Space of modular forms of level 825, weight 4, and Character 825.c

• Label: 825.4.c
• Level: $825$
• Weight: $4$
• Character orbit: 825.c
• Rep. character: $\chi_{825}(199,\cdot)$
• Character field: $\Q$
• Dimension: $92$
• Newform subspaces: $19$
• Sturm bound: $480$
• Trace bound: $14$

Defining parameters

Level:	\( N \)	\(=\)	\( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	825.c (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 5 \)
Character field:			\(\Q\)
Newform subspaces:			\( 19 \)
Sturm bound:			\(480\)
Trace bound:			\(14\)
Distinguishing \(T_p\):			\(2\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(825, [\chi])\).

	Total	New	Old
Modular forms	372	92	280
Cusp forms	348	92	256
Eisenstein series	24	0	24

Trace form

92 * q - 360 * q^4 + 24 * q^6 - 828 * q^9 + 472 * q^14 + 1624 * q^16 - 24 * q^19 + 48 * q^21 - 648 * q^24 + 512 * q^26 + 976 * q^29 + 848 * q^31 - 1712 * q^34 + 3240 * q^36 + 168 * q^39 - 1984 * q^41 - 1232 * q^44 + 1848 * q^46 - 5516 * q^49 + 1488 * q^51 - 216 * q^54 + 264 * q^56 + 2624 * q^59 - 5216 * q^61 - 4888 * q^64 + 528 * q^66 - 48 * q^69 - 1152 * q^71 + 9904 * q^74 + 152 * q^76 - 1024 * q^79 + 7452 * q^81 + 3744 * q^84 + 2616 * q^86 + 4768 * q^89 + 3064 * q^91 - 4088 * q^94 + 2808 * q^96

Decomposition of \(S_{4}^{\mathrm{new}}(825, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
825.4.c.a	$825$	$4$	825.c	5.b	$2$	$2$	$2$	$48.677$	\(\Q(\sqrt{-1}) \)	None					33.4.a.a	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+5 i q^{2}+3 i q^{3}-17 q^{4}-15 q^{6}+\cdots\)
825.4.c.b	$825$	$4$	825.c	5.b	$2$	$2$	$2$	$48.677$	\(\Q(\sqrt{-1}) \)	None		✓			825.4.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+5 i q^{2}-3 i q^{3}-17 q^{4}+15 q^{6}+\cdots\)
825.4.c.c	$825$	$4$	825.c	5.b	$2$	$2$	$2$	$48.677$	\(\Q(\sqrt{-1}) \)	None		✓			825.4.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+4 i q^{2}+3 i q^{3}-8 q^{4}-12 q^{6}+\cdots\)
825.4.c.d	$825$	$4$	825.c	5.b	$2$	$2$	$2$	$48.677$	\(\Q(\sqrt{-1}) \)	None		✓			825.4.a.c	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+3 i q^{2}-3 i q^{3}-q^{4}+9 q^{6}+7 i q^{7}+\cdots\)
825.4.c.e	$825$	$4$	825.c	5.b	$2$	$2$	$2$	$48.677$	\(\Q(\sqrt{-1}) \)	None					165.4.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+i q^{2}-3 i q^{3}+7 q^{4}+3 q^{6}+36 i q^{7}+\cdots\)
825.4.c.f	$825$	$4$	825.c	5.b	$2$	$2$	$2$	$48.677$	\(\Q(\sqrt{-1}) \)	None					33.4.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+i q^{2}-3 i q^{3}+7 q^{4}+3 q^{6}+26 i q^{7}+\cdots\)
825.4.c.g	$825$	$4$	825.c	5.b	$2$	$2$	$2$	$48.677$	\(\Q(\sqrt{-1}) \)	None					165.4.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-3 i q^{3}+8 q^{4}-2 i q^{7}-9 q^{9}+\cdots\)
825.4.c.h	$825$	$4$	825.c	5.b	$2$	$4$	$4$	$48.677$	\(\Q(i, \sqrt{97})\)	None					33.4.a.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{2}-3\beta _{2}q^{3}+(-17+\beta _{3})q^{4}+\cdots\)
825.4.c.i	$825$	$4$	825.c	5.b	$2$	$4$	$4$	$48.677$	\(\Q(i, \sqrt{33})\)	None					33.4.a.d	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{2}+3\beta _{2}q^{3}+(-1+\beta _{3})q^{4}+\cdots\)
825.4.c.j	$825$	$4$	825.c	5.b	$2$	$4$	$4$	$48.677$	\(\Q(i, \sqrt{17})\)	None					165.4.a.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{2}-3\beta _{2}q^{3}+(3+\beta _{3})q^{4}+(-3+\cdots)q^{6}+\cdots\)
825.4.c.k	$825$	$4$	825.c	5.b	$2$	$6$	$6$	$48.677$	6.0.\(\cdots\).1	None					165.4.a.e	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+(\beta _{1}+\beta _{3})q^{2}+3\beta _{3}q^{3}+(-10+\beta _{4}+\cdots)q^{4}+\cdots\)
825.4.c.l	$825$	$4$	825.c	5.b	$2$	$6$	$6$	$48.677$	6.0.2230106176.1	None					165.4.a.d	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(\beta _{1}-\beta _{2})q^{2}+3\beta _{2}q^{3}+(-7+\beta _{3}+\cdots)q^{4}+\cdots\)
825.4.c.m	$825$	$4$	825.c	5.b	$2$	$6$	$6$	$48.677$	6.0.245110336.2	None					165.4.a.g	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{2}q^{2}+3\beta _{1}q^{3}+(-6-2\beta _{3}+\beta _{5})q^{4}+\cdots\)
825.4.c.n	$825$	$4$	825.c	5.b	$2$	$6$	$6$	$48.677$	6.0.181494784.1	None		✓			825.4.a.o	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{3}q^{2}-3\beta _{4}q^{3}+(-3+2\beta _{2})q^{4}+\cdots\)
825.4.c.o	$825$	$4$	825.c	5.b	$2$	$6$	$6$	$48.677$	6.0.9935104.1	None					165.4.a.f	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{3}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{2}q^{2}+3\beta _{3}q^{3}+(2-\beta _{4})q^{4}+3\beta _{1}q^{6}+\cdots\)
825.4.c.p	$825$	$4$	825.c	5.b	$2$	$8$	$8$	$48.677$	\(\mathbb{Q}[x]/(x^{8} + \cdots)\)	None					165.4.a.h	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(\beta _{1}-\beta _{2})q^{2}+3\beta _{2}q^{3}+(-7+\beta _{3}+\cdots)q^{4}+\cdots\)
825.4.c.q	$825$	$4$	825.c	5.b	$2$	$8$	$8$	$48.677$	\(\mathbb{Q}[x]/(x^{8} + \cdots)\)	None		✓			825.4.a.u	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(\beta _{2}+\beta _{3})q^{2}+3\beta _{3}q^{3}+(3+\beta _{1})q^{4}+\cdots\)
825.4.c.r	$825$	$4$	825.c	5.b	$2$	$10$	$10$	$48.677$	\(\mathbb{Q}[x]/(x^{10} + \cdots)\)	None		✓			825.4.a.w	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+(\beta _{1}-\beta _{2})q^{2}+3\beta _{2}q^{3}+(-9-\beta _{3}+\cdots)q^{4}+\cdots\)
825.4.c.s	$825$	$4$	825.c	5.b	$2$	$10$	$10$	$48.677$	\(\mathbb{Q}[x]/(x^{10} + \cdots)\)	None		✓			825.4.a.x	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{2}+3\beta _{2}q^{3}+(-1+\beta _{3})q^{4}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(825, [\chi])\) into lower level spaces

\( S_{4}^{\mathrm{old}}(825, [\chi]) \simeq \) \(S_{4}^{\mathrm{new}}(15, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(25, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(55, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(75, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(165, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(275, [\chi])\)\(^{\oplus 2}\)