Space of modular forms of level 1300, weight 2, and Character 1300.bn

• Label: 1300.2.bn
• Level: $1300$
• Weight: $2$
• Character orbit: 1300.bn
• Rep. character: $\chi_{1300}(93,\cdot)$
• Character field: $\Q(\zeta_{12})$
• Dimension: $84$
• Newform subspaces: $5$
• Sturm bound: $420$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 1300 = 2^{2} \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1300.bn (of order \(12\) and degree \(4\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 65 \)
Character field:			\(\Q(\zeta_{12})\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(420\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(1300, [\chi])\).

	Total	New	Old
Modular forms	912	84	828
Cusp forms	768	84	684
Eisenstein series	144	0	144

Trace form

\( 84 q - 24 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(1300, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
1300.2.bn.a	$1300$	$2$	1300.bn	65.t	$12$	$4$	$1$	$10.381$	\(\Q(\zeta_{12})\)	None		$2$	$0$	\(0\)	\(-4\)	\(0\)	\(-2\)	$1$	$\mathrm{SU}(2)[C_{12}]$	\(q+(-1+\zeta_{12})q^{3}+(-1+2\zeta_{12}+\zeta_{12}^{2}+\cdots)q^{7}+\cdots\)
1300.2.bn.b	$1300$	$2$	1300.bn	65.t	$12$	$4$	$1$	$10.381$	\(\Q(\zeta_{12})\)	None		$2$	$0$	\(0\)	\(2\)	\(0\)	\(-4\)	$1$	$\mathrm{SU}(2)[C_{12}]$	\(q+(1-\zeta_{12}^{2}-\zeta_{12}^{3})q^{3}+(-2-\zeta_{12}+\cdots)q^{7}+\cdots\)
1300.2.bn.c	$1300$	$2$	1300.bn	65.t	$12$	$16$	$4$	$10.381$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}$	$\mathrm{SU}(2)[C_{12}]$	\(q+\beta _{13}q^{3}+(\beta _{11}-\beta _{13}-\beta _{14})q^{7}+\cdots\)
1300.2.bn.d	$1300$	$2$	1300.bn	65.t	$12$	$20$	$5$	$10.381$	\(\mathbb{Q}[x]/(x^{20} + \cdots)\)	None		$2$	$0$	\(0\)	\(2\)	\(0\)	\(6\)	$2^{4}$	$\mathrm{SU}(2)[C_{12}]$	\(q-\beta _{16}q^{3}+(1+\beta _{3}-\beta _{6}-\beta _{7}+\beta _{9}+\cdots)q^{7}+\cdots\)
1300.2.bn.e	$1300$	$2$	1300.bn	65.t	$12$	$40$	$10$	$10.381$		None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{12}]$

Decomposition of \(S_{2}^{\mathrm{old}}(1300, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(1300, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(65, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(130, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(260, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(325, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(650, [\chi])\)\(^{\oplus 2}\)