# Properties

 Label 1280.3.e.b Level $1280$ Weight $3$ Character orbit 1280.e Analytic conductor $34.877$ Analytic rank $0$ Dimension $2$ CM discriminant -20 Inner twists $4$

# 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: $$2$$ 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: $\mathrm{U}(1)[D_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + 4 i q^{3} -5 i q^{5} -4 q^{7} -7 q^{9} +O(q^{10})$$ $$q + 4 i q^{3} -5 i q^{5} -4 q^{7} -7 q^{9} + 20 q^{15} -16 i q^{21} + 44 q^{23} -25 q^{25} + 8 i q^{27} + 22 i q^{29} + 20 i q^{35} -62 q^{41} + 76 i q^{43} + 35 i q^{45} + 4 q^{47} -33 q^{49} + 58 i q^{61} + 28 q^{63} + 116 i q^{67} + 176 i q^{69} -100 i q^{75} -95 q^{81} -76 i q^{83} -88 q^{87} + 142 q^{89} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 8 q^{7} - 14 q^{9} + O(q^{10})$$ $$2 q - 8 q^{7} - 14 q^{9} + 40 q^{15} + 88 q^{23} - 50 q^{25} - 124 q^{41} + 8 q^{47} - 66 q^{49} + 56 q^{63} - 190 q^{81} - 176 q^{87} + 284 q^{89} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
639.1
 − 1.00000i 1.00000i
0 4.00000i 0 5.00000i 0 −4.00000 0 −7.00000 0
639.2 0 4.00000i 0 5.00000i 0 −4.00000 0 −7.00000 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
20.d odd 2 1 CM by $$\Q(\sqrt{-5})$$
8.b even 2 1 inner
40.e odd 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.b 2
4.b odd 2 1 1280.3.e.c 2
5.b even 2 1 1280.3.e.c 2
8.b even 2 1 inner 1280.3.e.b 2
8.d odd 2 1 1280.3.e.c 2
16.e even 4 1 20.3.d.b yes 1
16.e even 4 1 320.3.h.b 1
16.f odd 4 1 20.3.d.a 1
16.f odd 4 1 320.3.h.a 1
20.d odd 2 1 CM 1280.3.e.b 2
40.e odd 2 1 inner 1280.3.e.b 2
40.f even 2 1 1280.3.e.c 2
48.i odd 4 1 180.3.f.a 1
48.k even 4 1 180.3.f.b 1
80.i odd 4 1 100.3.b.c 2
80.i odd 4 1 1600.3.b.f 2
80.j even 4 1 100.3.b.c 2
80.j even 4 1 1600.3.b.f 2
80.k odd 4 1 20.3.d.b yes 1
80.k odd 4 1 320.3.h.b 1
80.q even 4 1 20.3.d.a 1
80.q even 4 1 320.3.h.a 1
80.s even 4 1 100.3.b.c 2
80.s even 4 1 1600.3.b.f 2
80.t odd 4 1 100.3.b.c 2
80.t odd 4 1 1600.3.b.f 2
240.t even 4 1 180.3.f.a 1
240.z odd 4 1 900.3.c.h 2
240.bb even 4 1 900.3.c.h 2
240.bd odd 4 1 900.3.c.h 2
240.bf even 4 1 900.3.c.h 2
240.bm odd 4 1 180.3.f.b 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.3.d.a 1 16.f odd 4 1
20.3.d.a 1 80.q even 4 1
20.3.d.b yes 1 16.e even 4 1
20.3.d.b yes 1 80.k odd 4 1
100.3.b.c 2 80.i odd 4 1
100.3.b.c 2 80.j even 4 1
100.3.b.c 2 80.s even 4 1
100.3.b.c 2 80.t odd 4 1
180.3.f.a 1 48.i odd 4 1
180.3.f.a 1 240.t even 4 1
180.3.f.b 1 48.k even 4 1
180.3.f.b 1 240.bm odd 4 1
320.3.h.a 1 16.f odd 4 1
320.3.h.a 1 80.q even 4 1
320.3.h.b 1 16.e even 4 1
320.3.h.b 1 80.k odd 4 1
900.3.c.h 2 240.z odd 4 1
900.3.c.h 2 240.bb even 4 1
900.3.c.h 2 240.bd odd 4 1
900.3.c.h 2 240.bf even 4 1
1280.3.e.b 2 1.a even 1 1 trivial
1280.3.e.b 2 8.b even 2 1 inner
1280.3.e.b 2 20.d odd 2 1 CM
1280.3.e.b 2 40.e odd 2 1 inner
1280.3.e.c 2 4.b odd 2 1
1280.3.e.c 2 5.b even 2 1
1280.3.e.c 2 8.d odd 2 1
1280.3.e.c 2 40.f even 2 1
1600.3.b.f 2 80.i odd 4 1
1600.3.b.f 2 80.j even 4 1
1600.3.b.f 2 80.s even 4 1
1600.3.b.f 2 80.t odd 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(1280, [\chi])$$:

 $$T_{3}^{2} + 16$$ $$T_{7} + 4$$ $$T_{11}$$ $$T_{13}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$16 + T^{2}$$
$5$ $$25 + T^{2}$$
$7$ $$( 4 + T )^{2}$$
$11$ $$T^{2}$$
$13$ $$T^{2}$$
$17$ $$T^{2}$$
$19$ $$T^{2}$$
$23$ $$( -44 + T )^{2}$$
$29$ $$484 + T^{2}$$
$31$ $$T^{2}$$
$37$ $$T^{2}$$
$41$ $$( 62 + T )^{2}$$
$43$ $$5776 + T^{2}$$
$47$ $$( -4 + T )^{2}$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$3364 + T^{2}$$
$67$ $$13456 + T^{2}$$
$71$ $$T^{2}$$
$73$ $$T^{2}$$
$79$ $$T^{2}$$
$83$ $$5776 + T^{2}$$
$89$ $$( -142 + T )^{2}$$
$97$ $$T^{2}$$