# Properties

 Label 320.6.a.c 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 20) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - 22 q^{3} + 25 q^{5} + 218 q^{7} + 241 q^{9}+O(q^{10})$$ q - 22 * q^3 + 25 * q^5 + 218 * q^7 + 241 * q^9 $$q - 22 q^{3} + 25 q^{5} + 218 q^{7} + 241 q^{9} + 480 q^{11} + 622 q^{13} - 550 q^{15} + 186 q^{17} + 1204 q^{19} - 4796 q^{21} - 3186 q^{23} + 625 q^{25} + 44 q^{27} - 5526 q^{29} + 9356 q^{31} - 10560 q^{33} + 5450 q^{35} - 5618 q^{37} - 13684 q^{39} - 14394 q^{41} + 370 q^{43} + 6025 q^{45} + 16146 q^{47} + 30717 q^{49} - 4092 q^{51} + 4374 q^{53} + 12000 q^{55} - 26488 q^{57} + 11748 q^{59} - 13202 q^{61} + 52538 q^{63} + 15550 q^{65} + 11542 q^{67} + 70092 q^{69} - 29532 q^{71} + 33698 q^{73} - 13750 q^{75} + 104640 q^{77} + 31208 q^{79} - 59531 q^{81} + 38466 q^{83} + 4650 q^{85} + 121572 q^{87} + 119514 q^{89} + 135596 q^{91} - 205832 q^{93} + 30100 q^{95} + 94658 q^{97} + 115680 q^{99}+O(q^{100})$$ q - 22 * q^3 + 25 * q^5 + 218 * q^7 + 241 * q^9 + 480 * q^11 + 622 * q^13 - 550 * q^15 + 186 * q^17 + 1204 * q^19 - 4796 * q^21 - 3186 * q^23 + 625 * q^25 + 44 * q^27 - 5526 * q^29 + 9356 * q^31 - 10560 * q^33 + 5450 * q^35 - 5618 * q^37 - 13684 * q^39 - 14394 * q^41 + 370 * q^43 + 6025 * q^45 + 16146 * q^47 + 30717 * q^49 - 4092 * q^51 + 4374 * q^53 + 12000 * q^55 - 26488 * q^57 + 11748 * q^59 - 13202 * q^61 + 52538 * q^63 + 15550 * q^65 + 11542 * q^67 + 70092 * q^69 - 29532 * q^71 + 33698 * q^73 - 13750 * q^75 + 104640 * q^77 + 31208 * q^79 - 59531 * q^81 + 38466 * q^83 + 4650 * q^85 + 121572 * q^87 + 119514 * q^89 + 135596 * q^91 - 205832 * q^93 + 30100 * q^95 + 94658 * q^97 + 115680 * 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 −22.0000 0 25.0000 0 218.000 0 241.000 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.c 1
4.b odd 2 1 320.6.a.n 1
8.b even 2 1 20.6.a.a 1
8.d odd 2 1 80.6.a.b 1
24.f even 2 1 720.6.a.l 1
24.h odd 2 1 180.6.a.e 1
40.e odd 2 1 400.6.a.m 1
40.f even 2 1 100.6.a.a 1
40.i odd 4 2 100.6.c.a 2
40.k even 4 2 400.6.c.c 2
56.h odd 2 1 980.6.a.b 1
120.i odd 2 1 900.6.a.b 1
120.w even 4 2 900.6.d.h 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.6.a.a 1 8.b even 2 1
80.6.a.b 1 8.d odd 2 1
100.6.a.a 1 40.f even 2 1
100.6.c.a 2 40.i odd 4 2
180.6.a.e 1 24.h odd 2 1
320.6.a.c 1 1.a even 1 1 trivial
320.6.a.n 1 4.b odd 2 1
400.6.a.m 1 40.e odd 2 1
400.6.c.c 2 40.k even 4 2
720.6.a.l 1 24.f even 2 1
900.6.a.b 1 120.i odd 2 1
900.6.d.h 2 120.w even 4 2
980.6.a.b 1 56.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3} + 22$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(320))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T + 22$$
$5$ $$T - 25$$
$7$ $$T - 218$$
$11$ $$T - 480$$
$13$ $$T - 622$$
$17$ $$T - 186$$
$19$ $$T - 1204$$
$23$ $$T + 3186$$
$29$ $$T + 5526$$
$31$ $$T - 9356$$
$37$ $$T + 5618$$
$41$ $$T + 14394$$
$43$ $$T - 370$$
$47$ $$T - 16146$$
$53$ $$T - 4374$$
$59$ $$T - 11748$$
$61$ $$T + 13202$$
$67$ $$T - 11542$$
$71$ $$T + 29532$$
$73$ $$T - 33698$$
$79$ $$T - 31208$$
$83$ $$T - 38466$$
$89$ $$T - 119514$$
$97$ $$T - 94658$$