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

• Label: 900.2.bf
• Level: $900$
• Weight: $2$
• Character orbit: 900.bf
• Rep. character: $\chi_{900}(7,\cdot)$
• Character field: $\Q(\zeta_{12})$
• Dimension: $416$
• Newform subspaces: $6$
• Sturm bound: $360$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	900.bf (of order \(12\) and degree \(4\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 180 \)
Character field:			\(\Q(\zeta_{12})\)
Newform subspaces:			\( 6 \)
Sturm bound:			\(360\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(7\)

Dimensions

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

	Total	New	Old
Modular forms	768	448	320
Cusp forms	672	416	256
Eisenstein series	96	32	64

Trace form

416 * q + 2 * q^2 - 8 * q^6 + 8 * q^8 - 2 * q^12 + 4 * q^13 - 4 * q^16 + 16 * q^17 + 36 * q^18 + 10 * q^22 + 48 * q^26 - 8 * q^28 - 18 * q^32 + 20 * q^33 + 8 * q^36 + 16 * q^37 + 34 * q^38 - 8 * q^41 - 34 * q^42 + 32 * q^46 + 22 * q^48 + 18 * q^52 + 64 * q^53 + 52 * q^56 + 48 * q^57 + 10 * q^58 - 8 * q^61 - 44 * q^62 - 24 * q^66 - 58 * q^68 - 78 * q^72 + 16 * q^73 + 4 * q^76 + 60 * q^77 - 114 * q^78 - 96 * q^81 - 76 * q^86 + 10 * q^88 - 52 * q^92 + 68 * q^93 - 128 * q^96 + 4 * q^97 - 164 * q^98

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.bf.a	$900$	$2$	900.bf	180.x	$12$	$8$	$2$	$7.187$	8.0.12960000.1	\(\Q(\sqrt{-5}) \)		✓			900.2.bf.a	$8$	$0$	\(-4\)	\(-2\)	\(0\)	\(-6\)	$2^{2}$	$\mathrm{U}(1)[D_{12}]$	\(q+(-1+\beta _{4}+\beta _{7})q^{2}+(\beta _{1}+\beta _{3})q^{3}+\cdots\)
900.2.bf.b	$900$	$2$	900.bf	180.x	$12$	$8$	$2$	$7.187$	8.0.12960000.1	\(\Q(\sqrt{-5}) \)		✓			900.2.bf.a	$8$	$0$	\(4\)	\(2\)	\(0\)	\(6\)	$2^{2}$	$\mathrm{U}(1)[D_{12}]$	\(q+(1-\beta _{4}+\beta _{7})q^{2}+(\beta _{1}+\beta _{2})q^{3}+2\beta _{1}q^{4}+\cdots\)
900.2.bf.c	$900$	$2$	900.bf	180.x	$12$	$16$	$4$	$7.187$	16.0.\(\cdots\).1	None		✓			900.2.bf.c	$16$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{12}]$	\(q+(\beta _{1}-\beta _{3}+\beta _{13})q^{2}+(-\beta _{9}-2\beta _{15})q^{3}+\cdots\)
900.2.bf.d	$900$	$2$	900.bf	180.x	$12$	$64$	$16$	$7.187$		None		✓			900.2.bf.d	$16$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{12}]$
900.2.bf.e	$900$	$2$	900.bf	180.x	$12$	$128$	$32$	$7.187$		None					180.2.x.a	$8$	$0$	\(2\)	\(0\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{12}]$
900.2.bf.f	$900$	$2$	900.bf	180.x	$12$	$192$	$48$	$7.187$		None		✓	✓		900.2.bf.f	$16$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{12}]$

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

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