Space of modular forms of level 8040, weight 2, and trivial character

• Label: 8040.2.a
• Level: $8040$
• Weight: $2$
• Character orbit: 8040.a
• Rep. character: $\chi_{8040}(1,\cdot)$
• Character field: $\Q$
• Dimension: $132$
• Newform subspaces: $29$
• Sturm bound: $3264$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 8040 = 2^{3} \cdot 3 \cdot 5 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8040.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 29 \)
Sturm bound:			\(3264\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(7\), \(11\), \(13\), \(17\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(8040))\).

	Total	New	Old
Modular forms	1648	132	1516
Cusp forms	1617	132	1485
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.

\(2\)	\(3\)	\(5\)	\(67\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	\(+\)	$+$	\(9\)
\(+\)	\(+\)	\(+\)	\(-\)	$-$	\(7\)
\(+\)	\(+\)	\(-\)	\(+\)	$-$	\(11\)
\(+\)	\(+\)	\(-\)	\(-\)	$+$	\(6\)
\(+\)	\(-\)	\(+\)	\(+\)	$-$	\(10\)
\(+\)	\(-\)	\(+\)	\(-\)	$+$	\(6\)
\(+\)	\(-\)	\(-\)	\(+\)	$+$	\(6\)
\(+\)	\(-\)	\(-\)	\(-\)	$-$	\(11\)
\(-\)	\(+\)	\(+\)	\(+\)	$-$	\(8\)
\(-\)	\(+\)	\(+\)	\(-\)	$+$	\(9\)
\(-\)	\(+\)	\(-\)	\(+\)	$+$	\(8\)
\(-\)	\(+\)	\(-\)	\(-\)	$-$	\(8\)
\(-\)	\(-\)	\(+\)	\(+\)	$+$	\(9\)
\(-\)	\(-\)	\(+\)	\(-\)	$-$	\(8\)
\(-\)	\(-\)	\(-\)	\(+\)	$-$	\(11\)
\(-\)	\(-\)	\(-\)	\(-\)	$+$	\(5\)
Plus space				\(+\)	\(58\)
Minus space				\(-\)	\(74\)

Trace form

\( 132 q + 132 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(8040))\) 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	67
8040.2.a.a	$8040$	$2$	8040.a	1.a	$1$	$1$	$1$	$64.200$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-1\)	\(-1\)	\(0\)		$+$	$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{5}+q^{9}-4q^{11}-2q^{13}+\cdots\)
8040.2.a.b	$8040$	$2$	8040.a	1.a	$1$	$1$	$1$	$64.200$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-1\)	\(-1\)	\(0\)		$+$	$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{5}+q^{9}+2q^{13}+q^{15}+2q^{17}+\cdots\)
8040.2.a.c	$8040$	$2$	8040.a	1.a	$1$	$1$	$1$	$64.200$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-1\)	\(1\)	\(-4\)		$-$	$+$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{5}-4q^{7}+q^{9}-4q^{11}+2q^{13}+\cdots\)
8040.2.a.d	$8040$	$2$	8040.a	1.a	$1$	$1$	$1$	$64.200$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-1\)	\(1\)	\(-2\)		$+$	$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{5}-2q^{7}+q^{9}-4q^{11}+6q^{13}+\cdots\)
8040.2.a.e	$8040$	$2$	8040.a	1.a	$1$	$1$	$1$	$64.200$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-1\)	\(1\)	\(-2\)		$+$	$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{5}-2q^{7}+q^{9}-2q^{13}-q^{15}+\cdots\)
8040.2.a.f	$8040$	$2$	8040.a	1.a	$1$	$1$	$1$	$64.200$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-1\)	\(1\)	\(2\)		$-$	$+$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{5}+2q^{7}+q^{9}+2q^{11}+2q^{13}+\cdots\)
8040.2.a.g	$8040$	$2$	8040.a	1.a	$1$	$1$	$1$	$64.200$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(1\)	\(-1\)	\(-4\)		$+$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{5}-4q^{7}+q^{9}-4q^{11}+2q^{13}+\cdots\)
8040.2.a.h	$8040$	$2$	8040.a	1.a	$1$	$1$	$1$	$64.200$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(1\)	\(-1\)	\(-2\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{5}-2q^{7}+q^{9}+6q^{13}-q^{15}+\cdots\)
8040.2.a.i	$8040$	$2$	8040.a	1.a	$1$	$1$	$1$	$64.200$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(1\)	\(-1\)	\(4\)		$+$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{5}+4q^{7}+q^{9}-4q^{11}+6q^{13}+\cdots\)
8040.2.a.j	$8040$	$2$	8040.a	1.a	$1$	$1$	$1$	$64.200$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(1\)	\(1\)	\(-4\)		$+$	$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{5}-4q^{7}+q^{9}-4q^{13}+q^{15}+\cdots\)
8040.2.a.k	$8040$	$2$	8040.a	1.a	$1$	$1$	$1$	$64.200$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(1\)	\(1\)	\(-1\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{5}-q^{7}+q^{9}+3q^{11}+2q^{13}+\cdots\)
8040.2.a.l	$8040$	$2$	8040.a	1.a	$1$	$1$	$1$	$64.200$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(1\)	\(1\)	\(4\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{5}+4q^{7}+q^{9}+6q^{11}-2q^{13}+\cdots\)
8040.2.a.m	$8040$	$2$	8040.a	1.a	$1$	$2$	$2$	$64.200$	\(\Q(\sqrt{17}) \)	None	✓	$1$	$0$	\(0\)	\(2\)	\(2\)	\(1\)		$+$	$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{5}+\beta q^{7}+q^{9}+(4-\beta )q^{11}+\cdots\)
8040.2.a.n	$8040$	$2$	8040.a	1.a	$1$	$5$	$5$	$64.200$	5.5.1034533.1	None	✓	$1$	$0$	\(0\)	\(-5\)	\(-5\)	\(-1\)		$+$	$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{5}+(-\beta _{1}-\beta _{4})q^{7}+q^{9}+\cdots\)
8040.2.a.o	$8040$	$2$	8040.a	1.a	$1$	$5$	$5$	$64.200$	5.5.630757.1	None	✓	$1$	$1$	\(0\)	\(-5\)	\(5\)	\(6\)		$+$	$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{5}+(1-\beta _{3})q^{7}+q^{9}+(\beta _{1}+\cdots)q^{11}+\cdots\)
8040.2.a.p	$8040$	$2$	8040.a	1.a	$1$	$5$	$5$	$64.200$	5.5.81589.1	None	✓	$1$	$1$	\(0\)	\(5\)	\(5\)	\(-7\)		$-$	$-$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{5}+(-1-\beta _{3}-\beta _{4})q^{7}+q^{9}+\cdots\)
8040.2.a.q	$8040$	$2$	8040.a	1.a	$1$	$6$	$6$	$64.200$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(-6\)	\(6\)	\(-2\)		$-$	$+$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{5}-\beta _{1}q^{7}+q^{9}+(-1-\beta _{4}+\cdots)q^{11}+\cdots\)
8040.2.a.r	$8040$	$2$	8040.a	1.a	$1$	$6$	$6$	$64.200$	6.6.15350572.1	None	✓	$1$	$1$	\(0\)	\(6\)	\(-6\)	\(-3\)		$+$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{5}+(-1+\beta _{4})q^{7}+q^{9}+(-\beta _{1}+\cdots)q^{11}+\cdots\)
8040.2.a.s	$8040$	$2$	8040.a	1.a	$1$	$6$	$6$	$64.200$	6.6.24199421.1	None	✓	$1$	$1$	\(0\)	\(6\)	\(6\)	\(-1\)		$+$	$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{5}+\beta _{4}q^{7}+q^{9}+(-\beta _{1}-\beta _{3}+\cdots)q^{11}+\cdots\)
8040.2.a.t	$8040$	$2$	8040.a	1.a	$1$	$7$	$7$	$64.200$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(7\)	\(-7\)	\(10\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{5}+(1+\beta _{3})q^{7}+q^{9}+(-\beta _{1}+\cdots)q^{11}+\cdots\)
8040.2.a.u	$8040$	$2$	8040.a	1.a	$1$	$8$	$8$	$64.200$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(-8\)	\(-8\)	\(1\)		$-$	$+$	$+$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{5}-\beta _{4}q^{7}+q^{9}+(1+\beta _{1}+\cdots)q^{11}+\cdots\)
8040.2.a.v	$8040$	$2$	8040.a	1.a	$1$	$8$	$8$	$64.200$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(-8\)	\(8\)	\(3\)		$-$	$+$	$-$	$-$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{5}-\beta _{7}q^{7}+q^{9}+(1+\beta _{4}+\cdots)q^{11}+\cdots\)
8040.2.a.w	$8040$	$2$	8040.a	1.a	$1$	$8$	$8$	$64.200$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(8\)	\(-8\)	\(0\)		$+$	$-$	$+$	$+$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{5}+\beta _{4}q^{7}+q^{9}+(1-\beta _{3}+\cdots)q^{11}+\cdots\)
8040.2.a.x	$8040$	$2$	8040.a	1.a	$1$	$8$	$8$	$64.200$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(8\)	\(8\)	\(5\)		$+$	$-$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{5}+(1-\beta _{1})q^{7}+q^{9}-\beta _{4}q^{11}+\cdots\)
8040.2.a.y	$8040$	$2$	8040.a	1.a	$1$	$9$	$9$	$64.200$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(-9\)	\(-9\)	\(-6\)		$-$	$+$	$+$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{5}+(-1-\beta _{6})q^{7}+q^{9}+(-\beta _{3}+\cdots)q^{11}+\cdots\)
8040.2.a.z	$8040$	$2$	8040.a	1.a	$1$	$9$	$9$	$64.200$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(-9\)	\(-9\)	\(4\)		$+$	$+$	$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{5}+\beta _{2}q^{7}+q^{9}-\beta _{4}q^{11}+\cdots\)
8040.2.a.ba	$8040$	$2$	8040.a	1.a	$1$	$9$	$9$	$64.200$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(9\)	\(-9\)	\(-7\)		$-$	$-$	$+$	$+$	$+$	$2^{4}$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{5}+(-1-\beta _{2})q^{7}+q^{9}+(1+\cdots)q^{11}+\cdots\)
8040.2.a.bb	$8040$	$2$	8040.a	1.a	$1$	$9$	$9$	$64.200$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(9\)	\(9\)	\(5\)		$-$	$-$	$-$	$+$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{5}+(1-\beta _{1})q^{7}+q^{9}+(-1+\cdots)q^{11}+\cdots\)
8040.2.a.bc	$8040$	$2$	8040.a	1.a	$1$	$10$	$10$	$64.200$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(-10\)	\(10\)	\(1\)		$+$	$+$	$-$	$+$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{5}-\beta _{9}q^{7}+q^{9}+(1-\beta _{7}+\cdots)q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(8040))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(8040)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(60))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(67))\)\(^{\oplus 16}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(120))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(134))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(201))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(268))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(335))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(402))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(536))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(670))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(804))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1005))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1340))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1608))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(2010))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(2680))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(4020))\)\(^{\oplus 2}\)