Space of modular forms of level 1216, weight 4, and trivial character

• Label: 1216.4.a
• Level: $1216$
• Weight: $4$
• Character orbit: 1216.a
• Rep. character: $\chi_{1216}(1,\cdot)$
• Character field: $\Q$
• Dimension: $108$
• Newform subspaces: $34$
• Sturm bound: $640$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 1216 = 2^{6} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1216.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 34 \)
Sturm bound:			\(640\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(3\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(1216))\).

	Total	New	Old
Modular forms	492	108	384
Cusp forms	468	108	360
Eisenstein series	24	0	24

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(2\)	\(19\)	Fricke	Dim.
\(+\)	\(+\)	\(+\)	\(28\)
\(+\)	\(-\)	\(-\)	\(25\)
\(-\)	\(+\)	\(-\)	\(26\)
\(-\)	\(-\)	\(+\)	\(29\)
Plus space		\(+\)	\(57\)
Minus space		\(-\)	\(51\)

Trace form

\( 108 q + 972 q^{9} + O(q^{10}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(1216))\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Inner twists	Rank*	Traces					A-L signs		Fricke sign	Coefficient ring index	Sato-Tate	$q$-expansion
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		2	19
1216.4.a.a	$1216$	$4$	1216.a	1.a	$1$	$1$	$1$	$71.746$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-5\)	\(12\)	\(-11\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-5q^{3}+12q^{5}-11q^{7}-2q^{9}-54q^{11}+\cdots\)
1216.4.a.b	$1216$	$4$	1216.a	1.a	$1$	$1$	$1$	$71.746$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-2\)	\(9\)	\(31\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}+9q^{5}+31q^{7}-23q^{9}+57q^{11}+\cdots\)
1216.4.a.c	$1216$	$4$	1216.a	1.a	$1$	$1$	$1$	$71.746$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-1\)	\(8\)	\(17\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+8q^{5}+17q^{7}-26q^{9}-70q^{11}+\cdots\)
1216.4.a.d	$1216$	$4$	1216.a	1.a	$1$	$1$	$1$	$71.746$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(1\)	\(8\)	\(-17\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+8q^{5}-17q^{7}-26q^{9}+70q^{11}+\cdots\)
1216.4.a.e	$1216$	$4$	1216.a	1.a	$1$	$1$	$1$	$71.746$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(2\)	\(9\)	\(-31\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}+9q^{5}-31q^{7}-23q^{9}-57q^{11}+\cdots\)
1216.4.a.f	$1216$	$4$	1216.a	1.a	$1$	$1$	$1$	$71.746$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(5\)	\(12\)	\(11\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+5q^{3}+12q^{5}+11q^{7}-2q^{9}+54q^{11}+\cdots\)
1216.4.a.g	$1216$	$4$	1216.a	1.a	$1$	$2$	$2$	$71.746$	\(\Q(\sqrt{73}) \)	None	✓	$1$	$0$	\(0\)	\(-9\)	\(9\)	\(-18\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-4-\beta )q^{3}+(3+3\beta )q^{5}+(-7-4\beta )q^{7}+\cdots\)
1216.4.a.h	$1216$	$4$	1216.a	1.a	$1$	$2$	$2$	$71.746$	\(\Q(\sqrt{33}) \)	None	✓	$1$	$1$	\(0\)	\(-5\)	\(5\)	\(30\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-2-\beta )q^{3}+(5-5\beta )q^{5}+(13+4\beta )q^{7}+\cdots\)
1216.4.a.i	$1216$	$4$	1216.a	1.a	$1$	$2$	$2$	$71.746$	\(\Q(\sqrt{93}) \)	None	✓	$1$	$1$	\(0\)	\(-2\)	\(4\)	\(-44\)		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q-q^{3}+(2+2\beta )q^{5}+(-22-\beta )q^{7}+\cdots\)
1216.4.a.j	$1216$	$4$	1216.a	1.a	$1$	$2$	$2$	$71.746$	\(\Q(\sqrt{177}) \)	None	✓	$1$	$1$	\(0\)	\(-1\)	\(-10\)	\(57\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{3}+(-6+2\beta )q^{5}+(28+\beta )q^{7}+\cdots\)
1216.4.a.k	$1216$	$4$	1216.a	1.a	$1$	$2$	$2$	$71.746$	\(\Q(\sqrt{57}) \)	None	✓	$1$	$1$	\(0\)	\(-1\)	\(5\)	\(6\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{3}+(3-\beta )q^{5}+(5-4\beta )q^{7}+(-13+\cdots)q^{9}+\cdots\)
1216.4.a.l	$1216$	$4$	1216.a	1.a	$1$	$2$	$2$	$71.746$	\(\Q(\sqrt{177}) \)	None	✓	$1$	$1$	\(0\)	\(1\)	\(-10\)	\(-57\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+(-6+2\beta )q^{5}+(-28-\beta )q^{7}+\cdots\)
1216.4.a.m	$1216$	$4$	1216.a	1.a	$1$	$2$	$2$	$71.746$	\(\Q(\sqrt{57}) \)	None	✓	$1$	$1$	\(0\)	\(1\)	\(5\)	\(-6\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+(3-\beta )q^{5}+(-5+4\beta )q^{7}+\cdots\)
1216.4.a.n	$1216$	$4$	1216.a	1.a	$1$	$2$	$2$	$71.746$	\(\Q(\sqrt{93}) \)	None	✓	$1$	$0$	\(0\)	\(2\)	\(4\)	\(44\)		$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+q^{3}+(2+2\beta )q^{5}+(22+\beta )q^{7}-26q^{9}+\cdots\)
1216.4.a.o	$1216$	$4$	1216.a	1.a	$1$	$2$	$2$	$71.746$	\(\Q(\sqrt{33}) \)	None	✓	$1$	$1$	\(0\)	\(5\)	\(5\)	\(-30\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(3-\beta )q^{3}+5\beta q^{5}+(-17+4\beta )q^{7}+\cdots\)
1216.4.a.p	$1216$	$4$	1216.a	1.a	$1$	$2$	$2$	$71.746$	\(\Q(\sqrt{73}) \)	None	✓	$1$	$0$	\(0\)	\(9\)	\(9\)	\(18\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(5-\beta )q^{3}+(6-3\beta )q^{5}+(11-4\beta )q^{7}+\cdots\)
1216.4.a.q	$1216$	$4$	1216.a	1.a	$1$	$3$	$3$	$71.746$	3.3.3221.1	None	✓	$1$	$0$	\(0\)	\(-5\)	\(-2\)	\(35\)		$-$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(-2-\beta _{2})q^{3}+(-1+\beta _{1}-2\beta _{2})q^{5}+\cdots\)
1216.4.a.r	$1216$	$4$	1216.a	1.a	$1$	$3$	$3$	$71.746$	3.3.7057.1	None	✓	$1$	$1$	\(0\)	\(-4\)	\(-7\)	\(7\)		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{2})q^{3}+(-2-2\beta _{1}-\beta _{2})q^{5}+\cdots\)
1216.4.a.s	$1216$	$4$	1216.a	1.a	$1$	$3$	$3$	$71.746$	3.3.3144.1	None	✓	$1$	$1$	\(0\)	\(-1\)	\(-14\)	\(-35\)		$+$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q-\beta _{2}q^{3}+(-4+\beta _{1}-\beta _{2})q^{5}+(-11+\cdots)q^{7}+\cdots\)
1216.4.a.t	$1216$	$4$	1216.a	1.a	$1$	$3$	$3$	$71.746$	3.3.35529.1	None	✓	$1$	$0$	\(0\)	\(-1\)	\(-9\)	\(44\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-\beta _{1}-\beta _{2})q^{3}+(-3-\beta _{2})q^{5}+(15+\cdots)q^{7}+\cdots\)
1216.4.a.u	$1216$	$4$	1216.a	1.a	$1$	$3$	$3$	$71.746$	3.3.3144.1	None	✓	$1$	$1$	\(0\)	\(1\)	\(-14\)	\(35\)		$-$	$+$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{2}q^{3}+(-4+\beta _{1}-\beta _{2})q^{5}+(11+\cdots)q^{7}+\cdots\)
1216.4.a.v	$1216$	$4$	1216.a	1.a	$1$	$3$	$3$	$71.746$	3.3.35529.1	None	✓	$1$	$0$	\(0\)	\(1\)	\(-9\)	\(-44\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(\beta _{1}+\beta _{2})q^{3}+(-3-\beta _{2})q^{5}+(-15+\cdots)q^{7}+\cdots\)
1216.4.a.w	$1216$	$4$	1216.a	1.a	$1$	$3$	$3$	$71.746$	3.3.7057.1	None	✓	$1$	$1$	\(0\)	\(4\)	\(-7\)	\(-7\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(1+\beta _{2})q^{3}+(-2-2\beta _{1}-\beta _{2})q^{5}+\cdots\)
1216.4.a.x	$1216$	$4$	1216.a	1.a	$1$	$3$	$3$	$71.746$	3.3.3221.1	None	✓	$1$	$0$	\(0\)	\(5\)	\(-2\)	\(-35\)		$+$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(2+\beta _{2})q^{3}+(-1+\beta _{1}-2\beta _{2})q^{5}+\cdots\)
1216.4.a.y	$1216$	$4$	1216.a	1.a	$1$	$5$	$5$	$71.746$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(-6\)	\(-5\)	\(-7\)		$+$	$-$	$-$	$2^{4}$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{2})q^{3}+(-1+\beta _{2}-\beta _{3})q^{5}+\cdots\)
1216.4.a.z	$1216$	$4$	1216.a	1.a	$1$	$5$	$5$	$71.746$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(-3\)	\(-27\)	\(-20\)		$+$	$+$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{3})q^{3}+(-5+\beta _{1})q^{5}+(-5+\cdots)q^{7}+\cdots\)
1216.4.a.ba	$1216$	$4$	1216.a	1.a	$1$	$5$	$5$	$71.746$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(-2\)	\(0\)	\(22\)		$+$	$+$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q-\beta _{2}q^{3}+(-\beta _{1}-\beta _{2})q^{5}+(4+\beta _{2}+\cdots)q^{7}+\cdots\)
1216.4.a.bb	$1216$	$4$	1216.a	1.a	$1$	$5$	$5$	$71.746$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(2\)	\(0\)	\(-22\)		$-$	$-$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q+\beta _{2}q^{3}+(-\beta _{1}-\beta _{2})q^{5}+(-4-\beta _{2}+\cdots)q^{7}+\cdots\)
1216.4.a.bc	$1216$	$4$	1216.a	1.a	$1$	$5$	$5$	$71.746$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(3\)	\(-27\)	\(20\)		$+$	$-$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q+(1+\beta _{3})q^{3}+(-5+\beta _{1})q^{5}+(5-\beta _{1}+\cdots)q^{7}+\cdots\)
1216.4.a.bd	$1216$	$4$	1216.a	1.a	$1$	$5$	$5$	$71.746$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(6\)	\(-5\)	\(7\)		$+$	$+$	$+$	$2^{4}$	$\mathrm{SU}(2)$	\(q+(1+\beta _{2})q^{3}+(-1+\beta _{2}-\beta _{3})q^{5}+\cdots\)
1216.4.a.be	$1216$	$4$	1216.a	1.a	$1$	$7$	$7$	$71.746$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(-9\)	\(5\)	\(28\)		$-$	$+$	$-$	$2^{6}$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{1})q^{3}+(1-\beta _{5})q^{5}+(4+\beta _{3}+\cdots)q^{7}+\cdots\)
1216.4.a.bf	$1216$	$4$	1216.a	1.a	$1$	$7$	$7$	$71.746$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(-3\)	\(17\)	\(42\)		$-$	$-$	$+$	$2^{6}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(2+\beta _{3})q^{5}+(6-\beta _{2}+\beta _{5}+\cdots)q^{7}+\cdots\)
1216.4.a.bg	$1216$	$4$	1216.a	1.a	$1$	$7$	$7$	$71.746$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(3\)	\(17\)	\(-42\)		$-$	$+$	$-$	$2^{6}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(2+\beta _{3})q^{5}+(-6+\beta _{2}-\beta _{5}+\cdots)q^{7}+\cdots\)
1216.4.a.bh	$1216$	$4$	1216.a	1.a	$1$	$7$	$7$	$71.746$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(9\)	\(5\)	\(-28\)		$-$	$-$	$+$	$2^{6}$	$\mathrm{SU}(2)$	\(q+(1+\beta _{1})q^{3}+(1-\beta _{5})q^{5}+(-4-\beta _{3}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(1216))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(1216)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(19))\)\(^{\oplus 7}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(38))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(64))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(76))\)\(^{\oplus 5}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(152))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(304))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(608))\)\(^{\oplus 2}\)