Properties

 Label 320.6.a.t Level $320$ Weight $6$ Character orbit 320.a Self dual yes Analytic conductor $51.323$ Analytic rank $1$ Dimension $2$ CM no Inner twists $2$

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: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ Defining polynomial: $$x^{2} - x - 1$$ x^2 - x - 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 160) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 2\sqrt{5}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - 3 \beta q^{3} + 25 q^{5} + 31 \beta q^{7} - 63 q^{9} +O(q^{10})$$ q - 3*b * q^3 + 25 * q^5 + 31*b * q^7 - 63 * q^9 $$q - 3 \beta q^{3} + 25 q^{5} + 31 \beta q^{7} - 63 q^{9} - 58 \beta q^{11} - 154 q^{13} - 75 \beta q^{15} + 178 q^{17} - 216 \beta q^{19} - 1860 q^{21} + 589 \beta q^{23} + 625 q^{25} + 918 \beta q^{27} - 4110 q^{29} + 706 \beta q^{31} + 3480 q^{33} + 775 \beta q^{35} - 7442 q^{37} + 462 \beta q^{39} + 7270 q^{41} + 4005 \beta q^{43} - 1575 q^{45} - 1657 \beta q^{47} + 2413 q^{49} - 534 \beta q^{51} - 32226 q^{53} - 1450 \beta q^{55} + 12960 q^{57} - 7612 \beta q^{59} - 26770 q^{61} - 1953 \beta q^{63} - 3850 q^{65} - 11137 \beta q^{67} - 35340 q^{69} + 12098 \beta q^{71} - 18534 q^{73} - 1875 \beta q^{75} - 35960 q^{77} - 19396 \beta q^{79} - 39771 q^{81} + 17585 \beta q^{83} + 4450 q^{85} + 12330 \beta q^{87} - 107590 q^{89} - 4774 \beta q^{91} - 42360 q^{93} - 5400 \beta q^{95} - 108838 q^{97} + 3654 \beta q^{99} +O(q^{100})$$ q - 3*b * q^3 + 25 * q^5 + 31*b * q^7 - 63 * q^9 - 58*b * q^11 - 154 * q^13 - 75*b * q^15 + 178 * q^17 - 216*b * q^19 - 1860 * q^21 + 589*b * q^23 + 625 * q^25 + 918*b * q^27 - 4110 * q^29 + 706*b * q^31 + 3480 * q^33 + 775*b * q^35 - 7442 * q^37 + 462*b * q^39 + 7270 * q^41 + 4005*b * q^43 - 1575 * q^45 - 1657*b * q^47 + 2413 * q^49 - 534*b * q^51 - 32226 * q^53 - 1450*b * q^55 + 12960 * q^57 - 7612*b * q^59 - 26770 * q^61 - 1953*b * q^63 - 3850 * q^65 - 11137*b * q^67 - 35340 * q^69 + 12098*b * q^71 - 18534 * q^73 - 1875*b * q^75 - 35960 * q^77 - 19396*b * q^79 - 39771 * q^81 + 17585*b * q^83 + 4450 * q^85 + 12330*b * q^87 - 107590 * q^89 - 4774*b * q^91 - 42360 * q^93 - 5400*b * q^95 - 108838 * q^97 + 3654*b * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 50 q^{5} - 126 q^{9}+O(q^{10})$$ 2 * q + 50 * q^5 - 126 * q^9 $$2 q + 50 q^{5} - 126 q^{9} - 308 q^{13} + 356 q^{17} - 3720 q^{21} + 1250 q^{25} - 8220 q^{29} + 6960 q^{33} - 14884 q^{37} + 14540 q^{41} - 3150 q^{45} + 4826 q^{49} - 64452 q^{53} + 25920 q^{57} - 53540 q^{61} - 7700 q^{65} - 70680 q^{69} - 37068 q^{73} - 71920 q^{77} - 79542 q^{81} + 8900 q^{85} - 215180 q^{89} - 84720 q^{93} - 217676 q^{97}+O(q^{100})$$ 2 * q + 50 * q^5 - 126 * q^9 - 308 * q^13 + 356 * q^17 - 3720 * q^21 + 1250 * q^25 - 8220 * q^29 + 6960 * q^33 - 14884 * q^37 + 14540 * q^41 - 3150 * q^45 + 4826 * q^49 - 64452 * q^53 + 25920 * q^57 - 53540 * q^61 - 7700 * q^65 - 70680 * q^69 - 37068 * q^73 - 71920 * q^77 - 79542 * q^81 + 8900 * q^85 - 215180 * q^89 - 84720 * q^93 - 217676 * q^97

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
 1.61803 −0.618034
0 −13.4164 0 25.0000 0 138.636 0 −63.0000 0
1.2 0 13.4164 0 25.0000 0 −138.636 0 −63.0000 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

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 320.6.a.t 2
4.b odd 2 1 inner 320.6.a.t 2
8.b even 2 1 160.6.a.b 2
8.d odd 2 1 160.6.a.b 2
40.e odd 2 1 800.6.a.i 2
40.f even 2 1 800.6.a.i 2
40.i odd 4 2 800.6.c.h 4
40.k even 4 2 800.6.c.h 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.6.a.b 2 8.b even 2 1
160.6.a.b 2 8.d odd 2 1
320.6.a.t 2 1.a even 1 1 trivial
320.6.a.t 2 4.b odd 2 1 inner
800.6.a.i 2 40.e odd 2 1
800.6.a.i 2 40.f even 2 1
800.6.c.h 4 40.i odd 4 2
800.6.c.h 4 40.k even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} - 180$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(320))$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - 180$$
$5$ $$(T - 25)^{2}$$
$7$ $$T^{2} - 19220$$
$11$ $$T^{2} - 67280$$
$13$ $$(T + 154)^{2}$$
$17$ $$(T - 178)^{2}$$
$19$ $$T^{2} - 933120$$
$23$ $$T^{2} - 6938420$$
$29$ $$(T + 4110)^{2}$$
$31$ $$T^{2} - 9968720$$
$37$ $$(T + 7442)^{2}$$
$41$ $$(T - 7270)^{2}$$
$43$ $$T^{2} - 320800500$$
$47$ $$T^{2} - 54912980$$
$53$ $$(T + 32226)^{2}$$
$59$ $$T^{2} - 1158850880$$
$61$ $$(T + 26770)^{2}$$
$67$ $$T^{2} - 2480655380$$
$71$ $$T^{2} - 2927232080$$
$73$ $$(T + 18534)^{2}$$
$79$ $$T^{2} - 7524096320$$
$83$ $$T^{2} - 6184644500$$
$89$ $$(T + 107590)^{2}$$
$97$ $$(T + 108838)^{2}$$