Space of modular forms of level 2420, weight 4, and trivial character

• Label: 2420.4.a
• Level: $2420$
• Weight: $4$
• Character orbit: 2420.a
• Rep. character: $\chi_{2420}(1,\cdot)$
• Character field: $\Q$
• Dimension: $109$
• Newform subspaces: $21$
• Sturm bound: $1584$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 2420 = 2^{2} \cdot 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2420.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 21 \)
Sturm bound:			\(1584\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(3\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(2420))\).

	Total	New	Old
Modular forms	1224	109	1115
Cusp forms	1152	109	1043
Eisenstein series	72	0	72

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

\(2\)	\(5\)	\(11\)	Fricke	Dim
\(-\)	\(+\)	\(+\)	$-$	\(24\)
\(-\)	\(+\)	\(-\)	$+$	\(30\)
\(-\)	\(-\)	\(+\)	$+$	\(30\)
\(-\)	\(-\)	\(-\)	$-$	\(25\)
Plus space			\(+\)	\(60\)
Minus space			\(-\)	\(49\)

Trace form

\( 109 q + 8 q^{3} + 5 q^{5} - 16 q^{7} + 973 q^{9} + O(q^{10}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(2420))\) 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	5	11
2420.4.a.a	$2420$	$4$	2420.a	1.a	$1$	$1$	$1$	$142.785$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-7\)	\(5\)	\(-17\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-7q^{3}+5q^{5}-17q^{7}+22q^{9}-2^{6}q^{13}+\cdots\)
2420.4.a.b	$2420$	$4$	2420.a	1.a	$1$	$1$	$1$	$142.785$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-7\)	\(5\)	\(17\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-7q^{3}+5q^{5}+17q^{7}+22q^{9}+2^{6}q^{13}+\cdots\)
2420.4.a.c	$2420$	$4$	2420.a	1.a	$1$	$1$	$1$	$142.785$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-5\)	\(5\)	\(-11\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-5q^{3}+5q^{5}-11q^{7}-2q^{9}+22q^{13}+\cdots\)
2420.4.a.d	$2420$	$4$	2420.a	1.a	$1$	$1$	$1$	$142.785$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(4\)	\(5\)	\(16\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+4q^{3}+5q^{5}+2^{4}q^{7}-11q^{9}-86q^{13}+\cdots\)
2420.4.a.e	$2420$	$4$	2420.a	1.a	$1$	$1$	$1$	$142.785$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(5\)	\(-5\)	\(19\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+5q^{3}-5q^{5}+19q^{7}-2q^{9}+62q^{13}+\cdots\)
2420.4.a.f	$2420$	$4$	2420.a	1.a	$1$	$1$	$1$	$142.785$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(8\)	\(5\)	\(-24\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+8q^{3}+5q^{5}-24q^{7}+37q^{9}+22q^{13}+\cdots\)
2420.4.a.g	$2420$	$4$	2420.a	1.a	$1$	$2$	$2$	$142.785$	\(\Q(\sqrt{97}) \)	None	✓	$1$	$1$	\(0\)	\(-9\)	\(10\)	\(15\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-4-\beta )q^{3}+5q^{5}+(8-\beta )q^{7}+(13+\cdots)q^{9}+\cdots\)
2420.4.a.h	$2420$	$4$	2420.a	1.a	$1$	$2$	$2$	$142.785$	\(\Q(\sqrt{6}) \)	None	✓	$1$	$0$	\(0\)	\(-8\)	\(-10\)	\(-36\)		$-$	$+$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(-4+\beta )q^{3}-5q^{5}+(-18+2\beta )q^{7}+\cdots\)
2420.4.a.i	$2420$	$4$	2420.a	1.a	$1$	$3$	$3$	$142.785$	3.3.9192.1	None	✓	$1$	$0$	\(0\)	\(-3\)	\(-15\)	\(5\)		$-$	$+$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{2})q^{3}-5q^{5}+(2+\beta _{1}-2\beta _{2})q^{7}+\cdots\)
2420.4.a.j	$2420$	$4$	2420.a	1.a	$1$	$3$	$3$	$142.785$	3.3.404.1	None	✓	$1$	$1$	\(0\)	\(7\)	\(15\)	\(-11\)		$-$	$-$	$-$	$-$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q+(2-\beta _{1})q^{3}+5q^{5}+(-3+2\beta _{1}+3\beta _{2})q^{7}+\cdots\)
2420.4.a.k	$2420$	$4$	2420.a	1.a	$1$	$3$	$3$	$142.785$	3.3.404.1	None	✓	$1$	$1$	\(0\)	\(7\)	\(15\)	\(11\)		$-$	$-$	$-$	$-$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q+(2-\beta _{1})q^{3}+5q^{5}+(3-2\beta _{1}-3\beta _{2})q^{7}+\cdots\)
2420.4.a.l	$2420$	$4$	2420.a	1.a	$1$	$4$	$4$	$142.785$	4.4.72684525.1	None	✓	$2$	$1$	\(0\)	\(6\)	\(-20\)	\(0\)		$-$	$+$	$+$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{3}-5q^{5}+\beta _{2}q^{7}+(-2+\cdots)q^{9}+\cdots\)
2420.4.a.m	$2420$	$4$	2420.a	1.a	$1$	$6$	$6$	$142.785$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(-30\)	\(-24\)		$-$	$+$	$-$	$+$	$2^{5}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}-5q^{5}+(-4+\beta _{1}-\beta _{4})q^{7}+\cdots\)
2420.4.a.n	$2420$	$4$	2420.a	1.a	$1$	$6$	$6$	$142.785$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(-30\)	\(24\)		$-$	$+$	$-$	$+$	$2^{5}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}-5q^{5}+(4-\beta _{1}+\beta _{4})q^{7}+\cdots\)
2420.4.a.o	$2420$	$4$	2420.a	1.a	$1$	$6$	$6$	$142.785$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	$2$	$0$	\(0\)	\(6\)	\(30\)	\(0\)		$-$	$-$	$+$	$+$	$2^{6}$	$\mathrm{SU}(2)$	\(q+(1+\beta _{1})q^{3}+5q^{5}-\beta _{2}q^{7}+(11+2\beta _{1}+\cdots)q^{9}+\cdots\)
2420.4.a.p	$2420$	$4$	2420.a	1.a	$1$	$8$	$8$	$142.785$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	$2$	$1$	\(0\)	\(0\)	\(-40\)	\(0\)		$-$	$+$	$+$	$-$	$2^{8}\cdot 3$	$\mathrm{SU}(2)$	\(q+\beta _{4}q^{3}-5q^{5}+(\beta _{1}+\beta _{6})q^{7}+(7+\beta _{2}+\cdots)q^{9}+\cdots\)
2420.4.a.q	$2420$	$4$	2420.a	1.a	$1$	$12$	$12$	$142.785$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(-1\)	\(60\)	\(-41\)		$-$	$-$	$-$	$-$	$2^{4}\cdot 11^{2}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+5q^{5}+(-3-\beta _{5})q^{7}+(9+\cdots)q^{9}+\cdots\)
2420.4.a.r	$2420$	$4$	2420.a	1.a	$1$	$12$	$12$	$142.785$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(-1\)	\(60\)	\(41\)		$-$	$-$	$+$	$+$	$2^{4}\cdot 11^{2}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+5q^{5}+(3+\beta _{5})q^{7}+(9-\beta _{1}+\cdots)q^{9}+\cdots\)
2420.4.a.s	$2420$	$4$	2420.a	1.a	$1$	$12$	$12$	$142.785$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None	✓	$2$	$0$	\(0\)	\(0\)	\(60\)	\(0\)		$-$	$-$	$+$	$+$	$2^{12}\cdot 3$	$\mathrm{SU}(2)$	\(q-\beta _{3}q^{3}+5q^{5}-\beta _{1}q^{7}+(13+\beta _{5}+\cdots)q^{9}+\cdots\)
2420.4.a.t	$2420$	$4$	2420.a	1.a	$1$	$12$	$12$	$142.785$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(3\)	\(-60\)	\(-41\)		$-$	$+$	$+$	$-$	$2^{4}\cdot 11^{3}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}-5q^{5}+(-3+\beta _{8})q^{7}+(9+\cdots)q^{9}+\cdots\)
2420.4.a.u	$2420$	$4$	2420.a	1.a	$1$	$12$	$12$	$142.785$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(3\)	\(-60\)	\(41\)		$-$	$+$	$-$	$+$	$2^{4}\cdot 11^{3}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}-5q^{5}+(3-\beta _{8})q^{7}+(9+\beta _{2}+\cdots)q^{9}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(2420))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(2420)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 9}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(22))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(44))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(55))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(110))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(121))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(220))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(242))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(484))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(605))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(1210))\)\(^{\oplus 2}\)