Space of modular forms of level 300, weight 2, and Character 300.i

• Label: 300.2.i
• Level: $300$
• Weight: $2$
• Character orbit: 300.i
• Rep. character: $\chi_{300}(257,\cdot)$
• Character field: $\Q(\zeta_{4})$
• Dimension: $12$
• Newform subspaces: $3$
• Sturm bound: $120$
• Trace bound: $21$

Defining parameters

Level:	\( N \)	\(=\)	\( 300 = 2^{2} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	300.i (of order \(4\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 15 \)
Character field:			\(\Q(i)\)
Newform subspaces:			\( 3 \)
Sturm bound:			\(120\)
Trace bound:			\(21\)
Distinguishing \(T_p\):			\(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(300, [\chi])\).

	Total	New	Old
Modular forms	156	12	144
Cusp forms	84	12	72
Eisenstein series	72	0	72

Trace form

\( 12 q - 2 q^{3} + 4 q^{7} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(300, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
300.2.i.a	$300$	$2$	300.i	15.e	$4$	$4$	$2$	$2.396$	\(\Q(i, \sqrt{5})\)	None		$4$	$0$	\(0\)	\(-2\)	\(0\)	\(4\)	$2$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-1+\beta _{3})q^{3}+(1-\beta _{2})q^{7}+(1+\beta _{1}+\cdots)q^{9}+\cdots\)
300.2.i.b	$300$	$2$	300.i	15.e	$4$	$4$	$2$	$2.396$	\(\Q(i, \sqrt{6})\)	\(\Q(\sqrt{-3}) \)		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{4}]$	\(q+\beta _{1}q^{3}+2\beta _{3}q^{7}+3\beta _{2}q^{9}+4\beta _{1}q^{13}+\cdots\)
300.2.i.c	$300$	$2$	300.i	15.e	$4$	$4$	$2$	$2.396$	\(\Q(i, \sqrt{6})\)	\(\Q(\sqrt{-3}) \)		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{4}]$	\(q+\beta _{1}q^{3}-3\beta _{3}q^{7}+3\beta _{2}q^{9}-\beta _{1}q^{13}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(300, [\chi])\) into lower level spaces

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