Space of modular forms of level 1040, weight 2, and trivial character

• Label: 1040.2.a
• Level: $1040$
• Weight: $2$
• Character orbit: 1040.a
• Rep. character: $\chi_{1040}(1,\cdot)$
• Character field: $\Q$
• Dimension: $24$
• Newform subspaces: $15$
• Sturm bound: $336$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 1040 = 2^{4} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1040.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 15 \)
Sturm bound:			\(336\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(3\), \(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1040))\).

	Total	New	Old
Modular forms	180	24	156
Cusp forms	157	24	133
Eisenstein series	23	0	23

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(2\)	\(5\)	\(13\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	$+$	\(3\)
\(+\)	\(+\)	\(-\)	$-$	\(4\)
\(+\)	\(-\)	\(+\)	$-$	\(4\)
\(+\)	\(-\)	\(-\)	$+$	\(1\)
\(-\)	\(+\)	\(+\)	$-$	\(3\)
\(-\)	\(+\)	\(-\)	$+$	\(2\)
\(-\)	\(-\)	\(+\)	$+$	\(2\)
\(-\)	\(-\)	\(-\)	$-$	\(5\)
Plus space			\(+\)	\(8\)
Minus space			\(-\)	\(16\)

Trace form

\( 24 q + 4 q^{7} + 32 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1040))\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Inner twists	Rank*	Traces					A-L signs			Fricke sign	Coefficient ring index	Sato-Tate	$q$-expansion
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		2	5	13
1040.2.a.a	$1040$	$2$	1040.a	1.a	$1$	$1$	$1$	$8.304$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-2\)	\(-1\)	\(-2\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}-q^{5}-2q^{7}+q^{9}-4q^{11}+\cdots\)
1040.2.a.b	$1040$	$2$	1040.a	1.a	$1$	$1$	$1$	$8.304$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-2\)	\(-1\)	\(4\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}-q^{5}+4q^{7}+q^{9}+2q^{11}+\cdots\)
1040.2.a.c	$1040$	$2$	1040.a	1.a	$1$	$1$	$1$	$8.304$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-2\)	\(1\)	\(0\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}+q^{5}+q^{9}-2q^{11}+q^{13}+\cdots\)
1040.2.a.d	$1040$	$2$	1040.a	1.a	$1$	$1$	$1$	$8.304$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(-1\)	\(0\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}-3q^{9}+4q^{11}-q^{13}-6q^{17}+\cdots\)
1040.2.a.e	$1040$	$2$	1040.a	1.a	$1$	$1$	$1$	$8.304$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(1\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}-3q^{9}+q^{13}+2q^{17}+8q^{19}+\cdots\)
1040.2.a.f	$1040$	$2$	1040.a	1.a	$1$	$1$	$1$	$8.304$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(2\)	\(-1\)	\(4\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}-q^{5}+4q^{7}+q^{9}-2q^{11}+\cdots\)
1040.2.a.g	$1040$	$2$	1040.a	1.a	$1$	$1$	$1$	$8.304$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(2\)	\(1\)	\(4\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}+q^{5}+4q^{7}+q^{9}+6q^{11}+\cdots\)
1040.2.a.h	$1040$	$2$	1040.a	1.a	$1$	$2$	$2$	$8.304$	\(\Q(\sqrt{3}) \)	None	✓	$1$	$1$	\(0\)	\(-2\)	\(-2\)	\(-4\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{3}-q^{5}-2q^{7}+(1-2\beta )q^{9}+\cdots\)
1040.2.a.i	$1040$	$2$	1040.a	1.a	$1$	$2$	$2$	$8.304$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$1$	\(0\)	\(-2\)	\(-2\)	\(0\)		$+$	$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(-1-\beta )q^{3}-q^{5}+(3+2\beta )q^{9}+(-3+\cdots)q^{11}+\cdots\)
1040.2.a.j	$1040$	$2$	1040.a	1.a	$1$	$2$	$2$	$8.304$	\(\Q(\sqrt{2}) \)	None	✓	$1$	$1$	\(0\)	\(0\)	\(2\)	\(-4\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+q^{5}+(-2-2\beta )q^{7}-q^{9}+\cdots\)
1040.2.a.k	$1040$	$2$	1040.a	1.a	$1$	$2$	$2$	$8.304$	\(\Q(\sqrt{6}) \)	None	✓	$1$	$0$	\(0\)	\(0\)	\(2\)	\(-4\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+q^{5}-2q^{7}+3q^{9}+(2+\beta )q^{11}+\cdots\)
1040.2.a.l	$1040$	$2$	1040.a	1.a	$1$	$2$	$2$	$8.304$	\(\Q(\sqrt{3}) \)	None	✓	$1$	$0$	\(0\)	\(2\)	\(-2\)	\(0\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{3}-q^{5}+2\beta q^{7}+(1+2\beta )q^{9}+\cdots\)
1040.2.a.m	$1040$	$2$	1040.a	1.a	$1$	$2$	$2$	$8.304$	\(\Q(\sqrt{3}) \)	None	✓	$1$	$0$	\(0\)	\(2\)	\(-2\)	\(0\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{3}-q^{5}-2\beta q^{7}+(1+2\beta )q^{9}+\cdots\)
1040.2.a.n	$1040$	$2$	1040.a	1.a	$1$	$2$	$2$	$8.304$	\(\Q(\sqrt{2}) \)	None	✓	$1$	$0$	\(0\)	\(4\)	\(2\)	\(4\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(2+\beta )q^{3}+q^{5}+2q^{7}+(3+4\beta )q^{9}+\cdots\)
1040.2.a.o	$1040$	$2$	1040.a	1.a	$1$	$3$	$3$	$8.304$	3.3.564.1	None	✓	$1$	$0$	\(0\)	\(-2\)	\(3\)	\(2\)		$-$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{2})q^{3}+q^{5}+(1+\beta _{1})q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1040))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1040)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(26))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(52))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(65))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(80))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(104))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(130))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(208))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(260))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(520))\)\(^{\oplus 2}\)