Space of modular forms of level 3216, weight 2, and trivial character

• Label: 3216.2.a
• Level: $3216$
• Weight: $2$
• Character orbit: 3216.a
• Rep. character: $\chi_{3216}(1,\cdot)$
• Character field: $\Q$
• Dimension: $66$
• Newform subspaces: $28$
• Sturm bound: $1088$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 3216 = 2^{4} \cdot 3 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3216.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 28 \)
Sturm bound:			\(1088\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(5\), \(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(3216))\).

	Total	New	Old
Modular forms	556	66	490
Cusp forms	533	66	467
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.

\(2\)	\(3\)	\(67\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	\(+\)	\(8\)
\(+\)	\(+\)	\(-\)	\(-\)	\(8\)
\(+\)	\(-\)	\(+\)	\(-\)	\(11\)
\(+\)	\(-\)	\(-\)	\(+\)	\(5\)
\(-\)	\(+\)	\(+\)	\(-\)	\(10\)
\(-\)	\(+\)	\(-\)	\(+\)	\(7\)
\(-\)	\(-\)	\(+\)	\(+\)	\(7\)
\(-\)	\(-\)	\(-\)	\(-\)	\(10\)
Plus space			\(+\)	\(27\)
Minus space			\(-\)	\(39\)

Trace form

\( 66 q + 4 q^{5} + 66 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(3216))\) 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	3	67
3216.2.a.a	$3216$	$2$	3216.a	1.a	$1$	$1$	$1$	$25.680$	\(\Q\)	None	✓				402.2.a.b	$1$	$0$	\(0\)	\(-1\)	\(-3\)	\(1\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-3q^{5}+q^{7}+q^{9}-4q^{13}+3q^{15}+\cdots\)
3216.2.a.b	$3216$	$2$	3216.a	1.a	$1$	$1$	$1$	$25.680$	\(\Q\)	None	✓				1608.2.a.b	$1$	$0$	\(0\)	\(-1\)	\(-2\)	\(2\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-2q^{5}+2q^{7}+q^{9}+4q^{11}+\cdots\)
3216.2.a.c	$3216$	$2$	3216.a	1.a	$1$	$1$	$1$	$25.680$	\(\Q\)	None	✓				804.2.a.d	$1$	$1$	\(0\)	\(-1\)	\(-1\)	\(3\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{5}+3q^{7}+q^{9}+2q^{11}-2q^{13}+\cdots\)
3216.2.a.d	$3216$	$2$	3216.a	1.a	$1$	$1$	$1$	$25.680$	\(\Q\)	None	✓				201.2.a.b	$1$	$0$	\(0\)	\(-1\)	\(-1\)	\(5\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{5}+5q^{7}+q^{9}+4q^{11}-4q^{13}+\cdots\)
3216.2.a.e	$3216$	$2$	3216.a	1.a	$1$	$1$	$1$	$25.680$	\(\Q\)	None	✓				402.2.a.c	$1$	$1$	\(0\)	\(-1\)	\(2\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+2q^{5}+q^{9}-4q^{11}-2q^{13}+\cdots\)
3216.2.a.f	$3216$	$2$	3216.a	1.a	$1$	$1$	$1$	$25.680$	\(\Q\)	None	✓				1608.2.a.a	$1$	$0$	\(0\)	\(1\)	\(-4\)	\(4\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-4q^{5}+4q^{7}+q^{9}+6q^{11}+\cdots\)
3216.2.a.g	$3216$	$2$	3216.a	1.a	$1$	$1$	$1$	$25.680$	\(\Q\)	None	✓				804.2.a.a	$1$	$1$	\(0\)	\(1\)	\(-3\)	\(-3\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-3q^{5}-3q^{7}+q^{9}+2q^{11}+\cdots\)
3216.2.a.h	$3216$	$2$	3216.a	1.a	$1$	$1$	$1$	$25.680$	\(\Q\)	None	✓				201.2.a.c	$1$	$0$	\(0\)	\(1\)	\(-3\)	\(3\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-3q^{5}+3q^{7}+q^{9}+4q^{13}+\cdots\)
3216.2.a.i	$3216$	$2$	3216.a	1.a	$1$	$1$	$1$	$25.680$	\(\Q\)	None	✓				804.2.a.b	$1$	$1$	\(0\)	\(1\)	\(0\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{9}+2q^{11}-4q^{13}-3q^{17}+\cdots\)
3216.2.a.j	$3216$	$2$	3216.a	1.a	$1$	$1$	$1$	$25.680$	\(\Q\)	None	✓				201.2.a.a	$1$	$0$	\(0\)	\(1\)	\(0\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{9}+6q^{11}+4q^{13}-7q^{17}+\cdots\)
3216.2.a.k	$3216$	$2$	3216.a	1.a	$1$	$1$	$1$	$25.680$	\(\Q\)	None	✓				402.2.a.a	$1$	$0$	\(0\)	\(1\)	\(1\)	\(3\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{5}+3q^{7}+q^{9}-4q^{13}+q^{15}+\cdots\)
3216.2.a.l	$3216$	$2$	3216.a	1.a	$1$	$1$	$1$	$25.680$	\(\Q\)	None	✓				402.2.a.d	$1$	$0$	\(0\)	\(1\)	\(2\)	\(-2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+2q^{5}-2q^{7}+q^{9}+4q^{11}+\cdots\)
3216.2.a.m	$3216$	$2$	3216.a	1.a	$1$	$1$	$1$	$25.680$	\(\Q\)	None	✓				804.2.a.c	$1$	$0$	\(0\)	\(1\)	\(4\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+4q^{5}+q^{9}+2q^{13}+4q^{15}+\cdots\)
3216.2.a.n	$3216$	$2$	3216.a	1.a	$1$	$2$	$2$	$25.680$	\(\Q(\sqrt{2}) \)	None	✓				1608.2.a.c	$1$	$1$	\(0\)	\(2\)	\(-2\)	\(2\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+(-1+\beta )q^{5}+q^{7}+q^{9}-2q^{11}+\cdots\)
3216.2.a.o	$3216$	$2$	3216.a	1.a	$1$	$2$	$2$	$25.680$	\(\Q(\sqrt{3}) \)	None	✓				402.2.a.e	$1$	$1$	\(0\)	\(2\)	\(0\)	\(-6\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+2\beta q^{5}+(-3-\beta )q^{7}+q^{9}+\cdots\)
3216.2.a.p	$3216$	$2$	3216.a	1.a	$1$	$2$	$2$	$25.680$	\(\Q(\sqrt{41}) \)	None	✓				402.2.a.f	$1$	$0$	\(0\)	\(2\)	\(1\)	\(1\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+\beta q^{5}+\beta q^{7}+q^{9}-4q^{11}+\cdots\)
3216.2.a.q	$3216$	$2$	3216.a	1.a	$1$	$3$	$3$	$25.680$	3.3.316.1	None	✓				1608.2.a.e	$1$	$1$	\(0\)	\(-3\)	\(-4\)	\(6\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+(-1-\beta _{1})q^{5}+(2-\beta _{2})q^{7}+\cdots\)
3216.2.a.r	$3216$	$2$	3216.a	1.a	$1$	$3$	$3$	$25.680$	3.3.148.1	None	✓				1608.2.a.f	$1$	$0$	\(0\)	\(-3\)	\(-3\)	\(1\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+(-1-\beta _{2})q^{5}+(\beta _{1}-\beta _{2})q^{7}+\cdots\)
3216.2.a.s	$3216$	$2$	3216.a	1.a	$1$	$3$	$3$	$25.680$	3.3.316.1	None	✓				402.2.a.g	$1$	$0$	\(0\)	\(-3\)	\(3\)	\(-1\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+(1-\beta _{2})q^{5}-\beta _{1}q^{7}+q^{9}-2\beta _{2}q^{11}+\cdots\)
3216.2.a.t	$3216$	$2$	3216.a	1.a	$1$	$3$	$3$	$25.680$	3.3.1076.1	None	✓		✓		804.2.a.e	$1$	$0$	\(0\)	\(3\)	\(-3\)	\(5\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+(-1-\beta _{1})q^{5}+(2+\beta _{1}-\beta _{2})q^{7}+\cdots\)
3216.2.a.u	$3216$	$2$	3216.a	1.a	$1$	$3$	$3$	$25.680$	3.3.148.1	None	✓		✓		201.2.a.d	$1$	$1$	\(0\)	\(3\)	\(1\)	\(-1\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+\beta _{1}q^{5}+(-\beta _{1}-\beta _{2})q^{7}+q^{9}+\cdots\)
3216.2.a.v	$3216$	$2$	3216.a	1.a	$1$	$3$	$3$	$25.680$	3.3.316.1	None	✓		✓		1608.2.a.d	$1$	$1$	\(0\)	\(3\)	\(2\)	\(-6\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+(1-\beta _{1})q^{5}+(-2+\beta _{2})q^{7}+\cdots\)
3216.2.a.w	$3216$	$2$	3216.a	1.a	$1$	$4$	$4$	$25.680$	4.4.19796.1	None	✓		✓		1608.2.a.h	$1$	$0$	\(0\)	\(-4\)	\(7\)	\(-5\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+(2-\beta _{3})q^{5}+(-2+\beta _{1}-\beta _{2}+\cdots)q^{7}+\cdots\)
3216.2.a.x	$3216$	$2$	3216.a	1.a	$1$	$4$	$4$	$25.680$	4.4.25492.1	None	✓				1608.2.a.g	$1$	$0$	\(0\)	\(4\)	\(3\)	\(3\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+(1+\beta _{2})q^{5}+(1+\beta _{1}-\beta _{3})q^{7}+\cdots\)
3216.2.a.y	$3216$	$2$	3216.a	1.a	$1$	$5$	$5$	$25.680$	5.5.1025428.1	None	✓		✓		201.2.a.e	$1$	$1$	\(0\)	\(-5\)	\(-3\)	\(-7\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+(-1+\beta _{4})q^{5}+(-1-\beta _{1}+\beta _{3}+\cdots)q^{7}+\cdots\)
3216.2.a.z	$3216$	$2$	3216.a	1.a	$1$	$5$	$5$	$25.680$	5.5.8212256.1	None	✓		✓		1608.2.a.i	$1$	$1$	\(0\)	\(-5\)	\(2\)	\(0\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+\beta _{1}q^{5}-\beta _{4}q^{7}+q^{9}-2q^{11}+\cdots\)
3216.2.a.ba	$3216$	$2$	3216.a	1.a	$1$	$5$	$5$	$25.680$	5.5.24571284.1	None	✓		✓		804.2.a.f	$1$	$0$	\(0\)	\(-5\)	\(3\)	\(-5\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+(1-\beta _{1})q^{5}+(-1+\beta _{2})q^{7}+\cdots\)
3216.2.a.bb	$3216$	$2$	3216.a	1.a	$1$	$6$	$6$	$25.680$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓		✓		1608.2.a.j	$1$	$0$	\(0\)	\(6\)	\(5\)	\(-3\)		$+$	$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+q^{3}+(1-\beta _{1})q^{5}+(-\beta _{1}+\beta _{3})q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(3216))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(3216)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(48))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(67))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(134))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(201))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(268))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(402))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(536))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(804))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1072))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1608))\)\(^{\oplus 2}\)