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

• Label: 900.2.d
• Level: $900$
• Weight: $2$
• Character orbit: 900.d
• Rep. character: $\chi_{900}(649,\cdot)$
• Character field: $\Q$
• Dimension: $8$
• Newform subspaces: $4$
• Sturm bound: $360$
• Trace bound: $19$

Defining parameters

Level:	\( N \)	\(=\)	\( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	900.d (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 5 \)
Character field:			\(\Q\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(360\)
Trace bound:			\(19\)
Distinguishing \(T_p\):			\(7\), \(11\)

Dimensions

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

	Total	New	Old
Modular forms	216	8	208
Cusp forms	144	8	136
Eisenstein series	72	0	72

Trace form

\( 8 q - 12 q^{11} - 4 q^{19} + 4 q^{31} - 12 q^{41} + 12 q^{49} + 12 q^{59} + 4 q^{61} + 24 q^{71} - 16 q^{79} - 12 q^{89} - 4 q^{91}+O(q^{100}) \)

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.d.a	$900$	$2$	900.d	5.b	$2$	$2$	$2$	$7.187$	\(\Q(\sqrt{-1}) \)	None					300.2.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+i q^{7}-6 q^{11}+5 i q^{13}-6 i q^{17}+\cdots\)
900.2.d.b	$900$	$2$	900.d	5.b	$2$	$2$	$2$	$7.187$	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-3}) \)					36.2.a.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{U}(1)[D_{2}]$	\(q+2\beta q^{7}+\beta q^{13}-8 q^{19}-4 q^{31}+\cdots\)
900.2.d.c	$900$	$2$	900.d	5.b	$2$	$2$	$2$	$7.187$	\(\Q(\sqrt{-1}) \)	None					20.2.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta q^{7}-\beta q^{13}+3\beta q^{17}+4 q^{19}+\cdots\)
900.2.d.d	$900$	$2$	900.d	5.b	$2$	$2$	$2$	$7.187$	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-3}) \)		✓			900.2.a.d	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q+i q^{7}-7 i q^{13}+7 q^{19}+11 q^{31}+\cdots\)

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

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