Space of modular forms of level 1350, weight 2, and Character 1350.q

• Label: 1350.2.q
• Level: $1350$
• Weight: $2$
• Character orbit: 1350.q
• Rep. character: $\chi_{1350}(143,\cdot)$
• Character field: $\Q(\zeta_{12})$
• Dimension: $72$
• Newform subspaces: $8$
• Sturm bound: $540$
• Trace bound: $31$

Defining parameters

Level:	\( N \)	\(=\)	\( 1350 = 2 \cdot 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1350.q (of order \(12\) and degree \(4\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 45 \)
Character field:			\(\Q(\zeta_{12})\)
Newform subspaces:			\( 8 \)
Sturm bound:			\(540\)
Trace bound:			\(31\)
Distinguishing \(T_p\):			\(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(1350, [\chi])\).

	Total	New	Old
Modular forms	1224	72	1152
Cusp forms	936	72	864
Eisenstein series	288	0	288

Trace form

72 * q - 48 * q^11 + 36 * q^16 - 24 * q^23 + 24 * q^37 + 36 * q^38 + 72 * q^41 + 48 * q^46 + 48 * q^47 + 24 * q^56 - 12 * q^58 + 24 * q^61 + 12 * q^67 - 36 * q^68 - 48 * q^77 + 48 * q^82 + 60 * q^83 + 36 * q^86 - 96 * q^91 - 24 * q^92 + 36 * q^97

Decomposition of \(S_{2}^{\mathrm{new}}(1350, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
1350.2.q.a	$1350$	$2$	1350.q	45.l	$12$	$8$	$2$	$10.780$	\(\Q(\zeta_{24})\)	None					450.2.p.c	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{12}]$	\(q+(\zeta_{24}^{3}-\zeta_{24}^{7})q^{2}+(\zeta_{24}^{2}-\zeta_{24}^{6}+\cdots)q^{4}+\cdots\)
1350.2.q.b	$1350$	$2$	1350.q	45.l	$12$	$8$	$2$	$10.780$	\(\Q(\zeta_{24})\)	None					450.2.p.d	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{12}]$	\(q+\zeta_{24}^{5}q^{2}+(-\zeta_{24}^{2}+\zeta_{24}^{6})q^{4}+\cdots\)
1350.2.q.c	$1350$	$2$	1350.q	45.l	$12$	$8$	$2$	$10.780$	\(\Q(\zeta_{24})\)	None					450.2.p.g	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{12}]$	\(q+\zeta_{24}q^{2}+\zeta_{24}^{2}q^{4}+\zeta_{24}^{3}q^{8}+(-4+\cdots)q^{11}+\cdots\)
1350.2.q.d	$1350$	$2$	1350.q	45.l	$12$	$8$	$2$	$10.780$	\(\Q(\zeta_{24})\)	None					450.2.p.c	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{12}]$	\(q+\zeta_{24}^{7}q^{2}-\zeta_{24}^{2}q^{4}+(-3\zeta_{24}+2\zeta_{24}^{3}+\cdots)q^{7}+\cdots\)
1350.2.q.e	$1350$	$2$	1350.q	45.l	$12$	$8$	$2$	$10.780$	\(\Q(\zeta_{24})\)	None					450.2.p.b	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{12}]$	\(q+\zeta_{24}^{7}q^{2}-\zeta_{24}^{2}q^{4}+(\zeta_{24}-\zeta_{24}^{5}+\cdots)q^{8}+\cdots\)
1350.2.q.f	$1350$	$2$	1350.q	45.l	$12$	$8$	$2$	$10.780$	\(\Q(\zeta_{24})\)	None					450.2.p.f	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{12}]$	\(q-\zeta_{24}^{5}q^{2}+(-\zeta_{24}^{2}+\zeta_{24}^{6})q^{4}+\cdots\)
1350.2.q.g	$1350$	$2$	1350.q	45.l	$12$	$8$	$2$	$10.780$	\(\Q(\zeta_{24})\)	None					90.2.l.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(8\)	$1$	$\mathrm{SU}(2)[C_{12}]$	\(q+\zeta_{24}^{7}q^{2}-\zeta_{24}^{2}q^{4}+(2\zeta_{24}^{2}+2\zeta_{24}^{3}+\cdots)q^{7}+\cdots\)
1350.2.q.h	$1350$	$2$	1350.q	45.l	$12$	$16$	$4$	$10.780$	16.0.\(\cdots\).9	None			✓		90.2.l.b	$4$	$0$	\(0\)	\(0\)	\(0\)	\(-8\)	$2^{2}$	$\mathrm{SU}(2)[C_{12}]$	\(q-\beta _{7}q^{2}-\beta _{9}q^{4}+(-1-2\beta _{1}+\beta _{2}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(1350, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(1350, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(45, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(90, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(135, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(225, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(270, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(450, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(675, [\chi])\)\(^{\oplus 2}\)