Space of modular forms of level 48, weight 22, and trivial character

• Label: 48.22.a
• Level: $48$
• Weight: $22$
• Character orbit: 48.a
• Rep. character: $\chi_{48}(1,\cdot)$
• Character field: $\Q$
• Dimension: $21$
• Newform subspaces: $12$
• Sturm bound: $176$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 48 = 2^{4} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 22 \)
Character orbit:	\([\chi]\)	\(=\)	48.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 12 \)
Sturm bound:			\(176\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{22}(\Gamma_0(48))\).

	Total	New	Old
Modular forms	174	21	153
Cusp forms	162	21	141
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.

\(2\)	\(3\)	Fricke		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(43\)	\(5\)	\(38\)		\(40\)	\(5\)	\(35\)		\(3\)	\(0\)	\(3\)
\(+\)	\(-\)	\(-\)		\(44\)	\(6\)	\(38\)		\(41\)	\(6\)	\(35\)		\(3\)	\(0\)	\(3\)
\(-\)	\(+\)	\(-\)		\(44\)	\(5\)	\(39\)		\(41\)	\(5\)	\(36\)		\(3\)	\(0\)	\(3\)
\(-\)	\(-\)	\(+\)		\(43\)	\(5\)	\(38\)		\(40\)	\(5\)	\(35\)		\(3\)	\(0\)	\(3\)
Plus space		\(+\)		\(86\)	\(10\)	\(76\)		\(80\)	\(10\)	\(70\)		\(6\)	\(0\)	\(6\)
Minus space		\(-\)		\(88\)	\(11\)	\(77\)		\(82\)	\(11\)	\(71\)		\(6\)	\(0\)	\(6\)

Trace form

21 * q + 59049 * q^3 + 20783558 * q^5 + 3921376 * q^7 + 73222472421 * q^9 - 67333320740 * q^11 + 107655263398 * q^13 - 1153300781250 * q^15 + 1740714497002 * q^17 + 82838787800292 * q^19 - 418452746476856 * q^23 + 2323074987996923 * q^25 + 205891132094649 * q^27 - 4677630608158002 * q^29 - 2640183941596200 * q^31 - 3713351244904404 * q^33 - 64834922527352640 * q^35 + 1988948121223214 * q^37 - 54414290793772674 * q^39 + 150420339888752034 * q^41 - 166623042337914148 * q^43 + 72467785831678758 * q^45 - 17462175811351488 * q^47 + 2303574344546220109 * q^49 + 1157438529986272722 * q^51 + 20360614157128454 * q^53 + 1720929500125000456 * q^55 - 1049319487303864812 * q^57 - 2701054252188509156 * q^59 - 1831629911373143834 * q^61 + 13672992667255776 * q^63 - 18327724207299793708 * q^65 - 15969864020757261660 * q^67 + 1975921403015956968 * q^69 + 29401064349889550264 * q^71 + 24759370509019929202 * q^73 + 97771745898438102567 * q^75 + 12805259299915171584 * q^77 - 196543372695926909080 * q^79 + 255310974640195504821 * q^81 - 523179303521190228124 * q^83 + 400956930034144704268 * q^85 - 348153015275892945882 * q^87 + 298080922501213062882 * q^89 + 84553229378675305280 * q^91 - 381306176747254396008 * q^93 + 336835535226733920632 * q^95 - 808921119743863851222 * q^97 - 234776772423761776740 * q^99

Decomposition of \(S_{22}^{\mathrm{new}}(\Gamma_0(48))\) 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	3
48.22.a.a	$48$	$22$	48.a	1.a	$1$	$1$	$1$	$134.149$	\(\Q\)	None	✓				6.22.a.c	$1$	$0$	\(0\)	\(-59049\)	\(-23245050\)	\(1322977768\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3^{10}q^{3}-23245050q^{5}+1322977768q^{7}+\cdots\)
48.22.a.b	$48$	$22$	48.a	1.a	$1$	$1$	$1$	$134.149$	\(\Q\)	None	✓				12.22.a.a	$1$	$0$	\(0\)	\(-59049\)	\(-11268090\)	\(-281914136\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3^{10}q^{3}-11268090q^{5}-281914136q^{7}+\cdots\)
48.22.a.c	$48$	$22$	48.a	1.a	$1$	$1$	$1$	$134.149$	\(\Q\)	None	✓				6.22.a.a	$1$	$0$	\(0\)	\(-59049\)	\(26444550\)	\(-166115864\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3^{10}q^{3}+26444550q^{5}-166115864q^{7}+\cdots\)
48.22.a.d	$48$	$22$	48.a	1.a	$1$	$1$	$1$	$134.149$	\(\Q\)	None	✓				3.22.a.b	$1$	$1$	\(0\)	\(59049\)	\(-41512770\)	\(-538429808\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3^{10}q^{3}-41512770q^{5}-538429808q^{7}+\cdots\)
48.22.a.e	$48$	$22$	48.a	1.a	$1$	$1$	$1$	$134.149$	\(\Q\)	None	✓				3.22.a.a	$1$	$1$	\(0\)	\(59049\)	\(3109950\)	\(-363303920\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3^{10}q^{3}+3109950q^{5}-363303920q^{7}+\cdots\)
48.22.a.f	$48$	$22$	48.a	1.a	$1$	$1$	$1$	$134.149$	\(\Q\)	None	✓				6.22.a.b	$1$	$1$	\(0\)	\(59049\)	\(12954174\)	\(479513104\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3^{10}q^{3}+12954174q^{5}+479513104q^{7}+\cdots\)
48.22.a.g	$48$	$22$	48.a	1.a	$1$	$2$	$2$	$134.149$	\(\Q(\sqrt{649}) \)	None	✓		✓		3.22.a.c	$1$	$0$	\(0\)	\(-118098\)	\(996876\)	\(-679896112\)		$-$	$+$	$-$	$2^{9}\cdot 3^{2}\cdot 7$	$\mathrm{SU}(2)$	\(q-3^{10}q^{3}+(498438+53\beta )q^{5}+(-339948056+\cdots)q^{7}+\cdots\)
48.22.a.h	$48$	$22$	48.a	1.a	$1$	$2$	$2$	$134.149$	\(\Q(\sqrt{537541}) \)	None	✓				24.22.a.a	$1$	$1$	\(0\)	\(-118098\)	\(21948620\)	\(659451408\)		$+$	$+$	$+$	$2^{7}\cdot 3^{2}\cdot 7$	$\mathrm{SU}(2)$	\(q-3^{10}q^{3}+(10974310-5\beta )q^{5}+(329725704+\cdots)q^{7}+\cdots\)
48.22.a.i	$48$	$22$	48.a	1.a	$1$	$2$	$2$	$134.149$	\(\mathbb{Q}[x]/(x^{2} - \cdots)\)	None	✓		✓		12.22.a.b	$1$	$1$	\(0\)	\(118098\)	\(28827900\)	\(-509669728\)		$-$	$-$	$+$	$2^{7}\cdot 3^{2}\cdot 5$	$\mathrm{SU}(2)$	\(q+3^{10}q^{3}+(14413950-5\beta )q^{5}+(-254834864+\cdots)q^{7}+\cdots\)
48.22.a.j	$48$	$22$	48.a	1.a	$1$	$3$	$3$	$134.149$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓		✓		24.22.a.d	$1$	$1$	\(0\)	\(-177147\)	\(5280498\)	\(-852542376\)		$+$	$+$	$+$	$2^{21}\cdot 3^{4}\cdot 7$	$\mathrm{SU}(2)$	\(q-3^{10}q^{3}+(1760166-\beta _{1})q^{5}+(-284180792+\cdots)q^{7}+\cdots\)
48.22.a.k	$48$	$22$	48.a	1.a	$1$	$3$	$3$	$134.149$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓				24.22.a.b	$1$	$0$	\(0\)	\(177147\)	\(-4833126\)	\(-271431024\)		$+$	$-$	$-$	$2^{21}\cdot 3^{4}\cdot 5\cdot 7$	$\mathrm{SU}(2)$	\(q+3^{10}q^{3}+(-1611042-\beta _{1})q^{5}+\cdots\)
48.22.a.l	$48$	$22$	48.a	1.a	$1$	$3$	$3$	$134.149$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓				24.22.a.c	$1$	$0$	\(0\)	\(177147\)	\(2080026\)	\(1205282064\)		$+$	$-$	$-$	$2^{21}\cdot 3^{4}\cdot 5\cdot 7$	$\mathrm{SU}(2)$	\(q+3^{10}q^{3}+(693342+\beta _{1})q^{5}+(401760688+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{22}^{\mathrm{old}}(\Gamma_0(48))\) into lower level spaces

\( S_{22}^{\mathrm{old}}(\Gamma_0(48)) \simeq \) \(S_{22}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 10}\)\(\oplus\)\(S_{22}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 8}\)\(\oplus\)\(S_{22}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 5}\)\(\oplus\)\(S_{22}^{\mathrm{new}}(\Gamma_0(4))\)\(^{\oplus 6}\)\(\oplus\)\(S_{22}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 4}\)\(\oplus\)\(S_{22}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 4}\)\(\oplus\)\(S_{22}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 3}\)\(\oplus\)\(S_{22}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 2}\)\(\oplus\)\(S_{22}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 2}\)