Space of modular forms of level 225, weight 4, and Character 225.k

• Label: 225.4.k
• Level: $225$
• Weight: $4$
• Character orbit: 225.k
• Rep. character: $\chi_{225}(49,\cdot)$
• Character field: $\Q(\zeta_{6})$
• Dimension: $104$
• Newform subspaces: $5$
• Sturm bound: $120$
• Trace bound: $4$

Defining parameters

Level:	\( N \)	\(=\)	\( 225 = 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	225.k (of order \(6\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 45 \)
Character field:			\(\Q(\zeta_{6})\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(120\)
Trace bound:			\(4\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(225, [\chi])\).

	Total	New	Old
Modular forms	192	112	80
Cusp forms	168	104	64
Eisenstein series	24	8	16

Trace form

104 * q + 202 * q^4 - 4 * q^6 + 64 * q^9 - 98 * q^11 - 6 * q^14 - 754 * q^16 + 8 * q^19 - 78 * q^21 - 510 * q^24 + 1192 * q^26 + 682 * q^29 + 34 * q^31 - 378 * q^34 - 128 * q^36 - 58 * q^39 - 502 * q^41 - 4664 * q^44 + 1104 * q^46 + 2358 * q^49 + 68 * q^51 + 6682 * q^54 - 2052 * q^56 + 1964 * q^59 + 34 * q^61 - 6512 * q^64 + 8254 * q^66 - 1026 * q^69 + 2240 * q^71 + 4612 * q^74 - 1214 * q^76 + 686 * q^79 + 440 * q^81 - 5682 * q^84 - 9494 * q^86 - 10272 * q^89 + 2372 * q^91 + 402 * q^94 - 17516 * q^96 - 136 * q^99

Decomposition of \(S_{4}^{\mathrm{new}}(225, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
225.4.k.a	$225$	$4$	225.k	45.j	$6$	$8$	$4$	$13.275$	8.0.303595776.1	None					45.4.e.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$3^{2}$	$\mathrm{SU}(2)[C_{6}]$	\(q-\beta _{7}q^{2}+(3\beta _{1}+\beta _{4}+2\beta _{5}+2\beta _{7})q^{3}+\cdots\)
225.4.k.b	$225$	$4$	225.k	45.j	$6$	$8$	$4$	$13.275$	8.0.303595776.1	None					9.4.c.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$3^{2}$	$\mathrm{SU}(2)[C_{6}]$	\(q+(\beta _{1}-\beta _{7})q^{2}+(\beta _{4}-\beta _{5}-\beta _{7})q^{3}+\cdots\)
225.4.k.c	$225$	$4$	225.k	45.j	$6$	$12$	$6$	$13.275$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None					45.4.e.b	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{6}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-\beta _{7}+\beta _{11})q^{2}+(\beta _{5}+\beta _{8}+\beta _{11})q^{3}+\cdots\)
225.4.k.d	$225$	$4$	225.k	45.j	$6$	$28$	$14$	$13.275$		None					45.4.e.c	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{6}]$
225.4.k.e	$225$	$4$	225.k	45.j	$6$	$48$	$24$	$13.275$		None		✓	✓		225.4.e.e	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{6}]$

Decomposition of \(S_{4}^{\mathrm{old}}(225, [\chi])\) into lower level spaces

\( S_{4}^{\mathrm{old}}(225, [\chi]) \simeq \) \(S_{4}^{\mathrm{new}}(45, [\chi])\)\(^{\oplus 2}\)