# Properties

 Label 1280.3.e.a Level $1280$ Weight $3$ Character orbit 1280.e Analytic conductor $34.877$ Analytic rank $0$ Dimension $2$ CM discriminant -4 Inner twists $4$

# 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: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 20) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -4 + 3 i ) q^{5} + 9 q^{9} +O(q^{10})$$ $$q + ( -4 + 3 i ) q^{5} + 9 q^{9} -24 q^{13} + 16 i q^{17} + ( 7 - 24 i ) q^{25} -42 i q^{29} + 24 q^{37} + 18 q^{41} + ( -36 + 27 i ) q^{45} -49 q^{49} -56 q^{53} -22 i q^{61} + ( 96 - 72 i ) q^{65} -96 i q^{73} + 81 q^{81} + ( -48 - 64 i ) q^{85} + 78 q^{89} -144 i q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 8 q^{5} + 18 q^{9} + O(q^{10})$$ $$2 q - 8 q^{5} + 18 q^{9} - 48 q^{13} + 14 q^{25} + 48 q^{37} + 36 q^{41} - 72 q^{45} - 98 q^{49} - 112 q^{53} + 192 q^{65} + 162 q^{81} - 96 q^{85} + 156 q^{89} + O(q^{100})$$

## 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
 − 1.00000i 1.00000i
0 0 0 −4.00000 3.00000i 0 0 0 9.00000 0
639.2 0 0 0 −4.00000 + 3.00000i 0 0 0 9.00000 0
 $$n$$: e.g. 2-40 or 990-1000 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 CM by $$\Q(\sqrt{-1})$$
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.a 2
4.b odd 2 1 CM 1280.3.e.a 2
5.b even 2 1 1280.3.e.d 2
8.b even 2 1 1280.3.e.d 2
8.d odd 2 1 1280.3.e.d 2
16.e even 4 1 20.3.d.c 2
16.e even 4 1 320.3.h.d 2
16.f odd 4 1 20.3.d.c 2
16.f odd 4 1 320.3.h.d 2
20.d odd 2 1 1280.3.e.d 2
40.e odd 2 1 inner 1280.3.e.a 2
40.f even 2 1 inner 1280.3.e.a 2
48.i odd 4 1 180.3.f.c 2
48.k even 4 1 180.3.f.c 2
80.i odd 4 1 100.3.b.a 1
80.i odd 4 1 1600.3.b.c 1
80.j even 4 1 100.3.b.b 1
80.j even 4 1 1600.3.b.a 1
80.k odd 4 1 20.3.d.c 2
80.k odd 4 1 320.3.h.d 2
80.q even 4 1 20.3.d.c 2
80.q even 4 1 320.3.h.d 2
80.s even 4 1 100.3.b.a 1
80.s even 4 1 1600.3.b.c 1
80.t odd 4 1 100.3.b.b 1
80.t odd 4 1 1600.3.b.a 1
240.t even 4 1 180.3.f.c 2
240.z odd 4 1 900.3.c.d 1
240.bb even 4 1 900.3.c.d 1
240.bd odd 4 1 900.3.c.a 1
240.bf even 4 1 900.3.c.a 1
240.bm odd 4 1 180.3.f.c 2

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

## Hecke kernels

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

 $$T_{3}$$ $$T_{7}$$ $$T_{11}$$ $$T_{13} + 24$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$25 + 8 T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$T^{2}$$
$13$ $$( 24 + T )^{2}$$
$17$ $$256 + T^{2}$$
$19$ $$T^{2}$$
$23$ $$T^{2}$$
$29$ $$1764 + T^{2}$$
$31$ $$T^{2}$$
$37$ $$( -24 + T )^{2}$$
$41$ $$( -18 + T )^{2}$$
$43$ $$T^{2}$$
$47$ $$T^{2}$$
$53$ $$( 56 + T )^{2}$$
$59$ $$T^{2}$$
$61$ $$484 + T^{2}$$
$67$ $$T^{2}$$
$71$ $$T^{2}$$
$73$ $$9216 + T^{2}$$
$79$ $$T^{2}$$
$83$ $$T^{2}$$
$89$ $$( -78 + T )^{2}$$
$97$ $$20736 + T^{2}$$