# Properties

 Label 1280.2.f.h 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: $$x^{4} + 3x^{2} + 1$$ 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.

 $$f(q)$$ $$=$$ $$q + (\beta_{3} + 1) q^{3} - \beta_{2} q^{5} + (\beta_{2} - 3 \beta_1) q^{7} + (2 \beta_{3} + 3) q^{9}+O(q^{10})$$ q + (b3 + 1) * q^3 - b2 * q^5 + (b2 - 3*b1) * q^7 + (2*b3 + 3) * q^9 $$q + (\beta_{3} + 1) q^{3} - \beta_{2} q^{5} + (\beta_{2} - 3 \beta_1) q^{7} + (2 \beta_{3} + 3) q^{9} + ( - \beta_{2} - 5 \beta_1) q^{15} + ( - 2 \beta_{2} + 2 \beta_1) q^{21} + ( - 3 \beta_{2} + \beta_1) q^{23} - 5 q^{25} + (2 \beta_{3} + 10) q^{27} + 6 \beta_1 q^{29} + ( - 3 \beta_{3} + 5) q^{35} + 2 \beta_{3} q^{41} + ( - \beta_{3} - 9) q^{43} + ( - 3 \beta_{2} - 10 \beta_1) q^{45} + (3 \beta_{2} + 7 \beta_1) q^{47} + (6 \beta_{3} - 7) q^{49} - 6 \beta_{2} q^{61} + ( - 3 \beta_{2} + \beta_1) q^{63} + (5 \beta_{3} - 3) q^{67} + ( - 2 \beta_{2} - 14 \beta_1) q^{69} + ( - 5 \beta_{3} - 5) q^{75} + (6 \beta_{3} + 11) q^{81} + ( - 3 \beta_{3} - 11) q^{83} + (6 \beta_{2} + 6 \beta_1) q^{87} + 6 q^{89}+O(q^{100})$$ q + (b3 + 1) * q^3 - b2 * q^5 + (b2 - 3*b1) * q^7 + (2*b3 + 3) * q^9 + (-b2 - 5*b1) * q^15 + (-2*b2 + 2*b1) * q^21 + (-3*b2 + b1) * q^23 - 5 * q^25 + (2*b3 + 10) * q^27 + 6*b1 * q^29 + (-3*b3 + 5) * q^35 + 2*b3 * q^41 + (-b3 - 9) * q^43 + (-3*b2 - 10*b1) * q^45 + (3*b2 + 7*b1) * q^47 + (6*b3 - 7) * q^49 - 6*b2 * q^61 + (-3*b2 + b1) * q^63 + (5*b3 - 3) * q^67 + (-2*b2 - 14*b1) * q^69 + (-5*b3 - 5) * q^75 + (6*b3 + 11) * q^81 + (-3*b3 - 11) * q^83 + (6*b2 + 6*b1) * q^87 + 6 * q^89 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 4 q^{3} + 12 q^{9}+O(q^{10})$$ 4 * q + 4 * q^3 + 12 * q^9 $$4 q + 4 q^{3} + 12 q^{9} - 20 q^{25} + 40 q^{27} + 20 q^{35} - 36 q^{43} - 28 q^{49} - 12 q^{67} - 20 q^{75} + 44 q^{81} - 44 q^{83} + 24 q^{89}+O(q^{100})$$ 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

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

 $$\beta_{1}$$ $$=$$ $$\nu^{3} + 2\nu$$ v^3 + 2*v $$\beta_{2}$$ $$=$$ $$\nu^{3} + 4\nu$$ v^3 + 4*v $$\beta_{3}$$ $$=$$ $$2\nu^{2} + 3$$ 2*v^2 + 3
 $$\nu$$ $$=$$ $$( \beta_{2} - \beta_1 ) / 2$$ (b2 - b1) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{3} - 3 ) / 2$$ (b3 - 3) / 2 $$\nu^{3}$$ $$=$$ $$-\beta_{2} + 2\beta_1$$ -b2 + 2*b1

## 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
 1.61803i − 1.61803i 0.618034i − 0.618034i
0 −1.23607 0 2.23607i 0 5.23607i 0 −1.47214 0
129.2 0 −1.23607 0 2.23607i 0 5.23607i 0 −1.47214 0
129.3 0 3.23607 0 2.23607i 0 0.763932i 0 7.47214 0
129.4 0 3.23607 0 2.23607i 0 0.763932i 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
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.h 4
4.b odd 2 1 1280.2.f.g 4
5.b even 2 1 1280.2.f.g 4
8.b even 2 1 1280.2.f.g 4
8.d odd 2 1 inner 1280.2.f.h 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.h 4
40.e odd 2 1 1280.2.f.g 4
40.f even 2 1 inner 1280.2.f.h 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.j 2
80.i odd 4 1 1600.2.a.z 2
80.j even 4 1 800.2.a.j 2
80.j even 4 1 1600.2.a.z 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.n 2
80.s even 4 1 1600.2.a.bd 2
80.t odd 4 1 800.2.a.n 2
80.t odd 4 1 1600.2.a.bd 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.cr 2
240.bb even 4 1 7200.2.a.cb 2
240.bd odd 4 1 7200.2.a.cb 2
240.bf even 4 1 7200.2.a.cr 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.i odd 4 1
800.2.a.j 2 80.j even 4 1
800.2.a.n 2 80.s even 4 1
800.2.a.n 2 80.t odd 4 1
1280.2.f.g 4 4.b odd 2 1
1280.2.f.g 4 5.b even 2 1
1280.2.f.g 4 8.b even 2 1
1280.2.f.g 4 40.e odd 2 1
1280.2.f.h 4 1.a even 1 1 trivial
1280.2.f.h 4 8.d odd 2 1 inner
1280.2.f.h 4 20.d odd 2 1 CM
1280.2.f.h 4 40.f even 2 1 inner
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.i odd 4 1
1600.2.a.z 2 80.j even 4 1
1600.2.a.bd 2 80.s even 4 1
1600.2.a.bd 2 80.t odd 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.bb even 4 1
7200.2.a.cb 2 240.bd odd 4 1
7200.2.a.cr 2 240.z odd 4 1
7200.2.a.cr 2 240.bf even 4 1

## Hecke kernels

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

 $$T_{3}^{2} - 2T_{3} - 4$$ T3^2 - 2*T3 - 4 $$T_{13}$$ T13

## Hecke characteristic polynomials

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