# Properties

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

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$10.2208514587$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{3})$$ Defining polynomial: $$x^{4} + 4 x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 320) 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.

 $$f(q)$$ $$=$$ $$q + \beta_{3} q^{3} + ( \beta_{1} + \beta_{2} ) q^{5} -\beta_{1} q^{7} -3 q^{9} +O(q^{10})$$ $$q + \beta_{3} q^{3} + ( \beta_{1} + \beta_{2} ) q^{5} -\beta_{1} q^{7} -3 q^{9} + 2 q^{11} + 4 \beta_{1} q^{13} + ( 3 \beta_{1} - 2 \beta_{2} ) q^{15} -2 \beta_{3} q^{17} -6 q^{19} + 2 \beta_{2} q^{21} + 5 \beta_{1} q^{23} + ( 1 + 2 \beta_{3} ) q^{25} + 4 \beta_{2} q^{29} -4 \beta_{2} q^{31} + 2 \beta_{3} q^{33} + ( 2 - \beta_{3} ) q^{35} + 2 \beta_{1} q^{37} -8 \beta_{2} q^{39} + 4 q^{41} -\beta_{3} q^{43} + ( -3 \beta_{1} - 3 \beta_{2} ) q^{45} + 3 \beta_{1} q^{47} + 5 q^{49} + 12 q^{51} + ( 2 \beta_{1} + 2 \beta_{2} ) q^{55} -6 \beta_{3} q^{57} -2 q^{59} -2 \beta_{2} q^{61} + 3 \beta_{1} q^{63} + ( -8 + 4 \beta_{3} ) q^{65} -\beta_{3} q^{67} -10 \beta_{2} q^{69} -4 \beta_{2} q^{71} + 2 \beta_{3} q^{73} + ( -12 + \beta_{3} ) q^{75} -2 \beta_{1} q^{77} + 4 \beta_{2} q^{79} -9 q^{81} -5 \beta_{3} q^{83} + ( -6 \beta_{1} + 4 \beta_{2} ) q^{85} + 12 \beta_{1} q^{87} + 2 q^{89} + 8 q^{91} -12 \beta_{1} q^{93} + ( -6 \beta_{1} - 6 \beta_{2} ) q^{95} -6 \beta_{3} q^{97} -6 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 12q^{9} + O(q^{10})$$ $$4q - 12q^{9} + 8q^{11} - 24q^{19} + 4q^{25} + 8q^{35} + 16q^{41} + 20q^{49} + 48q^{51} - 8q^{59} - 32q^{65} - 48q^{75} - 36q^{81} + 8q^{89} + 32q^{91} - 24q^{99} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{3} + 3 \nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} + 2$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} + 5 \nu$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} - \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} - 2$$ $$\nu^{3}$$ $$=$$ $$($$$$-3 \beta_{3} + 5 \beta_{1}$$$$)/2$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
769.1
 − 1.93185i − 0.517638i 1.93185i 0.517638i
0 2.44949i 0 −1.73205 + 1.41421i 0 1.41421i 0 −3.00000 0
769.2 0 2.44949i 0 1.73205 1.41421i 0 1.41421i 0 −3.00000 0
769.3 0 2.44949i 0 −1.73205 1.41421i 0 1.41421i 0 −3.00000 0
769.4 0 2.44949i 0 1.73205 + 1.41421i 0 1.41421i 0 −3.00000 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
5.b even 2 1 inner
8.d odd 2 1 inner
40.e odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1280.2.c.h 4
4.b odd 2 1 1280.2.c.g 4
5.b even 2 1 inner 1280.2.c.h 4
5.c odd 4 2 6400.2.a.cu 4
8.b even 2 1 1280.2.c.g 4
8.d odd 2 1 inner 1280.2.c.h 4
16.e even 4 2 320.2.f.b 8
16.f odd 4 2 320.2.f.b 8
20.d odd 2 1 1280.2.c.g 4
20.e even 4 2 6400.2.a.ct 4
40.e odd 2 1 inner 1280.2.c.h 4
40.f even 2 1 1280.2.c.g 4
40.i odd 4 2 6400.2.a.ct 4
40.k even 4 2 6400.2.a.cu 4
48.i odd 4 2 2880.2.d.g 8
48.k even 4 2 2880.2.d.g 8
80.i odd 4 2 1600.2.d.i 8
80.j even 4 2 1600.2.d.i 8
80.k odd 4 2 320.2.f.b 8
80.q even 4 2 320.2.f.b 8
80.s even 4 2 1600.2.d.i 8
80.t odd 4 2 1600.2.d.i 8
240.t even 4 2 2880.2.d.g 8
240.bm odd 4 2 2880.2.d.g 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
320.2.f.b 8 16.e even 4 2
320.2.f.b 8 16.f odd 4 2
320.2.f.b 8 80.k odd 4 2
320.2.f.b 8 80.q even 4 2
1280.2.c.g 4 4.b odd 2 1
1280.2.c.g 4 8.b even 2 1
1280.2.c.g 4 20.d odd 2 1
1280.2.c.g 4 40.f even 2 1
1280.2.c.h 4 1.a even 1 1 trivial
1280.2.c.h 4 5.b even 2 1 inner
1280.2.c.h 4 8.d odd 2 1 inner
1280.2.c.h 4 40.e odd 2 1 inner
1600.2.d.i 8 80.i odd 4 2
1600.2.d.i 8 80.j even 4 2
1600.2.d.i 8 80.s even 4 2
1600.2.d.i 8 80.t odd 4 2
2880.2.d.g 8 48.i odd 4 2
2880.2.d.g 8 48.k even 4 2
2880.2.d.g 8 240.t even 4 2
2880.2.d.g 8 240.bm odd 4 2
6400.2.a.ct 4 20.e even 4 2
6400.2.a.ct 4 40.i odd 4 2
6400.2.a.cu 4 5.c odd 4 2
6400.2.a.cu 4 40.k even 4 2

## Hecke kernels

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

 $$T_{3}^{2} + 6$$ $$T_{7}^{2} + 2$$ $$T_{11} - 2$$ $$T_{29}^{2} - 48$$ $$T_{31}^{2} - 48$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$( 6 + T^{2} )^{2}$$
$5$ $$25 - 2 T^{2} + T^{4}$$
$7$ $$( 2 + T^{2} )^{2}$$
$11$ $$( -2 + T )^{4}$$
$13$ $$( 32 + T^{2} )^{2}$$
$17$ $$( 24 + T^{2} )^{2}$$
$19$ $$( 6 + T )^{4}$$
$23$ $$( 50 + T^{2} )^{2}$$
$29$ $$( -48 + T^{2} )^{2}$$
$31$ $$( -48 + T^{2} )^{2}$$
$37$ $$( 8 + T^{2} )^{2}$$
$41$ $$( -4 + T )^{4}$$
$43$ $$( 6 + T^{2} )^{2}$$
$47$ $$( 18 + T^{2} )^{2}$$
$53$ $$T^{4}$$
$59$ $$( 2 + T )^{4}$$
$61$ $$( -12 + T^{2} )^{2}$$
$67$ $$( 6 + T^{2} )^{2}$$
$71$ $$( -48 + T^{2} )^{2}$$
$73$ $$( 24 + T^{2} )^{2}$$
$79$ $$( -48 + T^{2} )^{2}$$
$83$ $$( 150 + T^{2} )^{2}$$
$89$ $$( -2 + T )^{4}$$
$97$ $$( 216 + T^{2} )^{2}$$