Space of modular forms of level 48, weight 18, and trivial character

• Label: 48.18.a
• Level: $48$
• Weight: $18$
• Character orbit: 48.a
• Rep. character: $\chi_{48}(1,\cdot)$
• Character field: $\Q$
• Dimension: $17$
• Newform subspaces: $11$
• Sturm bound: $144$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 48 = 2^{4} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 18 \)
Character orbit:	\([\chi]\)	\(=\)	48.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 11 \)
Sturm bound:			\(144\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{18}(\Gamma_0(48))\).

	Total	New	Old
Modular forms	142	17	125
Cusp forms	130	17	113
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
\(+\)	\(+\)	\(+\)		\(35\)	\(4\)	\(31\)		\(32\)	\(4\)	\(28\)		\(3\)	\(0\)	\(3\)
\(+\)	\(-\)	\(-\)		\(36\)	\(5\)	\(31\)		\(33\)	\(5\)	\(28\)		\(3\)	\(0\)	\(3\)
\(-\)	\(+\)	\(-\)		\(36\)	\(4\)	\(32\)		\(33\)	\(4\)	\(29\)		\(3\)	\(0\)	\(3\)
\(-\)	\(-\)	\(+\)		\(35\)	\(4\)	\(31\)		\(32\)	\(4\)	\(28\)		\(3\)	\(0\)	\(3\)
Plus space		\(+\)		\(70\)	\(8\)	\(62\)		\(64\)	\(8\)	\(56\)		\(6\)	\(0\)	\(6\)
Minus space		\(-\)		\(72\)	\(9\)	\(63\)		\(66\)	\(9\)	\(57\)		\(6\)	\(0\)	\(6\)

Trace form

17 * q + 6561 * q^3 + 24478 * q^5 - 18920320 * q^7 + 731794257 * q^9 - 8886916 * q^11 + 1590277918 * q^13 - 5125781250 * q^15 - 7489125598 * q^17 + 173442716036 * q^19 + 809469569288 * q^23 + 3171041796143 * q^25 + 282429536481 * q^27 - 4567156984410 * q^29 - 5690007233416 * q^31 + 6573473222076 * q^33 + 10408489585920 * q^35 - 18852385192154 * q^37 - 59919910722306 * q^39 + 34757875502106 * q^41 - 58364106768004 * q^43 + 1053697636638 * q^45 + 232614811457280 * q^47 + 259713329935113 * q^49 + 380692081266498 * q^51 + 1457690427899038 * q^53 - 1599957486374904 * q^55 + 382613919321924 * q^57 - 1485635180914564 * q^59 - 2768793635268578 * q^61 - 814457736270720 * q^63 - 2131102491367868 * q^65 - 3699035952859516 * q^67 + 4407893800766856 * q^69 + 7629530017252600 * q^71 + 11630704751258410 * q^73 - 3015498763225953 * q^75 - 8744691589647360 * q^77 - 20065277152622968 * q^79 + 31501343210481297 * q^81 + 87090139050149764 * q^83 - 802456097379012 * q^85 + 20063432853978246 * q^87 - 35392862392888038 * q^89 + 84775934699193088 * q^91 - 29077136096636808 * q^93 - 131196236248819208 * q^95 + 17680546822699042 * q^97 - 382552593602436 * q^99

Decomposition of \(S_{18}^{\mathrm{new}}(\Gamma_0(48))\) 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
48.18.a.a	$48$	$18$	48.a	1.a	$1$	$1$	$1$	$87.947$	\(\Q\)	None	✓				6.18.a.b	$1$	$0$	\(0\)	\(-6561\)	\(-72186\)	\(8640184\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3^{8}q^{3}-72186q^{5}+8640184q^{7}+\cdots\)
48.18.a.b	$48$	$18$	48.a	1.a	$1$	$1$	$1$	$87.947$	\(\Q\)	None	✓				12.18.a.b	$1$	$0$	\(0\)	\(-6561\)	\(130950\)	\(14846776\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3^{8}q^{3}+130950q^{5}+14846776q^{7}+\cdots\)
48.18.a.c	$48$	$18$	48.a	1.a	$1$	$1$	$1$	$87.947$	\(\Q\)	None	✓				12.18.a.a	$1$	$1$	\(0\)	\(6561\)	\(-1608930\)	\(9417184\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3^{8}q^{3}-1608930q^{5}+9417184q^{7}+\cdots\)
48.18.a.d	$48$	$18$	48.a	1.a	$1$	$1$	$1$	$87.947$	\(\Q\)	None	✓				6.18.a.c	$1$	$1$	\(0\)	\(6561\)	\(-199650\)	\(-24959264\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3^{8}q^{3}-199650q^{5}-24959264q^{7}+\cdots\)
48.18.a.e	$48$	$18$	48.a	1.a	$1$	$1$	$1$	$87.947$	\(\Q\)	None	✓				3.18.a.a	$1$	$1$	\(0\)	\(6561\)	\(-163554\)	\(20846560\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3^{8}q^{3}-163554q^{5}+20846560q^{7}+\cdots\)
48.18.a.f	$48$	$18$	48.a	1.a	$1$	$1$	$1$	$87.947$	\(\Q\)	None	✓				6.18.a.a	$1$	$1$	\(0\)	\(6561\)	\(645150\)	\(-3974432\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3^{8}q^{3}+645150q^{5}-3974432q^{7}+\cdots\)
48.18.a.g	$48$	$18$	48.a	1.a	$1$	$2$	$2$	$87.947$	\(\Q(\sqrt{1131}) \)	None	✓				24.18.a.b	$1$	$1$	\(0\)	\(-13122\)	\(-1139764\)	\(-3021648\)		$+$	$+$	$+$	$2^{8}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q-3^{8}q^{3}+(-569882+13\beta )q^{5}+(-1510824+\cdots)q^{7}+\cdots\)
48.18.a.h	$48$	$18$	48.a	1.a	$1$	$2$	$2$	$87.947$	\(\Q(\sqrt{14569}) \)	None	✓		✓		3.18.a.b	$1$	$0$	\(0\)	\(-13122\)	\(382860\)	\(-24471568\)		$-$	$+$	$-$	$2^{7}\cdot 3$	$\mathrm{SU}(2)$	\(q-3^{8}q^{3}+(191430-55\beta )q^{5}+(-12235784+\cdots)q^{7}+\cdots\)
48.18.a.i	$48$	$18$	48.a	1.a	$1$	$2$	$2$	$87.947$	\(\mathbb{Q}[x]/(x^{2} - \cdots)\)	None	✓				24.18.a.c	$1$	$1$	\(0\)	\(-13122\)	\(1101004\)	\(-5453904\)		$+$	$+$	$+$	$2^{7}\cdot 3$	$\mathrm{SU}(2)$	\(q-3^{8}q^{3}+(550502+\beta )q^{5}+(-2726952+\cdots)q^{7}+\cdots\)
48.18.a.j	$48$	$18$	48.a	1.a	$1$	$2$	$2$	$87.947$	\(\Q(\sqrt{1022389}) \)	None	✓				24.18.a.a	$1$	$0$	\(0\)	\(13122\)	\(354940\)	\(7934208\)		$+$	$-$	$-$	$2^{7}\cdot 3$	$\mathrm{SU}(2)$	\(q+3^{8}q^{3}+(177470-5\beta )q^{5}+(3967104+\cdots)q^{7}+\cdots\)
48.18.a.k	$48$	$18$	48.a	1.a	$1$	$3$	$3$	$87.947$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓		✓		24.18.a.d	$1$	$0$	\(0\)	\(19683\)	\(593658\)	\(-18724416\)		$+$	$-$	$-$	$2^{21}\cdot 3^{4}$	$\mathrm{SU}(2)$	\(q+3^{8}q^{3}+(197886+\beta _{1})q^{5}+(-6241472+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{18}^{\mathrm{old}}(\Gamma_0(48))\) into lower level spaces

\( S_{18}^{\mathrm{old}}(\Gamma_0(48)) \simeq \) \(S_{18}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 10}\)\(\oplus\)\(S_{18}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 8}\)\(\oplus\)\(S_{18}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 5}\)\(\oplus\)\(S_{18}^{\mathrm{new}}(\Gamma_0(4))\)\(^{\oplus 6}\)\(\oplus\)\(S_{18}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 4}\)\(\oplus\)\(S_{18}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 4}\)\(\oplus\)\(S_{18}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 3}\)\(\oplus\)\(S_{18}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 2}\)\(\oplus\)\(S_{18}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 2}\)