# Properties

 Label 320.6.a.n Level 320 Weight 6 Character orbit 320.a Self dual yes Analytic conductor 51.323 Analytic rank 1 Dimension 1 CM no Inner twists 1

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: yes Analytic conductor: $$51.3228223402$$ 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

 $$f(q)$$ $$=$$ $$q + 22q^{3} + 25q^{5} - 218q^{7} + 241q^{9} + O(q^{10})$$ $$q + 22q^{3} + 25q^{5} - 218q^{7} + 241q^{9} - 480q^{11} + 622q^{13} + 550q^{15} + 186q^{17} - 1204q^{19} - 4796q^{21} + 3186q^{23} + 625q^{25} - 44q^{27} - 5526q^{29} - 9356q^{31} - 10560q^{33} - 5450q^{35} - 5618q^{37} + 13684q^{39} - 14394q^{41} - 370q^{43} + 6025q^{45} - 16146q^{47} + 30717q^{49} + 4092q^{51} + 4374q^{53} - 12000q^{55} - 26488q^{57} - 11748q^{59} - 13202q^{61} - 52538q^{63} + 15550q^{65} - 11542q^{67} + 70092q^{69} + 29532q^{71} + 33698q^{73} + 13750q^{75} + 104640q^{77} - 31208q^{79} - 59531q^{81} - 38466q^{83} + 4650q^{85} - 121572q^{87} + 119514q^{89} - 135596q^{91} - 205832q^{93} - 30100q^{95} + 94658q^{97} - 115680q^{99} + O(q^{100})$$

## 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
 0
0 22.0000 0 25.0000 0 −218.000 0 241.000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 320.6.a.n 1
4.b odd 2 1 320.6.a.c 1
8.b even 2 1 80.6.a.b 1
8.d odd 2 1 20.6.a.a 1
24.f even 2 1 180.6.a.e 1
24.h odd 2 1 720.6.a.l 1
40.e odd 2 1 100.6.a.a 1
40.f even 2 1 400.6.a.m 1
40.i odd 4 2 400.6.c.c 2
40.k even 4 2 100.6.c.a 2
56.e even 2 1 980.6.a.b 1
120.m even 2 1 900.6.a.b 1
120.q odd 4 2 900.6.d.h 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.6.a.a 1 8.d odd 2 1
80.6.a.b 1 8.b even 2 1
100.6.a.a 1 40.e odd 2 1
100.6.c.a 2 40.k even 4 2
180.6.a.e 1 24.f even 2 1
320.6.a.c 1 4.b odd 2 1
320.6.a.n 1 1.a even 1 1 trivial
400.6.a.m 1 40.f even 2 1
400.6.c.c 2 40.i odd 4 2
720.6.a.l 1 24.h odd 2 1
900.6.a.b 1 120.m even 2 1
900.6.d.h 2 120.q odd 4 2
980.6.a.b 1 56.e even 2 1

## Atkin-Lehner signs

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3} - 22$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(320))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 - 22 T + 243 T^{2}$$
$5$ $$1 - 25 T$$
$7$ $$1 + 218 T + 16807 T^{2}$$
$11$ $$1 + 480 T + 161051 T^{2}$$
$13$ $$1 - 622 T + 371293 T^{2}$$
$17$ $$1 - 186 T + 1419857 T^{2}$$
$19$ $$1 + 1204 T + 2476099 T^{2}$$
$23$ $$1 - 3186 T + 6436343 T^{2}$$
$29$ $$1 + 5526 T + 20511149 T^{2}$$
$31$ $$1 + 9356 T + 28629151 T^{2}$$
$37$ $$1 + 5618 T + 69343957 T^{2}$$
$41$ $$1 + 14394 T + 115856201 T^{2}$$
$43$ $$1 + 370 T + 147008443 T^{2}$$
$47$ $$1 + 16146 T + 229345007 T^{2}$$
$53$ $$1 - 4374 T + 418195493 T^{2}$$
$59$ $$1 + 11748 T + 714924299 T^{2}$$
$61$ $$1 + 13202 T + 844596301 T^{2}$$
$67$ $$1 + 11542 T + 1350125107 T^{2}$$
$71$ $$1 - 29532 T + 1804229351 T^{2}$$
$73$ $$1 - 33698 T + 2073071593 T^{2}$$
$79$ $$1 + 31208 T + 3077056399 T^{2}$$
$83$ $$1 + 38466 T + 3939040643 T^{2}$$
$89$ $$1 - 119514 T + 5584059449 T^{2}$$
$97$ $$1 - 94658 T + 8587340257 T^{2}$$