Space of modular forms of level 840, weight 4, and trivial character

• Label: 840.4.a
• Level: $840$
• Weight: $4$
• Character orbit: 840.a
• Rep. character: $\chi_{840}(1,\cdot)$
• Character field: $\Q$
• Dimension: $36$
• Newform subspaces: $19$
• Sturm bound: $768$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 840 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	840.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 19 \)
Sturm bound:			\(768\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(840))\).

	Total	New	Old
Modular forms	592	36	556
Cusp forms	560	36	524
Eisenstein series	32	0	32

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

\(2\)	\(3\)	\(5\)	\(7\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	\(+\)	$+$	\(2\)
\(+\)	\(+\)	\(+\)	\(-\)	$-$	\(2\)
\(+\)	\(+\)	\(-\)	\(+\)	$-$	\(2\)
\(+\)	\(+\)	\(-\)	\(-\)	$+$	\(3\)
\(+\)	\(-\)	\(+\)	\(+\)	$-$	\(2\)
\(+\)	\(-\)	\(+\)	\(-\)	$+$	\(3\)
\(+\)	\(-\)	\(-\)	\(+\)	$+$	\(3\)
\(+\)	\(-\)	\(-\)	\(-\)	$-$	\(1\)
\(-\)	\(+\)	\(+\)	\(+\)	$-$	\(3\)
\(-\)	\(+\)	\(+\)	\(-\)	$+$	\(2\)
\(-\)	\(+\)	\(-\)	\(+\)	$+$	\(2\)
\(-\)	\(+\)	\(-\)	\(-\)	$-$	\(2\)
\(-\)	\(-\)	\(+\)	\(+\)	$+$	\(2\)
\(-\)	\(-\)	\(+\)	\(-\)	$-$	\(2\)
\(-\)	\(-\)	\(-\)	\(+\)	$-$	\(2\)
\(-\)	\(-\)	\(-\)	\(-\)	$+$	\(3\)
Plus space				\(+\)	\(20\)
Minus space				\(-\)	\(16\)

Trace form

\( 36 q + 324 q^{9} + O(q^{10}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(840))\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Twist minimal	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	7
840.4.a.a	$840$	$4$	840.a	1.a	$1$	$1$	$1$	$49.562$	\(\Q\)	None	✓	✓	840.4.a.a	$1$	$0$	\(0\)	\(-3\)	\(-5\)	\(-7\)		$+$	$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}-5q^{5}-7q^{7}+9q^{9}-58q^{11}+\cdots\)
840.4.a.b	$840$	$4$	840.a	1.a	$1$	$1$	$1$	$49.562$	\(\Q\)	None	✓	✓	840.4.a.b	$1$	$0$	\(0\)	\(-3\)	\(-5\)	\(-7\)		$+$	$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}-5q^{5}-7q^{7}+9q^{9}-54q^{13}+\cdots\)
840.4.a.c	$840$	$4$	840.a	1.a	$1$	$1$	$1$	$49.562$	\(\Q\)	None	✓	✓	840.4.a.c	$1$	$1$	\(0\)	\(3\)	\(5\)	\(-7\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}+5q^{5}-7q^{7}+9q^{9}-44q^{11}+\cdots\)
840.4.a.d	$840$	$4$	840.a	1.a	$1$	$1$	$1$	$49.562$	\(\Q\)	None	✓	✓	840.4.a.d	$1$	$0$	\(0\)	\(3\)	\(5\)	\(-7\)		$+$	$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}+5q^{5}-7q^{7}+9q^{9}+20q^{11}+\cdots\)
840.4.a.e	$840$	$4$	840.a	1.a	$1$	$1$	$1$	$49.562$	\(\Q\)	None	✓	✓	840.4.a.e	$1$	$1$	\(0\)	\(3\)	\(5\)	\(-7\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}+5q^{5}-7q^{7}+9q^{9}+22q^{11}+\cdots\)
840.4.a.f	$840$	$4$	840.a	1.a	$1$	$1$	$1$	$49.562$	\(\Q\)	None	✓	✓	840.4.a.f	$1$	$1$	\(0\)	\(3\)	\(5\)	\(7\)		$+$	$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}+5q^{5}+7q^{7}+9q^{9}-2^{4}q^{11}+\cdots\)
840.4.a.g	$840$	$4$	840.a	1.a	$1$	$2$	$2$	$49.562$	\(\Q(\sqrt{29}) \)	None	✓	✓	840.4.a.g	$1$	$1$	\(0\)	\(-6\)	\(-10\)	\(14\)		$+$	$+$	$+$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q-3q^{3}-5q^{5}+7q^{7}+9q^{9}+(-14+\cdots)q^{11}+\cdots\)
840.4.a.h	$840$	$4$	840.a	1.a	$1$	$2$	$2$	$49.562$	\(\Q(\sqrt{61}) \)	None	✓	✓	840.4.a.h	$1$	$0$	\(0\)	\(-6\)	\(-10\)	\(14\)		$-$	$+$	$+$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q-3q^{3}-5q^{5}+7q^{7}+9q^{9}+(26-\beta )q^{11}+\cdots\)
840.4.a.i	$840$	$4$	840.a	1.a	$1$	$2$	$2$	$49.562$	\(\Q(\sqrt{6}) \)	None	✓	✓	840.4.a.i	$1$	$1$	\(0\)	\(-6\)	\(10\)	\(-14\)		$+$	$+$	$-$	$+$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q-3q^{3}+5q^{5}-7q^{7}+9q^{9}+(8+4\beta )q^{11}+\cdots\)
840.4.a.j	$840$	$4$	840.a	1.a	$1$	$2$	$2$	$49.562$	\(\Q(\sqrt{22}) \)	None	✓	✓	840.4.a.j	$1$	$0$	\(0\)	\(-6\)	\(10\)	\(-14\)		$-$	$+$	$-$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q-3q^{3}+5q^{5}-7q^{7}+9q^{9}+(24+2\beta )q^{11}+\cdots\)
840.4.a.k	$840$	$4$	840.a	1.a	$1$	$2$	$2$	$49.562$	\(\Q(\sqrt{401}) \)	None	✓	✓	840.4.a.k	$1$	$1$	\(0\)	\(-6\)	\(10\)	\(14\)		$-$	$+$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q-3q^{3}+5q^{5}+7q^{7}+9q^{9}+(-13+\cdots)q^{11}+\cdots\)
840.4.a.l	$840$	$4$	840.a	1.a	$1$	$2$	$2$	$49.562$	\(\Q(\sqrt{309}) \)	None	✓	✓	840.4.a.l	$1$	$0$	\(0\)	\(6\)	\(-10\)	\(-14\)		$-$	$-$	$+$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+3q^{3}-5q^{5}-7q^{7}+9q^{9}+(-2+\cdots)q^{11}+\cdots\)
840.4.a.m	$840$	$4$	840.a	1.a	$1$	$2$	$2$	$49.562$	\(\Q(\sqrt{21}) \)	None	✓	✓	840.4.a.m	$1$	$1$	\(0\)	\(6\)	\(-10\)	\(-14\)		$+$	$-$	$+$	$+$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+3q^{3}-5q^{5}-7q^{7}+9q^{9}+(14+3\beta )q^{11}+\cdots\)
840.4.a.n	$840$	$4$	840.a	1.a	$1$	$2$	$2$	$49.562$	\(\Q(\sqrt{193}) \)	None	✓	✓	840.4.a.n	$1$	$1$	\(0\)	\(6\)	\(-10\)	\(14\)		$-$	$-$	$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+3q^{3}-5q^{5}+7q^{7}+9q^{9}+(-17+\cdots)q^{11}+\cdots\)
840.4.a.o	$840$	$4$	840.a	1.a	$1$	$2$	$2$	$49.562$	\(\Q(\sqrt{2881}) \)	None	✓	✓	840.4.a.o	$1$	$0$	\(0\)	\(6\)	\(10\)	\(-14\)		$+$	$-$	$-$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+3q^{3}+5q^{5}-7q^{7}+9q^{9}+(-11+\cdots)q^{11}+\cdots\)
840.4.a.p	$840$	$4$	840.a	1.a	$1$	$3$	$3$	$49.562$	3.3.821313.1	None	✓	✓	840.4.a.p	$1$	$1$	\(0\)	\(-9\)	\(-15\)	\(-21\)		$-$	$+$	$+$	$+$	$-$	$2^{4}$	$\mathrm{SU}(2)$	\(q-3q^{3}-5q^{5}-7q^{7}+9q^{9}+(-2+\cdots)q^{11}+\cdots\)
840.4.a.q	$840$	$4$	840.a	1.a	$1$	$3$	$3$	$49.562$	3.3.661769.1	None	✓	✓	840.4.a.q	$1$	$0$	\(0\)	\(-9\)	\(15\)	\(21\)		$+$	$+$	$-$	$-$	$+$	$2^{4}$	$\mathrm{SU}(2)$	\(q-3q^{3}+5q^{5}+7q^{7}+9q^{9}+(1-\beta _{1}+\cdots)q^{11}+\cdots\)
840.4.a.r	$840$	$4$	840.a	1.a	$1$	$3$	$3$	$49.562$	3.3.851417.1	None	✓	✓	840.4.a.r	$1$	$0$	\(0\)	\(9\)	\(-15\)	\(21\)		$+$	$-$	$+$	$-$	$+$	$2^{4}$	$\mathrm{SU}(2)$	\(q+3q^{3}-5q^{5}+7q^{7}+9q^{9}+(5-\beta _{1}+\cdots)q^{11}+\cdots\)
840.4.a.s	$840$	$4$	840.a	1.a	$1$	$3$	$3$	$49.562$	3.3.79853.1	None	✓	✓	840.4.a.s	$1$	$0$	\(0\)	\(9\)	\(15\)	\(21\)		$-$	$-$	$-$	$-$	$+$	$2^{5}$	$\mathrm{SU}(2)$	\(q+3q^{3}+5q^{5}+7q^{7}+9q^{9}+(12+\beta _{1}+\cdots)q^{11}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(840))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(840)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 16}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 16}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(28))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(35))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(42))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(56))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(60))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(70))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(84))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(105))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(120))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(140))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(168))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(210))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(280))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(420))\)\(^{\oplus 2}\)