Space of modular forms of level 1216, weight 2, and Character 1216.i

• Label: 1216.2.i
• Level: $1216$
• Weight: $2$
• Character orbit: 1216.i
• Rep. character: $\chi_{1216}(577,\cdot)$
• Character field: $\Q(\zeta_{3})$
• Dimension: $76$
• Newform subspaces: $18$
• Sturm bound: $320$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 1216 = 2^{6} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1216.i (of order \(3\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 19 \)
Character field:			\(\Q(\zeta_{3})\)
Newform subspaces:			\( 18 \)
Sturm bound:			\(320\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(3\), \(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(1216, [\chi])\).

	Total	New	Old
Modular forms	344	84	260
Cusp forms	296	76	220
Eisenstein series	48	8	40

Trace form

\( 76 q + 2 q^{5} - 36 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(1216, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
1216.2.i.a	$1216$	$2$	1216.i	19.c	$3$	$2$	$1$	$9.710$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(-3\)	\(2\)	\(4\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-3+3\zeta_{6})q^{3}+(2-2\zeta_{6})q^{5}+2q^{7}+\cdots\)
1216.2.i.b	$1216$	$2$	1216.i	19.c	$3$	$2$	$1$	$9.710$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(-1\)	\(-4\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-1+\zeta_{6})q^{3}+(-4+4\zeta_{6})q^{5}+2\zeta_{6}q^{9}+\cdots\)
1216.2.i.c	$1216$	$2$	1216.i	19.c	$3$	$2$	$1$	$9.710$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(-1\)	\(-1\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-1+\zeta_{6})q^{3}+(-1+\zeta_{6})q^{5}+2\zeta_{6}q^{9}+\cdots\)
1216.2.i.d	$1216$	$2$	1216.i	19.c	$3$	$2$	$1$	$9.710$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(-1\)	\(0\)	\(8\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-1+\zeta_{6})q^{3}+4q^{7}+2\zeta_{6}q^{9}+3q^{11}+\cdots\)
1216.2.i.e	$1216$	$2$	1216.i	19.c	$3$	$2$	$1$	$9.710$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(-1\)	\(3\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-1+\zeta_{6})q^{3}+(3-3\zeta_{6})q^{5}+2\zeta_{6}q^{9}+\cdots\)
1216.2.i.f	$1216$	$2$	1216.i	19.c	$3$	$2$	$1$	$9.710$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(1\)	\(-4\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1-\zeta_{6})q^{3}+(-4+4\zeta_{6})q^{5}+2\zeta_{6}q^{9}+\cdots\)
1216.2.i.g	$1216$	$2$	1216.i	19.c	$3$	$2$	$1$	$9.710$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(1\)	\(-1\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1-\zeta_{6})q^{3}+(-1+\zeta_{6})q^{5}+2\zeta_{6}q^{9}+\cdots\)
1216.2.i.h	$1216$	$2$	1216.i	19.c	$3$	$2$	$1$	$9.710$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(1\)	\(0\)	\(-8\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1-\zeta_{6})q^{3}-4q^{7}+2\zeta_{6}q^{9}-3q^{11}+\cdots\)
1216.2.i.i	$1216$	$2$	1216.i	19.c	$3$	$2$	$1$	$9.710$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(1\)	\(3\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1-\zeta_{6})q^{3}+(3-3\zeta_{6})q^{5}+2\zeta_{6}q^{9}+\cdots\)
1216.2.i.j	$1216$	$2$	1216.i	19.c	$3$	$2$	$1$	$9.710$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(3\)	\(2\)	\(-4\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(3-3\zeta_{6})q^{3}+(2-2\zeta_{6})q^{5}-2q^{7}+\cdots\)
1216.2.i.k	$1216$	$2$	1216.i	19.c	$3$	$4$	$2$	$9.710$	\(\Q(\sqrt{-3}, \sqrt{7})\)	None		$2$	$0$	\(0\)	\(0\)	\(2\)	\(-4\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+\beta _{1}q^{3}+(1+\beta _{1}+\beta _{2})q^{5}+(-1-\beta _{3})q^{7}+\cdots\)
1216.2.i.l	$1216$	$2$	1216.i	19.c	$3$	$4$	$2$	$9.710$	\(\Q(\sqrt{-3}, \sqrt{7})\)	None		$2$	$0$	\(0\)	\(0\)	\(2\)	\(4\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+\beta _{1}q^{3}+(1-\beta _{1}+\beta _{2})q^{5}+(1-\beta _{3})q^{7}+\cdots\)
1216.2.i.m	$1216$	$2$	1216.i	19.c	$3$	$6$	$3$	$9.710$	6.0.2696112.1	None		$2$	$0$	\(0\)	\(-1\)	\(1\)	\(4\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-\beta _{1}+\beta _{5})q^{3}+\beta _{1}q^{5}+(1+\beta _{2}+\cdots)q^{7}+\cdots\)
1216.2.i.n	$1216$	$2$	1216.i	19.c	$3$	$6$	$3$	$9.710$	6.0.2696112.1	None		$2$	$0$	\(0\)	\(1\)	\(1\)	\(-4\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-\beta _{1}-\beta _{2}-\beta _{3}+\beta _{5})q^{3}+(-\beta _{1}+\cdots)q^{5}+\cdots\)
1216.2.i.o	$1216$	$2$	1216.i	19.c	$3$	$8$	$4$	$9.710$	8.0.\(\cdots\).5	None		$2$	$0$	\(0\)	\(-4\)	\(-2\)	\(-8\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-1-\beta _{2}+\beta _{4}-\beta _{6})q^{3}+(-1-\beta _{1}+\cdots)q^{5}+\cdots\)
1216.2.i.p	$1216$	$2$	1216.i	19.c	$3$	$8$	$4$	$9.710$	8.0.\(\cdots\).1	None		$4$	$0$	\(0\)	\(0\)	\(-2\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(\beta _{2}-\beta _{4})q^{3}-\beta _{1}q^{5}+(-2+2\beta _{1}+\cdots)q^{9}+\cdots\)
1216.2.i.q	$1216$	$2$	1216.i	19.c	$3$	$8$	$4$	$9.710$	8.0.\(\cdots\).5	None		$2$	$0$	\(0\)	\(4\)	\(-2\)	\(8\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-\beta _{2}+\beta _{4})q^{3}+(-\beta _{1}-\beta _{2})q^{5}+\cdots\)
1216.2.i.r	$1216$	$2$	1216.i	19.c	$3$	$12$	$6$	$9.710$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(2\)	\(0\)	$2^{4}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(\beta _{5}-\beta _{8})q^{3}+\beta _{10}q^{5}+(\beta _{2}-\beta _{5}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(1216, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(1216, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(38, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(76, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(152, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(304, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(608, [\chi])\)\(^{\oplus 2}\)