Space of modular forms of level 525, weight 4, and trivial character

• Label: 525.4.a
• Level: $525$
• Weight: $4$
• Character orbit: 525.a
• Rep. character: $\chi_{525}(1,\cdot)$
• Character field: $\Q$
• Dimension: $56$
• Newform subspaces: $24$
• Sturm bound: $320$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	525.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 24 \)
Sturm bound:			\(320\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(2\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(525))\).

	Total	New	Old
Modular forms	252	56	196
Cusp forms	228	56	172
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.

\(3\)	\(5\)	\(7\)	Fricke		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(34\)	\(6\)	\(28\)		\(31\)	\(6\)	\(25\)		\(3\)	\(0\)	\(3\)
\(+\)	\(+\)	\(-\)	\(-\)		\(31\)	\(8\)	\(23\)		\(28\)	\(8\)	\(20\)		\(3\)	\(0\)	\(3\)
\(+\)	\(-\)	\(+\)	\(-\)		\(30\)	\(7\)	\(23\)		\(27\)	\(7\)	\(20\)		\(3\)	\(0\)	\(3\)
\(+\)	\(-\)	\(-\)	\(+\)		\(31\)	\(7\)	\(24\)		\(28\)	\(7\)	\(21\)		\(3\)	\(0\)	\(3\)
\(-\)	\(+\)	\(+\)	\(-\)		\(29\)	\(5\)	\(24\)		\(26\)	\(5\)	\(21\)		\(3\)	\(0\)	\(3\)
\(-\)	\(+\)	\(-\)	\(+\)		\(32\)	\(9\)	\(23\)		\(29\)	\(9\)	\(20\)		\(3\)	\(0\)	\(3\)
\(-\)	\(-\)	\(+\)	\(+\)		\(33\)	\(9\)	\(24\)		\(30\)	\(9\)	\(21\)		\(3\)	\(0\)	\(3\)
\(-\)	\(-\)	\(-\)	\(-\)		\(32\)	\(5\)	\(27\)		\(29\)	\(5\)	\(24\)		\(3\)	\(0\)	\(3\)
Plus space			\(+\)		\(130\)	\(31\)	\(99\)		\(118\)	\(31\)	\(87\)		\(12\)	\(0\)	\(12\)
Minus space			\(-\)		\(122\)	\(25\)	\(97\)		\(110\)	\(25\)	\(85\)		\(12\)	\(0\)	\(12\)

Trace form

56 * q + 6 * q^2 + 194 * q^4 + 12 * q^6 + 14 * q^7 + 30 * q^8 + 504 * q^9 + 20 * q^11 - 24 * q^12 + 112 * q^13 - 70 * q^14 + 858 * q^16 - 56 * q^17 + 54 * q^18 + 40 * q^19 - 42 * q^21 - 152 * q^22 - 212 * q^23 + 324 * q^24 + 868 * q^26 + 70 * q^28 + 920 * q^29 - 72 * q^31 + 1086 * q^32 + 288 * q^33 + 644 * q^34 + 1746 * q^36 + 152 * q^37 - 712 * q^38 - 336 * q^39 + 176 * q^41 + 84 * q^42 + 240 * q^43 + 716 * q^44 + 2228 * q^46 - 824 * q^47 + 432 * q^48 + 2744 * q^49 + 276 * q^51 + 1636 * q^52 - 480 * q^53 + 108 * q^54 - 378 * q^56 + 192 * q^57 - 268 * q^58 - 952 * q^59 - 3832 * q^61 + 2520 * q^62 + 126 * q^63 + 3790 * q^64 + 1896 * q^66 - 168 * q^67 + 2596 * q^68 + 336 * q^69 - 1676 * q^71 + 270 * q^72 - 2520 * q^73 - 9280 * q^74 + 3672 * q^76 - 1176 * q^77 + 1464 * q^78 + 336 * q^79 + 4536 * q^81 + 1604 * q^82 - 4208 * q^83 - 504 * q^84 - 9340 * q^86 + 792 * q^87 - 4952 * q^88 - 5120 * q^89 - 1148 * q^91 - 1568 * q^92 - 816 * q^93 + 9904 * q^94 + 7764 * q^96 + 4520 * q^97 + 294 * q^98 + 180 * q^99

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(525))\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		3	5	7
525.4.a.a	$525$	$4$	525.a	1.a	$1$	$1$	$1$	$30.976$	\(\Q\)	None	✓				105.4.a.b	$1$	$1$	\(-5\)	\(3\)	\(0\)	\(-7\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-5q^{2}+3q^{3}+17q^{4}-15q^{6}-7q^{7}+\cdots\)
525.4.a.b	$525$	$4$	525.a	1.a	$1$	$1$	$1$	$30.976$	\(\Q\)	None	✓				21.4.a.b	$1$	$0$	\(-4\)	\(3\)	\(0\)	\(7\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-4q^{2}+3q^{3}+8q^{4}-12q^{6}+7q^{7}+\cdots\)
525.4.a.c	$525$	$4$	525.a	1.a	$1$	$1$	$1$	$30.976$	\(\Q\)	None	✓	✓			525.4.a.c	$1$	$0$	\(-3\)	\(-3\)	\(0\)	\(7\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{2}-3q^{3}+q^{4}+9q^{6}+7q^{7}+\cdots\)
525.4.a.d	$525$	$4$	525.a	1.a	$1$	$1$	$1$	$30.976$	\(\Q\)	None	✓	✓			525.4.a.d	$1$	$1$	\(-2\)	\(3\)	\(0\)	\(-7\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+3q^{3}-4q^{4}-6q^{6}-7q^{7}+\cdots\)
525.4.a.e	$525$	$4$	525.a	1.a	$1$	$1$	$1$	$30.976$	\(\Q\)	None	✓				105.4.a.a	$1$	$1$	\(0\)	\(3\)	\(0\)	\(-7\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}-8q^{4}-7q^{7}+9q^{9}+42q^{11}+\cdots\)
525.4.a.f	$525$	$4$	525.a	1.a	$1$	$1$	$1$	$30.976$	\(\Q\)	None	✓	✓			525.4.a.d	$1$	$0$	\(2\)	\(-3\)	\(0\)	\(7\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}-3q^{3}-4q^{4}-6q^{6}+7q^{7}+\cdots\)
525.4.a.g	$525$	$4$	525.a	1.a	$1$	$1$	$1$	$30.976$	\(\Q\)	None	✓				21.4.a.a	$1$	$1$	\(3\)	\(3\)	\(0\)	\(-7\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{2}+3q^{3}+q^{4}+9q^{6}-7q^{7}+\cdots\)
525.4.a.h	$525$	$4$	525.a	1.a	$1$	$1$	$1$	$30.976$	\(\Q\)	None	✓	✓			525.4.a.c	$1$	$1$	\(3\)	\(3\)	\(0\)	\(-7\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{2}+3q^{3}+q^{4}+9q^{6}-7q^{7}+\cdots\)
525.4.a.i	$525$	$4$	525.a	1.a	$1$	$2$	$2$	$30.976$	\(\Q(\sqrt{41}) \)	None	✓				105.4.a.g	$1$	$0$	\(-3\)	\(-6\)	\(0\)	\(-14\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta )q^{2}-3q^{3}+(3+3\beta )q^{4}+\cdots\)
525.4.a.j	$525$	$4$	525.a	1.a	$1$	$2$	$2$	$30.976$	\(\Q(\sqrt{17}) \)	None	✓	✓			525.4.a.j	$1$	$1$	\(-3\)	\(6\)	\(0\)	\(14\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta )q^{2}+3q^{3}+(-3+3\beta )q^{4}+\cdots\)
525.4.a.k	$525$	$4$	525.a	1.a	$1$	$2$	$2$	$30.976$	\(\Q(\sqrt{65}) \)	None	✓				105.4.a.f	$1$	$1$	\(-1\)	\(-6\)	\(0\)	\(14\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{2}-3q^{3}+(8+\beta )q^{4}+3\beta q^{6}+\cdots\)
525.4.a.l	$525$	$4$	525.a	1.a	$1$	$2$	$2$	$30.976$	\(\Q(\sqrt{2}) \)	None	✓				105.4.a.e	$1$	$0$	\(2\)	\(6\)	\(0\)	\(14\)		$-$	$+$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{2}+3q^{3}+(1+2\beta )q^{4}+(3+\cdots)q^{6}+\cdots\)
525.4.a.m	$525$	$4$	525.a	1.a	$1$	$2$	$2$	$30.976$	\(\Q(\sqrt{17}) \)	None	✓	✓			525.4.a.j	$1$	$0$	\(3\)	\(-6\)	\(0\)	\(-14\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{2}-3q^{3}+(-3+3\beta )q^{4}+\cdots\)
525.4.a.n	$525$	$4$	525.a	1.a	$1$	$2$	$2$	$30.976$	\(\Q(\sqrt{57}) \)	None	✓				21.4.a.c	$1$	$0$	\(3\)	\(-6\)	\(0\)	\(-14\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{2}-3q^{3}+(7+3\beta )q^{4}+(-3+\cdots)q^{6}+\cdots\)
525.4.a.o	$525$	$4$	525.a	1.a	$1$	$2$	$2$	$30.976$	\(\Q(\sqrt{5}) \)	None	✓				105.4.a.d	$1$	$1$	\(4\)	\(-6\)	\(0\)	\(14\)		$+$	$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(2+\beta )q^{2}-3q^{3}+(1+4\beta )q^{4}+(-6+\cdots)q^{6}+\cdots\)
525.4.a.p	$525$	$4$	525.a	1.a	$1$	$2$	$2$	$30.976$	\(\Q(\sqrt{17}) \)	None	✓				105.4.a.c	$1$	$0$	\(7\)	\(6\)	\(0\)	\(14\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(4-\beta )q^{2}+3q^{3}+(12-7\beta )q^{4}+(12+\cdots)q^{6}+\cdots\)
525.4.a.q	$525$	$4$	525.a	1.a	$1$	$3$	$3$	$30.976$	3.3.2292.1	None	✓		✓		105.4.d.a	$1$	$1$	\(-1\)	\(9\)	\(0\)	\(21\)		$-$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+3q^{3}+(1+\beta _{1}+\beta _{2})q^{4}+\cdots\)
525.4.a.r	$525$	$4$	525.a	1.a	$1$	$3$	$3$	$30.976$	3.3.2292.1	None	✓				105.4.d.a	$1$	$1$	\(1\)	\(-9\)	\(0\)	\(-21\)		$+$	$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}-3q^{3}+(1+\beta _{1}+\beta _{2})q^{4}+\cdots\)
525.4.a.s	$525$	$4$	525.a	1.a	$1$	$4$	$4$	$30.976$	\(\Q(\sqrt{39 +2 \sqrt{185}})\)	None	✓	✓	✓		525.4.a.s	$1$	$1$	\(-6\)	\(-12\)	\(0\)	\(28\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{1})q^{2}-3q^{3}+(3+3\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
525.4.a.t	$525$	$4$	525.a	1.a	$1$	$4$	$4$	$30.976$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	✓	✓		525.4.a.t	$1$	$1$	\(0\)	\(-12\)	\(0\)	\(-28\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}-3q^{3}+(4+\beta _{2})q^{4}+3\beta _{1}q^{6}+\cdots\)
525.4.a.u	$525$	$4$	525.a	1.a	$1$	$4$	$4$	$30.976$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	✓	✓		525.4.a.t	$1$	$0$	\(0\)	\(12\)	\(0\)	\(28\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+3q^{3}+(4+\beta _{2})q^{4}+3\beta _{1}q^{6}+\cdots\)
525.4.a.v	$525$	$4$	525.a	1.a	$1$	$4$	$4$	$30.976$	\(\Q(\sqrt{39 +2 \sqrt{185}})\)	None	✓	✓			525.4.a.s	$1$	$0$	\(6\)	\(12\)	\(0\)	\(-28\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta _{1})q^{2}+3q^{3}+(3+3\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
525.4.a.w	$525$	$4$	525.a	1.a	$1$	$5$	$5$	$30.976$	5.5.78066700.1	None	✓		✓		105.4.d.b	$1$	$0$	\(-1\)	\(15\)	\(0\)	\(-35\)		$-$	$-$	$+$	$+$	$2^{5}\cdot 5$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+3q^{3}+(5-\beta _{1}-\beta _{3})q^{4}+\cdots\)
525.4.a.x	$525$	$4$	525.a	1.a	$1$	$5$	$5$	$30.976$	5.5.78066700.1	None	✓		✓		105.4.d.b	$1$	$0$	\(1\)	\(-15\)	\(0\)	\(35\)		$+$	$-$	$-$	$+$	$2^{5}\cdot 5$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}-3q^{3}+(5-\beta _{1}-\beta _{3})q^{4}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(525))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(525)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(25))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(35))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(75))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(105))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(175))\)\(^{\oplus 2}\)