Space of modular forms of level 8036, weight 2, and trivial character

• Label: 8036.2.a
• Level: $8036$
• Weight: $2$
• Character orbit: 8036.a
• Rep. character: $\chi_{8036}(1,\cdot)$
• Character field: $\Q$
• Dimension: $136$
• Newform subspaces: $20$
• Sturm bound: $2352$
• Trace bound: $13$

Defining parameters

Level:	\( N \)	\(=\)	\( 8036 = 2^{2} \cdot 7^{2} \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8036.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 20 \)
Sturm bound:			\(2352\)
Trace bound:			\(13\)
Distinguishing \(T_p\):			\(3\), \(5\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(8036))\).

	Total	New	Old
Modular forms	1200	136	1064
Cusp forms	1153	136	1017
Eisenstein series	47	0	47

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

\(2\)	\(7\)	\(41\)	Fricke	Dim
\(-\)	\(+\)	\(+\)	$-$	\(37\)
\(-\)	\(+\)	\(-\)	$+$	\(29\)
\(-\)	\(-\)	\(+\)	$+$	\(29\)
\(-\)	\(-\)	\(-\)	$-$	\(41\)
Plus space			\(+\)	\(58\)
Minus space			\(-\)	\(78\)

Trace form

\( 136 q + 2 q^{3} + 132 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(8036))\) 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	7	41
8036.2.a.a	$8036$	$2$	8036.a	1.a	$1$	$1$	$1$	$64.168$	\(\Q\)	None	✓	$1$	$2$	\(0\)	\(-1\)	\(-1\)	\(0\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{5}-2q^{9}-5q^{11}-2q^{13}+\cdots\)
8036.2.a.b	$8036$	$2$	8036.a	1.a	$1$	$1$	$1$	$64.168$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-1\)	\(-1\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{5}-2q^{9}+3q^{11}+2q^{13}+\cdots\)
8036.2.a.c	$8036$	$2$	8036.a	1.a	$1$	$1$	$1$	$64.168$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-1\)	\(3\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+3q^{5}-2q^{9}+3q^{11}+4q^{13}+\cdots\)
8036.2.a.d	$8036$	$2$	8036.a	1.a	$1$	$1$	$1$	$64.168$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(1\)	\(-3\)	\(0\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-3q^{5}-2q^{9}+3q^{11}-4q^{13}+\cdots\)
8036.2.a.e	$8036$	$2$	8036.a	1.a	$1$	$1$	$1$	$64.168$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(1\)	\(1\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{5}-2q^{9}-5q^{11}+2q^{13}+\cdots\)
8036.2.a.f	$8036$	$2$	8036.a	1.a	$1$	$1$	$1$	$64.168$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(1\)	\(1\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{5}-2q^{9}+3q^{11}-2q^{13}+\cdots\)
8036.2.a.g	$8036$	$2$	8036.a	1.a	$1$	$2$	$2$	$64.168$	\(\Q(\sqrt{13}) \)	None	✓	$1$	$0$	\(0\)	\(3\)	\(3\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{3}+(1+\beta )q^{5}+(1+3\beta )q^{9}+\cdots\)
8036.2.a.h	$8036$	$2$	8036.a	1.a	$1$	$3$	$3$	$64.168$	\(\Q(\zeta_{18})^+\)	None	✓	$1$	$0$	\(0\)	\(3\)	\(0\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{3}+(2\beta _{1}-2\beta _{2})q^{5}+(-2\beta _{1}+\cdots)q^{9}+\cdots\)
8036.2.a.i	$8036$	$2$	8036.a	1.a	$1$	$4$	$4$	$64.168$	4.4.25808.1	None	✓	$1$	$0$	\(0\)	\(-2\)	\(-4\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(\beta _{1}-\beta _{2})q^{3}+(-2+\beta _{2}-\beta _{3})q^{5}+\cdots\)
8036.2.a.j	$8036$	$2$	8036.a	1.a	$1$	$5$	$5$	$64.168$	5.5.1935333.1	None	✓	$1$	$1$	\(0\)	\(-2\)	\(-3\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(-1-\beta _{4})q^{5}+(1+\beta _{1}+\beta _{2}+\cdots)q^{9}+\cdots\)
8036.2.a.k	$8036$	$2$	8036.a	1.a	$1$	$5$	$5$	$64.168$	5.5.287349.1	None	✓	$1$	$0$	\(0\)	\(-2\)	\(3\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{2}q^{3}+(1-\beta _{2}+\beta _{3})q^{5}+(\beta _{2}+\beta _{4})q^{9}+\cdots\)
8036.2.a.l	$8036$	$2$	8036.a	1.a	$1$	$5$	$5$	$64.168$	5.5.470117.1	None	✓	$1$	$1$	\(0\)	\(2\)	\(1\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(1-\beta _{1}+\beta _{2})q^{5}+(\beta _{1}+\beta _{2}+\cdots)q^{9}+\cdots\)
8036.2.a.m	$8036$	$2$	8036.a	1.a	$1$	$8$	$8$	$64.168$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+\beta _{5}q^{5}+(1+\beta _{2})q^{9}+(-1+\cdots)q^{11}+\cdots\)
8036.2.a.n	$8036$	$2$	8036.a	1.a	$1$	$8$	$8$	$64.168$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}-\beta _{5}q^{5}+(1+\beta _{2})q^{9}+(-1+\cdots)q^{11}+\cdots\)
8036.2.a.o	$8036$	$2$	8036.a	1.a	$1$	$10$	$10$	$64.168$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(-2\)	\(-4\)	\(0\)		$-$	$-$	$+$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}-\beta _{2}q^{5}+(1+\beta _{2}+\beta _{3})q^{9}+\cdots\)
8036.2.a.p	$8036$	$2$	8036.a	1.a	$1$	$10$	$10$	$64.168$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(2\)	\(4\)	\(0\)		$-$	$-$	$-$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+\beta _{2}q^{5}+(1+\beta _{2}+\beta _{3})q^{9}+\cdots\)
8036.2.a.q	$8036$	$2$	8036.a	1.a	$1$	$15$	$15$	$64.168$	\(\mathbb{Q}[x]/(x^{15} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(-1\)	\(3\)	\(0\)		$-$	$+$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}-\beta _{3}q^{5}+(2+\beta _{2})q^{9}+(1-\beta _{3}+\cdots)q^{11}+\cdots\)
8036.2.a.r	$8036$	$2$	8036.a	1.a	$1$	$15$	$15$	$64.168$	\(\mathbb{Q}[x]/(x^{15} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(1\)	\(-3\)	\(0\)		$-$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+\beta _{3}q^{5}+(2+\beta _{2})q^{9}+(1-\beta _{3}+\cdots)q^{11}+\cdots\)
8036.2.a.s	$8036$	$2$	8036.a	1.a	$1$	$20$	$20$	$64.168$	\(\mathbb{Q}[x]/(x^{20} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(-4\)	\(-8\)	\(0\)		$-$	$+$	$-$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+\beta _{11}q^{5}+(1+\beta _{2})q^{9}+(-1+\cdots)q^{11}+\cdots\)
8036.2.a.t	$8036$	$2$	8036.a	1.a	$1$	$20$	$20$	$64.168$	\(\mathbb{Q}[x]/(x^{20} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(4\)	\(8\)	\(0\)		$-$	$+$	$+$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}-\beta _{11}q^{5}+(1+\beta _{2})q^{9}+(-1+\cdots)q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(8036))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(8036)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(28))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(41))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(49))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(82))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(98))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(164))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(196))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(287))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(574))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1148))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(2009))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(4018))\)\(^{\oplus 2}\)