Space of modular forms of level 300, weight 5, and Character 300.k

• Label: 300.5.k
• Level: $300$
• Weight: $5$
• Character orbit: 300.k
• Rep. character: $\chi_{300}(157,\cdot)$
• Character field: $\Q(\zeta_{4})$
• Dimension: $24$
• Newform subspaces: $4$
• Sturm bound: $300$
• Trace bound: $11$

Defining parameters

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

Dimensions

The following table gives the dimensions of various subspaces of \(M_{5}(300, [\chi])\).

	Total	New	Old
Modular forms	516	24	492
Cusp forms	444	24	420
Eisenstein series	72	0	72

Trace form

\( 24 q + 140 q^{7} + O(q^{10}) \)

Decomposition of \(S_{5}^{\mathrm{new}}(300, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
300.5.k.a	$300$	$5$	300.k	5.c	$4$	$4$	$2$	$31.011$	\(\Q(i, \sqrt{6})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+3\beta _{3}q^{3}+14\beta _{1}q^{7}-3^{3}\beta _{2}q^{9}-114q^{11}+\cdots\)
300.5.k.b	$300$	$5$	300.k	5.c	$4$	$4$	$2$	$31.011$	\(\Q(i, \sqrt{6})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q-3\beta _{3}q^{3}+\beta _{1}q^{7}-3^{3}\beta _{2}q^{9}+6q^{11}+\cdots\)
300.5.k.c	$300$	$5$	300.k	5.c	$4$	$8$	$4$	$31.011$	8.0.\(\cdots\).1	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{10}\cdot 3^{2}\cdot 5^{4}$	$\mathrm{SU}(2)[C_{4}]$	\(q+3\beta _{3}q^{3}+(-\beta _{1}-\beta _{7})q^{7}-3^{3}\beta _{2}q^{9}+\cdots\)
300.5.k.d	$300$	$5$	300.k	5.c	$4$	$8$	$4$	$31.011$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(140\)	$2^{2}\cdot 3^{2}\cdot 5^{4}$	$\mathrm{SU}(2)[C_{4}]$	\(q+\beta _{3}q^{3}+(17+17\beta _{1}+4\beta _{2}+\beta _{6}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{5}^{\mathrm{old}}(300, [\chi])\) into lower level spaces

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