Space of modular forms of level 45, weight 5, and Character 45.g

• Label: 45.5.g
• Level: $45$
• Weight: $5$
• Character orbit: 45.g
• Rep. character: $\chi_{45}(28,\cdot)$
• Character field: $\Q(\zeta_{4})$
• Dimension: $18$
• Newform subspaces: $5$
• Sturm bound: $30$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 45 = 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	45.g (of order \(4\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 5 \)
Character field:			\(\Q(i)\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(30\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{5}(45, [\chi])\).

	Total	New	Old
Modular forms	56	22	34
Cusp forms	40	18	22
Eisenstein series	16	4	12

Trace form

18 * q + 2 * q^2 + 44 * q^5 + 108 * q^7 - 120 * q^8 - 186 * q^10 + 304 * q^11 + 318 * q^13 - 1296 * q^16 - 898 * q^17 - 984 * q^20 + 2016 * q^22 + 1892 * q^23 + 2526 * q^25 + 3580 * q^26 - 2808 * q^28 - 2424 * q^31 - 6268 * q^32 - 6340 * q^35 - 102 * q^37 + 7320 * q^38 + 5028 * q^40 + 6088 * q^41 - 8412 * q^43 - 6168 * q^46 - 9748 * q^47 - 4594 * q^50 + 27588 * q^52 + 5642 * q^53 + 336 * q^55 + 27600 * q^56 + 4080 * q^58 - 22320 * q^61 - 27536 * q^62 - 8518 * q^65 + 1668 * q^67 + 1948 * q^68 + 4920 * q^70 + 4600 * q^71 - 3942 * q^73 - 9120 * q^76 - 56 * q^77 - 4084 * q^80 - 33024 * q^82 + 21092 * q^83 + 7422 * q^85 - 8696 * q^86 - 20160 * q^88 + 72072 * q^91 + 27152 * q^92 - 13464 * q^95 + 23178 * q^97 - 44342 * q^98

Decomposition of \(S_{5}^{\mathrm{new}}(45, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
45.5.g.a	$45$	$5$	45.g	5.c	$4$	$2$	$1$	$4.652$	\(\Q(\sqrt{-1}) \)	None		✓			45.5.g.a	$2$	$0$	\(-10\)	\(0\)	\(50\)	\(80\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-5 i-5)q^{2}+34 i q^{4}+25 q^{5}+\cdots\)
45.5.g.b	$45$	$5$	45.g	5.c	$4$	$2$	$1$	$4.652$	\(\Q(\sqrt{-1}) \)	None					5.5.c.a	$2$	$0$	\(2\)	\(0\)	\(-40\)	\(-52\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(i+1)q^{2}-14 i q^{4}+(-15 i-20)q^{5}+\cdots\)
45.5.g.c	$45$	$5$	45.g	5.c	$4$	$2$	$1$	$4.652$	\(\Q(\sqrt{-1}) \)	None		✓			45.5.g.a	$2$	$0$	\(10\)	\(0\)	\(-50\)	\(80\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(5 i+5)q^{2}+34 i q^{4}-25 q^{5}+\cdots\)
45.5.g.d	$45$	$5$	45.g	5.c	$4$	$4$	$2$	$4.652$	\(\Q(i, \sqrt{10})\)	None		✓			45.5.g.d	$4$	$0$	\(0\)	\(0\)	\(0\)	\(-20\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+\beta _{1}q^{2}-11\beta _{2}q^{4}+(2\beta _{1}-11\beta _{3})q^{5}+\cdots\)
45.5.g.e	$45$	$5$	45.g	5.c	$4$	$8$	$4$	$4.652$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None			✓		15.5.f.a	$2$	$0$	\(0\)	\(0\)	\(84\)	\(20\)	$2^{3}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{4}]$	\(q+\beta _{3}q^{2}+(\beta _{1}-12\beta _{2}+\beta _{3}+\beta _{5})q^{4}+\cdots\)

Decomposition of \(S_{5}^{\mathrm{old}}(45, [\chi])\) into lower level spaces

\( S_{5}^{\mathrm{old}}(45, [\chi]) \simeq \) \(S_{5}^{\mathrm{new}}(5, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(15, [\chi])\)\(^{\oplus 2}\)