Space of modular forms of level 1800, weight 2, and Character 1800.k

• Label: 1800.2.k
• Level: $1800$
• Weight: $2$
• Character orbit: 1800.k
• Rep. character: $\chi_{1800}(901,\cdot)$
• Character field: $\Q$
• Dimension: $92$
• Newform subspaces: $21$
• Sturm bound: $720$
• Trace bound: $17$

Defining parameters

Level:	\( N \)	\(=\)	\( 1800 = 2^{3} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1800.k (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 8 \)
Character field:			\(\Q\)
Newform subspaces:			\( 21 \)
Sturm bound:			\(720\)
Trace bound:			\(17\)
Distinguishing \(T_p\):			\(7\), \(11\), \(17\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(1800, [\chi])\).

	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 + 4 * q^7 - 6 * q^16 - 4 * q^17 - 4 * q^22 - 12 * q^23 + 12 * q^28 + 8 * q^31 + 20 * q^32 - 34 * q^34 - 12 * q^38 + 4 * q^41 + 50 * q^44 + 32 * q^46 - 4 * q^47 + 68 * q^49 + 40 * q^52 - 8 * q^56 - 28 * q^58 + 36 * q^62 - 22 * q^64 + 8 * q^68 + 40 * q^71 + 20 * q^74 + 38 * q^76 + 8 * q^79 + 4 * q^82 + 40 * q^86 + 16 * q^88 - 20 * q^89 - 60 * q^92 + 80 * q^94 - 16 * q^97 + 12 * q^98

Decomposition of \(S_{2}^{\mathrm{new}}(1800, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
1800.2.k.a	$1800$	$2$	1800.k	8.b	$2$	$2$	$2$	$14.373$	\(\Q(\sqrt{-1}) \)	None					24.2.d.a	$2$	$0$	\(-2\)	\(0\)	\(0\)	\(4\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-i-1)q^{2}+2 i q^{4}+2 q^{7}+(-2 i+2)q^{8}+\cdots\)
1800.2.k.b	$1800$	$2$	1800.k	8.b	$2$	$2$	$2$	$14.373$	\(\Q(\sqrt{-1}) \)	None					200.2.d.a	$2$	$0$	\(-2\)	\(0\)	\(0\)	\(4\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-i-1)q^{2}+2 i q^{4}+2 q^{7}+(-2 i+2)q^{8}+\cdots\)
1800.2.k.c	$1800$	$2$	1800.k	8.b	$2$	$2$	$2$	$14.373$	\(\Q(\sqrt{-1}) \)	None					360.2.k.a	$2$	$0$	\(-2\)	\(0\)	\(0\)	\(8\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-i-1)q^{2}+2 i q^{4}+4 q^{7}+(-2 i+2)q^{8}+\cdots\)
1800.2.k.d	$1800$	$2$	1800.k	8.b	$2$	$2$	$2$	$14.373$	\(\Q(\sqrt{-7}) \)	None					200.2.d.b	$2$	$0$	\(-1\)	\(0\)	\(0\)	\(8\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-1+\beta )q^{2}+(-1-\beta )q^{4}+4q^{7}+\cdots\)
1800.2.k.e	$1800$	$2$	1800.k	8.b	$2$	$2$	$2$	$14.373$	\(\Q(\sqrt{-2}) \)	\(\Q(\sqrt{-6}) \)					72.2.d.a	$4$	$1$	\(0\)	\(0\)	\(0\)	\(-4\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q-\beta q^{2}-2q^{4}-2q^{7}+2\beta q^{8}-4\beta q^{11}+\cdots\)
1800.2.k.f	$1800$	$2$	1800.k	8.b	$2$	$2$	$2$	$14.373$	\(\Q(\sqrt{-7}) \)	None					200.2.d.b	$2$	$0$	\(1\)	\(0\)	\(0\)	\(-8\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta q^{2}+(-2+\beta )q^{4}-4q^{7}+(-2+\cdots)q^{8}+\cdots\)
1800.2.k.g	$1800$	$2$	1800.k	8.b	$2$	$2$	$2$	$14.373$	\(\Q(\sqrt{-1}) \)	None					120.2.k.a	$2$	$0$	\(2\)	\(0\)	\(0\)	\(-4\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-i+1)q^{2}-2 i q^{4}-2 q^{7}+(-2 i-2)q^{8}+\cdots\)
1800.2.k.h	$1800$	$2$	1800.k	8.b	$2$	$2$	$2$	$14.373$	\(\Q(\sqrt{-1}) \)	None					200.2.d.a	$2$	$0$	\(2\)	\(0\)	\(0\)	\(-4\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+(i+1)q^{2}+2 i q^{4}-2 q^{7}+(2 i-2)q^{8}+\cdots\)
1800.2.k.i	$1800$	$2$	1800.k	8.b	$2$	$2$	$2$	$14.373$	\(\Q(\sqrt{-1}) \)	None					360.2.k.a	$2$	$0$	\(2\)	\(0\)	\(0\)	\(8\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-i+1)q^{2}-2 i q^{4}+4 q^{7}+(-2 i-2)q^{8}+\cdots\)
1800.2.k.j	$1800$	$2$	1800.k	8.b	$2$	$4$	$4$	$14.373$	\(\Q(\zeta_{12})\)	None					40.2.d.a	$2$	$0$	\(-2\)	\(0\)	\(0\)	\(4\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+(\beta_{2}-\beta_1)q^{2}+(\beta_{3}+\beta_1)q^{4}+(-\beta_{3}+\beta_{2}-\beta_1+1)q^{7}+\cdots\)
1800.2.k.k	$1800$	$2$	1800.k	8.b	$2$	$4$	$4$	$14.373$	\(\Q(\sqrt{-2}, \sqrt{-3})\)	\(\Q(\sqrt{-6}) \)					360.2.d.d	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}$	$\mathrm{U}(1)[D_{2}]$	\(q-\beta _{1}q^{2}-2q^{4}-\beta _{3}q^{7}+2\beta _{1}q^{8}+\cdots\)
1800.2.k.l	$1800$	$2$	1800.k	8.b	$2$	$4$	$4$	$14.373$	\(\Q(\sqrt{3}, \sqrt{-5})\)	\(\Q(\sqrt{-15}) \)					360.2.d.c	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{U}(1)[D_{2}]$	\(q-\beta q^{2}+\beta ^{2}q^{4}-\beta ^{3}q^{8}+(-4-\beta ^{2}+\cdots)q^{16}+\cdots\)
1800.2.k.m	$1800$	$2$	1800.k	8.b	$2$	$4$	$4$	$14.373$	\(\Q(\sqrt{-2}, \sqrt{-3})\)	None					40.2.f.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{1}q^{2}+(1+\beta _{3})q^{4}+(-2\beta _{1}+\beta _{2}+\cdots)q^{7}+\cdots\)
1800.2.k.n	$1800$	$2$	1800.k	8.b	$2$	$4$	$4$	$14.373$	\(\Q(i, \sqrt{7})\)	None					360.2.k.d	$4$	$0$	\(0\)	\(0\)	\(0\)	\(-8\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{2}+(1+\beta _{2})q^{4}-2q^{7}+(\beta _{1}+\beta _{3})q^{8}+\cdots\)
1800.2.k.o	$1800$	$2$	1800.k	8.b	$2$	$4$	$4$	$14.373$	\(\Q(\sqrt{2}, \sqrt{-5})\)	\(\Q(\sqrt{-30}) \)					360.2.d.a	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{3}$	$\mathrm{U}(1)[D_{2}]$	\(q+\beta _{1}q^{2}+2q^{4}+2\beta _{1}q^{8}+\beta _{3}q^{11}+\cdots\)
1800.2.k.p	$1800$	$2$	1800.k	8.b	$2$	$6$	$6$	$14.373$	6.0.399424.1	None					120.2.k.b	$2$	$0$	\(2\)	\(0\)	\(0\)	\(-4\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{1}q^{2}+\beta _{2}q^{4}+(-1-\beta _{1}+\beta _{2}+\cdots)q^{7}+\cdots\)
1800.2.k.q	$1800$	$2$	1800.k	8.b	$2$	$8$	$8$	$14.373$	8.0.214798336.3	None					600.2.k.d	$2$	$0$	\(-2\)	\(0\)	\(0\)	\(-8\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{3}q^{2}+(-\beta _{4}-\beta _{5}+\beta _{6})q^{4}+(-1+\cdots)q^{7}+\cdots\)
1800.2.k.r	$1800$	$2$	1800.k	8.b	$2$	$8$	$8$	$14.373$	8.0.\(\cdots\).29	None		✓			1800.2.k.r	$4$	$0$	\(0\)	\(0\)	\(0\)	\(-8\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{2}+\beta _{2}q^{4}+(-1-\beta _{2}+\beta _{5}+\cdots)q^{7}+\cdots\)
1800.2.k.s	$1800$	$2$	1800.k	8.b	$2$	$8$	$8$	$14.373$	8.0.\(\cdots\).29	None		✓			1800.2.k.r	$4$	$0$	\(0\)	\(0\)	\(0\)	\(8\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{6}q^{2}-\beta _{5}q^{4}+(1+\beta _{2}-\beta _{5})q^{7}+\cdots\)
1800.2.k.t	$1800$	$2$	1800.k	8.b	$2$	$8$	$8$	$14.373$	8.0.214798336.3	None					600.2.k.d	$2$	$0$	\(2\)	\(0\)	\(0\)	\(8\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{2}q^{2}+(1+\beta _{1}-\beta _{3}-\beta _{6})q^{4}+(1+\cdots)q^{7}+\cdots\)
1800.2.k.u	$1800$	$2$	1800.k	8.b	$2$	$12$	$12$	$14.373$	12.0.\(\cdots\).1	None			✓		120.2.d.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{10}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{2}+\beta _{4}q^{4}+(-\beta _{1}-\beta _{8})q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(1800, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(1800, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(24, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(40, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(72, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(120, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(200, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(360, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(600, [\chi])\)\(^{\oplus 2}\)