Space of modular forms of level 300, weight 9, and Character 300.k

• Label: 300.9.k
• Level: $300$
• Weight: $9$
• Character orbit: 300.k
• Rep. character: $\chi_{300}(157,\cdot)$
• Character field: $\Q(\zeta_{4})$
• Dimension: $48$
• Newform subspaces: $5$
• Sturm bound: $540$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 300 = 2^{2} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	300.k (of order \(4\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 5 \)
Character field:			\(\Q(i)\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(540\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{9}(300, [\chi])\).

	Total	New	Old
Modular forms	996	48	948
Cusp forms	924	48	876
Eisenstein series	72	0	72

Trace form

\( 48 q - 4220 q^{7} + O(q^{10}) \)

Decomposition of \(S_{9}^{\mathrm{new}}(300, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
300.9.k.a	$300$	$9$	300.k	5.c	$4$	$4$	$2$	$122.214$	\(\Q(i, \sqrt{6})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+3^{3}\beta _{3}q^{3}+311\beta _{1}q^{7}-3^{7}\beta _{2}q^{9}+\cdots\)
300.9.k.b	$300$	$9$	300.k	5.c	$4$	$8$	$4$	$122.214$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{16}\cdot 3^{4}\cdot 5^{4}$	$\mathrm{SU}(2)[C_{4}]$	\(q-3^{3}\beta _{4}q^{3}+609\beta _{1}q^{7}-3^{7}\beta _{3}q^{9}+\cdots\)
300.9.k.c	$300$	$9$	300.k	5.c	$4$	$8$	$4$	$122.214$	8.0.\(\cdots\).190	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{10}\cdot 3^{2}\cdot 5^{4}$	$\mathrm{SU}(2)[C_{4}]$	\(q-3^{3}\beta _{3}q^{3}+(-7^{2}\beta _{1}-7\beta _{5})q^{7}-3^{7}\beta _{2}q^{9}+\cdots\)
300.9.k.d	$300$	$9$	300.k	5.c	$4$	$12$	$6$	$122.214$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{26}\cdot 3^{18}\cdot 5^{8}$	$\mathrm{SU}(2)[C_{4}]$	\(q-\beta _{4}q^{3}+(-2^{4}\beta _{5}+\beta _{7})q^{7}-3^{7}\beta _{3}q^{9}+\cdots\)
300.9.k.e	$300$	$9$	300.k	5.c	$4$	$16$	$8$	$122.214$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(-4220\)	$2^{26}\cdot 3^{20}\cdot 5^{18}$	$\mathrm{SU}(2)[C_{4}]$	\(q+\beta _{2}q^{3}+(-264-264\beta _{1}-5\beta _{3}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{9}^{\mathrm{old}}(300, [\chi])\) into lower level spaces

\( S_{9}^{\mathrm{old}}(300, [\chi]) \cong \) \(S_{9}^{\mathrm{new}}(5, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(10, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(15, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(20, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(25, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(30, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(50, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(60, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(75, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(100, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(150, [\chi])\)\(^{\oplus 2}\)