Space of modular forms of level 1280, weight 2, and Character 1280.o

• Label: 1280.2.o
• Level: $1280$
• Weight: $2$
• Character orbit: 1280.o
• Rep. character: $\chi_{1280}(127,\cdot)$
• Character field: $\Q(\zeta_{4})$
• Dimension: $88$
• Newform subspaces: $22$
• Sturm bound: $384$
• Trace bound: $13$

Defining parameters

Level:	\( N \)	\(=\)	\( 1280 = 2^{8} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1280.o (of order \(4\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 40 \)
Character field:			\(\Q(i)\)
Newform subspaces:			\( 22 \)
Sturm bound:			\(384\)
Trace bound:			\(13\)
Distinguishing \(T_p\):			\(3\), \(7\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(1280, [\chi])\).

	Total	New	Old
Modular forms	432	104	328
Cusp forms	336	88	248
Eisenstein series	96	16	80

Trace form

\( 88 q - 8 q^{17} + 8 q^{25} + 16 q^{33} + 16 q^{41} + 32 q^{57} - 8 q^{65} - 24 q^{73} - 40 q^{81} - 8 q^{97}+O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(1280, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
1280.2.o.a	$1280$	$2$	1280.o	40.k	$4$	$2$	$1$	$10.221$	\(\Q(\sqrt{-1}) \)	None					160.2.n.a	$2$	$1$	\(0\)	\(-4\)	\(-2\)	\(4\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(2 i-2)q^{3}+(-2 i-1)q^{5}+(-2 i+2)q^{7}+\cdots\)
1280.2.o.b	$1280$	$2$	1280.o	40.k	$4$	$2$	$1$	$10.221$	\(\Q(\sqrt{-1}) \)	None					160.2.n.a	$2$	$1$	\(0\)	\(-4\)	\(2\)	\(-4\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(2 i-2)q^{3}+(2 i+1)q^{5}+(2 i-2)q^{7}+\cdots\)
1280.2.o.c	$1280$	$2$	1280.o	40.k	$4$	$2$	$1$	$10.221$	\(\Q(\sqrt{-1}) \)	None					160.2.n.c	$2$	$1$	\(0\)	\(-2\)	\(-4\)	\(-6\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(i-1)q^{3}+(i-2)q^{5}+(3 i-3)q^{7}+\cdots\)
1280.2.o.d	$1280$	$2$	1280.o	40.k	$4$	$2$	$1$	$10.221$	\(\Q(\sqrt{-1}) \)	None					160.2.n.b	$2$	$0$	\(0\)	\(-2\)	\(-4\)	\(-2\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(i-1)q^{3}+(-i-2)q^{5}+(i-1)q^{7}+\cdots\)
1280.2.o.e	$1280$	$2$	1280.o	40.k	$4$	$2$	$1$	$10.221$	\(\Q(\sqrt{-1}) \)	None					160.2.n.b	$2$	$0$	\(0\)	\(-2\)	\(4\)	\(2\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(i-1)q^{3}+(i+2)q^{5}+(-i+1)q^{7}+\cdots\)
1280.2.o.f	$1280$	$2$	1280.o	40.k	$4$	$2$	$1$	$10.221$	\(\Q(\sqrt{-1}) \)	None					160.2.n.c	$2$	$0$	\(0\)	\(-2\)	\(4\)	\(6\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(i-1)q^{3}+(-i+2)q^{5}+(-3 i+3)q^{7}+\cdots\)
1280.2.o.g	$1280$	$2$	1280.o	40.k	$4$	$2$	$1$	$10.221$	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-1}) \)					20.2.e.a	$4$	$0$	\(0\)	\(0\)	\(-2\)	\(0\)	$1$	$\mathrm{U}(1)[D_{4}]$	\(q+(-2 i-1)q^{5}+3 i q^{9}+(i+1)q^{13}+\cdots\)
1280.2.o.h	$1280$	$2$	1280.o	40.k	$4$	$2$	$1$	$10.221$	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-1}) \)					80.2.n.a	$4$	$0$	\(0\)	\(0\)	\(-2\)	\(0\)	$1$	$\mathrm{U}(1)[D_{4}]$	\(q+(2 i-1)q^{5}+3 i q^{9}+(5 i+5)q^{13}+\cdots\)
1280.2.o.i	$1280$	$2$	1280.o	40.k	$4$	$2$	$1$	$10.221$	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-1}) \)					80.2.n.a	$4$	$0$	\(0\)	\(0\)	\(2\)	\(0\)	$1$	$\mathrm{U}(1)[D_{4}]$	\(q+(-2 i+1)q^{5}+3 i q^{9}+(-5 i-5)q^{13}+\cdots\)
1280.2.o.j	$1280$	$2$	1280.o	40.k	$4$	$2$	$1$	$10.221$	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-1}) \)					20.2.e.a	$4$	$0$	\(0\)	\(0\)	\(2\)	\(0\)	$1$	$\mathrm{U}(1)[D_{4}]$	\(q+(2 i+1)q^{5}+3 i q^{9}+(-i-1)q^{13}+\cdots\)
1280.2.o.k	$1280$	$2$	1280.o	40.k	$4$	$2$	$1$	$10.221$	\(\Q(\sqrt{-1}) \)	None					160.2.n.b	$2$	$1$	\(0\)	\(2\)	\(-4\)	\(2\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-i+1)q^{3}+(-i-2)q^{5}+(-i+1)q^{7}+\cdots\)
1280.2.o.l	$1280$	$2$	1280.o	40.k	$4$	$2$	$1$	$10.221$	\(\Q(\sqrt{-1}) \)	None					160.2.n.c	$2$	$0$	\(0\)	\(2\)	\(-4\)	\(6\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-i+1)q^{3}+(i-2)q^{5}+(-3 i+3)q^{7}+\cdots\)
1280.2.o.m	$1280$	$2$	1280.o	40.k	$4$	$2$	$1$	$10.221$	\(\Q(\sqrt{-1}) \)	None					160.2.n.c	$2$	$0$	\(0\)	\(2\)	\(4\)	\(-6\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-i+1)q^{3}+(-i+2)q^{5}+(3 i-3)q^{7}+\cdots\)
1280.2.o.n	$1280$	$2$	1280.o	40.k	$4$	$2$	$1$	$10.221$	\(\Q(\sqrt{-1}) \)	None					160.2.n.b	$2$	$0$	\(0\)	\(2\)	\(4\)	\(-2\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-i+1)q^{3}+(i+2)q^{5}+(i-1)q^{7}+\cdots\)
1280.2.o.o	$1280$	$2$	1280.o	40.k	$4$	$2$	$1$	$10.221$	\(\Q(\sqrt{-1}) \)	None					160.2.n.a	$2$	$0$	\(0\)	\(4\)	\(-2\)	\(-4\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-2 i+2)q^{3}+(-2 i-1)q^{5}+\cdots\)
1280.2.o.p	$1280$	$2$	1280.o	40.k	$4$	$2$	$1$	$10.221$	\(\Q(\sqrt{-1}) \)	None					160.2.n.a	$2$	$0$	\(0\)	\(4\)	\(2\)	\(4\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-2 i+2)q^{3}+(2 i+1)q^{5}+(-2 i+2)q^{7}+\cdots\)
1280.2.o.q	$1280$	$2$	1280.o	40.k	$4$	$4$	$2$	$10.221$	\(\Q(\zeta_{12})\)	None					80.2.n.b	$4$	$0$	\(0\)	\(0\)	\(-8\)	\(0\)	$2^{3}$	$\mathrm{SU}(2)[C_{4}]$	\(q-\beta_{2} q^{3}+(-\beta_1-2)q^{5}+\beta_{2} q^{7}+\cdots\)
1280.2.o.r	$1280$	$2$	1280.o	40.k	$4$	$4$	$2$	$10.221$	\(\Q(\zeta_{12})\)	None					80.2.n.b	$4$	$0$	\(0\)	\(0\)	\(8\)	\(0\)	$2^{3}$	$\mathrm{SU}(2)[C_{4}]$	\(q-\beta_{2} q^{3}+(\beta_1+2)q^{5}-\beta_{2} q^{7}+\cdots\)
1280.2.o.s	$1280$	$2$	1280.o	40.k	$4$	$12$	$6$	$10.221$	12.0.\(\cdots\).1	None					640.2.n.a	$2$	$0$	\(0\)	\(0\)	\(-4\)	\(-8\)	$2^{9}$	$\mathrm{SU}(2)[C_{4}]$	\(q-\beta _{2}q^{3}-\beta _{7}q^{5}+(-1-\beta _{2}+\beta _{4}+\cdots)q^{7}+\cdots\)
1280.2.o.t	$1280$	$2$	1280.o	40.k	$4$	$12$	$6$	$10.221$	12.0.\(\cdots\).1	None					640.2.n.a	$2$	$0$	\(0\)	\(0\)	\(-4\)	\(8\)	$2^{9}$	$\mathrm{SU}(2)[C_{4}]$	\(q+\beta _{3}q^{3}-\beta _{6}q^{5}+(1+\beta _{3}+\beta _{5}-\beta _{7}+\cdots)q^{7}+\cdots\)
1280.2.o.u	$1280$	$2$	1280.o	40.k	$4$	$12$	$6$	$10.221$	12.0.\(\cdots\).1	None					640.2.n.a	$2$	$0$	\(0\)	\(0\)	\(4\)	\(-8\)	$2^{9}$	$\mathrm{SU}(2)[C_{4}]$	\(q+\beta _{3}q^{3}+\beta _{6}q^{5}+(-1-\beta _{3}-\beta _{5}+\cdots)q^{7}+\cdots\)
1280.2.o.v	$1280$	$2$	1280.o	40.k	$4$	$12$	$6$	$10.221$	12.0.\(\cdots\).1	None					640.2.n.a	$2$	$0$	\(0\)	\(0\)	\(4\)	\(8\)	$2^{9}$	$\mathrm{SU}(2)[C_{4}]$	\(q-\beta _{2}q^{3}+\beta _{7}q^{5}+(1+\beta _{2}-\beta _{4}-\beta _{6}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(1280, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(1280, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(40, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(160, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(320, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(640, [\chi])\)\(^{\oplus 2}\)