Space of modular forms of level 3971, weight 2, and trivial character

• Label: 3971.2.a
• Level: $3971$
• Weight: $2$
• Character orbit: 3971.a
• Rep. character: $\chi_{3971}(1,\cdot)$
• Character field: $\Q$
• Dimension: $284$
• Newform subspaces: $24$
• Sturm bound: $760$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 3971 = 11 \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3971.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 24 \)
Sturm bound:			\(760\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(2\), \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(3971))\).

	Total	New	Old
Modular forms	400	284	116
Cusp forms	361	284	77
Eisenstein series	39	0	39

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

\(11\)	\(19\)	Fricke	Dim
\(+\)	\(+\)	$+$	\(63\)
\(+\)	\(-\)	$-$	\(78\)
\(-\)	\(+\)	$-$	\(83\)
\(-\)	\(-\)	$+$	\(60\)
Plus space		\(+\)	\(123\)
Minus space		\(-\)	\(161\)

Trace form

\( 284 q + q^{2} - q^{3} + 279 q^{4} + 7 q^{5} + 6 q^{6} - 6 q^{7} + 3 q^{8} + 289 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(3971))\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		11	19
3971.2.a.a	$3971$	$2$	3971.a	1.a	$1$	$1$	$1$	$31.709$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-1\)	\(-3\)	\(-4\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-2q^{4}-3q^{5}-4q^{7}-2q^{9}+\cdots\)
3971.2.a.b	$3971$	$2$	3971.a	1.a	$1$	$1$	$1$	$31.709$	\(\Q\)	None	✓	$1$	$1$	\(2\)	\(1\)	\(1\)	\(-2\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+q^{3}+2q^{4}+q^{5}+2q^{6}-2q^{7}+\cdots\)
3971.2.a.c	$3971$	$2$	3971.a	1.a	$1$	$2$	$2$	$31.709$	\(\Q(\sqrt{7}) \)	None	✓	$2$	$1$	\(0\)	\(0\)	\(-2\)	\(-8\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}-2q^{4}-q^{5}-4q^{7}+4q^{9}+\cdots\)
3971.2.a.d	$3971$	$2$	3971.a	1.a	$1$	$2$	$2$	$31.709$	\(\Q(\sqrt{2}) \)	None	✓	$1$	$0$	\(0\)	\(2\)	\(-2\)	\(-4\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+(1-\beta )q^{3}-q^{5}+(-2+\beta )q^{6}+\cdots\)
3971.2.a.e	$3971$	$2$	3971.a	1.a	$1$	$3$	$3$	$31.709$	\(\Q(\zeta_{18})^+\)	None	✓	$1$	$2$	\(-3\)	\(-6\)	\(-6\)	\(-6\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}-2q^{3}+(1-2\beta _{1}+\beta _{2})q^{4}+\cdots\)
3971.2.a.f	$3971$	$2$	3971.a	1.a	$1$	$3$	$3$	$31.709$	\(\Q(\zeta_{18})^+\)	None	✓	$1$	$1$	\(3\)	\(6\)	\(-6\)	\(-6\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+2q^{3}+(1-2\beta _{1}+\beta _{2})q^{4}+\cdots\)
3971.2.a.g	$3971$	$2$	3971.a	1.a	$1$	$4$	$4$	$31.709$	\(\Q(\sqrt{3}, \sqrt{7})\)	None	✓	$2$	$1$	\(0\)	\(0\)	\(2\)	\(2\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(\beta _{1}+\beta _{3})q^{2}-\beta _{1}q^{3}+q^{4}-\beta _{2}q^{5}+\cdots\)
3971.2.a.h	$3971$	$2$	3971.a	1.a	$1$	$5$	$5$	$31.709$	5.5.246832.1	None	✓	$1$	$1$	\(-2\)	\(-1\)	\(-5\)	\(6\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{2}q^{2}-\beta _{3}q^{3}+(\beta _{1}+\beta _{2}+\beta _{3}+\beta _{4})q^{4}+\cdots\)
3971.2.a.i	$3971$	$2$	3971.a	1.a	$1$	$7$	$7$	$31.709$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	$1$	$0$	\(1\)	\(-2\)	\(2\)	\(10\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+\beta _{2}q^{3}+(2+\beta _{2}+\beta _{3})q^{4}+\cdots\)
3971.2.a.j	$3971$	$2$	3971.a	1.a	$1$	$9$	$9$	$31.709$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓	$1$	$1$	\(-1\)	\(-4\)	\(0\)	\(-3\)		$-$	$-$	$+$	$3$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}-\beta _{3}q^{3}+(1+\beta _{2})q^{4}+\beta _{8}q^{5}+\cdots\)
3971.2.a.k	$3971$	$2$	3971.a	1.a	$1$	$9$	$9$	$31.709$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓	$1$	$1$	\(-1\)	\(0\)	\(0\)	\(-3\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}-\beta _{3}q^{3}+(1+\beta _{2})q^{4}+(\beta _{1}+\cdots)q^{5}+\cdots\)
3971.2.a.l	$3971$	$2$	3971.a	1.a	$1$	$9$	$9$	$31.709$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓	$1$	$0$	\(1\)	\(0\)	\(0\)	\(-3\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+\beta _{3}q^{3}+(1+\beta _{2})q^{4}+(\beta _{1}+\cdots)q^{5}+\cdots\)
3971.2.a.m	$3971$	$2$	3971.a	1.a	$1$	$9$	$9$	$31.709$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓	$1$	$0$	\(1\)	\(4\)	\(0\)	\(-3\)		$-$	$+$	$-$	$3$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+\beta _{3}q^{3}+(1+\beta _{2})q^{4}+\beta _{8}q^{5}+\cdots\)
3971.2.a.n	$3971$	$2$	3971.a	1.a	$1$	$10$	$10$	$31.709$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓	$1$	$1$	\(-2\)	\(1\)	\(-8\)	\(-7\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+\beta _{7}q^{3}+(-\beta _{5}+\beta _{6}+\beta _{7}+\cdots)q^{4}+\cdots\)
3971.2.a.o	$3971$	$2$	3971.a	1.a	$1$	$10$	$10$	$31.709$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓	$2$	$0$	\(0\)	\(0\)	\(2\)	\(6\)		$-$	$+$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+\beta _{4}q^{3}+(2+\beta _{2})q^{4}+\beta _{8}q^{5}+\cdots\)
3971.2.a.p	$3971$	$2$	3971.a	1.a	$1$	$10$	$10$	$31.709$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓	$1$	$1$	\(2\)	\(-1\)	\(-8\)	\(-7\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}-\beta _{7}q^{3}+(-\beta _{5}+\beta _{6}+\beta _{7}+\cdots)q^{4}+\cdots\)
3971.2.a.q	$3971$	$2$	3971.a	1.a	$1$	$18$	$18$	$31.709$	\(\mathbb{Q}[x]/(x^{18} - \cdots)\)	None	✓	$1$	$0$	\(-2\)	\(0\)	\(5\)	\(7\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}-\beta _{6}q^{3}+(1+\beta _{2})q^{4}-\beta _{4}q^{5}+\cdots\)
3971.2.a.r	$3971$	$2$	3971.a	1.a	$1$	$18$	$18$	$31.709$	\(\mathbb{Q}[x]/(x^{18} - \cdots)\)	None	✓	$1$	$0$	\(2\)	\(0\)	\(5\)	\(7\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+\beta _{6}q^{3}+(1+\beta _{2})q^{4}-\beta _{4}q^{5}+\cdots\)
3971.2.a.s	$3971$	$2$	3971.a	1.a	$1$	$21$	$21$	$31.709$		None	✓	$1$	$1$	\(-6\)	\(-15\)	\(6\)	\(6\)		$-$	$-$	$+$		$\mathrm{SU}(2)$
3971.2.a.t	$3971$	$2$	3971.a	1.a	$1$	$21$	$21$	$31.709$		None	✓	$1$	$0$	\(6\)	\(15\)	\(6\)	\(6\)		$-$	$+$	$-$		$\mathrm{SU}(2)$
3971.2.a.u	$3971$	$2$	3971.a	1.a	$1$	$24$	$24$	$31.709$		None	✓	$1$	$1$	\(-3\)	\(-3\)	\(0\)	\(0\)		$+$	$+$	$+$		$\mathrm{SU}(2)$
3971.2.a.v	$3971$	$2$	3971.a	1.a	$1$	$24$	$24$	$31.709$		None	✓	$2$	$1$	\(0\)	\(0\)	\(-10\)	\(-14\)		$+$	$+$	$+$		$\mathrm{SU}(2)$
3971.2.a.w	$3971$	$2$	3971.a	1.a	$1$	$24$	$24$	$31.709$		None	✓	$1$	$0$	\(3\)	\(3\)	\(0\)	\(0\)		$+$	$-$	$-$		$\mathrm{SU}(2)$
3971.2.a.x	$3971$	$2$	3971.a	1.a	$1$	$40$	$40$	$31.709$		None	✓	$2$	$0$	\(0\)	\(0\)	\(28\)	\(14\)		$-$	$+$	$-$		$\mathrm{SU}(2)$

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(3971))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(3971)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(19))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(209))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(361))\)\(^{\oplus 2}\)