Space of modular forms of level 3120, weight 2, and trivial character

• Label: 3120.2.a
• Level: $3120$
• Weight: $2$
• Character orbit: 3120.a
• Rep. character: $\chi_{3120}(1,\cdot)$
• Character field: $\Q$
• Dimension: $48$
• Newform subspaces: $36$
• Sturm bound: $1344$
• Trace bound: $13$

Defining parameters

Level:	\( N \)	\(=\)	\( 3120 = 2^{4} \cdot 3 \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3120.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 36 \)
Sturm bound:			\(1344\)
Trace bound:			\(13\)
Distinguishing \(T_p\):			\(7\), \(11\), \(17\), \(19\), \(31\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(3120))\).

	Total	New	Old
Modular forms	696	48	648
Cusp forms	649	48	601
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.

\(2\)	\(3\)	\(5\)	\(13\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	\(+\)	$+$	\(1\)
\(+\)	\(+\)	\(+\)	\(-\)	$-$	\(5\)
\(+\)	\(+\)	\(-\)	\(+\)	$-$	\(4\)
\(+\)	\(+\)	\(-\)	\(-\)	$+$	\(2\)
\(+\)	\(-\)	\(+\)	\(+\)	$-$	\(4\)
\(+\)	\(-\)	\(+\)	\(-\)	$+$	\(2\)
\(+\)	\(-\)	\(-\)	\(+\)	$+$	\(1\)
\(+\)	\(-\)	\(-\)	\(-\)	$-$	\(5\)
\(-\)	\(+\)	\(+\)	\(+\)	$-$	\(3\)
\(-\)	\(+\)	\(+\)	\(-\)	$+$	\(2\)
\(-\)	\(+\)	\(-\)	\(+\)	$+$	\(4\)
\(-\)	\(+\)	\(-\)	\(-\)	$-$	\(3\)
\(-\)	\(-\)	\(+\)	\(+\)	$+$	\(4\)
\(-\)	\(-\)	\(+\)	\(-\)	$-$	\(3\)
\(-\)	\(-\)	\(-\)	\(+\)	$-$	\(3\)
\(-\)	\(-\)	\(-\)	\(-\)	$+$	\(2\)
Plus space				\(+\)	\(18\)
Minus space				\(-\)	\(30\)

Trace form

\( 48 q - 8 q^{7} + 48 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(3120))\) 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
3120.2.a.a	$3120$	$2$	3120.a	1.a	$1$	$1$	$1$	$24.913$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-1\)	\(-1\)	\(-4\)		$-$	$+$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{5}-4q^{7}+q^{9}-q^{13}+q^{15}+\cdots\)
3120.2.a.b	$3120$	$2$	3120.a	1.a	$1$	$1$	$1$	$24.913$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-1\)	\(-1\)	\(-2\)		$-$	$+$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{5}-2q^{7}+q^{9}+q^{13}+q^{15}+\cdots\)
3120.2.a.c	$3120$	$2$	3120.a	1.a	$1$	$1$	$1$	$24.913$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-1\)	\(-1\)	\(0\)		$+$	$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{5}+q^{9}-q^{13}+q^{15}+2q^{17}+\cdots\)
3120.2.a.d	$3120$	$2$	3120.a	1.a	$1$	$1$	$1$	$24.913$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-1\)	\(-1\)	\(1\)		$-$	$+$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{5}+q^{7}+q^{9}-5q^{11}-q^{13}+\cdots\)
3120.2.a.e	$3120$	$2$	3120.a	1.a	$1$	$1$	$1$	$24.913$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-1\)	\(-1\)	\(1\)		$-$	$+$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{5}+q^{7}+q^{9}-3q^{11}+q^{13}+\cdots\)
3120.2.a.f	$3120$	$2$	3120.a	1.a	$1$	$1$	$1$	$24.913$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-1\)	\(-1\)	\(1\)		$+$	$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{5}+q^{7}+q^{9}+5q^{11}+q^{13}+\cdots\)
3120.2.a.g	$3120$	$2$	3120.a	1.a	$1$	$1$	$1$	$24.913$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-1\)	\(-1\)	\(2\)		$-$	$+$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{5}+2q^{7}+q^{9}+6q^{11}-q^{13}+\cdots\)
3120.2.a.h	$3120$	$2$	3120.a	1.a	$1$	$1$	$1$	$24.913$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-1\)	\(-1\)	\(4\)		$+$	$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{5}+4q^{7}+q^{9}-4q^{11}+q^{13}+\cdots\)
3120.2.a.i	$3120$	$2$	3120.a	1.a	$1$	$1$	$1$	$24.913$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-1\)	\(1\)	\(-4\)		$+$	$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{5}-4q^{7}+q^{9}+q^{13}-q^{15}+\cdots\)
3120.2.a.j	$3120$	$2$	3120.a	1.a	$1$	$1$	$1$	$24.913$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-1\)	\(1\)	\(-2\)		$-$	$+$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{5}-2q^{7}+q^{9}+q^{13}-q^{15}+\cdots\)
3120.2.a.k	$3120$	$2$	3120.a	1.a	$1$	$1$	$1$	$24.913$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-1\)	\(1\)	\(0\)		$-$	$+$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{5}+q^{9}-4q^{11}+q^{13}-q^{15}+\cdots\)
3120.2.a.l	$3120$	$2$	3120.a	1.a	$1$	$1$	$1$	$24.913$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-1\)	\(1\)	\(2\)		$+$	$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{5}+2q^{7}+q^{9}+q^{13}-q^{15}+\cdots\)
3120.2.a.m	$3120$	$2$	3120.a	1.a	$1$	$1$	$1$	$24.913$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-1\)	\(1\)	\(3\)		$+$	$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{5}+3q^{7}+q^{9}+3q^{11}-q^{13}+\cdots\)
3120.2.a.n	$3120$	$2$	3120.a	1.a	$1$	$1$	$1$	$24.913$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-1\)	\(1\)	\(3\)		$-$	$+$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{5}+3q^{7}+q^{9}+5q^{11}+q^{13}+\cdots\)
3120.2.a.o	$3120$	$2$	3120.a	1.a	$1$	$1$	$1$	$24.913$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(1\)	\(-1\)	\(-2\)		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{5}-2q^{7}+q^{9}-4q^{11}-q^{13}+\cdots\)
3120.2.a.p	$3120$	$2$	3120.a	1.a	$1$	$1$	$1$	$24.913$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(1\)	\(-1\)	\(0\)		$+$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{5}+q^{9}-4q^{11}+q^{13}-q^{15}+\cdots\)
3120.2.a.q	$3120$	$2$	3120.a	1.a	$1$	$1$	$1$	$24.913$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(1\)	\(-1\)	\(0\)		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{5}+q^{9}-q^{13}-q^{15}-6q^{17}+\cdots\)
3120.2.a.r	$3120$	$2$	3120.a	1.a	$1$	$1$	$1$	$24.913$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(1\)	\(-1\)	\(0\)		$+$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{5}+q^{9}+q^{13}-q^{15}-6q^{17}+\cdots\)
3120.2.a.s	$3120$	$2$	3120.a	1.a	$1$	$1$	$1$	$24.913$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(1\)	\(1\)	\(-4\)		$+$	$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{5}-4q^{7}+q^{9}-4q^{11}+q^{13}+\cdots\)
3120.2.a.t	$3120$	$2$	3120.a	1.a	$1$	$1$	$1$	$24.913$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(1\)	\(1\)	\(-3\)		$-$	$-$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{5}-3q^{7}+q^{9}-q^{11}+q^{13}+\cdots\)
3120.2.a.u	$3120$	$2$	3120.a	1.a	$1$	$1$	$1$	$24.913$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(1\)	\(1\)	\(-3\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{5}-3q^{7}+q^{9}+q^{11}-q^{13}+\cdots\)
3120.2.a.v	$3120$	$2$	3120.a	1.a	$1$	$1$	$1$	$24.913$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(1\)	\(1\)	\(-2\)		$+$	$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{5}-2q^{7}+q^{9}-4q^{11}-q^{13}+\cdots\)
3120.2.a.w	$3120$	$2$	3120.a	1.a	$1$	$1$	$1$	$24.913$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(1\)	\(1\)	\(0\)		$-$	$-$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{5}+q^{9}-4q^{11}+q^{13}+q^{15}+\cdots\)
3120.2.a.x	$3120$	$2$	3120.a	1.a	$1$	$1$	$1$	$24.913$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(1\)	\(1\)	\(0\)		$+$	$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{5}+q^{9}+4q^{11}+q^{13}+q^{15}+\cdots\)
3120.2.a.y	$3120$	$2$	3120.a	1.a	$1$	$1$	$1$	$24.913$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(1\)	\(1\)	\(2\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{5}+2q^{7}+q^{9}-4q^{11}-q^{13}+\cdots\)
3120.2.a.z	$3120$	$2$	3120.a	1.a	$1$	$1$	$1$	$24.913$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(1\)	\(1\)	\(2\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{5}+2q^{7}+q^{9}+2q^{11}-q^{13}+\cdots\)
3120.2.a.ba	$3120$	$2$	3120.a	1.a	$1$	$1$	$1$	$24.913$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(1\)	\(1\)	\(5\)		$+$	$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{5}+5q^{7}+q^{9}-q^{11}+q^{13}+\cdots\)
3120.2.a.bb	$3120$	$2$	3120.a	1.a	$1$	$2$	$2$	$24.913$	\(\Q(\sqrt{17}) \)	None	✓	$1$	$1$	\(0\)	\(-2\)	\(2\)	\(-3\)		$-$	$+$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{5}+(-1-\beta )q^{7}+q^{9}+(-1+\cdots)q^{11}+\cdots\)
3120.2.a.bc	$3120$	$2$	3120.a	1.a	$1$	$2$	$2$	$24.913$	\(\Q(\sqrt{2}) \)	None	✓	$1$	$1$	\(0\)	\(-2\)	\(2\)	\(0\)		$-$	$+$	$-$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{5}+\beta q^{7}+q^{9}-2\beta q^{11}+\cdots\)
3120.2.a.bd	$3120$	$2$	3120.a	1.a	$1$	$2$	$2$	$24.913$	\(\Q(\sqrt{33}) \)	None	✓	$1$	$0$	\(0\)	\(2\)	\(-2\)	\(-1\)		$+$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{5}-\beta q^{7}+q^{9}+\beta q^{11}-q^{13}+\cdots\)
3120.2.a.be	$3120$	$2$	3120.a	1.a	$1$	$2$	$2$	$24.913$	\(\Q(\sqrt{73}) \)	None	✓	$1$	$1$	\(0\)	\(2\)	\(-2\)	\(1\)		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{5}+\beta q^{7}+q^{9}-\beta q^{11}-q^{13}+\cdots\)
3120.2.a.bf	$3120$	$2$	3120.a	1.a	$1$	$2$	$2$	$24.913$	\(\Q(\sqrt{41}) \)	None	✓	$1$	$0$	\(0\)	\(2\)	\(-2\)	\(1\)		$+$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{5}+\beta q^{7}+q^{9}+(2+\beta )q^{11}+\cdots\)
3120.2.a.bg	$3120$	$2$	3120.a	1.a	$1$	$2$	$2$	$24.913$	\(\Q(\sqrt{17}) \)	None	✓	$1$	$0$	\(0\)	\(2\)	\(2\)	\(1\)		$+$	$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{5}+\beta q^{7}+q^{9}+(4-\beta )q^{11}+\cdots\)
3120.2.a.bh	$3120$	$2$	3120.a	1.a	$1$	$3$	$3$	$24.913$	3.3.1849.1	None	✓	$1$	$0$	\(0\)	\(-3\)	\(-3\)	\(-5\)		$+$	$+$	$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{5}+(-2+\beta _{1})q^{7}+q^{9}+(-1+\cdots)q^{11}+\cdots\)
3120.2.a.bi	$3120$	$2$	3120.a	1.a	$1$	$3$	$3$	$24.913$	3.3.940.1	None	✓	$1$	$0$	\(0\)	\(-3\)	\(3\)	\(-1\)		$+$	$+$	$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{5}-\beta _{2}q^{7}+q^{9}+(-2-\beta _{1}+\cdots)q^{11}+\cdots\)
3120.2.a.bj	$3120$	$2$	3120.a	1.a	$1$	$3$	$3$	$24.913$	3.3.316.1	None	✓	$1$	$0$	\(0\)	\(3\)	\(-3\)	\(-1\)		$-$	$-$	$+$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{5}+\beta _{1}q^{7}+q^{9}+\beta _{1}q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(3120))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(3120)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(26))\)\(^{\oplus 16}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(39))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(48))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(52))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(65))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(78))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(80))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(104))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(120))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(130))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(156))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(195))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(208))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(240))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(260))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(312))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(390))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(520))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(624))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(780))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1040))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1560))\)\(^{\oplus 2}\)