Space of modular forms of level 528, weight 6, and trivial character

• Label: 528.6.a
• Level: $528$
• Weight: $6$
• Character orbit: 528.a
• Rep. character: $\chi_{528}(1,\cdot)$
• Character field: $\Q$
• Dimension: $50$
• Newform subspaces: $26$
• Sturm bound: $576$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 528 = 2^{4} \cdot 3 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	528.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 26 \)
Sturm bound:			\(576\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(528))\).

	Total	New	Old
Modular forms	492	50	442
Cusp forms	468	50	418
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		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(59\)	\(6\)	\(53\)		\(56\)	\(6\)	\(50\)		\(3\)	\(0\)	\(3\)
\(+\)	\(+\)	\(-\)	\(-\)		\(64\)	\(7\)	\(57\)		\(61\)	\(7\)	\(54\)		\(3\)	\(0\)	\(3\)
\(+\)	\(-\)	\(+\)	\(-\)		\(61\)	\(6\)	\(55\)		\(58\)	\(6\)	\(52\)		\(3\)	\(0\)	\(3\)
\(+\)	\(-\)	\(-\)	\(+\)		\(62\)	\(5\)	\(57\)		\(59\)	\(5\)	\(54\)		\(3\)	\(0\)	\(3\)
\(-\)	\(+\)	\(+\)	\(-\)		\(64\)	\(7\)	\(57\)		\(61\)	\(7\)	\(54\)		\(3\)	\(0\)	\(3\)
\(-\)	\(+\)	\(-\)	\(+\)		\(59\)	\(6\)	\(53\)		\(56\)	\(6\)	\(50\)		\(3\)	\(0\)	\(3\)
\(-\)	\(-\)	\(+\)	\(+\)		\(62\)	\(6\)	\(56\)		\(59\)	\(6\)	\(53\)		\(3\)	\(0\)	\(3\)
\(-\)	\(-\)	\(-\)	\(-\)		\(61\)	\(7\)	\(54\)		\(58\)	\(7\)	\(51\)		\(3\)	\(0\)	\(3\)
Plus space			\(+\)		\(242\)	\(23\)	\(219\)		\(230\)	\(23\)	\(207\)		\(12\)	\(0\)	\(12\)
Minus space			\(-\)		\(250\)	\(27\)	\(223\)		\(238\)	\(27\)	\(211\)		\(12\)	\(0\)	\(12\)

Trace form

50 * q - 18 * q^3 - 76 * q^5 + 124 * q^7 + 4050 * q^9 + 244 * q^13 + 900 * q^15 - 404 * q^17 - 5596 * q^19 + 34366 * q^25 - 1458 * q^27 + 8564 * q^29 + 5456 * q^31 - 4776 * q^35 + 3764 * q^37 - 21636 * q^39 + 2476 * q^41 + 9244 * q^43 - 6156 * q^45 - 65256 * q^47 + 105418 * q^49 + 10440 * q^51 + 40244 * q^53 + 24200 * q^55 + 12888 * q^57 + 14480 * q^59 - 66396 * q^61 + 10044 * q^63 - 86904 * q^65 - 217400 * q^67 + 22320 * q^69 - 117256 * q^71 + 77748 * q^73 - 78750 * q^75 + 13100 * q^79 + 328050 * q^81 + 118824 * q^83 + 233368 * q^85 + 230532 * q^89 - 236520 * q^91 - 188184 * q^95 - 4884 * q^97

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(528))\) 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	11
528.6.a.a	$528$	$6$	528.a	1.a	$1$	$1$	$1$	$84.683$	\(\Q\)	None	✓				33.6.a.b	$1$	$0$	\(0\)	\(-9\)	\(-92\)	\(26\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-9q^{3}-92q^{5}+26q^{7}+3^{4}q^{9}-11^{2}q^{11}+\cdots\)
528.6.a.b	$528$	$6$	528.a	1.a	$1$	$1$	$1$	$84.683$	\(\Q\)	None	✓				132.6.a.c	$1$	$1$	\(0\)	\(-9\)	\(-56\)	\(-58\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-9q^{3}-56q^{5}-58q^{7}+3^{4}q^{9}+11^{2}q^{11}+\cdots\)
528.6.a.c	$528$	$6$	528.a	1.a	$1$	$1$	$1$	$84.683$	\(\Q\)	None	✓				66.6.a.c	$1$	$0$	\(0\)	\(-9\)	\(-14\)	\(130\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-9q^{3}-14q^{5}+130q^{7}+3^{4}q^{9}+\cdots\)
528.6.a.d	$528$	$6$	528.a	1.a	$1$	$1$	$1$	$84.683$	\(\Q\)	None	✓				132.6.a.d	$1$	$1$	\(0\)	\(-9\)	\(42\)	\(208\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-9q^{3}+42q^{5}+208q^{7}+3^{4}q^{9}+\cdots\)
528.6.a.e	$528$	$6$	528.a	1.a	$1$	$1$	$1$	$84.683$	\(\Q\)	None	✓				132.6.a.a	$1$	$1$	\(0\)	\(9\)	\(-36\)	\(-122\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+9q^{3}-6^{2}q^{5}-122q^{7}+3^{4}q^{9}+\cdots\)
528.6.a.f	$528$	$6$	528.a	1.a	$1$	$1$	$1$	$84.683$	\(\Q\)	None	✓				66.6.a.b	$1$	$0$	\(0\)	\(9\)	\(-14\)	\(-130\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+9q^{3}-14q^{5}-130q^{7}+3^{4}q^{9}+\cdots\)
528.6.a.g	$528$	$6$	528.a	1.a	$1$	$1$	$1$	$84.683$	\(\Q\)	None	✓				66.6.a.a	$1$	$1$	\(0\)	\(9\)	\(-14\)	\(112\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+9q^{3}-14q^{5}+112q^{7}+3^{4}q^{9}+\cdots\)
528.6.a.h	$528$	$6$	528.a	1.a	$1$	$1$	$1$	$84.683$	\(\Q\)	None	✓				132.6.a.b	$1$	$1$	\(0\)	\(9\)	\(22\)	\(52\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+9q^{3}+22q^{5}+52q^{7}+3^{4}q^{9}-11^{2}q^{11}+\cdots\)
528.6.a.i	$528$	$6$	528.a	1.a	$1$	$1$	$1$	$84.683$	\(\Q\)	None	✓				33.6.a.a	$1$	$1$	\(0\)	\(9\)	\(46\)	\(-148\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+9q^{3}+46q^{5}-148q^{7}+3^{4}q^{9}+\cdots\)
528.6.a.j	$528$	$6$	528.a	1.a	$1$	$1$	$1$	$84.683$	\(\Q\)	None	✓				66.6.a.d	$1$	$0$	\(0\)	\(9\)	\(50\)	\(-2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+9q^{3}+50q^{5}-2q^{7}+3^{4}q^{9}+11^{2}q^{11}+\cdots\)
528.6.a.k	$528$	$6$	528.a	1.a	$1$	$2$	$2$	$84.683$	\(\Q(\sqrt{5641}) \)	None	✓				264.6.a.b	$1$	$0$	\(0\)	\(-18\)	\(-50\)	\(-14\)		$+$	$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q-9q^{3}+(-5^{2}-\beta )q^{5}+(-7+3\beta )q^{7}+\cdots\)
528.6.a.l	$528$	$6$	528.a	1.a	$1$	$2$	$2$	$84.683$	\(\Q(\sqrt{3041}) \)	None	✓				264.6.a.c	$1$	$1$	\(0\)	\(-18\)	\(-30\)	\(128\)		$+$	$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q-9q^{3}+(-15-\beta )q^{5}+(2^{6}-2\beta )q^{7}+\cdots\)
528.6.a.m	$528$	$6$	528.a	1.a	$1$	$2$	$2$	$84.683$	\(\Q(\sqrt{8761}) \)	None	✓				66.6.a.e	$1$	$1$	\(0\)	\(-18\)	\(-14\)	\(-210\)		$-$	$+$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q-9q^{3}+(-7-\beta )q^{5}+(-105-\beta )q^{7}+\cdots\)
528.6.a.n	$528$	$6$	528.a	1.a	$1$	$2$	$2$	$84.683$	\(\Q(\sqrt{2161}) \)	None	✓				66.6.a.f	$1$	$0$	\(0\)	\(-18\)	\(50\)	\(-96\)		$-$	$+$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q-9q^{3}+(5^{2}-\beta )q^{5}+(-48-2\beta )q^{7}+\cdots\)
528.6.a.o	$528$	$6$	528.a	1.a	$1$	$2$	$2$	$84.683$	\(\Q(\sqrt{33}) \)	None	✓				33.6.a.e	$1$	$1$	\(0\)	\(-18\)	\(58\)	\(-146\)		$-$	$+$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q-9q^{3}+(29-5\beta )q^{5}+(-73+31\beta )q^{7}+\cdots\)
528.6.a.p	$528$	$6$	528.a	1.a	$1$	$2$	$2$	$84.683$	\(\Q(\sqrt{3961}) \)	None	✓				264.6.a.d	$1$	$0$	\(0\)	\(-18\)	\(70\)	\(66\)		$+$	$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q-9q^{3}+(35-\beta )q^{5}+(33-\beta )q^{7}+3^{4}q^{9}+\cdots\)
528.6.a.q	$528$	$6$	528.a	1.a	$1$	$2$	$2$	$84.683$	\(\Q(\sqrt{313}) \)	None	✓		✓		33.6.a.d	$1$	$1$	\(0\)	\(18\)	\(-38\)	\(18\)		$-$	$-$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+9q^{3}+(-19-5\beta )q^{5}+(9+\beta )q^{7}+\cdots\)
528.6.a.r	$528$	$6$	528.a	1.a	$1$	$2$	$2$	$84.683$	\(\Q(\sqrt{409}) \)	None	✓				264.6.a.a	$1$	$1$	\(0\)	\(18\)	\(-30\)	\(-48\)		$+$	$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+9q^{3}+(-15-3\beta )q^{5}+(-24-2\beta )q^{7}+\cdots\)
528.6.a.s	$528$	$6$	528.a	1.a	$1$	$2$	$2$	$84.683$	\(\Q(\sqrt{177}) \)	None	✓				33.6.a.c	$1$	$0$	\(0\)	\(18\)	\(58\)	\(286\)		$-$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+9q^{3}+(29-5\beta )q^{5}+(143-5\beta )q^{7}+\cdots\)
528.6.a.t	$528$	$6$	528.a	1.a	$1$	$3$	$3$	$84.683$	3.3.290873.1	None	✓		✓		264.6.a.h	$1$	$0$	\(0\)	\(-27\)	\(-44\)	\(38\)		$+$	$+$	$-$	$-$	$2^{4}$	$\mathrm{SU}(2)$	\(q-9q^{3}+(-15-\beta _{1})q^{5}+(11-4\beta _{1}+\cdots)q^{7}+\cdots\)
528.6.a.u	$528$	$6$	528.a	1.a	$1$	$3$	$3$	$84.683$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓		✓		132.6.a.f	$1$	$0$	\(0\)	\(-27\)	\(36\)	\(28\)		$-$	$+$	$+$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q-9q^{3}+(12+\beta _{1})q^{5}+(9-\beta _{1}+\beta _{2})q^{7}+\cdots\)
528.6.a.v	$528$	$6$	528.a	1.a	$1$	$3$	$3$	$84.683$	3.3.236105.1	None	✓		✓		264.6.a.f	$1$	$1$	\(0\)	\(27\)	\(-44\)	\(20\)		$+$	$-$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+9q^{3}+(-15-\beta _{2})q^{5}+(10+7\beta _{1}+\cdots)q^{7}+\cdots\)
528.6.a.w	$528$	$6$	528.a	1.a	$1$	$3$	$3$	$84.683$	3.3.2109273.1	None	✓				264.6.a.e	$1$	$0$	\(0\)	\(27\)	\(-44\)	\(42\)		$+$	$-$	$+$	$-$	$2^{4}$	$\mathrm{SU}(2)$	\(q+9q^{3}+(-15+\beta _{1})q^{5}+(14+\beta _{1}+\cdots)q^{7}+\cdots\)
528.6.a.x	$528$	$6$	528.a	1.a	$1$	$3$	$3$	$84.683$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓				264.6.a.g	$1$	$0$	\(0\)	\(27\)	\(20\)	\(-70\)		$+$	$-$	$+$	$-$	$2^{4}$	$\mathrm{SU}(2)$	\(q+9q^{3}+(7-\beta _{1})q^{5}+(-23-\beta _{2})q^{7}+\cdots\)
528.6.a.y	$528$	$6$	528.a	1.a	$1$	$3$	$3$	$84.683$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓		✓		132.6.a.e	$1$	$0$	\(0\)	\(27\)	\(36\)	\(52\)		$-$	$-$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+9q^{3}+(12-\beta _{1})q^{5}+(17+2\beta _{1}-\beta _{2})q^{7}+\cdots\)
528.6.a.z	$528$	$6$	528.a	1.a	$1$	$4$	$4$	$84.683$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓		✓		264.6.a.i	$1$	$1$	\(0\)	\(-36\)	\(-44\)	\(-38\)		$+$	$+$	$+$	$+$	$2^{5}$	$\mathrm{SU}(2)$	\(q-9q^{3}+(-11+\beta _{1})q^{5}+(-10-\beta _{3})q^{7}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(528))\) into lower level spaces

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