Space of modular forms of level 2480, weight 2, and trivial character

• Label: 2480.2.a
• Level: $2480$
• Weight: $2$
• Character orbit: 2480.a
• Rep. character: $\chi_{2480}(1,\cdot)$
• Character field: $\Q$
• Dimension: $60$
• Newform subspaces: $28$
• Sturm bound: $768$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 2480 = 2^{4} \cdot 5 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2480.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 28 \)
Sturm bound:			\(768\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(3\), \(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(2480))\).

	Total	New	Old
Modular forms	396	60	336
Cusp forms	373	60	313
Eisenstein series	23	0	23

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

\(2\)	\(5\)	\(31\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	\(+\)	\(5\)
\(+\)	\(+\)	\(-\)	\(-\)	\(11\)
\(+\)	\(-\)	\(+\)	\(-\)	\(10\)
\(+\)	\(-\)	\(-\)	\(+\)	\(4\)
\(-\)	\(+\)	\(+\)	\(-\)	\(7\)
\(-\)	\(+\)	\(-\)	\(+\)	\(7\)
\(-\)	\(-\)	\(+\)	\(+\)	\(8\)
\(-\)	\(-\)	\(-\)	\(-\)	\(8\)
Plus space			\(+\)	\(24\)
Minus space			\(-\)	\(36\)

Trace form

60 * q + 4 * q^7 + 68 * q^9 + 8 * q^11 + 8 * q^17 + 8 * q^21 + 8 * q^23 + 60 * q^25 - 8 * q^29 - 12 * q^35 - 16 * q^37 - 24 * q^39 + 8 * q^43 + 8 * q^45 + 20 * q^47 + 68 * q^49 + 16 * q^51 - 16 * q^53 + 60 * q^63 + 8 * q^65 + 28 * q^67 + 8 * q^69 - 40 * q^71 + 24 * q^73 - 16 * q^77 + 40 * q^79 + 68 * q^81 + 16 * q^83 + 24 * q^87 + 8 * q^89 + 48 * q^91 + 16 * q^95 + 24 * q^97 + 48 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(2480))\) 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	5	31
2480.2.a.a	$2480$	$2$	2480.a	1.a	$1$	$1$	$1$	$19.803$	\(\Q\)	None	✓				1240.2.a.g	$1$	$1$	\(0\)	\(-3\)	\(1\)	\(-2\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}+q^{5}-2q^{7}+6q^{9}+2q^{11}+\cdots\)
2480.2.a.b	$2480$	$2$	2480.a	1.a	$1$	$1$	$1$	$19.803$	\(\Q\)	None	✓				155.2.a.b	$1$	$2$	\(0\)	\(-2\)	\(-1\)	\(-4\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}-q^{5}-4q^{7}+q^{9}-4q^{11}+\cdots\)
2480.2.a.c	$2480$	$2$	2480.a	1.a	$1$	$1$	$1$	$19.803$	\(\Q\)	None	✓				310.2.a.b	$1$	$1$	\(0\)	\(-2\)	\(-1\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}-q^{5}+q^{9}-2q^{11}+2q^{15}+\cdots\)
2480.2.a.d	$2480$	$2$	2480.a	1.a	$1$	$1$	$1$	$19.803$	\(\Q\)	None	✓				1240.2.a.f	$1$	$1$	\(0\)	\(-2\)	\(-1\)	\(0\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}-q^{5}+q^{9}+4q^{13}+2q^{15}+\cdots\)
2480.2.a.e	$2480$	$2$	2480.a	1.a	$1$	$1$	$1$	$19.803$	\(\Q\)	None	✓				1240.2.a.e	$1$	$0$	\(0\)	\(-2\)	\(-1\)	\(4\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}-q^{5}+4q^{7}+q^{9}-4q^{11}+\cdots\)
2480.2.a.f	$2480$	$2$	2480.a	1.a	$1$	$1$	$1$	$19.803$	\(\Q\)	None	✓				620.2.a.c	$1$	$0$	\(0\)	\(-1\)	\(-1\)	\(4\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{5}+4q^{7}-2q^{9}+2q^{13}+\cdots\)
2480.2.a.g	$2480$	$2$	2480.a	1.a	$1$	$1$	$1$	$19.803$	\(\Q\)	None	✓				1240.2.a.c	$1$	$1$	\(0\)	\(0\)	\(-1\)	\(0\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}-3q^{9}-2q^{11}+2q^{13}+8q^{19}+\cdots\)
2480.2.a.h	$2480$	$2$	2480.a	1.a	$1$	$1$	$1$	$19.803$	\(\Q\)	None	✓				1240.2.a.d	$1$	$0$	\(0\)	\(0\)	\(1\)	\(0\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}-3q^{9}+4q^{11}+2q^{13}+6q^{17}+\cdots\)
2480.2.a.i	$2480$	$2$	2480.a	1.a	$1$	$1$	$1$	$19.803$	\(\Q\)	None	✓				620.2.a.b	$1$	$0$	\(0\)	\(0\)	\(1\)	\(2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}+2q^{7}-3q^{9}+4q^{11}-4q^{13}+\cdots\)
2480.2.a.j	$2480$	$2$	2480.a	1.a	$1$	$1$	$1$	$19.803$	\(\Q\)	None	✓				1240.2.a.a	$1$	$1$	\(0\)	\(1\)	\(-1\)	\(0\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{5}-2q^{9}-2q^{13}-q^{15}+\cdots\)
2480.2.a.k	$2480$	$2$	2480.a	1.a	$1$	$1$	$1$	$19.803$	\(\Q\)	None	✓				155.2.a.c	$1$	$1$	\(0\)	\(1\)	\(-1\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{5}-2q^{9}+4q^{11}-6q^{13}+\cdots\)
2480.2.a.l	$2480$	$2$	2480.a	1.a	$1$	$1$	$1$	$19.803$	\(\Q\)	None	✓				1240.2.a.b	$1$	$1$	\(0\)	\(1\)	\(1\)	\(-2\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{5}-2q^{7}-2q^{9}+2q^{11}+\cdots\)
2480.2.a.m	$2480$	$2$	2480.a	1.a	$1$	$1$	$1$	$19.803$	\(\Q\)	None	✓				155.2.a.a	$1$	$1$	\(0\)	\(1\)	\(1\)	\(2\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{5}+2q^{7}-2q^{9}-2q^{11}+\cdots\)
2480.2.a.n	$2480$	$2$	2480.a	1.a	$1$	$1$	$1$	$19.803$	\(\Q\)	None	✓				310.2.a.a	$1$	$0$	\(0\)	\(2\)	\(-1\)	\(4\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}-q^{5}+4q^{7}+q^{9}-4q^{13}+\cdots\)
2480.2.a.o	$2480$	$2$	2480.a	1.a	$1$	$1$	$1$	$19.803$	\(\Q\)	None	✓				620.2.a.a	$1$	$0$	\(0\)	\(3\)	\(1\)	\(2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}+q^{5}+2q^{7}+6q^{9}-2q^{11}+\cdots\)
2480.2.a.p	$2480$	$2$	2480.a	1.a	$1$	$2$	$2$	$19.803$	\(\Q(\sqrt{2}) \)	None	✓		✓		1240.2.a.i	$1$	$1$	\(0\)	\(0\)	\(2\)	\(-4\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+q^{5}-2q^{7}-q^{9}+(-2+\beta )q^{11}+\cdots\)
2480.2.a.q	$2480$	$2$	2480.a	1.a	$1$	$2$	$2$	$19.803$	\(\Q(\sqrt{6}) \)	None	✓				310.2.a.d	$1$	$0$	\(0\)	\(0\)	\(2\)	\(4\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+q^{5}+2q^{7}+3q^{9}+(-2+\beta )q^{11}+\cdots\)
2480.2.a.r	$2480$	$2$	2480.a	1.a	$1$	$2$	$2$	$19.803$	\(\Q(\sqrt{17}) \)	None	✓		✓		1240.2.a.h	$1$	$1$	\(0\)	\(1\)	\(-2\)	\(0\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}-q^{5}+(1+\beta )q^{9}+(2-2\beta )q^{11}+\cdots\)
2480.2.a.s	$2480$	$2$	2480.a	1.a	$1$	$2$	$2$	$19.803$	\(\Q(\sqrt{3}) \)	None	✓				310.2.a.c	$1$	$1$	\(0\)	\(2\)	\(-2\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{3}-q^{5}-2\beta q^{7}+(1+2\beta )q^{9}+\cdots\)
2480.2.a.t	$2480$	$2$	2480.a	1.a	$1$	$3$	$3$	$19.803$	3.3.756.1	None	✓		✓		620.2.a.d	$1$	$1$	\(0\)	\(-3\)	\(-3\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{3}-q^{5}+\beta _{2}q^{7}+(2-2\beta _{1}+\cdots)q^{9}+\cdots\)
2480.2.a.u	$2480$	$2$	2480.a	1.a	$1$	$3$	$3$	$19.803$	3.3.148.1	None	✓				310.2.a.e	$1$	$1$	\(0\)	\(-2\)	\(3\)	\(0\)		$-$	$-$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{3}+q^{5}+(-\beta _{1}+\beta _{2})q^{7}+\cdots\)
2480.2.a.v	$2480$	$2$	2480.a	1.a	$1$	$3$	$3$	$19.803$	3.3.148.1	None	✓				1240.2.a.j	$1$	$0$	\(0\)	\(3\)	\(3\)	\(2\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta _{2})q^{3}+q^{5}+(1-\beta _{1}-\beta _{2})q^{7}+\cdots\)
2480.2.a.w	$2480$	$2$	2480.a	1.a	$1$	$4$	$4$	$19.803$	4.4.25492.1	None	✓		✓		620.2.a.e	$1$	$1$	\(0\)	\(-3\)	\(4\)	\(-8\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{2})q^{3}+q^{5}+(-2+\beta _{2}+\beta _{3})q^{7}+\cdots\)
2480.2.a.x	$2480$	$2$	2480.a	1.a	$1$	$4$	$4$	$19.803$	4.4.8468.1	None	✓		✓		155.2.a.e	$1$	$0$	\(0\)	\(-1\)	\(4\)	\(-2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+q^{5}+(-1+\beta _{1}-2\beta _{2}+\beta _{3})q^{7}+\cdots\)
2480.2.a.y	$2480$	$2$	2480.a	1.a	$1$	$4$	$4$	$19.803$	4.4.112820.1	None	✓				1240.2.a.k	$1$	$0$	\(0\)	\(1\)	\(-4\)	\(-4\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}-q^{5}+(-1+\beta _{1}-\beta _{3})q^{7}+\cdots\)
2480.2.a.z	$2480$	$2$	2480.a	1.a	$1$	$4$	$4$	$19.803$	4.4.20308.1	None	✓		✓		155.2.a.d	$1$	$0$	\(0\)	\(1\)	\(-4\)	\(0\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{3}q^{3}-q^{5}-\beta _{2}q^{7}+(2-\beta _{1}-2\beta _{2}+\cdots)q^{9}+\cdots\)
2480.2.a.ba	$2480$	$2$	2480.a	1.a	$1$	$6$	$6$	$19.803$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓		✓		1240.2.a.m	$1$	$0$	\(0\)	\(1\)	\(6\)	\(2\)		$+$	$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+q^{5}+\beta _{2}q^{7}+(4-\beta _{3}+\beta _{4}+\cdots)q^{9}+\cdots\)
2480.2.a.bb	$2480$	$2$	2480.a	1.a	$1$	$6$	$6$	$19.803$	6.6.473125168.1	None	✓		✓		1240.2.a.l	$1$	$0$	\(0\)	\(3\)	\(-6\)	\(4\)		$+$	$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}-q^{5}+(-\beta _{2}-\beta _{4})q^{7}+(\beta _{1}+\cdots)q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(2480))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(2480)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(31))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(62))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(80))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(124))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(155))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(248))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(310))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(496))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(620))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1240))\)\(^{\oplus 2}\)