Space of modular forms of level 900, weight 6, and Character 900.d

• Label: 900.6.d
• Level: $900$
• Weight: $6$
• Character orbit: 900.d
• Rep. character: $\chi_{900}(649,\cdot)$
• Character field: $\Q$
• Dimension: $38$
• Newform subspaces: $14$
• Sturm bound: $1080$
• Trace bound: $19$

Defining parameters

Level:	\( N \)	\(=\)	\( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	900.d (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 5 \)
Character field:			\(\Q\)
Newform subspaces:			\( 14 \)
Sturm bound:			\(1080\)
Trace bound:			\(19\)
Distinguishing \(T_p\):			\(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(900, [\chi])\).

	Total	New	Old
Modular forms	936	38	898
Cusp forms	864	38	826
Eisenstein series	72	0	72

Trace form

\( 38 q + 276 q^{11} - 1156 q^{19} + 3924 q^{29} + 1060 q^{31} - 18948 q^{41} - 82746 q^{49} + 9012 q^{59} + 60928 q^{61} - 68664 q^{71} - 25312 q^{79} + 162852 q^{89} + 224924 q^{91}+O(q^{100}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(900, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
900.6.d.a	$900$	$6$	900.d	5.b	$2$	$2$	$2$	$144.345$	\(\Q(\sqrt{-1}) \)	None					4.6.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+44\beta q^{7}-540 q^{11}-209\beta q^{13}+\cdots\)
900.6.d.b	$900$	$6$	900.d	5.b	$2$	$2$	$2$	$144.345$	\(\Q(\sqrt{-1}) \)	None					60.6.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+22\beta q^{7}-216 q^{11}-385\beta q^{13}+\cdots\)
900.6.d.c	$900$	$6$	900.d	5.b	$2$	$2$	$2$	$144.345$	\(\Q(\sqrt{-1}) \)	None					60.6.a.d	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+28\beta q^{7}-156 q^{11}-175\beta q^{13}+\cdots\)
900.6.d.d	$900$	$6$	900.d	5.b	$2$	$2$	$2$	$144.345$	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-3}) \)		✓			900.6.a.c	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q+211 i q^{7}-427 i q^{13}-3143 q^{19}+\cdots\)
900.6.d.e	$900$	$6$	900.d	5.b	$2$	$2$	$2$	$144.345$	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-3}) \)					36.6.a.b	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{U}(1)[D_{2}]$	\(q+118\beta q^{7}-601\beta q^{13}+1432 q^{19}+\cdots\)
900.6.d.f	$900$	$6$	900.d	5.b	$2$	$2$	$2$	$144.345$	\(\Q(\sqrt{-1}) \)	None					60.6.a.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+122\beta q^{7}+144 q^{11}+25\beta q^{13}+\cdots\)
900.6.d.g	$900$	$6$	900.d	5.b	$2$	$2$	$2$	$144.345$	\(\Q(\sqrt{-1}) \)	None					300.6.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+91 i q^{7}+174 q^{11}+785 i q^{13}+\cdots\)
900.6.d.h	$900$	$6$	900.d	5.b	$2$	$2$	$2$	$144.345$	\(\Q(\sqrt{-1}) \)	None					20.6.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+109\beta q^{7}+480 q^{11}+311\beta q^{13}+\cdots\)
900.6.d.i	$900$	$6$	900.d	5.b	$2$	$2$	$2$	$144.345$	\(\Q(\sqrt{-1}) \)	None					60.6.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+8\beta q^{7}+564 q^{11}-185\beta q^{13}+\cdots\)
900.6.d.j	$900$	$6$	900.d	5.b	$2$	$4$	$4$	$144.345$	\(\Q(i, \sqrt{7})\)	None					300.6.a.g	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{6}\cdot 3^{2}\cdot 5^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-11\beta _{1}+\beta _{3})q^{7}+(-186-3\beta _{2}+\cdots)q^{11}+\cdots\)
900.6.d.k	$900$	$6$	900.d	5.b	$2$	$4$	$4$	$144.345$	\(\Q(i, \sqrt{241})\)	None					180.6.a.f	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{5}\cdot 3^{2}\cdot 5^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(\beta _{1}+4\beta _{2})q^{7}+(-120+\beta _{3})q^{11}+\cdots\)
900.6.d.l	$900$	$6$	900.d	5.b	$2$	$4$	$4$	$144.345$	\(\Q(i, \sqrt{94})\)	None		✓			900.6.a.l	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{5}\cdot 3^{2}\cdot 5^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+71\beta _{1}q^{7}+\beta _{3}q^{11}-137\beta _{1}q^{13}+\cdots\)
900.6.d.m	$900$	$6$	900.d	5.b	$2$	$4$	$4$	$144.345$	\(\Q(i, \sqrt{409})\)	None					100.6.a.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}\cdot 3^{2}\cdot 5^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-2\beta _{1}+4\beta _{2})q^{7}+(30+\beta _{3})q^{11}+\cdots\)
900.6.d.n	$900$	$6$	900.d	5.b	$2$	$4$	$4$	$144.345$	\(\Q(i, \sqrt{241})\)	None					180.6.a.f	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{5}\cdot 3^{2}\cdot 5^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(\beta _{1}+4\beta _{2})q^{7}+(120-\beta _{3})q^{11}+(8\beta _{1}+\cdots)q^{13}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(900, [\chi])\) into lower level spaces

\( S_{6}^{\mathrm{old}}(900, [\chi]) \simeq \) \(S_{6}^{\mathrm{new}}(5, [\chi])\)\(^{\oplus 18}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(10, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(15, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(20, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(25, [\chi])\)\(^{\oplus 9}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(30, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(45, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(50, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(60, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(75, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(90, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(100, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(150, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(180, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(225, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(300, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(450, [\chi])\)\(^{\oplus 2}\)