Space of modular forms of level 1040, weight 2, and Character 1040.df

• Label: 1040.2.df
• Level: $1040$
• Weight: $2$
• Character orbit: 1040.df
• Rep. character: $\chi_{1040}(49,\cdot)$
• Character field: $\Q(\zeta_{6})$
• Dimension: $80$
• Newform subspaces: $6$
• Sturm bound: $336$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 1040 = 2^{4} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1040.df (of order \(6\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 65 \)
Character field:			\(\Q(\zeta_{6})\)
Newform subspaces:			\( 6 \)
Sturm bound:			\(336\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(1040, [\chi])\).

	Total	New	Old
Modular forms	360	88	272
Cusp forms	312	80	232
Eisenstein series	48	8	40

Trace form

\( 80 q + 34 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(1040, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
1040.2.df.a	$1040$	$2$	1040.df	65.l	$6$	$8$	$4$	$8.304$	8.0.\(\cdots\).1	None		$2$	$0$	\(0\)	\(0\)	\(-3\)	\(5\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(\beta _{1}-\beta _{2})q^{3}+\beta _{7}q^{5}+(2-\beta _{3}-\beta _{4}+\cdots)q^{7}+\cdots\)
1040.2.df.b	$1040$	$2$	1040.df	65.l	$6$	$8$	$4$	$8.304$	8.0.49787136.1	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{6}]$	\(q+\beta _{2}q^{3}+(-1+\beta _{2}+\beta _{3}-\beta _{6})q^{5}+\cdots\)
1040.2.df.c	$1040$	$2$	1040.df	65.l	$6$	$8$	$4$	$8.304$	8.0.\(\cdots\).1	None		$2$	$0$	\(0\)	\(0\)	\(3\)	\(-5\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+\beta _{2}q^{3}-\beta _{7}q^{5}+(-\beta _{3}+\beta _{5}+\beta _{6}+\cdots)q^{7}+\cdots\)
1040.2.df.d	$1040$	$2$	1040.df	65.l	$6$	$16$	$8$	$8.304$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{8}$	$\mathrm{SU}(2)[C_{6}]$	\(q-\beta _{3}q^{3}-\beta _{12}q^{5}-\beta _{8}q^{7}+(1+\beta _{1}+\cdots)q^{9}+\cdots\)
1040.2.df.e	$1040$	$2$	1040.df	65.l	$6$	$20$	$10$	$8.304$	\(\mathbb{Q}[x]/(x^{20} + \cdots)\)	None		$2$	$0$	\(0\)	\(0\)	\(-3\)	\(-5\)	$2^{8}$	$\mathrm{SU}(2)[C_{6}]$	\(q+\beta _{2}q^{3}+(-\beta _{4}+\beta _{8})q^{5}+(2\beta _{6}+\beta _{8}+\cdots)q^{7}+\cdots\)
1040.2.df.f	$1040$	$2$	1040.df	65.l	$6$	$20$	$10$	$8.304$	\(\mathbb{Q}[x]/(x^{20} + \cdots)\)	None		$2$	$0$	\(0\)	\(0\)	\(3\)	\(5\)	$2^{8}$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-\beta _{1}-\beta _{2})q^{3}+(\beta _{4}-\beta _{8})q^{5}+(2\beta _{6}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(1040, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(1040, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(65, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(130, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(260, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(520, [\chi])\)\(^{\oplus 2}\)