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

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