Space of modular forms of level 144, weight 16, and trivial character

• Label: 144.16.a
• Level: $144$
• Weight: $16$
• Character orbit: 144.a
• Rep. character: $\chi_{144}(1,\cdot)$
• Character field: $\Q$
• Dimension: $37$
• Newform subspaces: $24$
• Sturm bound: $384$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 144 = 2^{4} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 16 \)
Character orbit:	\([\chi]\)	\(=\)	144.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 24 \)
Sturm bound:			\(384\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{16}(\Gamma_0(144))\).

	Total	New	Old
Modular forms	372	38	334
Cusp forms	348	37	311
Eisenstein series	24	1	23

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
\(+\)	\(+\)	\(+\)		\(94\)	\(8\)	\(86\)		\(88\)	\(8\)	\(80\)		\(6\)	\(0\)	\(6\)
\(+\)	\(-\)	\(-\)		\(92\)	\(11\)	\(81\)		\(86\)	\(11\)	\(75\)		\(6\)	\(0\)	\(6\)
\(-\)	\(+\)	\(-\)		\(92\)	\(7\)	\(85\)		\(86\)	\(7\)	\(79\)		\(6\)	\(0\)	\(6\)
\(-\)	\(-\)	\(+\)		\(94\)	\(12\)	\(82\)		\(88\)	\(11\)	\(77\)		\(6\)	\(1\)	\(5\)
Plus space		\(+\)		\(188\)	\(20\)	\(168\)		\(176\)	\(19\)	\(157\)		\(12\)	\(1\)	\(11\)
Minus space		\(-\)		\(184\)	\(18\)	\(166\)		\(172\)	\(18\)	\(154\)		\(12\)	\(0\)	\(12\)

Trace form

37 * q - 68380 * q^5 - 3234660 * q^7 - 11568464 * q^11 + 64314922 * q^13 + 761979172 * q^17 + 4597217128 * q^19 + 39929327440 * q^23 + 181456009367 * q^25 - 161806497948 * q^29 + 137383183708 * q^31 - 524802328464 * q^35 + 126972888822 * q^37 + 528864393492 * q^41 - 656301196600 * q^43 + 11450173286832 * q^47 + 24877639930301 * q^49 + 867296410388 * q^53 - 4318927093888 * q^55 + 853241730208 * q^59 - 1660693077602 * q^61 + 39200727103352 * q^65 - 42770128991120 * q^67 - 200285818426864 * q^71 + 83347432823638 * q^73 + 176555781377760 * q^77 + 75388048469756 * q^79 - 207498596339536 * q^83 + 125011818400472 * q^85 - 125015286617724 * q^89 - 988521799844040 * q^91 + 541156699173440 * q^95 - 263368713265570 * q^97

Decomposition of \(S_{16}^{\mathrm{new}}(\Gamma_0(144))\) 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
144.16.a.a	$144$	$16$	144.a	1.a	$1$	$1$	$1$	$205.479$	\(\Q\)	None	✓				8.16.a.a	$1$	$1$	\(0\)	\(0\)	\(-313358\)	\(2324616\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-313358q^{5}+2324616q^{7}-55249084q^{11}+\cdots\)
144.16.a.b	$144$	$16$	144.a	1.a	$1$	$1$	$1$	$205.479$	\(\Q\)	None	✓				3.16.a.a	$1$	$0$	\(0\)	\(0\)	\(-280710\)	\(1373344\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-280710q^{5}+1373344q^{7}+34031052q^{11}+\cdots\)
144.16.a.c	$144$	$16$	144.a	1.a	$1$	$1$	$1$	$205.479$	\(\Q\)	None	✓				18.16.a.c	$1$	$1$	\(0\)	\(0\)	\(-263040\)	\(-3585764\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-263040q^{5}-3585764q^{7}+65754624q^{11}+\cdots\)
144.16.a.d	$144$	$16$	144.a	1.a	$1$	$1$	$1$	$205.479$	\(\Q\)	None	✓				2.16.a.a	$1$	$0$	\(0\)	\(0\)	\(-90510\)	\(-56\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-90510q^{5}-56q^{7}-95889948q^{11}+\cdots\)
144.16.a.e	$144$	$16$	144.a	1.a	$1$	$1$	$1$	$205.479$	\(\Q\)	None	✓				6.16.a.c	$1$	$0$	\(0\)	\(0\)	\(-77646\)	\(-762104\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-77646q^{5}-762104q^{7}+48011172q^{11}+\cdots\)
144.16.a.f	$144$	$16$	144.a	1.a	$1$	$1$	$1$	$205.479$	\(\Q\)	None	✓				1.16.a.a	$1$	$0$	\(0\)	\(0\)	\(-52110\)	\(-2822456\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-52110q^{5}-2822456q^{7}+20586852q^{11}+\cdots\)
144.16.a.g	$144$	$16$	144.a	1.a	$1$	$1$	$1$	$205.479$	\(\Q\)	None	✓				12.16.a.a	$1$	$0$	\(0\)	\(0\)	\(-45702\)	\(-1217888\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-45702q^{5}-1217888q^{7}-26895924q^{11}+\cdots\)
144.16.a.h	$144$	$16$	144.a	1.a	$1$	$1$	$1$	$205.479$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓				9.16.a.b	$2$	$1$	\(0\)	\(0\)	\(0\)	\(-1244900\)		$-$	$+$	$-$	$1$	$N(\mathrm{U}(1))$	\(q-1244900q^{7}+397771850q^{13}+\cdots\)
144.16.a.i	$144$	$16$	144.a	1.a	$1$	$1$	$1$	$205.479$	\(\Q\)	None	✓				24.16.a.a	$1$	$1$	\(0\)	\(0\)	\(42010\)	\(-385728\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+42010q^{5}-385728q^{7}+20444300q^{11}+\cdots\)
144.16.a.j	$144$	$16$	144.a	1.a	$1$	$1$	$1$	$205.479$	\(\Q\)	None	✓				6.16.a.b	$1$	$0$	\(0\)	\(0\)	\(114810\)	\(3034528\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+114810q^{5}+3034528q^{7}-103451700q^{11}+\cdots\)
144.16.a.k	$144$	$16$	144.a	1.a	$1$	$1$	$1$	$205.479$	\(\Q\)	None	✓				4.16.a.a	$1$	$0$	\(0\)	\(0\)	\(132210\)	\(3585736\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+132210q^{5}+3585736q^{7}+47801700q^{11}+\cdots\)
144.16.a.l	$144$	$16$	144.a	1.a	$1$	$1$	$1$	$205.479$	\(\Q\)	None	✓				3.16.a.b	$1$	$0$	\(0\)	\(0\)	\(221490\)	\(2149000\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+221490q^{5}+2149000q^{7}+37169316q^{11}+\cdots\)
144.16.a.m	$144$	$16$	144.a	1.a	$1$	$1$	$1$	$205.479$	\(\Q\)	None	✓				8.16.a.b	$1$	$1$	\(0\)	\(0\)	\(251890\)	\(-1374072\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+251890q^{5}-1374072q^{7}-43286716q^{11}+\cdots\)
144.16.a.n	$144$	$16$	144.a	1.a	$1$	$1$	$1$	$205.479$	\(\Q\)	None	✓				18.16.a.c	$1$	$1$	\(0\)	\(0\)	\(263040\)	\(-3585764\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+263040q^{5}-3585764q^{7}-65754624q^{11}+\cdots\)
144.16.a.o	$144$	$16$	144.a	1.a	$1$	$1$	$1$	$205.479$	\(\Q\)	None	✓				6.16.a.a	$1$	$0$	\(0\)	\(0\)	\(314490\)	\(-2025056\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+314490q^{5}-2025056q^{7}+110255052q^{11}+\cdots\)
144.16.a.p	$144$	$16$	144.a	1.a	$1$	$2$	$2$	$205.479$	\(\Q(\sqrt{158701}) \)	None	✓				24.16.a.b	$1$	$1$	\(0\)	\(0\)	\(-291404\)	\(-1161216\)		$+$	$-$	$-$	$2^{7}\cdot 3$	$\mathrm{SU}(2)$	\(q+(-145702-\beta )q^{5}+(-580608+\cdots)q^{7}+\cdots\)
144.16.a.q	$144$	$16$	144.a	1.a	$1$	$2$	$2$	$205.479$	\(\Q(\sqrt{1285}) \)	None	✓				24.16.a.d	$1$	$1$	\(0\)	\(0\)	\(-82780\)	\(42192\)		$+$	$-$	$-$	$2^{9}\cdot 3\cdot 5$	$\mathrm{SU}(2)$	\(q+(-41390-\beta )q^{5}+(21096-23\beta )q^{7}+\cdots\)
144.16.a.r	$144$	$16$	144.a	1.a	$1$	$2$	$2$	$205.479$	\(\Q(\sqrt{8017}) \)	None	✓		✓		12.16.a.b	$1$	$0$	\(0\)	\(0\)	\(-69660\)	\(-2491504\)		$-$	$-$	$+$	$2^{7}\cdot 3^{2}\cdot 5$	$\mathrm{SU}(2)$	\(q+(-34830-\beta )q^{5}+(-1245752+\cdots)q^{7}+\cdots\)
144.16.a.s	$144$	$16$	144.a	1.a	$1$	$2$	$2$	$205.479$	\(\Q(\sqrt{286}) \)	None	✓				36.16.a.d	$2$	$1$	\(0\)	\(0\)	\(0\)	\(56\)		$-$	$+$	$-$	$2^{5}\cdot 3^{2}\cdot 5\cdot 7$	$\mathrm{SU}(2)$	\(q+\beta q^{5}+28q^{7}+524\beta q^{11}-196534q^{13}+\cdots\)
144.16.a.t	$144$	$16$	144.a	1.a	$1$	$2$	$2$	$205.479$	\(\Q(\sqrt{370}) \)	None	✓				9.16.a.e	$2$	$1$	\(0\)	\(0\)	\(0\)	\(5182520\)		$-$	$+$	$-$	$2^{5}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+29\beta q^{5}+2591260q^{7}-6820\beta q^{11}+\cdots\)
144.16.a.u	$144$	$16$	144.a	1.a	$1$	$2$	$2$	$205.479$	\(\Q(\sqrt{22}) \)	None	✓				24.16.a.c	$1$	$1$	\(0\)	\(0\)	\(18340\)	\(680400\)		$+$	$-$	$-$	$2^{7}\cdot 3\cdot 5$	$\mathrm{SU}(2)$	\(q+(9170+19\beta )q^{5}+(340200-203\beta )q^{7}+\cdots\)
144.16.a.v	$144$	$16$	144.a	1.a	$1$	$2$	$2$	$205.479$	\(\Q(\sqrt{58}) \)	None	✓				8.16.a.c	$1$	$1$	\(0\)	\(0\)	\(140260\)	\(-126192\)		$+$	$-$	$-$	$2^{7}\cdot 3^{2}\cdot 5$	$\mathrm{SU}(2)$	\(q+(70130+4\beta )q^{5}+(-63096+10\beta )q^{7}+\cdots\)
144.16.a.w	$144$	$16$	144.a	1.a	$1$	$4$	$4$	$205.479$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓				72.16.a.h	$1$	$0$	\(0\)	\(0\)	\(-26944\)	\(-412176\)		$+$	$+$	$+$	$2^{22}\cdot 3^{8}\cdot 5$	$\mathrm{SU}(2)$	\(q+(-6736+\beta _{1})q^{5}+(-103044-\beta _{1}+\cdots)q^{7}+\cdots\)
144.16.a.x	$144$	$16$	144.a	1.a	$1$	$4$	$4$	$205.479$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓				72.16.a.h	$1$	$0$	\(0\)	\(0\)	\(26944\)	\(-412176\)		$+$	$+$	$+$	$2^{22}\cdot 3^{8}\cdot 5$	$\mathrm{SU}(2)$	\(q+(6736-\beta _{1})q^{5}+(-103044-\beta _{1}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{16}^{\mathrm{old}}(\Gamma_0(144))\) into lower level spaces

\( S_{16}^{\mathrm{old}}(\Gamma_0(144)) \simeq \) \(S_{16}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 15}\)\(\oplus\)\(S_{16}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 12}\)\(\oplus\)\(S_{16}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 10}\)\(\oplus\)\(S_{16}^{\mathrm{new}}(\Gamma_0(4))\)\(^{\oplus 9}\)\(\oplus\)\(S_{16}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 8}\)\(\oplus\)\(S_{16}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 6}\)\(\oplus\)\(S_{16}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 5}\)\(\oplus\)\(S_{16}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 6}\)\(\oplus\)\(S_{16}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 3}\)\(\oplus\)\(S_{16}^{\mathrm{new}}(\Gamma_0(18))\)\(^{\oplus 4}\)\(\oplus\)\(S_{16}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 4}\)\(\oplus\)\(S_{16}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 3}\)\(\oplus\)\(S_{16}^{\mathrm{new}}(\Gamma_0(48))\)\(^{\oplus 2}\)\(\oplus\)\(S_{16}^{\mathrm{new}}(\Gamma_0(72))\)\(^{\oplus 2}\)