Space of modular forms of level 4140, weight 2, and trivial character

• Label: 4140.2.a
• Level: $4140$
• Weight: $2$
• Character orbit: 4140.a
• Rep. character: $\chi_{4140}(1,\cdot)$
• Character field: $\Q$
• Dimension: $38$
• Newform subspaces: $21$
• Sturm bound: $1728$
• Trace bound: $13$

Defining parameters

Level:	\( N \)	\(=\)	\( 4140 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4140.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 21 \)
Sturm bound:			\(1728\)
Trace bound:			\(13\)
Distinguishing \(T_p\):			\(7\), \(11\), \(13\), \(17\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(4140))\).

	Total	New	Old
Modular forms	888	38	850
Cusp forms	841	38	803
Eisenstein series	47	0	47

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		Total				Cusp				Eisenstein
						All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)	\(+\)		\(52\)	\(0\)	\(52\)		\(49\)	\(0\)	\(49\)		\(3\)	\(0\)	\(3\)
\(+\)	\(+\)	\(+\)	\(-\)	\(-\)		\(58\)	\(0\)	\(58\)		\(54\)	\(0\)	\(54\)		\(4\)	\(0\)	\(4\)
\(+\)	\(+\)	\(-\)	\(+\)	\(-\)		\(60\)	\(0\)	\(60\)		\(56\)	\(0\)	\(56\)		\(4\)	\(0\)	\(4\)
\(+\)	\(+\)	\(-\)	\(-\)	\(+\)		\(54\)	\(0\)	\(54\)		\(50\)	\(0\)	\(50\)		\(4\)	\(0\)	\(4\)
\(+\)	\(-\)	\(+\)	\(+\)	\(-\)		\(59\)	\(0\)	\(59\)		\(55\)	\(0\)	\(55\)		\(4\)	\(0\)	\(4\)
\(+\)	\(-\)	\(+\)	\(-\)	\(+\)		\(53\)	\(0\)	\(53\)		\(49\)	\(0\)	\(49\)		\(4\)	\(0\)	\(4\)
\(+\)	\(-\)	\(-\)	\(+\)	\(+\)		\(53\)	\(0\)	\(53\)		\(49\)	\(0\)	\(49\)		\(4\)	\(0\)	\(4\)
\(+\)	\(-\)	\(-\)	\(-\)	\(-\)		\(59\)	\(0\)	\(59\)		\(55\)	\(0\)	\(55\)		\(4\)	\(0\)	\(4\)
\(-\)	\(+\)	\(+\)	\(+\)	\(-\)		\(56\)	\(5\)	\(51\)		\(54\)	\(5\)	\(49\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(+\)	\(-\)	\(+\)		\(56\)	\(3\)	\(53\)		\(54\)	\(3\)	\(51\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(-\)	\(+\)	\(+\)		\(54\)	\(3\)	\(51\)		\(52\)	\(3\)	\(49\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(-\)	\(-\)	\(-\)		\(54\)	\(5\)	\(49\)		\(52\)	\(5\)	\(47\)		\(2\)	\(0\)	\(2\)
\(-\)	\(-\)	\(+\)	\(+\)	\(+\)		\(55\)	\(5\)	\(50\)		\(53\)	\(5\)	\(48\)		\(2\)	\(0\)	\(2\)
\(-\)	\(-\)	\(+\)	\(-\)	\(-\)		\(55\)	\(6\)	\(49\)		\(53\)	\(6\)	\(47\)		\(2\)	\(0\)	\(2\)
\(-\)	\(-\)	\(-\)	\(+\)	\(-\)		\(55\)	\(6\)	\(49\)		\(53\)	\(6\)	\(47\)		\(2\)	\(0\)	\(2\)
\(-\)	\(-\)	\(-\)	\(-\)	\(+\)		\(55\)	\(5\)	\(50\)		\(53\)	\(5\)	\(48\)		\(2\)	\(0\)	\(2\)
Plus space				\(+\)		\(432\)	\(16\)	\(416\)		\(409\)	\(16\)	\(393\)		\(23\)	\(0\)	\(23\)
Minus space				\(-\)		\(456\)	\(22\)	\(434\)		\(432\)	\(22\)	\(410\)		\(24\)	\(0\)	\(24\)

Trace form

38 * q - 4 * q^7 + 4 * q^17 + 8 * q^19 + 38 * q^25 + 10 * q^29 + 14 * q^31 - 6 * q^35 + 12 * q^37 - 10 * q^41 + 32 * q^43 - 12 * q^47 + 40 * q^49 - 4 * q^53 - 6 * q^59 - 4 * q^61 + 4 * q^65 + 4 * q^67 - 38 * q^71 + 12 * q^73 - 4 * q^77 - 28 * q^79 + 20 * q^83 - 10 * q^85 + 20 * q^89 + 12 * q^91 + 8 * q^95 - 8 * q^97

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(4140))\) 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
4140.2.a.a	$4140$	$2$	4140.a	1.a	$1$	$1$	$1$	$33.058$	\(\Q\)	None	✓				1380.2.a.d	$1$	$0$	\(0\)	\(0\)	\(-1\)	\(-4\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}-4q^{7}+2q^{13}-6q^{17}+2q^{19}+\cdots\)
4140.2.a.b	$4140$	$2$	4140.a	1.a	$1$	$1$	$1$	$33.058$	\(\Q\)	None	✓				1380.2.a.e	$1$	$1$	\(0\)	\(0\)	\(-1\)	\(-3\)		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}-3q^{7}+2q^{11}-2q^{13}+7q^{17}+\cdots\)
4140.2.a.c	$4140$	$2$	4140.a	1.a	$1$	$1$	$1$	$33.058$	\(\Q\)	None	✓	✓			4140.2.a.c	$1$	$1$	\(0\)	\(0\)	\(-1\)	\(-2\)		$-$	$+$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}-2q^{7}+4q^{11}-2q^{13}-6q^{17}+\cdots\)
4140.2.a.d	$4140$	$2$	4140.a	1.a	$1$	$1$	$1$	$33.058$	\(\Q\)	None	✓				460.2.a.a	$1$	$0$	\(0\)	\(0\)	\(-1\)	\(-2\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}-2q^{7}+4q^{11}+q^{13}-4q^{19}+\cdots\)
4140.2.a.e	$4140$	$2$	4140.a	1.a	$1$	$1$	$1$	$33.058$	\(\Q\)	None	✓				1380.2.a.a	$1$	$1$	\(0\)	\(0\)	\(1\)	\(-5\)		$-$	$-$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}-5q^{7}+4q^{13}+3q^{17}-4q^{19}+\cdots\)
4140.2.a.f	$4140$	$2$	4140.a	1.a	$1$	$1$	$1$	$33.058$	\(\Q\)	None	✓				460.2.a.c	$1$	$0$	\(0\)	\(0\)	\(1\)	\(-4\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}-4q^{7}+6q^{11}-q^{13}+2q^{19}+\cdots\)
4140.2.a.g	$4140$	$2$	4140.a	1.a	$1$	$1$	$1$	$33.058$	\(\Q\)	None	✓	✓			4140.2.a.c	$1$	$1$	\(0\)	\(0\)	\(1\)	\(-2\)		$-$	$+$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}-2q^{7}-4q^{11}-2q^{13}+6q^{17}+\cdots\)
4140.2.a.h	$4140$	$2$	4140.a	1.a	$1$	$1$	$1$	$33.058$	\(\Q\)	None	✓				460.2.a.b	$1$	$1$	\(0\)	\(0\)	\(1\)	\(-1\)		$-$	$-$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}-q^{7}-6q^{11}+6q^{13}-7q^{17}+\cdots\)
4140.2.a.i	$4140$	$2$	4140.a	1.a	$1$	$1$	$1$	$33.058$	\(\Q\)	None	✓				1380.2.a.c	$1$	$1$	\(0\)	\(0\)	\(1\)	\(-1\)		$-$	$-$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}-q^{7}-4q^{13}+3q^{17}-4q^{19}+\cdots\)
4140.2.a.j	$4140$	$2$	4140.a	1.a	$1$	$1$	$1$	$33.058$	\(\Q\)	None	✓				1380.2.a.b	$1$	$1$	\(0\)	\(0\)	\(1\)	\(0\)		$-$	$-$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}-6q^{13}-2q^{17}+6q^{19}+q^{23}+\cdots\)
4140.2.a.k	$4140$	$2$	4140.a	1.a	$1$	$1$	$1$	$33.058$	\(\Q\)	None	✓				460.2.a.d	$1$	$1$	\(0\)	\(0\)	\(1\)	\(2\)		$-$	$-$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}+2q^{7}-3q^{13}-4q^{17}-4q^{19}+\cdots\)
4140.2.a.l	$4140$	$2$	4140.a	1.a	$1$	$2$	$2$	$33.058$	\(\Q(\sqrt{6}) \)	None	✓	✓	✓		4140.2.a.l	$1$	$1$	\(0\)	\(0\)	\(-2\)	\(-2\)		$-$	$+$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}+(-1+\beta )q^{7}+\beta q^{11}+(2-\beta )q^{13}+\cdots\)
4140.2.a.m	$4140$	$2$	4140.a	1.a	$1$	$2$	$2$	$33.058$	\(\Q(\sqrt{17}) \)	None	✓				460.2.a.e	$1$	$1$	\(0\)	\(0\)	\(-2\)	\(1\)		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}+\beta q^{7}-2q^{11}+(-1-\beta )q^{13}+\cdots\)
4140.2.a.n	$4140$	$2$	4140.a	1.a	$1$	$2$	$2$	$33.058$	\(\Q(\sqrt{17}) \)	None	✓				1380.2.a.g	$1$	$1$	\(0\)	\(0\)	\(-2\)	\(1\)		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}+\beta q^{7}-2\beta q^{11}+2q^{13}+(-2+\cdots)q^{17}+\cdots\)
4140.2.a.o	$4140$	$2$	4140.a	1.a	$1$	$2$	$2$	$33.058$	\(\Q(\sqrt{3}) \)	None	✓				1380.2.a.h	$1$	$0$	\(0\)	\(0\)	\(-2\)	\(2\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}+(1+2\beta )q^{7}+(1+3\beta )q^{11}+(-5+\cdots)q^{13}+\cdots\)
4140.2.a.p	$4140$	$2$	4140.a	1.a	$1$	$2$	$2$	$33.058$	\(\Q(\sqrt{15}) \)	None	✓				1380.2.a.i	$1$	$0$	\(0\)	\(0\)	\(-2\)	\(6\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}+3q^{7}+(-1-\beta )q^{11}+(1-\beta )q^{13}+\cdots\)
4140.2.a.q	$4140$	$2$	4140.a	1.a	$1$	$2$	$2$	$33.058$	\(\Q(\sqrt{6}) \)	None	✓	✓	✓		4140.2.a.l	$1$	$1$	\(0\)	\(0\)	\(2\)	\(-2\)		$-$	$+$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}+(-1+\beta )q^{7}-\beta q^{11}+(2-\beta )q^{13}+\cdots\)
4140.2.a.r	$4140$	$2$	4140.a	1.a	$1$	$2$	$2$	$33.058$	\(\Q(\sqrt{6}) \)	None	✓				1380.2.a.f	$1$	$0$	\(0\)	\(0\)	\(2\)	\(2\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}+q^{7}+(2-\beta )q^{11}-\beta q^{13}+(1+\cdots)q^{17}+\cdots\)
4140.2.a.s	$4140$	$2$	4140.a	1.a	$1$	$3$	$3$	$33.058$	3.3.3144.1	None	✓		✓		1380.2.a.j	$1$	$0$	\(0\)	\(0\)	\(3\)	\(2\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}+(1-\beta _{1})q^{7}+(-1-\beta _{2})q^{11}+\cdots\)
4140.2.a.t	$4140$	$2$	4140.a	1.a	$1$	$5$	$5$	$33.058$	5.5.14345904.1	None	✓	✓	✓	✓	4140.2.a.t	$1$	$0$	\(0\)	\(0\)	\(-5\)	\(4\)		$-$	$+$	$+$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q-q^{5}+(1+\beta _{3})q^{7}+(-1+\beta _{4})q^{11}+\cdots\)
4140.2.a.u	$4140$	$2$	4140.a	1.a	$1$	$5$	$5$	$33.058$	5.5.14345904.1	None	✓	✓	✓	✓	4140.2.a.t	$1$	$0$	\(0\)	\(0\)	\(5\)	\(4\)		$-$	$+$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+q^{5}+(1+\beta _{3})q^{7}+(1-\beta _{4})q^{11}+(1+\cdots)q^{13}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(4140))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(4140)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(23))\)\(^{\oplus 18}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(46))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(69))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(90))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(92))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(115))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(138))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(180))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(207))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(230))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(276))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(345))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(414))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(460))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(690))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(828))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1035))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1380))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(2070))\)\(^{\oplus 2}\)