Space of modular forms of level 80, weight 4, and trivial character

• Label: 80.4.a
• Level: $80$
• Weight: $4$
• Character orbit: 80.a
• Rep. character: $\chi_{80}(1,\cdot)$
• Character field: $\Q$
• Dimension: $6$
• Newform subspaces: $6$
• Sturm bound: $48$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 80 = 2^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	80.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 6 \)
Sturm bound:			\(48\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(3\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(80))\).

	Total	New	Old
Modular forms	42	6	36
Cusp forms	30	6	24
Eisenstein series	12	0	12

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	Dim
\(+\)	\(+\)	\(+\)	\(2\)
\(+\)	\(-\)	\(-\)	\(1\)
\(-\)	\(+\)	\(-\)	\(1\)
\(-\)	\(-\)	\(+\)	\(2\)
Plus space		\(+\)	\(4\)
Minus space		\(-\)	\(2\)

Trace form

\( 6 q - 6 q^{3} + 50 q^{7} + 74 q^{9} + O(q^{10}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(80))\) 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
80.4.a.a	$80$	$4$	80.a	1.a	$1$	$1$	$1$	$4.720$	\(\Q\)	None	✓				40.4.a.c	$1$	$0$	\(0\)	\(-10\)	\(-5\)	\(18\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-10q^{3}-5q^{5}+18q^{7}+73q^{9}+2^{4}q^{11}+\cdots\)
80.4.a.b	$80$	$4$	80.a	1.a	$1$	$1$	$1$	$4.720$	\(\Q\)	None	✓		✓	✓	40.4.a.b	$1$	$1$	\(0\)	\(-4\)	\(5\)	\(-16\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-4q^{3}+5q^{5}-2^{4}q^{7}-11q^{9}-6^{2}q^{11}+\cdots\)
80.4.a.c	$80$	$4$	80.a	1.a	$1$	$1$	$1$	$4.720$	\(\Q\)	None	✓				20.4.a.a	$1$	$0$	\(0\)	\(-4\)	\(5\)	\(16\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-4q^{3}+5q^{5}+2^{4}q^{7}-11q^{9}+60q^{11}+\cdots\)
80.4.a.d	$80$	$4$	80.a	1.a	$1$	$1$	$1$	$4.720$	\(\Q\)	None	✓		✓	✓	5.4.a.a	$1$	$1$	\(0\)	\(-2\)	\(-5\)	\(-6\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}-5q^{5}-6q^{7}-23q^{9}-2^{5}q^{11}+\cdots\)
80.4.a.e	$80$	$4$	80.a	1.a	$1$	$1$	$1$	$4.720$	\(\Q\)	None	✓				40.4.a.a	$1$	$0$	\(0\)	\(6\)	\(-5\)	\(34\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+6q^{3}-5q^{5}+34q^{7}+9q^{9}-2^{4}q^{11}+\cdots\)
80.4.a.f	$80$	$4$	80.a	1.a	$1$	$1$	$1$	$4.720$	\(\Q\)	None	✓				10.4.a.a	$1$	$0$	\(0\)	\(8\)	\(5\)	\(4\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+8q^{3}+5q^{5}+4q^{7}+37q^{9}-12q^{11}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(80))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(80)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 5}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 2}\)