Properties

 Label 320.6.a.k 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 10) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

 $$f(q)$$ $$=$$ $$q + 6 q^{3} + 25 q^{5} + 118 q^{7} - 207 q^{9}+O(q^{10})$$ q + 6 * q^3 + 25 * q^5 + 118 * q^7 - 207 * q^9 $$q + 6 q^{3} + 25 q^{5} + 118 q^{7} - 207 q^{9} + 192 q^{11} - 1106 q^{13} + 150 q^{15} + 762 q^{17} - 2740 q^{19} + 708 q^{21} - 1566 q^{23} + 625 q^{25} - 2700 q^{27} - 5910 q^{29} + 6868 q^{31} + 1152 q^{33} + 2950 q^{35} + 5518 q^{37} - 6636 q^{39} - 378 q^{41} - 2434 q^{43} - 5175 q^{45} - 13122 q^{47} - 2883 q^{49} + 4572 q^{51} + 9174 q^{53} + 4800 q^{55} - 16440 q^{57} - 34980 q^{59} + 9838 q^{61} - 24426 q^{63} - 27650 q^{65} + 33722 q^{67} - 9396 q^{69} - 70212 q^{71} + 21986 q^{73} + 3750 q^{75} + 22656 q^{77} - 4520 q^{79} + 34101 q^{81} - 109074 q^{83} + 19050 q^{85} - 35460 q^{87} + 38490 q^{89} - 130508 q^{91} + 41208 q^{93} - 68500 q^{95} - 1918 q^{97} - 39744 q^{99}+O(q^{100})$$ q + 6 * q^3 + 25 * q^5 + 118 * q^7 - 207 * q^9 + 192 * q^11 - 1106 * q^13 + 150 * q^15 + 762 * q^17 - 2740 * q^19 + 708 * q^21 - 1566 * q^23 + 625 * q^25 - 2700 * q^27 - 5910 * q^29 + 6868 * q^31 + 1152 * q^33 + 2950 * q^35 + 5518 * q^37 - 6636 * q^39 - 378 * q^41 - 2434 * q^43 - 5175 * q^45 - 13122 * q^47 - 2883 * q^49 + 4572 * q^51 + 9174 * q^53 + 4800 * q^55 - 16440 * q^57 - 34980 * q^59 + 9838 * q^61 - 24426 * q^63 - 27650 * q^65 + 33722 * q^67 - 9396 * q^69 - 70212 * q^71 + 21986 * q^73 + 3750 * q^75 + 22656 * q^77 - 4520 * q^79 + 34101 * q^81 - 109074 * q^83 + 19050 * q^85 - 35460 * q^87 + 38490 * q^89 - 130508 * q^91 + 41208 * q^93 - 68500 * q^95 - 1918 * q^97 - 39744 * 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 6.00000 0 25.0000 0 118.000 0 −207.000 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.k 1
4.b odd 2 1 320.6.a.f 1
8.b even 2 1 80.6.a.c 1
8.d odd 2 1 10.6.a.c 1
24.f even 2 1 90.6.a.b 1
24.h odd 2 1 720.6.a.v 1
40.e odd 2 1 50.6.a.b 1
40.f even 2 1 400.6.a.i 1
40.i odd 4 2 400.6.c.i 2
40.k even 4 2 50.6.b.b 2
56.e even 2 1 490.6.a.k 1
120.m even 2 1 450.6.a.u 1
120.q odd 4 2 450.6.c.f 2

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

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 6$$
$5$ $$T - 25$$
$7$ $$T - 118$$
$11$ $$T - 192$$
$13$ $$T + 1106$$
$17$ $$T - 762$$
$19$ $$T + 2740$$
$23$ $$T + 1566$$
$29$ $$T + 5910$$
$31$ $$T - 6868$$
$37$ $$T - 5518$$
$41$ $$T + 378$$
$43$ $$T + 2434$$
$47$ $$T + 13122$$
$53$ $$T - 9174$$
$59$ $$T + 34980$$
$61$ $$T - 9838$$
$67$ $$T - 33722$$
$71$ $$T + 70212$$
$73$ $$T - 21986$$
$79$ $$T + 4520$$
$83$ $$T + 109074$$
$89$ $$T - 38490$$
$97$ $$T + 1918$$