Space of modular forms of level 4056, weight 2, and trivial character

• Label: 4056.2.a
• Level: $4056$
• Weight: $2$
• Character orbit: 4056.a
• Rep. character: $\chi_{4056}(1,\cdot)$
• Character field: $\Q$
• Dimension: $77$
• Newform subspaces: $35$
• Sturm bound: $1456$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 4056 = 2^{3} \cdot 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4056.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 35 \)
Sturm bound:			\(1456\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(4056))\).

	Total	New	Old
Modular forms	784	77	707
Cusp forms	673	77	596
Eisenstein series	111	0	111

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

\(2\)	\(3\)	\(13\)	Fricke		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(91\)	\(10\)	\(81\)		\(78\)	\(10\)	\(68\)		\(13\)	\(0\)	\(13\)
\(+\)	\(+\)	\(-\)	\(-\)		\(105\)	\(9\)	\(96\)		\(91\)	\(9\)	\(82\)		\(14\)	\(0\)	\(14\)
\(+\)	\(-\)	\(+\)	\(-\)		\(98\)	\(11\)	\(87\)		\(84\)	\(11\)	\(73\)		\(14\)	\(0\)	\(14\)
\(+\)	\(-\)	\(-\)	\(+\)		\(98\)	\(9\)	\(89\)		\(84\)	\(9\)	\(75\)		\(14\)	\(0\)	\(14\)
\(-\)	\(+\)	\(+\)	\(-\)		\(105\)	\(13\)	\(92\)		\(91\)	\(13\)	\(78\)		\(14\)	\(0\)	\(14\)
\(-\)	\(+\)	\(-\)	\(+\)		\(91\)	\(6\)	\(85\)		\(77\)	\(6\)	\(71\)		\(14\)	\(0\)	\(14\)
\(-\)	\(-\)	\(+\)	\(+\)		\(98\)	\(7\)	\(91\)		\(84\)	\(7\)	\(77\)		\(14\)	\(0\)	\(14\)
\(-\)	\(-\)	\(-\)	\(-\)		\(98\)	\(12\)	\(86\)		\(84\)	\(12\)	\(72\)		\(14\)	\(0\)	\(14\)
Plus space			\(+\)		\(378\)	\(32\)	\(346\)		\(323\)	\(32\)	\(291\)		\(55\)	\(0\)	\(55\)
Minus space			\(-\)		\(406\)	\(45\)	\(361\)		\(350\)	\(45\)	\(305\)		\(56\)	\(0\)	\(56\)

Trace form

77 * q + q^3 + 2 * q^5 + 4 * q^7 + 77 * q^9 - 4 * q^11 + 2 * q^15 - 2 * q^17 - 8 * q^19 + 4 * q^21 - 16 * q^23 + 75 * q^25 + q^27 - 6 * q^29 - 4 * q^31 - 4 * q^33 - 8 * q^35 + 14 * q^37 + 14 * q^41 + 12 * q^43 + 2 * q^45 + 16 * q^47 + 81 * q^49 + 2 * q^51 + 10 * q^53 + 8 * q^55 + 8 * q^57 + 20 * q^59 + 10 * q^61 + 4 * q^63 + 2 * q^73 - q^75 - 8 * q^77 + 28 * q^79 + 77 * q^81 - 20 * q^83 - 28 * q^85 + 10 * q^87 + 38 * q^89 - 4 * q^93 + 24 * q^95 + 10 * q^97 - 4 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(4056))\) 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	13
4056.2.a.a	$4056$	$2$	4056.a	1.a	$1$	$1$	$1$	$32.387$	\(\Q\)	None	✓				312.2.a.c	$1$	$1$	\(0\)	\(-1\)	\(-4\)	\(0\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-4q^{5}+q^{9}+2q^{11}+4q^{15}+\cdots\)
4056.2.a.b	$4056$	$2$	4056.a	1.a	$1$	$1$	$1$	$32.387$	\(\Q\)	None	✓				312.2.q.c	$1$	$0$	\(0\)	\(-1\)	\(-3\)	\(4\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-3q^{5}+4q^{7}+q^{9}-4q^{11}+\cdots\)
4056.2.a.c	$4056$	$2$	4056.a	1.a	$1$	$1$	$1$	$32.387$	\(\Q\)	None	✓				312.2.q.b	$1$	$1$	\(0\)	\(-1\)	\(-2\)	\(1\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-2q^{5}+q^{7}+q^{9}-6q^{11}+2q^{15}+\cdots\)
4056.2.a.d	$4056$	$2$	4056.a	1.a	$1$	$1$	$1$	$32.387$	\(\Q\)	None	✓				312.2.c.a	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(-2\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-2q^{7}+q^{9}-2q^{11}-2q^{17}+\cdots\)
4056.2.a.e	$4056$	$2$	4056.a	1.a	$1$	$1$	$1$	$32.387$	\(\Q\)	None	✓				312.2.c.a	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(2\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+2q^{7}+q^{9}+2q^{11}-2q^{17}+\cdots\)
4056.2.a.f	$4056$	$2$	4056.a	1.a	$1$	$1$	$1$	$32.387$	\(\Q\)	None	✓				312.2.a.b	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(4\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+4q^{7}+q^{9}+2q^{11}-6q^{17}+\cdots\)
4056.2.a.g	$4056$	$2$	4056.a	1.a	$1$	$1$	$1$	$32.387$	\(\Q\)	None	✓				312.2.a.a	$1$	$0$	\(0\)	\(-1\)	\(2\)	\(-4\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+2q^{5}-4q^{7}+q^{9}-2q^{15}+\cdots\)
4056.2.a.h	$4056$	$2$	4056.a	1.a	$1$	$1$	$1$	$32.387$	\(\Q\)	None	✓				312.2.q.b	$1$	$0$	\(0\)	\(-1\)	\(2\)	\(-1\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+2q^{5}-q^{7}+q^{9}+6q^{11}-2q^{15}+\cdots\)
4056.2.a.i	$4056$	$2$	4056.a	1.a	$1$	$1$	$1$	$32.387$	\(\Q\)	None	✓				24.2.a.a	$1$	$1$	\(0\)	\(-1\)	\(2\)	\(0\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+2q^{5}+q^{9}-4q^{11}-2q^{15}+\cdots\)
4056.2.a.j	$4056$	$2$	4056.a	1.a	$1$	$1$	$1$	$32.387$	\(\Q\)	None	✓				312.2.q.c	$1$	$1$	\(0\)	\(-1\)	\(3\)	\(-4\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+3q^{5}-4q^{7}+q^{9}+4q^{11}+\cdots\)
4056.2.a.k	$4056$	$2$	4056.a	1.a	$1$	$1$	$1$	$32.387$	\(\Q\)	None	✓				312.2.c.c	$1$	$1$	\(0\)	\(1\)	\(-4\)	\(-2\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-4q^{5}-2q^{7}+q^{9}+2q^{11}+\cdots\)
4056.2.a.l	$4056$	$2$	4056.a	1.a	$1$	$1$	$1$	$32.387$	\(\Q\)	None	✓				312.2.q.a	$1$	$1$	\(0\)	\(1\)	\(-3\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-3q^{5}+q^{9}-3q^{15}-q^{17}+\cdots\)
4056.2.a.m	$4056$	$2$	4056.a	1.a	$1$	$1$	$1$	$32.387$	\(\Q\)	None	✓				312.2.a.f	$1$	$0$	\(0\)	\(1\)	\(-2\)	\(0\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-2q^{5}+q^{9}-2q^{15}+2q^{17}+\cdots\)
4056.2.a.n	$4056$	$2$	4056.a	1.a	$1$	$1$	$1$	$32.387$	\(\Q\)	None	✓				312.2.c.b	$1$	$0$	\(0\)	\(1\)	\(-2\)	\(2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-2q^{5}+2q^{7}+q^{9}+4q^{11}+\cdots\)
4056.2.a.o	$4056$	$2$	4056.a	1.a	$1$	$1$	$1$	$32.387$	\(\Q\)	None	✓				312.2.a.e	$1$	$1$	\(0\)	\(1\)	\(0\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{9}-6q^{11}+2q^{17}+4q^{23}+\cdots\)
4056.2.a.p	$4056$	$2$	4056.a	1.a	$1$	$1$	$1$	$32.387$	\(\Q\)	None	✓				312.2.c.b	$1$	$1$	\(0\)	\(1\)	\(2\)	\(-2\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+2q^{5}-2q^{7}+q^{9}-4q^{11}+\cdots\)
4056.2.a.q	$4056$	$2$	4056.a	1.a	$1$	$1$	$1$	$32.387$	\(\Q\)	None	✓				312.2.q.a	$1$	$0$	\(0\)	\(1\)	\(3\)	\(0\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+3q^{5}+q^{9}+3q^{15}-q^{17}+\cdots\)
4056.2.a.r	$4056$	$2$	4056.a	1.a	$1$	$1$	$1$	$32.387$	\(\Q\)	None	✓				312.2.c.c	$1$	$0$	\(0\)	\(1\)	\(4\)	\(2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+4q^{5}+2q^{7}+q^{9}-2q^{11}+\cdots\)
4056.2.a.s	$4056$	$2$	4056.a	1.a	$1$	$1$	$1$	$32.387$	\(\Q\)	None	✓				312.2.a.d	$1$	$0$	\(0\)	\(1\)	\(4\)	\(4\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+4q^{5}+4q^{7}+q^{9}+2q^{11}+\cdots\)
4056.2.a.t	$4056$	$2$	4056.a	1.a	$1$	$2$	$2$	$32.387$	\(\Q(\sqrt{3}) \)	None	✓				312.2.bf.a	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(-2\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+\beta q^{5}+(-1+\beta )q^{7}+q^{9}+(-1+\cdots)q^{11}+\cdots\)
4056.2.a.u	$4056$	$2$	4056.a	1.a	$1$	$2$	$2$	$32.387$	\(\Q(\sqrt{3}) \)	None	✓				312.2.bf.a	$1$	$0$	\(0\)	\(-2\)	\(0\)	\(2\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+\beta q^{5}+(1+\beta )q^{7}+q^{9}+(1+\beta )q^{11}+\cdots\)
4056.2.a.v	$4056$	$2$	4056.a	1.a	$1$	$2$	$2$	$32.387$	\(\Q(\sqrt{13}) \)	None	✓				312.2.q.d	$1$	$1$	\(0\)	\(2\)	\(-2\)	\(2\)		$-$	$-$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{5}+(1-\beta )q^{7}+q^{9}+(-1+\cdots)q^{11}+\cdots\)
4056.2.a.w	$4056$	$2$	4056.a	1.a	$1$	$2$	$2$	$32.387$	\(\Q(\sqrt{13}) \)	None	✓				312.2.q.d	$1$	$0$	\(0\)	\(2\)	\(2\)	\(-2\)		$+$	$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{5}+(-1-\beta )q^{7}+q^{9}+(1+\cdots)q^{11}+\cdots\)
4056.2.a.x	$4056$	$2$	4056.a	1.a	$1$	$3$	$3$	$32.387$	\(\Q(\zeta_{14})^+\)	None	✓	✓	✓		4056.2.a.x	$1$	$1$	\(0\)	\(-3\)	\(-2\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+(-1-\beta _{2})q^{5}+(-1+2\beta _{1}+\cdots)q^{7}+\cdots\)
4056.2.a.y	$4056$	$2$	4056.a	1.a	$1$	$3$	$3$	$32.387$	3.3.837.1	None	✓				312.2.q.e	$1$	$1$	\(0\)	\(-3\)	\(0\)	\(-3\)		$+$	$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q-q^{3}+\beta _{2}q^{5}+(-1+\beta _{1})q^{7}+q^{9}+\cdots\)
4056.2.a.z	$4056$	$2$	4056.a	1.a	$1$	$3$	$3$	$32.387$	3.3.837.1	None	✓				312.2.q.e	$1$	$0$	\(0\)	\(-3\)	\(0\)	\(3\)		$-$	$+$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q-q^{3}-\beta _{2}q^{5}+(1-\beta _{1})q^{7}+q^{9}+(-\beta _{1}+\cdots)q^{11}+\cdots\)
4056.2.a.ba	$4056$	$2$	4056.a	1.a	$1$	$3$	$3$	$32.387$	\(\Q(\zeta_{14})^+\)	None	✓	✓			4056.2.a.x	$1$	$1$	\(0\)	\(-3\)	\(2\)	\(0\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+(1-\beta _{1})q^{5}+(-1+\beta _{1}-2\beta _{2})q^{7}+\cdots\)
4056.2.a.bb	$4056$	$2$	4056.a	1.a	$1$	$3$	$3$	$32.387$	\(\Q(\zeta_{14})^+\)	None	✓	✓			4056.2.a.bb	$1$	$1$	\(0\)	\(3\)	\(0\)	\(0\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+(1-2\beta _{1}+\beta _{2})q^{5}+(-1+2\beta _{1}+\cdots)q^{7}+\cdots\)
4056.2.a.bc	$4056$	$2$	4056.a	1.a	$1$	$3$	$3$	$32.387$	\(\Q(\zeta_{14})^+\)	None	✓	✓	✓		4056.2.a.bb	$1$	$1$	\(0\)	\(3\)	\(0\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+(-\beta _{1}-\beta _{2})q^{5}+(\beta _{1}+\beta _{2})q^{7}+\cdots\)
4056.2.a.bd	$4056$	$2$	4056.a	1.a	$1$	$4$	$4$	$32.387$	\(\Q(\sqrt{15 +4 \sqrt{3}})\)	None	✓		✓		312.2.bf.b	$1$	$1$	\(0\)	\(4\)	\(-4\)	\(-2\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+(-1+\beta _{1}-\beta _{2})q^{5}+(-1-\beta _{1}+\cdots)q^{7}+\cdots\)
4056.2.a.be	$4056$	$2$	4056.a	1.a	$1$	$4$	$4$	$32.387$	\(\Q(\sqrt{15 +4 \sqrt{3}})\)	None	✓				312.2.bf.b	$1$	$0$	\(0\)	\(4\)	\(4\)	\(2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+(1-\beta _{1}+\beta _{2})q^{5}+(1+\beta _{1}-\beta _{2}+\cdots)q^{7}+\cdots\)
4056.2.a.bf	$4056$	$2$	4056.a	1.a	$1$	$6$	$6$	$32.387$	6.6.27700337.1	None	✓	✓	✓		4056.2.a.bf	$1$	$0$	\(0\)	\(-6\)	\(-1\)	\(-7\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-\beta _{5}q^{5}+(-1-\beta _{1})q^{7}+q^{9}+\cdots\)
4056.2.a.bg	$4056$	$2$	4056.a	1.a	$1$	$6$	$6$	$32.387$	6.6.27700337.1	None	✓	✓	✓		4056.2.a.bf	$1$	$0$	\(0\)	\(-6\)	\(1\)	\(7\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+\beta _{5}q^{5}+(1+\beta _{1})q^{7}+q^{9}+(1+\cdots)q^{11}+\cdots\)
4056.2.a.bh	$4056$	$2$	4056.a	1.a	$1$	$6$	$6$	$32.387$	6.6.27700337.1	None	✓	✓	✓		4056.2.a.bh	$1$	$0$	\(0\)	\(6\)	\(-1\)	\(5\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+(\beta _{1}-\beta _{3})q^{5}+(1-\beta _{2}-\beta _{4}+\cdots)q^{7}+\cdots\)
4056.2.a.bi	$4056$	$2$	4056.a	1.a	$1$	$6$	$6$	$32.387$	6.6.27700337.1	None	✓	✓	✓		4056.2.a.bh	$1$	$0$	\(0\)	\(6\)	\(1\)	\(-5\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+(-\beta _{1}+\beta _{3})q^{5}+(-1+\beta _{2}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(4056))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(4056)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(26))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(39))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(52))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(78))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(104))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(156))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(169))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(312))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(338))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(507))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(676))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1014))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1352))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(2028))\)\(^{\oplus 2}\)