# Properties

 Label 320.4.a.r Level $320$ Weight $4$ Character orbit 320.a Self dual yes Analytic conductor $18.881$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$320 = 2^{6} \cdot 5$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 320.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$18.8806112018$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{13})$$ Defining polynomial: $$x^{2} - x - 3$$ x^2 - x - 3 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ 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 $$\beta = 2\sqrt{13}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta q^{3} + 5 q^{5} + \beta q^{7} + 25 q^{9} +O(q^{10})$$ q - b * q^3 + 5 * q^5 + b * q^7 + 25 * q^9 $$q - \beta q^{3} + 5 q^{5} + \beta q^{7} + 25 q^{9} - 6 \beta q^{11} - 34 q^{13} - 5 \beta q^{15} + 114 q^{17} - 52 q^{21} - 29 \beta q^{23} + 25 q^{25} + 2 \beta q^{27} + 26 q^{29} + 14 \beta q^{31} + 312 q^{33} + 5 \beta q^{35} + 150 q^{37} + 34 \beta q^{39} + 342 q^{41} + 63 \beta q^{43} + 125 q^{45} + 81 \beta q^{47} - 291 q^{49} - 114 \beta q^{51} + 262 q^{53} - 30 \beta q^{55} + 68 \beta q^{59} + 262 q^{61} + 25 \beta q^{63} - 170 q^{65} + 69 \beta q^{67} + 1508 q^{69} - 146 \beta q^{71} + 682 q^{73} - 25 \beta q^{75} - 312 q^{77} - 28 \beta q^{79} - 779 q^{81} - 21 \beta q^{83} + 570 q^{85} - 26 \beta q^{87} - 630 q^{89} - 34 \beta q^{91} - 728 q^{93} - 966 q^{97} - 150 \beta q^{99} +O(q^{100})$$ q - b * q^3 + 5 * q^5 + b * q^7 + 25 * q^9 - 6*b * q^11 - 34 * q^13 - 5*b * q^15 + 114 * q^17 - 52 * q^21 - 29*b * q^23 + 25 * q^25 + 2*b * q^27 + 26 * q^29 + 14*b * q^31 + 312 * q^33 + 5*b * q^35 + 150 * q^37 + 34*b * q^39 + 342 * q^41 + 63*b * q^43 + 125 * q^45 + 81*b * q^47 - 291 * q^49 - 114*b * q^51 + 262 * q^53 - 30*b * q^55 + 68*b * q^59 + 262 * q^61 + 25*b * q^63 - 170 * q^65 + 69*b * q^67 + 1508 * q^69 - 146*b * q^71 + 682 * q^73 - 25*b * q^75 - 312 * q^77 - 28*b * q^79 - 779 * q^81 - 21*b * q^83 + 570 * q^85 - 26*b * q^87 - 630 * q^89 - 34*b * q^91 - 728 * q^93 - 966 * q^97 - 150*b * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 10 q^{5} + 50 q^{9}+O(q^{10})$$ 2 * q + 10 * q^5 + 50 * q^9 $$2 q + 10 q^{5} + 50 q^{9} - 68 q^{13} + 228 q^{17} - 104 q^{21} + 50 q^{25} + 52 q^{29} + 624 q^{33} + 300 q^{37} + 684 q^{41} + 250 q^{45} - 582 q^{49} + 524 q^{53} + 524 q^{61} - 340 q^{65} + 3016 q^{69} + 1364 q^{73} - 624 q^{77} - 1558 q^{81} + 1140 q^{85} - 1260 q^{89} - 1456 q^{93} - 1932 q^{97}+O(q^{100})$$ 2 * q + 10 * q^5 + 50 * q^9 - 68 * q^13 + 228 * q^17 - 104 * q^21 + 50 * q^25 + 52 * q^29 + 624 * q^33 + 300 * q^37 + 684 * q^41 + 250 * q^45 - 582 * q^49 + 524 * q^53 + 524 * q^61 - 340 * q^65 + 3016 * q^69 + 1364 * q^73 - 624 * q^77 - 1558 * q^81 + 1140 * q^85 - 1260 * q^89 - 1456 * q^93 - 1932 * 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
 2.30278 −1.30278
0 −7.21110 0 5.00000 0 7.21110 0 25.0000 0
1.2 0 7.21110 0 5.00000 0 −7.21110 0 25.0000 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 320.4.a.r 2
4.b odd 2 1 inner 320.4.a.r 2
5.b even 2 1 1600.4.a.ci 2
8.b even 2 1 160.4.a.e 2
8.d odd 2 1 160.4.a.e 2
16.e even 4 2 1280.4.d.t 4
16.f odd 4 2 1280.4.d.t 4
20.d odd 2 1 1600.4.a.ci 2
24.f even 2 1 1440.4.a.bd 2
24.h odd 2 1 1440.4.a.bd 2
40.e odd 2 1 800.4.a.q 2
40.f even 2 1 800.4.a.q 2
40.i odd 4 2 800.4.c.h 4
40.k even 4 2 800.4.c.h 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.4.a.e 2 8.b even 2 1
160.4.a.e 2 8.d odd 2 1
320.4.a.r 2 1.a even 1 1 trivial
320.4.a.r 2 4.b odd 2 1 inner
800.4.a.q 2 40.e odd 2 1
800.4.a.q 2 40.f even 2 1
800.4.c.h 4 40.i odd 4 2
800.4.c.h 4 40.k even 4 2
1280.4.d.t 4 16.e even 4 2
1280.4.d.t 4 16.f odd 4 2
1440.4.a.bd 2 24.f even 2 1
1440.4.a.bd 2 24.h odd 2 1
1600.4.a.ci 2 5.b even 2 1
1600.4.a.ci 2 20.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(320))$$:

 $$T_{3}^{2} - 52$$ T3^2 - 52 $$T_{7}^{2} - 52$$ T7^2 - 52

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - 52$$
$5$ $$(T - 5)^{2}$$
$7$ $$T^{2} - 52$$
$11$ $$T^{2} - 1872$$
$13$ $$(T + 34)^{2}$$
$17$ $$(T - 114)^{2}$$
$19$ $$T^{2}$$
$23$ $$T^{2} - 43732$$
$29$ $$(T - 26)^{2}$$
$31$ $$T^{2} - 10192$$
$37$ $$(T - 150)^{2}$$
$41$ $$(T - 342)^{2}$$
$43$ $$T^{2} - 206388$$
$47$ $$T^{2} - 341172$$
$53$ $$(T - 262)^{2}$$
$59$ $$T^{2} - 240448$$
$61$ $$(T - 262)^{2}$$
$67$ $$T^{2} - 247572$$
$71$ $$T^{2} - 1108432$$
$73$ $$(T - 682)^{2}$$
$79$ $$T^{2} - 40768$$
$83$ $$T^{2} - 22932$$
$89$ $$(T + 630)^{2}$$
$97$ $$(T + 966)^{2}$$