# Properties

 Label 1280.2.d.g Level $1280$ Weight $2$ Character orbit 1280.d 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.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 20) 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.

 $$f(q)$$ $$=$$ $$q + 2 i q^{3} + i q^{5} + 2 q^{7} - q^{9} +O(q^{10})$$ $$q + 2 i q^{3} + i q^{5} + 2 q^{7} - q^{9} + 2 i q^{13} -2 q^{15} -6 q^{17} + 4 i q^{19} + 4 i q^{21} + 6 q^{23} - q^{25} + 4 i q^{27} + 6 i q^{29} + 4 q^{31} + 2 i q^{35} -2 i q^{37} -4 q^{39} -6 q^{41} -10 i q^{43} -i q^{45} + 6 q^{47} -3 q^{49} -12 i q^{51} + 6 i q^{53} -8 q^{57} + 12 i q^{59} + 2 i q^{61} -2 q^{63} -2 q^{65} -2 i q^{67} + 12 i q^{69} -12 q^{71} -2 q^{73} -2 i q^{75} -8 q^{79} -11 q^{81} -6 i q^{83} -6 i q^{85} -12 q^{87} + 6 q^{89} + 4 i q^{91} + 8 i q^{93} -4 q^{95} + 2 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 4q^{7} - 2q^{9} + O(q^{10})$$ $$2q + 4q^{7} - 2q^{9} - 4q^{15} - 12q^{17} + 12q^{23} - 2q^{25} + 8q^{31} - 8q^{39} - 12q^{41} + 12q^{47} - 6q^{49} - 16q^{57} - 4q^{63} - 4q^{65} - 24q^{71} - 4q^{73} - 16q^{79} - 22q^{81} - 24q^{87} + 12q^{89} - 8q^{95} + 4q^{97} + O(q^{100})$$

## 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 2.00000i 0 1.00000i 0 2.00000 0 −1.00000 0
641.2 0 2.00000i 0 1.00000i 0 2.00000 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
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.g 2
4.b odd 2 1 1280.2.d.c 2
8.b even 2 1 inner 1280.2.d.g 2
8.d odd 2 1 1280.2.d.c 2
16.e even 4 1 80.2.a.b 1
16.e even 4 1 320.2.a.a 1
16.f odd 4 1 20.2.a.a 1
16.f odd 4 1 320.2.a.f 1
48.i odd 4 1 720.2.a.h 1
48.i odd 4 1 2880.2.a.f 1
48.k even 4 1 180.2.a.a 1
48.k even 4 1 2880.2.a.m 1
80.i odd 4 1 400.2.c.b 2
80.i odd 4 1 1600.2.c.e 2
80.j even 4 1 100.2.c.a 2
80.j even 4 1 1600.2.c.d 2
80.k odd 4 1 100.2.a.a 1
80.k odd 4 1 1600.2.a.c 1
80.q even 4 1 400.2.a.c 1
80.q even 4 1 1600.2.a.w 1
80.s even 4 1 100.2.c.a 2
80.s even 4 1 1600.2.c.d 2
80.t odd 4 1 400.2.c.b 2
80.t odd 4 1 1600.2.c.e 2
112.j even 4 1 980.2.a.h 1
112.l odd 4 1 3920.2.a.h 1
112.u odd 12 2 980.2.i.i 2
112.v even 12 2 980.2.i.c 2
144.u even 12 2 1620.2.i.b 2
144.v odd 12 2 1620.2.i.h 2
176.i even 4 1 2420.2.a.a 1
176.l odd 4 1 9680.2.a.ba 1
208.l even 4 1 3380.2.f.b 2
208.o odd 4 1 3380.2.a.c 1
208.s even 4 1 3380.2.f.b 2
240.t even 4 1 900.2.a.b 1
240.z odd 4 1 900.2.d.c 2
240.bb even 4 1 3600.2.f.j 2
240.bd odd 4 1 900.2.d.c 2
240.bf even 4 1 3600.2.f.j 2
240.bm odd 4 1 3600.2.a.be 1
272.i odd 4 1 5780.2.c.a 2
272.k odd 4 1 5780.2.a.f 1
272.t odd 4 1 5780.2.c.a 2
304.m even 4 1 7220.2.a.f 1
336.v odd 4 1 8820.2.a.g 1
560.u odd 4 1 4900.2.e.f 2
560.be even 4 1 4900.2.a.e 1
560.bm odd 4 1 4900.2.e.f 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.2.a.a 1 16.f odd 4 1
80.2.a.b 1 16.e even 4 1
100.2.a.a 1 80.k odd 4 1
100.2.c.a 2 80.j even 4 1
100.2.c.a 2 80.s even 4 1
180.2.a.a 1 48.k even 4 1
320.2.a.a 1 16.e even 4 1
320.2.a.f 1 16.f odd 4 1
400.2.a.c 1 80.q even 4 1
400.2.c.b 2 80.i odd 4 1
400.2.c.b 2 80.t odd 4 1
720.2.a.h 1 48.i odd 4 1
900.2.a.b 1 240.t even 4 1
900.2.d.c 2 240.z odd 4 1
900.2.d.c 2 240.bd odd 4 1
980.2.a.h 1 112.j even 4 1
980.2.i.c 2 112.v even 12 2
980.2.i.i 2 112.u odd 12 2
1280.2.d.c 2 4.b odd 2 1
1280.2.d.c 2 8.d odd 2 1
1280.2.d.g 2 1.a even 1 1 trivial
1280.2.d.g 2 8.b even 2 1 inner
1600.2.a.c 1 80.k odd 4 1
1600.2.a.w 1 80.q even 4 1
1600.2.c.d 2 80.j even 4 1
1600.2.c.d 2 80.s even 4 1
1600.2.c.e 2 80.i odd 4 1
1600.2.c.e 2 80.t odd 4 1
1620.2.i.b 2 144.u even 12 2
1620.2.i.h 2 144.v odd 12 2
2420.2.a.a 1 176.i even 4 1
2880.2.a.f 1 48.i odd 4 1
2880.2.a.m 1 48.k even 4 1
3380.2.a.c 1 208.o odd 4 1
3380.2.f.b 2 208.l even 4 1
3380.2.f.b 2 208.s even 4 1
3600.2.a.be 1 240.bm odd 4 1
3600.2.f.j 2 240.bb even 4 1
3600.2.f.j 2 240.bf even 4 1
3920.2.a.h 1 112.l odd 4 1
4900.2.a.e 1 560.be even 4 1
4900.2.e.f 2 560.u odd 4 1
4900.2.e.f 2 560.bm odd 4 1
5780.2.a.f 1 272.k odd 4 1
5780.2.c.a 2 272.i odd 4 1
5780.2.c.a 2 272.t odd 4 1
7220.2.a.f 1 304.m even 4 1
8820.2.a.g 1 336.v odd 4 1
9680.2.a.ba 1 176.l odd 4 1

## Hecke kernels

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$4 + T^{2}$$
$5$ $$1 + T^{2}$$
$7$ $$( -2 + T )^{2}$$
$11$ $$T^{2}$$
$13$ $$4 + T^{2}$$
$17$ $$( 6 + T )^{2}$$
$19$ $$16 + T^{2}$$
$23$ $$( -6 + T )^{2}$$
$29$ $$36 + T^{2}$$
$31$ $$( -4 + T )^{2}$$
$37$ $$4 + T^{2}$$
$41$ $$( 6 + T )^{2}$$
$43$ $$100 + T^{2}$$
$47$ $$( -6 + T )^{2}$$
$53$ $$36 + T^{2}$$
$59$ $$144 + T^{2}$$
$61$ $$4 + T^{2}$$
$67$ $$4 + T^{2}$$
$71$ $$( 12 + T )^{2}$$
$73$ $$( 2 + T )^{2}$$
$79$ $$( 8 + T )^{2}$$
$83$ $$36 + T^{2}$$
$89$ $$( -6 + T )^{2}$$
$97$ $$( -2 + T )^{2}$$