Space of modular forms of level 528, weight 8, and trivial character

• Label: 528.8.a
• Level: $528$
• Weight: $8$
• Character orbit: 528.a
• Rep. character: $\chi_{528}(1,\cdot)$
• Character field: $\Q$
• Dimension: $70$
• Newform subspaces: $24$
• Sturm bound: $768$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 528 = 2^{4} \cdot 3 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	528.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 24 \)
Sturm bound:			\(768\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_0(528))\).

	Total	New	Old
Modular forms	684	70	614
Cusp forms	660	70	590
Eisenstein series	24	0	24

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

\(2\)	\(3\)	\(11\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	$+$	\(9\)
\(+\)	\(+\)	\(-\)	$-$	\(8\)
\(+\)	\(-\)	\(+\)	$-$	\(9\)
\(+\)	\(-\)	\(-\)	$+$	\(10\)
\(-\)	\(+\)	\(+\)	$-$	\(8\)
\(-\)	\(+\)	\(-\)	$+$	\(9\)
\(-\)	\(-\)	\(+\)	$+$	\(9\)
\(-\)	\(-\)	\(-\)	$-$	\(8\)
Plus space			\(+\)	\(37\)
Minus space			\(-\)	\(33\)

Trace form

\( 70 q + 54 q^{3} + 556 q^{5} + 2012 q^{7} + 51030 q^{9} + O(q^{10}) \)

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_0(528))\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	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	11
528.8.a.a	$528$	$8$	528.a	1.a	$1$	$1$	$1$	$164.939$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-27\)	\(-410\)	\(1028\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3^{3}q^{3}-410q^{5}+1028q^{7}+3^{6}q^{9}+\cdots\)
528.8.a.b	$528$	$8$	528.a	1.a	$1$	$1$	$1$	$164.939$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-27\)	\(-70\)	\(8\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3^{3}q^{3}-70q^{5}+8q^{7}+3^{6}q^{9}+11^{3}q^{11}+\cdots\)
528.8.a.c	$528$	$8$	528.a	1.a	$1$	$1$	$1$	$164.939$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(27\)	\(0\)	\(286\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3^{3}q^{3}+286q^{7}+3^{6}q^{9}+11^{3}q^{11}+\cdots\)
528.8.a.d	$528$	$8$	528.a	1.a	$1$	$2$	$2$	$164.939$	\(\Q(\sqrt{131905}) \)	None	✓	$1$	$1$	\(0\)	\(-54\)	\(-70\)	\(-1162\)		$-$	$+$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q-3^{3}q^{3}+(-35-\beta )q^{5}+(-581+3\beta )q^{7}+\cdots\)
528.8.a.e	$528$	$8$	528.a	1.a	$1$	$2$	$2$	$164.939$	\(\Q(\sqrt{3529}) \)	None	✓	$1$	$1$	\(0\)	\(-54\)	\(-70\)	\(896\)		$-$	$+$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q-3^{3}q^{3}+(-35-\beta )q^{5}+(448-6\beta )q^{7}+\cdots\)
528.8.a.f	$528$	$8$	528.a	1.a	$1$	$2$	$2$	$164.939$	\(\Q(\sqrt{177}) \)	None	✓	$1$	$0$	\(0\)	\(-54\)	\(-34\)	\(166\)		$-$	$+$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q-3^{3}q^{3}+(-17-19\beta )q^{5}+(83-77\beta )q^{7}+\cdots\)
528.8.a.g	$528$	$8$	528.a	1.a	$1$	$2$	$2$	$164.939$	\(\Q(\sqrt{3217}) \)	None	✓	$1$	$0$	\(0\)	\(-54\)	\(250\)	\(-1222\)		$-$	$+$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q-3^{3}q^{3}+(5^{3}-5\beta )q^{5}+(-611-3\beta )q^{7}+\cdots\)
528.8.a.h	$528$	$8$	528.a	1.a	$1$	$2$	$2$	$164.939$	\(\Q(\sqrt{97}) \)	None	✓	$1$	$0$	\(0\)	\(54\)	\(-194\)	\(418\)		$-$	$-$	$+$	$+$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q+3^{3}q^{3}+(-97-7\beta )q^{5}+(209+21\beta )q^{7}+\cdots\)
528.8.a.i	$528$	$8$	528.a	1.a	$1$	$2$	$2$	$164.939$	\(\Q(\sqrt{43465}) \)	None	✓	$1$	$1$	\(0\)	\(54\)	\(-70\)	\(-336\)		$-$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+3^{3}q^{3}+(-35-\beta )q^{5}+(-168-2\beta )q^{7}+\cdots\)
528.8.a.j	$528$	$8$	528.a	1.a	$1$	$2$	$2$	$164.939$	\(\Q(\sqrt{97}) \)	None	✓	$1$	$0$	\(0\)	\(54\)	\(-70\)	\(278\)		$-$	$-$	$+$	$+$	$2\cdot 5$	$\mathrm{SU}(2)$	\(q+3^{3}q^{3}+(-35-5\beta )q^{5}+(139-21\beta )q^{7}+\cdots\)
528.8.a.k	$528$	$8$	528.a	1.a	$1$	$2$	$2$	$164.939$	\(\Q(\sqrt{3193}) \)	None	✓	$1$	$1$	\(0\)	\(54\)	\(-70\)	\(762\)		$-$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+3^{3}q^{3}+(-35-7\beta )q^{5}+(381+19\beta )q^{7}+\cdots\)
528.8.a.l	$528$	$8$	528.a	1.a	$1$	$2$	$2$	$164.939$	\(\Q(\sqrt{175345}) \)	None	✓	$1$	$0$	\(0\)	\(54\)	\(250\)	\(822\)		$-$	$-$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+3^{3}q^{3}+(5^{3}-\beta )q^{5}+(411-\beta )q^{7}+\cdots\)
528.8.a.m	$528$	$8$	528.a	1.a	$1$	$3$	$3$	$164.939$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(-81\)	\(84\)	\(386\)		$+$	$+$	$-$	$-$	$2^{5}\cdot 3$	$\mathrm{SU}(2)$	\(q-3^{3}q^{3}+(28-\beta _{1})q^{5}+(129-\beta _{1}+\cdots)q^{7}+\cdots\)
528.8.a.n	$528$	$8$	528.a	1.a	$1$	$3$	$3$	$164.939$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(-81\)	\(180\)	\(922\)		$-$	$+$	$-$	$+$	$2^{4}$	$\mathrm{SU}(2)$	\(q-3^{3}q^{3}+(60-\beta _{1})q^{5}+(307-\beta _{1}+\cdots)q^{7}+\cdots\)
528.8.a.o	$528$	$8$	528.a	1.a	$1$	$3$	$3$	$164.939$	3.3.115512.1	None	✓	$1$	$1$	\(0\)	\(81\)	\(-444\)	\(-1614\)		$-$	$-$	$-$	$-$	$2^{6}\cdot 3$	$\mathrm{SU}(2)$	\(q+3^{3}q^{3}+(-148+2\beta _{1}+\beta _{2})q^{5}+\cdots\)
528.8.a.p	$528$	$8$	528.a	1.a	$1$	$3$	$3$	$164.939$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(81\)	\(180\)	\(-362\)		$-$	$-$	$+$	$+$	$2^{4}$	$\mathrm{SU}(2)$	\(q+3^{3}q^{3}+(60-5\beta _{1})q^{5}+(-11^{2}-6\beta _{1}+\cdots)q^{7}+\cdots\)
528.8.a.q	$528$	$8$	528.a	1.a	$1$	$4$	$4$	$164.939$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(-108\)	\(14\)	\(1106\)		$+$	$+$	$+$	$+$	$2^{6}\cdot 3$	$\mathrm{SU}(2)$	\(q-3^{3}q^{3}+(4-\beta _{2}+\beta _{3})q^{5}+(277+\beta _{1}+\cdots)q^{7}+\cdots\)
528.8.a.r	$528$	$8$	528.a	1.a	$1$	$4$	$4$	$164.939$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(-108\)	\(306\)	\(-890\)		$-$	$+$	$+$	$-$	$2^{7}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q-3^{3}q^{3}+(77+\beta _{1}+\beta _{2})q^{5}+(-221+\cdots)q^{7}+\cdots\)
528.8.a.s	$528$	$8$	528.a	1.a	$1$	$4$	$4$	$164.939$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(108\)	\(84\)	\(460\)		$+$	$-$	$+$	$-$	$2^{11}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+3^{3}q^{3}+(21+\beta _{1})q^{5}+(115-2\beta _{1}+\cdots)q^{7}+\cdots\)
528.8.a.t	$528$	$8$	528.a	1.a	$1$	$5$	$5$	$164.939$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(-135\)	\(14\)	\(244\)		$+$	$+$	$-$	$-$	$2^{12}\cdot 3^{2}\cdot 5$	$\mathrm{SU}(2)$	\(q-3^{3}q^{3}+(3+\beta _{1})q^{5}+(7^{2}-\beta _{1}-\beta _{2}+\cdots)q^{7}+\cdots\)
528.8.a.u	$528$	$8$	528.a	1.a	$1$	$5$	$5$	$164.939$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(-135\)	\(334\)	\(-476\)		$+$	$+$	$+$	$+$	$2^{9}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q-3^{3}q^{3}+(67+\beta _{1})q^{5}+(-95-\beta _{1}+\cdots)q^{7}+\cdots\)
528.8.a.v	$528$	$8$	528.a	1.a	$1$	$5$	$5$	$164.939$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(135\)	\(14\)	\(-260\)		$+$	$-$	$-$	$+$	$2^{11}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+3^{3}q^{3}+(3-\beta _{2})q^{5}+(-52+\beta _{1}+\cdots)q^{7}+\cdots\)
528.8.a.w	$528$	$8$	528.a	1.a	$1$	$5$	$5$	$164.939$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(135\)	\(14\)	\(-84\)		$+$	$-$	$+$	$-$	$2^{16}\cdot 3$	$\mathrm{SU}(2)$	\(q+3^{3}q^{3}+(3-\beta _{1})q^{5}+(-17+\beta _{2}+\cdots)q^{7}+\cdots\)
528.8.a.x	$528$	$8$	528.a	1.a	$1$	$5$	$5$	$164.939$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(135\)	\(334\)	\(636\)		$+$	$-$	$-$	$+$	$2^{11}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+3^{3}q^{3}+(67-\beta _{1})q^{5}+(127-\beta _{1}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_0(528))\) into lower level spaces

\( S_{8}^{\mathrm{old}}(\Gamma_0(528)) \cong \) \(S_{8}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 16}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 10}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 8}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 8}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 10}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 6}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(22))\)\(^{\oplus 8}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(33))\)\(^{\oplus 5}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(44))\)\(^{\oplus 6}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(48))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(66))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(88))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(132))\)\(^{\oplus 3}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(176))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(264))\)\(^{\oplus 2}\)