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		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(12\)	\(2\)	\(10\)		\(9\)	\(2\)	\(7\)		\(3\)	\(0\)	\(3\)
\(+\)	\(-\)	\(-\)		\(10\)	\(1\)	\(9\)		\(7\)	\(1\)	\(6\)		\(3\)	\(0\)	\(3\)
\(-\)	\(+\)	\(-\)		\(9\)	\(1\)	\(8\)		\(6\)	\(1\)	\(5\)		\(3\)	\(0\)	\(3\)
\(-\)	\(-\)	\(+\)		\(11\)	\(2\)	\(9\)		\(8\)	\(2\)	\(6\)		\(3\)	\(0\)	\(3\)
Plus space		\(+\)		\(23\)	\(4\)	\(19\)		\(17\)	\(4\)	\(13\)		\(6\)	\(0\)	\(6\)
Minus space		\(-\)		\(19\)	\(2\)	\(17\)		\(13\)	\(2\)	\(11\)		\(6\)	\(0\)	\(6\)

Trace form

6 * q - 6 * q^3 + 50 * q^7 + 74 * q^9 - 20 * q^11 + 30 * q^15 - 76 * q^17 + 192 * q^19 + 68 * q^21 - 26 * q^23 + 150 * q^25 - 84 * q^27 - 228 * q^29 - 372 * q^31 - 384 * q^33 - 210 * q^35 - 8 * q^37 - 156 * q^39 + 128 * q^41 - 326 * q^43 - 220 * q^45 + 98 * q^47 - 14 * q^49 + 484 * q^51 + 768 * q^53 + 220 * q^55 - 552 * q^57 - 616 * q^59 - 256 * q^61 + 1906 * q^63 - 140 * q^65 + 1374 * q^67 + 268 * q^69 - 468 * q^71 + 1044 * q^73 - 150 * q^75 + 1424 * q^77 + 120 * q^79 + 1178 * q^81 - 2942 * q^83 + 120 * q^85 - 3540 * q^87 - 900 * q^89 + 3908 * q^91 - 2248 * q^93 + 760 * q^95 + 1244 * q^97 + 1052 * q^99

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

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