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

• Label: 900.2.e
• Level: $900$
• Weight: $2$
• Character orbit: 900.e
• Rep. character: $\chi_{900}(251,\cdot)$
• Character field: $\Q$
• Dimension: $38$
• Newform subspaces: $7$
• Sturm bound: $360$
• Trace bound: $13$

Defining parameters

Level:	\( N \)	\(=\)	\( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	900.e (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 12 \)
Character field:			\(\Q\)
Newform subspaces:			\( 7 \)
Sturm bound:			\(360\)
Trace bound:			\(13\)
Distinguishing \(T_p\):			\(7\), \(13\)

Dimensions

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

	Total	New	Old
Modular forms	204	38	166
Cusp forms	156	38	118
Eisenstein series	48	0	48

Trace form

38 * q + 4 * q^4 - 8 * q^13 - 16 * q^16 + 16 * q^22 + 32 * q^28 - 4 * q^34 + 12 * q^37 + 40 * q^46 - 6 * q^49 - 8 * q^52 - 52 * q^58 + 36 * q^61 - 8 * q^64 - 16 * q^73 - 72 * q^76 + 4 * q^82 + 64 * q^88 + 8 * q^94 - 64 * q^97

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.e.a	$900$	$2$	900.e	12.b	$2$	$2$	$2$	$7.187$	\(\Q(\sqrt{-2}) \)	\(\Q(\sqrt{-1}) \)					180.2.h.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q-\beta q^{2}-2q^{4}+2\beta q^{8}-6q^{13}+4q^{16}+\cdots\)
900.2.e.b	$900$	$2$	900.e	12.b	$2$	$2$	$2$	$7.187$	\(\Q(\sqrt{-2}) \)	\(\Q(\sqrt{-1}) \)					36.2.b.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q+\beta q^{2}-2q^{4}-2\beta q^{8}+4q^{13}+4q^{16}+\cdots\)
900.2.e.c	$900$	$2$	900.e	12.b	$2$	$2$	$2$	$7.187$	\(\Q(\sqrt{-2}) \)	\(\Q(\sqrt{-1}) \)					180.2.h.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q-\beta q^{2}-2q^{4}+2\beta q^{8}+6q^{13}+4q^{16}+\cdots\)
900.2.e.d	$900$	$2$	900.e	12.b	$2$	$8$	$8$	$7.187$	8.0.18939904.2	None					180.2.e.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{3}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{5}q^{2}+(1-\beta _{3}-\beta _{4}-\beta _{7})q^{4}+(-2+\cdots)q^{7}+\cdots\)
900.2.e.e	$900$	$2$	900.e	12.b	$2$	$8$	$8$	$7.187$	8.0.64000000.1	None		✓			900.2.e.e	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{6}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{4}q^{2}-\beta _{2}q^{4}+(-1+\beta _{1}-\beta _{2}+\cdots)q^{7}+\cdots\)
900.2.e.f	$900$	$2$	900.e	12.b	$2$	$8$	$8$	$7.187$	8.0.207360000.2	None					180.2.h.b	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{5}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{2}+(1-\beta _{5})q^{4}+(\beta _{2}+\beta _{7})q^{7}+\cdots\)
900.2.e.g	$900$	$2$	900.e	12.b	$2$	$8$	$8$	$7.187$	8.0.64000000.1	None		✓			900.2.e.e	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{6}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{1}q^{2}+\beta _{4}q^{4}+(-\beta _{4}+\beta _{7})q^{7}+\cdots\)

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

\( S_{2}^{\mathrm{old}}(900, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(36, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(60, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(180, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(300, [\chi])\)\(^{\oplus 2}\)