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

• Label: 900.2.i
• Level: $900$
• Weight: $2$
• Character orbit: 900.i
• Rep. character: $\chi_{900}(301,\cdot)$
• Character field: $\Q(\zeta_{3})$
• Dimension: $38$
• Newform subspaces: $6$
• Sturm bound: $360$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	900.i (of order \(3\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 9 \)
Character field:			\(\Q(\zeta_{3})\)
Newform subspaces:			\( 6 \)
Sturm bound:			\(360\)
Trace bound:			\(3\)
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	396	38	358
Cusp forms	324	38	286
Eisenstein series	72	0	72

Trace form

38 * q - 2 * q^3 + q^7 + 4 * q^9 + 5 * q^11 + q^13 - 8 * q^19 + 19 * q^21 - 9 * q^23 - 2 * q^27 - 3 * q^29 + q^31 - 3 * q^33 - 8 * q^37 - 3 * q^39 + 17 * q^41 + q^43 + 15 * q^47 - 24 * q^49 + 34 * q^51 + 36 * q^53 + 26 * q^57 + 7 * q^59 + q^61 + 35 * q^63 + 13 * q^67 + 19 * q^69 - 20 * q^73 + 3 * q^77 + 13 * q^79 - 32 * q^81 - 39 * q^83 - 33 * q^87 - 88 * q^89 - 22 * q^91 - 61 * q^93 + 19 * q^97 + 5 * q^99

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.i.a	$900$	$2$	900.i	9.c	$3$	$2$	$1$	$7.187$	\(\Q(\sqrt{-3}) \)	None					180.2.i.a	$2$	$0$	\(0\)	\(-3\)	\(0\)	\(-1\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-2+\zeta_{6})q^{3}+(-1+\zeta_{6})q^{7}+(3+\cdots)q^{9}+\cdots\)
900.2.i.b	$900$	$2$	900.i	9.c	$3$	$2$	$1$	$7.187$	\(\Q(\sqrt{-3}) \)	None					36.2.e.a	$2$	$1$	\(0\)	\(0\)	\(0\)	\(-1\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1-2\zeta_{6})q^{3}+(-1+\zeta_{6})q^{7}-3q^{9}+\cdots\)
900.2.i.c	$900$	$2$	900.i	9.c	$3$	$6$	$3$	$7.187$	6.0.954288.1	None					180.2.i.b	$2$	$0$	\(0\)	\(1\)	\(0\)	\(3\)	$3$	$\mathrm{SU}(2)[C_{3}]$	\(q-\beta _{4}q^{3}+(-\beta _{1}-\beta _{2}-\beta _{3}-\beta _{5})q^{7}+\cdots\)
900.2.i.d	$900$	$2$	900.i	9.c	$3$	$8$	$4$	$7.187$	8.0.142635249.1	None		✓			900.2.i.d	$2$	$0$	\(0\)	\(-1\)	\(0\)	\(1\)	$3^{3}$	$\mathrm{SU}(2)[C_{3}]$	\(q+\beta _{3}q^{3}+(-\beta _{3}+\beta _{4}+\beta _{5}+\beta _{7})q^{7}+\cdots\)
900.2.i.e	$900$	$2$	900.i	9.c	$3$	$8$	$4$	$7.187$	8.0.142635249.1	None		✓			900.2.i.d	$2$	$0$	\(0\)	\(1\)	\(0\)	\(-1\)	$3^{3}$	$\mathrm{SU}(2)[C_{3}]$	\(q-\beta _{2}q^{3}+(2\beta _{2}+\beta _{5}-\beta _{6}-\beta _{7})q^{7}+\cdots\)
900.2.i.f	$900$	$2$	900.i	9.c	$3$	$12$	$6$	$7.187$	\(\mathbb{Q}[x]/(x^{12} + \cdots)\)	None			✓		180.2.r.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+\beta _{1}q^{3}+(\beta _{5}+\beta _{11})q^{7}+(-1+\beta _{2}+\cdots)q^{9}+\cdots\)

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

\( S_{2}^{\mathrm{old}}(900, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(18, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(36, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(45, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(90, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(180, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(225, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(450, [\chi])\)\(^{\oplus 2}\)