Space of modular forms of level 80, weight 22, and trivial character

• Label: 80.22.a
• Level: $80$
• Weight: $22$
• Character orbit: 80.a
• Rep. character: $\chi_{80}(1,\cdot)$
• Character field: $\Q$
• Dimension: $42$
• Newform subspaces: $12$
• Sturm bound: $264$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 80 = 2^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 22 \)
Character orbit:	\([\chi]\)	\(=\)	80.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 12 \)
Sturm bound:			\(264\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{22}(\Gamma_0(80))\).

	Total	New	Old
Modular forms	258	42	216
Cusp forms	246	42	204
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
\(+\)	\(+\)	\(+\)		\(63\)	\(10\)	\(53\)		\(60\)	\(10\)	\(50\)		\(3\)	\(0\)	\(3\)
\(+\)	\(-\)	\(-\)		\(65\)	\(11\)	\(54\)		\(62\)	\(11\)	\(51\)		\(3\)	\(0\)	\(3\)
\(-\)	\(+\)	\(-\)		\(66\)	\(11\)	\(55\)		\(63\)	\(11\)	\(52\)		\(3\)	\(0\)	\(3\)
\(-\)	\(-\)	\(+\)		\(64\)	\(10\)	\(54\)		\(61\)	\(10\)	\(51\)		\(3\)	\(0\)	\(3\)
Plus space		\(+\)		\(127\)	\(20\)	\(107\)		\(121\)	\(20\)	\(101\)		\(6\)	\(0\)	\(6\)
Minus space		\(-\)		\(131\)	\(22\)	\(109\)		\(125\)	\(22\)	\(103\)		\(6\)	\(0\)	\(6\)

Trace form

42 * q + 118098 * q^3 + 572793250 * q^7 + 136719429302 * q^9 - 67333320740 * q^11 - 1153300781250 * q^15 + 1958571155372 * q^17 + 69100898696256 * q^19 + 6324578107916 * q^21 - 161368512043562 * q^23 + 4005432128906250 * q^25 + 2794868852607708 * q^27 - 4434313278843084 * q^29 + 3970456894804236 * q^31 + 6871998957428352 * q^33 + 16551284121093750 * q^35 - 7245708527119384 * q^37 + 100126218287484612 * q^39 + 12059175947240576 * q^41 + 144847925861462018 * q^43 - 76478613320312500 * q^45 + 132446810053928834 * q^47 + 3062431372911388478 * q^49 - 190549532914769132 * q^51 - 520669382441074944 * q^53 - 1013180648476562500 * q^55 - 7568405612269422936 * q^57 - 9427406735318771848 * q^59 - 2643322503613029632 * q^61 + 3833521710685867762 * q^63 - 609612638632812500 * q^65 - 39641825912216479578 * q^67 + 45022316959477399204 * q^69 - 20619608192033752116 * q^71 + 76331318616888002028 * q^73 + 11262702941894531250 * q^75 - 49900379662404545488 * q^77 + 50573283627869819640 * q^79 + 451574524337577781526 * q^81 + 411506653493400060826 * q^83 + 161828597480390625000 * q^85 + 88714144165017399180 * q^87 + 171046214499839066820 * q^89 - 359426405062539289004 * q^91 - 1064310805727016810136 * q^93 - 478989551390703125000 * q^95 - 1320832698027442196348 * q^97 - 1832483538942082967764 * q^99

Decomposition of \(S_{22}^{\mathrm{new}}(\Gamma_0(80))\) 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
80.22.a.a	$80$	$22$	80.a	1.a	$1$	$1$	$1$	$223.582$	\(\Q\)	None	✓				10.22.a.a	$1$	$1$	\(0\)	\(21924\)	\(9765625\)	\(722753248\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+21924q^{3}+5^{10}q^{5}+722753248q^{7}+\cdots\)
80.22.a.b	$80$	$22$	80.a	1.a	$1$	$2$	$2$	$223.582$	\(\Q(\sqrt{474529}) \)	None	✓				10.22.a.c	$1$	$1$	\(0\)	\(-100308\)	\(19531250\)	\(-1328895316\)		$-$	$-$	$+$	$2^{2}\cdot 3\cdot 5\cdot 7$	$\mathrm{SU}(2)$	\(q+(-50154-\beta )q^{3}+5^{10}q^{5}+(-664447658+\cdots)q^{7}+\cdots\)
80.22.a.c	$80$	$22$	80.a	1.a	$1$	$2$	$2$	$223.582$	\(\Q(\sqrt{1179649}) \)	None	✓				10.22.a.d	$1$	$0$	\(0\)	\(-30972\)	\(-19531250\)	\(439959356\)		$-$	$+$	$-$	$2^{4}\cdot 3\cdot 5$	$\mathrm{SU}(2)$	\(q+(-15486-\beta )q^{3}-5^{10}q^{5}+(219979678+\cdots)q^{7}+\cdots\)
80.22.a.d	$80$	$22$	80.a	1.a	$1$	$2$	$2$	$223.582$	\(\Q(\sqrt{157921}) \)	None	✓				10.22.a.b	$1$	$0$	\(0\)	\(126692\)	\(-19531250\)	\(292598684\)		$-$	$+$	$-$	$2^{4}\cdot 5\cdot 7$	$\mathrm{SU}(2)$	\(q+(63346-\beta )q^{3}-5^{10}q^{5}+(146299342+\cdots)q^{7}+\cdots\)
80.22.a.e	$80$	$22$	80.a	1.a	$1$	$3$	$3$	$223.582$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓				5.22.a.a	$1$	$0$	\(0\)	\(-52194\)	\(-29296875\)	\(-684416558\)		$-$	$+$	$-$	$2^{12}\cdot 3\cdot 5$	$\mathrm{SU}(2)$	\(q+(-17398+\beta _{2})q^{3}-5^{10}q^{5}+(-228133339+\cdots)q^{7}+\cdots\)
80.22.a.f	$80$	$22$	80.a	1.a	$1$	$3$	$3$	$223.582$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓				20.22.a.a	$1$	$1$	\(0\)	\(170292\)	\(29296875\)	\(486402168\)		$-$	$-$	$+$	$2^{9}\cdot 3^{2}\cdot 5\cdot 7$	$\mathrm{SU}(2)$	\(q+(56764+\beta _{1})q^{3}+5^{10}q^{5}+(162134056+\cdots)q^{7}+\cdots\)
80.22.a.g	$80$	$22$	80.a	1.a	$1$	$4$	$4$	$223.582$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓		✓		5.22.a.b	$1$	$1$	\(0\)	\(-83240\)	\(39062500\)	\(-512613800\)		$-$	$-$	$+$	$2^{20}\cdot 3\cdot 5^{2}\cdot 7$	$\mathrm{SU}(2)$	\(q+(-20810-\beta _{1})q^{3}+5^{10}q^{5}+(-128153450+\cdots)q^{7}+\cdots\)
80.22.a.h	$80$	$22$	80.a	1.a	$1$	$4$	$4$	$223.582$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓		✓		20.22.a.b	$1$	$0$	\(0\)	\(83240\)	\(-39062500\)	\(19261720\)		$-$	$+$	$-$	$2^{24}\cdot 3^{2}\cdot 5^{2}\cdot 7$	$\mathrm{SU}(2)$	\(q+(20810+\beta _{1})q^{3}-5^{10}q^{5}+(4815430+\cdots)q^{7}+\cdots\)
80.22.a.i	$80$	$22$	80.a	1.a	$1$	$5$	$5$	$223.582$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓				40.22.a.c	$1$	$1$	\(0\)	\(-60022\)	\(-48828125\)	\(-208546506\)		$+$	$+$	$+$	$2^{37}\cdot 3^{2}\cdot 5^{4}\cdot 7^{2}$	$\mathrm{SU}(2)$	\(q+(-12004-\beta _{1})q^{3}-5^{10}q^{5}+(-41708704+\cdots)q^{7}+\cdots\)
80.22.a.j	$80$	$22$	80.a	1.a	$1$	$5$	$5$	$223.582$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓				40.22.a.b	$1$	$0$	\(0\)	\(29504\)	\(48828125\)	\(390224932\)		$+$	$-$	$-$	$2^{32}\cdot 3^{2}\cdot 5^{3}\cdot 7$	$\mathrm{SU}(2)$	\(q+(5901+\beta _{1})q^{3}+5^{10}q^{5}+(78044935+\cdots)q^{7}+\cdots\)
80.22.a.k	$80$	$22$	80.a	1.a	$1$	$5$	$5$	$223.582$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓				40.22.a.a	$1$	$1$	\(0\)	\(51354\)	\(-48828125\)	\(-419885818\)		$+$	$+$	$+$	$2^{37}\cdot 3^{7}\cdot 5^{3}\cdot 7$	$\mathrm{SU}(2)$	\(q+(10271-\beta _{1})q^{3}-5^{10}q^{5}+(-83977651+\cdots)q^{7}+\cdots\)
80.22.a.l	$80$	$22$	80.a	1.a	$1$	$6$	$6$	$223.582$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓		✓		40.22.a.d	$1$	$0$	\(0\)	\(-38172\)	\(58593750\)	\(1375951140\)		$+$	$-$	$-$	$2^{48}\cdot 3^{8}\cdot 5^{5}\cdot 7^{2}\cdot 17$	$\mathrm{SU}(2)$	\(q+(-6362-\beta _{1})q^{3}+5^{10}q^{5}+(229325190+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{22}^{\mathrm{old}}(\Gamma_0(80))\) into lower level spaces

\( S_{22}^{\mathrm{old}}(\Gamma_0(80)) \simeq \) \(S_{22}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 10}\)\(\oplus\)\(S_{22}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 8}\)\(\oplus\)\(S_{22}^{\mathrm{new}}(\Gamma_0(4))\)\(^{\oplus 6}\)\(\oplus\)\(S_{22}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 5}\)\(\oplus\)\(S_{22}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 4}\)\(\oplus\)\(S_{22}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 4}\)\(\oplus\)\(S_{22}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 2}\)\(\oplus\)\(S_{22}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 3}\)\(\oplus\)\(S_{22}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 2}\)