Space of modular forms of level 805, weight 2, and trivial character

• Label: 805.2.a
• Level: $805$
• Weight: $2$
• Character orbit: 805.a
• Rep. character: $\chi_{805}(1,\cdot)$
• Character field: $\Q$
• Dimension: $43$
• Newform subspaces: $13$
• Sturm bound: $192$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 805 = 5 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	805.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 13 \)
Sturm bound:			\(192\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(2\), \(3\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(805))\).

	Total	New	Old
Modular forms	100	43	57
Cusp forms	93	43	50
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.

\(5\)	\(7\)	\(23\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	$+$	\(4\)
\(+\)	\(+\)	\(-\)	$-$	\(9\)
\(+\)	\(-\)	\(+\)	$-$	\(4\)
\(+\)	\(-\)	\(-\)	$+$	\(5\)
\(-\)	\(+\)	\(+\)	$-$	\(4\)
\(-\)	\(+\)	\(-\)	$+$	\(5\)
\(-\)	\(-\)	\(+\)	$+$	\(4\)
\(-\)	\(-\)	\(-\)	$-$	\(8\)
Plus space			\(+\)	\(18\)
Minus space			\(-\)	\(25\)

Trace form

\( 43 q + q^{2} + 4 q^{3} + 37 q^{4} - q^{5} + 8 q^{6} - q^{7} + 9 q^{8} + 59 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(805))\) 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	7	23
805.2.a.a	$805$	$2$	805.a	1.a	$1$	$1$	$1$	$6.428$	\(\Q\)	None	✓	$1$	$0$	\(-1\)	\(0\)	\(-1\)	\(-1\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{4}-q^{5}-q^{7}+3q^{8}-3q^{9}+\cdots\)
805.2.a.b	$805$	$2$	805.a	1.a	$1$	$1$	$1$	$6.428$	\(\Q\)	None	✓	$1$	$0$	\(-1\)	\(0\)	\(-1\)	\(-1\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{4}-q^{5}-q^{7}+3q^{8}-3q^{9}+\cdots\)
805.2.a.c	$805$	$2$	805.a	1.a	$1$	$1$	$1$	$6.428$	\(\Q\)	None	✓	$1$	$1$	\(2\)	\(-1\)	\(-1\)	\(-1\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}-q^{3}+2q^{4}-q^{5}-2q^{6}-q^{7}+\cdots\)
805.2.a.d	$805$	$2$	805.a	1.a	$1$	$1$	$1$	$6.428$	\(\Q\)	None	✓	$1$	$0$	\(2\)	\(3\)	\(-1\)	\(-1\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+3q^{3}+2q^{4}-q^{5}+6q^{6}+\cdots\)
805.2.a.e	$805$	$2$	805.a	1.a	$1$	$2$	$2$	$6.428$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$0$	\(3\)	\(0\)	\(-2\)	\(-2\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{2}+(1-2\beta )q^{3}+3\beta q^{4}-q^{5}+\cdots\)
805.2.a.f	$805$	$2$	805.a	1.a	$1$	$3$	$3$	$6.428$	\(\Q(\zeta_{14})^+\)	None	✓	$1$	$1$	\(-2\)	\(0\)	\(-3\)	\(-3\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{2})q^{2}+(-1+2\beta _{1}-\beta _{2})q^{3}+\cdots\)
805.2.a.g	$805$	$2$	805.a	1.a	$1$	$4$	$4$	$6.428$	4.4.2777.1	None	✓	$1$	$1$	\(-2\)	\(-5\)	\(4\)	\(4\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{3})q^{2}+(-1-\beta _{1})q^{3}+(1+\cdots)q^{4}+\cdots\)
805.2.a.h	$805$	$2$	805.a	1.a	$1$	$4$	$4$	$6.428$	4.4.22545.1	None	✓	$1$	$0$	\(-1\)	\(0\)	\(-4\)	\(-4\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{2}q^{2}+(\beta _{1}+\beta _{3})q^{3}+(3+\beta _{3})q^{4}+\cdots\)
805.2.a.i	$805$	$2$	805.a	1.a	$1$	$4$	$4$	$6.428$	4.4.7537.1	None	✓	$1$	$0$	\(1\)	\(4\)	\(-4\)	\(4\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(1+\beta _{3})q^{3}+(1+\beta _{2})q^{4}+\cdots\)
805.2.a.j	$805$	$2$	805.a	1.a	$1$	$4$	$4$	$6.428$	4.4.2777.1	None	✓	$1$	$0$	\(3\)	\(6\)	\(4\)	\(-4\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+(1+\beta _{3})q^{3}+(1-\beta _{1}+\cdots)q^{4}+\cdots\)
805.2.a.k	$805$	$2$	805.a	1.a	$1$	$5$	$5$	$6.428$	5.5.122821.1	None	✓	$1$	$1$	\(-3\)	\(-6\)	\(5\)	\(-5\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+(-1-\beta _{3})q^{3}+(\beta _{3}+\cdots)q^{4}+\cdots\)
805.2.a.l	$805$	$2$	805.a	1.a	$1$	$5$	$5$	$6.428$	5.5.255877.1	None	✓	$1$	$1$	\(-1\)	\(-4\)	\(-5\)	\(5\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(-1-\beta _{4})q^{3}+(1+\beta _{2})q^{4}+\cdots\)
805.2.a.m	$805$	$2$	805.a	1.a	$1$	$8$	$8$	$6.428$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	$1$	$0$	\(1\)	\(7\)	\(8\)	\(8\)		$-$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(1+\beta _{6})q^{3}+(1+\beta _{2})q^{4}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(805))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(805)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(23))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(35))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(115))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(161))\)\(^{\oplus 2}\)