# Properties

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

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + 2 q^{3} + 25 q^{5} - 62 q^{7} - 239 q^{9}+O(q^{10})$$ q + 2 * q^3 + 25 * q^5 - 62 * q^7 - 239 * q^9 $$q + 2 q^{3} + 25 q^{5} - 62 q^{7} - 239 q^{9} + 144 q^{11} + 654 q^{13} + 50 q^{15} - 1190 q^{17} - 556 q^{19} - 124 q^{21} + 2182 q^{23} + 625 q^{25} - 964 q^{27} + 1578 q^{29} + 9660 q^{31} + 288 q^{33} - 1550 q^{35} + 3534 q^{37} + 1308 q^{39} + 7462 q^{41} + 7114 q^{43} - 5975 q^{45} - 28294 q^{47} - 12963 q^{49} - 2380 q^{51} + 13046 q^{53} + 3600 q^{55} - 1112 q^{57} + 37092 q^{59} - 39570 q^{61} + 14818 q^{63} + 16350 q^{65} + 56734 q^{67} + 4364 q^{69} + 45588 q^{71} + 11842 q^{73} + 1250 q^{75} - 8928 q^{77} + 94216 q^{79} + 56149 q^{81} + 31482 q^{83} - 29750 q^{85} + 3156 q^{87} - 94054 q^{89} - 40548 q^{91} + 19320 q^{93} - 13900 q^{95} + 23714 q^{97} - 34416 q^{99}+O(q^{100})$$ q + 2 * q^3 + 25 * q^5 - 62 * q^7 - 239 * q^9 + 144 * q^11 + 654 * q^13 + 50 * q^15 - 1190 * q^17 - 556 * q^19 - 124 * q^21 + 2182 * q^23 + 625 * q^25 - 964 * q^27 + 1578 * q^29 + 9660 * q^31 + 288 * q^33 - 1550 * q^35 + 3534 * q^37 + 1308 * q^39 + 7462 * q^41 + 7114 * q^43 - 5975 * q^45 - 28294 * q^47 - 12963 * q^49 - 2380 * q^51 + 13046 * q^53 + 3600 * q^55 - 1112 * q^57 + 37092 * q^59 - 39570 * q^61 + 14818 * q^63 + 16350 * q^65 + 56734 * q^67 + 4364 * q^69 + 45588 * q^71 + 11842 * q^73 + 1250 * q^75 - 8928 * q^77 + 94216 * q^79 + 56149 * q^81 + 31482 * q^83 - 29750 * q^85 + 3156 * q^87 - 94054 * q^89 - 40548 * q^91 + 19320 * q^93 - 13900 * q^95 + 23714 * q^97 - 34416 * 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 2.00000 0 25.0000 0 −62.0000 0 −239.000 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.i 1
4.b odd 2 1 320.6.a.h 1
8.b even 2 1 40.6.a.c 1
8.d odd 2 1 80.6.a.d 1
24.f even 2 1 720.6.a.t 1
24.h odd 2 1 360.6.a.f 1
40.e odd 2 1 400.6.a.h 1
40.f even 2 1 200.6.a.b 1
40.i odd 4 2 200.6.c.d 2
40.k even 4 2 400.6.c.k 2

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3} - 2$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(320))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 2$$
$5$ $$T - 25$$
$7$ $$T + 62$$
$11$ $$T - 144$$
$13$ $$T - 654$$
$17$ $$T + 1190$$
$19$ $$T + 556$$
$23$ $$T - 2182$$
$29$ $$T - 1578$$
$31$ $$T - 9660$$
$37$ $$T - 3534$$
$41$ $$T - 7462$$
$43$ $$T - 7114$$
$47$ $$T + 28294$$
$53$ $$T - 13046$$
$59$ $$T - 37092$$
$61$ $$T + 39570$$
$67$ $$T - 56734$$
$71$ $$T - 45588$$
$73$ $$T - 11842$$
$79$ $$T - 94216$$
$83$ $$T - 31482$$
$89$ $$T + 94054$$
$97$ $$T - 23714$$