Space of modular forms of level 900, weight 3, and Character 900.f

• Label: 900.3.f
• Level: $900$
• Weight: $3$
• Character orbit: 900.f
• Rep. character: $\chi_{900}(199,\cdot)$
• Character field: $\Q$
• Dimension: $88$
• Newform subspaces: $9$
• Sturm bound: $540$
• Trace bound: $26$

Defining parameters

Level:	\( N \)	\(=\)	\( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	900.f (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 20 \)
Character field:			\(\Q\)
Newform subspaces:			\( 9 \)
Sturm bound:			\(540\)
Trace bound:			\(26\)
Distinguishing \(T_p\):			\(7\), \(29\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(900, [\chi])\).

	Total	New	Old
Modular forms	384	92	292
Cusp forms	336	88	248
Eisenstein series	48	4	44

Trace form

\( 88 q + 6 q^{4} + O(q^{10}) \)

Decomposition of \(S_{3}^{\mathrm{new}}(900, [\chi])\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
900.3.f.a	$900$	$3$	900.f	20.d	$2$	$2$	$2$	$24.523$	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-1}) \)		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{U}(1)[D_{2}]$	\(q-iq^{2}-4q^{4}+4iq^{8}-5iq^{13}+2^{4}q^{16}+\cdots\)
900.3.f.b	$900$	$3$	900.f	20.d	$2$	$2$	$2$	$24.523$	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-1}) \)		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{U}(1)[D_{2}]$	\(q+iq^{2}-4q^{4}-4iq^{8}-5iq^{13}+2^{4}q^{16}+\cdots\)
900.3.f.c	$900$	$3$	900.f	20.d	$2$	$4$	$4$	$24.523$	\(\Q(\zeta_{12})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\zeta_{12}q^{2}+(2+\zeta_{12}^{3})q^{4}+(-4\zeta_{12}+\cdots)q^{7}+\cdots\)
900.3.f.d	$900$	$3$	900.f	20.d	$2$	$8$	$8$	$24.523$	8.0.\(\cdots\).3	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}\cdot 5^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{4}q^{2}+(-1-\beta _{1})q^{4}+(-2\beta _{4}+2\beta _{7})q^{7}+\cdots\)
900.3.f.e	$900$	$3$	900.f	20.d	$2$	$8$	$8$	$24.523$	\(\Q(\zeta_{20})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{12}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\zeta_{20}q^{2}+(1+\zeta_{20}^{4})q^{4}+(-3\zeta_{20}+\cdots)q^{7}+\cdots\)
900.3.f.f	$900$	$3$	900.f	20.d	$2$	$16$	$16$	$24.523$	\(\mathbb{Q}[x]/(x^{16} + \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{28}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{6}q^{2}+(-1-\beta _{8})q^{4}+(-\beta _{11}-\beta _{13}+\cdots)q^{7}+\cdots\)
900.3.f.g	$900$	$3$	900.f	20.d	$2$	$16$	$16$	$24.523$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{20}\cdot 5^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{1}q^{2}+(-1+\beta _{2})q^{4}-\beta _{11}q^{7}+\cdots\)
900.3.f.h	$900$	$3$	900.f	20.d	$2$	$16$	$16$	$24.523$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{20}\cdot 5^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{3}q^{2}+(1+\beta _{5})q^{4}+(\beta _{3}-\beta _{4}-\beta _{8}+\cdots)q^{7}+\cdots\)
900.3.f.i	$900$	$3$	900.f	20.d	$2$	$16$	$16$	$24.523$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{28}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{10}q^{2}+(2-\beta _{5})q^{4}-\beta _{8}q^{7}+(2\beta _{10}+\cdots)q^{8}+\cdots\)

Decomposition of \(S_{3}^{\mathrm{old}}(900, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(900, [\chi]) \cong \) \(S_{3}^{\mathrm{new}}(20, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(60, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(180, [\chi])\)\(^{\oplus 2}\)