Space of modular forms of level 80, weight 18, and trivial character

• Label: 80.18.a
• Level: $80$
• Weight: $18$
• Character orbit: 80.a
• Rep. character: $\chi_{80}(1,\cdot)$
• Character field: $\Q$
• Dimension: $34$
• Newform subspaces: $12$
• Sturm bound: $216$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 80 = 2^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 18 \)
Character orbit:	\([\chi]\)	\(=\)	80.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 12 \)
Sturm bound:			\(216\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{18}(\Gamma_0(80))\).

	Total	New	Old
Modular forms	210	34	176
Cusp forms	198	34	164
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
\(+\)	\(+\)	\(+\)		\(51\)	\(8\)	\(43\)		\(48\)	\(8\)	\(40\)		\(3\)	\(0\)	\(3\)
\(+\)	\(-\)	\(-\)		\(53\)	\(9\)	\(44\)		\(50\)	\(9\)	\(41\)		\(3\)	\(0\)	\(3\)
\(-\)	\(+\)	\(-\)		\(54\)	\(9\)	\(45\)		\(51\)	\(9\)	\(42\)		\(3\)	\(0\)	\(3\)
\(-\)	\(-\)	\(+\)		\(52\)	\(8\)	\(44\)		\(49\)	\(8\)	\(41\)		\(3\)	\(0\)	\(3\)
Plus space		\(+\)		\(103\)	\(16\)	\(87\)		\(97\)	\(16\)	\(81\)		\(6\)	\(0\)	\(6\)
Minus space		\(-\)		\(107\)	\(18\)	\(89\)		\(101\)	\(18\)	\(83\)		\(6\)	\(0\)	\(6\)

Trace form

34 * q + 13122 * q^3 - 26311038 * q^7 + 1303838558 * q^9 - 8886916 * q^11 - 5125781250 * q^15 - 11440434244 * q^17 + 124198518528 * q^19 + 128106889532 * q^21 - 192886477450 * q^23 + 5187988281250 * q^25 - 1072944978564 * q^27 - 2109538835516 * q^29 + 12212994561036 * q^31 - 12755999101440 * q^33 + 13511252343750 * q^35 - 25580528691832 * q^37 - 88287651588348 * q^39 + 13494272644608 * q^41 + 25855374155218 * q^43 + 16173373437500 * q^45 - 144083993364478 * q^47 + 878372789122918 * q^49 - 516170242462604 * q^51 + 1004512331781120 * q^53 - 334935751562500 * q^55 + 1051523564813448 * q^57 + 750646951007224 * q^59 - 2405598236425216 * q^61 + 157550826408946 * q^63 - 597853876562500 * q^65 + 3004322034088630 * q^67 + 9144674645450164 * q^69 + 11353554470784780 * q^71 + 1139779843478716 * q^73 + 2002258300781250 * q^75 - 28497332048966416 * q^77 + 9063323759512568 * q^79 + 30013967749357790 * q^81 - 12004324522330550 * q^83 - 24631759746875000 * q^85 - 162146926052551668 * q^87 + 67890556540659252 * q^89 + 38158280057612660 * q^91 + 77201349237705608 * q^93 - 53073634503125000 * q^95 + 124561650488555828 * q^97 + 224051316688375436 * q^99

Decomposition of \(S_{18}^{\mathrm{new}}(\Gamma_0(80))\) 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
80.18.a.a	$80$	$18$	80.a	1.a	$1$	$1$	$1$	$146.578$	\(\Q\)	None	✓				10.18.a.a	$1$	$1$	\(0\)	\(14976\)	\(390625\)	\(-14808668\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+14976q^{3}+5^{8}q^{5}-14808668q^{7}+\cdots\)
80.18.a.b	$80$	$18$	80.a	1.a	$1$	$2$	$2$	$146.578$	\(\Q(\sqrt{2941}) \)	None	✓				10.18.a.d	$1$	$0$	\(0\)	\(-17628\)	\(-781250\)	\(-27684196\)		$-$	$+$	$-$	$2^{5}\cdot 3\cdot 5$	$\mathrm{SU}(2)$	\(q+(-8814-\beta )q^{3}-5^{8}q^{5}+(-13842098+\cdots)q^{7}+\cdots\)
80.18.a.c	$80$	$18$	80.a	1.a	$1$	$2$	$2$	$146.578$	\(\Q(\sqrt{247521}) \)	None	✓				20.18.a.a	$1$	$1$	\(0\)	\(-10980\)	\(781250\)	\(10044860\)		$-$	$-$	$+$	$2^{2}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+(-5490-\beta )q^{3}+5^{8}q^{5}+(5022430+\cdots)q^{7}+\cdots\)
80.18.a.d	$80$	$18$	80.a	1.a	$1$	$2$	$2$	$146.578$	\(\Q(\sqrt{83281}) \)	None	✓				10.18.a.c	$1$	$1$	\(0\)	\(1308\)	\(781250\)	\(-603844\)		$-$	$-$	$+$	$2^{2}\cdot 3\cdot 5$	$\mathrm{SU}(2)$	\(q+(654-\beta )q^{3}+5^{8}q^{5}+(-301922+\cdots)q^{7}+\cdots\)
80.18.a.e	$80$	$18$	80.a	1.a	$1$	$2$	$2$	$146.578$	\(\Q(\sqrt{36061}) \)	None	✓				10.18.a.b	$1$	$0$	\(0\)	\(6308\)	\(-781250\)	\(-6543844\)		$-$	$+$	$-$	$2^{5}\cdot 5$	$\mathrm{SU}(2)$	\(q+(3154-\beta )q^{3}-5^{8}q^{5}+(-3271922+\cdots)q^{7}+\cdots\)
80.18.a.f	$80$	$18$	80.a	1.a	$1$	$2$	$2$	$146.578$	\(\Q(\sqrt{39}) \)	None	✓				5.18.a.a	$1$	$0$	\(0\)	\(10980\)	\(-781250\)	\(22820700\)		$-$	$+$	$-$	$2^{4}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+(5490+13\beta )q^{3}-5^{8}q^{5}+(11410350+\cdots)q^{7}+\cdots\)
80.18.a.g	$80$	$18$	80.a	1.a	$1$	$3$	$3$	$146.578$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓		✓		5.18.a.b	$1$	$1$	\(0\)	\(-15944\)	\(1171875\)	\(-2139308\)		$-$	$-$	$+$	$2^{9}\cdot 5$	$\mathrm{SU}(2)$	\(q+(-5315-\beta _{2})q^{3}+5^{8}q^{5}+(-712892+\cdots)q^{7}+\cdots\)
80.18.a.h	$80$	$18$	80.a	1.a	$1$	$3$	$3$	$146.578$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓		✓		20.18.a.b	$1$	$0$	\(0\)	\(2822\)	\(-1171875\)	\(7384698\)		$-$	$+$	$-$	$2^{12}\cdot 3^{2}\cdot 5$	$\mathrm{SU}(2)$	\(q+(941+\beta _{1})q^{3}-5^{8}q^{5}+(2461385+\cdots)q^{7}+\cdots\)
80.18.a.i	$80$	$18$	80.a	1.a	$1$	$4$	$4$	$146.578$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓				40.18.a.c	$1$	$0$	\(0\)	\(-4744\)	\(1562500\)	\(11958712\)		$+$	$-$	$-$	$2^{19}\cdot 3^{2}\cdot 5^{2}$	$\mathrm{SU}(2)$	\(q+(-1186+\beta _{1})q^{3}+5^{8}q^{5}+(2989678+\cdots)q^{7}+\cdots\)
80.18.a.j	$80$	$18$	80.a	1.a	$1$	$4$	$4$	$146.578$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓				40.18.a.b	$1$	$1$	\(0\)	\(936\)	\(-1562500\)	\(-15139688\)		$+$	$+$	$+$	$2^{20}\cdot 3^{5}\cdot 5^{2}$	$\mathrm{SU}(2)$	\(q+(234+\beta _{1})q^{3}-5^{8}q^{5}+(-3784922+\cdots)q^{7}+\cdots\)
80.18.a.k	$80$	$18$	80.a	1.a	$1$	$4$	$4$	$146.578$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓				40.18.a.a	$1$	$1$	\(0\)	\(9704\)	\(-1562500\)	\(-11287592\)		$+$	$+$	$+$	$2^{20}\cdot 3^{3}\cdot 5^{2}$	$\mathrm{SU}(2)$	\(q+(2426+\beta _{1})q^{3}-5^{8}q^{5}+(-2821898+\cdots)q^{7}+\cdots\)
80.18.a.l	$80$	$18$	80.a	1.a	$1$	$5$	$5$	$146.578$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓		✓		40.18.a.d	$1$	$0$	\(0\)	\(15384\)	\(1953125\)	\(-312868\)		$+$	$-$	$-$	$2^{32}\cdot 3^{4}\cdot 5^{3}$	$\mathrm{SU}(2)$	\(q+(3077+\beta _{1})q^{3}+5^{8}q^{5}+(-62507+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{18}^{\mathrm{old}}(\Gamma_0(80))\) into lower level spaces

\( S_{18}^{\mathrm{old}}(\Gamma_0(80)) \simeq \) \(S_{18}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 10}\)\(\oplus\)\(S_{18}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 8}\)\(\oplus\)\(S_{18}^{\mathrm{new}}(\Gamma_0(4))\)\(^{\oplus 6}\)\(\oplus\)\(S_{18}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 5}\)\(\oplus\)\(S_{18}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 4}\)\(\oplus\)\(S_{18}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 4}\)\(\oplus\)\(S_{18}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 2}\)\(\oplus\)\(S_{18}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 3}\)\(\oplus\)\(S_{18}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 2}\)