Space of modular forms of level 80, weight 12, and trivial character

• Label: 80.12.a
• Level: $80$
• Weight: $12$
• Character orbit: 80.a
• Rep. character: $\chi_{80}(1,\cdot)$
• Character field: $\Q$
• Dimension: $22$
• Newform subspaces: $13$
• Sturm bound: $144$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 80 = 2^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	80.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 13 \)
Sturm bound:			\(144\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{12}(\Gamma_0(80))\).

	Total	New	Old
Modular forms	138	22	116
Cusp forms	126	22	104
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
\(+\)	\(+\)	\(+\)	\(6\)
\(+\)	\(-\)	\(-\)	\(5\)
\(-\)	\(+\)	\(-\)	\(5\)
\(-\)	\(-\)	\(+\)	\(6\)
Plus space		\(+\)	\(12\)
Minus space		\(-\)	\(10\)

Trace form

22 * q - 486 * q^3 + 136306 * q^7 + 1091834 * q^9 + 540844 * q^11 + 1518750 * q^15 - 7412332 * q^17 - 18693312 * q^19 - 12566044 * q^21 + 31878310 * q^23 + 214843750 * q^25 - 181608852 * q^27 + 174163452 * q^29 + 169953420 * q^31 + 239902080 * q^33 - 315131250 * q^35 - 522762056 * q^37 + 1387157220 * q^39 - 390087808 * q^41 - 1206175974 * q^43 - 1090637500 * q^45 + 5151813986 * q^47 + 6427001250 * q^49 - 12077212892 * q^51 + 2286408960 * q^53 + 2013137500 * q^55 + 7151616024 * q^57 - 27916805992 * q^59 - 3760629504 * q^61 + 54633495922 * q^63 + 2812412500 * q^65 - 15213306050 * q^67 + 14513114860 * q^69 + 62195665836 * q^71 - 10025549708 * q^73 - 4746093750 * q^75 + 15729197072 * q^77 + 80404762232 * q^79 + 55863326474 * q^81 - 91513522910 * q^83 - 42238925000 * q^85 + 231495154476 * q^87 + 22052766108 * q^89 - 72943947516 * q^91 + 55143096056 * q^93 + 61902475000 * q^95 + 112138523004 * q^97 + 251845394780 * q^99

Decomposition of \(S_{12}^{\mathrm{new}}(\Gamma_0(80))\) 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
80.12.a.a	$80$	$12$	80.a	1.a	$1$	$1$	$1$	$61.467$	\(\Q\)	None	✓				10.12.a.b	$1$	$1$	\(0\)	\(-738\)	\(-3125\)	\(-25574\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-738q^{3}-5^{5}q^{5}-25574q^{7}+367497q^{9}+\cdots\)
80.12.a.b	$80$	$12$	80.a	1.a	$1$	$1$	$1$	$61.467$	\(\Q\)	None	✓				40.12.a.a	$1$	$1$	\(0\)	\(-520\)	\(3125\)	\(-15148\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-520q^{3}+5^{5}q^{5}-15148q^{7}+93253q^{9}+\cdots\)
80.12.a.c	$80$	$12$	80.a	1.a	$1$	$1$	$1$	$61.467$	\(\Q\)	None	✓				20.12.a.a	$1$	$1$	\(0\)	\(-306\)	\(-3125\)	\(32074\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-306q^{3}-5^{5}q^{5}+32074q^{7}-83511q^{9}+\cdots\)
80.12.a.d	$80$	$12$	80.a	1.a	$1$	$1$	$1$	$61.467$	\(\Q\)	None	✓				10.12.a.a	$1$	$0$	\(0\)	\(12\)	\(3125\)	\(14176\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+12q^{3}+5^{5}q^{5}+14176q^{7}-177003q^{9}+\cdots\)
80.12.a.e	$80$	$12$	80.a	1.a	$1$	$1$	$1$	$61.467$	\(\Q\)	None	✓				10.12.a.c	$1$	$1$	\(0\)	\(318\)	\(-3125\)	\(70714\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+318q^{3}-5^{5}q^{5}+70714q^{7}-76023q^{9}+\cdots\)
80.12.a.f	$80$	$12$	80.a	1.a	$1$	$1$	$1$	$61.467$	\(\Q\)	None	✓				5.12.a.a	$1$	$0$	\(0\)	\(792\)	\(3125\)	\(17556\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+792q^{3}+5^{5}q^{5}+17556q^{7}+450117q^{9}+\cdots\)
80.12.a.g	$80$	$12$	80.a	1.a	$1$	$2$	$2$	$61.467$	\(\Q(\sqrt{1969}) \)	None	✓				10.12.a.d	$1$	$0$	\(0\)	\(-604\)	\(6250\)	\(-14092\)		$-$	$-$	$+$	$2^{2}\cdot 5$	$\mathrm{SU}(2)$	\(q+(-302-\beta )q^{3}+5^{5}q^{5}+(-7046+\cdots)q^{7}+\cdots\)
80.12.a.h	$80$	$12$	80.a	1.a	$1$	$2$	$2$	$61.467$	\(\Q(\sqrt{57}) \)	None	✓				40.12.a.c	$1$	$1$	\(0\)	\(-252\)	\(6250\)	\(24212\)		$+$	$-$	$-$	$2^{2}\cdot 3\cdot 5$	$\mathrm{SU}(2)$	\(q+(-126-\beta )q^{3}+5^{5}q^{5}+(12106+\cdots)q^{7}+\cdots\)
80.12.a.i	$80$	$12$	80.a	1.a	$1$	$2$	$2$	$61.467$	\(\Q(\sqrt{46729}) \)	None	✓				20.12.a.b	$1$	$0$	\(0\)	\(-220\)	\(6250\)	\(-3340\)		$-$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(-110-\beta )q^{3}+5^{5}q^{5}+(-1670+\cdots)q^{7}+\cdots\)
80.12.a.j	$80$	$12$	80.a	1.a	$1$	$2$	$2$	$61.467$	\(\Q(\sqrt{151}) \)	None	✓		✓		5.12.a.b	$1$	$1$	\(0\)	\(220\)	\(-6250\)	\(-57900\)		$-$	$+$	$-$	$2^{5}$	$\mathrm{SU}(2)$	\(q+(110+\beta )q^{3}-5^{5}q^{5}+(-28950+\cdots)q^{7}+\cdots\)
80.12.a.k	$80$	$12$	80.a	1.a	$1$	$2$	$2$	$61.467$	\(\Q(\sqrt{42421}) \)	None	✓				40.12.a.b	$1$	$1$	\(0\)	\(792\)	\(6250\)	\(-5632\)		$+$	$-$	$-$	$2^{7}\cdot 5$	$\mathrm{SU}(2)$	\(q+396q^{3}+5^{5}q^{5}+(-2816+\beta )q^{7}+\cdots\)
80.12.a.l	$80$	$12$	80.a	1.a	$1$	$3$	$3$	$61.467$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓				40.12.a.e	$1$	$0$	\(0\)	\(-78\)	\(-9375\)	\(-1530\)		$+$	$+$	$+$	$2^{11}\cdot 3\cdot 5$	$\mathrm{SU}(2)$	\(q+(-26-\beta _{1})q^{3}-5^{5}q^{5}+(-510+\cdots)q^{7}+\cdots\)
80.12.a.m	$80$	$12$	80.a	1.a	$1$	$3$	$3$	$61.467$	3.3.229897.1	None	✓				40.12.a.d	$1$	$0$	\(0\)	\(98\)	\(-9375\)	\(100790\)		$+$	$+$	$+$	$2^{11}\cdot 5$	$\mathrm{SU}(2)$	\(q+(33+\beta _{1})q^{3}-5^{5}q^{5}+(33580-3^{3}\beta _{1}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{12}^{\mathrm{old}}(\Gamma_0(80))\) into lower level spaces

\( S_{12}^{\mathrm{old}}(\Gamma_0(80)) \simeq \) \(S_{12}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 10}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(4))\)\(^{\oplus 6}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 5}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 4}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 4}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 2}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 3}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 2}\)