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

• Label: 1300.2.bb
• Level: $1300$
• Weight: $2$
• Character orbit: 1300.bb
• Rep. character: $\chi_{1300}(549,\cdot)$
• Character field: $\Q(\zeta_{6})$
• Dimension: $44$
• Newform subspaces: $7$
• Sturm bound: $420$
• Trace bound: $19$

Defining parameters

Level:	\( N \)	\(=\)	\( 1300 = 2^{2} \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1300.bb (of order \(6\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 65 \)
Character field:			\(\Q(\zeta_{6})\)
Newform subspaces:			\( 7 \)
Sturm bound:			\(420\)
Trace bound:			\(19\)
Distinguishing \(T_p\):			\(3\), \(7\), \(19\)

Dimensions

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

	Total	New	Old
Modular forms	456	44	412
Cusp forms	384	44	340
Eisenstein series	72	0	72

Trace form

\( 44 q + 22 q^{9} + O(q^{10}) \)

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.bb.a	$1300$	$2$	1300.bb	65.n	$6$	$4$	$2$	$10.381$	\(\Q(\zeta_{12})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+\zeta_{12}q^{3}+(-\zeta_{12}+\zeta_{12}^{3})q^{7}-2\zeta_{12}^{2}q^{9}+\cdots\)
1300.2.bb.b	$1300$	$2$	1300.bb	65.n	$6$	$4$	$2$	$10.381$	\(\Q(\zeta_{12})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+\zeta_{12}q^{3}+(-\zeta_{12}+\zeta_{12}^{3})q^{7}-2\zeta_{12}^{2}q^{9}+\cdots\)
1300.2.bb.c	$1300$	$2$	1300.bb	65.n	$6$	$4$	$2$	$10.381$	\(\Q(\zeta_{12})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+\zeta_{12}q^{3}+(-5\zeta_{12}+5\zeta_{12}^{3})q^{7}+\cdots\)
1300.2.bb.d	$1300$	$2$	1300.bb	65.n	$6$	$4$	$2$	$10.381$	\(\Q(\zeta_{12})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+2\zeta_{12}q^{3}+(4\zeta_{12}-4\zeta_{12}^{3})q^{7}+\zeta_{12}^{2}q^{9}+\cdots\)
1300.2.bb.e	$1300$	$2$	1300.bb	65.n	$6$	$4$	$2$	$10.381$	\(\Q(\zeta_{12})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+3\zeta_{12}q^{3}+(-3\zeta_{12}+3\zeta_{12}^{3})q^{7}+\cdots\)
1300.2.bb.f	$1300$	$2$	1300.bb	65.n	$6$	$4$	$2$	$10.381$	\(\Q(\zeta_{12})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+3\zeta_{12}q^{3}+(\zeta_{12}-\zeta_{12}^{3})q^{7}+6\zeta_{12}^{2}q^{9}+\cdots\)
1300.2.bb.g	$1300$	$2$	1300.bb	65.n	$6$	$20$	$10$	$10.381$	\(\mathbb{Q}[x]/(x^{20} - \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$3^{4}$	$\mathrm{SU}(2)[C_{6}]$	\(q+\beta _{1}q^{3}+(\beta _{9}+\beta _{15})q^{7}+(1+\beta _{2}+\beta _{4}+\cdots)q^{9}+\cdots\)

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

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