Space of modular forms of level 529, weight 4, and trivial character

• Label: 529.4.a
• Level: $529$
• Weight: $4$
• Character orbit: 529.a
• Rep. character: $\chi_{529}(1,\cdot)$
• Character field: $\Q$
• Dimension: $116$
• Newform subspaces: $14$
• Sturm bound: $184$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 529 = 23^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	529.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 14 \)
Sturm bound:			\(184\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(2\), \(3\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(529))\).

	Total	New	Old
Modular forms	150	137	13
Cusp forms	126	116	10
Eisenstein series	24	21	3

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

\(23\)	Dim
\(+\)	\(61\)
\(-\)	\(55\)

Trace form

116 * q - 2 * q^3 + 424 * q^4 - 8 * q^5 - 8 * q^6 - 8 * q^7 + 42 * q^8 + 926 * q^9 + 58 * q^10 - 42 * q^11 + 86 * q^12 - 54 * q^13 + 128 * q^14 - 40 * q^15 + 1360 * q^16 - 18 * q^17 + 40 * q^18 - 26 * q^19 - 164 * q^20 - 220 * q^21 - 152 * q^22 + 166 * q^24 + 1830 * q^25 + 280 * q^26 - 56 * q^27 - 314 * q^28 - 322 * q^29 + 466 * q^30 + 46 * q^31 + 616 * q^32 + 588 * q^33 - 826 * q^34 + 684 * q^35 + 3252 * q^36 + 128 * q^37 - 888 * q^38 + 156 * q^39 + 170 * q^40 + 46 * q^41 - 560 * q^42 - 90 * q^43 - 694 * q^44 - 180 * q^45 + 840 * q^47 + 44 * q^48 + 1818 * q^49 - 904 * q^50 - 60 * q^51 - 2934 * q^52 + 644 * q^53 - 756 * q^54 + 1100 * q^55 + 2366 * q^56 - 1672 * q^57 + 2028 * q^58 + 2182 * q^59 + 924 * q^60 - 1576 * q^61 + 260 * q^62 + 1076 * q^63 + 3038 * q^64 - 1660 * q^65 + 58 * q^66 - 1414 * q^67 - 604 * q^68 + 3366 * q^70 + 378 * q^71 - 630 * q^72 - 1066 * q^73 - 1962 * q^74 - 2438 * q^75 + 3552 * q^76 - 1008 * q^77 + 2180 * q^78 + 2160 * q^79 - 2942 * q^80 + 3828 * q^81 + 1738 * q^82 - 106 * q^83 - 2454 * q^84 - 480 * q^85 - 742 * q^86 - 288 * q^87 - 412 * q^88 - 1882 * q^89 - 1760 * q^90 - 748 * q^91 - 66 * q^93 - 1558 * q^94 + 828 * q^95 - 1940 * q^96 - 98 * q^97 - 4394 * q^98 + 1566 * q^99

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(529))\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		23
529.4.a.a	$529$	$4$	529.a	1.a	$1$	$1$	$1$	$31.212$	\(\Q\)	None	✓				23.4.a.a	$1$	$1$	\(-2\)	\(-5\)	\(6\)	\(8\)		$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}-5q^{3}-4q^{4}+6q^{5}+10q^{6}+\cdots\)
529.4.a.b	$529$	$4$	529.a	1.a	$1$	$1$	$1$	$31.212$	\(\Q\)	\(\Q(\sqrt{-23}) \)	✓	✓			529.4.a.b	$2$	$1$	\(3\)	\(4\)	\(0\)	\(0\)		$-$	$-$	$1$	$N(\mathrm{U}(1))$	\(q+3q^{2}+4q^{3}+q^{4}+12q^{6}-21q^{8}+\cdots\)
529.4.a.c	$529$	$4$	529.a	1.a	$1$	$2$	$2$	$31.212$	\(\Q(\sqrt{85}) \)	None	✓	✓			529.4.a.c	$2$	$1$	\(-4\)	\(-2\)	\(0\)	\(0\)		$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q-2q^{2}-q^{3}-4q^{4}+\beta q^{5}+2q^{6}+\cdots\)
529.4.a.d	$529$	$4$	529.a	1.a	$1$	$2$	$2$	$31.212$	\(\Q(\sqrt{3}) \)	None	✓	✓			529.4.a.d	$1$	$1$	\(-4\)	\(-2\)	\(-16\)	\(-22\)		$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-2+\beta )q^{2}+(-1+3\beta )q^{3}+(-1+\cdots)q^{4}+\cdots\)
529.4.a.e	$529$	$4$	529.a	1.a	$1$	$2$	$2$	$31.212$	\(\Q(\sqrt{3}) \)	None	✓	✓			529.4.a.d	$1$	$1$	\(-4\)	\(-2\)	\(16\)	\(22\)		$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-2+\beta )q^{2}+(-1+3\beta )q^{3}+(-1+\cdots)q^{4}+\cdots\)
529.4.a.f	$529$	$4$	529.a	1.a	$1$	$2$	$2$	$31.212$	\(\Q(\sqrt{69}) \)	\(\Q(\sqrt{-23}) \)	✓	✓			529.4.a.f	$2$	$1$	\(-3\)	\(-4\)	\(0\)	\(0\)		$-$	$-$	$1$	$N(\mathrm{U}(1))$	\(q+(-1-\beta )q^{2}+(-3+2\beta )q^{3}+(10+\cdots)q^{4}+\cdots\)
529.4.a.g	$529$	$4$	529.a	1.a	$1$	$4$	$4$	$31.212$	4.4.334189.1	None	✓				23.4.a.b	$1$	$1$	\(2\)	\(7\)	\(-14\)	\(-16\)		$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta _{3})q^{2}+(1+\beta _{1}+\beta _{2})q^{3}+(6+\cdots)q^{4}+\cdots\)
529.4.a.h	$529$	$4$	529.a	1.a	$1$	$5$	$5$	$31.212$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	✓			529.4.a.h	$1$	$1$	\(0\)	\(-7\)	\(-4\)	\(14\)		$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(-1-\beta _{1}-\beta _{2})q^{3}+(3+\beta _{1}+\cdots)q^{4}+\cdots\)
529.4.a.i	$529$	$4$	529.a	1.a	$1$	$5$	$5$	$31.212$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	✓			529.4.a.h	$1$	$1$	\(0\)	\(-7\)	\(4\)	\(-14\)		$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(-1-\beta _{1}-\beta _{2})q^{3}+(3+\beta _{1}+\cdots)q^{4}+\cdots\)
529.4.a.j	$529$	$4$	529.a	1.a	$1$	$6$	$6$	$31.212$	6.6.\(\cdots\).1	None	✓	✓			529.4.a.j	$2$	$1$	\(0\)	\(-6\)	\(0\)	\(0\)		$-$	$-$	$3$	$\mathrm{SU}(2)$	\(q+\beta _{2}q^{2}+(-1-\beta _{2}+\beta _{3})q^{3}+(6-\beta _{2}+\cdots)q^{4}+\cdots\)
529.4.a.k	$529$	$4$	529.a	1.a	$1$	$12$	$12$	$31.212$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None	✓	✓			529.4.a.k	$2$	$0$	\(4\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{2}q^{2}-\beta _{7}q^{3}+(4+\beta _{2}-\beta _{9})q^{4}+\cdots\)
529.4.a.l	$529$	$4$	529.a	1.a	$1$	$24$	$24$	$31.212$		None	✓	✓			529.4.a.l	$2$	$0$	\(8\)	\(24\)	\(0\)	\(0\)		$+$	$+$		$\mathrm{SU}(2)$
529.4.a.m	$529$	$4$	529.a	1.a	$1$	$25$	$25$	$31.212$		None	✓		✓		23.4.c.a	$1$	$1$	\(0\)	\(-1\)	\(-51\)	\(-73\)		$-$	$-$		$\mathrm{SU}(2)$
529.4.a.n	$529$	$4$	529.a	1.a	$1$	$25$	$25$	$31.212$		None	✓		✓		23.4.c.a	$1$	$0$	\(0\)	\(-1\)	\(51\)	\(73\)		$+$	$+$		$\mathrm{SU}(2)$

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(529))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(529)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(23))\)\(^{\oplus 2}\)