Space of modular forms of level 3201, weight 2, and trivial character

• Label: 3201.2.a
• Level: $3201$
• Weight: $2$
• Character orbit: 3201.a
• Rep. character: $\chi_{3201}(1,\cdot)$
• Character field: $\Q$
• Dimension: $159$
• Newform subspaces: $23$
• Sturm bound: $784$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 3201 = 3 \cdot 11 \cdot 97 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3201.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 23 \)
Sturm bound:			\(784\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(2\), \(5\), \(7\), \(17\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(3201))\).

	Total	New	Old
Modular forms	396	159	237
Cusp forms	389	159	230
Eisenstein series	7	0	7

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

\(3\)	\(11\)	\(97\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	$+$	\(19\)
\(+\)	\(+\)	\(-\)	$-$	\(21\)
\(+\)	\(-\)	\(+\)	$-$	\(23\)
\(+\)	\(-\)	\(-\)	$+$	\(17\)
\(-\)	\(+\)	\(+\)	$-$	\(19\)
\(-\)	\(+\)	\(-\)	$+$	\(21\)
\(-\)	\(-\)	\(+\)	$+$	\(15\)
\(-\)	\(-\)	\(-\)	$-$	\(24\)
Plus space			\(+\)	\(72\)
Minus space			\(-\)	\(87\)

Trace form

\( 159 q - 3 q^{2} - q^{3} + 161 q^{4} - 6 q^{5} + 5 q^{6} - 8 q^{7} - 15 q^{8} + 159 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(3201))\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		3	11	97
3201.2.a.a	$3201$	$2$	3201.a	1.a	$1$	$1$	$1$	$25.560$	\(\Q\)	None	✓	$1$	$1$	\(-1\)	\(-1\)	\(0\)	\(2\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{3}-q^{4}+q^{6}+2q^{7}+3q^{8}+\cdots\)
3201.2.a.b	$3201$	$2$	3201.a	1.a	$1$	$1$	$1$	$25.560$	\(\Q\)	None	✓	$1$	$1$	\(-1\)	\(1\)	\(-2\)	\(-4\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{3}-q^{4}-2q^{5}-q^{6}-4q^{7}+\cdots\)
3201.2.a.c	$3201$	$2$	3201.a	1.a	$1$	$1$	$1$	$25.560$	\(\Q\)	None	✓	$1$	$1$	\(-1\)	\(1\)	\(0\)	\(2\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{3}-q^{4}-q^{6}+2q^{7}+3q^{8}+\cdots\)
3201.2.a.d	$3201$	$2$	3201.a	1.a	$1$	$1$	$1$	$25.560$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(1\)	\(-3\)	\(-4\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-2q^{4}-3q^{5}-4q^{7}+q^{9}+q^{11}+\cdots\)
3201.2.a.e	$3201$	$2$	3201.a	1.a	$1$	$1$	$1$	$25.560$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(1\)	\(1\)	\(4\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-2q^{4}+q^{5}+4q^{7}+q^{9}-q^{11}+\cdots\)
3201.2.a.f	$3201$	$2$	3201.a	1.a	$1$	$1$	$1$	$25.560$	\(\Q\)	None	✓	$1$	$0$	\(1\)	\(-1\)	\(0\)	\(-2\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{3}-q^{4}-q^{6}-2q^{7}-3q^{8}+\cdots\)
3201.2.a.g	$3201$	$2$	3201.a	1.a	$1$	$1$	$1$	$25.560$	\(\Q\)	None	✓	$1$	$0$	\(1\)	\(1\)	\(2\)	\(-2\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{3}-q^{4}+2q^{5}+q^{6}-2q^{7}+\cdots\)
3201.2.a.h	$3201$	$2$	3201.a	1.a	$1$	$1$	$1$	$25.560$	\(\Q\)	None	✓	$1$	$1$	\(2\)	\(-1\)	\(-3\)	\(2\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}-q^{3}+2q^{4}-3q^{5}-2q^{6}+\cdots\)
3201.2.a.i	$3201$	$2$	3201.a	1.a	$1$	$2$	$2$	$25.560$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$0$	\(0\)	\(2\)	\(2\)	\(-2\)		$-$	$+$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q-\beta q^{2}+q^{3}+3q^{4}+(1-\beta )q^{5}-\beta q^{6}+\cdots\)
3201.2.a.j	$3201$	$2$	3201.a	1.a	$1$	$2$	$2$	$25.560$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$0$	\(2\)	\(-2\)	\(-4\)	\(-2\)		$+$	$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{3}-q^{4}-2q^{5}-q^{6}+(-1+\cdots)q^{7}+\cdots\)
3201.2.a.k	$3201$	$2$	3201.a	1.a	$1$	$2$	$2$	$25.560$	\(\Q(\sqrt{7}) \)	None	✓	$1$	$1$	\(2\)	\(-2\)	\(-2\)	\(-4\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{3}-q^{4}+(-1-\beta )q^{5}-q^{6}+\cdots\)
3201.2.a.l	$3201$	$2$	3201.a	1.a	$1$	$2$	$2$	$25.560$	\(\Q(\sqrt{2}) \)	None	✓	$1$	$0$	\(2\)	\(-2\)	\(4\)	\(-4\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{2}-q^{3}+(1+2\beta )q^{4}+(2+\beta )q^{5}+\cdots\)
3201.2.a.m	$3201$	$2$	3201.a	1.a	$1$	$2$	$2$	$25.560$	\(\Q(\sqrt{2}) \)	None	✓	$1$	$0$	\(2\)	\(2\)	\(-4\)	\(-4\)		$-$	$+$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{3}-q^{4}-2q^{5}+q^{6}+(-2+\cdots)q^{7}+\cdots\)
3201.2.a.n	$3201$	$2$	3201.a	1.a	$1$	$3$	$3$	$25.560$	3.3.148.1	None	✓	$1$	$0$	\(-1\)	\(-3\)	\(-2\)	\(4\)		$+$	$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}-q^{3}+(1+\beta _{1}+\beta _{2})q^{4}+(-1+\cdots)q^{5}+\cdots\)
3201.2.a.o	$3201$	$2$	3201.a	1.a	$1$	$7$	$7$	$25.560$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(-7\)	\(5\)	\(7\)		$+$	$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q-\beta _{3}q^{2}-q^{3}+(2-\beta _{2})q^{4}+(1-\beta _{4}+\cdots)q^{5}+\cdots\)
3201.2.a.p	$3201$	$2$	3201.a	1.a	$1$	$12$	$12$	$25.560$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None	✓	$1$	$0$	\(-7\)	\(-12\)	\(-1\)	\(6\)		$+$	$-$	$+$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}-q^{3}+(2-\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
3201.2.a.q	$3201$	$2$	3201.a	1.a	$1$	$13$	$13$	$25.560$	\(\mathbb{Q}[x]/(x^{13} - \cdots)\)	None	✓	$1$	$1$	\(-4\)	\(13\)	\(-11\)	\(3\)		$-$	$-$	$+$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+q^{3}+(1+\beta _{1}+\beta _{2})q^{4}+(-1+\cdots)q^{5}+\cdots\)
3201.2.a.r	$3201$	$2$	3201.a	1.a	$1$	$13$	$13$	$25.560$	\(\mathbb{Q}[x]/(x^{13} - \cdots)\)	None	✓	$1$	$1$	\(-2\)	\(-13\)	\(1\)	\(-11\)		$+$	$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}-q^{3}+(1-\beta _{2}-\beta _{3}-\beta _{4}+\cdots)q^{4}+\cdots\)
3201.2.a.s	$3201$	$2$	3201.a	1.a	$1$	$13$	$13$	$25.560$	\(\mathbb{Q}[x]/(x^{13} - \cdots)\)	None	✓	$1$	$0$	\(4\)	\(13\)	\(3\)	\(9\)		$-$	$+$	$+$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+q^{3}+(1+\beta _{1}+\beta _{2})q^{4}+(-\beta _{4}+\cdots)q^{5}+\cdots\)
3201.2.a.t	$3201$	$2$	3201.a	1.a	$1$	$17$	$17$	$25.560$	\(\mathbb{Q}[x]/(x^{17} - \cdots)\)	None	✓	$1$	$0$	\(-3\)	\(-17\)	\(-10\)	\(9\)		$+$	$+$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}-q^{3}+(1+\beta _{2})q^{4}+(-1-\beta _{5}+\cdots)q^{5}+\cdots\)
3201.2.a.u	$3201$	$2$	3201.a	1.a	$1$	$19$	$19$	$25.560$	\(\mathbb{Q}[x]/(x^{19} - \cdots)\)	None	✓	$1$	$1$	\(1\)	\(-19\)	\(8\)	\(-7\)		$+$	$+$	$+$	$+$	$2^{4}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}-q^{3}+(1+\beta _{2})q^{4}-\beta _{17}q^{5}+\cdots\)
3201.2.a.v	$3201$	$2$	3201.a	1.a	$1$	$20$	$20$	$25.560$	\(\mathbb{Q}[x]/(x^{20} - \cdots)\)	None	✓	$1$	$1$	\(-4\)	\(20\)	\(-8\)	\(-15\)		$-$	$+$	$-$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+q^{3}+(1+\beta _{2})q^{4}+\beta _{18}q^{5}+\cdots\)
3201.2.a.w	$3201$	$2$	3201.a	1.a	$1$	$24$	$24$	$25.560$		None	✓	$1$	$0$	\(4\)	\(24\)	\(18\)	\(5\)		$-$	$-$	$-$	$-$		$\mathrm{SU}(2)$

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(3201))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(3201)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(33))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(97))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(291))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1067))\)\(^{\oplus 2}\)