Space of modular forms of level 20, weight 8, and trivial character

• Label: 20.8.a
• Level: $20$
• Weight: $8$
• Character orbit: 20.a
• Rep. character: $\chi_{20}(1,\cdot)$
• Character field: $\Q$
• Dimension: $3$
• Newform subspaces: $2$
• Sturm bound: $24$
• Trace bound: $1$

Defining parameters

Level:	\( N \)	\(=\)	\( 20 = 2^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	20.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 2 \)
Sturm bound:			\(24\)
Trace bound:			\(1\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_0(20))\).

	Total	New	Old
Modular forms	24	3	21
Cusp forms	18	3	15
Eisenstein series	6	0	6

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

\(2\)	\(5\)	Fricke		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(7\)	\(0\)	\(7\)		\(5\)	\(0\)	\(5\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(-\)		\(6\)	\(0\)	\(6\)		\(4\)	\(0\)	\(4\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(-\)		\(5\)	\(1\)	\(4\)		\(4\)	\(1\)	\(3\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(+\)		\(6\)	\(2\)	\(4\)		\(5\)	\(2\)	\(3\)		\(1\)	\(0\)	\(1\)
Plus space		\(+\)		\(13\)	\(2\)	\(11\)		\(10\)	\(2\)	\(8\)		\(3\)	\(0\)	\(3\)
Minus space		\(-\)		\(11\)	\(1\)	\(10\)		\(8\)	\(1\)	\(7\)		\(3\)	\(0\)	\(3\)

Trace form

3 * q - 26 * q^3 + 125 * q^5 + 954 * q^7 + 2707 * q^9 - 240 * q^11 + 9126 * q^13 - 1750 * q^15 + 6318 * q^17 - 19428 * q^19 - 93652 * q^21 + 43518 * q^23 + 46875 * q^25 - 159452 * q^27 + 35562 * q^29 - 261492 * q^31 + 799920 * q^33 + 295750 * q^35 - 191946 * q^37 - 432628 * q^39 - 1042662 * q^41 + 2083230 * q^43 + 876125 * q^45 - 1832862 * q^47 + 137199 * q^49 - 4286724 * q^51 + 3235902 * q^53 + 930000 * q^55 - 1347304 * q^57 - 1362996 * q^59 - 4346634 * q^61 + 7176506 * q^63 + 2154250 * q^65 + 290634 * q^67 - 662844 * q^69 - 805116 * q^71 + 3433686 * q^73 - 406250 * q^75 - 1616880 * q^77 + 5934696 * q^79 - 229433 * q^81 - 5962722 * q^83 + 575250 * q^85 - 3939564 * q^87 - 9031362 * q^89 + 16727892 * q^91 - 26418136 * q^93 - 7689500 * q^95 + 18000534 * q^97 + 746640 * q^99

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_0(20))\) 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
20.8.a.a	$20$	$8$	20.a	1.a	$1$	$1$	$1$	$6.248$	\(\Q\)	None	✓	✓	✓	✓	20.8.a.a	$1$	$1$	\(0\)	\(-6\)	\(-125\)	\(-706\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-6q^{3}-5^{3}q^{5}-706q^{7}-2151q^{9}+\cdots\)
20.8.a.b	$20$	$8$	20.a	1.a	$1$	$2$	$2$	$6.248$	\(\Q(\sqrt{1129}) \)	None	✓	✓	✓	✓	20.8.a.b	$1$	$0$	\(0\)	\(-20\)	\(250\)	\(1660\)		$-$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(-10-\beta )q^{3}+5^{3}q^{5}+(830+9\beta )q^{7}+\cdots\)

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_0(20))\) into lower level spaces

\( S_{8}^{\mathrm{old}}(\Gamma_0(20)) \simeq \) \(S_{8}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 3}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 2}\)