Space of modular forms of level 1216, weight 2, and trivial character

• Label: 1216.2.a
• Level: $1216$
• Weight: $2$
• Character orbit: 1216.a
• Rep. character: $\chi_{1216}(1,\cdot)$
• Character field: $\Q$
• Dimension: $36$
• Newform subspaces: $24$
• Sturm bound: $320$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 1216 = 2^{6} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1216.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 24 \)
Sturm bound:			\(320\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(3\), \(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1216))\).

	Total	New	Old
Modular forms	172	36	136
Cusp forms	149	36	113
Eisenstein series	23	0	23

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
\(+\)	\(+\)	\(+\)	\(8\)
\(+\)	\(-\)	\(-\)	\(11\)
\(-\)	\(+\)	\(-\)	\(10\)
\(-\)	\(-\)	\(+\)	\(7\)
Plus space		\(+\)	\(15\)
Minus space		\(-\)	\(21\)

Trace form

\( 36 q + 36 q^{9} - 8 q^{17} + 28 q^{25} - 16 q^{29} - 16 q^{37} - 8 q^{41} + 48 q^{45} + 36 q^{49} + 32 q^{53} + 48 q^{69} + 8 q^{73} + 8 q^{77} + 4 q^{81} + 24 q^{85} - 24 q^{89} - 40 q^{97}+O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1216))\) 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					A-L signs		Fricke sign	Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		2	19
1216.2.a.a	$1216$	$2$	1216.a	1.a	$1$	$1$	$1$	$9.710$	\(\Q\)	None	✓				608.2.a.a	$1$	$1$	\(0\)	\(-3\)	\(0\)	\(-1\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}-q^{7}+6q^{9}-2q^{11}+q^{13}+\cdots\)
1216.2.a.b	$1216$	$2$	1216.a	1.a	$1$	$1$	$1$	$9.710$	\(\Q\)	None	✓				19.2.a.a	$1$	$1$	\(0\)	\(-2\)	\(-3\)	\(1\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}-3q^{5}+q^{7}+q^{9}+3q^{11}+\cdots\)
1216.2.a.c	$1216$	$2$	1216.a	1.a	$1$	$1$	$1$	$9.710$	\(\Q\)	None	✓				76.2.a.a	$1$	$0$	\(0\)	\(-2\)	\(1\)	\(-3\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}+q^{5}-3q^{7}+q^{9}-5q^{11}+\cdots\)
1216.2.a.d	$1216$	$2$	1216.a	1.a	$1$	$1$	$1$	$9.710$	\(\Q\)	None	✓				152.2.a.a	$1$	$0$	\(0\)	\(-2\)	\(1\)	\(3\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}+q^{5}+3q^{7}+q^{9}-3q^{11}+\cdots\)
1216.2.a.e	$1216$	$2$	1216.a	1.a	$1$	$1$	$1$	$9.710$	\(\Q\)	None	✓				38.2.a.a	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(-1\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{7}-2q^{9}+6q^{11}-5q^{13}+\cdots\)
1216.2.a.f	$1216$	$2$	1216.a	1.a	$1$	$1$	$1$	$9.710$	\(\Q\)	None	✓				152.2.a.b	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(3\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+3q^{7}-2q^{9}-2q^{11}-q^{13}+\cdots\)
1216.2.a.g	$1216$	$2$	1216.a	1.a	$1$	$1$	$1$	$9.710$	\(\Q\)	None	✓				38.2.a.b	$1$	$0$	\(0\)	\(-1\)	\(4\)	\(-3\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+4q^{5}-3q^{7}-2q^{9}+2q^{11}+\cdots\)
1216.2.a.h	$1216$	$2$	1216.a	1.a	$1$	$1$	$1$	$9.710$	\(\Q\)	None	✓				608.2.a.d	$1$	$0$	\(0\)	\(0\)	\(-3\)	\(-5\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{5}-5q^{7}-3q^{9}+5q^{11}+4q^{13}+\cdots\)
1216.2.a.i	$1216$	$2$	1216.a	1.a	$1$	$1$	$1$	$9.710$	\(\Q\)	None	✓				608.2.a.d	$1$	$1$	\(0\)	\(0\)	\(-3\)	\(5\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{5}+5q^{7}-3q^{9}-5q^{11}+4q^{13}+\cdots\)
1216.2.a.j	$1216$	$2$	1216.a	1.a	$1$	$1$	$1$	$9.710$	\(\Q\)	None	✓				608.2.a.b	$1$	$1$	\(0\)	\(0\)	\(1\)	\(-1\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}-q^{7}-3q^{9}-3q^{11}+4q^{13}+\cdots\)
1216.2.a.k	$1216$	$2$	1216.a	1.a	$1$	$1$	$1$	$9.710$	\(\Q\)	None	✓				608.2.a.b	$1$	$0$	\(0\)	\(0\)	\(1\)	\(1\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}+q^{7}-3q^{9}+3q^{11}+4q^{13}+\cdots\)
1216.2.a.l	$1216$	$2$	1216.a	1.a	$1$	$1$	$1$	$9.710$	\(\Q\)	None	✓				152.2.a.b	$1$	$1$	\(0\)	\(1\)	\(0\)	\(-3\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-3q^{7}-2q^{9}+2q^{11}-q^{13}+\cdots\)
1216.2.a.m	$1216$	$2$	1216.a	1.a	$1$	$1$	$1$	$9.710$	\(\Q\)	None	✓				38.2.a.a	$1$	$1$	\(0\)	\(1\)	\(0\)	\(1\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{7}-2q^{9}-6q^{11}-5q^{13}+\cdots\)
1216.2.a.n	$1216$	$2$	1216.a	1.a	$1$	$1$	$1$	$9.710$	\(\Q\)	None	✓				38.2.a.b	$1$	$0$	\(0\)	\(1\)	\(4\)	\(3\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+4q^{5}+3q^{7}-2q^{9}-2q^{11}+\cdots\)
1216.2.a.o	$1216$	$2$	1216.a	1.a	$1$	$1$	$1$	$9.710$	\(\Q\)	None	✓				19.2.a.a	$1$	$1$	\(0\)	\(2\)	\(-3\)	\(-1\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}-3q^{5}-q^{7}+q^{9}-3q^{11}+\cdots\)
1216.2.a.p	$1216$	$2$	1216.a	1.a	$1$	$1$	$1$	$9.710$	\(\Q\)	None	✓				152.2.a.a	$1$	$0$	\(0\)	\(2\)	\(1\)	\(-3\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}+q^{5}-3q^{7}+q^{9}+3q^{11}+\cdots\)
1216.2.a.q	$1216$	$2$	1216.a	1.a	$1$	$1$	$1$	$9.710$	\(\Q\)	None	✓				76.2.a.a	$1$	$0$	\(0\)	\(2\)	\(1\)	\(3\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}+q^{5}+3q^{7}+q^{9}+5q^{11}+\cdots\)
1216.2.a.r	$1216$	$2$	1216.a	1.a	$1$	$1$	$1$	$9.710$	\(\Q\)	None	✓				608.2.a.a	$1$	$0$	\(0\)	\(3\)	\(0\)	\(1\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}+q^{7}+6q^{9}+2q^{11}+q^{13}+\cdots\)
1216.2.a.s	$1216$	$2$	1216.a	1.a	$1$	$2$	$2$	$9.710$	\(\Q(\sqrt{17}) \)	None	✓				608.2.a.g	$1$	$0$	\(0\)	\(-1\)	\(1\)	\(6\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{3}+(1-\beta )q^{5}+3q^{7}+(1+\beta )q^{9}+\cdots\)
1216.2.a.t	$1216$	$2$	1216.a	1.a	$1$	$2$	$2$	$9.710$	\(\Q(\sqrt{17}) \)	None	✓		✓		608.2.a.g	$1$	$1$	\(0\)	\(1\)	\(1\)	\(-6\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+(1-\beta )q^{5}-3q^{7}+(1+\beta )q^{9}+\cdots\)
1216.2.a.u	$1216$	$2$	1216.a	1.a	$1$	$3$	$3$	$9.710$	3.3.961.1	None	✓				152.2.a.c	$1$	$0$	\(0\)	\(-1\)	\(-1\)	\(4\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+\beta _{2}q^{5}+(2-\beta _{1}+\beta _{2})q^{7}+\cdots\)
1216.2.a.v	$1216$	$2$	1216.a	1.a	$1$	$3$	$3$	$9.710$	3.3.961.1	None	✓		✓		152.2.a.c	$1$	$0$	\(0\)	\(1\)	\(-1\)	\(-4\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+\beta _{2}q^{5}+(-2+\beta _{1}-\beta _{2})q^{7}+\cdots\)
1216.2.a.w	$1216$	$2$	1216.a	1.a	$1$	$4$	$4$	$9.710$	4.4.15317.1	None	✓		✓		608.2.a.i	$1$	$1$	\(0\)	\(-2\)	\(-1\)	\(-1\)		$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{2})q^{3}+\beta _{1}q^{5}+(\beta _{2}-\beta _{3})q^{7}+\cdots\)
1216.2.a.x	$1216$	$2$	1216.a	1.a	$1$	$4$	$4$	$9.710$	4.4.15317.1	None	✓		✓		608.2.a.i	$1$	$0$	\(0\)	\(2\)	\(-1\)	\(1\)		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(1+\beta _{2})q^{3}+(-\beta _{1}+\beta _{2})q^{5}-\beta _{3}q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1216))\) into lower level spaces

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