Space of modular forms of level 3330, weight 2, and Character 3330.d

• Label: 3330.2.d
• Level: $3330$
• Weight: $2$
• Character orbit: 3330.d
• Rep. character: $\chi_{3330}(1999,\cdot)$
• Character field: $\Q$
• Dimension: $90$
• Newform subspaces: $18$
• Sturm bound: $1368$
• Trace bound: $14$

Defining parameters

Level:	\( N \)	\(=\)	\( 3330 = 2 \cdot 3^{2} \cdot 5 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3330.d (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 5 \)
Character field:			\(\Q\)
Newform subspaces:			\( 18 \)
Sturm bound:			\(1368\)
Trace bound:			\(14\)
Distinguishing \(T_p\):			\(7\), \(11\), \(17\), \(29\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(3330, [\chi])\).

	Total	New	Old
Modular forms	700	90	610
Cusp forms	668	90	578
Eisenstein series	32	0	32

Trace form

90 * q - 90 * q^4 - 4 * q^5 + 4 * q^10 + 16 * q^11 + 4 * q^14 + 90 * q^16 + 12 * q^19 + 4 * q^20 - 8 * q^25 - 16 * q^29 - 8 * q^31 - 4 * q^40 + 4 * q^41 - 16 * q^44 - 12 * q^46 - 78 * q^49 - 8 * q^50 + 20 * q^55 - 4 * q^56 - 4 * q^59 - 90 * q^64 - 24 * q^65 - 8 * q^71 + 10 * q^74 - 12 * q^76 + 32 * q^79 - 4 * q^80 - 12 * q^85 + 52 * q^86 + 60 * q^89 - 112 * q^91 + 28 * q^94 - 4 * q^95

Decomposition of \(S_{2}^{\mathrm{new}}(3330, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
3330.2.d.a	$3330$	$2$	3330.d	5.b	$2$	$2$	$2$	$26.590$	\(\Q(\sqrt{-1}) \)	None					1110.2.d.e	$2$	$0$	\(0\)	\(0\)	\(-4\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+i q^{2}-q^{4}+(i-2)q^{5}+2 i q^{7}+\cdots\)
3330.2.d.b	$3330$	$2$	3330.d	5.b	$2$	$2$	$2$	$26.590$	\(\Q(\sqrt{-1}) \)	None					1110.2.d.d	$2$	$1$	\(0\)	\(0\)	\(-4\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-i q^{2}-q^{4}+(i-2)q^{5}+2 i q^{7}+\cdots\)
3330.2.d.c	$3330$	$2$	3330.d	5.b	$2$	$2$	$2$	$26.590$	\(\Q(\sqrt{-1}) \)	None					370.2.b.b	$2$	$1$	\(0\)	\(0\)	\(-2\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-i q^{2}-q^{4}+(-2 i-1)q^{5}+i q^{7}+\cdots\)
3330.2.d.d	$3330$	$2$	3330.d	5.b	$2$	$2$	$2$	$26.590$	\(\Q(\sqrt{-1}) \)	None					1110.2.d.c	$2$	$0$	\(0\)	\(0\)	\(-2\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-i q^{2}-q^{4}+(2 i-1)q^{5}+i q^{7}+\cdots\)
3330.2.d.e	$3330$	$2$	3330.d	5.b	$2$	$2$	$2$	$26.590$	\(\Q(\sqrt{-1}) \)	None					1110.2.d.b	$2$	$1$	\(0\)	\(0\)	\(2\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+i q^{2}-q^{4}+(-2 i+1)q^{5}+5 i q^{7}+\cdots\)
3330.2.d.f	$3330$	$2$	3330.d	5.b	$2$	$2$	$2$	$26.590$	\(\Q(\sqrt{-1}) \)	None					370.2.b.a	$2$	$0$	\(0\)	\(0\)	\(4\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+i q^{2}-q^{4}+(-i+2)q^{5}+2 i q^{7}+\cdots\)
3330.2.d.g	$3330$	$2$	3330.d	5.b	$2$	$2$	$2$	$26.590$	\(\Q(\sqrt{-1}) \)	None					1110.2.d.a	$2$	$0$	\(0\)	\(0\)	\(4\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-i q^{2}-q^{4}+(i+2)q^{5}+2 i q^{7}+\cdots\)
3330.2.d.h	$3330$	$2$	3330.d	5.b	$2$	$4$	$4$	$26.590$	\(\Q(i, \sqrt{6})\)	None					1110.2.d.h	$2$	$0$	\(0\)	\(0\)	\(-4\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{2}q^{2}-q^{4}+(-1-\beta _{2}-\beta _{3})q^{5}+\cdots\)
3330.2.d.i	$3330$	$2$	3330.d	5.b	$2$	$4$	$4$	$26.590$	\(\Q(\zeta_{12})\)	None		✓			3330.2.d.i	$2$	$0$	\(0\)	\(0\)	\(-4\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta_1 q^{2}-q^{4}+(-2\beta_1-1)q^{5}+\cdots\)
3330.2.d.j	$3330$	$2$	3330.d	5.b	$2$	$4$	$4$	$26.590$	\(\Q(i, \sqrt{5})\)	None					1110.2.d.f	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{2}-q^{4}-\beta _{2}q^{5}-2\beta _{1}q^{7}-\beta _{1}q^{8}+\cdots\)
3330.2.d.k	$3330$	$2$	3330.d	5.b	$2$	$4$	$4$	$26.590$	\(\Q(\zeta_{8})\)	None					1110.2.d.g	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-\zeta_{8}^{2}q^{2}-q^{4}+(\zeta_{8}+2\zeta_{8}^{3})q^{5}+(\zeta_{8}+\cdots)q^{7}+\cdots\)
3330.2.d.l	$3330$	$2$	3330.d	5.b	$2$	$4$	$4$	$26.590$	\(\Q(\zeta_{12})\)	None		✓			3330.2.d.i	$2$	$0$	\(0\)	\(0\)	\(4\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta_1 q^{2}-q^{4}+(2\beta_1+1)q^{5}+(-2\beta_{2}+\beta_1)q^{7}+\cdots\)
3330.2.d.m	$3330$	$2$	3330.d	5.b	$2$	$4$	$4$	$26.590$	\(\Q(i, \sqrt{6})\)	None					370.2.b.c	$2$	$0$	\(0\)	\(0\)	\(8\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{1}q^{2}-q^{4}+(2-\beta _{1})q^{5}+(2\beta _{1}-\beta _{2}+\cdots)q^{7}+\cdots\)
3330.2.d.n	$3330$	$2$	3330.d	5.b	$2$	$6$	$6$	$26.590$	6.0.5161984.1	None					1110.2.d.i	$2$	$0$	\(0\)	\(0\)	\(-2\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{4}q^{2}-q^{4}+(\beta _{1}+\beta _{3})q^{5}+(-\beta _{1}+\cdots)q^{7}+\cdots\)
3330.2.d.o	$3330$	$2$	3330.d	5.b	$2$	$8$	$8$	$26.590$	8.0.\(\cdots\).2	None					1110.2.d.j	$2$	$0$	\(0\)	\(0\)	\(2\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{3}q^{2}-q^{4}+(-\beta _{1}-\beta _{5})q^{5}+(-\beta _{2}+\cdots)q^{7}+\cdots\)
3330.2.d.p	$3330$	$2$	3330.d	5.b	$2$	$10$	$10$	$26.590$	10.0.\(\cdots\).1	None					370.2.b.d	$2$	$0$	\(0\)	\(0\)	\(-6\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{2}q^{2}-q^{4}+(-1+\beta _{1}+\beta _{3}-\beta _{4}+\cdots)q^{5}+\cdots\)
3330.2.d.q	$3330$	$2$	3330.d	5.b	$2$	$14$	$14$	$26.590$	\(\mathbb{Q}[x]/(x^{14} + \cdots)\)	None		✓			3330.2.d.q	$2$	$0$	\(0\)	\(0\)	\(-4\)	\(0\)	$2^{6}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{6}q^{2}-q^{4}+\beta _{9}q^{5}+(\beta _{6}-\beta _{7})q^{7}+\cdots\)
3330.2.d.r	$3330$	$2$	3330.d	5.b	$2$	$14$	$14$	$26.590$	\(\mathbb{Q}[x]/(x^{14} + \cdots)\)	None		✓			3330.2.d.q	$2$	$0$	\(0\)	\(0\)	\(4\)	\(0\)	$2^{6}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{6}q^{2}-q^{4}-\beta _{9}q^{5}+(\beta _{6}-\beta _{7})q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(3330, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(3330, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(30, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(45, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(90, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(185, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(370, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(555, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(1110, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(1665, [\chi])\)\(^{\oplus 2}\)