Space of modular forms of level 312, weight 2, and Character 312.bf

• Label: 312.2.bf
• Level: $312$
• Weight: $2$
• Character orbit: 312.bf
• Rep. character: $\chi_{312}(49,\cdot)$
• Character field: $\Q(\zeta_{6})$
• Dimension: $12$
• Newform subspaces: $2$
• Sturm bound: $112$
• Trace bound: $1$

Defining parameters

Level:	\( N \)	\(=\)	\( 312 = 2^{3} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	312.bf (of order \(6\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 13 \)
Character field:			\(\Q(\zeta_{6})\)
Newform subspaces:			\( 2 \)
Sturm bound:			\(112\)
Trace bound:			\(1\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(312, [\chi])\).

	Total	New	Old
Modular forms	128	12	116
Cusp forms	96	12	84
Eisenstein series	32	0	32

Trace form

12 * q - 2 * q^3 + 6 * q^7 - 6 * q^9 - 12 * q^11 - 6 * q^13 + 8 * q^17 + 12 * q^19 + 4 * q^23 + 4 * q^25 + 4 * q^27 + 4 * q^29 + 4 * q^35 + 12 * q^37 - 24 * q^41 + 6 * q^43 - 12 * q^45 + 12 * q^49 - 32 * q^51 - 16 * q^55 - 36 * q^59 + 18 * q^61 - 6 * q^63 - 28 * q^65 + 42 * q^67 - 8 * q^69 - 12 * q^71 + 6 * q^75 + 40 * q^77 - 4 * q^79 - 6 * q^81 + 12 * q^85 + 8 * q^87 + 34 * q^91 + 6 * q^93 - 40 * q^95 - 66 * q^97

Decomposition of \(S_{2}^{\mathrm{new}}(312, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
312.2.bf.a	$312$	$2$	312.bf	13.e	$6$	$4$	$2$	$2.491$	\(\Q(\zeta_{12})\)	None		✓			312.2.bf.a	$2$	$0$	\(0\)	\(2\)	\(0\)	\(6\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(1-\zeta_{12}^{2})q^{3}+(1-2\zeta_{12}^{2})q^{5}+(2+\cdots)q^{7}+\cdots\)
312.2.bf.b	$312$	$2$	312.bf	13.e	$6$	$8$	$4$	$2.491$	8.0.649638144.4	None		✓	✓		312.2.bf.b	$2$	$0$	\(0\)	\(-4\)	\(0\)	\(0\)	$2^{4}$	$\mathrm{SU}(2)[C_{6}]$	\(q+\beta _{2}q^{3}+(-\beta _{2}-\beta _{6})q^{5}+(1+\beta _{1}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(312, [\chi])\) into lower level spaces

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