# Properties

 Label 1280.3.e.j Level $1280$ Weight $3$ Character orbit 1280.e Analytic conductor $34.877$ Analytic rank $0$ Dimension $8$ CM no Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1280 = 2^{8} \cdot 5$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 1280.e (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$34.8774738381$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\Q(\zeta_{24})$$ Defining polynomial: $$x^{8} - x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{18}$$ Twist minimal: no (minimal twist has level 80) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{24}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 2 \zeta_{24} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} ) q^{3} + ( -2 \zeta_{24} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} - \zeta_{24}^{6} + 4 \zeta_{24}^{7} ) q^{5} + ( -6 \zeta_{24} - 6 \zeta_{24}^{3} + 6 \zeta_{24}^{5} ) q^{7} + q^{9} +O(q^{10})$$ $$q + ( 2 \zeta_{24} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} ) q^{3} + ( -2 \zeta_{24} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} - \zeta_{24}^{6} + 4 \zeta_{24}^{7} ) q^{5} + ( -6 \zeta_{24} - 6 \zeta_{24}^{3} + 6 \zeta_{24}^{5} ) q^{7} + q^{9} + ( 16 \zeta_{24}^{2} - 8 \zeta_{24}^{6} ) q^{11} + ( 4 \zeta_{24} + 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} - 8 \zeta_{24}^{7} ) q^{13} + ( -8 - 2 \zeta_{24} - 2 \zeta_{24}^{3} + 16 \zeta_{24}^{4} + 2 \zeta_{24}^{5} ) q^{15} + ( -8 \zeta_{24} + 8 \zeta_{24}^{3} - 8 \zeta_{24}^{5} - 16 \zeta_{24}^{7} ) q^{17} + ( 16 \zeta_{24}^{2} - 8 \zeta_{24}^{6} ) q^{19} + 24 \zeta_{24}^{6} q^{21} + ( 18 \zeta_{24} + 18 \zeta_{24}^{3} - 18 \zeta_{24}^{5} ) q^{23} + ( 23 + 4 \zeta_{24} - 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} + 8 \zeta_{24}^{7} ) q^{25} + ( 20 \zeta_{24} - 20 \zeta_{24}^{3} - 20 \zeta_{24}^{5} ) q^{27} + 22 \zeta_{24}^{6} q^{29} + ( 32 - 64 \zeta_{24}^{4} ) q^{31} + ( -16 \zeta_{24} + 16 \zeta_{24}^{3} - 16 \zeta_{24}^{5} - 32 \zeta_{24}^{7} ) q^{33} + ( -6 \zeta_{24} + 48 \zeta_{24}^{2} + 6 \zeta_{24}^{3} + 6 \zeta_{24}^{5} - 24 \zeta_{24}^{6} ) q^{35} + ( -20 \zeta_{24} - 20 \zeta_{24}^{3} - 20 \zeta_{24}^{5} + 40 \zeta_{24}^{7} ) q^{37} + ( 16 - 32 \zeta_{24}^{4} ) q^{39} -22 q^{41} + ( -42 \zeta_{24} + 42 \zeta_{24}^{3} + 42 \zeta_{24}^{5} ) q^{43} + ( -2 \zeta_{24} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} - \zeta_{24}^{6} + 4 \zeta_{24}^{7} ) q^{45} + ( 6 \zeta_{24} + 6 \zeta_{24}^{3} - 6 \zeta_{24}^{5} ) q^{47} + 23 q^{49} + ( -64 \zeta_{24}^{2} + 32 \zeta_{24}^{6} ) q^{51} + ( -12 \zeta_{24} - 12 \zeta_{24}^{3} - 12 \zeta_{24}^{5} + 24 \zeta_{24}^{7} ) q^{53} + ( 8 - 48 \zeta_{24} - 48 \zeta_{24}^{3} - 16 \zeta_{24}^{4} + 48 \zeta_{24}^{5} ) q^{55} + ( -16 \zeta_{24} + 16 \zeta_{24}^{3} - 16 \zeta_{24}^{5} - 32 \zeta_{24}^{7} ) q^{57} + ( 16 \zeta_{24}^{2} - 8 \zeta_{24}^{6} ) q^{59} -46 \zeta_{24}^{6} q^{61} + ( -6 \zeta_{24} - 6 \zeta_{24}^{3} + 6 \zeta_{24}^{5} ) q^{63} + ( -48 - 4 \zeta_{24} + 4 \zeta_{24}^{3} - 4 \zeta_{24}^{5} - 8 \zeta_{24}^{7} ) q^{65} + ( 42 \zeta_{24} - 42 \zeta_{24}^{3} - 42 \zeta_{24}^{5} ) q^{67} -72 \zeta_{24}^{6} q^{69} + ( 16 - 32 \zeta_{24}^{4} ) q^{71} + ( -32 \zeta_{24} + 32 \zeta_{24}^{3} - 32 \zeta_{24}^{5} - 64 \zeta_{24}^{7} ) q^{73} + ( 46 \zeta_{24} + 32 \zeta_{24}^{2} - 46 \zeta_{24}^{3} - 46 \zeta_{24}^{5} - 16 \zeta_{24}^{6} ) q^{75} + ( -48 \zeta_{24} - 48 \zeta_{24}^{3} - 48 \zeta_{24}^{5} + 96 \zeta_{24}^{7} ) q^{77} -71 q^{81} + ( -54 \zeta_{24} + 54 \zeta_{24}^{3} + 54 \zeta_{24}^{5} ) q^{83} + ( -8 \zeta_{24} - 8 \zeta_{24}^{3} - 8 \zeta_{24}^{5} + 96 \zeta_{24}^{6} + 16 \zeta_{24}^{7} ) q^{85} + ( 44 \zeta_{24} + 44 \zeta_{24}^{3} - 44 \zeta_{24}^{5} ) q^{87} -146 q^{89} + ( -96 \zeta_{24}^{2} + 48 \zeta_{24}^{6} ) q^{91} + ( -64 \zeta_{24} - 64 \zeta_{24}^{3} - 64 \zeta_{24}^{5} + 128 \zeta_{24}^{7} ) q^{93} + ( 8 - 48 \zeta_{24} - 48 \zeta_{24}^{3} - 16 \zeta_{24}^{4} + 48 \zeta_{24}^{5} ) q^{95} + ( -24 \zeta_{24} + 24 \zeta_{24}^{3} - 24 \zeta_{24}^{5} - 48 \zeta_{24}^{7} ) q^{97} + ( 16 \zeta_{24}^{2} - 8 \zeta_{24}^{6} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 8 q^{9} + O(q^{10})$$ $$8 q + 8 q^{9} + 184 q^{25} - 176 q^{41} + 184 q^{49} - 384 q^{65} - 568 q^{81} - 1168 q^{89} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1280\mathbb{Z}\right)^\times$$.

 $$n$$ $$257$$ $$261$$ $$511$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$-1$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
639.1
 0.965926 + 0.258819i 0.258819 − 0.965926i −0.258819 − 0.965926i −0.965926 + 0.258819i 0.258819 + 0.965926i 0.965926 − 0.258819i −0.965926 − 0.258819i −0.258819 + 0.965926i
0 2.82843i 0 −4.89898 1.00000i 0 −8.48528 0 1.00000 0
639.2 0 2.82843i 0 −4.89898 + 1.00000i 0 8.48528 0 1.00000 0
639.3 0 2.82843i 0 4.89898 1.00000i 0 −8.48528 0 1.00000 0
639.4 0 2.82843i 0 4.89898 + 1.00000i 0 8.48528 0 1.00000 0
639.5 0 2.82843i 0 −4.89898 1.00000i 0 8.48528 0 1.00000 0
639.6 0 2.82843i 0 −4.89898 + 1.00000i 0 −8.48528 0 1.00000 0
639.7 0 2.82843i 0 4.89898 1.00000i 0 8.48528 0 1.00000 0
639.8 0 2.82843i 0 4.89898 + 1.00000i 0 −8.48528 0 1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 639.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.b even 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner
20.d odd 2 1 inner
40.e odd 2 1 inner
40.f even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1280.3.e.j 8
4.b odd 2 1 inner 1280.3.e.j 8
5.b even 2 1 inner 1280.3.e.j 8
8.b even 2 1 inner 1280.3.e.j 8
8.d odd 2 1 inner 1280.3.e.j 8
16.e even 4 1 80.3.h.b 4
16.e even 4 1 320.3.h.e 4
16.f odd 4 1 80.3.h.b 4
16.f odd 4 1 320.3.h.e 4
20.d odd 2 1 inner 1280.3.e.j 8
40.e odd 2 1 inner 1280.3.e.j 8
40.f even 2 1 inner 1280.3.e.j 8
48.i odd 4 1 720.3.j.e 4
48.k even 4 1 720.3.j.e 4
80.i odd 4 1 400.3.b.h 4
80.i odd 4 1 1600.3.b.t 4
80.j even 4 1 400.3.b.h 4
80.j even 4 1 1600.3.b.t 4
80.k odd 4 1 80.3.h.b 4
80.k odd 4 1 320.3.h.e 4
80.q even 4 1 80.3.h.b 4
80.q even 4 1 320.3.h.e 4
80.s even 4 1 400.3.b.h 4
80.s even 4 1 1600.3.b.t 4
80.t odd 4 1 400.3.b.h 4
80.t odd 4 1 1600.3.b.t 4
240.t even 4 1 720.3.j.e 4
240.z odd 4 1 3600.3.e.bd 4
240.bb even 4 1 3600.3.e.bd 4
240.bd odd 4 1 3600.3.e.bd 4
240.bf even 4 1 3600.3.e.bd 4
240.bm odd 4 1 720.3.j.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
80.3.h.b 4 16.e even 4 1
80.3.h.b 4 16.f odd 4 1
80.3.h.b 4 80.k odd 4 1
80.3.h.b 4 80.q even 4 1
320.3.h.e 4 16.e even 4 1
320.3.h.e 4 16.f odd 4 1
320.3.h.e 4 80.k odd 4 1
320.3.h.e 4 80.q even 4 1
400.3.b.h 4 80.i odd 4 1
400.3.b.h 4 80.j even 4 1
400.3.b.h 4 80.s even 4 1
400.3.b.h 4 80.t odd 4 1
720.3.j.e 4 48.i odd 4 1
720.3.j.e 4 48.k even 4 1
720.3.j.e 4 240.t even 4 1
720.3.j.e 4 240.bm odd 4 1
1280.3.e.j 8 1.a even 1 1 trivial
1280.3.e.j 8 4.b odd 2 1 inner
1280.3.e.j 8 5.b even 2 1 inner
1280.3.e.j 8 8.b even 2 1 inner
1280.3.e.j 8 8.d odd 2 1 inner
1280.3.e.j 8 20.d odd 2 1 inner
1280.3.e.j 8 40.e odd 2 1 inner
1280.3.e.j 8 40.f even 2 1 inner
1600.3.b.t 4 80.i odd 4 1
1600.3.b.t 4 80.j even 4 1
1600.3.b.t 4 80.s even 4 1
1600.3.b.t 4 80.t odd 4 1
3600.3.e.bd 4 240.z odd 4 1
3600.3.e.bd 4 240.bb even 4 1
3600.3.e.bd 4 240.bd odd 4 1
3600.3.e.bd 4 240.bf even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(1280, [\chi])$$:

 $$T_{3}^{2} + 8$$ $$T_{7}^{2} - 72$$ $$T_{11}^{2} - 192$$ $$T_{13}^{2} - 96$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$( 8 + T^{2} )^{4}$$
$5$ $$( 625 - 46 T^{2} + T^{4} )^{2}$$
$7$ $$( -72 + T^{2} )^{4}$$
$11$ $$( -192 + T^{2} )^{4}$$
$13$ $$( -96 + T^{2} )^{4}$$
$17$ $$( 384 + T^{2} )^{4}$$
$19$ $$( -192 + T^{2} )^{4}$$
$23$ $$( -648 + T^{2} )^{4}$$
$29$ $$( 484 + T^{2} )^{4}$$
$31$ $$( 3072 + T^{2} )^{4}$$
$37$ $$( -2400 + T^{2} )^{4}$$
$41$ $$( 22 + T )^{8}$$
$43$ $$( 3528 + T^{2} )^{4}$$
$47$ $$( -72 + T^{2} )^{4}$$
$53$ $$( -864 + T^{2} )^{4}$$
$59$ $$( -192 + T^{2} )^{4}$$
$61$ $$( 2116 + T^{2} )^{4}$$
$67$ $$( 3528 + T^{2} )^{4}$$
$71$ $$( 768 + T^{2} )^{4}$$
$73$ $$( 6144 + T^{2} )^{4}$$
$79$ $$T^{8}$$
$83$ $$( 5832 + T^{2} )^{4}$$
$89$ $$( 146 + T )^{8}$$
$97$ $$( 3456 + T^{2} )^{4}$$