# Properties

 Label 320.6.a.l 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 40) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + 8 q^{3} - 25 q^{5} - 108 q^{7} - 179 q^{9}+O(q^{10})$$ q + 8 * q^3 - 25 * q^5 - 108 * q^7 - 179 * q^9 $$q + 8 q^{3} - 25 q^{5} - 108 q^{7} - 179 q^{9} + 604 q^{11} + 306 q^{13} - 200 q^{15} + 930 q^{17} + 1324 q^{19} - 864 q^{21} - 852 q^{23} + 625 q^{25} - 3376 q^{27} - 5902 q^{29} - 3320 q^{31} + 4832 q^{33} + 2700 q^{35} - 10774 q^{37} + 2448 q^{39} - 17958 q^{41} - 9264 q^{43} + 4475 q^{45} - 9796 q^{47} - 5143 q^{49} + 7440 q^{51} + 31434 q^{53} - 15100 q^{55} + 10592 q^{57} - 33228 q^{59} + 40210 q^{61} + 19332 q^{63} - 7650 q^{65} - 58864 q^{67} - 6816 q^{69} - 55312 q^{71} + 27258 q^{73} + 5000 q^{75} - 65232 q^{77} + 31456 q^{79} + 16489 q^{81} - 24552 q^{83} - 23250 q^{85} - 47216 q^{87} - 90854 q^{89} - 33048 q^{91} - 26560 q^{93} - 33100 q^{95} + 154706 q^{97} - 108116 q^{99}+O(q^{100})$$ q + 8 * q^3 - 25 * q^5 - 108 * q^7 - 179 * q^9 + 604 * q^11 + 306 * q^13 - 200 * q^15 + 930 * q^17 + 1324 * q^19 - 864 * q^21 - 852 * q^23 + 625 * q^25 - 3376 * q^27 - 5902 * q^29 - 3320 * q^31 + 4832 * q^33 + 2700 * q^35 - 10774 * q^37 + 2448 * q^39 - 17958 * q^41 - 9264 * q^43 + 4475 * q^45 - 9796 * q^47 - 5143 * q^49 + 7440 * q^51 + 31434 * q^53 - 15100 * q^55 + 10592 * q^57 - 33228 * q^59 + 40210 * q^61 + 19332 * q^63 - 7650 * q^65 - 58864 * q^67 - 6816 * q^69 - 55312 * q^71 + 27258 * q^73 + 5000 * q^75 - 65232 * q^77 + 31456 * q^79 + 16489 * q^81 - 24552 * q^83 - 23250 * q^85 - 47216 * q^87 - 90854 * q^89 - 33048 * q^91 - 26560 * q^93 - 33100 * q^95 + 154706 * q^97 - 108116 * 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 8.00000 0 −25.0000 0 −108.000 0 −179.000 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.l 1
4.b odd 2 1 320.6.a.e 1
8.b even 2 1 40.6.a.b 1
8.d odd 2 1 80.6.a.f 1
24.f even 2 1 720.6.a.h 1
24.h odd 2 1 360.6.a.b 1
40.e odd 2 1 400.6.a.f 1
40.f even 2 1 200.6.a.c 1
40.i odd 4 2 200.6.c.c 2
40.k even 4 2 400.6.c.h 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.6.a.b 1 8.b even 2 1
80.6.a.f 1 8.d odd 2 1
200.6.a.c 1 40.f even 2 1
200.6.c.c 2 40.i odd 4 2
320.6.a.e 1 4.b odd 2 1
320.6.a.l 1 1.a even 1 1 trivial
360.6.a.b 1 24.h odd 2 1
400.6.a.f 1 40.e odd 2 1
400.6.c.h 2 40.k even 4 2
720.6.a.h 1 24.f even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3} - 8$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(320))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 8$$
$5$ $$T + 25$$
$7$ $$T + 108$$
$11$ $$T - 604$$
$13$ $$T - 306$$
$17$ $$T - 930$$
$19$ $$T - 1324$$
$23$ $$T + 852$$
$29$ $$T + 5902$$
$31$ $$T + 3320$$
$37$ $$T + 10774$$
$41$ $$T + 17958$$
$43$ $$T + 9264$$
$47$ $$T + 9796$$
$53$ $$T - 31434$$
$59$ $$T + 33228$$
$61$ $$T - 40210$$
$67$ $$T + 58864$$
$71$ $$T + 55312$$
$73$ $$T - 27258$$
$79$ $$T - 31456$$
$83$ $$T + 24552$$
$89$ $$T + 90854$$
$97$ $$T - 154706$$