# Properties

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

# 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: $$2$$ 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.

 $$f(q)$$ $$=$$ $$q + 2 q^{3} + ( 2 - i ) q^{5} + 2 i q^{7} + q^{9} +O(q^{10})$$ $$q + 2 q^{3} + ( 2 - i ) q^{5} + 2 i q^{7} + q^{9} -4 i q^{11} + 4 q^{13} + ( 4 - 2 i ) q^{15} -4 i q^{19} + 4 i q^{21} + 2 i q^{23} + ( 3 - 4 i ) q^{25} -4 q^{27} + 2 i q^{29} -8 i q^{33} + ( 2 + 4 i ) q^{35} + 4 q^{37} + 8 q^{39} -2 q^{41} + 6 q^{43} + ( 2 - i ) q^{45} + 6 i q^{47} + 3 q^{49} + 4 q^{53} + ( -4 - 8 i ) q^{55} -8 i q^{57} + 12 i q^{59} + 10 i q^{61} + 2 i q^{63} + ( 8 - 4 i ) q^{65} -14 q^{67} + 4 i q^{69} -8 q^{71} -8 i q^{73} + ( 6 - 8 i ) q^{75} + 8 q^{77} -16 q^{79} -11 q^{81} + 2 q^{83} + 4 i q^{87} + 6 q^{89} + 8 i q^{91} + ( -4 - 8 i ) q^{95} -16 i q^{97} -4 i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 4q^{3} + 4q^{5} + 2q^{9} + O(q^{10})$$ $$2q + 4q^{3} + 4q^{5} + 2q^{9} + 8q^{13} + 8q^{15} + 6q^{25} - 8q^{27} + 4q^{35} + 8q^{37} + 16q^{39} - 4q^{41} + 12q^{43} + 4q^{45} + 6q^{49} + 8q^{53} - 8q^{55} + 16q^{65} - 28q^{67} - 16q^{71} + 12q^{75} + 16q^{77} - 32q^{79} - 22q^{81} + 4q^{83} + 12q^{89} - 8q^{95} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
129.1
 1.00000i − 1.00000i
0 2.00000 0 2.00000 1.00000i 0 2.00000i 0 1.00000 0
129.2 0 2.00000 0 2.00000 + 1.00000i 0 2.00000i 0 1.00000 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
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.f 2
4.b odd 2 1 1280.2.f.b 2
5.b even 2 1 1280.2.f.a 2
8.b even 2 1 1280.2.f.a 2
8.d odd 2 1 1280.2.f.e 2
16.e even 4 1 40.2.c.a 2
16.e even 4 1 320.2.c.c 2
16.f odd 4 1 80.2.c.a 2
16.f odd 4 1 320.2.c.b 2
20.d odd 2 1 1280.2.f.e 2
40.e odd 2 1 1280.2.f.b 2
40.f even 2 1 inner 1280.2.f.f 2
48.i odd 4 1 360.2.f.c 2
48.i odd 4 1 2880.2.f.h 2
48.k even 4 1 720.2.f.e 2
48.k even 4 1 2880.2.f.i 2
80.i odd 4 1 200.2.a.b 1
80.i odd 4 1 1600.2.a.f 1
80.j even 4 1 400.2.a.b 1
80.j even 4 1 1600.2.a.d 1
80.k odd 4 1 80.2.c.a 2
80.k odd 4 1 320.2.c.b 2
80.q even 4 1 40.2.c.a 2
80.q even 4 1 320.2.c.c 2
80.s even 4 1 400.2.a.g 1
80.s even 4 1 1600.2.a.u 1
80.t odd 4 1 200.2.a.d 1
80.t odd 4 1 1600.2.a.v 1
112.l odd 4 1 1960.2.g.b 2
240.t even 4 1 720.2.f.e 2
240.t even 4 1 2880.2.f.i 2
240.z odd 4 1 3600.2.a.bb 1
240.bb even 4 1 1800.2.a.j 1
240.bd odd 4 1 3600.2.a.k 1
240.bf even 4 1 1800.2.a.s 1
240.bm odd 4 1 360.2.f.c 2
240.bm odd 4 1 2880.2.f.h 2
560.r even 4 1 9800.2.a.d 1
560.bf odd 4 1 1960.2.g.b 2
560.bn even 4 1 9800.2.a.bf 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.2.c.a 2 16.e even 4 1
40.2.c.a 2 80.q even 4 1
80.2.c.a 2 16.f odd 4 1
80.2.c.a 2 80.k odd 4 1
200.2.a.b 1 80.i odd 4 1
200.2.a.d 1 80.t odd 4 1
320.2.c.b 2 16.f odd 4 1
320.2.c.b 2 80.k odd 4 1
320.2.c.c 2 16.e even 4 1
320.2.c.c 2 80.q even 4 1
360.2.f.c 2 48.i odd 4 1
360.2.f.c 2 240.bm odd 4 1
400.2.a.b 1 80.j even 4 1
400.2.a.g 1 80.s even 4 1
720.2.f.e 2 48.k even 4 1
720.2.f.e 2 240.t even 4 1
1280.2.f.a 2 5.b even 2 1
1280.2.f.a 2 8.b even 2 1
1280.2.f.b 2 4.b odd 2 1
1280.2.f.b 2 40.e odd 2 1
1280.2.f.e 2 8.d odd 2 1
1280.2.f.e 2 20.d odd 2 1
1280.2.f.f 2 1.a even 1 1 trivial
1280.2.f.f 2 40.f even 2 1 inner
1600.2.a.d 1 80.j even 4 1
1600.2.a.f 1 80.i odd 4 1
1600.2.a.u 1 80.s even 4 1
1600.2.a.v 1 80.t odd 4 1
1800.2.a.j 1 240.bb even 4 1
1800.2.a.s 1 240.bf even 4 1
1960.2.g.b 2 112.l odd 4 1
1960.2.g.b 2 560.bf odd 4 1
2880.2.f.h 2 48.i odd 4 1
2880.2.f.h 2 240.bm odd 4 1
2880.2.f.i 2 48.k even 4 1
2880.2.f.i 2 240.t even 4 1
3600.2.a.k 1 240.bd odd 4 1
3600.2.a.bb 1 240.z odd 4 1
9800.2.a.d 1 560.r even 4 1
9800.2.a.bf 1 560.bn even 4 1

## Hecke kernels

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

 $$T_{3} - 2$$ $$T_{13} - 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$( -2 + T )^{2}$$
$5$ $$5 - 4 T + T^{2}$$
$7$ $$4 + T^{2}$$
$11$ $$16 + T^{2}$$
$13$ $$( -4 + T )^{2}$$
$17$ $$T^{2}$$
$19$ $$16 + T^{2}$$
$23$ $$4 + T^{2}$$
$29$ $$4 + T^{2}$$
$31$ $$T^{2}$$
$37$ $$( -4 + T )^{2}$$
$41$ $$( 2 + T )^{2}$$
$43$ $$( -6 + T )^{2}$$
$47$ $$36 + T^{2}$$
$53$ $$( -4 + T )^{2}$$
$59$ $$144 + T^{2}$$
$61$ $$100 + T^{2}$$
$67$ $$( 14 + T )^{2}$$
$71$ $$( 8 + T )^{2}$$
$73$ $$64 + T^{2}$$
$79$ $$( 16 + T )^{2}$$
$83$ $$( -2 + T )^{2}$$
$89$ $$( -6 + T )^{2}$$
$97$ $$256 + T^{2}$$