Space of modular forms of level 3420, weight 2, and Character 3420.t

• Label: 3420.2.t
• Level: $3420$
• Weight: $2$
• Character orbit: 3420.t
• Rep. character: $\chi_{3420}(1261,\cdot)$
• Character field: $\Q(\zeta_{3})$
• Dimension: $64$
• Newform subspaces: $23$
• Sturm bound: $1440$
• Trace bound: $13$

Defining parameters

Level:	\( N \)	\(=\)	\( 3420 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3420.t (of order \(3\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 19 \)
Character field:			\(\Q(\zeta_{3})\)
Newform subspaces:			\( 23 \)
Sturm bound:			\(1440\)
Trace bound:			\(13\)
Distinguishing \(T_p\):			\(7\), \(11\), \(13\), \(17\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(3420, [\chi])\).

	Total	New	Old
Modular forms	1488	64	1424
Cusp forms	1392	64	1328
Eisenstein series	96	0	96

Trace form

\( 64 q - 4 q^{7} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(3420, [\chi])\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
3420.2.t.a	$3420$	$2$	3420.t	19.c	$3$	$2$	$1$	$27.309$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(0\)	\(-1\)	\(-10\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-1+\zeta_{6})q^{5}-5q^{7}-6q^{11}+\zeta_{6}q^{13}+\cdots\)
3420.2.t.b	$3420$	$2$	3420.t	19.c	$3$	$2$	$1$	$27.309$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(0\)	\(-1\)	\(-8\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-1+\zeta_{6})q^{5}-4q^{7}+3q^{11}-6\zeta_{6}q^{13}+\cdots\)
3420.2.t.c	$3420$	$2$	3420.t	19.c	$3$	$2$	$1$	$27.309$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(0\)	\(-1\)	\(-6\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-1+\zeta_{6})q^{5}-3q^{7}+2q^{11}+\zeta_{6}q^{13}+\cdots\)
3420.2.t.d	$3420$	$2$	3420.t	19.c	$3$	$2$	$1$	$27.309$	\(\Q(\sqrt{-3}) \)	None		$2$	$1$	\(0\)	\(0\)	\(-1\)	\(-4\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-1+\zeta_{6})q^{5}-2q^{7}-2q^{11}+4\zeta_{6}q^{13}+\cdots\)
3420.2.t.e	$3420$	$2$	3420.t	19.c	$3$	$2$	$1$	$27.309$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(0\)	\(-1\)	\(4\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-1+\zeta_{6})q^{5}+2q^{7}-3q^{11}+6\zeta_{6}q^{13}+\cdots\)
3420.2.t.f	$3420$	$2$	3420.t	19.c	$3$	$2$	$1$	$27.309$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(0\)	\(-1\)	\(4\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-1+\zeta_{6})q^{5}+2q^{7}+q^{11}-6\zeta_{6}q^{13}+\cdots\)
3420.2.t.g	$3420$	$2$	3420.t	19.c	$3$	$2$	$1$	$27.309$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(0\)	\(-1\)	\(4\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-1+\zeta_{6})q^{5}+2q^{7}+6q^{11}+(5+\cdots)q^{17}+\cdots\)
3420.2.t.h	$3420$	$2$	3420.t	19.c	$3$	$2$	$1$	$27.309$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(0\)	\(-1\)	\(4\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-1+\zeta_{6})q^{5}+2q^{7}+6q^{11}+4\zeta_{6}q^{13}+\cdots\)
3420.2.t.i	$3420$	$2$	3420.t	19.c	$3$	$2$	$1$	$27.309$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(0\)	\(-1\)	\(8\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-1+\zeta_{6})q^{5}+4q^{7}+2q^{11}-6\zeta_{6}q^{13}+\cdots\)
3420.2.t.j	$3420$	$2$	3420.t	19.c	$3$	$2$	$1$	$27.309$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(0\)	\(1\)	\(-4\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1-\zeta_{6})q^{5}-2q^{7}-3q^{11}+(-4+\cdots)q^{17}+\cdots\)
3420.2.t.k	$3420$	$2$	3420.t	19.c	$3$	$2$	$1$	$27.309$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(0\)	\(1\)	\(-4\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1-\zeta_{6})q^{5}-2q^{7}+q^{11}-4\zeta_{6}q^{13}+\cdots\)
3420.2.t.l	$3420$	$2$	3420.t	19.c	$3$	$2$	$1$	$27.309$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(0\)	\(1\)	\(-4\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1-\zeta_{6})q^{5}-2q^{7}+2q^{11}+(1-\zeta_{6})q^{17}+\cdots\)
3420.2.t.m	$3420$	$2$	3420.t	19.c	$3$	$2$	$1$	$27.309$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(0\)	\(1\)	\(-4\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1-\zeta_{6})q^{5}-2q^{7}+2q^{11}+4\zeta_{6}q^{13}+\cdots\)
3420.2.t.n	$3420$	$2$	3420.t	19.c	$3$	$2$	$1$	$27.309$	\(\Q(\sqrt{-3}) \)	None		$2$	$1$	\(0\)	\(0\)	\(1\)	\(-2\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1-\zeta_{6})q^{5}-q^{7}-2q^{11}-3\zeta_{6}q^{13}+\cdots\)
3420.2.t.o	$3420$	$2$	3420.t	19.c	$3$	$2$	$1$	$27.309$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(0\)	\(1\)	\(2\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1-\zeta_{6})q^{5}+q^{7}+6q^{11}-3\zeta_{6}q^{13}+\cdots\)
3420.2.t.p	$3420$	$2$	3420.t	19.c	$3$	$2$	$1$	$27.309$	\(\Q(\sqrt{-3}) \)	None		$2$	$1$	\(0\)	\(0\)	\(1\)	\(4\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1-\zeta_{6})q^{5}+2q^{7}-6q^{11}+(-5+\cdots)q^{17}+\cdots\)
3420.2.t.q	$3420$	$2$	3420.t	19.c	$3$	$2$	$1$	$27.309$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(0\)	\(1\)	\(8\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1-\zeta_{6})q^{5}+4q^{7}-2q^{11}+2\zeta_{6}q^{13}+\cdots\)
3420.2.t.r	$3420$	$2$	3420.t	19.c	$3$	$4$	$2$	$27.309$	\(\Q(\sqrt{-3}, \sqrt{5})\)	None		$2$	$0$	\(0\)	\(0\)	\(-2\)	\(-8\)	$2^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-1-\beta _{1})q^{5}+(-2-\beta _{3})q^{7}-2q^{11}+\cdots\)
3420.2.t.s	$3420$	$2$	3420.t	19.c	$3$	$4$	$2$	$27.309$	\(\Q(\sqrt{-3}, \sqrt{-5})\)	None		$2$	$0$	\(0\)	\(0\)	\(-2\)	\(8\)	$3$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-1+\beta _{1})q^{5}+2q^{7}-\beta _{3}q^{11}+(\beta _{1}+\cdots)q^{13}+\cdots\)
3420.2.t.t	$3420$	$2$	3420.t	19.c	$3$	$4$	$2$	$27.309$	\(\Q(\sqrt{-3}, \sqrt{5})\)	None		$2$	$0$	\(0\)	\(0\)	\(2\)	\(-8\)	$2^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1+\beta _{1})q^{5}+(-2-\beta _{3})q^{7}+2q^{11}+\cdots\)
3420.2.t.u	$3420$	$2$	3420.t	19.c	$3$	$4$	$2$	$27.309$	\(\Q(\sqrt{-3}, \sqrt{-5})\)	None		$2$	$0$	\(0\)	\(0\)	\(2\)	\(8\)	$3$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1-\beta _{1})q^{5}+2q^{7}-\beta _{3}q^{11}+(\beta _{1}+\cdots)q^{13}+\cdots\)
3420.2.t.v	$3420$	$2$	3420.t	19.c	$3$	$6$	$3$	$27.309$	6.0.1783323.2	None		$2$	$0$	\(0\)	\(0\)	\(-3\)	\(4\)	$3$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-1-\beta _{4})q^{5}+(1-\beta _{3})q^{7}+(-2+\cdots)q^{11}+\cdots\)
3420.2.t.w	$3420$	$2$	3420.t	19.c	$3$	$8$	$4$	$27.309$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None		$2$	$0$	\(0\)	\(0\)	\(4\)	\(0\)	$3$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1+\beta _{4})q^{5}+\beta _{7}q^{7}+(-1-2\beta _{3}+\cdots)q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(3420, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(3420, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(38, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(57, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(76, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(114, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(171, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(228, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(342, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(684, [\chi])\)\(^{\oplus 2}\)