Space of modular forms of level 312, weight 4, and Character 312.c

• Label: 312.4.c
• Level: $312$
• Weight: $4$
• Character orbit: 312.c
• Rep. character: $\chi_{312}(25,\cdot)$
• Character field: $\Q$
• Dimension: $22$
• Newform subspaces: $2$
• Sturm bound: $224$
• Trace bound: $1$

Defining parameters

Level:	\( N \)	\(=\)	\( 312 = 2^{3} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	312.c (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 13 \)
Character field:			\(\Q\)
Newform subspaces:			\( 2 \)
Sturm bound:			\(224\)
Trace bound:			\(1\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(312, [\chi])\).

	Total	New	Old
Modular forms	176	22	154
Cusp forms	160	22	138
Eisenstein series	16	0	16

Trace form

22 * q - 6 * q^3 + 198 * q^9 + 54 * q^13 + 20 * q^17 - 152 * q^23 - 794 * q^25 - 54 * q^27 + 436 * q^29 + 520 * q^35 - 198 * q^39 + 744 * q^43 - 1982 * q^49 + 492 * q^51 + 396 * q^53 + 576 * q^55 + 508 * q^61 - 1384 * q^65 - 552 * q^69 + 1674 * q^75 - 128 * q^77 - 2688 * q^79 + 1782 * q^81 + 1164 * q^87 - 360 * q^91 + 1016 * q^95

Decomposition of \(S_{4}^{\mathrm{new}}(312, [\chi])\) 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				Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
312.4.c.a	$312$	$4$	312.c	13.b	$2$	$10$	$10$	$18.409$	\(\mathbb{Q}[x]/(x^{10} + \cdots)\)	None		✓			312.4.c.a	$2$	$0$	\(0\)	\(30\)	\(0\)	\(0\)	$2^{25}$	$\mathrm{SU}(2)[C_{2}]$	\(q+3q^{3}+\beta _{1}q^{5}+\beta _{3}q^{7}+9q^{9}+(-\beta _{6}+\cdots)q^{11}+\cdots\)
312.4.c.b	$312$	$4$	312.c	13.b	$2$	$12$	$12$	$18.409$	\(\mathbb{Q}[x]/(x^{12} + \cdots)\)	None		✓	✓		312.4.c.b	$2$	$0$	\(0\)	\(-36\)	\(0\)	\(0\)	$2^{35}$	$\mathrm{SU}(2)[C_{2}]$	\(q-3q^{3}+\beta _{5}q^{5}+\beta _{7}q^{7}+9q^{9}-\beta _{8}q^{11}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(312, [\chi])\) into lower level spaces

\( S_{4}^{\mathrm{old}}(312, [\chi]) \simeq \) \(S_{4}^{\mathrm{new}}(13, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(26, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(39, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(52, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(78, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(104, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(156, [\chi])\)\(^{\oplus 2}\)