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

• Label: 1300.2.y
• Level: $1300$
• Weight: $2$
• Character orbit: 1300.y
• Rep. character: $\chi_{1300}(101,\cdot)$
• Character field: $\Q(\zeta_{6})$
• Dimension: $46$
• 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.y (of order \(6\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 13 \)
Character field:			\(\Q(\zeta_{6})\)
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	456	46	410
Cusp forms	384	46	338
Eisenstein series	72	0	72

Trace form

\( 46 q + q^{3} - 3 q^{7} - 28 q^{9} + O(q^{10}) \)

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.y.a	$1300$	$2$	1300.y	13.e	$6$	$2$	$1$	$10.381$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(-1\)	\(0\)	\(3\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-1+\zeta_{6})q^{3}+(2-\zeta_{6})q^{7}+2\zeta_{6}q^{9}+\cdots\)
1300.2.y.b	$1300$	$2$	1300.y	13.e	$6$	$8$	$4$	$10.381$	8.0.22581504.2	None		$2$	$0$	\(0\)	\(2\)	\(0\)	\(-6\)	$2^{4}\cdot 3$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-\beta _{4}+\beta _{6}-\beta _{7})q^{3}+(-1-\beta _{1}+\cdots)q^{7}+\cdots\)
1300.2.y.c	$1300$	$2$	1300.y	13.e	$6$	$10$	$5$	$10.381$	\(\mathbb{Q}[x]/(x^{10} + \cdots)\)	None		$2$	$0$	\(0\)	\(-1\)	\(0\)	\(0\)	$3^{2}$	$\mathrm{SU}(2)[C_{6}]$	\(q+\beta _{9}q^{3}+(-\beta _{4}+\beta _{6}-\beta _{7})q^{7}+(-2+\cdots)q^{9}+\cdots\)
1300.2.y.d	$1300$	$2$	1300.y	13.e	$6$	$10$	$5$	$10.381$	\(\mathbb{Q}[x]/(x^{10} + \cdots)\)	None		$2$	$0$	\(0\)	\(1\)	\(0\)	\(0\)	$3^{2}$	$\mathrm{SU}(2)[C_{6}]$	\(q+(\beta _{5}+\beta _{9})q^{3}+(-\beta _{6}-\beta _{8})q^{7}+(2\beta _{1}+\cdots)q^{9}+\cdots\)
1300.2.y.e	$1300$	$2$	1300.y	13.e	$6$	$16$	$8$	$10.381$	\(\mathbb{Q}[x]/(x^{16} + \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{12}$	$\mathrm{SU}(2)[C_{6}]$	\(q-\beta _{2}q^{3}+(\beta _{6}-\beta _{9})q^{7}+(-\beta _{3}+\beta _{7}+\cdots)q^{9}+\cdots\)

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

\( S_{2}^{\mathrm{old}}(1300, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(13, [\chi])\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(26, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(52, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(65, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(130, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(260, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(325, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(650, [\chi])\)\(^{\oplus 2}\)