Space of modular forms of level 288, weight 8, and trivial character

• Label: 288.8.a
• Level: $288$
• Weight: $8$
• Character orbit: 288.a
• Rep. character: $\chi_{288}(1,\cdot)$
• Character field: $\Q$
• Dimension: $35$
• Newform subspaces: $18$
• Sturm bound: $384$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 288 = 2^{5} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	288.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 18 \)
Sturm bound:			\(384\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(5\), \(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_0(288))\).

	Total	New	Old
Modular forms	352	35	317
Cusp forms	320	35	285
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
\(+\)	\(+\)	\(+\)		\(90\)	\(7\)	\(83\)		\(82\)	\(7\)	\(75\)		\(8\)	\(0\)	\(8\)
\(+\)	\(-\)	\(-\)		\(88\)	\(10\)	\(78\)		\(80\)	\(10\)	\(70\)		\(8\)	\(0\)	\(8\)
\(-\)	\(+\)	\(-\)		\(86\)	\(7\)	\(79\)		\(78\)	\(7\)	\(71\)		\(8\)	\(0\)	\(8\)
\(-\)	\(-\)	\(+\)		\(88\)	\(11\)	\(77\)		\(80\)	\(11\)	\(69\)		\(8\)	\(0\)	\(8\)
Plus space		\(+\)		\(178\)	\(18\)	\(160\)		\(162\)	\(18\)	\(144\)		\(16\)	\(0\)	\(16\)
Minus space		\(-\)		\(174\)	\(17\)	\(157\)		\(158\)	\(17\)	\(141\)		\(16\)	\(0\)	\(16\)

Trace form

35 * q - 278 * q^5 - 510 * q^13 - 19610 * q^17 + 550573 * q^25 + 67602 * q^29 + 675410 * q^37 - 1309394 * q^41 + 2571547 * q^49 - 4188502 * q^53 - 1428854 * q^61 + 1491596 * q^65 - 965658 * q^73 - 3745696 * q^77 - 371252 * q^85 + 11948574 * q^89 + 12237438 * q^97

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_0(288))\) 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
288.8.a.a	$288$	$8$	288.a	1.a	$1$	$1$	$1$	$89.967$	\(\Q\)	\(\Q(\sqrt{-1}) \)	✓	✓			288.8.a.a	$2$	$1$	\(0\)	\(0\)	\(-556\)	\(0\)		$-$	$+$	$-$	$1$	$N(\mathrm{U}(1))$	\(q-556q^{5}+8898q^{13}-5816q^{17}+\cdots\)
288.8.a.b	$288$	$8$	288.a	1.a	$1$	$1$	$1$	$89.967$	\(\Q\)	\(\Q(\sqrt{-1}) \)	✓				32.8.a.a	$2$	$1$	\(0\)	\(0\)	\(58\)	\(0\)		$+$	$-$	$-$	$1$	$N(\mathrm{U}(1))$	\(q+58q^{5}-8898q^{13}+40094q^{17}+\cdots\)
288.8.a.c	$288$	$8$	288.a	1.a	$1$	$1$	$1$	$89.967$	\(\Q\)	None	✓				96.8.a.a	$1$	$0$	\(0\)	\(0\)	\(70\)	\(-92\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+70q^{5}-92q^{7}+3124q^{11}+1174q^{13}+\cdots\)
288.8.a.d	$288$	$8$	288.a	1.a	$1$	$1$	$1$	$89.967$	\(\Q\)	None	✓				96.8.a.a	$1$	$1$	\(0\)	\(0\)	\(70\)	\(92\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+70q^{5}+92q^{7}-3124q^{11}+1174q^{13}+\cdots\)
288.8.a.e	$288$	$8$	288.a	1.a	$1$	$1$	$1$	$89.967$	\(\Q\)	\(\Q(\sqrt{-1}) \)	✓	✓			288.8.a.a	$2$	$0$	\(0\)	\(0\)	\(556\)	\(0\)		$+$	$+$	$+$	$1$	$N(\mathrm{U}(1))$	\(q+556q^{5}+8898q^{13}+5816q^{17}+\cdots\)
288.8.a.f	$288$	$8$	288.a	1.a	$1$	$2$	$2$	$89.967$	\(\Q(\sqrt{435}) \)	None	✓	✓			288.8.a.f	$2$	$0$	\(0\)	\(0\)	\(-280\)	\(0\)		$+$	$+$	$+$	$2^{4}\cdot 3$	$\mathrm{SU}(2)$	\(q-140q^{5}+\beta q^{7}+4\beta q^{11}-2238q^{13}+\cdots\)
288.8.a.g	$288$	$8$	288.a	1.a	$1$	$2$	$2$	$89.967$	\(\Q(\sqrt{46}) \)	None	✓				96.8.a.e	$1$	$1$	\(0\)	\(0\)	\(-196\)	\(-504\)		$+$	$-$	$-$	$2^{4}\cdot 3$	$\mathrm{SU}(2)$	\(q+(-98+\beta )q^{5}+(-252+3\beta )q^{7}+\cdots\)
288.8.a.h	$288$	$8$	288.a	1.a	$1$	$2$	$2$	$89.967$	\(\Q(\sqrt{46}) \)	None	✓				96.8.a.e	$1$	$1$	\(0\)	\(0\)	\(-196\)	\(504\)		$+$	$-$	$-$	$2^{4}\cdot 3$	$\mathrm{SU}(2)$	\(q+(-98+\beta )q^{5}+(252-3\beta )q^{7}+(-828+\cdots)q^{11}+\cdots\)
288.8.a.i	$288$	$8$	288.a	1.a	$1$	$2$	$2$	$89.967$	\(\Q(\sqrt{235}) \)	None	✓				96.8.a.d	$1$	$0$	\(0\)	\(0\)	\(-180\)	\(-1032\)		$-$	$-$	$+$	$2^{4}$	$\mathrm{SU}(2)$	\(q+(-90+\beta )q^{5}+(-516-5\beta )q^{7}+\cdots\)
288.8.a.j	$288$	$8$	288.a	1.a	$1$	$2$	$2$	$89.967$	\(\Q(\sqrt{235}) \)	None	✓				96.8.a.d	$1$	$1$	\(0\)	\(0\)	\(-180\)	\(1032\)		$+$	$-$	$-$	$2^{4}$	$\mathrm{SU}(2)$	\(q+(-90+\beta )q^{5}+(516+5\beta )q^{7}+(-1420+\cdots)q^{11}+\cdots\)
288.8.a.k	$288$	$8$	288.a	1.a	$1$	$2$	$2$	$89.967$	\(\Q(\sqrt{15}) \)	None	✓				32.8.a.c	$2$	$0$	\(0\)	\(0\)	\(-140\)	\(0\)		$-$	$-$	$+$	$2^{4}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q-70q^{5}+2\beta q^{7}+13\beta q^{11}+13758q^{13}+\cdots\)
288.8.a.l	$288$	$8$	288.a	1.a	$1$	$2$	$2$	$89.967$	\(\Q(\sqrt{6}) \)	None	✓				96.8.a.c	$1$	$0$	\(0\)	\(0\)	\(28\)	\(-936\)		$-$	$-$	$+$	$2^{4}\cdot 3$	$\mathrm{SU}(2)$	\(q+(14+3\beta )q^{5}+(-468+7\beta )q^{7}+(-1692+\cdots)q^{11}+\cdots\)
288.8.a.m	$288$	$8$	288.a	1.a	$1$	$2$	$2$	$89.967$	\(\Q(\sqrt{6}) \)	None	✓				96.8.a.c	$1$	$0$	\(0\)	\(0\)	\(28\)	\(936\)		$-$	$-$	$+$	$2^{4}\cdot 3$	$\mathrm{SU}(2)$	\(q+(14+3\beta )q^{5}+(468-7\beta )q^{7}+(1692+\cdots)q^{11}+\cdots\)
288.8.a.n	$288$	$8$	288.a	1.a	$1$	$2$	$2$	$89.967$	\(\Q(\sqrt{10}) \)	None	✓				32.8.a.b	$1$	$1$	\(0\)	\(0\)	\(180\)	\(-1248\)		$+$	$-$	$-$	$2^{4}$	$\mathrm{SU}(2)$	\(q+(90+8\beta )q^{5}+(-624-14\beta )q^{7}+\cdots\)
288.8.a.o	$288$	$8$	288.a	1.a	$1$	$2$	$2$	$89.967$	\(\Q(\sqrt{10}) \)	None	✓				32.8.a.b	$1$	$0$	\(0\)	\(0\)	\(180\)	\(1248\)		$-$	$-$	$+$	$2^{4}$	$\mathrm{SU}(2)$	\(q+(90+8\beta )q^{5}+(624+14\beta )q^{7}+(-4520+\cdots)q^{11}+\cdots\)
288.8.a.p	$288$	$8$	288.a	1.a	$1$	$2$	$2$	$89.967$	\(\Q(\sqrt{435}) \)	None	✓	✓			288.8.a.f	$2$	$1$	\(0\)	\(0\)	\(280\)	\(0\)		$-$	$+$	$-$	$2^{4}\cdot 3$	$\mathrm{SU}(2)$	\(q+140q^{5}+\beta q^{7}-4\beta q^{11}-2238q^{13}+\cdots\)
288.8.a.q	$288$	$8$	288.a	1.a	$1$	$4$	$4$	$89.967$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	✓	✓		288.8.a.q	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$+$	$2^{14}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{2}q^{5}+\beta _{3}q^{7}+(-160-\beta _{1})q^{11}+\cdots\)
288.8.a.r	$288$	$8$	288.a	1.a	$1$	$4$	$4$	$89.967$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	✓	✓		288.8.a.q	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$2^{14}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{2}q^{5}-\beta _{3}q^{7}+(160+\beta _{1})q^{11}+\cdots\)

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_0(288))\) into lower level spaces

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