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

• Label: 675.2.e
• Level: $675$
• Weight: $2$
• Character orbit: 675.e
• Rep. character: $\chi_{675}(226,\cdot)$
• Character field: $\Q(\zeta_{3})$
• Dimension: $32$
• Newform subspaces: $5$
• Sturm bound: $180$
• Trace bound: $2$

Defining parameters

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

Dimensions

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

	Total	New	Old
Modular forms	216	44	172
Cusp forms	144	32	112
Eisenstein series	72	12	60

Trace form

32 * q - 2 * q^2 - 12 * q^4 + 2 * q^7 + 6 * q^11 + 2 * q^13 - 6 * q^14 - 8 * q^16 + 4 * q^17 + 8 * q^19 - 6 * q^22 - 6 * q^23 + 24 * q^26 - 16 * q^28 + 6 * q^29 + 4 * q^31 - 22 * q^32 - 14 * q^34 - 4 * q^37 + 10 * q^38 + 6 * q^41 + 2 * q^43 - 72 * q^44 - 16 * q^46 - 20 * q^47 + 10 * q^49 - 10 * q^52 - 8 * q^53 - 48 * q^56 - 18 * q^58 + 42 * q^59 + 10 * q^61 + 84 * q^62 + 28 * q^64 + 8 * q^67 + 26 * q^68 - 24 * q^71 + 8 * q^73 + 48 * q^74 - 30 * q^76 - 6 * q^77 + 2 * q^79 + 48 * q^82 + 6 * q^83 - 12 * q^86 - 18 * q^88 + 60 * q^89 - 60 * q^91 - 36 * q^92 - 2 * q^94 - 16 * q^97 - 76 * q^98

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	Twist minimal	Largest	Maximal	Minimal twist	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
675.2.e.a	$675$	$2$	675.e	9.c	$3$	$2$	$1$	$5.390$	\(\Q(\sqrt{-3}) \)	None					45.2.e.a	$2$	$0$	\(-1\)	\(0\)	\(0\)	\(-3\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-1+\zeta_{6})q^{2}+\zeta_{6}q^{4}+(-3+3\zeta_{6})q^{7}+\cdots\)
675.2.e.b	$675$	$2$	675.e	9.c	$3$	$6$	$3$	$5.390$	6.0.954288.1	None					45.2.e.b	$2$	$0$	\(-1\)	\(0\)	\(0\)	\(5\)	$3$	$\mathrm{SU}(2)[C_{3}]$	\(q+(\beta _{4}-\beta _{5})q^{2}+(-2-\beta _{1}-\beta _{2}+2\beta _{3}+\cdots)q^{4}+\cdots\)
675.2.e.c	$675$	$2$	675.e	9.c	$3$	$8$	$4$	$5.390$	8.0.1223810289.2	None					225.2.e.c	$2$	$0$	\(-2\)	\(0\)	\(0\)	\(-1\)	$3$	$\mathrm{SU}(2)[C_{3}]$	\(q-\beta _{1}q^{2}+(\beta _{1}-\beta _{2}-\beta _{3}+\beta _{6})q^{4}+\cdots\)
675.2.e.d	$675$	$2$	675.e	9.c	$3$	$8$	$4$	$5.390$	\(\Q(\zeta_{24})\)	None					45.2.j.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$3^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q-\beta_{7} q^{2}-\beta_{2} q^{4}+(-\beta_{7}+2\beta_{5}-2\beta_{3})q^{7}+\cdots\)
675.2.e.e	$675$	$2$	675.e	9.c	$3$	$8$	$4$	$5.390$	8.0.1223810289.2	None					225.2.e.c	$2$	$0$	\(2\)	\(0\)	\(0\)	\(1\)	$3$	$\mathrm{SU}(2)[C_{3}]$	\(q+\beta _{1}q^{2}+(\beta _{1}-\beta _{2}-\beta _{3}+\beta _{6})q^{4}+\cdots\)

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

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