Space of modular forms of level 3211, weight 2, and trivial character

• Label: 3211.2.a
• Level: $3211$
• Weight: $2$
• Character orbit: 3211.a
• Rep. character: $\chi_{3211}(1,\cdot)$
• Character field: $\Q$
• Dimension: $232$
• Newform subspaces: $18$
• Sturm bound: $606$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 3211 = 13^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3211.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 18 \)
Sturm bound:			\(606\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(3211))\).

	Total	New	Old
Modular forms	316	232	84
Cusp forms	289	232	57
Eisenstein series	27	0	27

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

\(13\)	\(19\)	Fricke	Dim
\(+\)	\(+\)	$+$	\(52\)
\(+\)	\(-\)	$-$	\(66\)
\(-\)	\(+\)	$-$	\(63\)
\(-\)	\(-\)	$+$	\(51\)
Plus space		\(+\)	\(103\)
Minus space		\(-\)	\(129\)

Trace form

\( 232 q + q^{2} + 2 q^{3} + 233 q^{4} + q^{5} + 7 q^{7} - 3 q^{8} + 232 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(3211))\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		13	19
3211.2.a.a	$3211$	$2$	3211.a	1.a	$1$	$1$	$1$	$25.640$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-2\)	\(-3\)	\(1\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}-2q^{4}-3q^{5}+q^{7}+q^{9}-3q^{11}+\cdots\)
3211.2.a.b	$3211$	$2$	3211.a	1.a	$1$	$2$	$2$	$25.640$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$0$	\(-1\)	\(-2\)	\(-2\)	\(4\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{2}+(-2+2\beta )q^{3}+(-1+\beta )q^{4}+\cdots\)
3211.2.a.c	$3211$	$2$	3211.a	1.a	$1$	$3$	$3$	$25.640$	\(\Q(\zeta_{18})^+\)	None	✓	$1$	$1$	\(3\)	\(-3\)	\(3\)	\(3\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+(-1+\beta _{1}-\beta _{2})q^{3}+\cdots\)
3211.2.a.d	$3211$	$2$	3211.a	1.a	$1$	$4$	$4$	$25.640$	4.4.6809.1	None	✓	$1$	$0$	\(3\)	\(-1\)	\(8\)	\(2\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta _{3})q^{2}+(\beta _{1}+\beta _{3})q^{3}+(1-\beta _{2}+\cdots)q^{4}+\cdots\)
3211.2.a.e	$3211$	$2$	3211.a	1.a	$1$	$5$	$5$	$25.640$	5.5.288565.1	None	✓	$1$	$1$	\(-4\)	\(3\)	\(-3\)	\(1\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+(1-\beta _{2})q^{3}+(1-\beta _{1}+\cdots)q^{4}+\cdots\)
3211.2.a.f	$3211$	$2$	3211.a	1.a	$1$	$5$	$5$	$25.640$	5.5.2655049.1	None	✓	$1$	$0$	\(0\)	\(3\)	\(-2\)	\(-4\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(1+\beta _{3})q^{3}+(2+\beta _{2})q^{4}+\cdots\)
3211.2.a.g	$3211$	$2$	3211.a	1.a	$1$	$10$	$10$	$25.640$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓	$1$	$1$	\(-3\)	\(0\)	\(-8\)	\(-2\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}-\beta _{6}q^{3}+(1+\beta _{2})q^{4}+(-1+\cdots)q^{5}+\cdots\)
3211.2.a.h	$3211$	$2$	3211.a	1.a	$1$	$10$	$10$	$25.640$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓	$1$	$0$	\(3\)	\(0\)	\(8\)	\(2\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}-\beta _{6}q^{3}+(1+\beta _{2})q^{4}+(1+\beta _{4}+\cdots)q^{5}+\cdots\)
3211.2.a.i	$3211$	$2$	3211.a	1.a	$1$	$11$	$11$	$25.640$	\(\mathbb{Q}[x]/(x^{11} - \cdots)\)	None	✓	$1$	$1$	\(-4\)	\(0\)	\(-6\)	\(-2\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}-\beta _{5}q^{3}+(1+\beta _{2})q^{4}+(-1+\cdots)q^{5}+\cdots\)
3211.2.a.j	$3211$	$2$	3211.a	1.a	$1$	$11$	$11$	$25.640$	\(\mathbb{Q}[x]/(x^{11} - \cdots)\)	None	✓	$1$	$1$	\(-2\)	\(0\)	\(-10\)	\(-2\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}-\beta _{5}q^{3}+(1+\beta _{2})q^{4}+(-1+\cdots)q^{5}+\cdots\)
3211.2.a.k	$3211$	$2$	3211.a	1.a	$1$	$11$	$11$	$25.640$	\(\mathbb{Q}[x]/(x^{11} - \cdots)\)	None	✓	$1$	$0$	\(2\)	\(0\)	\(10\)	\(2\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}-\beta _{5}q^{3}+(1+\beta _{2})q^{4}+(1+\beta _{7}+\cdots)q^{5}+\cdots\)
3211.2.a.l	$3211$	$2$	3211.a	1.a	$1$	$11$	$11$	$25.640$	\(\mathbb{Q}[x]/(x^{11} - \cdots)\)	None	✓	$1$	$0$	\(4\)	\(0\)	\(6\)	\(2\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}-\beta _{5}q^{3}+(1+\beta _{2})q^{4}+(1+\beta _{8}+\cdots)q^{5}+\cdots\)
3211.2.a.m	$3211$	$2$	3211.a	1.a	$1$	$20$	$20$	$25.640$	\(\mathbb{Q}[x]/(x^{20} - \cdots)\)	None	✓	$1$	$1$	\(-6\)	\(0\)	\(-16\)	\(-4\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}-\beta _{11}q^{3}+(1+\beta _{2})q^{4}+(-1+\cdots)q^{5}+\cdots\)
3211.2.a.n	$3211$	$2$	3211.a	1.a	$1$	$20$	$20$	$25.640$	\(\mathbb{Q}[x]/(x^{20} - \cdots)\)	None	✓	$1$	$0$	\(6\)	\(0\)	\(16\)	\(4\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}-\beta _{11}q^{3}+(1+\beta _{2})q^{4}+(1+\cdots)q^{5}+\cdots\)
3211.2.a.o	$3211$	$2$	3211.a	1.a	$1$	$21$	$21$	$25.640$		None	✓	$1$	$1$	\(-1\)	\(-5\)	\(4\)	\(1\)		$+$	$+$	$+$		$\mathrm{SU}(2)$
3211.2.a.p	$3211$	$2$	3211.a	1.a	$1$	$21$	$21$	$25.640$		None	✓	$1$	$1$	\(1\)	\(-5\)	\(-4\)	\(-1\)		$-$	$-$	$+$		$\mathrm{SU}(2)$
3211.2.a.q	$3211$	$2$	3211.a	1.a	$1$	$33$	$33$	$25.640$		None	✓	$1$	$0$	\(-1\)	\(7\)	\(5\)	\(6\)		$+$	$-$	$-$		$\mathrm{SU}(2)$
3211.2.a.r	$3211$	$2$	3211.a	1.a	$1$	$33$	$33$	$25.640$		None	✓	$1$	$0$	\(1\)	\(7\)	\(-5\)	\(-6\)		$-$	$+$	$-$		$\mathrm{SU}(2)$

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(3211))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(3211)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(19))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(169))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(247))\)\(^{\oplus 2}\)