Space of modular forms of level 3328, weight 2, and trivial character

• Label: 3328.2.a
• Level: $3328$
• Weight: $2$
• Character orbit: 3328.a
• Rep. character: $\chi_{3328}(1,\cdot)$
• Character field: $\Q$
• Dimension: $96$
• Newform subspaces: $42$
• Sturm bound: $896$
• Trace bound: $9$

Defining parameters

Level:	\( N \)	\(=\)	\( 3328 = 2^{8} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3328.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 42 \)
Sturm bound:			\(896\)
Trace bound:			\(9\)
Distinguishing \(T_p\):			\(3\), \(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(3328))\).

	Total	New	Old
Modular forms	472	96	376
Cusp forms	425	96	329
Eisenstein series	47	0	47

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

\(2\)	\(13\)	Fricke		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(112\)	\(22\)	\(90\)		\(101\)	\(22\)	\(79\)		\(11\)	\(0\)	\(11\)
\(+\)	\(-\)	\(-\)		\(120\)	\(26\)	\(94\)		\(108\)	\(26\)	\(82\)		\(12\)	\(0\)	\(12\)
\(-\)	\(+\)	\(-\)		\(124\)	\(26\)	\(98\)		\(112\)	\(26\)	\(86\)		\(12\)	\(0\)	\(12\)
\(-\)	\(-\)	\(+\)		\(116\)	\(22\)	\(94\)		\(104\)	\(22\)	\(82\)		\(12\)	\(0\)	\(12\)
Plus space		\(+\)		\(228\)	\(44\)	\(184\)		\(205\)	\(44\)	\(161\)		\(23\)	\(0\)	\(23\)
Minus space		\(-\)		\(244\)	\(52\)	\(192\)		\(220\)	\(52\)	\(168\)		\(24\)	\(0\)	\(24\)

Trace form

\( 96 q + 96 q^{9} + 96 q^{25} + 96 q^{49} + 96 q^{81}+O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(3328))\) 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	13
3328.2.a.a	$3328$	$2$	3328.a	1.a	$1$	$1$	$1$	$26.574$	\(\Q\)	None	✓				1664.2.b.a	$1$	$2$	\(0\)	\(-3\)	\(-1\)	\(-3\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}-q^{5}-3q^{7}+6q^{9}-6q^{11}+\cdots\)
3328.2.a.b	$3328$	$2$	3328.a	1.a	$1$	$1$	$1$	$26.574$	\(\Q\)	None	✓				1664.2.b.a	$1$	$0$	\(0\)	\(-3\)	\(1\)	\(3\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}+q^{5}+3q^{7}+6q^{9}-6q^{11}+\cdots\)
3328.2.a.c	$3328$	$2$	3328.a	1.a	$1$	$1$	$1$	$26.574$	\(\Q\)	None	✓				104.2.b.a	$1$	$2$	\(0\)	\(-1\)	\(-3\)	\(-3\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-3q^{5}-3q^{7}-2q^{9}+q^{13}+\cdots\)
3328.2.a.d	$3328$	$2$	3328.a	1.a	$1$	$1$	$1$	$26.574$	\(\Q\)	None	✓				1664.2.b.b	$1$	$1$	\(0\)	\(-1\)	\(-1\)	\(-1\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{5}-q^{7}-2q^{9}-2q^{11}+q^{13}+\cdots\)
3328.2.a.e	$3328$	$2$	3328.a	1.a	$1$	$1$	$1$	$26.574$	\(\Q\)	None	✓				1664.2.b.b	$1$	$1$	\(0\)	\(-1\)	\(1\)	\(1\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{5}+q^{7}-2q^{9}-2q^{11}-q^{13}+\cdots\)
3328.2.a.f	$3328$	$2$	3328.a	1.a	$1$	$1$	$1$	$26.574$	\(\Q\)	None	✓				104.2.b.a	$1$	$0$	\(0\)	\(-1\)	\(3\)	\(3\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+3q^{5}+3q^{7}-2q^{9}-q^{13}+\cdots\)
3328.2.a.g	$3328$	$2$	3328.a	1.a	$1$	$1$	$1$	$26.574$	\(\Q\)	None	✓				104.2.b.a	$1$	$1$	\(0\)	\(1\)	\(-3\)	\(3\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-3q^{5}+3q^{7}-2q^{9}+q^{13}+\cdots\)
3328.2.a.h	$3328$	$2$	3328.a	1.a	$1$	$1$	$1$	$26.574$	\(\Q\)	None	✓				1664.2.b.b	$1$	$1$	\(0\)	\(1\)	\(-1\)	\(1\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{5}+q^{7}-2q^{9}+2q^{11}+q^{13}+\cdots\)
3328.2.a.i	$3328$	$2$	3328.a	1.a	$1$	$1$	$1$	$26.574$	\(\Q\)	None	✓				1664.2.b.b	$1$	$1$	\(0\)	\(1\)	\(1\)	\(-1\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{5}-q^{7}-2q^{9}+2q^{11}-q^{13}+\cdots\)
3328.2.a.j	$3328$	$2$	3328.a	1.a	$1$	$1$	$1$	$26.574$	\(\Q\)	None	✓				104.2.b.a	$1$	$1$	\(0\)	\(1\)	\(3\)	\(-3\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+3q^{5}-3q^{7}-2q^{9}-q^{13}+\cdots\)
3328.2.a.k	$3328$	$2$	3328.a	1.a	$1$	$1$	$1$	$26.574$	\(\Q\)	None	✓				1664.2.b.a	$1$	$0$	\(0\)	\(3\)	\(-1\)	\(3\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}-q^{5}+3q^{7}+6q^{9}+6q^{11}+\cdots\)
3328.2.a.l	$3328$	$2$	3328.a	1.a	$1$	$1$	$1$	$26.574$	\(\Q\)	None	✓				1664.2.b.a	$1$	$0$	\(0\)	\(3\)	\(1\)	\(-3\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}+q^{5}-3q^{7}+6q^{9}+6q^{11}+\cdots\)
3328.2.a.m	$3328$	$2$	3328.a	1.a	$1$	$2$	$2$	$26.574$	\(\Q(\sqrt{3}) \)	None	✓				104.2.b.b	$1$	$1$	\(0\)	\(-4\)	\(0\)	\(-6\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}-2\beta q^{5}+(-3+\beta )q^{7}+q^{9}+\cdots\)
3328.2.a.n	$3328$	$2$	3328.a	1.a	$1$	$2$	$2$	$26.574$	\(\Q(\sqrt{3}) \)	None	✓				104.2.b.b	$1$	$1$	\(0\)	\(-4\)	\(0\)	\(6\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}+2\beta q^{5}+(3-\beta )q^{7}+q^{9}+(-3+\cdots)q^{11}+\cdots\)
3328.2.a.o	$3328$	$2$	3328.a	1.a	$1$	$2$	$2$	$26.574$	\(\Q(\sqrt{2}) \)	None	✓				1664.2.b.e	$1$	$1$	\(0\)	\(-2\)	\(-2\)	\(-2\)		$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q-q^{3}+(-1+\beta )q^{5}+(-1+\beta )q^{7}+\cdots\)
3328.2.a.p	$3328$	$2$	3328.a	1.a	$1$	$2$	$2$	$26.574$	\(\Q(\sqrt{2}) \)	None	✓				832.2.b.a	$1$	$0$	\(0\)	\(-2\)	\(-2\)	\(6\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{3}+(-1-2\beta )q^{5}+(3-\beta )q^{7}+\cdots\)
3328.2.a.q	$3328$	$2$	3328.a	1.a	$1$	$2$	$2$	$26.574$	\(\Q(\sqrt{2}) \)	None	✓				832.2.b.a	$1$	$1$	\(0\)	\(-2\)	\(2\)	\(-6\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{3}+(1+2\beta )q^{5}+(-3+\beta )q^{7}+\cdots\)
3328.2.a.r	$3328$	$2$	3328.a	1.a	$1$	$2$	$2$	$26.574$	\(\Q(\sqrt{2}) \)	None	✓				1664.2.b.e	$1$	$0$	\(0\)	\(-2\)	\(2\)	\(2\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q-q^{3}+(1-\beta )q^{5}+(1-\beta )q^{7}-2q^{9}+\cdots\)
3328.2.a.s	$3328$	$2$	3328.a	1.a	$1$	$2$	$2$	$26.574$	\(\Q(\sqrt{2}) \)	None	✓				1664.2.b.h	$2$	$0$	\(0\)	\(0\)	\(-4\)	\(0\)		$-$	$+$	$-$	$3$	$\mathrm{SU}(2)$	\(q-2q^{5}-\beta q^{7}-3q^{9}-\beta q^{11}-q^{13}+\cdots\)
3328.2.a.t	$3328$	$2$	3328.a	1.a	$1$	$2$	$2$	$26.574$	\(\Q(\sqrt{2}) \)	None	✓				1664.2.b.f	$2$	$1$	\(0\)	\(0\)	\(-4\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2\beta q^{3}-2q^{5}-3\beta q^{7}+5q^{9}+\beta q^{11}+\cdots\)
3328.2.a.u	$3328$	$2$	3328.a	1.a	$1$	$2$	$2$	$26.574$	\(\Q(\sqrt{5}) \)	None	✓				1664.2.b.g	$2$	$1$	\(0\)	\(0\)	\(-2\)	\(0\)		$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q-\beta q^{3}-q^{5}-\beta q^{7}+2q^{9}+2\beta q^{11}+\cdots\)
3328.2.a.v	$3328$	$2$	3328.a	1.a	$1$	$2$	$2$	$26.574$	\(\Q(\sqrt{5}) \)	None	✓				1664.2.b.g	$2$	$1$	\(0\)	\(0\)	\(2\)	\(0\)		$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q-\beta q^{3}+q^{5}+\beta q^{7}+2q^{9}+2\beta q^{11}+\cdots\)
3328.2.a.w	$3328$	$2$	3328.a	1.a	$1$	$2$	$2$	$26.574$	\(\Q(\sqrt{2}) \)	None	✓				1664.2.b.h	$2$	$0$	\(0\)	\(0\)	\(4\)	\(0\)		$+$	$-$	$-$	$3$	$\mathrm{SU}(2)$	\(q+2q^{5}-\beta q^{7}-3q^{9}+\beta q^{11}+q^{13}+\cdots\)
3328.2.a.x	$3328$	$2$	3328.a	1.a	$1$	$2$	$2$	$26.574$	\(\Q(\sqrt{2}) \)	None	✓				1664.2.b.f	$2$	$0$	\(0\)	\(0\)	\(4\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2\beta q^{3}+2q^{5}+3\beta q^{7}+5q^{9}+\beta q^{11}+\cdots\)
3328.2.a.y	$3328$	$2$	3328.a	1.a	$1$	$2$	$2$	$26.574$	\(\Q(\sqrt{2}) \)	None	✓				832.2.b.a	$1$	$1$	\(0\)	\(2\)	\(-2\)	\(-6\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{3}+(-1+2\beta )q^{5}+(-3-\beta )q^{7}+\cdots\)
3328.2.a.z	$3328$	$2$	3328.a	1.a	$1$	$2$	$2$	$26.574$	\(\Q(\sqrt{2}) \)	None	✓				1664.2.b.e	$1$	$1$	\(0\)	\(2\)	\(-2\)	\(2\)		$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+q^{3}+(-1+\beta )q^{5}+(1-\beta )q^{7}-2q^{9}+\cdots\)
3328.2.a.ba	$3328$	$2$	3328.a	1.a	$1$	$2$	$2$	$26.574$	\(\Q(\sqrt{2}) \)	None	✓				1664.2.b.e	$1$	$0$	\(0\)	\(2\)	\(2\)	\(-2\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+q^{3}+(1+\beta )q^{5}+(-1-\beta )q^{7}-2q^{9}+\cdots\)
3328.2.a.bb	$3328$	$2$	3328.a	1.a	$1$	$2$	$2$	$26.574$	\(\Q(\sqrt{2}) \)	None	✓				832.2.b.a	$1$	$0$	\(0\)	\(2\)	\(2\)	\(6\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{3}+(1-2\beta )q^{5}+(3+\beta )q^{7}+\cdots\)
3328.2.a.bc	$3328$	$2$	3328.a	1.a	$1$	$2$	$2$	$26.574$	\(\Q(\sqrt{3}) \)	None	✓				104.2.b.b	$1$	$0$	\(0\)	\(4\)	\(0\)	\(-6\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}+2\beta q^{5}+(-3+\beta )q^{7}+q^{9}+\cdots\)
3328.2.a.bd	$3328$	$2$	3328.a	1.a	$1$	$2$	$2$	$26.574$	\(\Q(\sqrt{3}) \)	None	✓				104.2.b.b	$1$	$0$	\(0\)	\(4\)	\(0\)	\(6\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}-2\beta q^{5}+(3-\beta )q^{7}+q^{9}+(3+\cdots)q^{11}+\cdots\)
3328.2.a.be	$3328$	$2$	3328.a	1.a	$1$	$3$	$3$	$26.574$	3.3.316.1	None	✓				104.2.b.c	$1$	$1$	\(0\)	\(-1\)	\(-3\)	\(-1\)		$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{2}q^{3}-q^{5}+\beta _{2}q^{7}+(2-\beta _{1}-\beta _{2})q^{9}+\cdots\)
3328.2.a.bf	$3328$	$2$	3328.a	1.a	$1$	$3$	$3$	$26.574$	3.3.316.1	None	✓				104.2.b.c	$1$	$1$	\(0\)	\(-1\)	\(3\)	\(1\)		$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{2}q^{3}+q^{5}-\beta _{2}q^{7}+(2-\beta _{1}-\beta _{2})q^{9}+\cdots\)
3328.2.a.bg	$3328$	$2$	3328.a	1.a	$1$	$3$	$3$	$26.574$	3.3.316.1	None	✓				104.2.b.c	$1$	$0$	\(0\)	\(1\)	\(-3\)	\(1\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q-\beta _{2}q^{3}-q^{5}-\beta _{2}q^{7}+(2-\beta _{1}-\beta _{2})q^{9}+\cdots\)
3328.2.a.bh	$3328$	$2$	3328.a	1.a	$1$	$3$	$3$	$26.574$	3.3.316.1	None	✓				104.2.b.c	$1$	$0$	\(0\)	\(1\)	\(3\)	\(-1\)		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q-\beta _{2}q^{3}+q^{5}+\beta _{2}q^{7}+(2-\beta _{1}-\beta _{2})q^{9}+\cdots\)
3328.2.a.bi	$3328$	$2$	3328.a	1.a	$1$	$4$	$4$	$26.574$	4.4.13968.1	None	✓				832.2.b.c	$1$	$0$	\(0\)	\(-2\)	\(-2\)	\(0\)		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}-\beta _{1}q^{5}+(\beta _{1}-\beta _{3})q^{7}+(1+\cdots)q^{9}+\cdots\)
3328.2.a.bj	$3328$	$2$	3328.a	1.a	$1$	$4$	$4$	$26.574$	4.4.13968.1	None	✓				832.2.b.c	$1$	$1$	\(0\)	\(-2\)	\(2\)	\(0\)		$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+\beta _{1}q^{5}+(-\beta _{1}+\beta _{3})q^{7}+\cdots\)
3328.2.a.bk	$3328$	$2$	3328.a	1.a	$1$	$4$	$4$	$26.574$	4.4.13448.1	None	✓				1664.2.b.j	$2$	$0$	\(0\)	\(0\)	\(-2\)	\(0\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q-\beta _{3}q^{3}+\beta _{2}q^{5}-\beta _{1}q^{7}+(3-\beta _{2}+\cdots)q^{9}+\cdots\)
3328.2.a.bl	$3328$	$2$	3328.a	1.a	$1$	$4$	$4$	$26.574$	4.4.13448.1	None	✓				1664.2.b.j	$2$	$0$	\(0\)	\(0\)	\(2\)	\(0\)		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q-\beta _{3}q^{3}-\beta _{2}q^{5}+\beta _{1}q^{7}+(3-\beta _{2}+\cdots)q^{9}+\cdots\)
3328.2.a.bm	$3328$	$2$	3328.a	1.a	$1$	$4$	$4$	$26.574$	4.4.13968.1	None	✓		✓		832.2.b.c	$1$	$1$	\(0\)	\(2\)	\(-2\)	\(0\)		$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}-\beta _{1}q^{5}+(-\beta _{1}+\beta _{3})q^{7}+\cdots\)
3328.2.a.bn	$3328$	$2$	3328.a	1.a	$1$	$4$	$4$	$26.574$	4.4.13968.1	None	✓				832.2.b.c	$1$	$0$	\(0\)	\(2\)	\(2\)	\(0\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+\beta _{1}q^{5}+(\beta _{1}-\beta _{3})q^{7}+(1+\cdots)q^{9}+\cdots\)
3328.2.a.bo	$3328$	$2$	3328.a	1.a	$1$	$6$	$6$	$26.574$	6.6.10323968.1	None	✓		✓		1664.2.b.k	$2$	$1$	\(0\)	\(0\)	\(-8\)	\(0\)		$+$	$+$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(-1-\beta _{2})q^{5}+\beta _{5}q^{7}+\beta _{2}q^{9}+\cdots\)
3328.2.a.bp	$3328$	$2$	3328.a	1.a	$1$	$6$	$6$	$26.574$	6.6.10323968.1	None	✓		✓		1664.2.b.k	$2$	$0$	\(0\)	\(0\)	\(8\)	\(0\)		$+$	$-$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(1+\beta _{2})q^{5}-\beta _{5}q^{7}+\beta _{2}q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(3328))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(3328)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(26))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(52))\)\(^{\oplus 7}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(64))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(104))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(128))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(208))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(256))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(416))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(832))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1664))\)\(^{\oplus 2}\)