# Properties

 Label 320.4.a.f Level $320$ Weight $4$ Character orbit 320.a Self dual yes Analytic conductor $18.881$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# 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: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 160) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

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

## 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
 0
0 −2.00000 0 5.00000 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

## Atkin-Lehner signs

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

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

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

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

## Hecke kernels

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

 $$T_{3} + 2$$ T3 + 2 $$T_{7} + 6$$ T7 + 6

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T + 2$$
$5$ $$T - 5$$
$7$ $$T + 6$$
$11$ $$T - 60$$
$13$ $$T + 50$$
$17$ $$T + 30$$
$19$ $$T - 40$$
$23$ $$T + 178$$
$29$ $$T + 166$$
$31$ $$T + 20$$
$37$ $$T + 10$$
$41$ $$T + 250$$
$43$ $$T - 142$$
$47$ $$T + 214$$
$53$ $$T + 490$$
$59$ $$T + 800$$
$61$ $$T + 250$$
$67$ $$T + 774$$
$71$ $$T + 100$$
$73$ $$T + 230$$
$79$ $$T - 1320$$
$83$ $$T - 982$$
$89$ $$T - 874$$
$97$ $$T + 310$$