# Properties

 Label 1280.3.g.e Level $1280$ Weight $3$ Character orbit 1280.g Analytic conductor $34.877$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$34.8774738381$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\Q(\zeta_{20})$$ Defining polynomial: $$x^{8} - x^{6} + x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{16}$$ Twist minimal: no (minimal twist has level 20) 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 $$\zeta_{20}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -4 \zeta_{20} + 2 \zeta_{20}^{3} - 2 \zeta_{20}^{5} + 2 \zeta_{20}^{7} ) q^{3} + ( 2 \zeta_{20}^{3} - \zeta_{20}^{5} + 2 \zeta_{20}^{7} ) q^{5} + ( 4 - 8 \zeta_{20}^{2} + 2 \zeta_{20}^{4} - 6 \zeta_{20}^{6} ) q^{7} + ( 3 + 4 \zeta_{20}^{4} - 4 \zeta_{20}^{6} ) q^{9} +O(q^{10})$$ $$q + ( -4 \zeta_{20} + 2 \zeta_{20}^{3} - 2 \zeta_{20}^{5} + 2 \zeta_{20}^{7} ) q^{3} + ( 2 \zeta_{20}^{3} - \zeta_{20}^{5} + 2 \zeta_{20}^{7} ) q^{5} + ( 4 - 8 \zeta_{20}^{2} + 2 \zeta_{20}^{4} - 6 \zeta_{20}^{6} ) q^{7} + ( 3 + 4 \zeta_{20}^{4} - 4 \zeta_{20}^{6} ) q^{9} + ( -8 \zeta_{20} + 12 \zeta_{20}^{3} - 4 \zeta_{20}^{5} - 4 \zeta_{20}^{7} ) q^{11} + ( -4 \zeta_{20}^{3} + 6 \zeta_{20}^{5} - 4 \zeta_{20}^{7} ) q^{13} + ( 4 - 8 \zeta_{20}^{2} + 2 \zeta_{20}^{4} - 6 \zeta_{20}^{6} ) q^{15} + ( -14 - 16 \zeta_{20}^{4} + 16 \zeta_{20}^{6} ) q^{17} + ( 8 \zeta_{20}^{3} - 8 \zeta_{20}^{7} ) q^{19} + ( 20 \zeta_{20}^{3} + 20 \zeta_{20}^{7} ) q^{21} + ( 4 - 8 \zeta_{20}^{2} - 2 \zeta_{20}^{4} - 10 \zeta_{20}^{6} ) q^{23} -5 q^{25} + ( 24 \zeta_{20} - 20 \zeta_{20}^{3} + 12 \zeta_{20}^{5} - 4 \zeta_{20}^{7} ) q^{27} + ( -8 \zeta_{20}^{3} + 6 \zeta_{20}^{5} - 8 \zeta_{20}^{7} ) q^{29} + ( 8 - 16 \zeta_{20}^{2} - 12 \zeta_{20}^{4} - 28 \zeta_{20}^{6} ) q^{31} + ( 8 - 24 \zeta_{20}^{4} + 24 \zeta_{20}^{6} ) q^{33} + ( 20 \zeta_{20} - 10 \zeta_{20}^{3} + 10 \zeta_{20}^{5} - 10 \zeta_{20}^{7} ) q^{35} + ( 20 \zeta_{20}^{3} - 6 \zeta_{20}^{5} + 20 \zeta_{20}^{7} ) q^{37} + ( -8 + 16 \zeta_{20}^{2} - 12 \zeta_{20}^{4} + 4 \zeta_{20}^{6} ) q^{39} + ( 34 + 12 \zeta_{20}^{4} - 12 \zeta_{20}^{6} ) q^{41} + ( -12 \zeta_{20} + 14 \zeta_{20}^{3} - 6 \zeta_{20}^{5} - 2 \zeta_{20}^{7} ) q^{43} + ( 2 \zeta_{20}^{3} + 9 \zeta_{20}^{5} + 2 \zeta_{20}^{7} ) q^{45} + ( 12 - 24 \zeta_{20}^{2} + 38 \zeta_{20}^{4} + 14 \zeta_{20}^{6} ) q^{47} + ( -11 - 20 \zeta_{20}^{4} + 20 \zeta_{20}^{6} ) q^{49} + ( 56 \zeta_{20} + 4 \zeta_{20}^{3} + 28 \zeta_{20}^{5} - 60 \zeta_{20}^{7} ) q^{51} + ( 20 \zeta_{20}^{3} - 54 \zeta_{20}^{5} + 20 \zeta_{20}^{7} ) q^{53} + ( 16 - 32 \zeta_{20}^{2} + 28 \zeta_{20}^{4} - 4 \zeta_{20}^{6} ) q^{55} + ( -16 - 32 \zeta_{20}^{4} + 32 \zeta_{20}^{6} ) q^{57} + ( 48 \zeta_{20} + 24 \zeta_{20}^{5} - 48 \zeta_{20}^{7} ) q^{59} + ( 52 \zeta_{20}^{3} - 58 \zeta_{20}^{5} + 52 \zeta_{20}^{7} ) q^{61} + ( 4 - 8 \zeta_{20}^{2} - 18 \zeta_{20}^{4} - 26 \zeta_{20}^{6} ) q^{63} + ( 6 - 8 \zeta_{20}^{4} + 8 \zeta_{20}^{6} ) q^{65} + ( -44 \zeta_{20} - 18 \zeta_{20}^{3} - 22 \zeta_{20}^{5} + 62 \zeta_{20}^{7} ) q^{67} + ( 28 \zeta_{20}^{3} + 16 \zeta_{20}^{5} + 28 \zeta_{20}^{7} ) q^{69} + ( 40 - 80 \zeta_{20}^{2} + 44 \zeta_{20}^{4} - 36 \zeta_{20}^{6} ) q^{71} + ( -98 - 64 \zeta_{20}^{4} + 64 \zeta_{20}^{6} ) q^{73} + ( 20 \zeta_{20} - 10 \zeta_{20}^{3} + 10 \zeta_{20}^{5} - 10 \zeta_{20}^{7} ) q^{75} + ( 40 \zeta_{20}^{3} - 80 \zeta_{20}^{5} + 40 \zeta_{20}^{7} ) q^{77} + ( -32 + 64 \zeta_{20}^{2} + 8 \zeta_{20}^{4} + 72 \zeta_{20}^{6} ) q^{79} + ( -83 - 28 \zeta_{20}^{4} + 28 \zeta_{20}^{6} ) q^{81} + ( -52 \zeta_{20} + 50 \zeta_{20}^{3} - 26 \zeta_{20}^{5} + 2 \zeta_{20}^{7} ) q^{83} + ( -12 \zeta_{20}^{3} - 34 \zeta_{20}^{5} - 12 \zeta_{20}^{7} ) q^{85} + ( -16 + 32 \zeta_{20}^{2} - 12 \zeta_{20}^{4} + 20 \zeta_{20}^{6} ) q^{87} + ( -18 - 80 \zeta_{20}^{4} + 80 \zeta_{20}^{6} ) q^{89} + ( -24 \zeta_{20} + 28 \zeta_{20}^{3} - 12 \zeta_{20}^{5} - 4 \zeta_{20}^{7} ) q^{91} + ( 72 \zeta_{20}^{3} + 64 \zeta_{20}^{5} + 72 \zeta_{20}^{7} ) q^{93} + ( 8 - 16 \zeta_{20}^{2} + 24 \zeta_{20}^{4} + 8 \zeta_{20}^{6} ) q^{95} + ( -54 + 24 \zeta_{20}^{4} - 24 \zeta_{20}^{6} ) q^{97} + ( 40 \zeta_{20} - 44 \zeta_{20}^{3} + 20 \zeta_{20}^{5} + 4 \zeta_{20}^{7} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 8q^{9} + O(q^{10})$$ $$8q + 8q^{9} - 48q^{17} - 40q^{25} + 160q^{33} + 224q^{41} - 8q^{49} + 80q^{65} - 528q^{73} - 552q^{81} + 176q^{89} - 528q^{97} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1151.1
 0.951057 − 0.309017i 0.951057 + 0.309017i 0.587785 − 0.809017i 0.587785 + 0.809017i −0.587785 − 0.809017i −0.587785 + 0.809017i −0.951057 − 0.309017i −0.951057 + 0.309017i
0 −3.80423 0 2.23607i 0 8.50651i 0 5.47214 0
1151.2 0 −3.80423 0 2.23607i 0 8.50651i 0 5.47214 0
1151.3 0 −2.35114 0 2.23607i 0 5.25731i 0 −3.47214 0
1151.4 0 −2.35114 0 2.23607i 0 5.25731i 0 −3.47214 0
1151.5 0 2.35114 0 2.23607i 0 5.25731i 0 −3.47214 0
1151.6 0 2.35114 0 2.23607i 0 5.25731i 0 −3.47214 0
1151.7 0 3.80423 0 2.23607i 0 8.50651i 0 5.47214 0
1151.8 0 3.80423 0 2.23607i 0 8.50651i 0 5.47214 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1151.8 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
8.b even 2 1 inner
8.d odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1280.3.g.e 8
4.b odd 2 1 inner 1280.3.g.e 8
8.b even 2 1 inner 1280.3.g.e 8
8.d odd 2 1 inner 1280.3.g.e 8
16.e even 4 1 20.3.b.a 4
16.e even 4 1 320.3.b.c 4
16.f odd 4 1 20.3.b.a 4
16.f odd 4 1 320.3.b.c 4
48.i odd 4 1 180.3.c.a 4
48.i odd 4 1 2880.3.e.e 4
48.k even 4 1 180.3.c.a 4
48.k even 4 1 2880.3.e.e 4
80.i odd 4 1 100.3.d.b 8
80.i odd 4 1 1600.3.h.n 8
80.j even 4 1 100.3.d.b 8
80.j even 4 1 1600.3.h.n 8
80.k odd 4 1 100.3.b.f 4
80.k odd 4 1 1600.3.b.s 4
80.q even 4 1 100.3.b.f 4
80.q even 4 1 1600.3.b.s 4
80.s even 4 1 100.3.d.b 8
80.s even 4 1 1600.3.h.n 8
80.t odd 4 1 100.3.d.b 8
80.t odd 4 1 1600.3.h.n 8
240.t even 4 1 900.3.c.k 4
240.z odd 4 1 900.3.f.e 8
240.bb even 4 1 900.3.f.e 8
240.bd odd 4 1 900.3.f.e 8
240.bf even 4 1 900.3.f.e 8
240.bm odd 4 1 900.3.c.k 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.3.b.a 4 16.e even 4 1
20.3.b.a 4 16.f odd 4 1
100.3.b.f 4 80.k odd 4 1
100.3.b.f 4 80.q even 4 1
100.3.d.b 8 80.i odd 4 1
100.3.d.b 8 80.j even 4 1
100.3.d.b 8 80.s even 4 1
100.3.d.b 8 80.t odd 4 1
180.3.c.a 4 48.i odd 4 1
180.3.c.a 4 48.k even 4 1
320.3.b.c 4 16.e even 4 1
320.3.b.c 4 16.f odd 4 1
900.3.c.k 4 240.t even 4 1
900.3.c.k 4 240.bm odd 4 1
900.3.f.e 8 240.z odd 4 1
900.3.f.e 8 240.bb even 4 1
900.3.f.e 8 240.bd odd 4 1
900.3.f.e 8 240.bf even 4 1
1280.3.g.e 8 1.a even 1 1 trivial
1280.3.g.e 8 4.b odd 2 1 inner
1280.3.g.e 8 8.b even 2 1 inner
1280.3.g.e 8 8.d odd 2 1 inner
1600.3.b.s 4 80.k odd 4 1
1600.3.b.s 4 80.q even 4 1
1600.3.h.n 8 80.i odd 4 1
1600.3.h.n 8 80.j even 4 1
1600.3.h.n 8 80.s even 4 1
1600.3.h.n 8 80.t odd 4 1
2880.3.e.e 4 48.i odd 4 1
2880.3.e.e 4 48.k even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} - 20 T_{3}^{2} + 80$$ acting on $$S_{3}^{\mathrm{new}}(1280, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$( 80 - 20 T^{2} + T^{4} )^{2}$$
$5$ $$( 5 + T^{2} )^{4}$$
$7$ $$( 2000 + 100 T^{2} + T^{4} )^{2}$$
$11$ $$( 1280 - 400 T^{2} + T^{4} )^{2}$$
$13$ $$( 16 + 72 T^{2} + T^{4} )^{2}$$
$17$ $$( -284 + 12 T + T^{2} )^{4}$$
$19$ $$( 20480 - 320 T^{2} + T^{4} )^{2}$$
$23$ $$( 80 + 260 T^{2} + T^{4} )^{2}$$
$29$ $$( 5776 + 168 T^{2} + T^{4} )^{2}$$
$31$ $$( 154880 + 2320 T^{2} + T^{4} )^{2}$$
$37$ $$( 234256 + 1032 T^{2} + T^{4} )^{2}$$
$41$ $$( 604 - 56 T + T^{2} )^{4}$$
$43$ $$( 2000 - 500 T^{2} + T^{4} )^{2}$$
$47$ $$( 3561680 + 4100 T^{2} + T^{4} )^{2}$$
$53$ $$( 2062096 + 4872 T^{2} + T^{4} )^{2}$$
$59$ $$( 1658880 - 5760 T^{2} + T^{4} )^{2}$$
$61$ $$( 5550736 + 8808 T^{2} + T^{4} )^{2}$$
$67$ $$( 19920080 - 10420 T^{2} + T^{4} )^{2}$$
$71$ $$( 10138880 + 8080 T^{2} + T^{4} )^{2}$$
$73$ $$( -764 + 132 T + T^{2} )^{4}$$
$79$ $$( 2478080 + 13120 T^{2} + T^{4} )^{2}$$
$83$ $$( 2620880 - 6260 T^{2} + T^{4} )^{2}$$
$89$ $$( -7516 - 44 T + T^{2} )^{4}$$
$97$ $$( 3636 + 132 T + T^{2} )^{4}$$