Space of modular forms of level 300, weight 7, and Character 300.b

• Label: 300.7.b
• Level: $300$
• Weight: $7$
• Character orbit: 300.b
• Rep. character: $\chi_{300}(149,\cdot)$
• Character field: $\Q$
• Dimension: $36$
• Newform subspaces: $5$
• Sturm bound: $420$
• Trace bound: $19$

Defining parameters

Level:	\( N \)	\(=\)	\( 300 = 2^{2} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 7 \)
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:			\(420\)
Trace bound:			\(19\)
Distinguishing \(T_p\):			\(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{7}(300, [\chi])\).

	Total	New	Old
Modular forms	378	36	342
Cusp forms	342	36	306
Eisenstein series	36	0	36

Trace form

\( 36 q - 1744 q^{9} + O(q^{10}) \)

Decomposition of \(S_{7}^{\mathrm{new}}(300, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
300.7.b.a	$300$	$7$	300.b	15.d	$2$	$2$	$2$	$69.016$	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-3}) \)		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q+3^{3}iq^{3}+683iq^{7}-3^{6}q^{9}-3527iq^{13}+\cdots\)
300.7.b.b	$300$	$7$	300.b	15.d	$2$	$2$	$2$	$69.016$	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-3}) \)		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q-3^{3}iq^{3}+397iq^{7}-3^{6}q^{9}-4033iq^{13}+\cdots\)
300.7.b.c	$300$	$7$	300.b	15.d	$2$	$4$	$4$	$69.016$	\(\Q(i, \sqrt{5})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{8}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-2\beta _{1}-\beta _{2})q^{3}-11^{2}\beta _{1}q^{7}+(711+\cdots)q^{9}+\cdots\)
300.7.b.d	$300$	$7$	300.b	15.d	$2$	$12$	$12$	$69.016$	\(\mathbb{Q}[x]/(x^{12} + \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{15}\cdot 3^{8}\cdot 5^{10}\cdot 7^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-2\beta _{2}-\beta _{3})q^{3}+(12\beta _{2}+\beta _{3}-\beta _{11})q^{7}+\cdots\)
300.7.b.e	$300$	$7$	300.b	15.d	$2$	$16$	$16$	$69.016$	\(\mathbb{Q}[x]/(x^{16} + \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{30}\cdot 3^{18}\cdot 5^{26}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{3}q^{3}+(\beta _{4}-\beta _{5})q^{7}+(-187+\beta _{8}+\cdots)q^{9}+\cdots\)

Decomposition of \(S_{7}^{\mathrm{old}}(300, [\chi])\) into lower level spaces

\( S_{7}^{\mathrm{old}}(300, [\chi]) \cong \) \(S_{7}^{\mathrm{new}}(15, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(30, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(60, [\chi])\)\(^{\oplus 2}\)