Space of modular forms of level 1680, weight 2, and trivial character

• Label: 1680.2.a
• Level: $1680$
• Weight: $2$
• Character orbit: 1680.a
• Rep. character: $\chi_{1680}(1,\cdot)$
• Character field: $\Q$
• Dimension: $24$
• Newform subspaces: $22$
• Sturm bound: $768$
• Trace bound: $17$

Defining parameters

Level:	\( N \)	\(=\)	\( 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1680.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 22 \)
Sturm bound:			\(768\)
Trace bound:			\(17\)
Distinguishing \(T_p\):			\(11\), \(13\), \(17\), \(19\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1680))\).

	Total	New	Old
Modular forms	408	24	384
Cusp forms	361	24	337
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.

\(2\)	\(3\)	\(5\)	\(7\)	Fricke		Total				Cusp				Eisenstein
						All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)	\(+\)		\(21\)	\(1\)	\(20\)		\(19\)	\(1\)	\(18\)		\(2\)	\(0\)	\(2\)
\(+\)	\(+\)	\(+\)	\(-\)	\(-\)		\(28\)	\(2\)	\(26\)		\(25\)	\(2\)	\(23\)		\(3\)	\(0\)	\(3\)
\(+\)	\(+\)	\(-\)	\(+\)	\(-\)		\(28\)	\(2\)	\(26\)		\(25\)	\(2\)	\(23\)		\(3\)	\(0\)	\(3\)
\(+\)	\(+\)	\(-\)	\(-\)	\(+\)		\(25\)	\(1\)	\(24\)		\(22\)	\(1\)	\(21\)		\(3\)	\(0\)	\(3\)
\(+\)	\(-\)	\(+\)	\(+\)	\(-\)		\(25\)	\(2\)	\(23\)		\(22\)	\(2\)	\(20\)		\(3\)	\(0\)	\(3\)
\(+\)	\(-\)	\(+\)	\(-\)	\(+\)		\(24\)	\(1\)	\(23\)		\(21\)	\(1\)	\(20\)		\(3\)	\(0\)	\(3\)
\(+\)	\(-\)	\(-\)	\(+\)	\(+\)		\(28\)	\(1\)	\(27\)		\(25\)	\(1\)	\(24\)		\(3\)	\(0\)	\(3\)
\(+\)	\(-\)	\(-\)	\(-\)	\(-\)		\(25\)	\(2\)	\(23\)		\(22\)	\(2\)	\(20\)		\(3\)	\(0\)	\(3\)
\(-\)	\(+\)	\(+\)	\(+\)	\(-\)		\(30\)	\(2\)	\(28\)		\(27\)	\(2\)	\(25\)		\(3\)	\(0\)	\(3\)
\(-\)	\(+\)	\(+\)	\(-\)	\(+\)		\(23\)	\(0\)	\(23\)		\(20\)	\(0\)	\(20\)		\(3\)	\(0\)	\(3\)
\(-\)	\(+\)	\(-\)	\(+\)	\(+\)		\(23\)	\(2\)	\(21\)		\(20\)	\(2\)	\(18\)		\(3\)	\(0\)	\(3\)
\(-\)	\(+\)	\(-\)	\(-\)	\(-\)		\(26\)	\(2\)	\(24\)		\(23\)	\(2\)	\(21\)		\(3\)	\(0\)	\(3\)
\(-\)	\(-\)	\(+\)	\(+\)	\(+\)		\(26\)	\(2\)	\(24\)		\(23\)	\(2\)	\(21\)		\(3\)	\(0\)	\(3\)
\(-\)	\(-\)	\(+\)	\(-\)	\(-\)		\(27\)	\(2\)	\(25\)		\(24\)	\(2\)	\(22\)		\(3\)	\(0\)	\(3\)
\(-\)	\(-\)	\(-\)	\(+\)	\(-\)		\(23\)	\(2\)	\(21\)		\(20\)	\(2\)	\(18\)		\(3\)	\(0\)	\(3\)
\(-\)	\(-\)	\(-\)	\(-\)	\(+\)		\(26\)	\(0\)	\(26\)		\(23\)	\(0\)	\(23\)		\(3\)	\(0\)	\(3\)
Plus space				\(+\)		\(196\)	\(8\)	\(188\)		\(173\)	\(8\)	\(165\)		\(23\)	\(0\)	\(23\)
Minus space				\(-\)		\(212\)	\(16\)	\(196\)		\(188\)	\(16\)	\(172\)		\(24\)	\(0\)	\(24\)

Trace form

24 * q - 4 * q^7 + 24 * q^9 - 8 * q^11 - 4 * q^15 - 8 * q^19 - 8 * q^23 + 24 * q^25 + 16 * q^29 - 8 * q^31 + 16 * q^37 + 16 * q^43 + 48 * q^47 + 24 * q^49 - 8 * q^51 + 16 * q^53 + 48 * q^59 + 16 * q^61 - 4 * q^63 + 16 * q^67 + 16 * q^69 + 40 * q^71 + 16 * q^77 + 32 * q^79 + 24 * q^81 + 48 * q^83 + 16 * q^85 - 32 * q^89 - 32 * q^97 - 8 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1680))\) 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	7
1680.2.a.a	$1680$	$2$	1680.a	1.a	$1$	$1$	$1$	$13.415$	\(\Q\)	None	✓				420.2.a.c	$1$	$0$	\(0\)	\(-1\)	\(-1\)	\(-1\)		$-$	$+$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{5}-q^{7}+q^{9}-6q^{11}-4q^{13}+\cdots\)
1680.2.a.b	$1680$	$2$	1680.a	1.a	$1$	$1$	$1$	$13.415$	\(\Q\)	None	✓				210.2.a.d	$1$	$0$	\(0\)	\(-1\)	\(-1\)	\(-1\)		$-$	$+$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{5}-q^{7}+q^{9}+2q^{13}+q^{15}+\cdots\)
1680.2.a.c	$1680$	$2$	1680.a	1.a	$1$	$1$	$1$	$13.415$	\(\Q\)	None	✓		✓	✓	840.2.a.i	$1$	$1$	\(0\)	\(-1\)	\(-1\)	\(-1\)		$+$	$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{5}-q^{7}+q^{9}+2q^{13}+q^{15}+\cdots\)
1680.2.a.d	$1680$	$2$	1680.a	1.a	$1$	$1$	$1$	$13.415$	\(\Q\)	None	✓				840.2.a.h	$1$	$0$	\(0\)	\(-1\)	\(-1\)	\(1\)		$+$	$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{5}+q^{7}+q^{9}+6q^{13}+q^{15}+\cdots\)
1680.2.a.e	$1680$	$2$	1680.a	1.a	$1$	$1$	$1$	$13.415$	\(\Q\)	None	✓				840.2.a.g	$1$	$0$	\(0\)	\(-1\)	\(-1\)	\(1\)		$+$	$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{5}+q^{7}+q^{9}+4q^{11}-6q^{13}+\cdots\)
1680.2.a.f	$1680$	$2$	1680.a	1.a	$1$	$1$	$1$	$13.415$	\(\Q\)	None	✓				105.2.a.a	$1$	$1$	\(0\)	\(-1\)	\(1\)	\(-1\)		$-$	$+$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{5}-q^{7}+q^{9}-6q^{13}-q^{15}+\cdots\)
1680.2.a.g	$1680$	$2$	1680.a	1.a	$1$	$1$	$1$	$13.415$	\(\Q\)	None	✓				210.2.a.b	$1$	$1$	\(0\)	\(-1\)	\(1\)	\(-1\)		$-$	$+$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{5}-q^{7}+q^{9}+2q^{13}-q^{15}+\cdots\)
1680.2.a.h	$1680$	$2$	1680.a	1.a	$1$	$1$	$1$	$13.415$	\(\Q\)	None	✓		✓	✓	840.2.a.j	$1$	$1$	\(0\)	\(-1\)	\(1\)	\(1\)		$+$	$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{5}+q^{7}+q^{9}-4q^{11}-2q^{13}+\cdots\)
1680.2.a.i	$1680$	$2$	1680.a	1.a	$1$	$1$	$1$	$13.415$	\(\Q\)	None	✓				420.2.a.d	$1$	$0$	\(0\)	\(-1\)	\(1\)	\(1\)		$-$	$+$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{5}+q^{7}+q^{9}-2q^{11}+4q^{13}+\cdots\)
1680.2.a.j	$1680$	$2$	1680.a	1.a	$1$	$1$	$1$	$13.415$	\(\Q\)	None	✓				210.2.a.e	$1$	$0$	\(0\)	\(-1\)	\(1\)	\(1\)		$-$	$+$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{5}+q^{7}+q^{9}+4q^{11}-2q^{13}+\cdots\)
1680.2.a.k	$1680$	$2$	1680.a	1.a	$1$	$1$	$1$	$13.415$	\(\Q\)	None	✓				840.2.a.c	$1$	$0$	\(0\)	\(1\)	\(-1\)	\(-1\)		$+$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{5}-q^{7}+q^{9}-2q^{13}-q^{15}+\cdots\)
1680.2.a.l	$1680$	$2$	1680.a	1.a	$1$	$1$	$1$	$13.415$	\(\Q\)	None	✓				840.2.a.b	$1$	$0$	\(0\)	\(1\)	\(-1\)	\(-1\)		$+$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{5}-q^{7}+q^{9}+4q^{11}+2q^{13}+\cdots\)
1680.2.a.m	$1680$	$2$	1680.a	1.a	$1$	$1$	$1$	$13.415$	\(\Q\)	None	✓		✓	✓	840.2.a.a	$1$	$1$	\(0\)	\(1\)	\(-1\)	\(1\)		$+$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{5}+q^{7}+q^{9}-4q^{11}-2q^{13}+\cdots\)
1680.2.a.n	$1680$	$2$	1680.a	1.a	$1$	$1$	$1$	$13.415$	\(\Q\)	None	✓				420.2.a.a	$1$	$0$	\(0\)	\(1\)	\(-1\)	\(1\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{5}+q^{7}+q^{9}-2q^{11}+4q^{13}+\cdots\)
1680.2.a.o	$1680$	$2$	1680.a	1.a	$1$	$1$	$1$	$13.415$	\(\Q\)	None	✓				210.2.a.a	$1$	$0$	\(0\)	\(1\)	\(-1\)	\(1\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{5}+q^{7}+q^{9}+4q^{11}-2q^{13}+\cdots\)
1680.2.a.p	$1680$	$2$	1680.a	1.a	$1$	$1$	$1$	$13.415$	\(\Q\)	None	✓		✓	✓	840.2.a.f	$1$	$1$	\(0\)	\(1\)	\(1\)	\(-1\)		$+$	$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{5}-q^{7}+q^{9}-4q^{11}-2q^{13}+\cdots\)
1680.2.a.q	$1680$	$2$	1680.a	1.a	$1$	$1$	$1$	$13.415$	\(\Q\)	None	✓				210.2.a.c	$1$	$0$	\(0\)	\(1\)	\(1\)	\(-1\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{5}-q^{7}+q^{9}-4q^{11}-2q^{13}+\cdots\)
1680.2.a.r	$1680$	$2$	1680.a	1.a	$1$	$1$	$1$	$13.415$	\(\Q\)	None	✓				420.2.a.b	$1$	$0$	\(0\)	\(1\)	\(1\)	\(-1\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{5}-q^{7}+q^{9}+2q^{11}+4q^{13}+\cdots\)
1680.2.a.s	$1680$	$2$	1680.a	1.a	$1$	$1$	$1$	$13.415$	\(\Q\)	None	✓				840.2.a.e	$1$	$0$	\(0\)	\(1\)	\(1\)	\(1\)		$+$	$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{5}+q^{7}+q^{9}+2q^{13}+q^{15}+\cdots\)
1680.2.a.t	$1680$	$2$	1680.a	1.a	$1$	$1$	$1$	$13.415$	\(\Q\)	None	✓				840.2.a.d	$1$	$0$	\(0\)	\(1\)	\(1\)	\(1\)		$+$	$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{5}+q^{7}+q^{9}+4q^{11}-2q^{13}+\cdots\)
1680.2.a.u	$1680$	$2$	1680.a	1.a	$1$	$2$	$2$	$13.415$	\(\Q(\sqrt{2}) \)	None	✓		✓	✓	840.2.a.k	$1$	$0$	\(0\)	\(-2\)	\(2\)	\(-2\)		$+$	$+$	$-$	$+$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{5}-q^{7}+q^{9}+\beta q^{11}+2q^{13}+\cdots\)
1680.2.a.v	$1680$	$2$	1680.a	1.a	$1$	$2$	$2$	$13.415$	\(\Q(\sqrt{5}) \)	None	✓		✓	✓	105.2.a.b	$1$	$1$	\(0\)	\(2\)	\(-2\)	\(-2\)		$-$	$-$	$+$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{5}-q^{7}+q^{9}+(-2-\beta )q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1680))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1680)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 16}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(35))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(42))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(48))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(56))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(70))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(80))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(84))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(105))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(112))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(120))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(140))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(168))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(210))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(240))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(280))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(336))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(420))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(560))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(840))\)\(^{\oplus 2}\)