Space of modular forms of level 1350, weight 3, and Character 1350.b

• Label: 1350.3.b
• Level: $1350$
• Weight: $3$
• Character orbit: 1350.b
• Rep. character: $\chi_{1350}(1349,\cdot)$
• Character field: $\Q$
• Dimension: $48$
• Newform subspaces: $9$
• Sturm bound: $810$
• Trace bound: $19$

Defining parameters

Level:	\( N \)	\(=\)	\( 1350 = 2 \cdot 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1350.b (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 15 \)
Character field:			\(\Q\)
Newform subspaces:			\( 9 \)
Sturm bound:			\(810\)
Trace bound:			\(19\)
Distinguishing \(T_p\):			\(7\), \(11\), \(17\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(1350, [\chi])\).

	Total	New	Old
Modular forms	576	48	528
Cusp forms	504	48	456
Eisenstein series	72	0	72

Trace form

\( 48 q + 96 q^{4} + 192 q^{16} - 36 q^{19} - 84 q^{31} - 192 q^{34} - 144 q^{46} - 36 q^{49} + 300 q^{61} + 384 q^{64} - 72 q^{76} - 96 q^{79} + 396 q^{91} + 288 q^{94}+O(q^{100}) \)

Decomposition of \(S_{3}^{\mathrm{new}}(1350, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
1350.3.b.a	$1350$	$3$	1350.b	15.d	$2$	$4$	$4$	$36.785$	\(\Q(\zeta_{8})\)	None		✓			1350.3.d.l	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta_{3} q^{2}+2 q^{4}+(-3\beta_{2}+4\beta_1)q^{7}+\cdots\)
1350.3.b.b	$1350$	$3$	1350.b	15.d	$2$	$4$	$4$	$36.785$	\(\Q(\zeta_{8})\)	None					54.3.b.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta_{3} q^{2}+2 q^{4}+5\beta_1 q^{7}-2\beta_{3} q^{8}+\cdots\)
1350.3.b.c	$1350$	$3$	1350.b	15.d	$2$	$4$	$4$	$36.785$	\(\Q(\zeta_{8})\)	None		✓			1350.3.d.c	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta_{3} q^{2}+2 q^{4}+10\beta_1 q^{7}+2\beta_{3} q^{8}+\cdots\)
1350.3.b.d	$1350$	$3$	1350.b	15.d	$2$	$4$	$4$	$36.785$	\(\Q(\zeta_{8})\)	None		✓			1350.3.d.e	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta_{3} q^{2}+2 q^{4}+5\beta_1 q^{7}+2\beta_{3} q^{8}+\cdots\)
1350.3.b.e	$1350$	$3$	1350.b	15.d	$2$	$4$	$4$	$36.785$	\(\Q(\zeta_{8})\)	None		✓			1350.3.d.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta_{3} q^{2}+2 q^{4}+11\beta_1 q^{7}-2\beta_{3} q^{8}+\cdots\)
1350.3.b.f	$1350$	$3$	1350.b	15.d	$2$	$4$	$4$	$36.785$	\(\Q(\zeta_{8})\)	None		✓			1350.3.d.l	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta_{3} q^{2}+2 q^{4}+(-3\beta_{2}+4\beta_1)q^{7}+\cdots\)
1350.3.b.g	$1350$	$3$	1350.b	15.d	$2$	$8$	$8$	$36.785$	8.0.40960000.1	None					270.3.d.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{10}\cdot 3^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{2}q^{2}+2q^{4}+(-2\beta _{3}-\beta _{4})q^{7}+\cdots\)
1350.3.b.h	$1350$	$3$	1350.b	15.d	$2$	$8$	$8$	$36.785$	8.0.40960000.1	None					270.3.d.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{10}\cdot 3^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{3}q^{2}+2q^{4}+\beta _{1}q^{7}-2\beta _{3}q^{8}+\cdots\)
1350.3.b.i	$1350$	$3$	1350.b	15.d	$2$	$8$	$8$	$36.785$	8.0.40960000.1	None					270.3.d.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{10}\cdot 3^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{2}q^{2}+2q^{4}+(-2\beta _{3}-\beta _{4})q^{7}+\cdots\)

Decomposition of \(S_{3}^{\mathrm{old}}(1350, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(1350, [\chi]) \simeq \) \(S_{3}^{\mathrm{new}}(15, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(30, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(45, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(75, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(90, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(135, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(150, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(225, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(270, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(450, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(675, [\chi])\)\(^{\oplus 2}\)