Space of modular forms of level 900, weight 4, and Character 900.d

• Label: 900.4.d
• Level: $900$
• Weight: $4$
• Character orbit: 900.d
• Rep. character: $\chi_{900}(649,\cdot)$
• Character field: $\Q$
• Dimension: $22$
• Newform subspaces: $11$
• Sturm bound: $720$
• Trace bound: $19$

Defining parameters

Level:	\( N \)	\(=\)	\( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	900.d (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 5 \)
Character field:			\(\Q\)
Newform subspaces:			\( 11 \)
Sturm bound:			\(720\)
Trace bound:			\(19\)
Distinguishing \(T_p\):			\(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(900, [\chi])\).

	Total	New	Old
Modular forms	576	22	554
Cusp forms	504	22	482
Eisenstein series	72	0	72

Trace form

\( 22 q + O(q^{10}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(900, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
900.4.d.a	$900$	$4$	900.d	5.b	$2$	$2$	$2$	$53.102$	\(\Q(\sqrt{-1}) \)	None					100.4.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+26 i q^{7}-45 q^{11}-44 i q^{13}-117 i q^{17}+\cdots\)
900.4.d.b	$900$	$4$	900.d	5.b	$2$	$2$	$2$	$53.102$	\(\Q(\sqrt{-1}) \)	None					60.4.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+16\beta q^{7}-36 q^{11}+5\beta q^{13}+39\beta q^{17}+\cdots\)
900.4.d.c	$900$	$4$	900.d	5.b	$2$	$2$	$2$	$53.102$	\(\Q(\sqrt{-1}) \)	None					12.4.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+4\beta q^{7}-36 q^{11}+5\beta q^{13}-9\beta q^{17}+\cdots\)
900.4.d.d	$900$	$4$	900.d	5.b	$2$	$2$	$2$	$53.102$	\(\Q(\sqrt{-1}) \)	None					180.4.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta q^{7}-30 q^{11}+2\beta q^{13}-45\beta q^{17}+\cdots\)
900.4.d.e	$900$	$4$	900.d	5.b	$2$	$2$	$2$	$53.102$	\(\Q(\sqrt{-1}) \)	None					300.4.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+13 i q^{7}-6 q^{11}+5 i q^{13}-78 i q^{17}+\cdots\)
900.4.d.f	$900$	$4$	900.d	5.b	$2$	$2$	$2$	$53.102$	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-3}) \)		✓			900.4.a.e	$4$	$1$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q+17 i q^{7}+19 i q^{13}-107 q^{19}+\cdots\)
900.4.d.g	$900$	$4$	900.d	5.b	$2$	$2$	$2$	$53.102$	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-3}) \)		✓			900.4.a.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q+37 i q^{7}+89 i q^{13}+163 q^{19}+\cdots\)
900.4.d.h	$900$	$4$	900.d	5.b	$2$	$2$	$2$	$53.102$	\(\Q(\sqrt{-1}) \)	None					60.4.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+14\beta q^{7}+24 q^{11}-35\beta q^{13}+51\beta q^{17}+\cdots\)
900.4.d.i	$900$	$4$	900.d	5.b	$2$	$2$	$2$	$53.102$	\(\Q(\sqrt{-1}) \)	None					180.4.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta q^{7}+30 q^{11}+2\beta q^{13}+45\beta q^{17}+\cdots\)
900.4.d.j	$900$	$4$	900.d	5.b	$2$	$2$	$2$	$53.102$	\(\Q(\sqrt{-1}) \)	None					300.4.a.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+7 i q^{7}+54 q^{11}-55 i q^{13}+18 i q^{17}+\cdots\)
900.4.d.k	$900$	$4$	900.d	5.b	$2$	$2$	$2$	$53.102$	\(\Q(\sqrt{-1}) \)	None					20.4.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+8\beta q^{7}+60 q^{11}+43\beta q^{13}+9\beta q^{17}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(900, [\chi])\) into lower level spaces

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