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

• Label: 900.2.h
• Level: $900$
• Weight: $2$
• Character orbit: 900.h
• Rep. character: $\chi_{900}(899,\cdot)$
• Character field: $\Q$
• Dimension: $36$
• Newform subspaces: $4$
• Sturm bound: $360$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	900.h (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 60 \)
Character field:			\(\Q\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(360\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(7\), \(17\)

Dimensions

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

	Total	New	Old
Modular forms	204	36	168
Cusp forms	156	36	120
Eisenstein series	48	0	48

Trace form

\( 36 q - 8 q^{4} + 8 q^{16} + 48 q^{34} - 40 q^{46} + 52 q^{49} + 72 q^{61} - 104 q^{64} - 24 q^{76} + 24 q^{94}+O(q^{100}) \)

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.h.a	$900$	$2$	900.h	60.h	$2$	$4$	$4$	$7.187$	\(\Q(\zeta_{8})\)	\(\Q(\sqrt{-1}) \)					36.2.b.a	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{U}(1)[D_{2}]$	\(q-\beta_{3} q^{2}+2 q^{4}-2\beta_{3} q^{8}+2\beta_1 q^{13}+\cdots\)
900.2.h.b	$900$	$2$	900.h	60.h	$2$	$8$	$8$	$7.187$	8.0.18939904.2	None					180.2.e.a	$4$	$0$	\(-4\)	\(0\)	\(0\)	\(0\)	$2^{5}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-1-\beta _{3})q^{2}+(\beta _{3}+\beta _{5})q^{4}+(-2\beta _{4}+\cdots)q^{7}+\cdots\)
900.2.h.c	$900$	$2$	900.h	60.h	$2$	$8$	$8$	$7.187$	8.0.18939904.2	None					180.2.e.a	$4$	$0$	\(4\)	\(0\)	\(0\)	\(0\)	$2^{5}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(1+\beta _{3})q^{2}+(\beta _{3}+\beta _{5})q^{4}+(-2\beta _{4}+\cdots)q^{7}+\cdots\)
900.2.h.d	$900$	$2$	900.h	60.h	$2$	$16$	$16$	$7.187$	\(\Q(\zeta_{40})\)	None		✓	✓		900.2.e.e	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{16}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta_{9} q^{2}+\beta_{3} q^{4}+(\beta_{10}+\beta_{5})q^{7}+\cdots\)

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

\( S_{2}^{\mathrm{old}}(900, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(60, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(180, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(300, [\chi])\)\(^{\oplus 2}\)