# Properties

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

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## 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: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 40) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + i q^{5} + 4 q^{7} + 3 q^{9} +O(q^{10})$$ $$q + i q^{5} + 4 q^{7} + 3 q^{9} + 4 i q^{11} + 2 i q^{13} + 2 q^{17} -4 i q^{19} -4 q^{23} - q^{25} + 2 i q^{29} -8 q^{31} + 4 i q^{35} + 6 i q^{37} + 6 q^{41} -8 i q^{43} + 3 i q^{45} + 4 q^{47} + 9 q^{49} + 6 i q^{53} -4 q^{55} -4 i q^{59} + 2 i q^{61} + 12 q^{63} -2 q^{65} -8 i q^{67} + 6 q^{73} + 16 i q^{77} + 9 q^{81} + 16 i q^{83} + 2 i q^{85} + 6 q^{89} + 8 i q^{91} + 4 q^{95} -14 q^{97} + 12 i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 8q^{7} + 6q^{9} + O(q^{10})$$ $$2q + 8q^{7} + 6q^{9} + 4q^{17} - 8q^{23} - 2q^{25} - 16q^{31} + 12q^{41} + 8q^{47} + 18q^{49} - 8q^{55} + 24q^{63} - 4q^{65} + 12q^{73} + 18q^{81} + 12q^{89} + 8q^{95} - 28q^{97} + O(q^{100})$$

## 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
 − 1.00000i 1.00000i
0 0 0 1.00000i 0 4.00000 0 3.00000 0
641.2 0 0 0 1.00000i 0 4.00000 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
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.j 2
4.b odd 2 1 1280.2.d.a 2
8.b even 2 1 inner 1280.2.d.j 2
8.d odd 2 1 1280.2.d.a 2
16.e even 4 1 40.2.a.a 1
16.e even 4 1 320.2.a.c 1
16.f odd 4 1 80.2.a.a 1
16.f odd 4 1 320.2.a.d 1
48.i odd 4 1 360.2.a.a 1
48.i odd 4 1 2880.2.a.t 1
48.k even 4 1 720.2.a.e 1
48.k even 4 1 2880.2.a.bg 1
80.i odd 4 1 200.2.c.b 2
80.i odd 4 1 1600.2.c.k 2
80.j even 4 1 400.2.c.d 2
80.j even 4 1 1600.2.c.m 2
80.k odd 4 1 400.2.a.e 1
80.k odd 4 1 1600.2.a.k 1
80.q even 4 1 200.2.a.c 1
80.q even 4 1 1600.2.a.o 1
80.s even 4 1 400.2.c.d 2
80.s even 4 1 1600.2.c.m 2
80.t odd 4 1 200.2.c.b 2
80.t odd 4 1 1600.2.c.k 2
112.j even 4 1 3920.2.a.s 1
112.l odd 4 1 1960.2.a.g 1
112.w even 12 2 1960.2.q.h 2
112.x odd 12 2 1960.2.q.i 2
144.w odd 12 2 3240.2.q.x 2
144.x even 12 2 3240.2.q.k 2
176.i even 4 1 9680.2.a.q 1
176.l odd 4 1 4840.2.a.f 1
208.p even 4 1 6760.2.a.i 1
240.t even 4 1 3600.2.a.h 1
240.z odd 4 1 3600.2.f.t 2
240.bb even 4 1 1800.2.f.a 2
240.bd odd 4 1 3600.2.f.t 2
240.bf even 4 1 1800.2.f.a 2
240.bm odd 4 1 1800.2.a.v 1
560.bf odd 4 1 9800.2.a.x 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.2.a.a 1 16.e even 4 1
80.2.a.a 1 16.f odd 4 1
200.2.a.c 1 80.q even 4 1
200.2.c.b 2 80.i odd 4 1
200.2.c.b 2 80.t odd 4 1
320.2.a.c 1 16.e even 4 1
320.2.a.d 1 16.f odd 4 1
360.2.a.a 1 48.i odd 4 1
400.2.a.e 1 80.k odd 4 1
400.2.c.d 2 80.j even 4 1
400.2.c.d 2 80.s even 4 1
720.2.a.e 1 48.k even 4 1
1280.2.d.a 2 4.b odd 2 1
1280.2.d.a 2 8.d odd 2 1
1280.2.d.j 2 1.a even 1 1 trivial
1280.2.d.j 2 8.b even 2 1 inner
1600.2.a.k 1 80.k odd 4 1
1600.2.a.o 1 80.q even 4 1
1600.2.c.k 2 80.i odd 4 1
1600.2.c.k 2 80.t odd 4 1
1600.2.c.m 2 80.j even 4 1
1600.2.c.m 2 80.s even 4 1
1800.2.a.v 1 240.bm odd 4 1
1800.2.f.a 2 240.bb even 4 1
1800.2.f.a 2 240.bf even 4 1
1960.2.a.g 1 112.l odd 4 1
1960.2.q.h 2 112.w even 12 2
1960.2.q.i 2 112.x odd 12 2
2880.2.a.t 1 48.i odd 4 1
2880.2.a.bg 1 48.k even 4 1
3240.2.q.k 2 144.x even 12 2
3240.2.q.x 2 144.w odd 12 2
3600.2.a.h 1 240.t even 4 1
3600.2.f.t 2 240.z odd 4 1
3600.2.f.t 2 240.bd odd 4 1
3920.2.a.s 1 112.j even 4 1
4840.2.a.f 1 176.l odd 4 1
6760.2.a.i 1 208.p even 4 1
9680.2.a.q 1 176.i even 4 1
9800.2.a.x 1 560.bf odd 4 1

## Hecke kernels

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

 $$T_{3}$$ $$T_{7} - 4$$ $$T_{11}^{2} + 16$$ $$T_{31} + 8$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$1 + T^{2}$$
$7$ $$( -4 + T )^{2}$$
$11$ $$16 + T^{2}$$
$13$ $$4 + T^{2}$$
$17$ $$( -2 + T )^{2}$$
$19$ $$16 + T^{2}$$
$23$ $$( 4 + T )^{2}$$
$29$ $$4 + T^{2}$$
$31$ $$( 8 + T )^{2}$$
$37$ $$36 + T^{2}$$
$41$ $$( -6 + T )^{2}$$
$43$ $$64 + T^{2}$$
$47$ $$( -4 + T )^{2}$$
$53$ $$36 + T^{2}$$
$59$ $$16 + T^{2}$$
$61$ $$4 + T^{2}$$
$67$ $$64 + T^{2}$$
$71$ $$T^{2}$$
$73$ $$( -6 + T )^{2}$$
$79$ $$T^{2}$$
$83$ $$256 + T^{2}$$
$89$ $$( -6 + T )^{2}$$
$97$ $$( 14 + T )^{2}$$
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