Space of modular forms of level 75, weight 18, and Character 75.b

• Label: 75.18.b
• Level: $75$
• Weight: $18$
• Character orbit: 75.b
• Rep. character: $\chi_{75}(49,\cdot)$
• Character field: $\Q$
• Dimension: $52$
• Newform subspaces: $8$
• Sturm bound: $180$
• Trace bound: $4$

Defining parameters

Level:	\( N \)	\(=\)	\( 75 = 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 18 \)
Character orbit:	\([\chi]\)	\(=\)	75.b (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 5 \)
Character field:			\(\Q\)
Newform subspaces:			\( 8 \)
Sturm bound:			\(180\)
Trace bound:			\(4\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{18}(75, [\chi])\).

	Total	New	Old
Modular forms	176	52	124
Cusp forms	164	52	112
Eisenstein series	12	0	12

Trace form

52 * q - 3648556 * q^4 + 1600884 * q^6 - 2238429492 * q^9 - 1334042212 * q^11 - 52886269524 * q^14 + 185795789620 * q^16 + 63473515666 * q^19 + 216718608618 * q^21 + 221962750308 * q^24 - 8048009054116 * q^26 - 5262333260848 * q^29 + 21760455035762 * q^31 - 35070740962632 * q^34 + 157058372184876 * q^36 - 79755435863946 * q^39 + 219681308023132 * q^41 - 408048442430008 * q^44 + 503971248729672 * q^46 - 1721572419609450 * q^49 + 914558825458296 * q^51 - 68912806901364 * q^54 + 8762233055880480 * q^56 - 1587117603353156 * q^59 + 2821916084622710 * q^61 - 4689251373899284 * q^64 + 14701071546828432 * q^66 - 4522231273258764 * q^69 + 21681238806885664 * q^71 + 8646948486335576 * q^74 + 21423188343296096 * q^76 + 11628828618767680 * q^79 + 96357049820295732 * q^81 - 41644683665959896 * q^84 - 229548209623088692 * q^86 + 107684569360458516 * q^89 - 91144169859367238 * q^91 - 99176816964416616 * q^94 - 256993421676812892 * q^96 + 57426142902186852 * q^99

Decomposition of \(S_{18}^{\mathrm{new}}(75, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
75.18.b.a	$75$	$18$	75.b	5.b	$2$	$2$	$2$	$137.417$	\(\Q(\sqrt{-1}) \)	None					3.18.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+204 i q^{2}+6561 i q^{3}+89456 q^{4}+\cdots\)
75.18.b.b	$75$	$18$	75.b	5.b	$2$	$4$	$4$	$137.417$	\(\Q(i, \sqrt{849})\)	None					15.18.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}\cdot 3^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-178\beta _{1}+\beta _{2})q^{2}-3^{8}\beta _{1}q^{3}+(-175688+\cdots)q^{4}+\cdots\)
75.18.b.c	$75$	$18$	75.b	5.b	$2$	$4$	$4$	$137.417$	\(\mathbb{Q}[x]/(x^{4} + \cdots)\)	None					3.18.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-297\beta _{1}+\beta _{2})q^{2}+3^{8}\beta _{1}q^{3}+(-88258+\cdots)q^{4}+\cdots\)
75.18.b.d	$75$	$18$	75.b	5.b	$2$	$6$	$6$	$137.417$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None					15.18.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{14}\cdot 3^{2}\cdot 5^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-147\beta _{1}-\beta _{3})q^{2}+3^{8}\beta _{1}q^{3}+(-99318+\cdots)q^{4}+\cdots\)
75.18.b.e	$75$	$18$	75.b	5.b	$2$	$6$	$6$	$137.417$	\(\mathbb{Q}[x]/(x^{6} + \cdots)\)	None					15.18.a.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}\cdot 3^{2}\cdot 5^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(\beta _{1}+84\beta _{3})q^{2}+3^{8}\beta _{3}q^{3}+(2408+\cdots)q^{4}+\cdots\)
75.18.b.f	$75$	$18$	75.b	5.b	$2$	$8$	$8$	$137.417$	\(\mathbb{Q}[x]/(x^{8} + \cdots)\)	None					15.18.a.d	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{14}\cdot 3^{8}\cdot 5^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(\beta _{1}-8\beta _{5})q^{2}+3^{8}\beta _{5}q^{3}+(-109856+\cdots)q^{4}+\cdots\)
75.18.b.g	$75$	$18$	75.b	5.b	$2$	$10$	$10$	$137.417$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None		✓			75.18.a.g	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{26}\cdot 3^{6}\cdot 5^{8}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-41\beta _{1}+\beta _{2})q^{2}+3^{8}\beta _{1}q^{3}+(-52431+\cdots)q^{4}+\cdots\)
75.18.b.h	$75$	$18$	75.b	5.b	$2$	$12$	$12$	$137.417$	\(\mathbb{Q}[x]/(x^{12} + \cdots)\)	None		✓	✓		75.18.a.i	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{26}\cdot 3^{10}\cdot 5^{14}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-56\beta _{6}+\beta _{7})q^{2}-3^{8}\beta _{6}q^{3}+(-65542+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{18}^{\mathrm{old}}(75, [\chi])\) into lower level spaces

\( S_{18}^{\mathrm{old}}(75, [\chi]) \simeq \) \(S_{18}^{\mathrm{new}}(5, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{18}^{\mathrm{new}}(15, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{18}^{\mathrm{new}}(25, [\chi])\)\(^{\oplus 2}\)