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

• Label: 312.2.g
• Level: $312$
• Weight: $2$
• Character orbit: 312.g
• Rep. character: $\chi_{312}(157,\cdot)$
• Character field: $\Q$
• Dimension: $24$
• 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.g (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 8 \)
Character field:			\(\Q\)
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	60	24	36
Cusp forms	52	24	28
Eisenstein series	8	0	8

Trace form

24 * q + 4 * q^2 + 4 * q^6 + 8 * q^7 - 8 * q^8 - 24 * q^9 - 12 * q^10 - 4 * q^12 + 8 * q^14 - 8 * q^15 - 4 * q^16 - 4 * q^18 + 12 * q^20 - 4 * q^22 - 16 * q^23 - 4 * q^24 - 24 * q^25 + 32 * q^28 + 12 * q^30 - 8 * q^31 + 4 * q^32 - 16 * q^34 - 16 * q^38 + 8 * q^39 + 20 * q^40 - 4 * q^44 + 32 * q^46 + 48 * q^47 - 16 * q^48 + 24 * q^49 - 12 * q^50 - 4 * q^52 - 4 * q^54 - 16 * q^55 - 56 * q^56 - 16 * q^58 + 4 * q^60 + 16 * q^62 - 8 * q^63 - 48 * q^64 - 20 * q^66 + 32 * q^68 + 72 * q^70 + 16 * q^71 + 8 * q^72 - 56 * q^74 + 40 * q^76 - 68 * q^80 + 24 * q^81 + 12 * q^82 + 16 * q^84 + 64 * q^86 + 4 * q^88 + 12 * q^90 - 8 * q^92 - 44 * q^94 - 48 * q^95 + 24 * q^96 - 16 * q^97 - 52 * q^98

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.g.a	$312$	$2$	312.g	8.b	$2$	$8$	$8$	$2.491$	\(\Q(\zeta_{20})\)	None		✓			312.2.g.a	$2$	$0$	\(2\)	\(0\)	\(0\)	\(4\)	$2^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta_{7} q^{2}-\beta_{3} q^{3}+\beta_1 q^{4}+(\beta_{7}+\beta_{4}-\beta_{3}+\cdots-1)q^{5}+\cdots\)
312.2.g.b	$312$	$2$	312.g	8.b	$2$	$16$	$16$	$2.491$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None		✓	✓		312.2.g.b	$2$	$0$	\(2\)	\(0\)	\(0\)	\(4\)	$2^{7}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{10}q^{2}+\beta _{9}q^{3}-\beta _{2}q^{4}+\beta _{14}q^{5}+\cdots\)

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

\( S_{2}^{\mathrm{old}}(312, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(24, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(104, [\chi])\)\(^{\oplus 2}\)