Space of modular forms of level 900, weight 3, and Character 900.c

• Label: 900.3.c
• Level: $900$
• Weight: $3$
• Character orbit: 900.c
• Rep. character: $\chi_{900}(451,\cdot)$
• Character field: $\Q$
• Dimension: $92$
• Newform subspaces: $21$
• Sturm bound: $540$
• Trace bound: $13$

Defining parameters

Level:	\( N \)	\(=\)	\( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	900.c (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 4 \)
Character field:			\(\Q\)
Newform subspaces:			\( 21 \)
Sturm bound:			\(540\)
Trace bound:			\(13\)
Distinguishing \(T_p\):			\(7\), \(13\), \(17\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(900, [\chi])\).

	Total	New	Old
Modular forms	384	98	286
Cusp forms	336	92	244
Eisenstein series	48	6	42

Trace form

92 * q + 2 * q^4 - 12 * q^8 - 24 * q^14 - 22 * q^16 - 4 * q^17 - 8 * q^22 - 44 * q^26 + 56 * q^28 - 20 * q^29 + 20 * q^32 - 6 * q^34 - 88 * q^37 + 112 * q^38 + 4 * q^41 - 90 * q^44 + 84 * q^46 - 556 * q^49 - 72 * q^52 + 28 * q^53 + 420 * q^56 - 56 * q^58 + 48 * q^61 - 112 * q^62 - 118 * q^64 - 360 * q^68 + 88 * q^73 - 92 * q^74 - 78 * q^76 + 48 * q^77 + 72 * q^82 - 276 * q^86 + 352 * q^88 + 220 * q^89 + 216 * q^92 - 120 * q^94 - 152 * q^97 + 760 * q^98

Decomposition of \(S_{3}^{\mathrm{new}}(900, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
900.3.c.a	$900$	$3$	900.c	4.b	$2$	$1$	$1$	$24.523$	\(\Q\)	\(\Q(\sqrt{-1}) \)	✓				20.3.d.c	$2$	$0$	\(-2\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q-2q^{2}+4q^{4}-8q^{8}-24q^{13}+2^{4}q^{16}+\cdots\)
900.3.c.b	$900$	$3$	900.c	4.b	$2$	$1$	$1$	$24.523$	\(\Q\)	\(\Q(\sqrt{-1}) \)	✓				36.3.d.a	$2$	$0$	\(-2\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q-2q^{2}+4q^{4}-8q^{8}+10q^{13}+2^{4}q^{16}+\cdots\)
900.3.c.c	$900$	$3$	900.c	4.b	$2$	$1$	$1$	$24.523$	\(\Q\)	\(\Q(\sqrt{-1}) \)	✓				36.3.d.a	$2$	$0$	\(2\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q+2q^{2}+4q^{4}+8q^{8}+10q^{13}+2^{4}q^{16}+\cdots\)
900.3.c.d	$900$	$3$	900.c	4.b	$2$	$1$	$1$	$24.523$	\(\Q\)	\(\Q(\sqrt{-1}) \)	✓				20.3.d.c	$2$	$0$	\(2\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q+2q^{2}+4q^{4}+8q^{8}+24q^{13}+2^{4}q^{16}+\cdots\)
900.3.c.e	$900$	$3$	900.c	4.b	$2$	$2$	$2$	$24.523$	\(\Q(\sqrt{-3}) \)	None					12.3.d.a	$2$	$0$	\(-2\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+(\beta-1)q^{2}+(-2\beta-2)q^{4}-4\beta q^{7}+\cdots\)
900.3.c.f	$900$	$3$	900.c	4.b	$2$	$2$	$2$	$24.523$	\(\Q(\sqrt{-3}) \)	None					60.3.f.a	$2$	$0$	\(-2\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-\beta-1)q^{2}+(2\beta-2)q^{4}-6\beta q^{7}+\cdots\)
900.3.c.g	$900$	$3$	900.c	4.b	$2$	$2$	$2$	$24.523$	\(\Q(\sqrt{-15}) \)	\(\Q(\sqrt{-15}) \)					180.3.f.f	$4$	$0$	\(-1\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q-\beta q^{2}+(-4+\beta )q^{4}+(4+3\beta )q^{8}+\cdots\)
900.3.c.h	$900$	$3$	900.c	4.b	$2$	$2$	$2$	$24.523$	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-5}) \)					20.3.d.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{U}(1)[D_{2}]$	\(q-\beta q^{2}-4 q^{4}+2\beta q^{7}+4\beta q^{8}+\cdots\)
900.3.c.i	$900$	$3$	900.c	4.b	$2$	$2$	$2$	$24.523$	\(\Q(\sqrt{-15}) \)	\(\Q(\sqrt{-15}) \)					180.3.f.f	$4$	$0$	\(1\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q+(1-\beta )q^{2}+(-3-\beta )q^{4}+(-7+3\beta )q^{8}+\cdots\)
900.3.c.j	$900$	$3$	900.c	4.b	$2$	$2$	$2$	$24.523$	\(\Q(\sqrt{-3}) \)	None					60.3.f.a	$2$	$0$	\(2\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-\beta+1)q^{2}+(-2\beta-2)q^{4}-6\beta q^{7}+\cdots\)
900.3.c.k	$900$	$3$	900.c	4.b	$2$	$4$	$4$	$24.523$	\(\Q(\zeta_{10})\)	None					20.3.b.a	$2$	$0$	\(-2\)	\(0\)	\(0\)	\(0\)	$2^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta_1 q^{2}+(\beta_{3}+\beta_{2}+\beta_1-1)q^{4}+\cdots\)
900.3.c.l	$900$	$3$	900.c	4.b	$2$	$4$	$4$	$24.523$	4.0.8405.1	None					100.3.b.d	$2$	$0$	\(-1\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{2}q^{2}+(1-\beta _{1}-\beta _{2}-\beta _{3})q^{4}+(2\beta _{1}+\cdots)q^{7}+\cdots\)
900.3.c.m	$900$	$3$	900.c	4.b	$2$	$4$	$4$	$24.523$	4.0.8405.1	None					100.3.b.d	$2$	$0$	\(1\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{2}q^{2}+(1+\beta _{1}+\beta _{2}-\beta _{3})q^{4}+(2\beta _{1}+\cdots)q^{7}+\cdots\)
900.3.c.n	$900$	$3$	900.c	4.b	$2$	$8$	$8$	$24.523$	8.0.4069419264.1	None					300.3.c.e	$2$	$0$	\(-2\)	\(0\)	\(0\)	\(0\)	$2^{8}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{4}q^{2}+(-1+\beta _{7})q^{4}+(-\beta _{5}-\beta _{6}+\cdots)q^{7}+\cdots\)
900.3.c.o	$900$	$3$	900.c	4.b	$2$	$8$	$8$	$24.523$	8.0.\(\cdots\).1	None					180.3.c.c	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{12}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{2}+(-2+\beta _{3})q^{4}+(-\beta _{3}-\beta _{7})q^{7}+\cdots\)
900.3.c.p	$900$	$3$	900.c	4.b	$2$	$8$	$8$	$24.523$	8.0.\(\cdots\).1	None		✓			900.3.c.p	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{8}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{6}q^{2}+(1+\beta _{7})q^{4}-\beta _{2}q^{7}+(-\beta _{5}+\cdots)q^{8}+\cdots\)
900.3.c.q	$900$	$3$	900.c	4.b	$2$	$8$	$8$	$24.523$	8.0.\(\cdots\).1	None		✓			900.3.c.p	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{8}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{1}q^{2}+\beta _{2}q^{4}-\beta _{4}q^{7}-\beta _{3}q^{8}+\cdots\)
900.3.c.r	$900$	$3$	900.c	4.b	$2$	$8$	$8$	$24.523$	8.0.6080256576.2	None					60.3.f.b	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{12}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{2}q^{2}+(1+\beta _{7})q^{4}+(\beta _{2}+\beta _{4})q^{7}+\cdots\)
900.3.c.s	$900$	$3$	900.c	4.b	$2$	$8$	$8$	$24.523$	8.0.\(\cdots\).9	None					180.3.f.d	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{12}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(\beta _{1}-\beta _{2})q^{2}+(2-\beta _{5})q^{4}-\beta _{3}q^{7}+\cdots\)
900.3.c.t	$900$	$3$	900.c	4.b	$2$	$8$	$8$	$24.523$	8.0.4069419264.1	None					300.3.c.e	$2$	$0$	\(2\)	\(0\)	\(0\)	\(0\)	$2^{8}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{2}q^{2}+(-1+\beta _{2}-\beta _{3}+\beta _{5})q^{4}+\cdots\)
900.3.c.u	$900$	$3$	900.c	4.b	$2$	$8$	$8$	$24.523$	8.0.85100625.1	None					60.3.c.a	$2$	$0$	\(4\)	\(0\)	\(0\)	\(0\)	$2^{12}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(1-\beta _{5})q^{2}+(1-\beta _{4})q^{4}+(-1-\beta _{3}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{3}^{\mathrm{old}}(900, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(900, [\chi]) \simeq \) \(S_{3}^{\mathrm{new}}(12, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(20, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(36, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(60, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(100, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(180, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(300, [\chi])\)\(^{\oplus 2}\)