Space of modular forms of level 2040, weight 2, and trivial character

• Label: 2040.2.a
• Level: $2040$
• Weight: $2$
• Character orbit: 2040.a
• Rep. character: $\chi_{2040}(1,\cdot)$
• Character field: $\Q$
• Dimension: $32$
• Newform subspaces: $24$
• Sturm bound: $864$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 2040 = 2^{3} \cdot 3 \cdot 5 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2040.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 24 \)
Sturm bound:			\(864\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(7\), \(11\), \(13\), \(19\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(2040))\).

	Total	New	Old
Modular forms	448	32	416
Cusp forms	417	32	385
Eisenstein series	31	0	31

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

\(2\)	\(3\)	\(5\)	\(17\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	\(+\)	$+$	\(2\)
\(+\)	\(+\)	\(+\)	\(-\)	$-$	\(2\)
\(+\)	\(+\)	\(-\)	\(+\)	$-$	\(3\)
\(+\)	\(+\)	\(-\)	\(-\)	$+$	\(2\)
\(+\)	\(-\)	\(+\)	\(+\)	$-$	\(2\)
\(+\)	\(-\)	\(+\)	\(-\)	$+$	\(1\)
\(+\)	\(-\)	\(-\)	\(+\)	$+$	\(1\)
\(+\)	\(-\)	\(-\)	\(-\)	$-$	\(3\)
\(-\)	\(+\)	\(+\)	\(+\)	$-$	\(3\)
\(-\)	\(+\)	\(+\)	\(-\)	$+$	\(2\)
\(-\)	\(+\)	\(-\)	\(+\)	$+$	\(1\)
\(-\)	\(+\)	\(-\)	\(-\)	$-$	\(3\)
\(-\)	\(-\)	\(+\)	\(+\)	$+$	\(2\)
\(-\)	\(-\)	\(+\)	\(-\)	$-$	\(2\)
\(-\)	\(-\)	\(-\)	\(+\)	$-$	\(2\)
\(-\)	\(-\)	\(-\)	\(-\)	$+$	\(1\)
Plus space				\(+\)	\(12\)
Minus space				\(-\)	\(20\)

Trace form

\( 32 q - 4 q^{3} - 8 q^{7} + 32 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(2040))\) 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	3	5	17
2040.2.a.a	$2040$	$2$	2040.a	1.a	$1$	$1$	$1$	$16.289$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-1\)	\(-1\)	\(-3\)		$+$	$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{5}-3q^{7}+q^{9}+3q^{11}+4q^{13}+\cdots\)
2040.2.a.b	$2040$	$2$	2040.a	1.a	$1$	$1$	$1$	$16.289$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-1\)	\(-1\)	\(0\)		$-$	$+$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{5}+q^{9}-4q^{11}+6q^{13}+\cdots\)
2040.2.a.c	$2040$	$2$	2040.a	1.a	$1$	$1$	$1$	$16.289$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-1\)	\(-1\)	\(0\)		$+$	$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{5}+q^{9}-2q^{13}+q^{15}-q^{17}+\cdots\)
2040.2.a.d	$2040$	$2$	2040.a	1.a	$1$	$1$	$1$	$16.289$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-1\)	\(1\)	\(-4\)		$-$	$+$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{5}-4q^{7}+q^{9}+4q^{11}+2q^{13}+\cdots\)
2040.2.a.e	$2040$	$2$	2040.a	1.a	$1$	$1$	$1$	$16.289$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-1\)	\(1\)	\(-3\)		$+$	$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{5}-3q^{7}+q^{9}-3q^{11}+4q^{13}+\cdots\)
2040.2.a.f	$2040$	$2$	2040.a	1.a	$1$	$1$	$1$	$16.289$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-1\)	\(1\)	\(0\)		$+$	$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{5}+q^{9}-2q^{13}-q^{15}+q^{17}+\cdots\)
2040.2.a.g	$2040$	$2$	2040.a	1.a	$1$	$1$	$1$	$16.289$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-1\)	\(1\)	\(3\)		$-$	$+$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{5}+3q^{7}+q^{9}-q^{11}-6q^{13}+\cdots\)
2040.2.a.h	$2040$	$2$	2040.a	1.a	$1$	$1$	$1$	$16.289$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(1\)	\(-1\)	\(-4\)		$+$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{5}-4q^{7}+q^{9}+2q^{13}-q^{15}+\cdots\)
2040.2.a.i	$2040$	$2$	2040.a	1.a	$1$	$1$	$1$	$16.289$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(1\)	\(-1\)	\(-3\)		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{5}-3q^{7}+q^{9}+5q^{11}-2q^{13}+\cdots\)
2040.2.a.j	$2040$	$2$	2040.a	1.a	$1$	$1$	$1$	$16.289$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(1\)	\(-1\)	\(-1\)		$+$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{5}-q^{7}+q^{9}-5q^{11}+4q^{13}+\cdots\)
2040.2.a.k	$2040$	$2$	2040.a	1.a	$1$	$1$	$1$	$16.289$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(1\)	\(-1\)	\(0\)		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{5}+q^{9}-4q^{11}-2q^{13}+\cdots\)
2040.2.a.l	$2040$	$2$	2040.a	1.a	$1$	$1$	$1$	$16.289$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(1\)	\(-1\)	\(1\)		$+$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{5}+q^{7}+q^{9}+5q^{11}+2q^{13}+\cdots\)
2040.2.a.m	$2040$	$2$	2040.a	1.a	$1$	$1$	$1$	$16.289$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(1\)	\(1\)	\(-3\)		$-$	$-$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{5}-3q^{7}+q^{9}-q^{11}-6q^{13}+\cdots\)
2040.2.a.n	$2040$	$2$	2040.a	1.a	$1$	$1$	$1$	$16.289$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(1\)	\(1\)	\(-2\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{5}-2q^{7}+q^{9}+q^{15}-q^{17}+\cdots\)
2040.2.a.o	$2040$	$2$	2040.a	1.a	$1$	$1$	$1$	$16.289$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(1\)	\(1\)	\(-1\)		$+$	$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{5}-q^{7}+q^{9}-3q^{11}-4q^{13}+\cdots\)
2040.2.a.p	$2040$	$2$	2040.a	1.a	$1$	$1$	$1$	$16.289$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(1\)	\(1\)	\(2\)		$+$	$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{5}+2q^{7}+q^{9}+4q^{11}+4q^{13}+\cdots\)
2040.2.a.q	$2040$	$2$	2040.a	1.a	$1$	$1$	$1$	$16.289$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(1\)	\(1\)	\(3\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{5}+3q^{7}+q^{9}+5q^{11}+q^{15}+\cdots\)
2040.2.a.r	$2040$	$2$	2040.a	1.a	$1$	$2$	$2$	$16.289$	\(\Q(\sqrt{17}) \)	None	✓	$1$	$1$	\(0\)	\(-2\)	\(-2\)	\(-1\)		$-$	$+$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{5}-\beta q^{7}+q^{9}+\beta q^{11}-2q^{13}+\cdots\)
2040.2.a.s	$2040$	$2$	2040.a	1.a	$1$	$2$	$2$	$16.289$	\(\Q(\sqrt{73}) \)	None	✓	$1$	$0$	\(0\)	\(-2\)	\(-2\)	\(-1\)		$-$	$+$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{5}-\beta q^{7}+q^{9}+(2-\beta )q^{11}+\cdots\)
2040.2.a.t	$2040$	$2$	2040.a	1.a	$1$	$2$	$2$	$16.289$	\(\Q(\sqrt{17}) \)	None	✓	$1$	$0$	\(0\)	\(-2\)	\(-2\)	\(5\)		$+$	$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{5}+(3-\beta )q^{7}+q^{9}+(1-\beta )q^{11}+\cdots\)
2040.2.a.u	$2040$	$2$	2040.a	1.a	$1$	$2$	$2$	$16.289$	\(\Q(\sqrt{33}) \)	None	✓	$1$	$0$	\(0\)	\(-2\)	\(2\)	\(1\)		$-$	$+$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{5}+\beta q^{7}+q^{9}+(-4+\beta )q^{11}+\cdots\)
2040.2.a.v	$2040$	$2$	2040.a	1.a	$1$	$2$	$2$	$16.289$	\(\Q(\sqrt{17}) \)	None	✓	$1$	$0$	\(0\)	\(2\)	\(-2\)	\(-1\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{5}-\beta q^{7}+q^{9}-\beta q^{11}+(-2+\cdots)q^{13}+\cdots\)
2040.2.a.w	$2040$	$2$	2040.a	1.a	$1$	$2$	$2$	$16.289$	\(\Q(\sqrt{33}) \)	None	✓	$1$	$0$	\(0\)	\(2\)	\(2\)	\(1\)		$+$	$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{5}+\beta q^{7}+q^{9}-\beta q^{11}-2q^{13}+\cdots\)
2040.2.a.x	$2040$	$2$	2040.a	1.a	$1$	$3$	$3$	$16.289$	3.3.316.1	None	✓	$1$	$0$	\(0\)	\(-3\)	\(3\)	\(3\)		$+$	$+$	$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{5}+(1+\beta _{1})q^{7}+q^{9}+(2-2\beta _{1}+\cdots)q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(2040))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(2040)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(17))\)\(^{\oplus 16}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(34))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(51))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(60))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(68))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(85))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(102))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(120))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(136))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(170))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(204))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(255))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(340))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(408))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(510))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(680))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1020))\)\(^{\oplus 2}\)