Space of modular forms of level 525, weight 4, and Character 525.d

• Label: 525.4.d
• Level: $525$
• Weight: $4$
• Character orbit: 525.d
• Rep. character: $\chi_{525}(274,\cdot)$
• Character field: $\Q$
• Dimension: $56$
• Newform subspaces: $15$
• Sturm bound: $320$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	525.d (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 5 \)
Character field:			\(\Q\)
Newform subspaces:			\( 15 \)
Sturm bound:			\(320\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(2\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(525, [\chi])\).

	Total	New	Old
Modular forms	252	56	196
Cusp forms	228	56	172
Eisenstein series	24	0	24

Trace form

56 * q - 216 * q^4 + 24 * q^6 - 504 * q^9 + 1016 * q^16 + 96 * q^19 + 84 * q^21 - 648 * q^24 + 184 * q^26 + 560 * q^29 + 96 * q^31 + 160 * q^34 + 1944 * q^36 - 576 * q^39 - 2112 * q^41 - 1580 * q^44 + 2044 * q^46 - 2744 * q^49 + 2304 * q^51 - 216 * q^54 + 84 * q^56 + 112 * q^59 - 4208 * q^61 - 5988 * q^64 - 1296 * q^66 + 2304 * q^69 - 1616 * q^71 + 8708 * q^74 - 8952 * q^76 - 6904 * q^79 + 4536 * q^81 - 1008 * q^84 - 1460 * q^86 + 4288 * q^89 + 1456 * q^91 + 1216 * q^94 + 7968 * q^96

Decomposition of \(S_{4}^{\mathrm{new}}(525, [\chi])\) 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				Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
525.4.d.a	$525$	$4$	525.d	5.b	$2$	$2$	$2$	$30.976$	\(\Q(\sqrt{-1}) \)	None					105.4.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+5 i q^{2}+3 i q^{3}-17 q^{4}-15 q^{6}+\cdots\)
525.4.d.b	$525$	$4$	525.d	5.b	$2$	$2$	$2$	$30.976$	\(\Q(\sqrt{-1}) \)	None					21.4.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+4 i q^{2}+3 i q^{3}-8 q^{4}-12 q^{6}+\cdots\)
525.4.d.c	$525$	$4$	525.d	5.b	$2$	$2$	$2$	$30.976$	\(\Q(\sqrt{-1}) \)	None					21.4.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+3 i q^{2}-3 i q^{3}-q^{4}+9 q^{6}-7 i q^{7}+\cdots\)
525.4.d.d	$525$	$4$	525.d	5.b	$2$	$2$	$2$	$30.976$	\(\Q(\sqrt{-1}) \)	None		✓			525.4.a.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+3 i q^{2}-3 i q^{3}-q^{4}+9 q^{6}-7 i q^{7}+\cdots\)
525.4.d.e	$525$	$4$	525.d	5.b	$2$	$2$	$2$	$30.976$	\(\Q(\sqrt{-1}) \)	None		✓			525.4.a.d	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+2 i q^{2}+3 i q^{3}+4 q^{4}-6 q^{6}+\cdots\)
525.4.d.f	$525$	$4$	525.d	5.b	$2$	$2$	$2$	$30.976$	\(\Q(\sqrt{-1}) \)	None					105.4.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-3 i q^{3}+8 q^{4}-7 i q^{7}-9 q^{9}+\cdots\)
525.4.d.g	$525$	$4$	525.d	5.b	$2$	$4$	$4$	$30.976$	\(\Q(i, \sqrt{57})\)	None					21.4.a.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+(\beta _{1}-\beta _{2})q^{2}-3\beta _{2}q^{3}+(-10+3\beta _{3})q^{4}+\cdots\)
525.4.d.h	$525$	$4$	525.d	5.b	$2$	$4$	$4$	$30.976$	\(\Q(i, \sqrt{65})\)	None					105.4.a.f	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{2}+3\beta _{2}q^{3}+(-9+\beta _{3})q^{4}+\cdots\)
525.4.d.i	$525$	$4$	525.d	5.b	$2$	$4$	$4$	$30.976$	\(\Q(i, \sqrt{17})\)	None					105.4.a.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+(\beta _{1}+4\beta _{2})q^{2}-3\beta _{2}q^{3}+(-5-7\beta _{3})q^{4}+\cdots\)
525.4.d.j	$525$	$4$	525.d	5.b	$2$	$4$	$4$	$30.976$	\(\Q(i, \sqrt{41})\)	None					105.4.a.g	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+(\beta _{1}-\beta _{2})q^{2}+3\beta _{2}q^{3}+(-6+3\beta _{3})q^{4}+\cdots\)
525.4.d.k	$525$	$4$	525.d	5.b	$2$	$4$	$4$	$30.976$	\(\Q(i, \sqrt{5})\)	None					105.4.a.d	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-2\beta _{1}+\beta _{2})q^{2}-3\beta _{1}q^{3}+(-1+\cdots)q^{4}+\cdots\)
525.4.d.l	$525$	$4$	525.d	5.b	$2$	$4$	$4$	$30.976$	\(\Q(\zeta_{8})\)	None					105.4.a.e	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{3}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(\beta_{2}+\beta_1)q^{2}-3\beta_1 q^{3}+(-2\beta_{3}-1)q^{4}+\cdots\)
525.4.d.m	$525$	$4$	525.d	5.b	$2$	$4$	$4$	$30.976$	\(\Q(i, \sqrt{17})\)	None		✓			525.4.a.j	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+(\beta _{1}-\beta _{2})q^{2}-3\beta _{2}q^{3}+3\beta _{3}q^{4}+\cdots\)
525.4.d.n	$525$	$4$	525.d	5.b	$2$	$8$	$8$	$30.976$	\(\mathbb{Q}[x]/(x^{8} + \cdots)\)	None		✓			525.4.a.t	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{2}-3\beta _{3}q^{3}+(-4+\beta _{2})q^{4}+\cdots\)
525.4.d.o	$525$	$4$	525.d	5.b	$2$	$8$	$8$	$30.976$	\(\mathbb{Q}[x]/(x^{8} + \cdots)\)	None		✓			525.4.a.s	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+(\beta _{1}+\beta _{4})q^{2}-3\beta _{4}q^{3}+(-5+\beta _{2}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(525, [\chi])\) into lower level spaces

\( S_{4}^{\mathrm{old}}(525, [\chi]) \simeq \) \(S_{4}^{\mathrm{new}}(15, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(25, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(35, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(75, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(105, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(175, [\chi])\)\(^{\oplus 2}\)