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

• Label: 1350.2.c
• Level: $1350$
• Weight: $2$
• Character orbit: 1350.c
• Rep. character: $\chi_{1350}(649,\cdot)$
• Character field: $\Q$
• Dimension: $24$
• Newform subspaces: $12$
• Sturm bound: $540$
• Trace bound: $29$

Defining parameters

Level:	\( N \)	\(=\)	\( 1350 = 2 \cdot 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1350.c (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 5 \)
Character field:			\(\Q\)
Newform subspaces:			\( 12 \)
Sturm bound:			\(540\)
Trace bound:			\(29\)
Distinguishing \(T_p\):			\(7\), \(11\), \(13\), \(17\), \(29\)

Dimensions

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

	Total	New	Old
Modular forms	306	24	282
Cusp forms	234	24	210
Eisenstein series	72	0	72

Trace form

\( 24 q - 24 q^{4} + 24 q^{16} - 12 q^{19} + 24 q^{34} + 48 q^{49} - 12 q^{61} - 24 q^{64} + 12 q^{76} + 36 q^{79} + 96 q^{91} - 72 q^{94}+O(q^{100}) \)

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.c.a	$1350$	$2$	1350.c	5.b	$2$	$2$	$2$	$10.780$	\(\Q(\sqrt{-1}) \)	None					270.2.a.a	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+i q^{2}-q^{4}+2 i q^{7}-i q^{8}-3 q^{11}+\cdots\)
1350.2.c.b	$1350$	$2$	1350.c	5.b	$2$	$2$	$2$	$10.780$	\(\Q(\sqrt{-1}) \)	None					54.2.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+i q^{2}-q^{4}+i q^{7}-i q^{8}-3 q^{11}+\cdots\)
1350.2.c.c	$1350$	$2$	1350.c	5.b	$2$	$2$	$2$	$10.780$	\(\Q(\sqrt{-1}) \)	None					270.2.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-i q^{2}-q^{4}+2 i q^{7}+i q^{8}-3 q^{11}+\cdots\)
1350.2.c.d	$1350$	$2$	1350.c	5.b	$2$	$2$	$2$	$10.780$	\(\Q(\sqrt{-1}) \)	None		✓			1350.2.a.e	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-i q^{2}-q^{4}+2 i q^{7}+i q^{8}-3 q^{11}+\cdots\)
1350.2.c.e	$1350$	$2$	1350.c	5.b	$2$	$2$	$2$	$10.780$	\(\Q(\sqrt{-1}) \)	None		✓			1350.2.a.a	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-i q^{2}-q^{4}+4 i q^{7}+i q^{8}-3 q^{11}+\cdots\)
1350.2.c.f	$1350$	$2$	1350.c	5.b	$2$	$2$	$2$	$10.780$	\(\Q(\sqrt{-1}) \)	None		✓			1350.2.a.f	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+i q^{2}-q^{4}+i q^{7}-i q^{8}+2 i q^{13}+\cdots\)
1350.2.c.g	$1350$	$2$	1350.c	5.b	$2$	$2$	$2$	$10.780$	\(\Q(\sqrt{-1}) \)	None		✓			1350.2.a.f	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-i q^{2}-q^{4}+i q^{7}+i q^{8}+2 i q^{13}+\cdots\)
1350.2.c.h	$1350$	$2$	1350.c	5.b	$2$	$2$	$2$	$10.780$	\(\Q(\sqrt{-1}) \)	None		✓			1350.2.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+i q^{2}-q^{4}+4 i q^{7}-i q^{8}+3 q^{11}+\cdots\)
1350.2.c.i	$1350$	$2$	1350.c	5.b	$2$	$2$	$2$	$10.780$	\(\Q(\sqrt{-1}) \)	None		✓			1350.2.a.e	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+i q^{2}-q^{4}+2 i q^{7}-i q^{8}+3 q^{11}+\cdots\)
1350.2.c.j	$1350$	$2$	1350.c	5.b	$2$	$2$	$2$	$10.780$	\(\Q(\sqrt{-1}) \)	None					270.2.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+i q^{2}-q^{4}+2 i q^{7}-i q^{8}+3 q^{11}+\cdots\)
1350.2.c.k	$1350$	$2$	1350.c	5.b	$2$	$2$	$2$	$10.780$	\(\Q(\sqrt{-1}) \)	None					54.2.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-i q^{2}-q^{4}+i q^{7}+i q^{8}+3 q^{11}+\cdots\)
1350.2.c.l	$1350$	$2$	1350.c	5.b	$2$	$2$	$2$	$10.780$	\(\Q(\sqrt{-1}) \)	None					270.2.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-i q^{2}-q^{4}+2 i q^{7}+i q^{8}+3 q^{11}+\cdots\)

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

\( S_{2}^{\mathrm{old}}(1350, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(30, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(45, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(50, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(75, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(90, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(135, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(150, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(225, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(270, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(450, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(675, [\chi])\)\(^{\oplus 2}\)