# Properties

 Label 320.6.a.d 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

 $$f(q)$$ $$=$$ $$q - 18 q^{3} + 25 q^{5} - 242 q^{7} + 81 q^{9}+O(q^{10})$$ q - 18 * q^3 + 25 * q^5 - 242 * q^7 + 81 * q^9 $$q - 18 q^{3} + 25 q^{5} - 242 q^{7} + 81 q^{9} + 656 q^{11} + 206 q^{13} - 450 q^{15} + 1690 q^{17} - 1364 q^{19} + 4356 q^{21} - 2198 q^{23} + 625 q^{25} + 2916 q^{27} + 2218 q^{29} + 1700 q^{31} - 11808 q^{33} - 6050 q^{35} + 846 q^{37} - 3708 q^{39} - 1818 q^{41} + 10534 q^{43} + 2025 q^{45} - 12074 q^{47} + 41757 q^{49} - 30420 q^{51} - 32586 q^{53} + 16400 q^{55} + 24552 q^{57} + 8668 q^{59} + 34670 q^{61} - 19602 q^{63} + 5150 q^{65} - 47566 q^{67} + 39564 q^{69} - 948 q^{71} - 63102 q^{73} - 11250 q^{75} - 158752 q^{77} - 46536 q^{79} - 72171 q^{81} - 88778 q^{83} + 42250 q^{85} - 39924 q^{87} - 104934 q^{89} - 49852 q^{91} - 30600 q^{93} - 34100 q^{95} - 36254 q^{97} + 53136 q^{99}+O(q^{100})$$ q - 18 * q^3 + 25 * q^5 - 242 * q^7 + 81 * q^9 + 656 * q^11 + 206 * q^13 - 450 * q^15 + 1690 * q^17 - 1364 * q^19 + 4356 * q^21 - 2198 * q^23 + 625 * q^25 + 2916 * q^27 + 2218 * q^29 + 1700 * q^31 - 11808 * q^33 - 6050 * q^35 + 846 * q^37 - 3708 * q^39 - 1818 * q^41 + 10534 * q^43 + 2025 * q^45 - 12074 * q^47 + 41757 * q^49 - 30420 * q^51 - 32586 * q^53 + 16400 * q^55 + 24552 * q^57 + 8668 * q^59 + 34670 * q^61 - 19602 * q^63 + 5150 * q^65 - 47566 * q^67 + 39564 * q^69 - 948 * q^71 - 63102 * q^73 - 11250 * q^75 - 158752 * q^77 - 46536 * q^79 - 72171 * q^81 - 88778 * q^83 + 42250 * q^85 - 39924 * q^87 - 104934 * q^89 - 49852 * q^91 - 30600 * q^93 - 34100 * q^95 - 36254 * q^97 + 53136 * 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 −18.0000 0 25.0000 0 −242.000 0 81.0000 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 320.6.a.d 1
4.b odd 2 1 320.6.a.m 1
8.b even 2 1 80.6.a.g 1
8.d odd 2 1 40.6.a.a 1
24.f even 2 1 360.6.a.i 1
24.h odd 2 1 720.6.a.k 1
40.e odd 2 1 200.6.a.d 1
40.f even 2 1 400.6.a.b 1
40.i odd 4 2 400.6.c.e 2
40.k even 4 2 200.6.c.b 2

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3} + 18$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(320))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T + 18$$
$5$ $$T - 25$$
$7$ $$T + 242$$
$11$ $$T - 656$$
$13$ $$T - 206$$
$17$ $$T - 1690$$
$19$ $$T + 1364$$
$23$ $$T + 2198$$
$29$ $$T - 2218$$
$31$ $$T - 1700$$
$37$ $$T - 846$$
$41$ $$T + 1818$$
$43$ $$T - 10534$$
$47$ $$T + 12074$$
$53$ $$T + 32586$$
$59$ $$T - 8668$$
$61$ $$T - 34670$$
$67$ $$T + 47566$$
$71$ $$T + 948$$
$73$ $$T + 63102$$
$79$ $$T + 46536$$
$83$ $$T + 88778$$
$89$ $$T + 104934$$
$97$ $$T + 36254$$