# Properties

 Label 320.4.a.k 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 20) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + 4 q^{3} - 5 q^{5} + 16 q^{7} - 11 q^{9}+O(q^{10})$$ q + 4 * q^3 - 5 * q^5 + 16 * q^7 - 11 * q^9 $$q + 4 q^{3} - 5 q^{5} + 16 q^{7} - 11 q^{9} - 60 q^{11} - 86 q^{13} - 20 q^{15} + 18 q^{17} + 44 q^{19} + 64 q^{21} - 48 q^{23} + 25 q^{25} - 152 q^{27} + 186 q^{29} - 176 q^{31} - 240 q^{33} - 80 q^{35} - 254 q^{37} - 344 q^{39} + 186 q^{41} - 100 q^{43} + 55 q^{45} - 168 q^{47} - 87 q^{49} + 72 q^{51} + 498 q^{53} + 300 q^{55} + 176 q^{57} - 252 q^{59} + 58 q^{61} - 176 q^{63} + 430 q^{65} - 1036 q^{67} - 192 q^{69} - 168 q^{71} + 506 q^{73} + 100 q^{75} - 960 q^{77} - 272 q^{79} - 311 q^{81} + 948 q^{83} - 90 q^{85} + 744 q^{87} - 1014 q^{89} - 1376 q^{91} - 704 q^{93} - 220 q^{95} - 766 q^{97} + 660 q^{99}+O(q^{100})$$ q + 4 * q^3 - 5 * q^5 + 16 * q^7 - 11 * q^9 - 60 * q^11 - 86 * q^13 - 20 * q^15 + 18 * q^17 + 44 * q^19 + 64 * q^21 - 48 * q^23 + 25 * q^25 - 152 * q^27 + 186 * q^29 - 176 * q^31 - 240 * q^33 - 80 * q^35 - 254 * q^37 - 344 * q^39 + 186 * q^41 - 100 * q^43 + 55 * q^45 - 168 * q^47 - 87 * q^49 + 72 * q^51 + 498 * q^53 + 300 * q^55 + 176 * q^57 - 252 * q^59 + 58 * q^61 - 176 * q^63 + 430 * q^65 - 1036 * q^67 - 192 * q^69 - 168 * q^71 + 506 * q^73 + 100 * q^75 - 960 * q^77 - 272 * q^79 - 311 * q^81 + 948 * q^83 - 90 * q^85 + 744 * q^87 - 1014 * q^89 - 1376 * q^91 - 704 * q^93 - 220 * q^95 - 766 * q^97 + 660 * 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 4.00000 0 −5.00000 0 16.0000 0 −11.0000 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.k 1
4.b odd 2 1 320.4.a.d 1
5.b even 2 1 1600.4.a.p 1
8.b even 2 1 80.4.a.c 1
8.d odd 2 1 20.4.a.a 1
16.e even 4 2 1280.4.d.c 2
16.f odd 4 2 1280.4.d.n 2
20.d odd 2 1 1600.4.a.bl 1
24.f even 2 1 180.4.a.a 1
24.h odd 2 1 720.4.a.k 1
40.e odd 2 1 100.4.a.a 1
40.f even 2 1 400.4.a.o 1
40.i odd 4 2 400.4.c.j 2
40.k even 4 2 100.4.c.a 2
56.e even 2 1 980.4.a.c 1
56.k odd 6 2 980.4.i.e 2
56.m even 6 2 980.4.i.n 2
72.l even 6 2 1620.4.i.j 2
72.p odd 6 2 1620.4.i.d 2
88.g even 2 1 2420.4.a.d 1
120.m even 2 1 900.4.a.m 1
120.q odd 4 2 900.4.d.k 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.4.a.a 1 8.d odd 2 1
80.4.a.c 1 8.b even 2 1
100.4.a.a 1 40.e odd 2 1
100.4.c.a 2 40.k even 4 2
180.4.a.a 1 24.f even 2 1
320.4.a.d 1 4.b odd 2 1
320.4.a.k 1 1.a even 1 1 trivial
400.4.a.o 1 40.f even 2 1
400.4.c.j 2 40.i odd 4 2
720.4.a.k 1 24.h odd 2 1
900.4.a.m 1 120.m even 2 1
900.4.d.k 2 120.q odd 4 2
980.4.a.c 1 56.e even 2 1
980.4.i.e 2 56.k odd 6 2
980.4.i.n 2 56.m even 6 2
1280.4.d.c 2 16.e even 4 2
1280.4.d.n 2 16.f odd 4 2
1600.4.a.p 1 5.b even 2 1
1600.4.a.bl 1 20.d odd 2 1
1620.4.i.d 2 72.p odd 6 2
1620.4.i.j 2 72.l even 6 2
2420.4.a.d 1 88.g even 2 1

## Hecke kernels

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

 $$T_{3} - 4$$ T3 - 4 $$T_{7} - 16$$ T7 - 16

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 4$$
$5$ $$T + 5$$
$7$ $$T - 16$$
$11$ $$T + 60$$
$13$ $$T + 86$$
$17$ $$T - 18$$
$19$ $$T - 44$$
$23$ $$T + 48$$
$29$ $$T - 186$$
$31$ $$T + 176$$
$37$ $$T + 254$$
$41$ $$T - 186$$
$43$ $$T + 100$$
$47$ $$T + 168$$
$53$ $$T - 498$$
$59$ $$T + 252$$
$61$ $$T - 58$$
$67$ $$T + 1036$$
$71$ $$T + 168$$
$73$ $$T - 506$$
$79$ $$T + 272$$
$83$ $$T - 948$$
$89$ $$T + 1014$$
$97$ $$T + 766$$