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

• Label: 315.2.d
• Level: $315$
• Weight: $2$
• Character orbit: 315.d
• Rep. character: $\chi_{315}(64,\cdot)$
• Character field: $\Q$
• Dimension: $14$
• Newform subspaces: $5$
• Sturm bound: $96$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	315.d (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 5 \)
Character field:			\(\Q\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(96\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(2\), \(11\), \(29\)

Dimensions

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

	Total	New	Old
Modular forms	56	14	42
Cusp forms	40	14	26
Eisenstein series	16	0	16

Trace form

14 * q - 8 * q^4 - 8 * q^10 + 6 * q^11 - 4 * q^14 + 12 * q^16 + 20 * q^20 + 10 * q^25 - 28 * q^26 - 10 * q^29 - 28 * q^31 - 4 * q^34 - 2 * q^35 + 4 * q^40 - 12 * q^41 + 20 * q^44 + 8 * q^46 - 14 * q^49 + 24 * q^50 + 4 * q^55 + 24 * q^56 + 36 * q^59 + 8 * q^61 - 52 * q^64 - 22 * q^65 + 8 * q^70 - 16 * q^71 - 72 * q^74 + 16 * q^76 + 10 * q^79 - 60 * q^80 + 22 * q^85 + 16 * q^86 + 40 * q^89 - 2 * q^91 + 92 * q^94 - 16 * 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.d.a	$315$	$2$	315.d	5.b	$2$	$2$	$2$	$2.515$	\(\Q(\sqrt{-1}) \)	None					35.2.b.a	$2$	$0$	\(0\)	\(0\)	\(4\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+2 i q^{2}-2 q^{4}+(-i+2)q^{5}+i q^{7}+\cdots\)
315.2.d.b	$315$	$2$	315.d	5.b	$2$	$2$	$2$	$2.515$	\(\Q(\sqrt{-1}) \)	None		✓			315.2.d.b	$2$	$0$	\(0\)	\(0\)	\(-4\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+i q^{2}+q^{4}+(i-2)q^{5}+i q^{7}+\cdots\)
315.2.d.c	$315$	$2$	315.d	5.b	$2$	$2$	$2$	$2.515$	\(\Q(\sqrt{-1}) \)	None					105.2.d.a	$2$	$0$	\(0\)	\(0\)	\(-2\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+i q^{2}+q^{4}+(-2 i-1)q^{5}-i q^{7}+\cdots\)
315.2.d.d	$315$	$2$	315.d	5.b	$2$	$2$	$2$	$2.515$	\(\Q(\sqrt{-1}) \)	None		✓			315.2.d.b	$2$	$0$	\(0\)	\(0\)	\(4\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+i q^{2}+q^{4}+(i+2)q^{5}-i q^{7}+\cdots\)
315.2.d.e	$315$	$2$	315.d	5.b	$2$	$6$	$6$	$2.515$	6.0.350464.1	None			✓		105.2.d.b	$2$	$0$	\(0\)	\(0\)	\(-2\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{1}q^{2}+(-1+\beta _{3}+\beta _{5})q^{4}+(-\beta _{1}+\cdots)q^{5}+\cdots\)

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

\( S_{2}^{\mathrm{old}}(315, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(35, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(45, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(105, [\chi])\)\(^{\oplus 2}\)