# Properties

 Label 320.6.a.o 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

 $$f(q)$$ $$=$$ $$q + 24q^{3} - 25q^{5} + 172q^{7} + 333q^{9} + O(q^{10})$$ $$q + 24q^{3} - 25q^{5} + 172q^{7} + 333q^{9} + 132q^{11} + 946q^{13} - 600q^{15} - 222q^{17} + 500q^{19} + 4128q^{21} - 3564q^{23} + 625q^{25} + 2160q^{27} - 2190q^{29} - 2312q^{31} + 3168q^{33} - 4300q^{35} + 11242q^{37} + 22704q^{39} + 1242q^{41} + 20624q^{43} - 8325q^{45} - 6588q^{47} + 12777q^{49} - 5328q^{51} + 21066q^{53} - 3300q^{55} + 12000q^{57} + 7980q^{59} - 16622q^{61} + 57276q^{63} - 23650q^{65} + 1808q^{67} - 85536q^{69} + 24528q^{71} + 20474q^{73} + 15000q^{75} + 22704q^{77} + 46240q^{79} - 29079q^{81} - 51576q^{83} + 5550q^{85} - 52560q^{87} - 110310q^{89} + 162712q^{91} - 55488q^{93} - 12500q^{95} - 78382q^{97} + 43956q^{99} + O(q^{100})$$

## 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 24.0000 0 −25.0000 0 172.000 0 333.000 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 320.6.a.o 1
4.b odd 2 1 320.6.a.b 1
8.b even 2 1 80.6.a.a 1
8.d odd 2 1 10.6.a.b 1
24.f even 2 1 90.6.a.d 1
24.h odd 2 1 720.6.a.j 1
40.e odd 2 1 50.6.a.d 1
40.f even 2 1 400.6.a.n 1
40.i odd 4 2 400.6.c.b 2
40.k even 4 2 50.6.b.a 2
56.e even 2 1 490.6.a.a 1
120.m even 2 1 450.6.a.l 1
120.q odd 4 2 450.6.c.h 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.6.a.b 1 8.d odd 2 1
50.6.a.d 1 40.e odd 2 1
50.6.b.a 2 40.k even 4 2
80.6.a.a 1 8.b even 2 1
90.6.a.d 1 24.f even 2 1
320.6.a.b 1 4.b odd 2 1
320.6.a.o 1 1.a even 1 1 trivial
400.6.a.n 1 40.f even 2 1
400.6.c.b 2 40.i odd 4 2
450.6.a.l 1 120.m even 2 1
450.6.c.h 2 120.q odd 4 2
490.6.a.a 1 56.e even 2 1
720.6.a.j 1 24.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3} - 24$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(320))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 - 24 T + 243 T^{2}$$
$5$ $$1 + 25 T$$
$7$ $$1 - 172 T + 16807 T^{2}$$
$11$ $$1 - 132 T + 161051 T^{2}$$
$13$ $$1 - 946 T + 371293 T^{2}$$
$17$ $$1 + 222 T + 1419857 T^{2}$$
$19$ $$1 - 500 T + 2476099 T^{2}$$
$23$ $$1 + 3564 T + 6436343 T^{2}$$
$29$ $$1 + 2190 T + 20511149 T^{2}$$
$31$ $$1 + 2312 T + 28629151 T^{2}$$
$37$ $$1 - 11242 T + 69343957 T^{2}$$
$41$ $$1 - 1242 T + 115856201 T^{2}$$
$43$ $$1 - 20624 T + 147008443 T^{2}$$
$47$ $$1 + 6588 T + 229345007 T^{2}$$
$53$ $$1 - 21066 T + 418195493 T^{2}$$
$59$ $$1 - 7980 T + 714924299 T^{2}$$
$61$ $$1 + 16622 T + 844596301 T^{2}$$
$67$ $$1 - 1808 T + 1350125107 T^{2}$$
$71$ $$1 - 24528 T + 1804229351 T^{2}$$
$73$ $$1 - 20474 T + 2073071593 T^{2}$$
$79$ $$1 - 46240 T + 3077056399 T^{2}$$
$83$ $$1 + 51576 T + 3939040643 T^{2}$$
$89$ $$1 + 110310 T + 5584059449 T^{2}$$
$97$ $$1 + 78382 T + 8587340257 T^{2}$$