Space of modular forms of level 1216, weight 3, and Character 1216.e

• Label: 1216.3.e
• Level: $1216$
• Weight: $3$
• Character orbit: 1216.e
• Rep. character: $\chi_{1216}(1025,\cdot)$
• Character field: $\Q$
• Dimension: $78$
• Newform subspaces: $16$
• Sturm bound: $480$
• Trace bound: $17$

Defining parameters

Level:	\( N \)	\(=\)	\( 1216 = 2^{6} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1216.e (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 19 \)
Character field:			\(\Q\)
Newform subspaces:			\( 16 \)
Sturm bound:			\(480\)
Trace bound:			\(17\)
Distinguishing \(T_p\):			\(3\), \(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(1216, [\chi])\).

	Total	New	Old
Modular forms	332	82	250
Cusp forms	308	78	230
Eisenstein series	24	4	20

Trace form

\( 78 q + 4 q^{5} - 226 q^{9} + O(q^{10}) \)

Decomposition of \(S_{3}^{\mathrm{new}}(1216, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
1216.3.e.a	$1216$	$3$	1216.e	19.b	$2$	$1$	$1$	$33.134$	\(\Q\)	\(\Q(\sqrt{-19}) \)	✓	$2$	$0$	\(0\)	\(0\)	\(9\)	\(-5\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q+9q^{5}-5q^{7}+9q^{9}-3q^{11}+15q^{17}+\cdots\)
1216.3.e.b	$1216$	$3$	1216.e	19.b	$2$	$1$	$1$	$33.134$	\(\Q\)	\(\Q(\sqrt{-19}) \)	✓	$2$	$0$	\(0\)	\(0\)	\(9\)	\(5\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q+9q^{5}+5q^{7}+9q^{9}+3q^{11}+15q^{17}+\cdots\)
1216.3.e.c	$1216$	$3$	1216.e	19.b	$2$	$2$	$2$	$33.134$	\(\Q(\sqrt{-2}) \)	None		$2$	$0$	\(0\)	\(0\)	\(-14\)	\(-22\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta q^{3}-7q^{5}-11q^{7}-23q^{9}+3q^{11}+\cdots\)
1216.3.e.d	$1216$	$3$	1216.e	19.b	$2$	$2$	$2$	$33.134$	\(\Q(\sqrt{-2}) \)	None		$2$	$0$	\(0\)	\(0\)	\(-14\)	\(22\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta q^{3}-7q^{5}+11q^{7}-23q^{9}-3q^{11}+\cdots\)
1216.3.e.e	$1216$	$3$	1216.e	19.b	$2$	$2$	$2$	$33.134$	\(\Q(\sqrt{57}) \)	\(\Q(\sqrt{-19}) \)	✓	$2$	$0$	\(0\)	\(0\)	\(-9\)	\(-5\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q+(-4-\beta )q^{5}+(-4+3\beta )q^{7}+9q^{9}+\cdots\)
1216.3.e.f	$1216$	$3$	1216.e	19.b	$2$	$2$	$2$	$33.134$	\(\Q(\sqrt{57}) \)	\(\Q(\sqrt{-19}) \)	✓	$2$	$0$	\(0\)	\(0\)	\(-9\)	\(5\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q+(-4-\beta )q^{5}+(4-3\beta )q^{7}+9q^{9}+\cdots\)
1216.3.e.g	$1216$	$3$	1216.e	19.b	$2$	$2$	$2$	$33.134$	\(\Q(\sqrt{-13}) \)	None		$2$	$0$	\(0\)	\(0\)	\(-8\)	\(-10\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta q^{3}-4q^{5}-5q^{7}-4q^{9}+10q^{11}+\cdots\)
1216.3.e.h	$1216$	$3$	1216.e	19.b	$2$	$2$	$2$	$33.134$	\(\Q(\sqrt{-13}) \)	None		$2$	$0$	\(0\)	\(0\)	\(-8\)	\(10\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta q^{3}-4q^{5}+5q^{7}-4q^{9}-10q^{11}+\cdots\)
1216.3.e.i	$1216$	$3$	1216.e	19.b	$2$	$2$	$2$	$33.134$	\(\Q(\sqrt{-2}) \)	None		$2$	$0$	\(0\)	\(0\)	\(2\)	\(-10\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta q^{3}+q^{5}-5q^{7}+q^{9}+5q^{11}+\cdots\)
1216.3.e.j	$1216$	$3$	1216.e	19.b	$2$	$2$	$2$	$33.134$	\(\Q(\sqrt{-2}) \)	None		$2$	$0$	\(0\)	\(0\)	\(2\)	\(10\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta q^{3}+q^{5}+5q^{7}+q^{9}-5q^{11}+\cdots\)
1216.3.e.k	$1216$	$3$	1216.e	19.b	$2$	$2$	$2$	$33.134$	\(\Q(\sqrt{-29}) \)	None		$2$	$0$	\(0\)	\(0\)	\(8\)	\(-2\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta q^{3}+4q^{5}-q^{7}-20q^{9}-14q^{11}+\cdots\)
1216.3.e.l	$1216$	$3$	1216.e	19.b	$2$	$2$	$2$	$33.134$	\(\Q(\sqrt{-29}) \)	None		$2$	$0$	\(0\)	\(0\)	\(8\)	\(2\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta q^{3}+4q^{5}+q^{7}-20q^{9}+14q^{11}+\cdots\)
1216.3.e.m	$1216$	$3$	1216.e	19.b	$2$	$8$	$8$	$33.134$	\(\mathbb{Q}[x]/(x^{8} + \cdots)\)	None		$2$	$0$	\(0\)	\(0\)	\(14\)	\(-6\)	$2^{5}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{3}+(2+\beta _{4})q^{5}+(-1+\beta _{3})q^{7}+\cdots\)
1216.3.e.n	$1216$	$3$	1216.e	19.b	$2$	$8$	$8$	$33.134$	\(\mathbb{Q}[x]/(x^{8} + \cdots)\)	None		$2$	$0$	\(0\)	\(0\)	\(14\)	\(6\)	$2^{5}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{3}+(2+\beta _{4})q^{5}+(1-\beta _{3})q^{7}+\cdots\)
1216.3.e.o	$1216$	$3$	1216.e	19.b	$2$	$20$	$20$	$33.134$	\(\mathbb{Q}[x]/(x^{20} + \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{27}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{11}q^{3}+\beta _{2}q^{5}+\beta _{3}q^{7}+(-3+\beta _{5}+\cdots)q^{9}+\cdots\)
1216.3.e.p	$1216$	$3$	1216.e	19.b	$2$	$20$	$20$	$33.134$	\(\mathbb{Q}[x]/(x^{20} - \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{27}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{7}q^{3}-\beta _{6}q^{5}+\beta _{1}q^{7}+(-3+\beta _{4}+\cdots)q^{9}+\cdots\)

Decomposition of \(S_{3}^{\mathrm{old}}(1216, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(1216, [\chi]) \cong \) \(S_{3}^{\mathrm{new}}(19, [\chi])\)\(^{\oplus 7}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(38, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(76, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(152, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(304, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(608, [\chi])\)\(^{\oplus 2}\)