# Properties

 Label 320.6.a.p Level 320 Weight 6 Character orbit 320.a Self dual yes Analytic conductor 51.323 Analytic rank 0 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: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 10) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + 26q^{3} + 25q^{5} - 22q^{7} + 433q^{9} + O(q^{10})$$ $$q + 26q^{3} + 25q^{5} - 22q^{7} + 433q^{9} + 768q^{11} + 46q^{13} + 650q^{15} + 378q^{17} - 1100q^{19} - 572q^{21} - 1986q^{23} + 625q^{25} + 4940q^{27} + 5610q^{29} - 3988q^{31} + 19968q^{33} - 550q^{35} + 142q^{37} + 1196q^{39} + 1542q^{41} + 5026q^{43} + 10825q^{45} + 24738q^{47} - 16323q^{49} + 9828q^{51} + 14166q^{53} + 19200q^{55} - 28600q^{57} - 28380q^{59} - 5522q^{61} - 9526q^{63} + 1150q^{65} + 24742q^{67} - 51636q^{69} + 42372q^{71} - 52126q^{73} + 16250q^{75} - 16896q^{77} - 39640q^{79} + 23221q^{81} + 59826q^{83} + 9450q^{85} + 145860q^{87} + 57690q^{89} - 1012q^{91} - 103688q^{93} - 27500q^{95} - 144382q^{97} + 332544q^{99} + O(q^{100})$$

## 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 26.0000 0 25.0000 0 −22.0000 0 433.000 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.p 1
4.b odd 2 1 320.6.a.a 1
8.b even 2 1 10.6.a.a 1
8.d odd 2 1 80.6.a.h 1
24.f even 2 1 720.6.a.r 1
24.h odd 2 1 90.6.a.f 1
40.e odd 2 1 400.6.a.a 1
40.f even 2 1 50.6.a.g 1
40.i odd 4 2 50.6.b.d 2
40.k even 4 2 400.6.c.a 2
56.h odd 2 1 490.6.a.j 1
120.i odd 2 1 450.6.a.h 1
120.w even 4 2 450.6.c.o 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.6.a.a 1 8.b even 2 1
50.6.a.g 1 40.f even 2 1
50.6.b.d 2 40.i odd 4 2
80.6.a.h 1 8.d odd 2 1
90.6.a.f 1 24.h odd 2 1
320.6.a.a 1 4.b odd 2 1
320.6.a.p 1 1.a even 1 1 trivial
400.6.a.a 1 40.e odd 2 1
400.6.c.a 2 40.k even 4 2
450.6.a.h 1 120.i odd 2 1
450.6.c.o 2 120.w even 4 2
490.6.a.j 1 56.h odd 2 1
720.6.a.r 1 24.f 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 $$T_{3} - 26$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(320))$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 
$3$ $$1 - 26 T + 243 T^{2}$$
$5$ $$1 - 25 T$$
$7$ $$1 + 22 T + 16807 T^{2}$$
$11$ $$1 - 768 T + 161051 T^{2}$$
$13$ $$1 - 46 T + 371293 T^{2}$$
$17$ $$1 - 378 T + 1419857 T^{2}$$
$19$ $$1 + 1100 T + 2476099 T^{2}$$
$23$ $$1 + 1986 T + 6436343 T^{2}$$
$29$ $$1 - 5610 T + 20511149 T^{2}$$
$31$ $$1 + 3988 T + 28629151 T^{2}$$
$37$ $$1 - 142 T + 69343957 T^{2}$$
$41$ $$1 - 1542 T + 115856201 T^{2}$$
$43$ $$1 - 5026 T + 147008443 T^{2}$$
$47$ $$1 - 24738 T + 229345007 T^{2}$$
$53$ $$1 - 14166 T + 418195493 T^{2}$$
$59$ $$1 + 28380 T + 714924299 T^{2}$$
$61$ $$1 + 5522 T + 844596301 T^{2}$$
$67$ $$1 - 24742 T + 1350125107 T^{2}$$
$71$ $$1 - 42372 T + 1804229351 T^{2}$$
$73$ $$1 + 52126 T + 2073071593 T^{2}$$
$79$ $$1 + 39640 T + 3077056399 T^{2}$$
$83$ $$1 - 59826 T + 3939040643 T^{2}$$
$89$ $$1 - 57690 T + 5584059449 T^{2}$$
$97$ $$1 + 144382 T + 8587340257 T^{2}$$