Space of modular forms of level 3936, weight 2, and trivial character

• Label: 3936.2.a
• Level: $3936$
• Weight: $2$
• Character orbit: 3936.a
• Rep. character: $\chi_{3936}(1,\cdot)$
• Character field: $\Q$
• Dimension: $80$
• Newform subspaces: $22$
• Sturm bound: $1344$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 3936 = 2^{5} \cdot 3 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3936.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 22 \)
Sturm bound:			\(1344\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(3936))\).

	Total	New	Old
Modular forms	688	80	608
Cusp forms	657	80	577
Eisenstein series	31	0	31

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

\(2\)	\(3\)	\(41\)	Fricke		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(82\)	\(10\)	\(72\)		\(79\)	\(10\)	\(69\)		\(3\)	\(0\)	\(3\)
\(+\)	\(+\)	\(-\)	\(-\)		\(88\)	\(11\)	\(77\)		\(84\)	\(11\)	\(73\)		\(4\)	\(0\)	\(4\)
\(+\)	\(-\)	\(+\)	\(-\)		\(88\)	\(12\)	\(76\)		\(84\)	\(12\)	\(72\)		\(4\)	\(0\)	\(4\)
\(+\)	\(-\)	\(-\)	\(+\)		\(82\)	\(7\)	\(75\)		\(78\)	\(7\)	\(71\)		\(4\)	\(0\)	\(4\)
\(-\)	\(+\)	\(+\)	\(-\)		\(90\)	\(10\)	\(80\)		\(86\)	\(10\)	\(76\)		\(4\)	\(0\)	\(4\)
\(-\)	\(+\)	\(-\)	\(+\)		\(84\)	\(9\)	\(75\)		\(80\)	\(9\)	\(71\)		\(4\)	\(0\)	\(4\)
\(-\)	\(-\)	\(+\)	\(+\)		\(84\)	\(8\)	\(76\)		\(80\)	\(8\)	\(72\)		\(4\)	\(0\)	\(4\)
\(-\)	\(-\)	\(-\)	\(-\)		\(90\)	\(13\)	\(77\)		\(86\)	\(13\)	\(73\)		\(4\)	\(0\)	\(4\)
Plus space			\(+\)		\(332\)	\(34\)	\(298\)		\(317\)	\(34\)	\(283\)		\(15\)	\(0\)	\(15\)
Minus space			\(-\)		\(356\)	\(46\)	\(310\)		\(340\)	\(46\)	\(294\)		\(16\)	\(0\)	\(16\)

Trace form

\( 80 q + 80 q^{9} + 16 q^{13} + 16 q^{21} + 64 q^{25} - 16 q^{33} + 16 q^{37} + 80 q^{49} - 32 q^{53} + 16 q^{57} - 16 q^{61} + 64 q^{77} + 80 q^{81} + 96 q^{85} - 32 q^{89} + 16 q^{93} + 32 q^{97}+O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(3936))\) 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	3	41
3936.2.a.a	$3936$	$2$	3936.a	1.a	$1$	$1$	$1$	$31.429$	\(\Q\)	None	✓	✓			3936.2.a.a	$1$	$1$	\(0\)	\(-1\)	\(-2\)	\(4\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-2q^{5}+4q^{7}+q^{9}-4q^{11}+\cdots\)
3936.2.a.b	$3936$	$2$	3936.a	1.a	$1$	$1$	$1$	$31.429$	\(\Q\)	None	✓	✓			3936.2.a.b	$1$	$1$	\(0\)	\(-1\)	\(1\)	\(-2\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{5}-2q^{7}+q^{9}+2q^{11}-5q^{13}+\cdots\)
3936.2.a.c	$3936$	$2$	3936.a	1.a	$1$	$1$	$1$	$31.429$	\(\Q\)	None	✓	✓			3936.2.a.c	$1$	$0$	\(0\)	\(-1\)	\(2\)	\(-2\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+2q^{5}-2q^{7}+q^{9}-4q^{11}+\cdots\)
3936.2.a.d	$3936$	$2$	3936.a	1.a	$1$	$1$	$1$	$31.429$	\(\Q\)	None	✓	✓			3936.2.a.a	$1$	$0$	\(0\)	\(1\)	\(-2\)	\(-4\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-2q^{5}-4q^{7}+q^{9}+4q^{11}+\cdots\)
3936.2.a.e	$3936$	$2$	3936.a	1.a	$1$	$1$	$1$	$31.429$	\(\Q\)	None	✓	✓			3936.2.a.b	$1$	$0$	\(0\)	\(1\)	\(1\)	\(2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{5}+2q^{7}+q^{9}-2q^{11}-5q^{13}+\cdots\)
3936.2.a.f	$3936$	$2$	3936.a	1.a	$1$	$1$	$1$	$31.429$	\(\Q\)	None	✓	✓			3936.2.a.c	$1$	$0$	\(0\)	\(1\)	\(2\)	\(2\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+2q^{5}+2q^{7}+q^{9}+4q^{11}+\cdots\)
3936.2.a.g	$3936$	$2$	3936.a	1.a	$1$	$3$	$3$	$31.429$	3.3.148.1	None	✓	✓			3936.2.a.g	$1$	$1$	\(0\)	\(-3\)	\(-2\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+(-1+\beta _{1})q^{5}+\beta _{2}q^{7}+q^{9}+\cdots\)
3936.2.a.h	$3936$	$2$	3936.a	1.a	$1$	$3$	$3$	$31.429$	3.3.148.1	None	✓	✓			3936.2.a.g	$1$	$1$	\(0\)	\(3\)	\(-2\)	\(0\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+(-1+\beta _{1})q^{5}-\beta _{2}q^{7}+q^{9}+\cdots\)
3936.2.a.i	$3936$	$2$	3936.a	1.a	$1$	$4$	$4$	$31.429$	4.4.15188.1	None	✓	✓			3936.2.a.i	$1$	$1$	\(0\)	\(-4\)	\(-4\)	\(6\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+(-1-\beta _{3})q^{5}+(1-\beta _{2})q^{7}+\cdots\)
3936.2.a.j	$3936$	$2$	3936.a	1.a	$1$	$4$	$4$	$31.429$	4.4.202932.1	None	✓	✓	✓		3936.2.a.j	$1$	$1$	\(0\)	\(-4\)	\(-2\)	\(-2\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+(-1-\beta _{2})q^{5}+(-1+\beta _{1})q^{7}+\cdots\)
3936.2.a.k	$3936$	$2$	3936.a	1.a	$1$	$4$	$4$	$31.429$	4.4.17428.1	None	✓	✓			3936.2.a.k	$1$	$0$	\(0\)	\(-4\)	\(-1\)	\(4\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-\beta _{1}q^{5}+(1-\beta _{1}+\beta _{2})q^{7}+q^{9}+\cdots\)
3936.2.a.l	$3936$	$2$	3936.a	1.a	$1$	$4$	$4$	$31.429$	4.4.8468.1	None	✓	✓			3936.2.a.l	$1$	$0$	\(0\)	\(-4\)	\(1\)	\(0\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-\beta _{3}q^{5}+(\beta _{2}-\beta _{3})q^{7}+q^{9}+\cdots\)
3936.2.a.m	$3936$	$2$	3936.a	1.a	$1$	$4$	$4$	$31.429$	4.4.15188.1	None	✓	✓			3936.2.a.i	$1$	$1$	\(0\)	\(4\)	\(-4\)	\(-6\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+(-1-\beta _{3})q^{5}+(-1+\beta _{2})q^{7}+\cdots\)
3936.2.a.n	$3936$	$2$	3936.a	1.a	$1$	$4$	$4$	$31.429$	4.4.202932.1	None	✓	✓			3936.2.a.j	$1$	$0$	\(0\)	\(4\)	\(-2\)	\(2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+(-1-\beta _{2})q^{5}+(1-\beta _{1})q^{7}+\cdots\)
3936.2.a.o	$3936$	$2$	3936.a	1.a	$1$	$4$	$4$	$31.429$	4.4.17428.1	None	✓	✓	✓		3936.2.a.k	$1$	$1$	\(0\)	\(4\)	\(-1\)	\(-4\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-\beta _{1}q^{5}+(-1+\beta _{1}-\beta _{2})q^{7}+\cdots\)
3936.2.a.p	$3936$	$2$	3936.a	1.a	$1$	$4$	$4$	$31.429$	4.4.8468.1	None	✓	✓			3936.2.a.l	$1$	$1$	\(0\)	\(4\)	\(1\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-\beta _{3}q^{5}+(-\beta _{2}+\beta _{3})q^{7}+q^{9}+\cdots\)
3936.2.a.q	$3936$	$2$	3936.a	1.a	$1$	$5$	$5$	$31.429$	5.5.6852676.1	None	✓	✓	✓		3936.2.a.q	$1$	$0$	\(0\)	\(-5\)	\(-2\)	\(0\)		$-$	$+$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q-q^{3}-\beta _{1}q^{5}+\beta _{4}q^{7}+q^{9}+(1-\beta _{1}+\cdots)q^{11}+\cdots\)
3936.2.a.r	$3936$	$2$	3936.a	1.a	$1$	$5$	$5$	$31.429$	5.5.6852676.1	None	✓	✓			3936.2.a.q	$1$	$0$	\(0\)	\(5\)	\(-2\)	\(0\)		$+$	$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+q^{3}-\beta _{1}q^{5}-\beta _{4}q^{7}+q^{9}+(-1+\cdots)q^{11}+\cdots\)
3936.2.a.s	$3936$	$2$	3936.a	1.a	$1$	$6$	$6$	$31.429$	6.6.121506628.1	None	✓	✓	✓		3936.2.a.s	$1$	$1$	\(0\)	\(-6\)	\(3\)	\(-8\)		$+$	$+$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q-q^{3}-\beta _{4}q^{5}+(-1+\beta _{3})q^{7}+q^{9}+\cdots\)
3936.2.a.t	$3936$	$2$	3936.a	1.a	$1$	$6$	$6$	$31.429$	6.6.121506628.1	None	✓	✓	✓		3936.2.a.s	$1$	$0$	\(0\)	\(6\)	\(3\)	\(8\)		$+$	$-$	$+$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+q^{3}-\beta _{4}q^{5}+(1-\beta _{3})q^{7}+q^{9}+(-\beta _{2}+\cdots)q^{11}+\cdots\)
3936.2.a.u	$3936$	$2$	3936.a	1.a	$1$	$7$	$7$	$31.429$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	✓	✓		3936.2.a.u	$1$	$0$	\(0\)	\(-7\)	\(6\)	\(-8\)		$+$	$+$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q-q^{3}+(1+\beta _{1})q^{5}+(-1-\beta _{2})q^{7}+\cdots\)
3936.2.a.v	$3936$	$2$	3936.a	1.a	$1$	$7$	$7$	$31.429$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	✓	✓		3936.2.a.u	$1$	$0$	\(0\)	\(7\)	\(6\)	\(8\)		$-$	$-$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+q^{3}+(1+\beta _{1})q^{5}+(1+\beta _{2})q^{7}+q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(3936))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(3936)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(41))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(48))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(82))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(96))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(123))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(164))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(246))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(328))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(492))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(656))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(984))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1312))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1968))\)\(^{\oplus 2}\)