Space of modular forms of level 588, weight 8, and trivial character

• Label: 588.8.a
• Level: $588$
• Weight: $8$
• Character orbit: 588.a
• Rep. character: $\chi_{588}(1,\cdot)$
• Character field: $\Q$
• Dimension: $48$
• Newform subspaces: $14$
• Sturm bound: $896$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 588 = 2^{2} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	588.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 14 \)
Sturm bound:			\(896\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_0(588))\).

	Total	New	Old
Modular forms	808	48	760
Cusp forms	760	48	712
Eisenstein series	48	0	48

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(2\)	\(3\)	\(7\)	Fricke		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(104\)	\(0\)	\(104\)		\(96\)	\(0\)	\(96\)		\(8\)	\(0\)	\(8\)
\(+\)	\(+\)	\(-\)	\(-\)		\(101\)	\(0\)	\(101\)		\(93\)	\(0\)	\(93\)		\(8\)	\(0\)	\(8\)
\(+\)	\(-\)	\(+\)	\(-\)		\(100\)	\(0\)	\(100\)		\(92\)	\(0\)	\(92\)		\(8\)	\(0\)	\(8\)
\(+\)	\(-\)	\(-\)	\(+\)		\(103\)	\(0\)	\(103\)		\(95\)	\(0\)	\(95\)		\(8\)	\(0\)	\(8\)
\(-\)	\(+\)	\(+\)	\(-\)		\(100\)	\(12\)	\(88\)		\(96\)	\(12\)	\(84\)		\(4\)	\(0\)	\(4\)
\(-\)	\(+\)	\(-\)	\(+\)		\(100\)	\(12\)	\(88\)		\(96\)	\(12\)	\(84\)		\(4\)	\(0\)	\(4\)
\(-\)	\(-\)	\(+\)	\(+\)		\(100\)	\(13\)	\(87\)		\(96\)	\(13\)	\(83\)		\(4\)	\(0\)	\(4\)
\(-\)	\(-\)	\(-\)	\(-\)		\(100\)	\(11\)	\(89\)		\(96\)	\(11\)	\(85\)		\(4\)	\(0\)	\(4\)
Plus space			\(+\)		\(407\)	\(25\)	\(382\)		\(383\)	\(25\)	\(358\)		\(24\)	\(0\)	\(24\)
Minus space			\(-\)		\(401\)	\(23\)	\(378\)		\(377\)	\(23\)	\(354\)		\(24\)	\(0\)	\(24\)

Trace form

48 * q - 112 * q^5 + 34992 * q^9 + 5572 * q^11 - 2128 * q^13 - 3780 * q^15 + 8008 * q^17 + 1400 * q^19 + 86160 * q^23 + 769712 * q^25 - 324068 * q^29 - 284704 * q^31 - 102816 * q^33 - 724462 * q^37 + 128142 * q^39 - 1090488 * q^41 + 1767486 * q^43 - 81648 * q^45 - 137928 * q^47 + 874800 * q^51 - 3335036 * q^53 - 3869544 * q^55 - 577854 * q^57 - 834904 * q^59 - 338464 * q^61 - 2326132 * q^65 - 3364910 * q^67 - 4008312 * q^69 + 3096940 * q^71 - 4378864 * q^73 + 3770928 * q^75 + 9362202 * q^79 + 25509168 * q^81 - 22918112 * q^83 + 16344368 * q^85 + 4539024 * q^87 + 14917560 * q^89 - 5375430 * q^93 - 10453468 * q^95 - 33327616 * q^97 + 4061988 * q^99

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_0(588))\) 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	7
588.8.a.a	$588$	$8$	588.a	1.a	$1$	$1$	$1$	$183.682$	\(\Q\)	None	✓				12.8.a.b	$1$	$0$	\(0\)	\(-27\)	\(-270\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3^{3}q^{3}-270q^{5}+3^{6}q^{9}-5724q^{11}+\cdots\)
588.8.a.b	$588$	$8$	588.a	1.a	$1$	$1$	$1$	$183.682$	\(\Q\)	None	✓				84.8.a.b	$1$	$0$	\(0\)	\(-27\)	\(240\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3^{3}q^{3}+240q^{5}+3^{6}q^{9}+702q^{11}+\cdots\)
588.8.a.c	$588$	$8$	588.a	1.a	$1$	$1$	$1$	$183.682$	\(\Q\)	None	✓				84.8.a.a	$1$	$1$	\(0\)	\(27\)	\(-100\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3^{3}q^{3}-10^{2}q^{5}+3^{6}q^{9}+2774q^{11}+\cdots\)
588.8.a.d	$588$	$8$	588.a	1.a	$1$	$1$	$1$	$183.682$	\(\Q\)	None	✓				12.8.a.a	$1$	$1$	\(0\)	\(27\)	\(378\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3^{3}q^{3}+378q^{5}+3^{6}q^{9}-2484q^{11}+\cdots\)
588.8.a.e	$588$	$8$	588.a	1.a	$1$	$2$	$2$	$183.682$	\(\Q(\sqrt{21961}) \)	None	✓				84.8.a.d	$1$	$0$	\(0\)	\(-54\)	\(-96\)	\(0\)		$-$	$+$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q-3^{3}q^{3}+(-48-\beta )q^{5}+3^{6}q^{9}+(2070+\cdots)q^{11}+\cdots\)
588.8.a.f	$588$	$8$	588.a	1.a	$1$	$2$	$2$	$183.682$	\(\Q(\sqrt{3649}) \)	None	✓				84.8.a.c	$1$	$1$	\(0\)	\(54\)	\(-264\)	\(0\)		$-$	$-$	$-$	$-$	$2^{2}\cdot 3$	$\mathrm{SU}(2)$	\(q+3^{3}q^{3}+(-132-\beta )q^{5}+3^{6}q^{9}+\cdots\)
588.8.a.g	$588$	$8$	588.a	1.a	$1$	$3$	$3$	$183.682$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓	✓			588.8.a.g	$1$	$0$	\(0\)	\(-81\)	\(-254\)	\(0\)		$-$	$+$	$-$	$+$	$2^{4}$	$\mathrm{SU}(2)$	\(q-3^{3}q^{3}+(-85-\beta _{1})q^{5}+3^{6}q^{9}+\cdots\)
588.8.a.h	$588$	$8$	588.a	1.a	$1$	$3$	$3$	$183.682$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓	✓			588.8.a.g	$1$	$1$	\(0\)	\(81\)	\(254\)	\(0\)		$-$	$-$	$-$	$-$	$2^{4}$	$\mathrm{SU}(2)$	\(q+3^{3}q^{3}+(85+\beta _{1})q^{5}+3^{6}q^{9}+(-438+\cdots)q^{11}+\cdots\)
588.8.a.i	$588$	$8$	588.a	1.a	$1$	$4$	$4$	$183.682$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓				84.8.i.a	$1$	$1$	\(0\)	\(-108\)	\(196\)	\(0\)		$-$	$+$	$+$	$-$	$2^{3}\cdot 3^{3}\cdot 7$	$\mathrm{SU}(2)$	\(q-3^{3}q^{3}+(7^{2}+\beta _{2})q^{5}+3^{6}q^{9}+(-98+\cdots)q^{11}+\cdots\)
588.8.a.j	$588$	$8$	588.a	1.a	$1$	$4$	$4$	$183.682$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓		✓		84.8.i.a	$1$	$1$	\(0\)	\(108\)	\(-196\)	\(0\)		$-$	$-$	$-$	$-$	$2^{3}\cdot 3^{3}\cdot 7$	$\mathrm{SU}(2)$	\(q+3^{3}q^{3}+(-7^{2}-\beta _{2})q^{5}+3^{6}q^{9}+\cdots\)
588.8.a.k	$588$	$8$	588.a	1.a	$1$	$5$	$5$	$183.682$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓		✓		84.8.i.b	$1$	$0$	\(0\)	\(-135\)	\(198\)	\(0\)		$-$	$+$	$-$	$+$	$2^{6}\cdot 3^{4}\cdot 7^{2}$	$\mathrm{SU}(2)$	\(q-3^{3}q^{3}+(40+\beta _{1})q^{5}+3^{6}q^{9}+(1460+\cdots)q^{11}+\cdots\)
588.8.a.l	$588$	$8$	588.a	1.a	$1$	$5$	$5$	$183.682$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓				84.8.i.b	$1$	$0$	\(0\)	\(135\)	\(-198\)	\(0\)		$-$	$-$	$+$	$+$	$2^{6}\cdot 3^{4}\cdot 7^{2}$	$\mathrm{SU}(2)$	\(q+3^{3}q^{3}+(-40-\beta _{1})q^{5}+3^{6}q^{9}+\cdots\)
588.8.a.m	$588$	$8$	588.a	1.a	$1$	$8$	$8$	$183.682$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	✓	✓		588.8.a.m	$1$	$1$	\(0\)	\(-216\)	\(0\)	\(0\)		$-$	$+$	$+$	$-$	$2^{17}\cdot 3^{4}\cdot 7^{6}$	$\mathrm{SU}(2)$	\(q-3^{3}q^{3}+\beta _{2}q^{5}+3^{6}q^{9}+(13\beta _{1}+\beta _{5}+\cdots)q^{11}+\cdots\)
588.8.a.n	$588$	$8$	588.a	1.a	$1$	$8$	$8$	$183.682$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	✓	✓		588.8.a.m	$1$	$0$	\(0\)	\(216\)	\(0\)	\(0\)		$-$	$-$	$+$	$+$	$2^{17}\cdot 3^{4}\cdot 7^{6}$	$\mathrm{SU}(2)$	\(q+3^{3}q^{3}-\beta _{2}q^{5}+3^{6}q^{9}+(13\beta _{1}+\beta _{5}+\cdots)q^{11}+\cdots\)

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_0(588))\) into lower level spaces

\( S_{8}^{\mathrm{old}}(\Gamma_0(588)) \simeq \) \(S_{8}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 12}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 9}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 6}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 12}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 3}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 8}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 6}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(28))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(42))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(49))\)\(^{\oplus 6}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(84))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(98))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(147))\)\(^{\oplus 3}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(196))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(294))\)\(^{\oplus 2}\)