}-q^{13}+\cdots\)
312.2.a.d	$312$	$2$	312.a	1.a	$1$	$1$	$1$	$2.491$	\(\Q\)	None	✓	✓	✓	✓	312.2.a.d	$1$	$1$	\(0\)	\(1\)	\(-4\)	\(-4\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-4q^{5}-4q^{7}+q^{9}-2q^{11}+\cdots\)
312.2.a.e	$312$	$2$	312.a	1.a	$1$	$1$	$1$	$2.491$	\(\Q\)	None	✓	✓	✓	✓	312.2.a.e	$1$	$0$	\(0\)	\(1\)	\(0\)	\(0\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{9}+6q^{11}-q^{13}+2q^{17}+\cdots\)
312.2.a.f	$312$	$2$	312.a	1.a	$1$	$1$	$1$	$2.491$	\(\Q\)	None	✓	✓	✓	✓	312.2.a.f	$1$	$0$	\(0\)	\(1\)	\(2\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+2q^{5}+q^{9}+q^{13}+2q^{15}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(312))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(312)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(26))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(39))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(52))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(78))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(104))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(156))\)\(^{\oplus 2}\)