Space of modular forms of level 2070, weight 2, and trivial character

• Label: 2070.2.a
• Level: $2070$
• Weight: $2$
• Character orbit: 2070.a
• Rep. character: $\chi_{2070}(1,\cdot)$
• Character field: $\Q$
• Dimension: $34$
• Newform subspaces: $26$
• Sturm bound: $864$
• Trace bound: $13$

Defining parameters

Level:	$N$	$=$	$2070 = 2 \cdot 3^{2} \cdot 5 \cdot 23$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	2070.a (trivial)
Character field:			$\Q$
Newform subspaces:			$26$
Sturm bound:			$864$
Trace bound:			$13$
Distinguishing $T_p$:			$7$, $11$, $13$, $17$, $29$

Dimensions

The following table gives the dimensions of various subspaces of $M_{2}(\Gamma_0(2070))$.

	Total	New	Old
Modular forms	448	34	414
Cusp forms	417	34	383
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.

$2$	$3$	$5$	$23$	Fricke	Dim
$+$	$+$	$+$	$+$	$+$	$2$
$+$	$+$	$+$	$-$	$-$	$1$
$+$	$+$	$-$	$+$	$-$	$1$
$+$	$+$	$-$	$-$	$+$	$2$
$+$	$-$	$+$	$+$	$-$	$4$
$+$	$-$	$+$	$-$	$+$	$2$
$+$	$-$	$-$	$+$	$+$	$2$
$+$	$-$	$-$	$-$	$-$	$4$
$-$	$+$	$+$	$+$	$-$	$2$
$-$	$+$	$+$	$-$	$+$	$1$
$-$	$+$	$-$	$+$	$+$	$1$
$-$	$+$	$-$	$-$	$-$	$2$
$-$	$-$	$+$	$+$	$+$	$1$
$-$	$-$	$+$	$-$	$-$	$4$
$-$	$-$	$-$	$+$	$-$	$4$
$-$	$-$	$-$	$-$	$+$	$1$
Plus space				$+$	$12$
Minus space				$-$	$22$

Trace form

34 * q - 2 * q^2 + 34 * q^4 - 2 * q^8 - 12 * q^11 + 4 * q^13 + 8 * q^14 + 34 * q^16 + 4 * q^17 + 12 * q^19 + 20 * q^22 + 34 * q^25 + 4 * q^29 - 12 * q^31 - 2 * q^32 + 12 * q^34 - 32 * q^37 + 4 * q^38 + 8 * q^41 - 12 * q^43 - 12 * q^44 + 24 * q^47 + 46 * q^49 - 2 * q^50 + 4 * q^52 + 32 * q^53 + 8 * q^56 + 4 * q^58 + 24 * q^59 + 24 * q^61 + 34 * q^64 - 8 * q^65 + 44 * q^67 + 4 * q^68 + 4 * q^70 - 20 * q^71 + 28 * q^73 + 32 * q^74 + 12 * q^76 + 56 * q^77 + 24 * q^79 + 20 * q^82 + 20 * q^83 + 32 * q^85 - 4 * q^86 + 20 * q^88 + 20 * q^89 + 32 * q^91 - 8 * q^94 + 8 * q^95 - 4 * q^97 - 18 * q^98

Decomposition of $S_{2}^{\mathrm{new}}(\Gamma_0(2070))$ 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	5	23
2070.2.a.a	$2070$	$2$	2070.a	1.a	$1$	$1$	$1$	$16.529$	$\Q$	None	✓	✓	✓	✓	2070.2.a.a	$1$	$0$	$-1$	$0$	$-1$	$-4$		$+$	$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+q^{4}-q^{5}-4q^{7}-q^{8}+q^{10}+\cdots$
2070.2.a.b	$2070$	$2$	2070.a	1.a	$1$	$1$	$1$	$16.529$	$\Q$	None	✓				690.2.a.k	$1$	$1$	$-1$	$0$	$-1$	$0$		$+$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+q^{4}-q^{5}-q^{8}+q^{10}-4q^{11}+\cdots$
2070.2.a.c	$2070$	$2$	2070.a	1.a	$1$	$1$	$1$	$16.529$	$\Q$	None	✓				690.2.a.j	$1$	$1$	$-1$	$0$	$-1$	$0$		$+$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+q^{4}-q^{5}-q^{8}+q^{10}+2q^{11}+\cdots$
2070.2.a.d	$2070$	$2$	2070.a	1.a	$1$	$1$	$1$	$16.529$	$\Q$	None	✓	✓			2070.2.a.d	$1$	$1$	$-1$	$0$	$1$	$-4$		$+$	$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+q^{4}+q^{5}-4q^{7}-q^{8}-q^{10}+\cdots$
2070.2.a.e	$2070$	$2$	2070.a	1.a	$1$	$1$	$1$	$16.529$	$\Q$	None	✓				690.2.a.g	$1$	$1$	$-1$	$0$	$1$	$-2$		$+$	$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+q^{4}+q^{5}-2q^{7}-q^{8}-q^{10}+\cdots$
2070.2.a.f	$2070$	$2$	2070.a	1.a	$1$	$1$	$1$	$16.529$	$\Q$	None	✓	✓	✓	✓	2070.2.a.f	$1$	$0$	$-1$	$0$	$1$	$-2$		$+$	$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+q^{4}+q^{5}-2q^{7}-q^{8}-q^{10}+\cdots$
2070.2.a.g	$2070$	$2$	2070.a	1.a	$1$	$1$	$1$	$16.529$	$\Q$	None	✓				690.2.a.i	$1$	$1$	$-1$	$0$	$1$	$0$		$+$	$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+q^{4}+q^{5}-q^{8}-q^{10}-2q^{11}+\cdots$
2070.2.a.h	$2070$	$2$	2070.a	1.a	$1$	$1$	$1$	$16.529$	$\Q$	None	✓				690.2.a.h	$1$	$0$	$-1$	$0$	$1$	$0$		$+$	$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+q^{4}+q^{5}-q^{8}-q^{10}+6q^{13}+\cdots$
2070.2.a.i	$2070$	$2$	2070.a	1.a	$1$	$1$	$1$	$16.529$	$\Q$	None	✓	✓			2070.2.a.i	$1$	$1$	$-1$	$0$	$1$	$2$		$+$	$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+q^{4}+q^{5}+2q^{7}-q^{8}-q^{10}+\cdots$
2070.2.a.j	$2070$	$2$	2070.a	1.a	$1$	$1$	$1$	$16.529$	$\Q$	None	✓	✓			2070.2.a.d	$1$	$0$	$1$	$0$	$-1$	$-4$		$-$	$+$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+q^{4}-q^{5}-4q^{7}+q^{8}-q^{10}+\cdots$
2070.2.a.k	$2070$	$2$	2070.a	1.a	$1$	$1$	$1$	$16.529$	$\Q$	None	✓				690.2.a.c	$1$	$0$	$1$	$0$	$-1$	$-2$		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+q^{4}-q^{5}-2q^{7}+q^{8}-q^{10}+\cdots$
2070.2.a.l	$2070$	$2$	2070.a	1.a	$1$	$1$	$1$	$16.529$	$\Q$	None	✓	✓	✓	✓	2070.2.a.f	$1$	$1$	$1$	$0$	$-1$	$-2$		$-$	$+$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+q^{4}-q^{5}-2q^{7}+q^{8}-q^{10}+\cdots$
2070.2.a.m	$2070$	$2$	2070.a	1.a	$1$	$1$	$1$	$16.529$	$\Q$	None	✓		✓	✓	690.2.a.f	$1$	$1$	$1$	$0$	$-1$	$0$		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+q^{4}-q^{5}+q^{8}-q^{10}-4q^{11}+\cdots$
2070.2.a.n	$2070$	$2$	2070.a	1.a	$1$	$1$	$1$	$16.529$	$\Q$	None	✓	✓			2070.2.a.i	$1$	$0$	$1$	$0$	$-1$	$2$		$-$	$+$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+q^{4}-q^{5}+2q^{7}+q^{8}-q^{10}+\cdots$
2070.2.a.o	$2070$	$2$	2070.a	1.a	$1$	$1$	$1$	$16.529$	$\Q$	None	✓				690.2.a.d	$1$	$0$	$1$	$0$	$-1$	$4$		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+q^{4}-q^{5}+4q^{7}+q^{8}-q^{10}+\cdots$
2070.2.a.p	$2070$	$2$	2070.a	1.a	$1$	$1$	$1$	$16.529$	$\Q$	None	✓	✓	✓	✓	2070.2.a.a	$1$	$1$	$1$	$0$	$1$	$-4$		$-$	$+$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+q^{4}+q^{5}-4q^{7}+q^{8}+q^{10}+\cdots$
2070.2.a.q	$2070$	$2$	2070.a	1.a	$1$	$1$	$1$	$16.529$	$\Q$	None	✓		✓	✓	690.2.a.a	$1$	$1$	$1$	$0$	$1$	$-2$		$-$	$-$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+q^{4}+q^{5}-2q^{7}+q^{8}+q^{10}+\cdots$
2070.2.a.r	$2070$	$2$	2070.a	1.a	$1$	$1$	$1$	$16.529$	$\Q$	None	✓				690.2.a.e	$1$	$0$	$1$	$0$	$1$	$0$		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+q^{4}+q^{5}+q^{8}+q^{10}+4q^{11}+\cdots$
2070.2.a.s	$2070$	$2$	2070.a	1.a	$1$	$1$	$1$	$16.529$	$\Q$	None	✓				690.2.a.b	$1$	$0$	$1$	$0$	$1$	$4$		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+q^{4}+q^{5}+4q^{7}+q^{8}+q^{10}+\cdots$
2070.2.a.t	$2070$	$2$	2070.a	1.a	$1$	$2$	$2$	$16.529$	$\Q(\sqrt{17})$	None	✓				690.2.a.l	$1$	$0$	$-2$	$0$	$-2$	$-2$		$+$	$-$	$+$	$+$	$-$	$2$	$\mathrm{SU}(2)$	$q-q^{2}+q^{4}-q^{5}+(-1-\beta )q^{7}-q^{8}+\cdots$
2070.2.a.u	$2070$	$2$	2070.a	1.a	$1$	$2$	$2$	$16.529$	$\Q(\sqrt{5})$	None	✓				230.2.a.c	$1$	$0$	$-2$	$0$	$-2$	$1$		$+$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+q^{4}-q^{5}+\beta q^{7}-q^{8}+q^{10}+\cdots$
2070.2.a.v	$2070$	$2$	2070.a	1.a	$1$	$2$	$2$	$16.529$	$\Q(\sqrt{2})$	None	✓	✓	✓	✓	2070.2.a.v	$1$	$1$	$-2$	$0$	$-2$	$4$		$+$	$+$	$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	$q-q^{2}+q^{4}-q^{5}+(2+\beta )q^{7}-q^{8}+\cdots$
2070.2.a.w	$2070$	$2$	2070.a	1.a	$1$	$2$	$2$	$16.529$	$\Q(\sqrt{13})$	None	✓		✓		230.2.a.b	$1$	$0$	$2$	$0$	$-2$	$3$		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+q^{4}-q^{5}+(1+\beta )q^{7}+q^{8}+\cdots$
2070.2.a.x	$2070$	$2$	2070.a	1.a	$1$	$2$	$2$	$16.529$	$\Q(\sqrt{21})$	None	✓		✓		230.2.a.a	$1$	$0$	$2$	$0$	$2$	$1$		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+q^{4}+q^{5}+\beta q^{7}+q^{8}+q^{10}+\cdots$
2070.2.a.y	$2070$	$2$	2070.a	1.a	$1$	$2$	$2$	$16.529$	$\Q(\sqrt{2})$	None	✓	✓	✓	✓	2070.2.a.v	$1$	$0$	$2$	$0$	$2$	$4$		$-$	$+$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	$q+q^{2}+q^{4}+q^{5}+(2+\beta )q^{7}+q^{8}+\cdots$
2070.2.a.z	$2070$	$2$	2070.a	1.a	$1$	$3$	$3$	$16.529$	3.3.1101.1	None	✓		✓		230.2.a.d	$1$	$0$	$-3$	$0$	$3$	$3$		$+$	$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+q^{4}+q^{5}+(1-\beta _{1}-\beta _{2})q^{7}+\cdots$

Decomposition of $S_{2}^{\mathrm{old}}(\Gamma_0(2070))$ into lower level spaces

$S_{2}^{\mathrm{old}}(\Gamma_0(2070)) \simeq$ $S_{2}^{\mathrm{new}}(\Gamma_0(15))$$^{\oplus 8}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(23))$$^{\oplus 12}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(30))$$^{\oplus 4}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(45))$$^{\oplus 4}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(46))$$^{\oplus 6}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(69))$$^{\oplus 8}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(90))$$^{\oplus 2}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(115))$$^{\oplus 6}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(138))$$^{\oplus 4}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(207))$$^{\oplus 4}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(230))$$^{\oplus 3}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(345))$$^{\oplus 4}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(414))$$^{\oplus 2}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(690))$$^{\oplus 2}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(1035))$$^{\oplus 2}$