Space of modular forms of level 528, weight 4, and trivial character

• Label: 528.4.a
• Level: $528$
• Weight: $4$
• Character orbit: 528.a
• Rep. character: $\chi_{528}(1,\cdot)$
• Character field: $\Q$
• Dimension: $30$
• Newform subspaces: $20$
• Sturm bound: $384$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 528 = 2^{4} \cdot 3 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	528.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 20 \)
Sturm bound:			\(384\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(528))\).

	Total	New	Old
Modular forms	300	30	270
Cusp forms	276	30	246
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
\(+\)	\(+\)	\(+\)	\(+\)	\(4\)
\(+\)	\(+\)	\(-\)	\(-\)	\(3\)
\(+\)	\(-\)	\(+\)	\(-\)	\(4\)
\(+\)	\(-\)	\(-\)	\(+\)	\(5\)
\(-\)	\(+\)	\(+\)	\(-\)	\(3\)
\(-\)	\(+\)	\(-\)	\(+\)	\(4\)
\(-\)	\(-\)	\(+\)	\(+\)	\(4\)
\(-\)	\(-\)	\(-\)	\(-\)	\(3\)
Plus space			\(+\)	\(17\)
Minus space			\(-\)	\(13\)

Trace form

30 * q + 6 * q^3 - 4 * q^5 - 36 * q^7 + 270 * q^9 + 92 * q^13 - 60 * q^15 + 52 * q^17 - 204 * q^19 + 706 * q^25 + 54 * q^27 + 284 * q^29 - 48 * q^31 - 456 * q^35 - 132 * q^37 - 420 * q^39 - 236 * q^41 - 84 * q^43 - 36 * q^45 + 408 * q^47 + 1830 * q^49 + 744 * q^51 + 572 * q^53 - 440 * q^55 + 168 * q^57 - 688 * q^59 + 1068 * q^61 - 324 * q^63 + 696 * q^65 + 1896 * q^67 + 528 * q^69 + 1208 * q^71 + 524 * q^73 + 1050 * q^75 + 3180 * q^79 + 2430 * q^81 + 456 * q^83 + 1160 * q^85 - 900 * q^89 + 3128 * q^91 + 7656 * q^95 - 2188 * q^97

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(528))\) 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
528.4.a.a	$528$	$4$	528.a	1.a	$1$	$1$	$1$	$31.153$	\(\Q\)	None	✓				33.4.a.a	$1$	$0$	\(0\)	\(-3\)	\(-14\)	\(32\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}-14q^{5}+2^{5}q^{7}+9q^{9}+11q^{11}+\cdots\)
528.4.a.b	$528$	$4$	528.a	1.a	$1$	$1$	$1$	$31.153$	\(\Q\)	None	✓				264.4.a.d	$1$	$1$	\(0\)	\(-3\)	\(-6\)	\(8\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}-6q^{5}+8q^{7}+9q^{9}+11q^{11}+\cdots\)
528.4.a.c	$528$	$4$	528.a	1.a	$1$	$1$	$1$	$31.153$	\(\Q\)	None	✓				264.4.a.c	$1$	$0$	\(0\)	\(-3\)	\(-6\)	\(14\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}-6q^{5}+14q^{7}+9q^{9}-11q^{11}+\cdots\)
528.4.a.d	$528$	$4$	528.a	1.a	$1$	$1$	$1$	$31.153$	\(\Q\)	None	✓				66.4.a.a	$1$	$1$	\(0\)	\(-3\)	\(0\)	\(-14\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}-14q^{7}+9q^{9}-11q^{11}+80q^{13}+\cdots\)
528.4.a.e	$528$	$4$	528.a	1.a	$1$	$1$	$1$	$31.153$	\(\Q\)	None	✓				132.4.a.d	$1$	$0$	\(0\)	\(-3\)	\(10\)	\(-8\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}+10q^{5}-8q^{7}+9q^{9}+11q^{11}+\cdots\)
528.4.a.f	$528$	$4$	528.a	1.a	$1$	$1$	$1$	$31.153$	\(\Q\)	None	✓				264.4.a.a	$1$	$1$	\(0\)	\(3\)	\(-18\)	\(28\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}-18q^{5}+28q^{7}+9q^{9}-11q^{11}+\cdots\)
528.4.a.g	$528$	$4$	528.a	1.a	$1$	$1$	$1$	$31.153$	\(\Q\)	None	✓				132.4.a.a	$1$	$0$	\(0\)	\(3\)	\(-12\)	\(-14\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}-12q^{5}-14q^{7}+9q^{9}-11q^{11}+\cdots\)
528.4.a.h	$528$	$4$	528.a	1.a	$1$	$1$	$1$	$31.153$	\(\Q\)	None	✓				33.4.a.b	$1$	$0$	\(0\)	\(3\)	\(-4\)	\(26\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}-4q^{5}+26q^{7}+9q^{9}-11q^{11}+\cdots\)
528.4.a.i	$528$	$4$	528.a	1.a	$1$	$1$	$1$	$31.153$	\(\Q\)	None	✓				132.4.a.b	$1$	$1$	\(0\)	\(3\)	\(0\)	\(-2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}-2q^{7}+9q^{9}+11q^{11}-88q^{13}+\cdots\)
528.4.a.j	$528$	$4$	528.a	1.a	$1$	$1$	$1$	$31.153$	\(\Q\)	None	✓				66.4.a.b	$1$	$0$	\(0\)	\(3\)	\(10\)	\(-16\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}+10q^{5}-2^{4}q^{7}+9q^{9}-11q^{11}+\cdots\)
528.4.a.k	$528$	$4$	528.a	1.a	$1$	$1$	$1$	$31.153$	\(\Q\)	None	✓				264.4.a.b	$1$	$1$	\(0\)	\(3\)	\(12\)	\(-22\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}+12q^{5}-22q^{7}+9q^{9}-11q^{11}+\cdots\)
528.4.a.l	$528$	$4$	528.a	1.a	$1$	$1$	$1$	$31.153$	\(\Q\)	None	✓				132.4.a.c	$1$	$0$	\(0\)	\(3\)	\(22\)	\(20\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}+22q^{5}+20q^{7}+9q^{9}-11q^{11}+\cdots\)
528.4.a.m	$528$	$4$	528.a	1.a	$1$	$2$	$2$	$31.153$	\(\Q(\sqrt{185}) \)	None	✓		✓		264.4.a.g	$1$	$1$	\(0\)	\(-6\)	\(-6\)	\(-22\)		$+$	$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q-3q^{3}+(-3-\beta )q^{5}+(-11+\beta )q^{7}+\cdots\)
528.4.a.n	$528$	$4$	528.a	1.a	$1$	$2$	$2$	$31.153$	\(\Q(\sqrt{97}) \)	None	✓		✓		66.4.a.c	$1$	$0$	\(0\)	\(-6\)	\(10\)	\(2\)		$-$	$+$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q-3q^{3}+(5-\beta )q^{5}+(1-3\beta )q^{7}+9q^{9}+\cdots\)
528.4.a.o	$528$	$4$	528.a	1.a	$1$	$2$	$2$	$31.153$	\(\Q(\sqrt{33}) \)	None	✓		✓		33.4.a.d	$1$	$1$	\(0\)	\(-6\)	\(16\)	\(-2\)		$-$	$+$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q-3q^{3}+(8+2\beta )q^{5}+(-1-\beta )q^{7}+\cdots\)
528.4.a.p	$528$	$4$	528.a	1.a	$1$	$2$	$2$	$31.153$	\(\Q(\sqrt{97}) \)	None	✓		✓		33.4.a.c	$1$	$1$	\(0\)	\(6\)	\(-14\)	\(-24\)		$-$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+3q^{3}+(-7-\beta )q^{5}+(-12+2\beta )q^{7}+\cdots\)
528.4.a.q	$528$	$4$	528.a	1.a	$1$	$2$	$2$	$31.153$	\(\Q(\sqrt{137}) \)	None	✓				264.4.a.f	$1$	$0$	\(0\)	\(6\)	\(-6\)	\(-16\)		$+$	$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+3q^{3}+(-3-\beta )q^{5}+(-8-2\beta )q^{7}+\cdots\)
528.4.a.r	$528$	$4$	528.a	1.a	$1$	$2$	$2$	$31.153$	\(\Q(\sqrt{17}) \)	None	✓		✓		264.4.a.e	$1$	$1$	\(0\)	\(6\)	\(-6\)	\(-10\)		$+$	$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+3q^{3}+(-3-\beta )q^{5}+(-5-\beta )q^{7}+\cdots\)
528.4.a.s	$528$	$4$	528.a	1.a	$1$	$3$	$3$	$31.153$	3.3.123209.1	None	✓		✓		264.4.a.i	$1$	$0$	\(0\)	\(-9\)	\(4\)	\(-28\)		$+$	$+$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q-3q^{3}+(1-\beta _{1})q^{5}+(-10-\beta _{1}-\beta _{2})q^{7}+\cdots\)
528.4.a.t	$528$	$4$	528.a	1.a	$1$	$3$	$3$	$31.153$	3.3.142161.1	None	✓		✓		264.4.a.h	$1$	$0$	\(0\)	\(9\)	\(4\)	\(12\)		$+$	$-$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+3q^{3}+(1-\beta _{1})q^{5}+(4-\beta _{2})q^{7}+9q^{9}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(528))\) into lower level spaces

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