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

• Label: 300.5.g
• Level: $300$
• Weight: $5$
• Character orbit: 300.g
• Rep. character: $\chi_{300}(101,\cdot)$
• Character field: $\Q$
• Dimension: $25$
• Newform subspaces: $8$
• Sturm bound: $300$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 300 = 2^{2} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	300.g (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 3 \)
Character field:			\(\Q\)
Newform subspaces:			\( 8 \)
Sturm bound:			\(300\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(7\)

Dimensions

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

	Total	New	Old
Modular forms	258	25	233
Cusp forms	222	25	197
Eisenstein series	36	0	36

Trace form

\( 25 q - q^{3} - 14 q^{7} - 25 q^{9} + O(q^{10}) \)

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.g.a	$300$	$5$	300.g	3.b	$2$	$1$	$1$	$31.011$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓	$2$	$0$	\(0\)	\(-9\)	\(0\)	\(-71\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q-9q^{3}-71q^{7}+3^{4}q^{9}-191q^{13}+\cdots\)
300.5.g.b	$300$	$5$	300.g	3.b	$2$	$1$	$1$	$31.011$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓	$2$	$0$	\(0\)	\(-9\)	\(0\)	\(94\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q-9q^{3}+94q^{7}+3^{4}q^{9}-146q^{13}+\cdots\)
300.5.g.c	$300$	$5$	300.g	3.b	$2$	$1$	$1$	$31.011$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓	$2$	$0$	\(0\)	\(9\)	\(0\)	\(71\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q+9q^{3}+71q^{7}+3^{4}q^{9}+191q^{13}+\cdots\)
300.5.g.d	$300$	$5$	300.g	3.b	$2$	$2$	$2$	$31.011$	\(\Q(\sqrt{-5}) \)	None		$2$	$0$	\(0\)	\(-12\)	\(0\)	\(-148\)	$3$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-6+\beta )q^{3}-74q^{7}+(-9-12\beta )q^{9}+\cdots\)
300.5.g.e	$300$	$5$	300.g	3.b	$2$	$4$	$4$	$31.011$	4.0.9254440.1	None		$2$	$0$	\(0\)	\(-5\)	\(0\)	\(-70\)	$2\cdot 3^{2}\cdot 5$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-1+\beta _{1})q^{3}+(-18-2\beta _{1}-\beta _{3})q^{7}+\cdots\)
300.5.g.f	$300$	$5$	300.g	3.b	$2$	$4$	$4$	$31.011$	4.0.9254440.1	None		$2$	$0$	\(0\)	\(5\)	\(0\)	\(70\)	$2\cdot 3^{2}\cdot 5$	$\mathrm{SU}(2)[C_{2}]$	\(q+(1-\beta _{1})q^{3}+(18+2\beta _{1}+\beta _{3})q^{7}+\cdots\)
300.5.g.g	$300$	$5$	300.g	3.b	$2$	$4$	$4$	$31.011$	\(\Q(\sqrt{-5}, \sqrt{34})\)	None		$2$	$0$	\(0\)	\(20\)	\(0\)	\(40\)	$3^{2}\cdot 5$	$\mathrm{SU}(2)[C_{2}]$	\(q+(5+\beta _{1})q^{3}+(10-\beta _{1}-\beta _{2})q^{7}+(-14+\cdots)q^{9}+\cdots\)
300.5.g.h	$300$	$5$	300.g	3.b	$2$	$8$	$8$	$31.011$	\(\mathbb{Q}[x]/(x^{8} + \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{7}\cdot 3^{2}\cdot 5^{6}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{2}q^{3}-\beta _{1}q^{7}+(-7-\beta _{5})q^{9}+\beta _{6}q^{11}+\cdots\)

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

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