Space of modular forms of level 900, weight 2, and Character 900.r

• Label: 900.2.r
• Level: $900$
• Weight: $2$
• Character orbit: 900.r
• Rep. character: $\chi_{900}(551,\cdot)$
• Character field: $\Q(\zeta_{6})$
• Dimension: $216$
• Newform subspaces: $7$
• Sturm bound: $360$
• Trace bound: $12$

Defining parameters

Level:	\( N \)	\(=\)	\( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	900.r (of order \(6\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 36 \)
Character field:			\(\Q(\zeta_{6})\)
Newform subspaces:			\( 7 \)
Sturm bound:			\(360\)
Trace bound:			\(12\)
Distinguishing \(T_p\):			\(7\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(900, [\chi])\).

	Total	New	Old
Modular forms	384	240	144
Cusp forms	336	216	120
Eisenstein series	48	24	24

Trace form

216 * q + 3 * q^2 + q^4 - 11 * q^6 + 2 * q^9 + 16 * q^12 + 2 * q^13 + 18 * q^14 - 3 * q^16 - 2 * q^18 - 10 * q^21 - 3 * q^22 - 7 * q^24 + 12 * q^28 - 6 * q^29 + 33 * q^32 + 20 * q^33 + q^34 + 7 * q^36 + 8 * q^37 - 33 * q^38 + 12 * q^41 + 36 * q^42 - 28 * q^46 - 9 * q^48 + 74 * q^49 + 2 * q^52 - 19 * q^54 - 30 * q^56 + 6 * q^57 + 14 * q^58 - 6 * q^61 + 10 * q^64 - 6 * q^66 - 33 * q^68 + 22 * q^69 - 39 * q^72 + 20 * q^73 - 78 * q^74 + 15 * q^76 - 18 * q^77 - 26 * q^78 + 34 * q^81 + 26 * q^82 - 116 * q^84 - 33 * q^86 + 21 * q^88 - 84 * q^92 + 2 * q^93 + 16 * q^94 - 44 * q^96 + 8 * q^97

Decomposition of \(S_{2}^{\mathrm{new}}(900, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
900.2.r.a	$900$	$2$	900.r	36.h	$6$	$8$	$4$	$7.187$	8.0.3317760000.3	\(\Q(\sqrt{-5}) \)					180.2.n.c	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{6}]$	\(q-\beta _{5}q^{2}+(\beta _{1}-\beta _{3}-\beta _{5})q^{3}-2\beta _{4}q^{4}+\cdots\)
900.2.r.b	$900$	$2$	900.r	36.h	$6$	$8$	$4$	$7.187$	\(\Q(\zeta_{24})\)	None					180.2.n.a	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{6}]$	\(q-\beta_{3} q^{2}+(\beta_{4}-\beta_{2})q^{3}+(-2\beta_1+2)q^{4}+\cdots\)
900.2.r.c	$900$	$2$	900.r	36.h	$6$	$8$	$4$	$7.187$	8.0.170772624.1	None					36.2.h.a	$4$	$0$	\(3\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+\beta _{1}q^{2}+(-1+\beta _{2}+\beta _{3}-\beta _{4}-\beta _{7})q^{3}+\cdots\)
900.2.r.d	$900$	$2$	900.r	36.h	$6$	$48$	$24$	$7.187$		None		✓			900.2.r.d	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{6}]$
900.2.r.e	$900$	$2$	900.r	36.h	$6$	$48$	$24$	$7.187$		None		✓			900.2.r.d	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{6}]$
900.2.r.f	$900$	$2$	900.r	36.h	$6$	$48$	$24$	$7.187$		None					180.2.q.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{6}]$
900.2.r.g	$900$	$2$	900.r	36.h	$6$	$48$	$24$	$7.187$		None					180.2.n.d	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{6}]$

Decomposition of \(S_{2}^{\mathrm{old}}(900, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(900, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(36, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(180, [\chi])\)\(^{\oplus 2}\)