Space of modular forms of level 5040, weight 2, and Character 5040.t

• Label: 5040.2.t
• Level: $5040$
• Weight: $2$
• Character orbit: 5040.t
• Rep. character: $\chi_{5040}(1009,\cdot)$
• Character field: $\Q$
• Dimension: $90$
• Newform subspaces: $28$
• Sturm bound: $2304$
• Trace bound: $19$

Defining parameters

Level:	\( N \)	\(=\)	\( 5040 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5040.t (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 5 \)
Character field:			\(\Q\)
Newform subspaces:			\( 28 \)
Sturm bound:			\(2304\)
Trace bound:			\(19\)
Distinguishing \(T_p\):			\(11\), \(13\), \(17\), \(19\), \(29\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(5040, [\chi])\).

	Total	New	Old
Modular forms	1200	90	1110
Cusp forms	1104	90	1014
Eisenstein series	96	0	96

Trace form

\( 90 q + 2 q^{5} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(5040, [\chi])\) 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				Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
5040.2.t.a	$5040$	$2$	5040.t	5.b	$2$	$2$	$2$	$40.245$	\(\Q(\sqrt{-1}) \)	None					280.2.g.a	$2$	$0$	\(0\)	\(0\)	\(-4\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+(i-2)q^{5}-i q^{7}-q^{11}-i q^{13}+\cdots\)
5040.2.t.b	$5040$	$2$	5040.t	5.b	$2$	$2$	$2$	$40.245$	\(\Q(\sqrt{-1}) \)	None					630.2.g.a	$2$	$0$	\(0\)	\(0\)	\(-4\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-i-2)q^{5}-i q^{7}-4 i q^{13}+\cdots\)
5040.2.t.c	$5040$	$2$	5040.t	5.b	$2$	$2$	$2$	$40.245$	\(\Q(\sqrt{-1}) \)	None					315.2.d.b	$2$	$0$	\(0\)	\(0\)	\(-4\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+(i-2)q^{5}-i q^{7}+4 i q^{13}+2 i q^{17}+\cdots\)
5040.2.t.d	$5040$	$2$	5040.t	5.b	$2$	$2$	$2$	$40.245$	\(\Q(\sqrt{-1}) \)	None					420.2.k.b	$2$	$0$	\(0\)	\(0\)	\(-4\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-i-2)q^{5}+i q^{7}+4 q^{11}+2 i q^{13}+\cdots\)
5040.2.t.e	$5040$	$2$	5040.t	5.b	$2$	$2$	$2$	$40.245$	\(\Q(\sqrt{-1}) \)	None					105.2.d.a	$2$	$0$	\(0\)	\(0\)	\(-2\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+(2 i-1)q^{5}-i q^{7}-6 q^{11}-2 i q^{13}+\cdots\)
5040.2.t.f	$5040$	$2$	5040.t	5.b	$2$	$2$	$2$	$40.245$	\(\Q(\sqrt{-1}) \)	None					2520.2.t.b	$2$	$0$	\(0\)	\(0\)	\(-2\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-2 i-1)q^{5}+i q^{7}-2 q^{11}+\cdots\)
5040.2.t.g	$5040$	$2$	5040.t	5.b	$2$	$2$	$2$	$40.245$	\(\Q(\sqrt{-1}) \)	None					140.2.e.b	$2$	$0$	\(0\)	\(0\)	\(-2\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-2 i-1)q^{5}-i q^{7}-4 i q^{13}+\cdots\)
5040.2.t.h	$5040$	$2$	5040.t	5.b	$2$	$2$	$2$	$40.245$	\(\Q(\sqrt{-1}) \)	None					210.2.g.b	$2$	$0$	\(0\)	\(0\)	\(-2\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-2 i-1)q^{5}-i q^{7}+2 q^{11}+\cdots\)
5040.2.t.i	$5040$	$2$	5040.t	5.b	$2$	$2$	$2$	$40.245$	\(\Q(\sqrt{-1}) \)	None					630.2.g.b	$2$	$0$	\(0\)	\(0\)	\(-2\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-2 i-1)q^{5}-i q^{7}+6 q^{11}+\cdots\)
5040.2.t.j	$5040$	$2$	5040.t	5.b	$2$	$2$	$2$	$40.245$	\(\Q(\sqrt{-1}) \)	None					630.2.g.b	$2$	$1$	\(0\)	\(0\)	\(2\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+(2 i+1)q^{5}-i q^{7}-6 q^{11}+2 i q^{13}+\cdots\)
5040.2.t.k	$5040$	$2$	5040.t	5.b	$2$	$2$	$2$	$40.245$	\(\Q(\sqrt{-1}) \)	None					210.2.g.a	$2$	$0$	\(0\)	\(0\)	\(2\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+(2 i+1)q^{5}+i q^{7}-2 q^{11}+2 i q^{13}+\cdots\)
5040.2.t.l	$5040$	$2$	5040.t	5.b	$2$	$2$	$2$	$40.245$	\(\Q(\sqrt{-1}) \)	None					840.2.t.b	$2$	$1$	\(0\)	\(0\)	\(2\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-2 i+1)q^{5}+i q^{7}+2 q^{11}+\cdots\)
5040.2.t.m	$5040$	$2$	5040.t	5.b	$2$	$2$	$2$	$40.245$	\(\Q(\sqrt{-1}) \)	None					2520.2.t.b	$2$	$1$	\(0\)	\(0\)	\(2\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+(2 i+1)q^{5}+i q^{7}+2 q^{11}-2 i q^{13}+\cdots\)
5040.2.t.n	$5040$	$2$	5040.t	5.b	$2$	$2$	$2$	$40.245$	\(\Q(\sqrt{-1}) \)	None					840.2.t.a	$2$	$0$	\(0\)	\(0\)	\(2\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+(2 i+1)q^{5}+i q^{7}+2 q^{11}-2 i q^{13}+\cdots\)
5040.2.t.o	$5040$	$2$	5040.t	5.b	$2$	$2$	$2$	$40.245$	\(\Q(\sqrt{-1}) \)	None					420.2.k.a	$2$	$0$	\(0\)	\(0\)	\(4\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+(i+2)q^{5}-i q^{7}-4 q^{11}+6 i q^{13}+\cdots\)
5040.2.t.p	$5040$	$2$	5040.t	5.b	$2$	$2$	$2$	$40.245$	\(\Q(\sqrt{-1}) \)	None					35.2.b.a	$2$	$0$	\(0\)	\(0\)	\(4\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+(i+2)q^{5}+i q^{7}-3 q^{11}-i q^{13}+\cdots\)
5040.2.t.q	$5040$	$2$	5040.t	5.b	$2$	$2$	$2$	$40.245$	\(\Q(\sqrt{-1}) \)	None					630.2.g.a	$2$	$0$	\(0\)	\(0\)	\(4\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-i+2)q^{5}+i q^{7}+4 i q^{13}+\cdots\)
5040.2.t.r	$5040$	$2$	5040.t	5.b	$2$	$2$	$2$	$40.245$	\(\Q(\sqrt{-1}) \)	None					315.2.d.b	$2$	$0$	\(0\)	\(0\)	\(4\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+(i+2)q^{5}+i q^{7}-4 i q^{13}+2 i q^{17}+\cdots\)
5040.2.t.s	$5040$	$2$	5040.t	5.b	$2$	$2$	$2$	$40.245$	\(\Q(\sqrt{-1}) \)	None					140.2.e.a	$2$	$0$	\(0\)	\(0\)	\(4\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-i+2)q^{5}+i q^{7}+3 q^{11}+i q^{13}+\cdots\)
5040.2.t.t	$5040$	$2$	5040.t	5.b	$2$	$4$	$4$	$40.245$	\(\Q(i, \sqrt{6})\)	None					70.2.c.a	$2$	$0$	\(0\)	\(0\)	\(-4\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-1-\beta _{2}-\beta _{3})q^{5}+\beta _{2}q^{7}+(2\beta _{1}+\cdots)q^{11}+\cdots\)
5040.2.t.u	$5040$	$2$	5040.t	5.b	$2$	$4$	$4$	$40.245$	\(\Q(i, \sqrt{5})\)	None					840.2.t.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{3}q^{5}+\beta _{1}q^{7}-2q^{11}+2\beta _{2}q^{13}+\cdots\)
5040.2.t.v	$5040$	$2$	5040.t	5.b	$2$	$6$	$6$	$40.245$	6.0.350464.1	None					105.2.d.b	$2$	$0$	\(0\)	\(0\)	\(-2\)	\(0\)	$2^{3}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{2}q^{5}+\beta _{1}q^{7}+2q^{11}+(2\beta _{1}-\beta _{2}+\cdots)q^{13}+\cdots\)
5040.2.t.w	$5040$	$2$	5040.t	5.b	$2$	$6$	$6$	$40.245$	6.0.350464.1	None					2520.2.t.h	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{3}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{3}q^{5}+\beta _{1}q^{7}+(-2+\beta _{2}-\beta _{3}+\cdots)q^{11}+\cdots\)
5040.2.t.x	$5040$	$2$	5040.t	5.b	$2$	$6$	$6$	$40.245$	6.0.350464.1	None					2520.2.t.h	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{3}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{3}q^{5}+\beta _{1}q^{7}+(2-\beta _{2}+\beta _{3}+\beta _{4}+\cdots)q^{11}+\cdots\)
5040.2.t.y	$5040$	$2$	5040.t	5.b	$2$	$6$	$6$	$40.245$	6.0.5161984.1	None					280.2.g.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-\beta _{1}+\beta _{5})q^{5}-\beta _{4}q^{7}+(2-\beta _{1}+\cdots)q^{11}+\cdots\)
5040.2.t.z	$5040$	$2$	5040.t	5.b	$2$	$6$	$6$	$40.245$	6.0.350464.1	None					840.2.t.d	$2$	$0$	\(0\)	\(0\)	\(2\)	\(0\)	$2^{3}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{5}q^{5}+\beta _{1}q^{7}+(-\beta _{2}-\beta _{3}-\beta _{4}+\cdots)q^{11}+\cdots\)
5040.2.t.ba	$5040$	$2$	5040.t	5.b	$2$	$6$	$6$	$40.245$	6.0.350464.1	None					840.2.t.e	$2$	$0$	\(0\)	\(0\)	\(2\)	\(0\)	$2^{3}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{5}q^{5}-\beta _{1}q^{7}+(\beta _{2}-\beta _{3}-\beta _{4}-\beta _{5})q^{11}+\cdots\)
5040.2.t.bb	$5040$	$2$	5040.t	5.b	$2$	$8$	$8$	$40.245$	8.0.49787136.1	None			✓		1260.2.k.e	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{8}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{7}q^{5}+\beta _{1}q^{7}+(\beta _{3}+\beta _{4}+\beta _{5}+\beta _{7})q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(5040, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(5040, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(30, [\chi])\)\(^{\oplus 16}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(35, [\chi])\)\(^{\oplus 15}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(40, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(45, [\chi])\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(60, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(70, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(80, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(90, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(105, [\chi])\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(120, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(140, [\chi])\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(180, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(210, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(240, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(280, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(315, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(360, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(420, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(560, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(630, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(720, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(840, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(1260, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(1680, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(2520, [\chi])\)\(^{\oplus 2}\)