Space of modular forms of level 3969, weight 2, and trivial character

• Label: 3969.2.a
• Level: $3969$
• Weight: $2$
• Character orbit: 3969.a
• Rep. character: $\chi_{3969}(1,\cdot)$
• Character field: $\Q$
• Dimension: $154$
• Newform subspaces: $36$
• Sturm bound: $1008$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 3969 = 3^{4} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3969.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 36 \)
Sturm bound:			\(1008\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(2\), \(5\), \(11\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(3969))\).

	Total	New	Old
Modular forms	552	174	378
Cusp forms	457	154	303
Eisenstein series	95	20	75

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

\(3\)	\(7\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	\(34\)
\(+\)	\(-\)	\(-\)	\(42\)
\(-\)	\(+\)	\(-\)	\(42\)
\(-\)	\(-\)	\(+\)	\(36\)
Plus space		\(+\)	\(70\)
Minus space		\(-\)	\(84\)

Trace form

154 * q + 142 * q^4 - 6 * q^10 - 10 * q^13 + 130 * q^16 + 8 * q^19 + 36 * q^22 + 112 * q^25 - 4 * q^31 + 6 * q^34 - 10 * q^37 - 18 * q^40 + 20 * q^43 + 36 * q^46 + 2 * q^52 + 18 * q^58 - 22 * q^61 + 106 * q^64 + 44 * q^67 + 38 * q^73 + 56 * q^76 - 4 * q^79 + 48 * q^82 - 30 * q^85 + 96 * q^88 + 60 * q^94 - 4 * q^97

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(3969))\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		3	7
3969.2.a.a	$3969$	$2$	3969.a	1.a	$1$	$1$	$1$	$31.693$	\(\Q\)	None	✓				63.2.g.a	$1$	$1$	\(-1\)	\(0\)	\(-1\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{4}-q^{5}+3q^{8}+q^{10}-5q^{11}+\cdots\)
3969.2.a.b	$3969$	$2$	3969.a	1.a	$1$	$1$	$1$	$31.693$	\(\Q\)	None	✓				567.2.a.a	$1$	$0$	\(-1\)	\(0\)	\(-1\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{4}-q^{5}+3q^{8}+q^{10}-2q^{11}+\cdots\)
3969.2.a.c	$3969$	$2$	3969.a	1.a	$1$	$1$	$1$	$31.693$	\(\Q\)	None	✓				63.2.g.a	$1$	$0$	\(-1\)	\(0\)	\(1\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{4}+q^{5}+3q^{8}-q^{10}-5q^{11}+\cdots\)
3969.2.a.d	$3969$	$2$	3969.a	1.a	$1$	$1$	$1$	$31.693$	\(\Q\)	None	✓				63.2.g.a	$1$	$1$	\(1\)	\(0\)	\(-1\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{4}-q^{5}-3q^{8}-q^{10}+5q^{11}+\cdots\)
3969.2.a.e	$3969$	$2$	3969.a	1.a	$1$	$1$	$1$	$31.693$	\(\Q\)	None	✓				567.2.a.a	$1$	$0$	\(1\)	\(0\)	\(1\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{4}+q^{5}-3q^{8}+q^{10}+2q^{11}+\cdots\)
3969.2.a.f	$3969$	$2$	3969.a	1.a	$1$	$1$	$1$	$31.693$	\(\Q\)	None	✓				63.2.g.a	$1$	$0$	\(1\)	\(0\)	\(1\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{4}+q^{5}-3q^{8}+q^{10}+5q^{11}+\cdots\)
3969.2.a.g	$3969$	$2$	3969.a	1.a	$1$	$2$	$2$	$31.693$	\(\Q(\sqrt{15}) \)	None	✓	✓			3969.2.a.g	$2$	$0$	\(-2\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{4}+\beta q^{5}+3q^{8}-\beta q^{10}+\cdots\)
3969.2.a.h	$3969$	$2$	3969.a	1.a	$1$	$2$	$2$	$31.693$	\(\Q(\sqrt{21}) \)	\(\Q(\sqrt{-7}) \)	✓	✓			3969.2.a.h	$2$	$0$	\(-1\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$	$1$	$N(\mathrm{U}(1))$	\(q-\beta q^{2}+(3+\beta )q^{4}+(-5-2\beta )q^{8}+\cdots\)
3969.2.a.i	$3969$	$2$	3969.a	1.a	$1$	$2$	$2$	$31.693$	\(\Q(\sqrt{3}) \)	None	✓				81.2.a.a	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+q^{4}+\beta q^{5}-\beta q^{8}+3q^{10}+\cdots\)
3969.2.a.j	$3969$	$2$	3969.a	1.a	$1$	$2$	$2$	$31.693$	\(\Q(\sqrt{21}) \)	\(\Q(\sqrt{-7}) \)	✓	✓			3969.2.a.h	$2$	$0$	\(1\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$	$1$	$N(\mathrm{U}(1))$	\(q+\beta q^{2}+(3+\beta )q^{4}+(5+2\beta )q^{8}+(1+\cdots)q^{11}+\cdots\)
3969.2.a.k	$3969$	$2$	3969.a	1.a	$1$	$2$	$2$	$31.693$	\(\Q(\sqrt{15}) \)	None	✓	✓			3969.2.a.g	$2$	$0$	\(2\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{4}+\beta q^{5}-3q^{8}+\beta q^{10}+\cdots\)
3969.2.a.l	$3969$	$2$	3969.a	1.a	$1$	$3$	$3$	$31.693$	\(\Q(\zeta_{18})^+\)	None	✓				63.2.f.a	$1$	$1$	\(-3\)	\(0\)	\(3\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+(1-2\beta _{1}+\beta _{2})q^{4}+\cdots\)
3969.2.a.m	$3969$	$2$	3969.a	1.a	$1$	$3$	$3$	$31.693$	3.3.321.1	None	✓				63.2.f.b	$1$	$0$	\(-1\)	\(0\)	\(5\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(1+\beta _{1}+\beta _{2})q^{4}+(2+\beta _{2})q^{5}+\cdots\)
3969.2.a.n	$3969$	$2$	3969.a	1.a	$1$	$3$	$3$	$31.693$	3.3.621.1	None	✓				567.2.a.e	$1$	$0$	\(0\)	\(0\)	\(-3\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(2+\beta _{1}+\beta _{2})q^{4}+(-1-\beta _{2})q^{5}+\cdots\)
3969.2.a.o	$3969$	$2$	3969.a	1.a	$1$	$3$	$3$	$31.693$	3.3.621.1	None	✓				567.2.a.e	$1$	$0$	\(0\)	\(0\)	\(3\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(2+\beta _{1}+\beta _{2})q^{4}+(1+\beta _{2})q^{5}+\cdots\)
3969.2.a.p	$3969$	$2$	3969.a	1.a	$1$	$3$	$3$	$31.693$	3.3.321.1	None	✓				63.2.f.b	$1$	$1$	\(1\)	\(0\)	\(-5\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(1+\beta _{1}+\beta _{2})q^{4}+(-2-\beta _{2})q^{5}+\cdots\)
3969.2.a.q	$3969$	$2$	3969.a	1.a	$1$	$3$	$3$	$31.693$	\(\Q(\zeta_{18})^+\)	None	✓				63.2.f.a	$1$	$0$	\(3\)	\(0\)	\(-3\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+(1-2\beta _{1}+\beta _{2})q^{4}+(-1+\cdots)q^{5}+\cdots\)
3969.2.a.r	$3969$	$2$	3969.a	1.a	$1$	$4$	$4$	$31.693$	\(\Q(\sqrt{2}, \sqrt{5})\)	None	✓	✓			3969.2.a.r	$2$	$1$	\(-2\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{2})q^{2}-\beta _{2}q^{4}-\beta _{1}q^{5}+(1+\cdots)q^{8}+\cdots\)
3969.2.a.s	$3969$	$2$	3969.a	1.a	$1$	$4$	$4$	$31.693$	4.4.14013.1	None	✓				567.2.e.c	$1$	$1$	\(-1\)	\(0\)	\(-2\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(1+\beta _{2})q^{4}+(-1+\beta _{1}+\beta _{2}+\cdots)q^{5}+\cdots\)
3969.2.a.t	$3969$	$2$	3969.a	1.a	$1$	$4$	$4$	$31.693$	4.4.14013.1	None	✓				567.2.e.c	$1$	$0$	\(-1\)	\(0\)	\(2\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(1+\beta _{2})q^{4}+(1-\beta _{1}-\beta _{2}+\cdots)q^{5}+\cdots\)
3969.2.a.u	$3969$	$2$	3969.a	1.a	$1$	$4$	$4$	$31.693$	\(\Q(\sqrt{3}, \sqrt{7})\)	None	✓				567.2.a.i	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{3}q^{2}-\beta _{2}q^{4}+2\beta _{1}q^{5}+(\beta _{1}+\beta _{3})q^{8}+\cdots\)
3969.2.a.v	$3969$	$2$	3969.a	1.a	$1$	$4$	$4$	$31.693$	\(\Q(\sqrt{3}, \sqrt{7})\)	\(\Q(\sqrt{-7}) \)	✓	✓			3969.2.a.v	$4$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$1$	$N(\mathrm{U}(1))$	\(q+\beta _{3}q^{2}-\beta _{2}q^{4}+(\beta _{1}+\beta _{3})q^{8}+(-\beta _{1}+\cdots)q^{11}+\cdots\)
3969.2.a.w	$3969$	$2$	3969.a	1.a	$1$	$4$	$4$	$31.693$	4.4.14013.1	None	✓				567.2.e.c	$1$	$0$	\(1\)	\(0\)	\(-2\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(1+\beta _{2})q^{4}+(-1+\beta _{1}+\beta _{2}+\cdots)q^{5}+\cdots\)
3969.2.a.x	$3969$	$2$	3969.a	1.a	$1$	$4$	$4$	$31.693$	4.4.14013.1	None	✓				567.2.e.c	$1$	$1$	\(1\)	\(0\)	\(2\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(1+\beta _{2})q^{4}+(1-\beta _{1}-\beta _{2}+\cdots)q^{5}+\cdots\)
3969.2.a.y	$3969$	$2$	3969.a	1.a	$1$	$4$	$4$	$31.693$	\(\Q(\sqrt{2}, \sqrt{5})\)	None	✓	✓			3969.2.a.r	$2$	$1$	\(2\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{2})q^{2}-\beta _{2}q^{4}-\beta _{1}q^{5}+(-1+\cdots)q^{8}+\cdots\)
3969.2.a.z	$3969$	$2$	3969.a	1.a	$1$	$5$	$5$	$31.693$	5.5.574857.1	None	✓				63.2.g.b	$1$	$1$	\(-2\)	\(0\)	\(-4\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(1+\beta _{2})q^{4}+(-1-\beta _{4})q^{5}+\cdots\)
3969.2.a.ba	$3969$	$2$	3969.a	1.a	$1$	$5$	$5$	$31.693$	5.5.574857.1	None	✓				63.2.g.b	$1$	$0$	\(-2\)	\(0\)	\(4\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(1+\beta _{2})q^{4}+(1+\beta _{4})q^{5}+\cdots\)
3969.2.a.bb	$3969$	$2$	3969.a	1.a	$1$	$5$	$5$	$31.693$	5.5.574857.1	None	✓				63.2.g.b	$1$	$1$	\(2\)	\(0\)	\(-4\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(1+\beta _{2})q^{4}+(-1-\beta _{4})q^{5}+\cdots\)
3969.2.a.bc	$3969$	$2$	3969.a	1.a	$1$	$5$	$5$	$31.693$	5.5.574857.1	None	✓				63.2.g.b	$1$	$0$	\(2\)	\(0\)	\(4\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(1+\beta _{2})q^{4}+(1+\beta _{4})q^{5}+\cdots\)
3969.2.a.bd	$3969$	$2$	3969.a	1.a	$1$	$6$	$6$	$31.693$	6.6.59351616.1	None	✓				441.2.f.g	$2$	$1$	\(-2\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$3$	$\mathrm{SU}(2)$	\(q-\beta _{2}q^{2}+(1+\beta _{2}+\beta _{4})q^{4}-\beta _{1}q^{5}+\cdots\)
3969.2.a.be	$3969$	$2$	3969.a	1.a	$1$	$6$	$6$	$31.693$	6.6.59351616.1	None	✓		✓		441.2.f.g	$2$	$0$	\(2\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$	$3$	$\mathrm{SU}(2)$	\(q+\beta _{2}q^{2}+(1+\beta _{2}+\beta _{4})q^{4}+\beta _{1}q^{5}+\cdots\)
3969.2.a.bf	$3969$	$2$	3969.a	1.a	$1$	$8$	$8$	$31.693$	8.8.\(\cdots\).1	None	✓		✓		567.2.e.g	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$3$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(1+\beta _{2})q^{4}+(-\beta _{1}+\beta _{3}+\cdots)q^{5}+\cdots\)
3969.2.a.bg	$3969$	$2$	3969.a	1.a	$1$	$8$	$8$	$31.693$	8.8.\(\cdots\).1	None	✓				567.2.e.g	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$3$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(1+\beta _{2})q^{4}+(\beta _{1}-\beta _{3})q^{5}+\cdots\)
3969.2.a.bh	$3969$	$2$	3969.a	1.a	$1$	$12$	$12$	$31.693$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None	✓		✓		441.2.f.h	$2$	$1$	\(-4\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{2}q^{2}+(1+\beta _{8})q^{4}-\beta _{10}q^{5}+(-1+\cdots)q^{8}+\cdots\)
3969.2.a.bi	$3969$	$2$	3969.a	1.a	$1$	$12$	$12$	$31.693$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None	✓				441.2.f.h	$2$	$0$	\(4\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{2}q^{2}+(1+\beta _{8})q^{4}-\beta _{10}q^{5}+(1+\cdots)q^{8}+\cdots\)
3969.2.a.bj	$3969$	$2$	3969.a	1.a	$1$	$16$	$16$	$31.693$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None	✓	✓	✓		3969.2.a.bj	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$2^{6}$	$\mathrm{SU}(2)$	\(q-\beta _{5}q^{2}+(2-\beta _{12})q^{4}+(\beta _{3}-\beta _{9})q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(3969))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(3969)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(27))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(49))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(63))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(81))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(147))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(189))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(441))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(567))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1323))\)\(^{\oplus 2}\)