Space of modular forms of level 300, weight 6, and Character 300.d

• Label: 300.6.d
• Level: $300$
• Weight: $6$
• Character orbit: 300.d
• Rep. character: $\chi_{300}(49,\cdot)$
• Character field: $\Q$
• Dimension: $14$
• Newform subspaces: $6$
• Sturm bound: $360$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 300 = 2^{2} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	300.d (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 5 \)
Character field:			\(\Q\)
Newform subspaces:			\( 6 \)
Sturm bound:			\(360\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(300, [\chi])\).

	Total	New	Old
Modular forms	318	14	304
Cusp forms	282	14	268
Eisenstein series	36	0	36

Trace form

\( 14 q - 1134 q^{9} + O(q^{10}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(300, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
300.6.d.a	$300$	$6$	300.d	5.b	$2$	$2$	$2$	$48.115$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-9iq^{3}+2^{4}iq^{7}-3^{4}q^{9}-564q^{11}+\cdots\)
300.6.d.b	$300$	$6$	300.d	5.b	$2$	$2$	$2$	$48.115$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-9iq^{3}+91iq^{7}-3^{4}q^{9}-174q^{11}+\cdots\)
300.6.d.c	$300$	$6$	300.d	5.b	$2$	$2$	$2$	$48.115$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+9iq^{3}+244iq^{7}-3^{4}q^{9}-12^{2}q^{11}+\cdots\)
300.6.d.d	$300$	$6$	300.d	5.b	$2$	$2$	$2$	$48.115$	\(\Q(\sqrt{-1}) \)	None		$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-9iq^{3}+56iq^{7}-3^{4}q^{9}+156q^{11}+\cdots\)
300.6.d.e	$300$	$6$	300.d	5.b	$2$	$2$	$2$	$48.115$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+9iq^{3}+44iq^{7}-3^{4}q^{9}+6^{3}q^{11}+\cdots\)
300.6.d.f	$300$	$6$	300.d	5.b	$2$	$4$	$4$	$48.115$	\(\Q(i, \sqrt{7})\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{6}\cdot 3^{2}\cdot 5^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+9\beta _{1}q^{3}+(-11\beta _{1}+\beta _{3})q^{7}-3^{4}q^{9}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(300, [\chi])\) into lower level spaces

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