Space of modular forms of level 80, weight 20, and trivial character

• Label: 80.20.a
• Level: $80$
• Weight: $20$
• Character orbit: 80.a
• Rep. character: $\chi_{80}(1,\cdot)$
• Character field: $\Q$
• Dimension: $38$
• Newform subspaces: $12$
• Sturm bound: $240$
• Trace bound: $3$

Defining parameters

Level:	$N$	$=$	$80 = 2^{4} \cdot 5$
Weight:	$k$	$=$	$20$
Character orbit:	$[\chi]$	$=$	80.a (trivial)
Character field:			$\Q$
Newform subspaces:			$12$
Sturm bound:			$240$
Trace bound:			$3$
Distinguishing $T_p$:			$3$

Dimensions

The following table gives the dimensions of various subspaces of $M_{20}(\Gamma_0(80))$.

	Total	New	Old
Modular forms	234	38	196
Cusp forms	222	38	184
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
$+$	$+$	$+$		$60$	$10$	$50$		$57$	$10$	$47$		$3$	$0$	$3$
$+$	$-$	$-$		$58$	$9$	$49$		$55$	$9$	$46$		$3$	$0$	$3$
$-$	$+$	$-$		$57$	$9$	$48$		$54$	$9$	$45$		$3$	$0$	$3$
$-$	$-$	$+$		$59$	$10$	$49$		$56$	$10$	$46$		$3$	$0$	$3$
Plus space		$+$		$119$	$20$	$99$		$113$	$20$	$93$		$6$	$0$	$6$
Minus space		$-$		$115$	$18$	$97$		$109$	$18$	$91$		$6$	$0$	$6$

Trace form

38 * q - 39366 * q^3 + 287433906 * q^7 + 16340925866 * q^9 + 11401439596 * q^11 + 76886718750 * q^15 - 673857170572 * q^17 + 1879708750272 * q^19 - 1653150431356 * q^21 + 23790953521510 * q^23 + 144958496093750 * q^25 - 27279354872532 * q^27 + 124451708838108 * q^29 - 482260381805940 * q^31 + 316983675876480 * q^33 - 472893832031250 * q^35 + 1112248744900984 * q^37 - 2458083252666780 * q^39 + 154919101386368 * q^41 - 18090927956503174 * q^43 - 4372753210937500 * q^45 + 86844302458466 * q^47 + 70071259408630610 * q^49 + 37727458164805732 * q^51 - 14188990897424640 * q^53 + 18421466335937500 * q^55 - 75209697994315176 * q^57 + 47809811805792152 * q^59 + 108024624000680704 * q^61 + 180070450841744242 * q^63 + 104518175570312500 * q^65 - 581725188794625250 * q^67 - 344313800816604980 * q^69 + 420660335542370604 * q^71 - 883450134375920428 * q^73 - 150169372558593750 * q^75 + 1652723978311121552 * q^77 - 3033532924172813192 * q^79 + 8223488129561090426 * q^81 + 678033983776945090 * q^83 + 230512381296875000 * q^85 + 14369436088363681836 * q^87 - 2509236628651721028 * q^89 - 14512207336095152444 * q^91 - 885373520362684744 * q^93 + 5041995277796875000 * q^95 + 4407015956479735324 * q^97 + 2630392002223261340 * q^99

Decomposition of $S_{20}^{\mathrm{new}}(\Gamma_0(80))$ 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
80.20.a.a	$80$	$20$	80.a	1.a	$1$	$1$	$1$	$183.053$	$\Q$	None	✓				10.20.a.b	$1$	$0$	$0$	$-38628$	$1953125$	$144185776$		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q-38628q^{3}+5^{9}q^{5}+144185776q^{7}+\cdots$
80.20.a.b	$80$	$20$	80.a	1.a	$1$	$1$	$1$	$183.053$	$\Q$	None	✓				10.20.a.c	$1$	$1$	$0$	$-24642$	$-1953125$	$171901114$		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q-24642q^{3}-5^{9}q^{5}+171901114q^{7}+\cdots$
80.20.a.c	$80$	$20$	80.a	1.a	$1$	$1$	$1$	$183.053$	$\Q$	None	✓				10.20.a.a	$1$	$1$	$0$	$26622$	$-1953125$	$39884026$		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+26622q^{3}-5^{9}q^{5}+39884026q^{7}+\cdots$
80.20.a.d	$80$	$20$	80.a	1.a	$1$	$2$	$2$	$183.053$	$\mathbb{Q}[x]/(x^{2} - \cdots)$	None	✓				10.20.a.d	$1$	$0$	$0$	$-33724$	$3906250$	$-83061292$		$-$	$-$	$+$	$2^{2}\cdot 5$	$\mathrm{SU}(2)$	$q+(-16862-\beta )q^{3}+5^{9}q^{5}+(-41530646+\cdots)q^{7}+\cdots$
80.20.a.e	$80$	$20$	80.a	1.a	$1$	$3$	$3$	$183.053$	$\mathbb{Q}[x]/(x^{3} - \cdots)$	None	✓				20.20.a.a	$1$	$1$	$0$	$-34086$	$-5859375$	$115130574$		$-$	$+$	$-$	$2^{11}\cdot 3^{3}\cdot 5$	$\mathrm{SU}(2)$	$q+(-11362-\beta _{1})q^{3}-5^{9}q^{5}+(38376858+\cdots)q^{7}+\cdots$
80.20.a.f	$80$	$20$	80.a	1.a	$1$	$3$	$3$	$183.053$	$\mathbb{Q}[x]/(x^{3} - \cdots)$	None	✓				5.20.a.a	$1$	$0$	$0$	$73452$	$5859375$	$54910456$		$-$	$-$	$+$	$2^{9}\cdot 3\cdot 5$	$\mathrm{SU}(2)$	$q+(24485+\beta _{1}+3\beta _{2})q^{3}+5^{9}q^{5}+\cdots$
80.20.a.g	$80$	$20$	80.a	1.a	$1$	$4$	$4$	$183.053$	$\mathbb{Q}[x]/(x^{4} - \cdots)$	None	✓		✓		5.20.a.b	$1$	$1$	$0$	$-3080$	$-7812500$	$-214021400$		$-$	$+$	$-$	$2^{24}\cdot 3^{2}\cdot 5^{2}$	$\mathrm{SU}(2)$	$q+(-770-\beta _{1})q^{3}-5^{9}q^{5}+(-53505350+\cdots)q^{7}+\cdots$
80.20.a.h	$80$	$20$	80.a	1.a	$1$	$4$	$4$	$183.053$	$\mathbb{Q}[x]/(x^{4} - \cdots)$	None	✓		✓		20.20.a.b	$1$	$0$	$0$	$3080$	$7812500$	$-148222040$		$-$	$-$	$+$	$2^{19}\cdot 3^{3}\cdot 5^{2}$	$\mathrm{SU}(2)$	$q+(770-\beta _{1})q^{3}+5^{9}q^{5}+(-37055510+\cdots)q^{7}+\cdots$
80.20.a.i	$80$	$20$	80.a	1.a	$1$	$4$	$4$	$183.053$	$\mathbb{Q}[x]/(x^{4} - \cdots)$	None	✓				40.20.a.a	$1$	$1$	$0$	$46152$	$7812500$	$-59073944$		$+$	$-$	$-$	$2^{18}\cdot 3^{4}\cdot 5^{2}$	$\mathrm{SU}(2)$	$q+(11538-\beta _{1})q^{3}+5^{9}q^{5}+(-14768486+\cdots)q^{7}+\cdots$
80.20.a.j	$80$	$20$	80.a	1.a	$1$	$5$	$5$	$183.053$	$\mathbb{Q}[x]/(x^{5} - \cdots)$	None	✓		✓		40.20.a.d	$1$	$1$	$0$	$-50332$	$9765625$	$113917176$		$+$	$-$	$-$	$2^{35}\cdot 3^{3}\cdot 5^{3}$	$\mathrm{SU}(2)$	$q+(-10066+\beta _{1})q^{3}+5^{9}q^{5}+(22783495+\cdots)q^{7}+\cdots$
80.20.a.k	$80$	$20$	80.a	1.a	$1$	$5$	$5$	$183.053$	$\mathbb{Q}[x]/(x^{5} - \cdots)$	None	✓				40.20.a.c	$1$	$0$	$0$	$-21298$	$-9765625$	$88371034$		$+$	$+$	$+$	$2^{37}\cdot 3^{4}\cdot 5^{3}$	$\mathrm{SU}(2)$	$q+(-4260+\beta _{1})q^{3}-5^{9}q^{5}+(17674394+\cdots)q^{7}+\cdots$
80.20.a.l	$80$	$20$	80.a	1.a	$1$	$5$	$5$	$183.053$	$\mathbb{Q}[x]/(x^{5} - \cdots)$	None	✓				40.20.a.b	$1$	$0$	$0$	$17118$	$-9765625$	$63512426$		$+$	$+$	$+$	$2^{37}\cdot 3^{4}\cdot 5^{3}$	$\mathrm{SU}(2)$	$q+(3424-\beta _{1})q^{3}-5^{9}q^{5}+(12703124+\cdots)q^{7}+\cdots$

Decomposition of $S_{20}^{\mathrm{old}}(\Gamma_0(80))$ into lower level spaces

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