# Properties

 Label 1280.4.d.f Level $1280$ Weight $4$ Character orbit 1280.d Analytic conductor $75.522$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$1280 = 2^{8} \cdot 5$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1280.d (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$75.5224448073$$ 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 160) 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.

 $$f(q)$$ $$=$$ $$q + 2 i q^{3} -5 i q^{5} -6 q^{7} + 23 q^{9} +O(q^{10})$$ $$q + 2 i q^{3} -5 i q^{5} -6 q^{7} + 23 q^{9} + 60 i q^{11} -50 i q^{13} + 10 q^{15} -30 q^{17} -40 i q^{19} -12 i q^{21} -178 q^{23} -25 q^{25} + 100 i q^{27} -166 i q^{29} + 20 q^{31} -120 q^{33} + 30 i q^{35} + 10 i q^{37} + 100 q^{39} + 250 q^{41} + 142 i q^{43} -115 i q^{45} + 214 q^{47} -307 q^{49} -60 i q^{51} + 490 i q^{53} + 300 q^{55} + 80 q^{57} -800 i q^{59} -250 i q^{61} -138 q^{63} -250 q^{65} + 774 i q^{67} -356 i q^{69} -100 q^{71} + 230 q^{73} -50 i q^{75} -360 i q^{77} -1320 q^{79} + 421 q^{81} -982 i q^{83} + 150 i q^{85} + 332 q^{87} -874 q^{89} + 300 i q^{91} + 40 i q^{93} -200 q^{95} -310 q^{97} + 1380 i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 12q^{7} + 46q^{9} + O(q^{10})$$ $$2q - 12q^{7} + 46q^{9} + 20q^{15} - 60q^{17} - 356q^{23} - 50q^{25} + 40q^{31} - 240q^{33} + 200q^{39} + 500q^{41} + 428q^{47} - 614q^{49} + 600q^{55} + 160q^{57} - 276q^{63} - 500q^{65} - 200q^{71} + 460q^{73} - 2640q^{79} + 842q^{81} + 664q^{87} - 1748q^{89} - 400q^{95} - 620q^{97} + O(q^{100})$$

## 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 5.00000i 0 −6.00000 0 23.0000 0
641.2 0 2.00000i 0 5.00000i 0 −6.00000 0 23.0000 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
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 1280.4.d.f 2
4.b odd 2 1 1280.4.d.k 2
8.b even 2 1 inner 1280.4.d.f 2
8.d odd 2 1 1280.4.d.k 2
16.e even 4 1 160.4.a.a 1
16.e even 4 1 320.4.a.i 1
16.f odd 4 1 160.4.a.b yes 1
16.f odd 4 1 320.4.a.f 1
48.i odd 4 1 1440.4.a.o 1
48.k even 4 1 1440.4.a.n 1
80.i odd 4 1 800.4.c.f 2
80.j even 4 1 800.4.c.e 2
80.k odd 4 1 800.4.a.d 1
80.k odd 4 1 1600.4.a.bj 1
80.q even 4 1 800.4.a.h 1
80.q even 4 1 1600.4.a.r 1
80.s even 4 1 800.4.c.e 2
80.t odd 4 1 800.4.c.f 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.4.a.a 1 16.e even 4 1
160.4.a.b yes 1 16.f odd 4 1
320.4.a.f 1 16.f odd 4 1
320.4.a.i 1 16.e even 4 1
800.4.a.d 1 80.k odd 4 1
800.4.a.h 1 80.q even 4 1
800.4.c.e 2 80.j even 4 1
800.4.c.e 2 80.s even 4 1
800.4.c.f 2 80.i odd 4 1
800.4.c.f 2 80.t odd 4 1
1280.4.d.f 2 1.a even 1 1 trivial
1280.4.d.f 2 8.b even 2 1 inner
1280.4.d.k 2 4.b odd 2 1
1280.4.d.k 2 8.d odd 2 1
1440.4.a.n 1 48.k even 4 1
1440.4.a.o 1 48.i odd 4 1
1600.4.a.r 1 80.q even 4 1
1600.4.a.bj 1 80.k odd 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(1280, [\chi])$$:

 $$T_{3}^{2} + 4$$ $$T_{7} + 6$$ $$T_{11}^{2} + 3600$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$4 + T^{2}$$
$5$ $$25 + T^{2}$$
$7$ $$( 6 + T )^{2}$$
$11$ $$3600 + T^{2}$$
$13$ $$2500 + T^{2}$$
$17$ $$( 30 + T )^{2}$$
$19$ $$1600 + T^{2}$$
$23$ $$( 178 + T )^{2}$$
$29$ $$27556 + T^{2}$$
$31$ $$( -20 + T )^{2}$$
$37$ $$100 + T^{2}$$
$41$ $$( -250 + T )^{2}$$
$43$ $$20164 + T^{2}$$
$47$ $$( -214 + T )^{2}$$
$53$ $$240100 + T^{2}$$
$59$ $$640000 + T^{2}$$
$61$ $$62500 + T^{2}$$
$67$ $$599076 + T^{2}$$
$71$ $$( 100 + T )^{2}$$
$73$ $$( -230 + T )^{2}$$
$79$ $$( 1320 + T )^{2}$$
$83$ $$964324 + T^{2}$$
$89$ $$( 874 + T )^{2}$$
$97$ $$( 310 + T )^{2}$$
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