Space of modular forms of level 804, weight 2, and trivial character

• Label: 804.2.a
• Level: $804$
• Weight: $2$
• Character orbit: 804.a
• Rep. character: $\chi_{804}(1,\cdot)$
• Character field: $\Q$
• Dimension: $12$
• Newform subspaces: $6$
• Sturm bound: $272$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 804 = 2^{2} \cdot 3 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	804.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 6 \)
Sturm bound:			\(272\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(804))\).

	Total	New	Old
Modular forms	142	12	130
Cusp forms	131	12	119
Eisenstein series	11	0	11

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

\(2\)	\(3\)	\(67\)	Fricke	Dim
\(-\)	\(+\)	\(+\)	$-$	\(4\)
\(-\)	\(+\)	\(-\)	$+$	\(2\)
\(-\)	\(-\)	\(+\)	$+$	\(1\)
\(-\)	\(-\)	\(-\)	$-$	\(5\)
Plus space			\(+\)	\(3\)
Minus space			\(-\)	\(9\)

Trace form

\( 12 q + 12 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(804))\) 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	3	67
804.2.a.a	$804$	$2$	804.a	1.a	$1$	$1$	$1$	$6.420$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-1\)	\(-3\)	\(3\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-3q^{5}+3q^{7}+q^{9}-2q^{11}+\cdots\)
804.2.a.b	$804$	$2$	804.a	1.a	$1$	$1$	$1$	$6.420$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{9}-2q^{11}-4q^{13}-3q^{17}+\cdots\)
804.2.a.c	$804$	$2$	804.a	1.a	$1$	$1$	$1$	$6.420$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-1\)	\(4\)	\(0\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+4q^{5}+q^{9}+2q^{13}-4q^{15}+\cdots\)
804.2.a.d	$804$	$2$	804.a	1.a	$1$	$1$	$1$	$6.420$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(1\)	\(-1\)	\(-3\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{5}-3q^{7}+q^{9}-2q^{11}-2q^{13}+\cdots\)
804.2.a.e	$804$	$2$	804.a	1.a	$1$	$3$	$3$	$6.420$	3.3.1076.1	None	✓	$1$	$0$	\(0\)	\(-3\)	\(-3\)	\(-5\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+(-1-\beta _{1})q^{5}+(-2-\beta _{1}+\beta _{2})q^{7}+\cdots\)
804.2.a.f	$804$	$2$	804.a	1.a	$1$	$5$	$5$	$6.420$	5.5.24571284.1	None	✓	$1$	$0$	\(0\)	\(5\)	\(3\)	\(5\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+(1-\beta _{1})q^{5}+(1-\beta _{2})q^{7}+q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(804))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(804)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(67))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(134))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(201))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(268))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(402))\)\(^{\oplus 2}\)