# Properties

 Label 320.4.a.q Level $320$ Weight $4$ Character orbit 320.a Self dual yes Analytic conductor $18.881$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$320 = 2^{6} \cdot 5$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 320.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$18.8806112018$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ Defining polynomial: $$x^{2} - x - 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 160) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 2\sqrt{5}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta q^{3} + 5 q^{5} -7 \beta q^{7} -7 q^{9} +O(q^{10})$$ $$q -\beta q^{3} + 5 q^{5} -7 \beta q^{7} -7 q^{9} + 2 \beta q^{11} + 62 q^{13} -5 \beta q^{15} -46 q^{17} -24 \beta q^{19} + 140 q^{21} + 43 \beta q^{23} + 25 q^{25} + 34 \beta q^{27} + 90 q^{29} -34 \beta q^{31} -40 q^{33} -35 \beta q^{35} + 214 q^{37} -62 \beta q^{39} -10 q^{41} + 15 \beta q^{43} -35 q^{45} + 89 \beta q^{47} + 637 q^{49} + 46 \beta q^{51} + 678 q^{53} + 10 \beta q^{55} + 480 q^{57} + 92 \beta q^{59} -250 q^{61} + 49 \beta q^{63} + 310 q^{65} -11 \beta q^{67} -860 q^{69} -82 \beta q^{71} + 522 q^{73} -25 \beta q^{75} -280 q^{77} + 196 \beta q^{79} -491 q^{81} -85 \beta q^{83} -230 q^{85} -90 \beta q^{87} + 970 q^{89} -434 \beta q^{91} + 680 q^{93} -120 \beta q^{95} -934 q^{97} -14 \beta q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 10 q^{5} - 14 q^{9} + O(q^{10})$$ $$2 q + 10 q^{5} - 14 q^{9} + 124 q^{13} - 92 q^{17} + 280 q^{21} + 50 q^{25} + 180 q^{29} - 80 q^{33} + 428 q^{37} - 20 q^{41} - 70 q^{45} + 1274 q^{49} + 1356 q^{53} + 960 q^{57} - 500 q^{61} + 620 q^{65} - 1720 q^{69} + 1044 q^{73} - 560 q^{77} - 982 q^{81} - 460 q^{85} + 1940 q^{89} + 1360 q^{93} - 1868 q^{97} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 1.61803 −0.618034
0 −4.47214 0 5.00000 0 −31.3050 0 −7.00000 0
1.2 0 4.47214 0 5.00000 0 31.3050 0 −7.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$5$$ $$-1$$

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 320.4.a.q 2
4.b odd 2 1 inner 320.4.a.q 2
5.b even 2 1 1600.4.a.cg 2
8.b even 2 1 160.4.a.d 2
8.d odd 2 1 160.4.a.d 2
16.e even 4 2 1280.4.d.v 4
16.f odd 4 2 1280.4.d.v 4
20.d odd 2 1 1600.4.a.cg 2
24.f even 2 1 1440.4.a.bb 2
24.h odd 2 1 1440.4.a.bb 2
40.e odd 2 1 800.4.a.o 2
40.f even 2 1 800.4.a.o 2
40.i odd 4 2 800.4.c.l 4
40.k even 4 2 800.4.c.l 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.4.a.d 2 8.b even 2 1
160.4.a.d 2 8.d odd 2 1
320.4.a.q 2 1.a even 1 1 trivial
320.4.a.q 2 4.b odd 2 1 inner
800.4.a.o 2 40.e odd 2 1
800.4.a.o 2 40.f even 2 1
800.4.c.l 4 40.i odd 4 2
800.4.c.l 4 40.k even 4 2
1280.4.d.v 4 16.e even 4 2
1280.4.d.v 4 16.f odd 4 2
1440.4.a.bb 2 24.f even 2 1
1440.4.a.bb 2 24.h odd 2 1
1600.4.a.cg 2 5.b even 2 1
1600.4.a.cg 2 20.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(320))$$:

 $$T_{3}^{2} - 20$$ $$T_{7}^{2} - 980$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$-20 + T^{2}$$
$5$ $$( -5 + T )^{2}$$
$7$ $$-980 + T^{2}$$
$11$ $$-80 + T^{2}$$
$13$ $$( -62 + T )^{2}$$
$17$ $$( 46 + T )^{2}$$
$19$ $$-11520 + T^{2}$$
$23$ $$-36980 + T^{2}$$
$29$ $$( -90 + T )^{2}$$
$31$ $$-23120 + T^{2}$$
$37$ $$( -214 + T )^{2}$$
$41$ $$( 10 + T )^{2}$$
$43$ $$-4500 + T^{2}$$
$47$ $$-158420 + T^{2}$$
$53$ $$( -678 + T )^{2}$$
$59$ $$-169280 + T^{2}$$
$61$ $$( 250 + T )^{2}$$
$67$ $$-2420 + T^{2}$$
$71$ $$-134480 + T^{2}$$
$73$ $$( -522 + T )^{2}$$
$79$ $$-768320 + T^{2}$$
$83$ $$-144500 + T^{2}$$
$89$ $$( -970 + T )^{2}$$
$97$ $$( 934 + T )^{2}$$