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

• Label: 1300.2.c
• Level: $1300$
• Weight: $2$
• Character orbit: 1300.c
• Rep. character: $\chi_{1300}(1249,\cdot)$
• Character field: $\Q$
• Dimension: $18$
• Newform subspaces: $6$
• Sturm bound: $420$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 1300 = 2^{2} \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1300.c (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 5 \)
Character field:			\(\Q\)
Newform subspaces:			\( 6 \)
Sturm bound:			\(420\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(3\), \(7\)

Dimensions

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

	Total	New	Old
Modular forms	228	18	210
Cusp forms	192	18	174
Eisenstein series	36	0	36

Trace form

\( 18 q - 30 q^{9} + O(q^{10}) \)

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.c.a	$1300$	$2$	1300.c	5.b	$2$	$2$	$2$	$10.381$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+2iq^{3}-2iq^{7}-q^{9}+4q^{11}-iq^{13}+\cdots\)
1300.2.c.b	$1300$	$2$	1300.c	5.b	$2$	$2$	$2$	$10.381$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+iq^{3}+2iq^{7}+2q^{9}-2q^{11}+iq^{13}+\cdots\)
1300.2.c.c	$1300$	$2$	1300.c	5.b	$2$	$2$	$2$	$10.381$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+2iq^{7}+3q^{9}-2q^{11}-iq^{13}-6iq^{17}+\cdots\)
1300.2.c.d	$1300$	$2$	1300.c	5.b	$2$	$2$	$2$	$10.381$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+3iq^{7}+3q^{9}+3q^{11}+iq^{13}+iq^{17}+\cdots\)
1300.2.c.e	$1300$	$2$	1300.c	5.b	$2$	$4$	$4$	$10.381$	\(\Q(i, \sqrt{33})\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{3}+(-\beta _{1}-\beta _{2})q^{7}+(-6+\beta _{3})q^{9}+\cdots\)
1300.2.c.f	$1300$	$2$	1300.c	5.b	$2$	$6$	$6$	$10.381$	6.0.5089536.1	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{3}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-\beta _{1}-\beta _{4})q^{3}+(-\beta _{4}-\beta _{5})q^{7}+\cdots\)

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

\( S_{2}^{\mathrm{old}}(1300, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(50, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(65, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(100, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(130, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(260, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(325, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(650, [\chi])\)\(^{\oplus 2}\)