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	Dim
$+$	$+$	$+$	$+$	$10$
$+$	$+$	$-$	$-$	$9$
$+$	$-$	$+$	$-$	$11$
$+$	$-$	$-$	$+$	$9$
$-$	$+$	$+$	$-$	$13$
$-$	$+$	$-$	$+$	$6$
$-$	$-$	$+$	$+$	$7$
$-$	$-$	$-$	$-$	$12$
Plus space			$+$	$32$
Minus space			$-$	$45$

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	$1$	$4$	$4$	$32.387$	4.4.25488.1	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	$1$	$4$	$4$	$32.387$	4.4.25488.1	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}$