# Properties

 Label 1280.3.e.e Level $1280$ Weight $3$ Character orbit 1280.e Analytic conductor $34.877$ Analytic rank $0$ Dimension $4$ CM discriminant -20 Inner twists $8$

# 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: $$4$$ Coefficient field: $$\Q(i, \sqrt{5})$$ Defining polynomial: $$x^{4} + 3 x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{29}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: no (minimal twist has level 80) 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.

 $$f(q)$$ $$=$$ $$q -\beta_{2} q^{3} + 5 \beta_{1} q^{5} + 3 \beta_{3} q^{7} -11 q^{9} +O(q^{10})$$ $$q -\beta_{2} q^{3} + 5 \beta_{1} q^{5} + 3 \beta_{3} q^{7} -11 q^{9} + 5 \beta_{3} q^{15} -60 \beta_{1} q^{21} + 3 \beta_{3} q^{23} -25 q^{25} + 2 \beta_{2} q^{27} + 22 \beta_{1} q^{29} + 15 \beta_{2} q^{35} + 62 q^{41} + 9 \beta_{2} q^{43} -55 \beta_{1} q^{45} + 21 \beta_{3} q^{47} + 131 q^{49} -58 \beta_{1} q^{61} -33 \beta_{3} q^{63} + 15 \beta_{2} q^{67} -60 \beta_{1} q^{69} + 25 \beta_{2} q^{75} -59 q^{81} -33 \beta_{2} q^{83} + 22 \beta_{3} q^{87} + 142 q^{89} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 44 q^{9} + O(q^{10})$$ $$4 q - 44 q^{9} - 100 q^{25} + 248 q^{41} + 524 q^{49} - 236 q^{81} + 568 q^{89} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{3} + 2 \nu$$ $$\beta_{2}$$ $$=$$ $$2 \nu^{3} + 8 \nu$$ $$\beta_{3}$$ $$=$$ $$4 \nu^{2} + 6$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} - 2 \beta_{1}$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} - 6$$$$)/4$$ $$\nu^{3}$$ $$=$$ $$($$$$-\beta_{2} + 4 \beta_{1}$$$$)/2$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
639.1
 1.61803i 0.618034i − 0.618034i − 1.61803i
0 4.47214i 0 5.00000i 0 −13.4164 0 −11.0000 0
639.2 0 4.47214i 0 5.00000i 0 13.4164 0 −11.0000 0
639.3 0 4.47214i 0 5.00000i 0 13.4164 0 −11.0000 0
639.4 0 4.47214i 0 5.00000i 0 −13.4164 0 −11.0000 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 $$\Q(\sqrt{-5})$$
4.b odd 2 1 inner
5.b even 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner
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.e 4
4.b odd 2 1 inner 1280.3.e.e 4
5.b even 2 1 inner 1280.3.e.e 4
8.b even 2 1 inner 1280.3.e.e 4
8.d odd 2 1 inner 1280.3.e.e 4
16.e even 4 1 80.3.h.a 2
16.e even 4 1 320.3.h.c 2
16.f odd 4 1 80.3.h.a 2
16.f odd 4 1 320.3.h.c 2
20.d odd 2 1 CM 1280.3.e.e 4
40.e odd 2 1 inner 1280.3.e.e 4
40.f even 2 1 inner 1280.3.e.e 4
48.i odd 4 1 720.3.j.a 2
48.k even 4 1 720.3.j.a 2
80.i odd 4 1 400.3.b.b 2
80.i odd 4 1 1600.3.b.d 2
80.j even 4 1 400.3.b.b 2
80.j even 4 1 1600.3.b.d 2
80.k odd 4 1 80.3.h.a 2
80.k odd 4 1 320.3.h.c 2
80.q even 4 1 80.3.h.a 2
80.q even 4 1 320.3.h.c 2
80.s even 4 1 400.3.b.b 2
80.s even 4 1 1600.3.b.d 2
80.t odd 4 1 400.3.b.b 2
80.t odd 4 1 1600.3.b.d 2
240.t even 4 1 720.3.j.a 2
240.z odd 4 1 3600.3.e.o 2
240.bb even 4 1 3600.3.e.o 2
240.bd odd 4 1 3600.3.e.o 2
240.bf even 4 1 3600.3.e.o 2
240.bm odd 4 1 720.3.j.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
80.3.h.a 2 16.e even 4 1
80.3.h.a 2 16.f odd 4 1
80.3.h.a 2 80.k odd 4 1
80.3.h.a 2 80.q even 4 1
320.3.h.c 2 16.e even 4 1
320.3.h.c 2 16.f odd 4 1
320.3.h.c 2 80.k odd 4 1
320.3.h.c 2 80.q even 4 1
400.3.b.b 2 80.i odd 4 1
400.3.b.b 2 80.j even 4 1
400.3.b.b 2 80.s even 4 1
400.3.b.b 2 80.t odd 4 1
720.3.j.a 2 48.i odd 4 1
720.3.j.a 2 48.k even 4 1
720.3.j.a 2 240.t even 4 1
720.3.j.a 2 240.bm odd 4 1
1280.3.e.e 4 1.a even 1 1 trivial
1280.3.e.e 4 4.b odd 2 1 inner
1280.3.e.e 4 5.b even 2 1 inner
1280.3.e.e 4 8.b even 2 1 inner
1280.3.e.e 4 8.d odd 2 1 inner
1280.3.e.e 4 20.d odd 2 1 CM
1280.3.e.e 4 40.e odd 2 1 inner
1280.3.e.e 4 40.f even 2 1 inner
1600.3.b.d 2 80.i odd 4 1
1600.3.b.d 2 80.j even 4 1
1600.3.b.d 2 80.s even 4 1
1600.3.b.d 2 80.t odd 4 1
3600.3.e.o 2 240.z odd 4 1
3600.3.e.o 2 240.bb even 4 1
3600.3.e.o 2 240.bd odd 4 1
3600.3.e.o 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 $$S_{3}^{\mathrm{new}}(1280, [\chi])$$:

 $$T_{3}^{2} + 20$$ $$T_{7}^{2} - 180$$ $$T_{11}$$ $$T_{13}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$( 20 + T^{2} )^{2}$$
$5$ $$( 25 + T^{2} )^{2}$$
$7$ $$( -180 + T^{2} )^{2}$$
$11$ $$T^{4}$$
$13$ $$T^{4}$$
$17$ $$T^{4}$$
$19$ $$T^{4}$$
$23$ $$( -180 + T^{2} )^{2}$$
$29$ $$( 484 + T^{2} )^{2}$$
$31$ $$T^{4}$$
$37$ $$T^{4}$$
$41$ $$( -62 + T )^{4}$$
$43$ $$( 1620 + T^{2} )^{2}$$
$47$ $$( -8820 + T^{2} )^{2}$$
$53$ $$T^{4}$$
$59$ $$T^{4}$$
$61$ $$( 3364 + T^{2} )^{2}$$
$67$ $$( 4500 + T^{2} )^{2}$$
$71$ $$T^{4}$$
$73$ $$T^{4}$$
$79$ $$T^{4}$$
$83$ $$( 21780 + T^{2} )^{2}$$
$89$ $$( -142 + T )^{4}$$
$97$ $$T^{4}$$