Space of modular forms of level 315, weight 2, and Character 315.bf

• Label: 315.2.bf
• Level: $315$
• Weight: $2$
• Character orbit: 315.bf
• Rep. character: $\chi_{315}(109,\cdot)$
• Character field: $\Q(\zeta_{6})$
• Dimension: $36$
• Newform subspaces: $3$
• Sturm bound: $96$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	315.bf (of order \(6\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 35 \)
Character field:			\(\Q(\zeta_{6})\)
Newform subspaces:			\( 3 \)
Sturm bound:			\(96\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(315, [\chi])\).

	Total	New	Old
Modular forms	112	44	68
Cusp forms	80	36	44
Eisenstein series	32	8	24

Trace form

36 * q + 14 * q^4 + 16 * q^14 - 14 * q^16 + 4 * q^19 + 4 * q^20 - 14 * q^25 + 16 * q^26 + 4 * q^29 - 28 * q^31 - 72 * q^34 + 26 * q^35 + 28 * q^40 - 36 * q^41 - 20 * q^44 - 18 * q^46 - 14 * q^49 + 24 * q^50 + 8 * q^55 - 90 * q^56 - 24 * q^59 - 22 * q^61 + 84 * q^64 - 38 * q^65 - 50 * q^70 + 64 * q^71 - 24 * q^74 + 56 * q^76 - 36 * q^79 - 54 * q^80 + 62 * q^86 - 34 * q^89 + 116 * q^91 + 10 * q^95

Decomposition of \(S_{2}^{\mathrm{new}}(315, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
315.2.bf.a	$315$	$2$	315.bf	35.j	$6$	$4$	$2$	$2.515$	\(\Q(\zeta_{12})\)	None					35.2.j.a	$4$	$0$	\(0\)	\(0\)	\(2\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+\zeta_{12}q^{2}-\zeta_{12}^{2}q^{4}+(1-2\zeta_{12}-\zeta_{12}^{2}+\cdots)q^{5}+\cdots\)
315.2.bf.b	$315$	$2$	315.bf	35.j	$6$	$16$	$8$	$2.515$	16.0.\(\cdots\).1	None					105.2.q.a	$4$	$0$	\(0\)	\(0\)	\(-2\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-\beta _{5}+\beta _{6}-\beta _{15})q^{2}+(1-\beta _{2}+\beta _{7}+\cdots)q^{4}+\cdots\)
315.2.bf.c	$315$	$2$	315.bf	35.j	$6$	$16$	$8$	$2.515$	\(\mathbb{Q}[x]/(x^{16} + \cdots)\)	None		✓			315.2.bf.c	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{6}]$	\(q+\beta _{9}q^{2}+(1+\beta _{5}-\beta _{8})q^{4}+(\beta _{1}+\beta _{9}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(315, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(315, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(35, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(105, [\chi])\)\(^{\oplus 2}\)