Space of modular forms of level 819, weight 2, and Character 819.do

• Label: 819.2.do
• Level: $819$
• Weight: $2$
• Character orbit: 819.do
• Rep. character: $\chi_{819}(361,\cdot)$
• Character field: $\Q(\zeta_{6})$
• Dimension: $90$
• Newform subspaces: $8$
• Sturm bound: $224$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 819 = 3^{2} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	819.do (of order \(6\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 91 \)
Character field:			\(\Q(\zeta_{6})\)
Newform subspaces:			\( 8 \)
Sturm bound:			\(224\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(819, [\chi])\).

	Total	New	Old
Modular forms	240	98	142
Cusp forms	208	90	118
Eisenstein series	32	8	24

Trace form

\( 90 q + 3 q^{2} + 45 q^{4} - 4 q^{7} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(819, [\chi])\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
819.2.do.a	$819$	$2$	819.do	91.u	$6$	$2$	$1$	$6.540$	\(\Q(\sqrt{-3}) \)	None		$2$	$1$	\(-3\)	\(0\)	\(-6\)	\(1\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-2+\zeta_{6})q^{2}+(1-\zeta_{6})q^{4}+(-2+\cdots)q^{5}+\cdots\)
819.2.do.b	$819$	$2$	819.do	91.u	$6$	$2$	$1$	$6.540$	\(\Q(\sqrt{-3}) \)	\(\Q(\sqrt{-3}) \)		$4$	$1$	\(0\)	\(0\)	\(0\)	\(-5\)	$1$	$\mathrm{U}(1)[D_{6}]$	\(q+(-2+2\zeta_{6})q^{4}+(-3+\zeta_{6})q^{7}+(-3+\cdots)q^{13}+\cdots\)
819.2.do.c	$819$	$2$	819.do	91.u	$6$	$2$	$1$	$6.540$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(3\)	\(0\)	\(3\)	\(-5\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(2-\zeta_{6})q^{2}+(1-\zeta_{6})q^{4}+(1+\zeta_{6})q^{5}+\cdots\)
819.2.do.d	$819$	$2$	819.do	91.u	$6$	$4$	$2$	$6.540$	\(\Q(\sqrt{-3}, \sqrt{-7})\)	None		$2$	$0$	\(-3\)	\(0\)	\(6\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-1+\beta _{3})q^{2}+(\beta _{1}+\beta _{2}-2\beta _{3})q^{4}+\cdots\)
819.2.do.e	$819$	$2$	819.do	91.u	$6$	$12$	$6$	$6.540$	12.0.\(\cdots\).1	None		$2$	$0$	\(0\)	\(0\)	\(-3\)	\(3\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+\beta _{10}q^{2}+(\beta _{1}+\beta _{4}-\beta _{7}+\beta _{11})q^{4}+\cdots\)
819.2.do.f	$819$	$2$	819.do	91.u	$6$	$12$	$6$	$6.540$	12.0.\(\cdots\).1	None		$2$	$0$	\(3\)	\(0\)	\(-6\)	\(3\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(\beta _{1}+\beta _{6}-\beta _{8})q^{2}+(-1+\beta _{1}-\beta _{2}+\cdots)q^{4}+\cdots\)
819.2.do.g	$819$	$2$	819.do	91.u	$6$	$20$	$10$	$6.540$	\(\mathbb{Q}[x]/(x^{20} + \cdots)\)	None		$2$	$0$	\(3\)	\(0\)	\(6\)	\(-5\)	$3$	$\mathrm{SU}(2)[C_{6}]$	\(q-\beta _{3}q^{2}+(\beta _{11}+\beta _{17})q^{4}-\beta _{8}q^{5}+\cdots\)
819.2.do.h	$819$	$2$	819.do	91.u	$6$	$36$	$18$	$6.540$		None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(4\)		$\mathrm{SU}(2)[C_{6}]$

Decomposition of \(S_{2}^{\mathrm{old}}(819, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(819, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(91, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(273, [\chi])\)\(^{\oplus 2}\)