Space of modular forms of level 1045, weight 2, and trivial character

• Label: 1045.2.a
• Level: $1045$
• Weight: $2$
• Character orbit: 1045.a
• Rep. character: $\chi_{1045}(1,\cdot)$
• Character field: $\Q$
• Dimension: $59$
• Newform subspaces: $11$
• Sturm bound: $240$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 1045 = 5 \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1045.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 11 \)
Sturm bound:			\(240\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1045))\).

	Total	New	Old
Modular forms	124	59	65
Cusp forms	117	59	58
Eisenstein series	7	0	7

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

\(5\)	\(11\)	\(19\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	$+$	\(7\)
\(+\)	\(+\)	\(-\)	$-$	\(9\)
\(+\)	\(-\)	\(+\)	$-$	\(7\)
\(+\)	\(-\)	\(-\)	$+$	\(5\)
\(-\)	\(+\)	\(+\)	$-$	\(8\)
\(-\)	\(+\)	\(-\)	$+$	\(6\)
\(-\)	\(-\)	\(+\)	$+$	\(8\)
\(-\)	\(-\)	\(-\)	$-$	\(9\)
Plus space			\(+\)	\(26\)
Minus space			\(-\)	\(33\)

Trace form

\( 59 q - 3 q^{2} - 4 q^{3} + 53 q^{4} + 3 q^{5} + 20 q^{6} - 8 q^{7} + 9 q^{8} + 55 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1045))\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		5	11	19
1045.2.a.a	$1045$	$2$	1045.a	1.a	$1$	$1$	$1$	$8.344$	\(\Q\)	None	✓	$1$	$1$	\(-1\)	\(-2\)	\(-1\)	\(-2\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-2q^{3}-q^{4}-q^{5}+2q^{6}-2q^{7}+\cdots\)
1045.2.a.b	$1045$	$2$	1045.a	1.a	$1$	$1$	$1$	$8.344$	\(\Q\)	None	✓	$1$	$1$	\(1\)	\(0\)	\(1\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{4}+q^{5}-3q^{8}-3q^{9}+q^{10}+\cdots\)
1045.2.a.c	$1045$	$2$	1045.a	1.a	$1$	$2$	$2$	$8.344$	\(\Q(\sqrt{2}) \)	None	✓	$1$	$0$	\(2\)	\(4\)	\(2\)	\(0\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{2}+2q^{3}+(1+2\beta )q^{4}+q^{5}+\cdots\)
1045.2.a.d	$1045$	$2$	1045.a	1.a	$1$	$5$	$5$	$8.344$	\(\Q(\zeta_{22})^+\)	None	✓	$1$	$1$	\(-3\)	\(-7\)	\(5\)	\(-11\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1}-\beta _{4})q^{2}+(-2-\beta _{2}+\beta _{3}+\cdots)q^{3}+\cdots\)
1045.2.a.e	$1045$	$2$	1045.a	1.a	$1$	$5$	$5$	$8.344$	5.5.36497.1	None	✓	$1$	$1$	\(-1\)	\(-3\)	\(-5\)	\(3\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{4}q^{2}+(-1+\beta _{1})q^{3}+\beta _{3}q^{4}-q^{5}+\cdots\)
1045.2.a.f	$1045$	$2$	1045.a	1.a	$1$	$6$	$6$	$8.344$	6.6.7281497.1	None	✓	$1$	$1$	\(-2\)	\(-1\)	\(-6\)	\(5\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{4}q^{2}+\beta _{3}q^{3}+(\beta _{4}+\beta _{5})q^{4}-q^{5}+\cdots\)
1045.2.a.g	$1045$	$2$	1045.a	1.a	$1$	$6$	$6$	$8.344$	6.6.131947641.1	None	✓	$1$	$0$	\(0\)	\(3\)	\(6\)	\(5\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(1-\beta _{3})q^{3}+(2+\beta _{2}-\beta _{3}+\cdots)q^{4}+\cdots\)
1045.2.a.h	$1045$	$2$	1045.a	1.a	$1$	$7$	$7$	$8.344$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	$1$	$0$	\(1\)	\(3\)	\(-7\)	\(-1\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+\beta _{5}q^{3}+(1+\beta _{2})q^{4}-q^{5}+\cdots\)
1045.2.a.i	$1045$	$2$	1045.a	1.a	$1$	$8$	$8$	$8.344$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	$1$	$1$	\(-6\)	\(-7\)	\(8\)	\(-11\)		$-$	$-$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+(-1-\beta _{6})q^{3}+(2+\cdots)q^{4}+\cdots\)
1045.2.a.j	$1045$	$2$	1045.a	1.a	$1$	$9$	$9$	$8.344$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓	$1$	$0$	\(3\)	\(3\)	\(-9\)	\(-9\)		$+$	$+$	$-$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+\beta _{5}q^{3}+(1-\beta _{1}-\beta _{8})q^{4}+\cdots\)
1045.2.a.k	$1045$	$2$	1045.a	1.a	$1$	$9$	$9$	$8.344$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓	$1$	$0$	\(3\)	\(3\)	\(9\)	\(13\)		$-$	$-$	$-$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+\beta _{5}q^{3}+(1+\beta _{2})q^{4}+q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1045))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1045)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(19))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(55))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(95))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(209))\)\(^{\oplus 2}\)