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

• Label: 1300.2.f
• Level: $1300$
• Weight: $2$
• Character orbit: 1300.f
• Rep. character: $\chi_{1300}(701,\cdot)$
• Character field: $\Q$
• Dimension: $22$
• Newform subspaces: $6$
• 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.f (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 13 \)
Character field:			\(\Q\)
Newform subspaces:			\( 6 \)
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	228	22	206
Cusp forms	192	22	170
Eisenstein series	36	0	36

Trace form

\( 22 q + 18 q^{9} + O(q^{10}) \)

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.f.a	$1300$	$2$	1300.f	13.b	$2$	$2$	$2$	$10.381$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(-4\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q-2q^{3}-\zeta_{6}q^{7}+q^{9}+3\zeta_{6}q^{11}+(1+\cdots)q^{13}+\cdots\)
1300.2.f.b	$1300$	$2$	1300.f	13.b	$2$	$2$	$2$	$10.381$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(-2\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-q^{3}-\zeta_{6}q^{7}-2q^{9}+(-1+\zeta_{6})q^{13}+\cdots\)
1300.2.f.c	$1300$	$2$	1300.f	13.b	$2$	$2$	$2$	$10.381$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(2\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+q^{3}-\zeta_{6}q^{7}-2q^{9}+(1+\zeta_{6})q^{13}+\cdots\)
1300.2.f.d	$1300$	$2$	1300.f	13.b	$2$	$2$	$2$	$10.381$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(4\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+2q^{3}+\zeta_{6}q^{7}+q^{9}+3\zeta_{6}q^{11}+(-1+\cdots)q^{13}+\cdots\)
1300.2.f.e	$1300$	$2$	1300.f	13.b	$2$	$6$	$6$	$10.381$	6.0.9144576.1	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2\cdot 3$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{3}q^{3}+\beta _{5}q^{7}+(1-\beta _{4})q^{9}+(\beta _{2}+\cdots)q^{11}+\cdots\)
1300.2.f.f	$1300$	$2$	1300.f	13.b	$2$	$8$	$8$	$10.381$	8.0.796594176.1	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{8}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{3}q^{3}+(\beta _{2}-\beta _{4}-\beta _{5})q^{7}+(2+\beta _{7})q^{9}+\cdots\)

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

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