Space of modular forms of level 2700, weight 2, and Character 2700.bf

• Label: 2700.2.bf
• Level: $2700$
• Weight: $2$
• Character orbit: 2700.bf
• Rep. character: $\chi_{2700}(557,\cdot)$
• Character field: $\Q(\zeta_{12})$
• Dimension: $72$
• Newform subspaces: $6$
• Sturm bound: $1080$
• Trace bound: $13$

Defining parameters

Level:	\( N \)	\(=\)	\( 2700 = 2^{2} \cdot 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2700.bf (of order \(12\) and degree \(4\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 45 \)
Character field:			\(\Q(\zeta_{12})\)
Newform subspaces:			\( 6 \)
Sturm bound:			\(1080\)
Trace bound:			\(13\)
Distinguishing \(T_p\):			\(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(2700, [\chi])\).

	Total	New	Old
Modular forms	2376	72	2304
Cusp forms	1944	72	1872
Eisenstein series	432	0	432

Trace form

\( 72 q + 24 q^{11} - 24 q^{23} - 12 q^{37} - 72 q^{41} - 42 q^{47} + 24 q^{61} - 6 q^{67} + 96 q^{77} + 60 q^{83} + 48 q^{91} + 18 q^{97}+O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(2700, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
2700.2.bf.a	$2700$	$2$	2700.bf	45.l	$12$	$4$	$1$	$21.560$	\(\Q(\zeta_{12})\)	None					900.2.be.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-6\)	$1$	$\mathrm{SU}(2)[C_{12}]$	\(q+(-2-2\zeta_{12}+\zeta_{12}^{2}+\zeta_{12}^{3})q^{7}+\cdots\)
2700.2.bf.b	$2700$	$2$	2700.bf	45.l	$12$	$4$	$1$	$21.560$	\(\Q(\zeta_{12})\)	None					900.2.be.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{12}]$	\(q+(-1-\zeta_{12}+2\zeta_{12}^{2}-\zeta_{12}^{3})q^{7}+\cdots\)
2700.2.bf.c	$2700$	$2$	2700.bf	45.l	$12$	$4$	$1$	$21.560$	\(\Q(\zeta_{12})\)	None					900.2.be.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{12}]$	\(q+(1+\zeta_{12}-2\zeta_{12}^{2}+\zeta_{12}^{3})q^{7}+(1+\cdots)q^{11}+\cdots\)
2700.2.bf.d	$2700$	$2$	2700.bf	45.l	$12$	$4$	$1$	$21.560$	\(\Q(\zeta_{12})\)	None					900.2.be.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(6\)	$1$	$\mathrm{SU}(2)[C_{12}]$	\(q+(2+2\zeta_{12}-\zeta_{12}^{2}-\zeta_{12}^{3})q^{7}+(1+\cdots)q^{11}+\cdots\)
2700.2.bf.e	$2700$	$2$	2700.bf	45.l	$12$	$24$	$6$	$21.560$		None					180.2.w.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{12}]$
2700.2.bf.f	$2700$	$2$	2700.bf	45.l	$12$	$32$	$8$	$21.560$		None			✓		900.2.be.f	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{12}]$

Decomposition of \(S_{2}^{\mathrm{old}}(2700, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(2700, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(45, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(90, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(135, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(180, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(225, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(270, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(450, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(540, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(675, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(900, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(1350, [\chi])\)\(^{\oplus 2}\)