Space of modular forms of level 675, weight 2, and Character 675.f

• Label: 675.2.f
• Level: $675$
• Weight: $2$
• Character orbit: 675.f
• Rep. character: $\chi_{675}(107,\cdot)$
• Character field: $\Q(\zeta_{4})$
• Dimension: $48$
• Newform subspaces: $9$
• Sturm bound: $180$
• Trace bound: $31$

Defining parameters

Level:	\( N \)	\(=\)	\( 675 = 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	675.f (of order \(4\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 15 \)
Character field:			\(\Q(i)\)
Newform subspaces:			\( 9 \)
Sturm bound:			\(180\)
Trace bound:			\(31\)
Distinguishing \(T_p\):			\(2\), \(7\), \(29\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(675, [\chi])\).

	Total	New	Old
Modular forms	216	48	168
Cusp forms	144	48	96
Eisenstein series	72	0	72

Trace form

\( 48 q - 4 q^{7} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(675, [\chi])\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
675.2.f.a	$675$	$2$	675.f	15.e	$4$	$4$	$2$	$5.390$	\(\Q(i, \sqrt{6})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+\beta _{1}q^{2}+\beta _{2}q^{4}+2\beta _{3}q^{7}-\beta _{3}q^{8}+\cdots\)
675.2.f.b	$675$	$2$	675.f	15.e	$4$	$4$	$2$	$5.390$	\(\Q(i, \sqrt{6})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+\beta _{1}q^{2}+\beta _{2}q^{4}+\beta _{3}q^{7}-\beta _{3}q^{8}+\cdots\)
675.2.f.c	$675$	$2$	675.f	15.e	$4$	$4$	$2$	$5.390$	\(\Q(i, \sqrt{6})\)	\(\Q(\sqrt{-3}) \)		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{4}]$	\(q+2\beta _{2}q^{4}+2\beta _{1}q^{7}-3\beta _{3}q^{13}-4q^{16}+\cdots\)
675.2.f.d	$675$	$2$	675.f	15.e	$4$	$4$	$2$	$5.390$	\(\Q(i, \sqrt{6})\)	\(\Q(\sqrt{-3}) \)		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$3^{2}$	$\mathrm{U}(1)[D_{4}]$	\(q+2\beta _{2}q^{4}-\beta _{1}q^{7}-\beta _{3}q^{13}-4q^{16}+\cdots\)
675.2.f.e	$675$	$2$	675.f	15.e	$4$	$4$	$2$	$5.390$	\(\Q(i, \sqrt{6})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+\beta _{1}q^{2}+\beta _{2}q^{4}-\beta _{3}q^{7}-\beta _{3}q^{8}+\cdots\)
675.2.f.f	$675$	$2$	675.f	15.e	$4$	$4$	$2$	$5.390$	\(\Q(i, \sqrt{6})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+\beta _{1}q^{2}+\beta _{2}q^{4}-2\beta _{3}q^{7}-\beta _{3}q^{8}+\cdots\)
675.2.f.g	$675$	$2$	675.f	15.e	$4$	$8$	$4$	$5.390$	8.0.\(\cdots\).8	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(-8\)	$2$	$\mathrm{SU}(2)[C_{4}]$	\(q+\beta _{1}q^{2}+(2\beta _{2}+\beta _{4}+\beta _{5})q^{4}+(-1+\cdots)q^{7}+\cdots\)
675.2.f.h	$675$	$2$	675.f	15.e	$4$	$8$	$4$	$5.390$	8.0.3317760000.9	\(\Q(\sqrt{-15}) \)		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$3^{4}$	$\mathrm{U}(1)[D_{4}]$	\(q+\beta _{4}q^{2}+(3\beta _{1}+\beta _{2})q^{4}+(-3\beta _{3}-\beta _{5}+\cdots)q^{8}+\cdots\)
675.2.f.i	$675$	$2$	675.f	15.e	$4$	$8$	$4$	$5.390$	8.0.\(\cdots\).1	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(4\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+\beta _{1}q^{2}+(\beta _{4}+\beta _{6})q^{4}+(-\beta _{2}-\beta _{4}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(675, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(675, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(45, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(75, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(135, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(225, [\chi])\)\(^{\oplus 2}\)