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

• Label: 900.3.l
• Level: $900$
• Weight: $3$
• Character orbit: 900.l
• Rep. character: $\chi_{900}(757,\cdot)$
• Character field: $\Q(\zeta_{4})$
• Dimension: $30$
• Newform subspaces: $8$
• Sturm bound: $540$
• Trace bound: $31$

Defining parameters

Level:	\( N \)	\(=\)	\( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	900.l (of order \(4\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 5 \)
Character field:			\(\Q(i)\)
Newform subspaces:			\( 8 \)
Sturm bound:			\(540\)
Trace bound:			\(31\)
Distinguishing \(T_p\):			\(7\), \(11\)

Dimensions

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

	Total	New	Old
Modular forms	792	30	762
Cusp forms	648	30	618
Eisenstein series	144	0	144

Trace form

\( 30 q - 2 q^{7} + O(q^{10}) \)

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.l.a	$900$	$3$	900.l	5.c	$4$	$2$	$1$	$24.523$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(14\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(7+7i)q^{7}-10q^{11}+(-9+9i)q^{13}+\cdots\)
900.3.l.b	$900$	$3$	900.l	5.c	$4$	$4$	$2$	$24.523$	\(\Q(i, \sqrt{6})\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(-20\)	$2$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-5+5\beta _{1}+\beta _{2}-\beta _{3})q^{7}+(4+5\beta _{3})q^{11}+\cdots\)
900.3.l.c	$900$	$3$	900.l	5.c	$4$	$4$	$2$	$24.523$	\(\Q(i, \sqrt{6})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{4}]$	\(q+\beta _{1}q^{7}-6q^{11}+5\beta _{3}q^{13}-6\beta _{1}q^{17}+\cdots\)
900.3.l.d	$900$	$3$	900.l	5.c	$4$	$4$	$2$	$24.523$	\(\Q(i, \sqrt{6})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+3\beta _{1}q^{7}-6q^{11}-5\beta _{3}q^{13}-18\beta _{1}q^{17}+\cdots\)
900.3.l.e	$900$	$3$	900.l	5.c	$4$	$4$	$2$	$24.523$	\(\Q(i, \sqrt{6})\)	\(\Q(\sqrt{-3}) \)		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{4}]$	\(q+3\beta _{1}q^{7}+7\beta _{3}q^{13}+11\beta _{2}q^{19}+\cdots\)
900.3.l.f	$900$	$3$	900.l	5.c	$4$	$4$	$2$	$24.523$	\(\Q(i, \sqrt{6})\)	\(\Q(\sqrt{-3}) \)		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{6}$	$\mathrm{U}(1)[D_{4}]$	\(q+\beta _{1}q^{7}-\beta _{3}q^{13}+26\beta _{2}q^{19}+46q^{31}+\cdots\)
900.3.l.g	$900$	$3$	900.l	5.c	$4$	$4$	$2$	$24.523$	\(\Q(i, \sqrt{6})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+4\beta _{1}q^{7}+15q^{11}-2\beta _{3}q^{13}-3\beta _{1}q^{17}+\cdots\)
900.3.l.h	$900$	$3$	900.l	5.c	$4$	$4$	$2$	$24.523$	\(\Q(i, \sqrt{10})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(4\)	$2\cdot 5^{2}$	$\mathrm{SU}(2)[C_{4}]$	\(q+(1-\beta _{1})q^{7}-\beta _{3}q^{11}+(6+6\beta _{1})q^{13}+\cdots\)

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

\( S_{3}^{\mathrm{old}}(900, [\chi]) \cong \) \(S_{3}^{\mathrm{new}}(10, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(15, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(20, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(25, [\chi])\)\(^{\oplus 9}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(30, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(45, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(50, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(60, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(75, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(90, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(100, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(150, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(180, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(225, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(300, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(450, [\chi])\)\(^{\oplus 2}\)