Space of modular forms of level 80, weight 6, and trivial character

• Label: 80.6.a
• Level: $80$
• Weight: $6$
• Character orbit: 80.a
• Rep. character: $\chi_{80}(1,\cdot)$
• Character field: $\Q$
• Dimension: $10$
• Newform subspaces: $9$
• Sturm bound: $72$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 80 = 2^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	80.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 9 \)
Sturm bound:			\(72\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(80))\).

	Total	New	Old
Modular forms	66	10	56
Cusp forms	54	10	44
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
\(+\)	\(+\)	\(+\)		\(15\)	\(2\)	\(13\)		\(12\)	\(2\)	\(10\)		\(3\)	\(0\)	\(3\)
\(+\)	\(-\)	\(-\)		\(17\)	\(3\)	\(14\)		\(14\)	\(3\)	\(11\)		\(3\)	\(0\)	\(3\)
\(-\)	\(+\)	\(-\)		\(18\)	\(3\)	\(15\)		\(15\)	\(3\)	\(12\)		\(3\)	\(0\)	\(3\)
\(-\)	\(-\)	\(+\)		\(16\)	\(2\)	\(14\)		\(13\)	\(2\)	\(11\)		\(3\)	\(0\)	\(3\)
Plus space		\(+\)		\(31\)	\(4\)	\(27\)		\(25\)	\(4\)	\(21\)		\(6\)	\(0\)	\(6\)
Minus space		\(-\)		\(35\)	\(6\)	\(29\)		\(29\)	\(6\)	\(23\)		\(6\)	\(0\)	\(6\)

Trace form

10 * q + 18 * q^3 - 222 * q^7 + 854 * q^9 + 604 * q^11 - 450 * q^15 + 1004 * q^17 + 4416 * q^19 - 820 * q^21 - 4010 * q^23 + 6250 * q^25 + 9564 * q^27 + 8052 * q^29 - 756 * q^31 - 1920 * q^33 + 7350 * q^35 - 10648 * q^37 + 42948 * q^39 - 10624 * q^41 - 41918 * q^43 + 5900 * q^45 - 19582 * q^47 + 45022 * q^49 - 50540 * q^51 - 42240 * q^53 - 12100 * q^55 - 60888 * q^57 - 37000 * q^59 + 36608 * q^61 + 754 * q^63 + 21100 * q^65 + 70630 * q^67 - 84764 * q^69 - 64308 * q^71 - 106196 * q^73 + 11250 * q^75 + 67376 * q^77 - 22792 * q^79 + 88118 * q^81 + 229210 * q^83 - 66200 * q^85 - 34932 * q^87 - 138108 * q^89 + 24788 * q^91 + 329192 * q^93 - 72200 * q^95 + 40772 * q^97 - 32596 * q^99

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(80))\) 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
80.6.a.a	$80$	$6$	80.a	1.a	$1$	$1$	$1$	$12.831$	\(\Q\)	None	✓				10.6.a.b	$1$	$1$	\(0\)	\(-24\)	\(25\)	\(172\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-24q^{3}+5^{2}q^{5}+172q^{7}+333q^{9}+\cdots\)
80.6.a.b	$80$	$6$	80.a	1.a	$1$	$1$	$1$	$12.831$	\(\Q\)	None	✓				20.6.a.a	$1$	$0$	\(0\)	\(-22\)	\(-25\)	\(-218\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-22q^{3}-5^{2}q^{5}-218q^{7}+241q^{9}+\cdots\)
80.6.a.c	$80$	$6$	80.a	1.a	$1$	$1$	$1$	$12.831$	\(\Q\)	None	✓				10.6.a.c	$1$	$0$	\(0\)	\(-6\)	\(-25\)	\(118\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-6q^{3}-5^{2}q^{5}+118q^{7}-207q^{9}+\cdots\)
80.6.a.d	$80$	$6$	80.a	1.a	$1$	$1$	$1$	$12.831$	\(\Q\)	None	✓				40.6.a.c	$1$	$1$	\(0\)	\(2\)	\(-25\)	\(62\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}-5^{2}q^{5}+62q^{7}-239q^{9}+\cdots\)
80.6.a.e	$80$	$6$	80.a	1.a	$1$	$1$	$1$	$12.831$	\(\Q\)	None	✓				5.6.a.a	$1$	$1$	\(0\)	\(4\)	\(25\)	\(-192\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+4q^{3}+5^{2}q^{5}-192q^{7}-227q^{9}+\cdots\)
80.6.a.f	$80$	$6$	80.a	1.a	$1$	$1$	$1$	$12.831$	\(\Q\)	None	✓				40.6.a.b	$1$	$0$	\(0\)	\(8\)	\(25\)	\(108\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+8q^{3}+5^{2}q^{5}+108q^{7}-179q^{9}+\cdots\)
80.6.a.g	$80$	$6$	80.a	1.a	$1$	$1$	$1$	$12.831$	\(\Q\)	None	✓				40.6.a.a	$1$	$1$	\(0\)	\(18\)	\(-25\)	\(-242\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+18q^{3}-5^{2}q^{5}-242q^{7}+3^{4}q^{9}+\cdots\)
80.6.a.h	$80$	$6$	80.a	1.a	$1$	$1$	$1$	$12.831$	\(\Q\)	None	✓				10.6.a.a	$1$	$0$	\(0\)	\(26\)	\(-25\)	\(22\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+26q^{3}-5^{2}q^{5}+22q^{7}+433q^{9}+\cdots\)
80.6.a.i	$80$	$6$	80.a	1.a	$1$	$2$	$2$	$12.831$	\(\Q(\sqrt{129}) \)	None	✓		✓		40.6.a.d	$1$	$0$	\(0\)	\(12\)	\(50\)	\(-52\)		$+$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(6+\beta )q^{3}+5^{2}q^{5}+(-26+3\beta )q^{7}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(80))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_0(80)) \simeq \) \(S_{6}^{\mathrm{new}}(\Gamma_0(4))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 5}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 2}\)