Space of modular forms of level 288, weight 10, and trivial character

• Label: 288.10.a
• Level: $288$
• Weight: $10$
• Character orbit: 288.a
• Rep. character: $\chi_{288}(1,\cdot)$
• Character field: $\Q$
• Dimension: $45$
• Newform subspaces: $19$
• Sturm bound: $480$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 288 = 2^{5} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	288.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 19 \)
Sturm bound:			\(480\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(5\), \(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{10}(\Gamma_0(288))\).

	Total	New	Old
Modular forms	448	45	403
Cusp forms	416	45	371
Eisenstein series	32	0	32

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
\(+\)	\(+\)	\(+\)		\(110\)	\(9\)	\(101\)		\(102\)	\(9\)	\(93\)		\(8\)	\(0\)	\(8\)
\(+\)	\(-\)	\(-\)		\(112\)	\(14\)	\(98\)		\(104\)	\(14\)	\(90\)		\(8\)	\(0\)	\(8\)
\(-\)	\(+\)	\(-\)		\(114\)	\(9\)	\(105\)		\(106\)	\(9\)	\(97\)		\(8\)	\(0\)	\(8\)
\(-\)	\(-\)	\(+\)		\(112\)	\(13\)	\(99\)		\(104\)	\(13\)	\(91\)		\(8\)	\(0\)	\(8\)
Plus space		\(+\)		\(222\)	\(22\)	\(200\)		\(206\)	\(22\)	\(184\)		\(16\)	\(0\)	\(16\)
Minus space		\(-\)		\(226\)	\(23\)	\(203\)		\(210\)	\(23\)	\(187\)		\(16\)	\(0\)	\(16\)

Trace form

45 * q + 718 * q^5 + 280774 * q^13 - 63782 * q^17 + 19398219 * q^25 + 4941270 * q^29 + 19110150 * q^37 + 2908354 * q^41 + 322529701 * q^49 + 106622510 * q^53 + 283524478 * q^61 - 639269452 * q^65 - 852908790 * q^73 - 391542496 * q^77 - 1297880444 * q^85 + 1477625778 * q^89 - 2143826078 * q^97

Decomposition of \(S_{10}^{\mathrm{new}}(\Gamma_0(288))\) 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
288.10.a.a	$288$	$10$	288.a	1.a	$1$	$1$	$1$	$148.330$	\(\Q\)	\(\Q(\sqrt{-1}) \)	✓				32.10.a.a	$2$	$0$	\(0\)	\(0\)	\(-2398\)	\(0\)		$+$	$-$	$-$	$1$	$N(\mathrm{U}(1))$	\(q-2398q^{5}+112806q^{13}-554882q^{17}+\cdots\)
288.10.a.b	$288$	$10$	288.a	1.a	$1$	$1$	$1$	$148.330$	\(\Q\)	\(\Q(\sqrt{-1}) \)	✓	✓			288.10.a.b	$2$	$0$	\(0\)	\(0\)	\(-1436\)	\(0\)		$-$	$+$	$-$	$1$	$N(\mathrm{U}(1))$	\(q-1436q^{5}-112806q^{13}-407992q^{17}+\cdots\)
288.10.a.c	$288$	$10$	288.a	1.a	$1$	$1$	$1$	$148.330$	\(\Q\)	\(\Q(\sqrt{-1}) \)	✓	✓			288.10.a.b	$2$	$1$	\(0\)	\(0\)	\(1436\)	\(0\)		$+$	$+$	$+$	$1$	$N(\mathrm{U}(1))$	\(q+1436q^{5}-112806q^{13}+407992q^{17}+\cdots\)
288.10.a.d	$288$	$10$	288.a	1.a	$1$	$2$	$2$	$148.330$	\(\Q(\sqrt{106}) \)	None	✓				32.10.a.b	$1$	$1$	\(0\)	\(0\)	\(-1404\)	\(-2784\)		$-$	$-$	$+$	$2^{4}$	$\mathrm{SU}(2)$	\(q+(-702+8\beta )q^{5}+(-1392+10\beta )q^{7}+\cdots\)
288.10.a.e	$288$	$10$	288.a	1.a	$1$	$2$	$2$	$148.330$	\(\Q(\sqrt{106}) \)	None	✓				32.10.a.b	$1$	$0$	\(0\)	\(0\)	\(-1404\)	\(2784\)		$+$	$-$	$-$	$2^{4}$	$\mathrm{SU}(2)$	\(q+(-702+8\beta )q^{5}+(1392-10\beta )q^{7}+\cdots\)
288.10.a.f	$288$	$10$	288.a	1.a	$1$	$2$	$2$	$148.330$	\(\Q(\sqrt{46}) \)	None	✓				96.10.a.c	$1$	$0$	\(0\)	\(0\)	\(-596\)	\(-5304\)		$+$	$-$	$-$	$2^{4}\cdot 3$	$\mathrm{SU}(2)$	\(q+(-298+\beta )q^{5}+(-2652-17\beta )q^{7}+\cdots\)
288.10.a.g	$288$	$10$	288.a	1.a	$1$	$2$	$2$	$148.330$	\(\Q(\sqrt{46}) \)	None	✓				96.10.a.c	$1$	$0$	\(0\)	\(0\)	\(-596\)	\(5304\)		$+$	$-$	$-$	$2^{4}\cdot 3$	$\mathrm{SU}(2)$	\(q+(-298+\beta )q^{5}+(2652+17\beta )q^{7}+\cdots\)
288.10.a.h	$288$	$10$	288.a	1.a	$1$	$2$	$2$	$148.330$	\(\Q(\sqrt{134}) \)	None	✓				96.10.a.b	$1$	$1$	\(0\)	\(0\)	\(-244\)	\(-3816\)		$-$	$-$	$+$	$2^{4}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+(-122+\beta )q^{5}+(-1908+\beta )q^{7}+\cdots\)
288.10.a.i	$288$	$10$	288.a	1.a	$1$	$2$	$2$	$148.330$	\(\Q(\sqrt{134}) \)	None	✓				96.10.a.b	$1$	$1$	\(0\)	\(0\)	\(-244\)	\(3816\)		$-$	$-$	$+$	$2^{4}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+(-122+\beta )q^{5}+(1908-\beta )q^{7}+(-1332+\cdots)q^{11}+\cdots\)
288.10.a.j	$288$	$10$	288.a	1.a	$1$	$2$	$2$	$148.330$	\(\Q(\sqrt{7}) \)	None	✓				32.10.a.d	$2$	$1$	\(0\)	\(0\)	\(68\)	\(0\)		$-$	$-$	$+$	$2^{4}\cdot 3$	$\mathrm{SU}(2)$	\(q+34q^{5}-86\beta q^{7}-83\beta q^{11}-62682q^{13}+\cdots\)
288.10.a.k	$288$	$10$	288.a	1.a	$1$	$2$	$2$	$148.330$	\(\Q(\sqrt{2059}) \)	None	✓				96.10.a.a	$1$	$0$	\(0\)	\(0\)	\(1404\)	\(-3528\)		$+$	$-$	$-$	$2^{4}$	$\mathrm{SU}(2)$	\(q+(702+\beta )q^{5}+(-42^{2}+7\beta )q^{7}+(-15620+\cdots)q^{11}+\cdots\)
288.10.a.l	$288$	$10$	288.a	1.a	$1$	$2$	$2$	$148.330$	\(\Q(\sqrt{2059}) \)	None	✓				96.10.a.a	$1$	$1$	\(0\)	\(0\)	\(1404\)	\(3528\)		$-$	$-$	$+$	$2^{4}$	$\mathrm{SU}(2)$	\(q+(702+\beta )q^{5}+(42^{2}-7\beta )q^{7}+(15620+\cdots)q^{11}+\cdots\)
288.10.a.m	$288$	$10$	288.a	1.a	$1$	$2$	$2$	$148.330$	\(\Q(\sqrt{5}) \)	None	✓				32.10.a.c	$2$	$0$	\(0\)	\(0\)	\(4420\)	\(0\)		$+$	$-$	$-$	$2^{6}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+2210q^{5}+14\beta q^{7}-113\beta q^{11}+\cdots\)
288.10.a.n	$288$	$10$	288.a	1.a	$1$	$3$	$3$	$148.330$	3.3.723301.1	None	✓		✓		96.10.a.g	$1$	$1$	\(0\)	\(0\)	\(154\)	\(-7564\)		$-$	$-$	$+$	$2^{15}\cdot 3$	$\mathrm{SU}(2)$	\(q+(51-\beta _{2})q^{5}+(-2521-\beta _{1})q^{7}+\cdots\)
288.10.a.o	$288$	$10$	288.a	1.a	$1$	$3$	$3$	$148.330$	3.3.723301.1	None	✓		✓		96.10.a.g	$1$	$0$	\(0\)	\(0\)	\(154\)	\(7564\)		$+$	$-$	$-$	$2^{15}\cdot 3$	$\mathrm{SU}(2)$	\(q+(51-\beta _{2})q^{5}+(2521+\beta _{1})q^{7}+(1986+\cdots)q^{11}+\cdots\)
288.10.a.p	$288$	$10$	288.a	1.a	$1$	$4$	$4$	$148.330$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	✓			288.10.a.p	$2$	$1$	\(0\)	\(0\)	\(-4288\)	\(0\)		$+$	$+$	$+$	$2^{16}\cdot 3^{5}\cdot 5$	$\mathrm{SU}(2)$	\(q+(-1072+\beta _{3})q^{5}-\beta _{1}q^{7}+(3\beta _{1}+\cdots)q^{11}+\cdots\)
288.10.a.q	$288$	$10$	288.a	1.a	$1$	$4$	$4$	$148.330$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	✓			288.10.a.q	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$+$	$2^{14}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{5}+(-\beta _{1}-\beta _{2})q^{7}+(-20960+\cdots)q^{11}+\cdots\)
288.10.a.r	$288$	$10$	288.a	1.a	$1$	$4$	$4$	$148.330$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	✓			288.10.a.q	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$2^{14}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{5}+(\beta _{1}+\beta _{2})q^{7}+(20960-\beta _{3})q^{11}+\cdots\)
288.10.a.s	$288$	$10$	288.a	1.a	$1$	$4$	$4$	$148.330$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	✓			288.10.a.p	$2$	$0$	\(0\)	\(0\)	\(4288\)	\(0\)		$-$	$+$	$-$	$2^{16}\cdot 3^{5}\cdot 5$	$\mathrm{SU}(2)$	\(q+(1072-\beta _{3})q^{5}-\beta _{1}q^{7}+(-3\beta _{1}+\cdots)q^{11}+\cdots\)

Decomposition of \(S_{10}^{\mathrm{old}}(\Gamma_0(288))\) into lower level spaces

\( S_{10}^{\mathrm{old}}(\Gamma_0(288)) \simeq \) \(S_{10}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 15}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 12}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(4))\)\(^{\oplus 12}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 10}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 9}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 6}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 8}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 6}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(18))\)\(^{\oplus 5}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 6}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 3}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 4}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(48))\)\(^{\oplus 4}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(72))\)\(^{\oplus 3}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(96))\)\(^{\oplus 2}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(144))\)\(^{\oplus 2}\)