Space of modular forms of level 4680, weight 2, and trivial character

• Label: 4680.2.a
• Level: $4680$
• Weight: $2$
• Character orbit: 4680.a
• Rep. character: $\chi_{4680}(1,\cdot)$
• Character field: $\Q$
• Dimension: $60$
• Newform subspaces: $38$
• Sturm bound: $2016$
• Trace bound: $19$

Defining parameters

Level:	\( N \)	\(=\)	\( 4680 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4680.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 38 \)
Sturm bound:			\(2016\)
Trace bound:			\(19\)
Distinguishing \(T_p\):			\(7\), \(11\), \(17\), \(19\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(4680))\).

	Total	New	Old
Modular forms	1040	60	980
Cusp forms	977	60	917
Eisenstein series	63	0	63

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

\(2\)	\(3\)	\(5\)	\(13\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	\(+\)	$+$	\(3\)
\(+\)	\(+\)	\(+\)	\(-\)	$-$	\(3\)
\(+\)	\(+\)	\(-\)	\(+\)	$-$	\(4\)
\(+\)	\(+\)	\(-\)	\(-\)	$+$	\(2\)
\(+\)	\(-\)	\(+\)	\(+\)	$-$	\(5\)
\(+\)	\(-\)	\(+\)	\(-\)	$+$	\(4\)
\(+\)	\(-\)	\(-\)	\(+\)	$+$	\(4\)
\(+\)	\(-\)	\(-\)	\(-\)	$-$	\(6\)
\(-\)	\(+\)	\(+\)	\(+\)	$-$	\(4\)
\(-\)	\(+\)	\(+\)	\(-\)	$+$	\(2\)
\(-\)	\(+\)	\(-\)	\(+\)	$+$	\(3\)
\(-\)	\(+\)	\(-\)	\(-\)	$-$	\(3\)
\(-\)	\(-\)	\(+\)	\(+\)	$+$	\(4\)
\(-\)	\(-\)	\(+\)	\(-\)	$-$	\(4\)
\(-\)	\(-\)	\(-\)	\(+\)	$-$	\(4\)
\(-\)	\(-\)	\(-\)	\(-\)	$+$	\(5\)
Plus space				\(+\)	\(27\)
Minus space				\(-\)	\(33\)

Trace form

\( 60 q + 2 q^{5} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(4680))\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	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	13
4680.2.a.a	$4680$	$2$	4680.a	1.a	$1$	$1$	$1$	$37.370$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(-1\)	\(-5\)		$+$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}-5q^{7}-q^{11}+q^{13}+3q^{17}+\cdots\)
4680.2.a.b	$4680$	$2$	4680.a	1.a	$1$	$1$	$1$	$37.370$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(-1\)	\(-3\)		$-$	$+$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}-3q^{7}-3q^{11}+q^{13}+3q^{17}+\cdots\)
4680.2.a.c	$4680$	$2$	4680.a	1.a	$1$	$1$	$1$	$37.370$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(-1\)	\(-3\)		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}-3q^{7}+3q^{11}-q^{13}+q^{17}+\cdots\)
4680.2.a.d	$4680$	$2$	4680.a	1.a	$1$	$1$	$1$	$37.370$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(-1\)	\(-2\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}-2q^{7}+q^{13}+4q^{17}+6q^{19}+\cdots\)
4680.2.a.e	$4680$	$2$	4680.a	1.a	$1$	$1$	$1$	$37.370$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(-1\)	\(0\)		$+$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}-2q^{11}+q^{13}-2q^{17}+2q^{19}+\cdots\)
4680.2.a.f	$4680$	$2$	4680.a	1.a	$1$	$1$	$1$	$37.370$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(-1\)	\(0\)		$-$	$+$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}+q^{13}+6q^{17}-8q^{19}-6q^{23}+\cdots\)
4680.2.a.g	$4680$	$2$	4680.a	1.a	$1$	$1$	$1$	$37.370$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(-1\)	\(0\)		$+$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}+4q^{11}+q^{13}-2q^{17}-4q^{19}+\cdots\)
4680.2.a.h	$4680$	$2$	4680.a	1.a	$1$	$1$	$1$	$37.370$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(-1\)	\(2\)		$-$	$+$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}+2q^{7}-4q^{11}-q^{13}-8q^{17}+\cdots\)
4680.2.a.i	$4680$	$2$	4680.a	1.a	$1$	$1$	$1$	$37.370$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(-1\)	\(2\)		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}+2q^{7}-4q^{11}-q^{13}+2q^{19}+\cdots\)
4680.2.a.j	$4680$	$2$	4680.a	1.a	$1$	$1$	$1$	$37.370$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(-1\)	\(3\)		$+$	$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}+3q^{7}+5q^{11}-q^{13}-3q^{17}+\cdots\)
4680.2.a.k	$4680$	$2$	4680.a	1.a	$1$	$1$	$1$	$37.370$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(-1\)	\(4\)		$+$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}+4q^{7}-4q^{11}+q^{13}-6q^{17}+\cdots\)
4680.2.a.l	$4680$	$2$	4680.a	1.a	$1$	$1$	$1$	$37.370$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(-1\)	\(4\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}+4q^{7}+q^{13}-2q^{17}+q^{25}+\cdots\)
4680.2.a.m	$4680$	$2$	4680.a	1.a	$1$	$1$	$1$	$37.370$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(1\)	\(-4\)		$-$	$-$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}-4q^{7}-4q^{11}+q^{13}+6q^{17}+\cdots\)
4680.2.a.n	$4680$	$2$	4680.a	1.a	$1$	$1$	$1$	$37.370$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(1\)	\(-3\)		$+$	$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}-3q^{7}+3q^{11}+q^{13}-3q^{17}+\cdots\)
4680.2.a.o	$4680$	$2$	4680.a	1.a	$1$	$1$	$1$	$37.370$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(1\)	\(-1\)		$-$	$-$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}-q^{7}+5q^{11}+q^{13}-3q^{17}+\cdots\)
4680.2.a.p	$4680$	$2$	4680.a	1.a	$1$	$1$	$1$	$37.370$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(1\)	\(0\)		$-$	$-$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}-4q^{11}+q^{13}+2q^{17}-4q^{19}+\cdots\)
4680.2.a.q	$4680$	$2$	4680.a	1.a	$1$	$1$	$1$	$37.370$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(1\)	\(0\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}-q^{13}-2q^{17}-4q^{23}+q^{25}+\cdots\)
4680.2.a.r	$4680$	$2$	4680.a	1.a	$1$	$1$	$1$	$37.370$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(1\)	\(0\)		$+$	$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}+q^{13}-6q^{17}-8q^{19}+6q^{23}+\cdots\)
4680.2.a.s	$4680$	$2$	4680.a	1.a	$1$	$1$	$1$	$37.370$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(1\)	\(0\)		$+$	$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}+q^{13}+6q^{17}+4q^{19}+q^{25}+\cdots\)
4680.2.a.t	$4680$	$2$	4680.a	1.a	$1$	$1$	$1$	$37.370$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(1\)	\(0\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}+4q^{11}-q^{13}+6q^{17}+4q^{19}+\cdots\)
4680.2.a.u	$4680$	$2$	4680.a	1.a	$1$	$1$	$1$	$37.370$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(1\)	\(2\)		$+$	$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}+2q^{7}+4q^{11}-q^{13}+8q^{17}+\cdots\)
4680.2.a.v	$4680$	$2$	4680.a	1.a	$1$	$1$	$1$	$37.370$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(1\)	\(3\)		$-$	$+$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}+3q^{7}-5q^{11}-q^{13}+3q^{17}+\cdots\)
4680.2.a.w	$4680$	$2$	4680.a	1.a	$1$	$2$	$2$	$37.370$	\(\Q(\sqrt{2}) \)	None	✓	$1$	$0$	\(0\)	\(0\)	\(-2\)	\(-4\)		$+$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}-2q^{7}-3\beta q^{11}-q^{13}+(2+2\beta )q^{17}+\cdots\)
4680.2.a.x	$4680$	$2$	4680.a	1.a	$1$	$2$	$2$	$37.370$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$1$	\(0\)	\(0\)	\(-2\)	\(-4\)		$+$	$+$	$+$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q-q^{5}-2q^{7}-q^{13}+(2-\beta )q^{17}+2q^{19}+\cdots\)
4680.2.a.y	$4680$	$2$	4680.a	1.a	$1$	$2$	$2$	$37.370$	\(\Q(\sqrt{17}) \)	None	✓	$1$	$0$	\(0\)	\(0\)	\(-2\)	\(-1\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}-\beta q^{7}+(4-\beta )q^{11}+q^{13}+(2+\cdots)q^{17}+\cdots\)
4680.2.a.z	$4680$	$2$	4680.a	1.a	$1$	$2$	$2$	$37.370$	\(\Q(\sqrt{6}) \)	None	✓	$1$	$1$	\(0\)	\(0\)	\(-2\)	\(4\)		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}+2q^{7}+(2-\beta )q^{11}-q^{13}+(-2+\cdots)q^{17}+\cdots\)
4680.2.a.ba	$4680$	$2$	4680.a	1.a	$1$	$2$	$2$	$37.370$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$1$	\(0\)	\(0\)	\(2\)	\(-4\)		$-$	$+$	$-$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+q^{5}-2q^{7}-q^{13}+(-2-\beta )q^{17}+\cdots\)
4680.2.a.bb	$4680$	$2$	4680.a	1.a	$1$	$2$	$2$	$37.370$	\(\Q(\sqrt{41}) \)	None	✓	$1$	$1$	\(0\)	\(0\)	\(2\)	\(-1\)		$+$	$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}-\beta q^{7}+(2+\beta )q^{11}-q^{13}+(-2+\cdots)q^{17}+\cdots\)
4680.2.a.bc	$4680$	$2$	4680.a	1.a	$1$	$2$	$2$	$37.370$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$1$	\(0\)	\(0\)	\(2\)	\(0\)		$+$	$-$	$-$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+q^{5}+(-3-\beta )q^{11}-q^{13}-2q^{17}+\cdots\)
4680.2.a.bd	$4680$	$2$	4680.a	1.a	$1$	$2$	$2$	$37.370$	\(\Q(\sqrt{3}) \)	None	✓	$1$	$1$	\(0\)	\(0\)	\(2\)	\(0\)		$-$	$-$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}+2\beta q^{7}+(-1-\beta )q^{11}+q^{13}+\cdots\)
4680.2.a.be	$4680$	$2$	4680.a	1.a	$1$	$2$	$2$	$37.370$	\(\Q(\sqrt{3}) \)	None	✓	$1$	$0$	\(0\)	\(0\)	\(2\)	\(0\)		$+$	$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}+2\beta q^{7}+(3+\beta )q^{11}+q^{13}+\cdots\)
4680.2.a.bf	$4680$	$2$	4680.a	1.a	$1$	$2$	$2$	$37.370$	\(\Q(\sqrt{33}) \)	None	✓	$1$	$0$	\(0\)	\(0\)	\(2\)	\(1\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}+\beta q^{7}+\beta q^{11}-q^{13}+(-2+\cdots)q^{17}+\cdots\)
4680.2.a.bg	$4680$	$2$	4680.a	1.a	$1$	$3$	$3$	$37.370$	3.3.1016.1	None	✓	$1$	$0$	\(0\)	\(0\)	\(-3\)	\(-1\)		$-$	$+$	$+$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q-q^{5}-\beta _{2}q^{7}+(2-\beta _{1})q^{11}-q^{13}+\cdots\)
4680.2.a.bh	$4680$	$2$	4680.a	1.a	$1$	$3$	$3$	$37.370$	3.3.940.1	None	✓	$1$	$0$	\(0\)	\(0\)	\(-3\)	\(1\)		$+$	$-$	$+$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q-q^{5}+\beta _{2}q^{7}+(-2-\beta _{1})q^{11}-q^{13}+\cdots\)
4680.2.a.bi	$4680$	$2$	4680.a	1.a	$1$	$3$	$3$	$37.370$	3.3.621.1	None	✓	$1$	$0$	\(0\)	\(0\)	\(-3\)	\(3\)		$+$	$+$	$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q-q^{5}+(1+\beta _{2})q^{7}+(-1+\beta _{1}+\beta _{2})q^{11}+\cdots\)
4680.2.a.bj	$4680$	$2$	4680.a	1.a	$1$	$3$	$3$	$37.370$	3.3.1016.1	None	✓	$1$	$0$	\(0\)	\(0\)	\(3\)	\(-1\)		$+$	$+$	$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+q^{5}-\beta _{2}q^{7}+(-2+\beta _{1})q^{11}-q^{13}+\cdots\)
4680.2.a.bk	$4680$	$2$	4680.a	1.a	$1$	$3$	$3$	$37.370$	3.3.621.1	None	✓	$1$	$0$	\(0\)	\(0\)	\(3\)	\(3\)		$-$	$+$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+q^{5}+(1+\beta _{2})q^{7}+(1-\beta _{1}-\beta _{2})q^{11}+\cdots\)
4680.2.a.bl	$4680$	$2$	4680.a	1.a	$1$	$3$	$3$	$37.370$	3.3.1849.1	None	✓	$1$	$0$	\(0\)	\(0\)	\(3\)	\(5\)		$+$	$-$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+q^{5}+(2-\beta _{1})q^{7}+(-1-\beta _{2})q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(4680))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(4680)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 16}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(26))\)\(^{\oplus 18}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(39))\)\(^{\oplus 16}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(52))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(60))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(65))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(72))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(78))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(90))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(104))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(117))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(120))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(130))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(156))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(180))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(195))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(234))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(260))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(312))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(360))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(390))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(468))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(520))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(585))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(780))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(936))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1170))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1560))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(2340))\)\(^{\oplus 2}\)