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

• Label: 312.2.q
• Level: $312$
• Weight: $2$
• Character orbit: 312.q
• Rep. character: $\chi_{312}(217,\cdot)$
• Character field: $\Q(\zeta_{3})$
• Dimension: $16$
• Newform subspaces: $5$
• Sturm bound: $112$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 312 = 2^{3} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	312.q (of order \(3\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 13 \)
Character field:			\(\Q(\zeta_{3})\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(112\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(5\), \(7\)

Dimensions

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

	Total	New	Old
Modular forms	128	16	112
Cusp forms	96	16	80
Eisenstein series	32	0	32

Trace form

16 * q + 2 * q^3 + 4 * q^5 - 2 * q^7 - 8 * q^9 + 4 * q^11 + 4 * q^13 - 6 * q^17 + 8 * q^19 + 4 * q^21 + 4 * q^23 + 28 * q^25 - 4 * q^27 - 2 * q^29 + 12 * q^31 + 4 * q^35 + 2 * q^37 - 8 * q^39 + 26 * q^41 + 14 * q^43 - 2 * q^45 - 56 * q^47 - 34 * q^49 - 32 * q^51 - 36 * q^53 - 8 * q^55 - 8 * q^57 - 20 * q^59 - 2 * q^63 + 14 * q^65 - 26 * q^67 - 8 * q^69 + 4 * q^71 + 40 * q^73 + 22 * q^75 - 40 * q^77 + 52 * q^79 - 8 * q^81 - 16 * q^83 - 6 * q^85 - 16 * q^87 - 28 * q^89 - 30 * q^91 - 2 * q^93 + 40 * q^95 + 2 * q^97 - 8 * q^99

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.q.a	$312$	$2$	312.q	13.c	$3$	$2$	$1$	$2.491$	\(\Q(\sqrt{-3}) \)	None		✓			312.2.q.a	$2$	$0$	\(0\)	\(-1\)	\(6\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-1+\zeta_{6})q^{3}+3q^{5}-\zeta_{6}q^{9}+(3+\cdots)q^{13}+\cdots\)
312.2.q.b	$312$	$2$	312.q	13.c	$3$	$2$	$1$	$2.491$	\(\Q(\sqrt{-3}) \)	None		✓			312.2.q.b	$2$	$0$	\(0\)	\(1\)	\(-4\)	\(-1\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1-\zeta_{6})q^{3}-2q^{5}-\zeta_{6}q^{7}-\zeta_{6}q^{9}+\cdots\)
312.2.q.c	$312$	$2$	312.q	13.c	$3$	$2$	$1$	$2.491$	\(\Q(\sqrt{-3}) \)	None		✓			312.2.q.c	$2$	$0$	\(0\)	\(1\)	\(6\)	\(4\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1-\zeta_{6})q^{3}+3q^{5}+4\zeta_{6}q^{7}-\zeta_{6}q^{9}+\cdots\)
312.2.q.d	$312$	$2$	312.q	13.c	$3$	$4$	$2$	$2.491$	\(\Q(\sqrt{-3}, \sqrt{13})\)	None		✓			312.2.q.d	$2$	$0$	\(0\)	\(-2\)	\(-4\)	\(-2\)	$2^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-1+\beta _{1})q^{3}-q^{5}+(-\beta _{1}-\beta _{2}+\cdots)q^{7}+\cdots\)
312.2.q.e	$312$	$2$	312.q	13.c	$3$	$6$	$3$	$2.491$	6.0.2101707.2	None		✓	✓		312.2.q.e	$2$	$0$	\(0\)	\(3\)	\(0\)	\(-3\)	$2^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1+\beta _{3})q^{3}+\beta _{2}q^{5}+(-2-\beta _{4}+2\beta _{5})q^{7}+\cdots\)

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

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