Space of modular forms of level 806, weight 2, and trivial character

• Label: 806.2.a
• Level: $806$
• Weight: $2$
• Character orbit: 806.a
• Rep. character: $\chi_{806}(1,\cdot)$
• Character field: $\Q$
• Dimension: $29$
• Newform subspaces: $12$
• Sturm bound: $224$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 806 = 2 \cdot 13 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	806.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 12 \)
Sturm bound:			\(224\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(3\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(806))\).

	Total	New	Old
Modular forms	116	29	87
Cusp forms	109	29	80
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.

\(2\)	\(13\)	\(31\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	$+$	\(2\)
\(+\)	\(+\)	\(-\)	$-$	\(5\)
\(+\)	\(-\)	\(+\)	$-$	\(5\)
\(+\)	\(-\)	\(-\)	$+$	\(2\)
\(-\)	\(+\)	\(+\)	$-$	\(6\)
\(-\)	\(+\)	\(-\)	$+$	\(2\)
\(-\)	\(-\)	\(+\)	$+$	\(2\)
\(-\)	\(-\)	\(-\)	$-$	\(5\)
Plus space			\(+\)	\(8\)
Minus space			\(-\)	\(21\)

Trace form

\( 29 q + q^{2} + 29 q^{4} + 6 q^{5} + 8 q^{6} + q^{8} + 29 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(806))\) 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	13	31
806.2.a.a	$806$	$2$	806.a	1.a	$1$	$1$	$1$	$6.436$	\(\Q\)	None	✓	$1$	$1$	\(-1\)	\(-1\)	\(-1\)	\(1\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{3}+q^{4}-q^{5}+q^{6}+q^{7}+\cdots\)
806.2.a.b	$806$	$2$	806.a	1.a	$1$	$1$	$1$	$6.436$	\(\Q\)	None	✓	$1$	$1$	\(-1\)	\(1\)	\(1\)	\(-3\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{3}+q^{4}+q^{5}-q^{6}-3q^{7}+\cdots\)
806.2.a.c	$806$	$2$	806.a	1.a	$1$	$1$	$1$	$6.436$	\(\Q\)	None	✓	$1$	$1$	\(1\)	\(-3\)	\(1\)	\(-3\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-3q^{3}+q^{4}+q^{5}-3q^{6}-3q^{7}+\cdots\)
806.2.a.d	$806$	$2$	806.a	1.a	$1$	$1$	$1$	$6.436$	\(\Q\)	None	✓	$1$	$0$	\(1\)	\(-1\)	\(1\)	\(3\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{3}+q^{4}+q^{5}-q^{6}+3q^{7}+\cdots\)
806.2.a.e	$806$	$2$	806.a	1.a	$1$	$1$	$1$	$6.436$	\(\Q\)	None	✓	$1$	$1$	\(1\)	\(1\)	\(-3\)	\(-3\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{3}+q^{4}-3q^{5}+q^{6}-3q^{7}+\cdots\)
806.2.a.f	$806$	$2$	806.a	1.a	$1$	$1$	$1$	$6.436$	\(\Q\)	None	✓	$1$	$0$	\(1\)	\(1\)	\(3\)	\(-1\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{3}+q^{4}+3q^{5}+q^{6}-q^{7}+\cdots\)
806.2.a.g	$806$	$2$	806.a	1.a	$1$	$2$	$2$	$6.436$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$1$	\(-2\)	\(0\)	\(-2\)	\(-4\)		$+$	$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q-q^{2}-\beta q^{3}+q^{4}-q^{5}+\beta q^{6}+(-2+\cdots)q^{7}+\cdots\)
806.2.a.h	$806$	$2$	806.a	1.a	$1$	$2$	$2$	$6.436$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$1$	\(2\)	\(-2\)	\(0\)	\(-4\)		$-$	$+$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{3}+q^{4}+\beta q^{5}-q^{6}+(-2+\cdots)q^{7}+\cdots\)
806.2.a.i	$806$	$2$	806.a	1.a	$1$	$3$	$3$	$6.436$	\(\Q(\zeta_{14})^+\)	None	✓	$1$	$0$	\(3\)	\(3\)	\(-5\)	\(4\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+(1-\beta _{1}-\beta _{2})q^{3}+q^{4}+(-2+\cdots)q^{5}+\cdots\)
806.2.a.j	$806$	$2$	806.a	1.a	$1$	$5$	$5$	$6.436$	5.5.1772453.1	None	✓	$1$	$0$	\(-5\)	\(-3\)	\(5\)	\(8\)		$+$	$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q-q^{2}+(-1-\beta _{2})q^{3}+q^{4}+(1+\beta _{1}+\cdots)q^{5}+\cdots\)
806.2.a.k	$806$	$2$	806.a	1.a	$1$	$5$	$5$	$6.436$	5.5.170701.1	None	✓	$1$	$0$	\(-5\)	\(-1\)	\(3\)	\(-2\)		$+$	$+$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q-q^{2}-\beta _{3}q^{3}+q^{4}+(1-\beta _{1}-\beta _{2}+\cdots)q^{5}+\cdots\)
806.2.a.l	$806$	$2$	806.a	1.a	$1$	$6$	$6$	$6.436$	6.6.58446133.1	None	✓	$1$	$0$	\(6\)	\(5\)	\(3\)	\(4\)		$-$	$+$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+q^{2}+(1-\beta _{2})q^{3}+q^{4}+\beta _{3}q^{5}+(1+\cdots)q^{6}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(806))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(806)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(26))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(31))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(62))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(403))\)\(^{\oplus 2}\)