Space of modular forms of level 225, weight 2, and Character 225.h

• Label: 225.2.h
• Level: $225$
• Weight: $2$
• Character orbit: 225.h
• Rep. character: $\chi_{225}(46,\cdot)$
• Character field: $\Q(\zeta_{5})$
• Dimension: $44$
• Newform subspaces: $5$
• Sturm bound: $60$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 225 = 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	225.h (of order \(5\) and degree \(4\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 25 \)
Character field:			\(\Q(\zeta_{5})\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(60\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(225, [\chi])\).

	Total	New	Old
Modular forms	136	52	84
Cusp forms	104	44	60
Eisenstein series	32	8	24

Trace form

44 * q + 4 * q^2 - 12 * q^4 + 11 * q^5 - 10 * q^7 - 7 * q^8 - 4 * q^10 - 4 * q^11 - 3 * q^13 + 11 * q^14 + 8 * q^16 + 2 * q^17 + 7 * q^19 - 21 * q^20 - 14 * q^22 - 25 * q^23 - 11 * q^25 + 44 * q^26 + 5 * q^28 - q^29 + 15 * q^31 - 30 * q^32 - 20 * q^34 + 25 * q^35 - 5 * q^37 + 21 * q^38 + 7 * q^40 - 2 * q^41 - 54 * q^43 - 32 * q^44 - 3 * q^46 + 18 * q^47 - 14 * q^49 - 76 * q^50 - 12 * q^52 - 45 * q^53 - 36 * q^55 + 18 * q^58 - 12 * q^59 + 25 * q^61 - 20 * q^62 - 7 * q^64 - 12 * q^65 - 42 * q^67 + 128 * q^68 - 20 * q^70 - 16 * q^71 + 45 * q^73 + 66 * q^74 - 36 * q^76 + 60 * q^77 + 31 * q^79 + 169 * q^80 + 120 * q^82 + 17 * q^83 - 22 * q^85 - 39 * q^86 + 82 * q^88 - 28 * q^89 - 8 * q^91 - 10 * q^92 - 27 * q^94 - 91 * q^95 + 48 * q^97 - 53 * q^98

Decomposition of \(S_{2}^{\mathrm{new}}(225, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
225.2.h.a	$225$	$2$	225.h	25.d	$5$	$4$	$1$	$1.797$	\(\Q(\zeta_{10})\)	None					75.2.g.a	$2$	$0$	\(1\)	\(0\)	\(-5\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{5}]$	\(q+(1-\zeta_{10}+\zeta_{10}^{2}-\zeta_{10}^{3})q^{2}+\zeta_{10}^{3}q^{4}+\cdots\)
225.2.h.b	$225$	$2$	225.h	25.d	$5$	$4$	$1$	$1.797$	\(\Q(\zeta_{10})\)	None					25.2.d.a	$2$	$0$	\(2\)	\(0\)	\(5\)	\(-2\)	$1$	$\mathrm{SU}(2)[C_{5}]$	\(q+(\zeta_{10}-\zeta_{10}^{2})q^{2}+(-1+\zeta_{10}+\zeta_{10}^{3})q^{4}+\cdots\)
225.2.h.c	$225$	$2$	225.h	25.d	$5$	$8$	$2$	$1.797$	8.0.26265625.1	None					75.2.g.b	$2$	$0$	\(1\)	\(0\)	\(5\)	\(4\)	$5$	$\mathrm{SU}(2)[C_{5}]$	\(q+(\beta _{3}+\beta _{7})q^{2}+(-1+2\beta _{1}+2\beta _{3}+\cdots)q^{4}+\cdots\)
225.2.h.d	$225$	$2$	225.h	25.d	$5$	$12$	$3$	$1.797$	\(\mathbb{Q}[x]/(x^{12} + \cdots)\)	None					75.2.g.c	$2$	$0$	\(0\)	\(0\)	\(6\)	\(-12\)	$1$	$\mathrm{SU}(2)[C_{5}]$	\(q+\beta _{2}q^{2}+(-1+\beta _{1}-\beta _{5}-2\beta _{8}+\beta _{9}+\cdots)q^{4}+\cdots\)
225.2.h.e	$225$	$2$	225.h	25.d	$5$	$16$	$4$	$1.797$	16.0.\(\cdots\).3	None		✓	✓		225.2.h.e	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$5^{2}$	$\mathrm{SU}(2)[C_{5}]$	\(q+(\beta _{3}-\beta _{10}+\beta _{12}-\beta _{14})q^{2}+(-\beta _{8}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(225, [\chi])\) into lower level spaces

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