Space of modular forms of level 1040, weight 6, and trivial character

• Label: 1040.6.a
• Level: $1040$
• Weight: $6$
• Character orbit: 1040.a
• Rep. character: $\chi_{1040}(1,\cdot)$
• Character field: $\Q$
• Dimension: $120$
• Newform subspaces: $27$
• Sturm bound: $1008$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 1040 = 2^{4} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	1040.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 27 \)
Sturm bound:			\(1008\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(1040))\).

	Total	New	Old
Modular forms	852	120	732
Cusp forms	828	120	708
Eisenstein series	24	0	24

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
\(+\)	\(+\)	\(+\)	\(+\)	\(15\)
\(+\)	\(+\)	\(-\)	\(-\)	\(16\)
\(+\)	\(-\)	\(+\)	\(-\)	\(16\)
\(+\)	\(-\)	\(-\)	\(+\)	\(13\)
\(-\)	\(+\)	\(+\)	\(-\)	\(15\)
\(-\)	\(+\)	\(-\)	\(+\)	\(14\)
\(-\)	\(-\)	\(+\)	\(+\)	\(14\)
\(-\)	\(-\)	\(-\)	\(-\)	\(17\)
Plus space			\(+\)	\(56\)
Minus space			\(-\)	\(64\)

Trace form

120 * q + 196 * q^7 + 9632 * q^9 - 1208 * q^11 + 900 * q^15 - 2008 * q^17 - 2360 * q^19 + 1640 * q^21 + 1672 * q^23 + 75000 * q^25 - 16104 * q^29 - 7160 * q^31 + 3840 * q^33 + 21296 * q^37 + 11608 * q^41 + 32072 * q^43 - 11800 * q^45 + 44180 * q^47 + 288120 * q^49 + 26800 * q^51 + 66968 * q^53 + 96000 * q^57 - 60648 * q^61 - 81860 * q^63 + 7292 * q^67 + 43160 * q^69 + 83056 * q^71 + 107256 * q^73 - 42472 * q^77 - 104400 * q^79 + 742304 * q^81 - 458420 * q^83 + 132400 * q^85 - 262728 * q^87 + 214272 * q^89 - 232584 * q^97 - 121248 * q^99

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(1040))\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Twist minimal	Largest	Maximal	Minimal twist	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
1040.6.a.a	$1040$	$6$	1040.a	1.a	$1$	$1$	$1$	$166.799$	\(\Q\)	None	✓				65.6.a.a	$1$	$0$	\(0\)	\(-6\)	\(-25\)	\(244\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-6q^{3}-5^{2}q^{5}+244q^{7}-207q^{9}+\cdots\)
1040.6.a.b	$1040$	$6$	1040.a	1.a	$1$	$1$	$1$	$166.799$	\(\Q\)	None	✓				260.6.a.a	$1$	$0$	\(0\)	\(14\)	\(-25\)	\(212\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+14q^{3}-5^{2}q^{5}+212q^{7}-47q^{9}+\cdots\)
1040.6.a.c	$1040$	$6$	1040.a	1.a	$1$	$2$	$2$	$166.799$	\(\Q(\sqrt{145}) \)	None	✓				130.6.a.c	$1$	$0$	\(0\)	\(-6\)	\(-50\)	\(-130\)		$-$	$+$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(-3-\beta )q^{3}-5^{2}q^{5}+(-65+\beta )q^{7}+\cdots\)
1040.6.a.d	$1040$	$6$	1040.a	1.a	$1$	$2$	$2$	$166.799$	\(\Q(\sqrt{19}) \)	None	✓				130.6.a.b	$1$	$1$	\(0\)	\(-6\)	\(-50\)	\(208\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-3+3\beta )q^{3}-5^{2}q^{5}+(104+10\beta )q^{7}+\cdots\)
1040.6.a.e	$1040$	$6$	1040.a	1.a	$1$	$2$	$2$	$166.799$	\(\Q(\sqrt{10}) \)	None	✓				130.6.a.a	$1$	$1$	\(0\)	\(4\)	\(50\)	\(-300\)		$-$	$-$	$+$	$+$	$5$	$\mathrm{SU}(2)$	\(q+(2+\beta )q^{3}+5^{2}q^{5}+(-150+4\beta )q^{7}+\cdots\)
1040.6.a.f	$1040$	$6$	1040.a	1.a	$1$	$2$	$2$	$166.799$	\(\Q(\sqrt{51}) \)	None	✓				260.6.a.c	$1$	$1$	\(0\)	\(6\)	\(-50\)	\(140\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(3+3\beta )q^{3}-5^{2}q^{5}+(70+8\beta )q^{7}+\cdots\)
1040.6.a.g	$1040$	$6$	1040.a	1.a	$1$	$2$	$2$	$166.799$	\(\Q(\sqrt{67}) \)	None	✓				260.6.a.b	$1$	$1$	\(0\)	\(14\)	\(-50\)	\(-164\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(7+\beta )q^{3}-5^{2}q^{5}+(-82-4\beta )q^{7}+\cdots\)
1040.6.a.h	$1040$	$6$	1040.a	1.a	$1$	$2$	$2$	$166.799$	\(\Q(\sqrt{14}) \)	None	✓				130.6.a.e	$1$	$1$	\(0\)	\(16\)	\(50\)	\(252\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(8+3\beta )q^{3}+5^{2}q^{5}+(126-20\beta )q^{7}+\cdots\)
1040.6.a.i	$1040$	$6$	1040.a	1.a	$1$	$2$	$2$	$166.799$	\(\Q(\sqrt{235}) \)	None	✓				130.6.a.d	$1$	$1$	\(0\)	\(26\)	\(-50\)	\(-160\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(13+\beta )q^{3}-5^{2}q^{5}+(-80-6\beta )q^{7}+\cdots\)
1040.6.a.j	$1040$	$6$	1040.a	1.a	$1$	$3$	$3$	$166.799$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓				130.6.a.g	$1$	$0$	\(0\)	\(8\)	\(-75\)	\(-42\)		$-$	$+$	$+$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(3+\beta _{1})q^{3}-5^{2}q^{5}+(-12+6\beta _{1}+\cdots)q^{7}+\cdots\)
1040.6.a.k	$1040$	$6$	1040.a	1.a	$1$	$3$	$3$	$166.799$	3.3.49857.1	None	✓				65.6.a.b	$1$	$0$	\(0\)	\(16\)	\(75\)	\(208\)		$-$	$-$	$-$	$-$	$2^{4}$	$\mathrm{SU}(2)$	\(q+(5+\beta _{2})q^{3}+5^{2}q^{5}+(72+2\beta _{1}-6\beta _{2})q^{7}+\cdots\)
1040.6.a.l	$1040$	$6$	1040.a	1.a	$1$	$3$	$3$	$166.799$	3.3.1458804.1	None	✓				130.6.a.f	$1$	$0$	\(0\)	\(22\)	\(75\)	\(234\)		$-$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(7+\beta _{1})q^{3}+5^{2}q^{5}+(76+7\beta _{1}+\beta _{2})q^{7}+\cdots\)
1040.6.a.m	$1040$	$6$	1040.a	1.a	$1$	$4$	$4$	$166.799$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓				130.6.a.h	$1$	$0$	\(0\)	\(-20\)	\(100\)	\(-62\)		$-$	$-$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(-5+\beta _{1})q^{3}+5^{2}q^{5}+(-2^{4}-\beta _{3})q^{7}+\cdots\)
1040.6.a.n	$1040$	$6$	1040.a	1.a	$1$	$4$	$4$	$166.799$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓				260.6.a.e	$1$	$0$	\(0\)	\(-12\)	\(-100\)	\(-126\)		$-$	$+$	$+$	$-$	$2^{8}$	$\mathrm{SU}(2)$	\(q+(-3-\beta _{1})q^{3}-5^{2}q^{5}+(-2^{5}+6\beta _{1}+\cdots)q^{7}+\cdots\)
1040.6.a.o	$1040$	$6$	1040.a	1.a	$1$	$4$	$4$	$166.799$	4.4.1878612.1	None	✓				65.6.a.c	$1$	$0$	\(0\)	\(4\)	\(-100\)	\(136\)		$-$	$+$	$+$	$-$	$2^{6}$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1}+\beta _{2})q^{3}-5^{2}q^{5}+(34+4\beta _{1}+\cdots)q^{7}+\cdots\)
1040.6.a.p	$1040$	$6$	1040.a	1.a	$1$	$4$	$4$	$166.799$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓				260.6.a.d	$1$	$1$	\(0\)	\(20\)	\(100\)	\(24\)		$-$	$-$	$+$	$+$	$2^{4}$	$\mathrm{SU}(2)$	\(q+(5+\beta _{3})q^{3}+5^{2}q^{5}+(6-\beta _{1}+2\beta _{2}+\cdots)q^{7}+\cdots\)
1040.6.a.q	$1040$	$6$	1040.a	1.a	$1$	$6$	$6$	$166.799$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓		✓		65.6.a.d	$1$	$1$	\(0\)	\(-38\)	\(-150\)	\(-220\)		$-$	$+$	$-$	$+$	$2^{11}\cdot 3$	$\mathrm{SU}(2)$	\(q+(-6-\beta _{1})q^{3}-5^{2}q^{5}+(-37+\beta _{1}+\cdots)q^{7}+\cdots\)
1040.6.a.r	$1040$	$6$	1040.a	1.a	$1$	$6$	$6$	$166.799$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓		✓		65.6.a.e	$1$	$1$	\(0\)	\(-20\)	\(150\)	\(-172\)		$-$	$-$	$+$	$+$	$2^{11}$	$\mathrm{SU}(2)$	\(q+(-3+\beta _{1})q^{3}+5^{2}q^{5}+(-28+\beta _{3}+\cdots)q^{7}+\cdots\)
1040.6.a.s	$1040$	$6$	1040.a	1.a	$1$	$6$	$6$	$166.799$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓				520.6.a.a	$1$	$1$	\(0\)	\(-12\)	\(150\)	\(146\)		$+$	$-$	$-$	$+$	$2^{8}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+(-2+\beta _{1})q^{3}+5^{2}q^{5}+(5^{2}-3\beta _{1}+\cdots)q^{7}+\cdots\)
1040.6.a.t	$1040$	$6$	1040.a	1.a	$1$	$7$	$7$	$166.799$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓		✓		520.6.a.c	$1$	$1$	\(0\)	\(-26\)	\(175\)	\(-146\)		$+$	$-$	$-$	$+$	$2^{11}\cdot 3$	$\mathrm{SU}(2)$	\(q+(-4+\beta _{1})q^{3}+5^{2}q^{5}+(-21-\beta _{2}+\cdots)q^{7}+\cdots\)
1040.6.a.u	$1040$	$6$	1040.a	1.a	$1$	$7$	$7$	$166.799$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓				520.6.a.b	$1$	$1$	\(0\)	\(-2\)	\(-175\)	\(-54\)		$+$	$+$	$+$	$+$	$2^{14}\cdot 3$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}-5^{2}q^{5}+(-8-\beta _{4})q^{7}+(74+\cdots)q^{9}+\cdots\)
1040.6.a.v	$1040$	$6$	1040.a	1.a	$1$	$7$	$7$	$166.799$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓		✓		260.6.a.f	$1$	$0$	\(0\)	\(2\)	\(175\)	\(-86\)		$-$	$-$	$-$	$-$	$2^{11}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+5^{2}q^{5}+(-13+\beta _{1}-\beta _{3}+\cdots)q^{7}+\cdots\)
1040.6.a.w	$1040$	$6$	1040.a	1.a	$1$	$8$	$8$	$166.799$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓		✓		520.6.a.h	$1$	$1$	\(0\)	\(-36\)	\(-200\)	\(54\)		$+$	$+$	$+$	$+$	$2^{19}\cdot 3$	$\mathrm{SU}(2)$	\(q+(-4-\beta _{1})q^{3}-5^{2}q^{5}+(7-\beta _{1}+\beta _{5}+\cdots)q^{7}+\cdots\)
1040.6.a.x	$1040$	$6$	1040.a	1.a	$1$	$8$	$8$	$166.799$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓				520.6.a.g	$1$	$0$	\(0\)	\(-12\)	\(200\)	\(80\)		$+$	$-$	$+$	$-$	$2^{15}\cdot 3$	$\mathrm{SU}(2)$	\(q+(-2+\beta _{1})q^{3}+5^{2}q^{5}+(10+\beta _{1}+\cdots)q^{7}+\cdots\)
1040.6.a.y	$1040$	$6$	1040.a	1.a	$1$	$8$	$8$	$166.799$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓				520.6.a.f	$1$	$0$	\(0\)	\(0\)	\(-200\)	\(184\)		$+$	$+$	$-$	$-$	$2^{15}\cdot 3\cdot 5$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}-5^{2}q^{5}+(23+2\beta _{1}+\beta _{3}+\cdots)q^{7}+\cdots\)
1040.6.a.z	$1040$	$6$	1040.a	1.a	$1$	$8$	$8$	$166.799$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓				520.6.a.e	$1$	$0$	\(0\)	\(16\)	\(-200\)	\(-184\)		$+$	$+$	$-$	$-$	$2^{15}\cdot 3$	$\mathrm{SU}(2)$	\(q+(2-\beta _{1})q^{3}-5^{2}q^{5}+(-23+\beta _{2}+\cdots)q^{7}+\cdots\)
1040.6.a.ba	$1040$	$6$	1040.a	1.a	$1$	$8$	$8$	$166.799$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓				520.6.a.d	$1$	$0$	\(0\)	\(28\)	\(200\)	\(-80\)		$+$	$-$	$+$	$-$	$2^{15}$	$\mathrm{SU}(2)$	\(q+(4-\beta _{1})q^{3}+5^{2}q^{5}+(-11+\beta _{1}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(1040))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_0(1040)) \simeq \) \(S_{6}^{\mathrm{new}}(\Gamma_0(4))\)\(^{\oplus 12}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 10}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 8}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 8}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(13))\)\(^{\oplus 10}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(26))\)\(^{\oplus 8}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(52))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(65))\)\(^{\oplus 5}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(80))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(104))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(130))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(208))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(260))\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(520))\)\(^{\oplus 2}\)