Space of modular forms of level 525, weight 6, and Character 525.d

• Label: 525.6.d
• Level: $525$
• Weight: $6$
• Character orbit: 525.d
• Rep. character: $\chi_{525}(274,\cdot)$
• Character field: $\Q$
• Dimension: $88$
• Newform subspaces: $16$
• Sturm bound: $480$
• Trace bound: $6$

Defining parameters

Level:	\( N \)	\(=\)	\( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	525.d (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 5 \)
Character field:			\(\Q\)
Newform subspaces:			\( 16 \)
Sturm bound:			\(480\)
Trace bound:			\(6\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(525, [\chi])\).

	Total	New	Old
Modular forms	412	88	324
Cusp forms	388	88	300
Eisenstein series	24	0	24

Trace form

88 * q - 1384 * q^4 - 360 * q^6 - 7128 * q^9 + 28280 * q^16 + 1208 * q^19 - 1764 * q^21 + 1080 * q^24 + 6856 * q^26 + 14120 * q^29 - 4640 * q^31 - 53136 * q^34 + 112104 * q^36 + 46656 * q^39 + 59232 * q^41 - 19724 * q^44 - 32884 * q^46 - 211288 * q^49 - 20736 * q^51 + 29160 * q^54 + 32340 * q^56 - 60536 * q^59 + 227152 * q^61 - 552804 * q^64 - 221616 * q^66 + 115128 * q^69 + 214192 * q^71 - 471436 * q^74 + 266888 * q^76 + 262304 * q^79 + 577368 * q^81 + 84672 * q^84 - 92180 * q^86 - 348776 * q^89 - 132496 * q^91 + 232352 * q^94 - 249120 * q^96

Decomposition of \(S_{6}^{\mathrm{new}}(525, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
525.6.d.a	$525$	$6$	525.d	5.b	$2$	$2$	$2$	$84.202$	\(\Q(\sqrt{-1}) \)	None					21.6.a.d	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+10 i q^{2}-9 i q^{3}-68 q^{4}+90 q^{6}+\cdots\)
525.6.d.b	$525$	$6$	525.d	5.b	$2$	$2$	$2$	$84.202$	\(\Q(\sqrt{-1}) \)	None					21.6.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+6 i q^{2}-9 i q^{3}-4 q^{4}+54 q^{6}+\cdots\)
525.6.d.c	$525$	$6$	525.d	5.b	$2$	$2$	$2$	$84.202$	\(\Q(\sqrt{-1}) \)	None					21.6.a.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+5 i q^{2}-9 i q^{3}+7 q^{4}+45 q^{6}+\cdots\)
525.6.d.d	$525$	$6$	525.d	5.b	$2$	$2$	$2$	$84.202$	\(\Q(\sqrt{-1}) \)	None					21.6.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+i q^{2}+9 i q^{3}+31 q^{4}-9 q^{6}+\cdots\)
525.6.d.e	$525$	$6$	525.d	5.b	$2$	$4$	$4$	$84.202$	\(\Q(i, \sqrt{65})\)	None					105.6.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+(\beta _{1}+7\beta _{2})q^{2}+9\beta _{2}q^{3}+(-20-13\beta _{3})q^{4}+\cdots\)
525.6.d.f	$525$	$6$	525.d	5.b	$2$	$4$	$4$	$84.202$	\(\Q(i, \sqrt{233})\)	None					105.6.a.e	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{2}-9\beta _{2}q^{3}+(-3^{3}+\beta _{3})q^{4}+\cdots\)
525.6.d.g	$525$	$6$	525.d	5.b	$2$	$4$	$4$	$84.202$	\(\Q(i, \sqrt{5})\)	None					105.6.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-4\beta _{1}+\beta _{2})q^{2}-9\beta _{1}q^{3}+(-4+\cdots)q^{4}+\cdots\)
525.6.d.h	$525$	$6$	525.d	5.b	$2$	$4$	$4$	$84.202$	\(\Q(\zeta_{8})\)	None					105.6.a.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{5}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(\beta_{2}+2\beta_1)q^{2}-9\beta_1 q^{3}+(-4\beta_{3}-4)q^{4}+\cdots\)
525.6.d.i	$525$	$6$	525.d	5.b	$2$	$4$	$4$	$84.202$	\(\Q(i, \sqrt{65})\)	None					105.6.a.f	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+(\beta _{1}-\beta _{2})q^{2}-9\beta _{2}q^{3}+(12+3\beta _{3})q^{4}+\cdots\)
525.6.d.j	$525$	$6$	525.d	5.b	$2$	$4$	$4$	$84.202$	\(\Q(i, \sqrt{73})\)	None					105.6.a.d	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{2}-9\beta _{2}q^{3}+(13+\beta _{3})q^{4}+(-9+\cdots)q^{6}+\cdots\)
525.6.d.k	$525$	$6$	525.d	5.b	$2$	$8$	$8$	$84.202$	\(\mathbb{Q}[x]/(x^{8} + \cdots)\)	None					105.6.a.g	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{6}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(\beta _{1}-2\beta _{3})q^{2}-9\beta _{3}q^{3}+(-29+\beta _{2}+\cdots)q^{4}+\cdots\)
525.6.d.l	$525$	$6$	525.d	5.b	$2$	$8$	$8$	$84.202$	\(\mathbb{Q}[x]/(x^{8} + \cdots)\)	None					105.6.a.h	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{8}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(\beta _{1}-2\beta _{2})q^{2}+9\beta _{2}q^{3}+(-30+3\beta _{3}+\cdots)q^{4}+\cdots\)
525.6.d.m	$525$	$6$	525.d	5.b	$2$	$8$	$8$	$84.202$	\(\mathbb{Q}[x]/(x^{8} + \cdots)\)	None		✓			525.6.a.k	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(\beta _{1}-4\beta _{2})q^{2}-9\beta _{2}q^{3}+(-14+\beta _{3}+\cdots)q^{4}+\cdots\)
525.6.d.n	$525$	$6$	525.d	5.b	$2$	$8$	$8$	$84.202$	\(\mathbb{Q}[x]/(x^{8} + \cdots)\)	None		✓			525.6.a.n	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}\cdot 5^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(\beta _{1}-\beta _{4})q^{2}-9\beta _{4}q^{3}+(-12-\beta _{3}+\cdots)q^{4}+\cdots\)
525.6.d.o	$525$	$6$	525.d	5.b	$2$	$12$	$12$	$84.202$	\(\mathbb{Q}[x]/(x^{12} + \cdots)\)	None		✓			525.6.a.q	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}\cdot 3^{2}\cdot 5^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(\beta _{1}+\beta _{5})q^{2}+9\beta _{5}q^{3}+(-22+\beta _{3}+\cdots)q^{4}+\cdots\)
525.6.d.p	$525$	$6$	525.d	5.b	$2$	$12$	$12$	$84.202$	\(\mathbb{Q}[x]/(x^{12} + \cdots)\)	None		✓			525.6.a.r	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}\cdot 5^{6}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(\beta _{1}+\beta _{6})q^{2}-9\beta _{6}q^{3}+(-22-\beta _{2}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(525, [\chi])\) into lower level spaces

\( S_{6}^{\mathrm{old}}(525, [\chi]) \simeq \) \(S_{6}^{\mathrm{new}}(5, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(15, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(25, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(35, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(75, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(105, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(175, [\chi])\)\(^{\oplus 2}\)