Space of modular forms of level 3510, weight 2, and Character 3510.j

• Label: 3510.2.j
• Level: $3510$
• Weight: $2$
• Character orbit: 3510.j
• Rep. character: $\chi_{3510}(1171,\cdot)$
• Character field: $\Q(\zeta_{3})$
• Dimension: $96$
• Newform subspaces: $14$
• Sturm bound: $1512$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 3510 = 2 \cdot 3^{3} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3510.j (of order \(3\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 9 \)
Character field:			\(\Q(\zeta_{3})\)
Newform subspaces:			\( 14 \)
Sturm bound:			\(1512\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(3510, [\chi])\).

	Total	New	Old
Modular forms	1560	96	1464
Cusp forms	1464	96	1368
Eisenstein series	96	0	96

Trace form

96 * q + 4 * q^2 - 48 * q^4 + 4 * q^5 - 8 * q^8 - 4 * q^11 - 4 * q^14 - 48 * q^16 + 8 * q^17 - 24 * q^19 + 4 * q^20 + 12 * q^22 - 24 * q^23 - 48 * q^25 - 4 * q^29 + 4 * q^32 + 12 * q^34 - 4 * q^38 + 40 * q^41 + 12 * q^43 + 8 * q^44 - 24 * q^46 - 36 * q^49 + 4 * q^50 + 64 * q^53 - 4 * q^56 - 44 * q^59 + 12 * q^61 + 96 * q^64 + 8 * q^65 + 12 * q^67 - 4 * q^68 + 12 * q^70 - 128 * q^71 - 24 * q^73 - 8 * q^74 + 12 * q^76 + 56 * q^77 - 8 * q^80 - 24 * q^82 + 4 * q^86 + 12 * q^88 - 8 * q^89 - 24 * q^92 + 12 * q^94 + 12 * q^97 - 72 * q^98

Decomposition of \(S_{2}^{\mathrm{new}}(3510, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
3510.2.j.a	$3510$	$2$	3510.j	9.c	$3$	$2$	$1$	$28.027$	\(\Q(\sqrt{-3}) \)	None					1170.2.j.f	$2$	$0$	\(-1\)	\(0\)	\(-1\)	\(1\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-1+\zeta_{6})q^{2}-\zeta_{6}q^{4}-\zeta_{6}q^{5}+(1+\cdots)q^{7}+\cdots\)
3510.2.j.b	$3510$	$2$	3510.j	9.c	$3$	$2$	$1$	$28.027$	\(\Q(\sqrt{-3}) \)	None					1170.2.j.d	$2$	$0$	\(-1\)	\(0\)	\(-1\)	\(3\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-1+\zeta_{6})q^{2}-\zeta_{6}q^{4}-\zeta_{6}q^{5}+(3+\cdots)q^{7}+\cdots\)
3510.2.j.c	$3510$	$2$	3510.j	9.c	$3$	$2$	$1$	$28.027$	\(\Q(\sqrt{-3}) \)	None					1170.2.j.e	$2$	$0$	\(-1\)	\(0\)	\(1\)	\(-2\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-1+\zeta_{6})q^{2}-\zeta_{6}q^{4}+\zeta_{6}q^{5}+(-2+\cdots)q^{7}+\cdots\)
3510.2.j.d	$3510$	$2$	3510.j	9.c	$3$	$2$	$1$	$28.027$	\(\Q(\sqrt{-3}) \)	None					1170.2.j.a	$2$	$0$	\(1\)	\(0\)	\(-1\)	\(-5\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1-\zeta_{6})q^{2}-\zeta_{6}q^{4}-\zeta_{6}q^{5}+(-5+\cdots)q^{7}+\cdots\)
3510.2.j.e	$3510$	$2$	3510.j	9.c	$3$	$2$	$1$	$28.027$	\(\Q(\sqrt{-3}) \)	None					1170.2.j.c	$2$	$0$	\(1\)	\(0\)	\(-1\)	\(-2\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1-\zeta_{6})q^{2}-\zeta_{6}q^{4}-\zeta_{6}q^{5}+(-2+\cdots)q^{7}+\cdots\)
3510.2.j.f	$3510$	$2$	3510.j	9.c	$3$	$2$	$1$	$28.027$	\(\Q(\sqrt{-3}) \)	None					1170.2.j.b	$2$	$0$	\(1\)	\(0\)	\(-1\)	\(2\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1-\zeta_{6})q^{2}-\zeta_{6}q^{4}-\zeta_{6}q^{5}+(2-2\zeta_{6})q^{7}+\cdots\)
3510.2.j.g	$3510$	$2$	3510.j	9.c	$3$	$4$	$2$	$28.027$	\(\Q(\zeta_{12})\)	None					1170.2.j.g	$2$	$0$	\(-2\)	\(0\)	\(-2\)	\(0\)	$3$	$\mathrm{SU}(2)[C_{3}]$	\(q-\beta_1 q^{2}+(\beta_1-1)q^{4}+(\beta_1-1)q^{5}+\cdots\)
3510.2.j.h	$3510$	$2$	3510.j	9.c	$3$	$8$	$4$	$28.027$	8.0.856615824.2	None					1170.2.j.h	$2$	$0$	\(4\)	\(0\)	\(-4\)	\(7\)	$2^{2}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1-\beta _{1})q^{2}-\beta _{1}q^{4}-\beta _{1}q^{5}+(2-2\beta _{1}+\cdots)q^{7}+\cdots\)
3510.2.j.i	$3510$	$2$	3510.j	9.c	$3$	$10$	$5$	$28.027$	10.0.\(\cdots\).1	None					1170.2.j.j	$2$	$0$	\(-5\)	\(0\)	\(5\)	\(-5\)	$3^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q+\beta _{5}q^{2}+(-1-\beta _{5})q^{4}+(1+\beta _{5})q^{5}+\cdots\)
3510.2.j.j	$3510$	$2$	3510.j	9.c	$3$	$10$	$5$	$28.027$	10.0.\(\cdots\).1	None					1170.2.j.i	$2$	$0$	\(5\)	\(0\)	\(-5\)	\(-4\)	$3^{4}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1-\beta _{1})q^{2}-\beta _{1}q^{4}-\beta _{1}q^{5}+(-1+\cdots)q^{7}+\cdots\)
3510.2.j.k	$3510$	$2$	3510.j	9.c	$3$	$12$	$6$	$28.027$	\(\mathbb{Q}[x]/(x^{12} + \cdots)\)	None					1170.2.j.l	$2$	$0$	\(-6\)	\(0\)	\(-6\)	\(-2\)	$3^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q-\beta _{3}q^{2}+(-1+\beta _{3})q^{4}+(-1+\beta _{3}+\cdots)q^{5}+\cdots\)
3510.2.j.l	$3510$	$2$	3510.j	9.c	$3$	$12$	$6$	$28.027$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None					1170.2.j.m	$2$	$0$	\(-6\)	\(0\)	\(6\)	\(3\)	$3^{6}$	$\mathrm{SU}(2)[C_{3}]$	\(q+\beta _{4}q^{2}+(-1-\beta _{4})q^{4}+(1+\beta _{4})q^{5}+\cdots\)
3510.2.j.m	$3510$	$2$	3510.j	9.c	$3$	$12$	$6$	$28.027$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None					1170.2.j.k	$2$	$0$	\(6\)	\(0\)	\(6\)	\(5\)	$3^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q-\beta _{4}q^{2}+(-1-\beta _{4})q^{4}+(1+\beta _{4})q^{5}+\cdots\)
3510.2.j.n	$3510$	$2$	3510.j	9.c	$3$	$16$	$8$	$28.027$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None			✓		1170.2.j.n	$2$	$0$	\(8\)	\(0\)	\(8\)	\(-1\)	$3^{7}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1+\beta _{3})q^{2}+\beta _{3}q^{4}-\beta _{3}q^{5}+(\beta _{1}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(3510, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(3510, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(18, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(45, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(54, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(90, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(117, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(135, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(234, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(270, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(351, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(585, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(702, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(1170, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(1755, [\chi])\)\(^{\oplus 2}\)