Space of modular forms of level 225, weight 5, and Character 225.g

• Label: 225.5.g
• Level: $225$
• Weight: $5$
• Character orbit: 225.g
• Rep. character: $\chi_{225}(82,\cdot)$
• Character field: $\Q(\zeta_{4})$
• Dimension: $58$
• Newform subspaces: $14$
• Sturm bound: $150$
• Trace bound: $16$

Defining parameters

Level:	\( N \)	\(=\)	\( 225 = 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	225.g (of order \(4\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 5 \)
Character field:			\(\Q(i)\)
Newform subspaces:			\( 14 \)
Sturm bound:			\(150\)
Trace bound:			\(16\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{5}(225, [\chi])\).

	Total	New	Old
Modular forms	264	62	202
Cusp forms	216	58	158
Eisenstein series	48	4	44

Trace form

58 * q - 2 * q^2 - 108 * q^7 + 120 * q^8 - 596 * q^11 - 318 * q^13 - 2284 * q^16 + 898 * q^17 - 2016 * q^22 - 1892 * q^23 - 4268 * q^26 + 2808 * q^28 + 3016 * q^31 + 6268 * q^32 + 102 * q^37 - 7320 * q^38 - 16484 * q^41 + 8412 * q^43 - 2616 * q^46 + 9748 * q^47 - 27588 * q^52 - 5642 * q^53 - 31800 * q^56 - 4080 * q^58 + 24488 * q^61 + 27536 * q^62 - 1668 * q^67 - 1948 * q^68 + 14752 * q^71 + 3942 * q^73 - 2380 * q^76 + 56 * q^77 + 33024 * q^82 - 21092 * q^83 - 5096 * q^86 + 20160 * q^88 - 32256 * q^91 - 27152 * q^92 - 23178 * q^97 + 44342 * q^98

Decomposition of \(S_{5}^{\mathrm{new}}(225, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
225.5.g.a	$225$	$5$	225.g	5.c	$4$	$2$	$1$	$23.258$	\(\Q(\sqrt{-1}) \)	None					45.5.g.a	$2$	$0$	\(-10\)	\(0\)	\(0\)	\(-80\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-5 i-5)q^{2}+34 i q^{4}+(-40 i-40)q^{7}+\cdots\)
225.5.g.b	$225$	$5$	225.g	5.c	$4$	$2$	$1$	$23.258$	\(\Q(\sqrt{-1}) \)	None					5.5.c.a	$2$	$0$	\(-2\)	\(0\)	\(0\)	\(52\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-i-1)q^{2}-14 i q^{4}+(26 i+26)q^{7}+\cdots\)
225.5.g.c	$225$	$5$	225.g	5.c	$4$	$2$	$1$	$23.258$	\(\Q(\sqrt{-1}) \)	None					45.5.g.a	$2$	$0$	\(10\)	\(0\)	\(0\)	\(-80\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(5 i+5)q^{2}+34 i q^{4}+(-40 i-40)q^{7}+\cdots\)
225.5.g.d	$225$	$5$	225.g	5.c	$4$	$4$	$2$	$23.258$	\(\Q(i, \sqrt{6})\)	None					75.5.f.a	$2$	$0$	\(-12\)	\(0\)	\(0\)	\(-72\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-3+3\beta _{2}-2\beta _{3})q^{2}+(12\beta _{1}-14\beta _{2}+\cdots)q^{4}+\cdots\)
225.5.g.e	$225$	$5$	225.g	5.c	$4$	$4$	$2$	$23.258$	\(\Q(i, \sqrt{6})\)	None					75.5.f.b	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+\beta _{1}q^{2}-13\beta _{2}q^{4}-30\beta _{1}q^{7}-29\beta _{3}q^{8}+\cdots\)
225.5.g.f	$225$	$5$	225.g	5.c	$4$	$4$	$2$	$23.258$	\(\Q(i, \sqrt{6})\)	None					25.5.c.b	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+3\beta _{1}q^{2}+11\beta _{2}q^{4}-28\beta _{1}q^{7}-15\beta _{3}q^{8}+\cdots\)
225.5.g.g	$225$	$5$	225.g	5.c	$4$	$4$	$2$	$23.258$	\(\Q(i, \sqrt{6})\)	None					75.5.f.c	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+2\beta _{1}q^{2}-4\beta _{2}q^{4}-17\beta _{1}q^{7}-40\beta _{3}q^{8}+\cdots\)
225.5.g.h	$225$	$5$	225.g	5.c	$4$	$4$	$2$	$23.258$	\(\Q(i, \sqrt{6})\)	\(\Q(\sqrt{-3}) \)		✓			225.5.g.h	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$5^{2}$	$\mathrm{U}(1)[D_{4}]$	\(q-2^{4}\beta _{2}q^{4}+11\beta _{1}q^{7}+3\beta _{3}q^{13}+\cdots\)
225.5.g.i	$225$	$5$	225.g	5.c	$4$	$4$	$2$	$23.258$	\(\Q(i, \sqrt{30})\)	\(\Q(\sqrt{-15}) \)		✓			225.5.g.i	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{4}]$	\(q+\beta _{1}q^{2}-\beta _{2}q^{4}-17\beta _{3}q^{8}+239q^{16}+\cdots\)
225.5.g.j	$225$	$5$	225.g	5.c	$4$	$4$	$2$	$23.258$	\(\Q(i, \sqrt{21})\)	None					25.5.c.c	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{3}$	$\mathrm{SU}(2)[C_{4}]$	\(q-\beta _{3}q^{2}-26\beta _{1}q^{4}-9\beta _{3}q^{7}-10\beta _{2}q^{8}+\cdots\)
225.5.g.k	$225$	$5$	225.g	5.c	$4$	$4$	$2$	$23.258$	\(\Q(i, \sqrt{10})\)	None					45.5.g.d	$4$	$0$	\(0\)	\(0\)	\(0\)	\(20\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+\beta _{1}q^{2}-11\beta _{2}q^{4}+(5+5\beta _{2})q^{7}+\cdots\)
225.5.g.l	$225$	$5$	225.g	5.c	$4$	$4$	$2$	$23.258$	\(\Q(i, \sqrt{6})\)	None					75.5.f.a	$2$	$0$	\(12\)	\(0\)	\(0\)	\(72\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(3-3\beta _{2}+2\beta _{3})q^{2}+(12\beta _{1}-14\beta _{2}+\cdots)q^{4}+\cdots\)
225.5.g.m	$225$	$5$	225.g	5.c	$4$	$8$	$4$	$23.258$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None					15.5.f.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-20\)	$2^{4}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{4}]$	\(q+\beta _{1}q^{2}+(-\beta _{1}+12\beta _{2}-\beta _{4}+\beta _{6}+\cdots)q^{4}+\cdots\)
225.5.g.n	$225$	$5$	225.g	5.c	$4$	$8$	$4$	$23.258$	\(\Q(i, \sqrt{6}, \sqrt{10})\)	None		✓			225.5.g.n	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{6}\cdot 3^{2}\cdot 5^{4}$	$\mathrm{SU}(2)[C_{4}]$	\(q-\beta _{5}q^{2}-14\beta _{3}q^{4}-7\beta _{2}q^{7}-2\beta _{4}q^{8}+\cdots\)

Decomposition of \(S_{5}^{\mathrm{old}}(225, [\chi])\) into lower level spaces

\( S_{5}^{\mathrm{old}}(225, [\chi]) \simeq \) \(S_{5}^{\mathrm{new}}(5, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(15, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(25, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(45, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(75, [\chi])\)\(^{\oplus 2}\)