Space of modular forms of level 820, weight 2, and trivial character

• Label: 820.2.a
• Level: $820$
• Weight: $2$
• Character orbit: 820.a
• Rep. character: $\chi_{820}(1,\cdot)$
• Character field: $\Q$
• Dimension: $12$
• Newform subspaces: $4$
• Sturm bound: $252$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 820 = 2^{2} \cdot 5 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	820.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(252\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(820))\).

	Total	New	Old
Modular forms	132	12	120
Cusp forms	121	12	109
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.

\(2\)	\(5\)	\(41\)	Fricke		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(13\)	\(0\)	\(13\)		\(12\)	\(0\)	\(12\)		\(1\)	\(0\)	\(1\)
\(+\)	\(+\)	\(-\)	\(-\)		\(19\)	\(0\)	\(19\)		\(17\)	\(0\)	\(17\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(+\)	\(-\)		\(16\)	\(0\)	\(16\)		\(14\)	\(0\)	\(14\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(-\)	\(+\)		\(18\)	\(0\)	\(18\)		\(16\)	\(0\)	\(16\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(+\)	\(-\)		\(20\)	\(4\)	\(16\)		\(19\)	\(4\)	\(15\)		\(1\)	\(0\)	\(1\)
\(-\)	\(+\)	\(-\)	\(+\)		\(14\)	\(2\)	\(12\)		\(13\)	\(2\)	\(11\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(+\)	\(+\)		\(17\)	\(2\)	\(15\)		\(16\)	\(2\)	\(14\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(-\)	\(-\)		\(15\)	\(4\)	\(11\)		\(14\)	\(4\)	\(10\)		\(1\)	\(0\)	\(1\)
Plus space			\(+\)		\(62\)	\(4\)	\(58\)		\(57\)	\(4\)	\(53\)		\(5\)	\(0\)	\(5\)
Minus space			\(-\)		\(70\)	\(8\)	\(62\)		\(64\)	\(8\)	\(56\)		\(6\)	\(0\)	\(6\)

Trace form

12 * q - 4 * q^7 + 4 * q^9 - 4 * q^11 - 4 * q^15 + 12 * q^17 - 8 * q^19 + 12 * q^21 + 4 * q^23 + 12 * q^25 + 20 * q^31 + 12 * q^33 + 4 * q^35 - 20 * q^37 + 28 * q^39 - 4 * q^43 + 28 * q^47 + 8 * q^49 + 36 * q^53 + 8 * q^55 - 4 * q^57 - 20 * q^59 - 16 * q^61 - 8 * q^63 - 12 * q^67 + 36 * q^69 + 28 * q^71 - 24 * q^73 + 12 * q^77 - 12 * q^79 + 12 * q^81 + 24 * q^83 - 8 * q^85 - 20 * q^87 + 4 * q^89 - 36 * q^91 - 20 * q^93 + 12 * q^97 - 64 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(820))\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Twist minimal	Largest	Maximal	Minimal twist	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	5	41
820.2.a.a	$820$	$2$	820.a	1.a	$1$	$2$	$2$	$6.548$	\(\Q(\sqrt{5}) \)	None	✓	✓	✓	✓	820.2.a.a	$1$	$1$	\(0\)	\(-2\)	\(2\)	\(-1\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{5}-\beta q^{7}-2q^{9}+(-1+4\beta )q^{11}+\cdots\)
820.2.a.b	$820$	$2$	820.a	1.a	$1$	$2$	$2$	$6.548$	\(\Q(\sqrt{13}) \)	None	✓	✓	✓	✓	820.2.a.b	$1$	$1$	\(0\)	\(2\)	\(-2\)	\(-3\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{5}+(-1-\beta )q^{7}-2q^{9}+(-3+\cdots)q^{11}+\cdots\)
820.2.a.c	$820$	$2$	820.a	1.a	$1$	$4$	$4$	$6.548$	\(\Q(\sqrt{5}, \sqrt{13})\)	None	✓	✓	✓	✓	820.2.a.c	$1$	$0$	\(0\)	\(0\)	\(-4\)	\(-1\)		$-$	$+$	$+$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}-q^{5}+(\beta _{1}-\beta _{2})q^{7}+(1+\beta _{2}+\cdots)q^{9}+\cdots\)
820.2.a.d	$820$	$2$	820.a	1.a	$1$	$4$	$4$	$6.548$	4.4.13068.1	None	✓	✓	✓	✓	820.2.a.d	$1$	$0$	\(0\)	\(0\)	\(4\)	\(1\)		$-$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q-\beta _{2}q^{3}+q^{5}+\beta _{3}q^{7}+(1-\beta _{1}+\beta _{3})q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(820))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(820)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(41))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(82))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(164))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(205))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(410))\)\(^{\oplus 2}\)