Space of modular forms of level 315, weight 4, and Character 315.j

• Label: 315.4.j
• Level: $315$
• Weight: $4$
• Character orbit: 315.j
• Rep. character: $\chi_{315}(46,\cdot)$
• Character field: $\Q(\zeta_{3})$
• Dimension: $80$
• Newform subspaces: $10$
• Sturm bound: $192$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	315.j (of order \(3\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 7 \)
Character field:			\(\Q(\zeta_{3})\)
Newform subspaces:			\( 10 \)
Sturm bound:			\(192\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(2\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(315, [\chi])\).

	Total	New	Old
Modular forms	304	80	224
Cusp forms	272	80	192
Eisenstein series	32	0	32

Trace form

80 * q + 2 * q^2 - 170 * q^4 - 10 * q^5 + 20 * q^7 + 36 * q^8 + 20 * q^10 - 38 * q^11 - 352 * q^13 - 206 * q^14 - 718 * q^16 - 140 * q^17 - 154 * q^19 + 200 * q^20 - 368 * q^22 - 80 * q^23 - 1000 * q^25 - 438 * q^26 - 6 * q^28 - 692 * q^29 - 160 * q^31 - 882 * q^32 - 1576 * q^34 - 110 * q^35 - 468 * q^37 + 152 * q^38 + 240 * q^40 + 1400 * q^41 + 1712 * q^43 + 522 * q^44 - 22 * q^46 - 992 * q^47 - 390 * q^49 - 100 * q^50 + 1348 * q^52 - 2156 * q^53 - 1120 * q^55 + 876 * q^56 - 1530 * q^58 - 188 * q^59 + 910 * q^61 + 6000 * q^62 + 7444 * q^64 + 530 * q^65 + 216 * q^67 - 1708 * q^68 + 520 * q^70 - 944 * q^71 + 12 * q^73 - 3246 * q^74 + 4916 * q^76 - 1048 * q^77 + 1400 * q^79 - 960 * q^80 + 5486 * q^82 + 7568 * q^83 - 1280 * q^85 + 3324 * q^86 + 5176 * q^88 - 294 * q^89 - 2716 * q^91 + 6204 * q^92 - 6814 * q^94 - 300 * q^95 - 6576 * q^97 - 4486 * q^98

Decomposition of \(S_{4}^{\mathrm{new}}(315, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
315.4.j.a	$315$	$4$	315.j	7.c	$3$	$2$	$1$	$18.586$	\(\Q(\sqrt{-3}) \)	None					105.4.i.a	$2$	$0$	\(3\)	\(0\)	\(-5\)	\(-28\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+3\zeta_{6}q^{2}+(-1+\zeta_{6})q^{4}-5\zeta_{6}q^{5}+\cdots\)
315.4.j.b	$315$	$4$	315.j	7.c	$3$	$2$	$1$	$18.586$	\(\Q(\sqrt{-3}) \)	None					35.4.e.a	$2$	$0$	\(3\)	\(0\)	\(5\)	\(-28\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+3\zeta_{6}q^{2}+(-1+\zeta_{6})q^{4}+5\zeta_{6}q^{5}+\cdots\)
315.4.j.c	$315$	$4$	315.j	7.c	$3$	$4$	$2$	$18.586$	\(\Q(\sqrt{2}, \sqrt{-3})\)	None					35.4.e.b	$2$	$0$	\(-6\)	\(0\)	\(10\)	\(22\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(\beta _{1}+3\beta _{2}+\beta _{3})q^{2}+(-3-6\beta _{1}+\cdots)q^{4}+\cdots\)
315.4.j.d	$315$	$4$	315.j	7.c	$3$	$4$	$2$	$18.586$	\(\Q(\sqrt{2}, \sqrt{-3})\)	None					105.4.i.b	$2$	$0$	\(-4\)	\(0\)	\(-10\)	\(50\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-\beta _{1}+2\beta _{2}-\beta _{3})q^{2}+(2+4\beta _{1}+\cdots)q^{4}+\cdots\)
315.4.j.e	$315$	$4$	315.j	7.c	$3$	$6$	$3$	$18.586$	6.0.646154928.2	None					105.4.i.c	$2$	$0$	\(3\)	\(0\)	\(15\)	\(-2\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1-\beta _{1}-\beta _{2}-\beta _{3})q^{2}+(-\beta _{1}+\beta _{4}+\cdots)q^{4}+\cdots\)
315.4.j.f	$315$	$4$	315.j	7.c	$3$	$10$	$5$	$18.586$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None					105.4.i.e	$2$	$0$	\(-1\)	\(0\)	\(-25\)	\(56\)	$2^{2}\cdot 7$	$\mathrm{SU}(2)[C_{3}]$	\(q-\beta _{1}q^{2}+(\beta _{1}-\beta _{2}+\beta _{3}-4\beta _{4}+\beta _{8}+\cdots)q^{4}+\cdots\)
315.4.j.g	$315$	$4$	315.j	7.c	$3$	$10$	$5$	$18.586$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None					35.4.e.c	$2$	$0$	\(1\)	\(0\)	\(-25\)	\(-62\)	$2^{4}$	$\mathrm{SU}(2)[C_{3}]$	\(q+\beta _{1}q^{2}+(-7-\beta _{4}+7\beta _{6}+\beta _{8})q^{4}+\cdots\)
315.4.j.h	$315$	$4$	315.j	7.c	$3$	$10$	$5$	$18.586$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None					105.4.i.d	$2$	$0$	\(3\)	\(0\)	\(25\)	\(-32\)	$2^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-\beta _{1}-\beta _{4})q^{2}+(-5-5\beta _{4}+\beta _{7}+\cdots)q^{4}+\cdots\)
315.4.j.i	$315$	$4$	315.j	7.c	$3$	$16$	$8$	$18.586$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None		✓			315.4.j.i	$2$	$0$	\(-4\)	\(0\)	\(40\)	\(22\)	$2^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(\beta _{1}+\beta _{6})q^{2}+(-6-\beta _{1}+\beta _{4}-\beta _{5}+\cdots)q^{4}+\cdots\)
315.4.j.j	$315$	$4$	315.j	7.c	$3$	$16$	$8$	$18.586$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None		✓			315.4.j.i	$2$	$0$	\(4\)	\(0\)	\(-40\)	\(22\)	$2^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-\beta _{1}-\beta _{6})q^{2}+(-6-\beta _{1}+\beta _{4}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(315, [\chi])\) into lower level spaces

\( S_{4}^{\mathrm{old}}(315, [\chi]) \simeq \) \(S_{4}^{\mathrm{new}}(7, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(35, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(63, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(105, [\chi])\)\(^{\oplus 2}\)