Space of modular forms of level 900, weight 5, and Character 900.l

• Label: 900.5.l
• Level: $900$
• Weight: $5$
• Character orbit: 900.l
• Rep. character: $\chi_{900}(757,\cdot)$
• Character field: $\Q(\zeta_{4})$
• Dimension: $60$
• Newform subspaces: $11$
• Sturm bound: $900$
• Trace bound: $31$

Defining parameters

Level:	\( N \)	\(=\)	\( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	900.l (of order \(4\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 5 \)
Character field:			\(\Q(i)\)
Newform subspaces:			\( 11 \)
Sturm bound:			\(900\)
Trace bound:			\(31\)
Distinguishing \(T_p\):			\(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{5}(900, [\chi])\).

	Total	New	Old
Modular forms	1512	60	1452
Cusp forms	1368	60	1308
Eisenstein series	144	0	144

Trace form

60 * q + 50 * q^7 - 24 * q^11 - 60 * q^17 + 510 * q^23 - 948 * q^31 - 60 * q^37 + 3288 * q^41 - 870 * q^43 + 2550 * q^47 + 5880 * q^53 - 52 * q^61 - 5350 * q^67 + 4764 * q^71 - 8540 * q^73 + 18780 * q^77 + 32670 * q^83 + 6276 * q^91 - 17100 * q^97

Decomposition of \(S_{5}^{\mathrm{new}}(900, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
900.5.l.a	$900$	$5$	900.l	5.c	$4$	$4$	$2$	$93.033$	\(\Q(i, \sqrt{241})\)	None					20.5.f.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-110\)	$2$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-26+26\beta _{1}+3\beta _{3})q^{7}+(80-5\beta _{1}+\cdots)q^{11}+\cdots\)
900.5.l.b	$900$	$5$	900.l	5.c	$4$	$4$	$2$	$93.033$	\(\Q(i, \sqrt{69})\)	None					100.5.f.b	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{3}$	$\mathrm{SU}(2)[C_{4}]$	\(q-7\beta _{3}q^{7}-180q^{11}+4\beta _{2}q^{13}-6^{2}\beta _{3}q^{17}+\cdots\)
900.5.l.c	$900$	$5$	900.l	5.c	$4$	$4$	$2$	$93.033$	\(\Q(i, \sqrt{6})\)	None					100.5.f.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+12\beta _{1}q^{7}-45q^{11}-134\beta _{3}q^{13}+\cdots\)
900.5.l.d	$900$	$5$	900.l	5.c	$4$	$4$	$2$	$93.033$	\(\Q(i, \sqrt{6})\)	None					300.5.k.b	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+\beta _{1}q^{7}-6q^{11}-35\beta _{3}q^{13}-6\beta _{1}q^{17}+\cdots\)
900.5.l.e	$900$	$5$	900.l	5.c	$4$	$4$	$2$	$93.033$	\(\Q(i, \sqrt{6})\)	\(\Q(\sqrt{-3}) \)		✓			900.5.l.e	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{4}]$	\(q+39\beta _{1}q^{7}-161\beta _{3}q^{13}-601\beta _{2}q^{19}+\cdots\)
900.5.l.f	$900$	$5$	900.l	5.c	$4$	$4$	$2$	$93.033$	\(\Q(i, \sqrt{6})\)	\(\Q(\sqrt{-3}) \)		✓			900.5.l.f	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{8}$	$\mathrm{U}(1)[D_{4}]$	\(q+\beta _{1}q^{7}+11\beta _{3}q^{13}-46\beta _{2}q^{19}+\cdots\)
900.5.l.g	$900$	$5$	900.l	5.c	$4$	$4$	$2$	$93.033$	\(\Q(i, \sqrt{6})\)	None					300.5.k.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{4}]$	\(q+7\beta _{1}q^{7}+114q^{11}-5^{2}\beta _{3}q^{13}+\cdots\)
900.5.l.h	$900$	$5$	900.l	5.c	$4$	$8$	$4$	$93.033$	\(\Q(i, \sqrt{6}, \sqrt{58})\)	None		✓			900.5.l.h	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{10}\cdot 3^{4}\cdot 5^{4}$	$\mathrm{SU}(2)[C_{4}]$	\(q+19\beta _{1}q^{7}+\beta _{5}q^{11}-71\beta _{3}q^{13}+\cdots\)
900.5.l.i	$900$	$5$	900.l	5.c	$4$	$8$	$4$	$93.033$	\(\Q(i, \sqrt{6}, \sqrt{22})\)	None					300.5.k.c	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{10}\cdot 3^{2}\cdot 5^{4}$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-\beta _{1}-\beta _{7})q^{7}+(54-\beta _{5})q^{11}+(-75\beta _{3}+\cdots)q^{13}+\cdots\)
900.5.l.j	$900$	$5$	900.l	5.c	$4$	$8$	$4$	$93.033$	\(\mathbb{Q}[x]/(x^{8} + \cdots)\)	None					180.5.l.c	$4$	$0$	\(0\)	\(0\)	\(0\)	\(20\)	$2^{5}\cdot 3^{4}\cdot 5^{4}$	$\mathrm{SU}(2)[C_{4}]$	\(q+(2-2\beta _{1}-\beta _{6})q^{7}+\beta _{2}q^{11}+(-8+\cdots)q^{13}+\cdots\)
900.5.l.k	$900$	$5$	900.l	5.c	$4$	$8$	$4$	$93.033$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None					60.5.k.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(140\)	$2^{6}\cdot 3^{2}\cdot 5^{4}$	$\mathrm{SU}(2)[C_{4}]$	\(q+(18-17\beta _{1}+\beta _{2}-\beta _{5})q^{7}+(-33+\cdots)q^{11}+\cdots\)

Decomposition of \(S_{5}^{\mathrm{old}}(900, [\chi])\) into lower level spaces

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