Space of modular forms of level 4212, weight 2, and Character 4212.i

• Label: 4212.2.i
• Level: $4212$
• Weight: $2$
• Character orbit: 4212.i
• Rep. character: $\chi_{4212}(1405,\cdot)$
• Character field: $\Q(\zeta_{3})$
• Dimension: $96$
• Newform subspaces: $25$
• Sturm bound: $1512$
• Trace bound: $17$

Defining parameters

Level:	\( N \)	\(=\)	\( 4212 = 2^{2} \cdot 3^{4} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4212.i (of order \(3\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 9 \)
Character field:			\(\Q(\zeta_{3})\)
Newform subspaces:			\( 25 \)
Sturm bound:			\(1512\)
Trace bound:			\(17\)
Distinguishing \(T_p\):			\(5\), \(7\), \(17\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(4212, [\chi])\).

	Total	New	Old
Modular forms	1584	96	1488
Cusp forms	1440	96	1344
Eisenstein series	144	0	144

Trace form

\( 96 q - 24 q^{19} - 60 q^{25} - 12 q^{31} + 24 q^{37} + 12 q^{43} - 24 q^{49} + 144 q^{55} + 24 q^{61} - 24 q^{73} - 60 q^{79}+O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(4212, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
4212.2.i.a	$4212$	$2$	4212.i	9.c	$3$	$2$	$1$	$33.633$	\(\Q(\sqrt{-3}) \)	None					468.2.a.a	$2$	$0$	\(0\)	\(0\)	\(-4\)	\(-4\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q-4\zeta_{6}q^{5}+(-4+4\zeta_{6})q^{7}+(4-4\zeta_{6})q^{11}+\cdots\)
4212.2.i.b	$4212$	$2$	4212.i	9.c	$3$	$2$	$1$	$33.633$	\(\Q(\sqrt{-3}) \)	None					156.2.a.a	$2$	$0$	\(0\)	\(0\)	\(-4\)	\(2\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q-4\zeta_{6}q^{5}+(2-2\zeta_{6})q^{7}+(-4+4\zeta_{6})q^{11}+\cdots\)
4212.2.i.c	$4212$	$2$	4212.i	9.c	$3$	$2$	$1$	$33.633$	\(\Q(\sqrt{-3}) \)	None					1404.2.a.a	$2$	$1$	\(0\)	\(0\)	\(-3\)	\(-2\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q-3\zeta_{6}q^{5}+(-2+2\zeta_{6})q^{7}-\zeta_{6}q^{13}+\cdots\)
4212.2.i.d	$4212$	$2$	4212.i	9.c	$3$	$2$	$1$	$33.633$	\(\Q(\sqrt{-3}) \)	None					52.2.a.a	$2$	$0$	\(0\)	\(0\)	\(-2\)	\(2\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q-2\zeta_{6}q^{5}+(2-2\zeta_{6})q^{7}+(2-2\zeta_{6})q^{11}+\cdots\)
4212.2.i.e	$4212$	$2$	4212.i	9.c	$3$	$2$	$1$	$33.633$	\(\Q(\sqrt{-3}) \)	None					1404.2.a.b	$2$	$0$	\(0\)	\(0\)	\(-1\)	\(2\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q-\zeta_{6}q^{5}+(2-2\zeta_{6})q^{7}+(4-4\zeta_{6})q^{11}+\cdots\)
4212.2.i.f	$4212$	$2$	4212.i	9.c	$3$	$2$	$1$	$33.633$	\(\Q(\sqrt{-3}) \)	None					156.2.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-2\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-2+2\zeta_{6})q^{7}-\zeta_{6}q^{13}-6q^{17}+\cdots\)
4212.2.i.g	$4212$	$2$	4212.i	9.c	$3$	$2$	$1$	$33.633$	\(\Q(\sqrt{-3}) \)	None					156.2.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-2\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-2+2\zeta_{6})q^{7}-\zeta_{6}q^{13}+6q^{17}+\cdots\)
4212.2.i.h	$4212$	$2$	4212.i	9.c	$3$	$2$	$1$	$33.633$	\(\Q(\sqrt{-3}) \)	None					1404.2.a.b	$2$	$0$	\(0\)	\(0\)	\(1\)	\(2\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+\zeta_{6}q^{5}+(2-2\zeta_{6})q^{7}+(-4+4\zeta_{6})q^{11}+\cdots\)
4212.2.i.i	$4212$	$2$	4212.i	9.c	$3$	$2$	$1$	$33.633$	\(\Q(\sqrt{-3}) \)	None					52.2.a.a	$2$	$1$	\(0\)	\(0\)	\(2\)	\(2\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+2\zeta_{6}q^{5}+(2-2\zeta_{6})q^{7}+(-2+2\zeta_{6})q^{11}+\cdots\)
4212.2.i.j	$4212$	$2$	4212.i	9.c	$3$	$2$	$1$	$33.633$	\(\Q(\sqrt{-3}) \)	None					1404.2.a.a	$2$	$0$	\(0\)	\(0\)	\(3\)	\(-2\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+3\zeta_{6}q^{5}+(-2+2\zeta_{6})q^{7}-\zeta_{6}q^{13}+\cdots\)
4212.2.i.k	$4212$	$2$	4212.i	9.c	$3$	$2$	$1$	$33.633$	\(\Q(\sqrt{-3}) \)	None					468.2.a.a	$2$	$0$	\(0\)	\(0\)	\(4\)	\(-4\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+4\zeta_{6}q^{5}+(-4+4\zeta_{6})q^{7}+(-4+4\zeta_{6})q^{11}+\cdots\)
4212.2.i.l	$4212$	$2$	4212.i	9.c	$3$	$2$	$1$	$33.633$	\(\Q(\sqrt{-3}) \)	None					156.2.a.a	$2$	$0$	\(0\)	\(0\)	\(4\)	\(2\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+4\zeta_{6}q^{5}+(2-2\zeta_{6})q^{7}+(4-4\zeta_{6})q^{11}+\cdots\)
4212.2.i.m	$4212$	$2$	4212.i	9.c	$3$	$4$	$2$	$33.633$	\(\Q(\sqrt{-3}, \sqrt{13})\)	None					1404.2.a.e	$2$	$0$	\(0\)	\(0\)	\(-3\)	\(-2\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-1-\beta _{1}-\beta _{2})q^{5}+\beta _{2}q^{7}+(1-\beta _{1}+\cdots)q^{11}+\cdots\)
4212.2.i.n	$4212$	$2$	4212.i	9.c	$3$	$4$	$2$	$33.633$	\(\Q(\sqrt{-3}, \sqrt{13})\)	None					1404.2.a.f	$2$	$0$	\(0\)	\(0\)	\(-1\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q-\beta _{1}q^{5}+(-2+2\beta _{1}-\beta _{2}+2\beta _{3})q^{7}+\cdots\)
4212.2.i.o	$4212$	$2$	4212.i	9.c	$3$	$4$	$2$	$33.633$	\(\Q(\sqrt{-3}, \sqrt{13})\)	None		✓			4212.2.a.c	$2$	$0$	\(0\)	\(0\)	\(-1\)	\(3\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q-\beta _{1}q^{5}+(1-\beta _{1}-\beta _{2}-\beta _{3})q^{7}+(-1+\cdots)q^{11}+\cdots\)
4212.2.i.p	$4212$	$2$	4212.i	9.c	$3$	$4$	$2$	$33.633$	\(\Q(\zeta_{12})\)	None		✓			4212.2.a.d	$4$	$0$	\(0\)	\(0\)	\(0\)	\(2\)	$3$	$\mathrm{SU}(2)[C_{3}]$	\(q-2\beta_{2} q^{5}+(-\beta_1+1)q^{7}+(-3\beta_{3}+3\beta_{2})q^{11}+\cdots\)
4212.2.i.q	$4212$	$2$	4212.i	9.c	$3$	$4$	$2$	$33.633$	\(\Q(\sqrt{-3}, \sqrt{7})\)	None					1404.2.a.h	$4$	$0$	\(0\)	\(0\)	\(0\)	\(4\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+\beta _{1}q^{5}-2\beta _{2}q^{7}+(2\beta _{1}+2\beta _{3})q^{11}+\cdots\)
4212.2.i.r	$4212$	$2$	4212.i	9.c	$3$	$4$	$2$	$33.633$	\(\Q(\sqrt{-3}, \sqrt{7})\)	None					1404.2.a.g	$4$	$0$	\(0\)	\(0\)	\(0\)	\(4\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+\beta _{1}q^{5}-2\beta _{2}q^{7}+(-2\beta _{1}-2\beta _{3})q^{11}+\cdots\)
4212.2.i.s	$4212$	$2$	4212.i	9.c	$3$	$4$	$2$	$33.633$	\(\Q(\sqrt{-3}, \sqrt{13})\)	None					1404.2.a.f	$2$	$0$	\(0\)	\(0\)	\(1\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+\beta _{1}q^{5}+(-2+2\beta _{1}-\beta _{2}+2\beta _{3})q^{7}+\cdots\)
4212.2.i.t	$4212$	$2$	4212.i	9.c	$3$	$4$	$2$	$33.633$	\(\Q(\sqrt{-3}, \sqrt{13})\)	None		✓			4212.2.a.c	$2$	$0$	\(0\)	\(0\)	\(1\)	\(3\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+\beta _{1}q^{5}+(1-\beta _{1}-\beta _{2}-\beta _{3})q^{7}+(1+\cdots)q^{11}+\cdots\)
4212.2.i.u	$4212$	$2$	4212.i	9.c	$3$	$4$	$2$	$33.633$	\(\Q(\sqrt{-3}, \sqrt{13})\)	None					1404.2.a.e	$2$	$0$	\(0\)	\(0\)	\(3\)	\(-2\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1+\beta _{1}+\beta _{2})q^{5}+\beta _{2}q^{7}+(-1+\beta _{1}+\cdots)q^{11}+\cdots\)
4212.2.i.v	$4212$	$2$	4212.i	9.c	$3$	$8$	$4$	$33.633$	8.0.\(\cdots\).1	None		✓			4212.2.a.f	$2$	$0$	\(0\)	\(0\)	\(-3\)	\(-1\)	$3^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-\beta _{1}+\beta _{3}+\beta _{4})q^{5}+(-1+\beta _{1}+\cdots)q^{7}+\cdots\)
4212.2.i.w	$4212$	$2$	4212.i	9.c	$3$	$8$	$4$	$33.633$	8.0.49787136.1	None		✓			4212.2.a.g	$4$	$0$	\(0\)	\(0\)	\(0\)	\(2\)	$3^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q+\beta _{3}q^{5}-\beta _{4}q^{7}+\beta _{2}q^{11}+(1-\beta _{5}+\cdots)q^{13}+\cdots\)
4212.2.i.x	$4212$	$2$	4212.i	9.c	$3$	$8$	$4$	$33.633$	8.0.\(\cdots\).1	None		✓			4212.2.a.f	$2$	$0$	\(0\)	\(0\)	\(3\)	\(-1\)	$3^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(\beta _{1}-\beta _{3}-\beta _{4})q^{5}+(-1+\beta _{1}-\beta _{3}+\cdots)q^{7}+\cdots\)
4212.2.i.y	$4212$	$2$	4212.i	9.c	$3$	$12$	$6$	$33.633$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None		✓	✓		4212.2.a.l	$4$	$0$	\(0\)	\(0\)	\(0\)	\(-8\)	$3^{3}$	$\mathrm{SU}(2)[C_{3}]$	\(q+\beta _{2}q^{5}+(2\beta _{4}+\beta _{6}-\beta _{7})q^{7}+(\beta _{3}+\cdots)q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(4212, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(4212, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(18, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(36, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(54, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(81, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(108, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(117, [\chi])\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(162, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(234, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(324, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(351, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(468, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(702, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(1053, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(1404, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(2106, [\chi])\)\(^{\oplus 2}\)