Space of modular forms of level 144, weight 4, and trivial character

• Label: 144.4.a
• Level: $144$
• Weight: $4$
• Character orbit: 144.a
• Rep. character: $\chi_{144}(1,\cdot)$
• Character field: $\Q$
• Dimension: $7$
• Newform subspaces: $7$
• Sturm bound: $96$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 144 = 2^{4} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	144.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 7 \)
Sturm bound:			\(96\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(144))\).

	Total	New	Old
Modular forms	84	8	76
Cusp forms	60	7	53
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	Dim
\(+\)	\(+\)	\(+\)	\(2\)
\(+\)	\(-\)	\(-\)	\(2\)
\(-\)	\(+\)	\(-\)	\(1\)
\(-\)	\(-\)	\(+\)	\(2\)
Plus space		\(+\)	\(4\)
Minus space		\(-\)	\(3\)

Trace form

7 * q + 12 * q^7 - 24 * q^11 + 22 * q^13 - 24 * q^17 + 160 * q^19 + 192 * q^23 + 197 * q^25 + 144 * q^29 - 164 * q^31 - 624 * q^35 - 30 * q^37 - 216 * q^41 + 464 * q^43 + 1104 * q^47 - 241 * q^49 - 240 * q^53 - 1168 * q^55 - 1416 * q^59 - 518 * q^61 + 672 * q^65 + 904 * q^67 + 2016 * q^71 - 614 * q^73 + 288 * q^77 - 2404 * q^79 - 2952 * q^83 + 992 * q^85 - 648 * q^89 + 1176 * q^91 + 3120 * q^95 - 430 * q^97

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(144))\) 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
144.4.a.a	$144$	$4$	144.a	1.a	$1$	$1$	$1$	$8.496$	\(\Q\)	None	✓				72.4.a.a	$1$	$0$	\(0\)	\(0\)	\(-16\)	\(12\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2^{4}q^{5}+12q^{7}+2^{6}q^{11}+58q^{13}+\cdots\)
144.4.a.b	$144$	$4$	144.a	1.a	$1$	$1$	$1$	$8.496$	\(\Q\)	None	✓				24.4.a.a	$1$	$1$	\(0\)	\(0\)	\(-14\)	\(24\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-14q^{5}+24q^{7}-28q^{11}-74q^{13}+\cdots\)
144.4.a.c	$144$	$4$	144.a	1.a	$1$	$1$	$1$	$8.496$	\(\Q\)	None	✓				6.4.a.a	$1$	$0$	\(0\)	\(0\)	\(-6\)	\(16\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-6q^{5}+2^{4}q^{7}+12q^{11}+38q^{13}+\cdots\)
144.4.a.d	$144$	$4$	144.a	1.a	$1$	$1$	$1$	$8.496$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓		✓	✓	9.4.a.a	$2$	$1$	\(0\)	\(0\)	\(0\)	\(-20\)		$-$	$+$	$-$	$1$	$N(\mathrm{U}(1))$	\(q-20q^{7}-70q^{13}-56q^{19}-5^{3}q^{25}+\cdots\)
144.4.a.e	$144$	$4$	144.a	1.a	$1$	$1$	$1$	$8.496$	\(\Q\)	None	✓				8.4.a.a	$1$	$1$	\(0\)	\(0\)	\(2\)	\(-24\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{5}-24q^{7}-44q^{11}+22q^{13}+\cdots\)
144.4.a.f	$144$	$4$	144.a	1.a	$1$	$1$	$1$	$8.496$	\(\Q\)	None	✓				72.4.a.a	$1$	$0$	\(0\)	\(0\)	\(16\)	\(12\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2^{4}q^{5}+12q^{7}-2^{6}q^{11}+58q^{13}+\cdots\)
144.4.a.g	$144$	$4$	144.a	1.a	$1$	$1$	$1$	$8.496$	\(\Q\)	None	✓				12.4.a.a	$1$	$0$	\(0\)	\(0\)	\(18\)	\(-8\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+18q^{5}-8q^{7}+6^{2}q^{11}-10q^{13}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(144))\) into lower level spaces

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