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

• Label: 300.9.b
• Level: $300$
• Weight: $9$
• Character orbit: 300.b
• Rep. character: $\chi_{300}(149,\cdot)$
• Character field: $\Q$
• Dimension: $48$
• Newform subspaces: $5$
• Sturm bound: $540$
• Trace bound: $19$

Defining parameters

Level:	\( N \)	\(=\)	\( 300 = 2^{2} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	300.b (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 15 \)
Character field:			\(\Q\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(540\)
Trace bound:			\(19\)
Distinguishing \(T_p\):			\(7\)

Dimensions

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

	Total	New	Old
Modular forms	498	48	450
Cusp forms	462	48	414
Eisenstein series	36	0	36

Trace form

\( 48 q + 21602 q^{9} + O(q^{10}) \)

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.b.a	$300$	$9$	300.b	15.d	$2$	$2$	$2$	$122.214$	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-3}) \)		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q-3^{4}iq^{3}+239iq^{7}-3^{8}q^{9}+20641iq^{13}+\cdots\)
300.9.b.b	$300$	$9$	300.b	15.d	$2$	$2$	$2$	$122.214$	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-3}) \)		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q-3^{4}iq^{3}+4034iq^{7}-3^{8}q^{9}+35806iq^{13}+\cdots\)
300.9.b.c	$300$	$9$	300.b	15.d	$2$	$4$	$4$	$122.214$	\(\Q(i, \sqrt{110})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{5}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-5^{2}\beta _{1}+\beta _{2})q^{3}+1547\beta _{1}q^{7}+\cdots\)
300.9.b.d	$300$	$9$	300.b	15.d	$2$	$20$	$20$	$122.214$	\(\mathbb{Q}[x]/(x^{20} + \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{42}\cdot 3^{34}\cdot 5^{38}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{2}q^{3}+(-11\beta _{2}+11\beta _{3}+\beta _{10}+\cdots)q^{7}+\cdots\)
300.9.b.e	$300$	$9$	300.b	15.d	$2$	$20$	$20$	$122.214$	\(\mathbb{Q}[x]/(x^{20} - \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{34}\cdot 3^{34}\cdot 5^{30}\cdot 7^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{2}q^{3}+(-\beta _{1}+2\beta _{2}+\beta _{7})q^{7}+(1245+\cdots)q^{9}+\cdots\)

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

\( S_{9}^{\mathrm{old}}(300, [\chi]) \simeq \) \(S_{9}^{\mathrm{new}}(15, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(30, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(60, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(75, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(150, [\chi])\)\(^{\oplus 2}\)