# Properties

 Label 1280.2.d.m Level $1280$ Weight $2$ Character orbit 1280.d Analytic conductor $10.221$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1280 = 2^{8} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1280.d (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: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 640) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( -\beta_{1} + \beta_{2} ) q^{3} + \beta_{1} q^{5} + ( 1 + \beta_{3} ) q^{7} + ( -3 + 2 \beta_{3} ) q^{9} +O(q^{10})$$ $$q + ( -\beta_{1} + \beta_{2} ) q^{3} + \beta_{1} q^{5} + ( 1 + \beta_{3} ) q^{7} + ( -3 + 2 \beta_{3} ) q^{9} + 2 \beta_{1} q^{11} -2 \beta_{2} q^{13} + ( 1 - \beta_{3} ) q^{15} + 2 \beta_{3} q^{17} -2 \beta_{2} q^{19} + 4 \beta_{1} q^{21} + ( 7 - \beta_{3} ) q^{23} - q^{25} + ( 10 \beta_{1} - 2 \beta_{2} ) q^{27} -2 \beta_{1} q^{29} + ( 2 + 2 \beta_{3} ) q^{31} + ( 2 - 2 \beta_{3} ) q^{33} + ( \beta_{1} + \beta_{2} ) q^{35} + ( 2 \beta_{1} - 4 \beta_{2} ) q^{37} + ( 10 - 2 \beta_{3} ) q^{39} + ( -8 - 2 \beta_{3} ) q^{41} + ( \beta_{1} + 3 \beta_{2} ) q^{43} + ( -3 \beta_{1} + 2 \beta_{2} ) q^{45} + ( -5 - \beta_{3} ) q^{47} + ( -1 + 2 \beta_{3} ) q^{49} + ( 10 \beta_{1} - 2 \beta_{2} ) q^{51} + ( 4 \beta_{1} - 2 \beta_{2} ) q^{53} -2 q^{55} + ( 10 - 2 \beta_{3} ) q^{57} + ( 4 \beta_{1} + 2 \beta_{2} ) q^{59} -6 \beta_{1} q^{61} + ( 7 - \beta_{3} ) q^{63} + 2 \beta_{3} q^{65} + ( \beta_{1} + 3 \beta_{2} ) q^{67} + ( -12 \beta_{1} + 8 \beta_{2} ) q^{69} + ( -2 + 2 \beta_{3} ) q^{71} -2 \beta_{3} q^{73} + ( \beta_{1} - \beta_{2} ) q^{75} + ( 2 \beta_{1} + 2 \beta_{2} ) q^{77} + ( 4 + 4 \beta_{3} ) q^{79} + ( 11 - 6 \beta_{3} ) q^{81} + ( 3 \beta_{1} - 3 \beta_{2} ) q^{83} + 2 \beta_{2} q^{85} + ( -2 + 2 \beta_{3} ) q^{87} + ( 6 + 4 \beta_{3} ) q^{89} + ( -10 \beta_{1} - 2 \beta_{2} ) q^{91} + 8 \beta_{1} q^{93} + 2 \beta_{3} q^{95} + ( -12 - 2 \beta_{3} ) q^{97} + ( -6 \beta_{1} + 4 \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 4 q^{7} - 12 q^{9} + O(q^{10})$$ $$4 q + 4 q^{7} - 12 q^{9} + 4 q^{15} + 28 q^{23} - 4 q^{25} + 8 q^{31} + 8 q^{33} + 40 q^{39} - 32 q^{41} - 20 q^{47} - 4 q^{49} - 8 q^{55} + 40 q^{57} + 28 q^{63} - 8 q^{71} + 16 q^{79} + 44 q^{81} - 8 q^{87} + 24 q^{89} - 48 q^{97} + O(q^{100})$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
641.1
 − 1.61803i − 0.618034i 0.618034i 1.61803i
0 3.23607i 0 1.00000i 0 −1.23607 0 −7.47214 0
641.2 0 1.23607i 0 1.00000i 0 3.23607 0 1.47214 0
641.3 0 1.23607i 0 1.00000i 0 3.23607 0 1.47214 0
641.4 0 3.23607i 0 1.00000i 0 −1.23607 0 −7.47214 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
8.b 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.d.m 4
4.b odd 2 1 1280.2.d.k 4
8.b even 2 1 inner 1280.2.d.m 4
8.d odd 2 1 1280.2.d.k 4
16.e even 4 1 640.2.a.i 2
16.e even 4 1 640.2.a.l yes 2
16.f odd 4 1 640.2.a.j yes 2
16.f odd 4 1 640.2.a.k yes 2
48.i odd 4 1 5760.2.a.bw 2
48.i odd 4 1 5760.2.a.ch 2
48.k even 4 1 5760.2.a.cd 2
48.k even 4 1 5760.2.a.ci 2
80.i odd 4 1 3200.2.c.u 4
80.i odd 4 1 3200.2.c.x 4
80.j even 4 1 3200.2.c.v 4
80.j even 4 1 3200.2.c.w 4
80.k odd 4 1 3200.2.a.be 2
80.k odd 4 1 3200.2.a.bk 2
80.q even 4 1 3200.2.a.bf 2
80.q even 4 1 3200.2.a.bl 2
80.s even 4 1 3200.2.c.v 4
80.s even 4 1 3200.2.c.w 4
80.t odd 4 1 3200.2.c.u 4
80.t odd 4 1 3200.2.c.x 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
640.2.a.i 2 16.e even 4 1
640.2.a.j yes 2 16.f odd 4 1
640.2.a.k yes 2 16.f odd 4 1
640.2.a.l yes 2 16.e even 4 1
1280.2.d.k 4 4.b odd 2 1
1280.2.d.k 4 8.d odd 2 1
1280.2.d.m 4 1.a even 1 1 trivial
1280.2.d.m 4 8.b even 2 1 inner
3200.2.a.be 2 80.k odd 4 1
3200.2.a.bf 2 80.q even 4 1
3200.2.a.bk 2 80.k odd 4 1
3200.2.a.bl 2 80.q even 4 1
3200.2.c.u 4 80.i odd 4 1
3200.2.c.u 4 80.t odd 4 1
3200.2.c.v 4 80.j even 4 1
3200.2.c.v 4 80.s even 4 1
3200.2.c.w 4 80.j even 4 1
3200.2.c.w 4 80.s even 4 1
3200.2.c.x 4 80.i odd 4 1
3200.2.c.x 4 80.t odd 4 1
5760.2.a.bw 2 48.i odd 4 1
5760.2.a.cd 2 48.k even 4 1
5760.2.a.ch 2 48.i odd 4 1
5760.2.a.ci 2 48.k even 4 1

## Hecke kernels

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

 $$T_{3}^{4} + 12 T_{3}^{2} + 16$$ $$T_{7}^{2} - 2 T_{7} - 4$$ $$T_{11}^{2} + 4$$ $$T_{31}^{2} - 4 T_{31} - 16$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$16 + 12 T^{2} + T^{4}$$
$5$ $$( 1 + T^{2} )^{2}$$
$7$ $$( -4 - 2 T + T^{2} )^{2}$$
$11$ $$( 4 + T^{2} )^{2}$$
$13$ $$( 20 + T^{2} )^{2}$$
$17$ $$( -20 + T^{2} )^{2}$$
$19$ $$( 20 + T^{2} )^{2}$$
$23$ $$( 44 - 14 T + T^{2} )^{2}$$
$29$ $$( 4 + T^{2} )^{2}$$
$31$ $$( -16 - 4 T + T^{2} )^{2}$$
$37$ $$5776 + 168 T^{2} + T^{4}$$
$41$ $$( 44 + 16 T + T^{2} )^{2}$$
$43$ $$1936 + 92 T^{2} + T^{4}$$
$47$ $$( 20 + 10 T + T^{2} )^{2}$$
$53$ $$16 + 72 T^{2} + T^{4}$$
$59$ $$16 + 72 T^{2} + T^{4}$$
$61$ $$( 36 + T^{2} )^{2}$$
$67$ $$1936 + 92 T^{2} + T^{4}$$
$71$ $$( -16 + 4 T + T^{2} )^{2}$$
$73$ $$( -20 + T^{2} )^{2}$$
$79$ $$( -64 - 8 T + T^{2} )^{2}$$
$83$ $$1296 + 108 T^{2} + T^{4}$$
$89$ $$( -44 - 12 T + T^{2} )^{2}$$
$97$ $$( 124 + 24 T + T^{2} )^{2}$$