# Properties

 Label 1280.2.f.g Level $1280$ Weight $2$ Character orbit 1280.f Analytic conductor $10.221$ Analytic rank $0$ Dimension $4$ CM discriminant -20 Inner twists $4$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$10.2208514587$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{5})$$ Defining polynomial: $$x^{4} + 3 x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 160) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -1 - \beta_{3} ) q^{3} -\beta_{2} q^{5} + ( 3 \beta_{1} - \beta_{2} ) q^{7} + ( 3 + 2 \beta_{3} ) q^{9} +O(q^{10})$$ $$q + ( -1 - \beta_{3} ) q^{3} -\beta_{2} q^{5} + ( 3 \beta_{1} - \beta_{2} ) q^{7} + ( 3 + 2 \beta_{3} ) q^{9} + ( 5 \beta_{1} + \beta_{2} ) q^{15} + ( 2 \beta_{1} - 2 \beta_{2} ) q^{21} + ( -\beta_{1} + 3 \beta_{2} ) q^{23} -5 q^{25} + ( -10 - 2 \beta_{3} ) q^{27} + 6 \beta_{1} q^{29} + ( -5 + 3 \beta_{3} ) q^{35} + 2 \beta_{3} q^{41} + ( 9 + \beta_{3} ) q^{43} + ( -10 \beta_{1} - 3 \beta_{2} ) q^{45} + ( -7 \beta_{1} - 3 \beta_{2} ) q^{47} + ( -7 + 6 \beta_{3} ) q^{49} -6 \beta_{2} q^{61} + ( -\beta_{1} + 3 \beta_{2} ) q^{63} + ( 3 - 5 \beta_{3} ) q^{67} + ( -14 \beta_{1} - 2 \beta_{2} ) q^{69} + ( 5 + 5 \beta_{3} ) q^{75} + ( 11 + 6 \beta_{3} ) q^{81} + ( 11 + 3 \beta_{3} ) q^{83} + ( -6 \beta_{1} - 6 \beta_{2} ) q^{87} + 6 q^{89} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{3} + 12 q^{9} + O(q^{10})$$ $$4 q - 4 q^{3} + 12 q^{9} - 20 q^{25} - 40 q^{27} - 20 q^{35} + 36 q^{43} - 28 q^{49} + 12 q^{67} + 20 q^{75} + 44 q^{81} + 44 q^{83} + 24 q^{89} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 3 x^{2} + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{3} + 2 \nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{3} + 4 \nu$$ $$\beta_{3}$$ $$=$$ $$2 \nu^{2} + 3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} - \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} - 3$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$-\beta_{2} + 2 \beta_{1}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
129.1
 0.618034i − 0.618034i 1.61803i − 1.61803i
0 −3.23607 0 2.23607i 0 0.763932i 0 7.47214 0
129.2 0 −3.23607 0 2.23607i 0 0.763932i 0 7.47214 0
129.3 0 1.23607 0 2.23607i 0 5.23607i 0 −1.47214 0
129.4 0 1.23607 0 2.23607i 0 5.23607i 0 −1.47214 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
20.d odd 2 1 CM by $$\Q(\sqrt{-5})$$
8.d 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 1280.2.f.g 4
4.b odd 2 1 1280.2.f.h 4
5.b even 2 1 1280.2.f.h 4
8.b even 2 1 1280.2.f.h 4
8.d odd 2 1 inner 1280.2.f.g 4
16.e even 4 1 160.2.c.b 4
16.e even 4 1 320.2.c.d 4
16.f odd 4 1 160.2.c.b 4
16.f odd 4 1 320.2.c.d 4
20.d odd 2 1 CM 1280.2.f.g 4
40.e odd 2 1 1280.2.f.h 4
40.f even 2 1 inner 1280.2.f.g 4
48.i odd 4 1 1440.2.f.i 4
48.i odd 4 1 2880.2.f.w 4
48.k even 4 1 1440.2.f.i 4
48.k even 4 1 2880.2.f.w 4
80.i odd 4 1 800.2.a.n 2
80.i odd 4 1 1600.2.a.bd 2
80.j even 4 1 800.2.a.n 2
80.j even 4 1 1600.2.a.bd 2
80.k odd 4 1 160.2.c.b 4
80.k odd 4 1 320.2.c.d 4
80.q even 4 1 160.2.c.b 4
80.q even 4 1 320.2.c.d 4
80.s even 4 1 800.2.a.j 2
80.s even 4 1 1600.2.a.z 2
80.t odd 4 1 800.2.a.j 2
80.t odd 4 1 1600.2.a.z 2
240.t even 4 1 1440.2.f.i 4
240.t even 4 1 2880.2.f.w 4
240.z odd 4 1 7200.2.a.cb 2
240.bb even 4 1 7200.2.a.cr 2
240.bd odd 4 1 7200.2.a.cr 2
240.bf even 4 1 7200.2.a.cb 2
240.bm odd 4 1 1440.2.f.i 4
240.bm odd 4 1 2880.2.f.w 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.2.c.b 4 16.e even 4 1
160.2.c.b 4 16.f odd 4 1
160.2.c.b 4 80.k odd 4 1
160.2.c.b 4 80.q even 4 1
320.2.c.d 4 16.e even 4 1
320.2.c.d 4 16.f odd 4 1
320.2.c.d 4 80.k odd 4 1
320.2.c.d 4 80.q even 4 1
800.2.a.j 2 80.s even 4 1
800.2.a.j 2 80.t odd 4 1
800.2.a.n 2 80.i odd 4 1
800.2.a.n 2 80.j even 4 1
1280.2.f.g 4 1.a even 1 1 trivial
1280.2.f.g 4 8.d odd 2 1 inner
1280.2.f.g 4 20.d odd 2 1 CM
1280.2.f.g 4 40.f even 2 1 inner
1280.2.f.h 4 4.b odd 2 1
1280.2.f.h 4 5.b even 2 1
1280.2.f.h 4 8.b even 2 1
1280.2.f.h 4 40.e odd 2 1
1440.2.f.i 4 48.i odd 4 1
1440.2.f.i 4 48.k even 4 1
1440.2.f.i 4 240.t even 4 1
1440.2.f.i 4 240.bm odd 4 1
1600.2.a.z 2 80.s even 4 1
1600.2.a.z 2 80.t odd 4 1
1600.2.a.bd 2 80.i odd 4 1
1600.2.a.bd 2 80.j even 4 1
2880.2.f.w 4 48.i odd 4 1
2880.2.f.w 4 48.k even 4 1
2880.2.f.w 4 240.t even 4 1
2880.2.f.w 4 240.bm odd 4 1
7200.2.a.cb 2 240.z odd 4 1
7200.2.a.cb 2 240.bf even 4 1
7200.2.a.cr 2 240.bb even 4 1
7200.2.a.cr 2 240.bd odd 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1280, [\chi])$$:

 $$T_{3}^{2} + 2 T_{3} - 4$$ $$T_{13}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$( -4 + 2 T + T^{2} )^{2}$$
$5$ $$( 5 + T^{2} )^{2}$$
$7$ $$16 + 28 T^{2} + T^{4}$$
$11$ $$T^{4}$$
$13$ $$T^{4}$$
$17$ $$T^{4}$$
$19$ $$T^{4}$$
$23$ $$1936 + 92 T^{2} + T^{4}$$
$29$ $$( 36 + T^{2} )^{2}$$
$31$ $$T^{4}$$
$37$ $$T^{4}$$
$41$ $$( -20 + T^{2} )^{2}$$
$43$ $$( 76 - 18 T + T^{2} )^{2}$$
$47$ $$16 + 188 T^{2} + T^{4}$$
$53$ $$T^{4}$$
$59$ $$T^{4}$$
$61$ $$( 180 + T^{2} )^{2}$$
$67$ $$( -116 - 6 T + T^{2} )^{2}$$
$71$ $$T^{4}$$
$73$ $$T^{4}$$
$79$ $$T^{4}$$
$83$ $$( 76 - 22 T + T^{2} )^{2}$$
$89$ $$( -6 + T )^{4}$$
$97$ $$T^{4}$$