# Properties

 Label 320.4.a.s Level $320$ Weight $4$ Character orbit 320.a Self dual yes Analytic conductor $18.881$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# 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: $$\Q(\sqrt{6})$$ Defining polynomial: $$x^{2} - 6$$ x^2 - 6 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ 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 $$\beta = 2\sqrt{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta + 4) q^{3} - 5 q^{5} + (5 \beta - 4) q^{7} + (8 \beta + 13) q^{9}+O(q^{10})$$ q + (b + 4) * q^3 - 5 * q^5 + (5*b - 4) * q^7 + (8*b + 13) * q^9 $$q + (\beta + 4) q^{3} - 5 q^{5} + (5 \beta - 4) q^{7} + (8 \beta + 13) q^{9} + (6 \beta + 32) q^{11} + ( - 8 \beta - 6) q^{13} + ( - 5 \beta - 20) q^{15} + ( - 24 \beta + 2) q^{17} + ( - 8 \beta + 104) q^{19} + (16 \beta + 104) q^{21} + (11 \beta - 60) q^{23} + 25 q^{25} + (18 \beta + 136) q^{27} + (16 \beta + 146) q^{29} + (6 \beta - 88) q^{31} + (56 \beta + 272) q^{33} + ( - 25 \beta + 20) q^{35} + ( - 16 \beta + 178) q^{37} + ( - 38 \beta - 216) q^{39} + ( - 40 \beta + 50) q^{41} + (37 \beta - 188) q^{43} + ( - 40 \beta - 65) q^{45} + (13 \beta + 140) q^{47} + ( - 40 \beta + 273) q^{49} + ( - 94 \beta - 568) q^{51} + ( - 24 \beta - 158) q^{53} + ( - 30 \beta - 160) q^{55} + (72 \beta + 224) q^{57} + ( - 20 \beta + 360) q^{59} + (32 \beta + 634) q^{61} + (33 \beta + 908) q^{63} + (40 \beta + 30) q^{65} + (47 \beta - 372) q^{67} + ( - 16 \beta + 24) q^{69} + ( - 218 \beta + 24) q^{71} + ( - 120 \beta - 470) q^{73} + (25 \beta + 100) q^{75} + (136 \beta + 592) q^{77} + (172 \beta - 16) q^{79} + ( - 8 \beta + 625) q^{81} + ( - 47 \beta - 796) q^{83} + (120 \beta - 10) q^{85} + (210 \beta + 968) q^{87} + (48 \beta - 390) q^{89} + (2 \beta - 936) q^{91} + ( - 64 \beta - 208) q^{93} + (40 \beta - 520) q^{95} + ( - 40 \beta + 610) q^{97} + (334 \beta + 1568) q^{99}+O(q^{100})$$ q + (b + 4) * q^3 - 5 * q^5 + (5*b - 4) * q^7 + (8*b + 13) * q^9 + (6*b + 32) * q^11 + (-8*b - 6) * q^13 + (-5*b - 20) * q^15 + (-24*b + 2) * q^17 + (-8*b + 104) * q^19 + (16*b + 104) * q^21 + (11*b - 60) * q^23 + 25 * q^25 + (18*b + 136) * q^27 + (16*b + 146) * q^29 + (6*b - 88) * q^31 + (56*b + 272) * q^33 + (-25*b + 20) * q^35 + (-16*b + 178) * q^37 + (-38*b - 216) * q^39 + (-40*b + 50) * q^41 + (37*b - 188) * q^43 + (-40*b - 65) * q^45 + (13*b + 140) * q^47 + (-40*b + 273) * q^49 + (-94*b - 568) * q^51 + (-24*b - 158) * q^53 + (-30*b - 160) * q^55 + (72*b + 224) * q^57 + (-20*b + 360) * q^59 + (32*b + 634) * q^61 + (33*b + 908) * q^63 + (40*b + 30) * q^65 + (47*b - 372) * q^67 + (-16*b + 24) * q^69 + (-218*b + 24) * q^71 + (-120*b - 470) * q^73 + (25*b + 100) * q^75 + (136*b + 592) * q^77 + (172*b - 16) * q^79 + (-8*b + 625) * q^81 + (-47*b - 796) * q^83 + (120*b - 10) * q^85 + (210*b + 968) * q^87 + (48*b - 390) * q^89 + (2*b - 936) * q^91 + (-64*b - 208) * q^93 + (40*b - 520) * q^95 + (-40*b + 610) * q^97 + (334*b + 1568) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 8 q^{3} - 10 q^{5} - 8 q^{7} + 26 q^{9}+O(q^{10})$$ 2 * q + 8 * q^3 - 10 * q^5 - 8 * q^7 + 26 * q^9 $$2 q + 8 q^{3} - 10 q^{5} - 8 q^{7} + 26 q^{9} + 64 q^{11} - 12 q^{13} - 40 q^{15} + 4 q^{17} + 208 q^{19} + 208 q^{21} - 120 q^{23} + 50 q^{25} + 272 q^{27} + 292 q^{29} - 176 q^{31} + 544 q^{33} + 40 q^{35} + 356 q^{37} - 432 q^{39} + 100 q^{41} - 376 q^{43} - 130 q^{45} + 280 q^{47} + 546 q^{49} - 1136 q^{51} - 316 q^{53} - 320 q^{55} + 448 q^{57} + 720 q^{59} + 1268 q^{61} + 1816 q^{63} + 60 q^{65} - 744 q^{67} + 48 q^{69} + 48 q^{71} - 940 q^{73} + 200 q^{75} + 1184 q^{77} - 32 q^{79} + 1250 q^{81} - 1592 q^{83} - 20 q^{85} + 1936 q^{87} - 780 q^{89} - 1872 q^{91} - 416 q^{93} - 1040 q^{95} + 1220 q^{97} + 3136 q^{99}+O(q^{100})$$ 2 * q + 8 * q^3 - 10 * q^5 - 8 * q^7 + 26 * q^9 + 64 * q^11 - 12 * q^13 - 40 * q^15 + 4 * q^17 + 208 * q^19 + 208 * q^21 - 120 * q^23 + 50 * q^25 + 272 * q^27 + 292 * q^29 - 176 * q^31 + 544 * q^33 + 40 * q^35 + 356 * q^37 - 432 * q^39 + 100 * q^41 - 376 * q^43 - 130 * q^45 + 280 * q^47 + 546 * q^49 - 1136 * q^51 - 316 * q^53 - 320 * q^55 + 448 * q^57 + 720 * q^59 + 1268 * q^61 + 1816 * q^63 + 60 * q^65 - 744 * q^67 + 48 * q^69 + 48 * q^71 - 940 * q^73 + 200 * q^75 + 1184 * q^77 - 32 * q^79 + 1250 * q^81 - 1592 * q^83 - 20 * q^85 + 1936 * q^87 - 780 * q^89 - 1872 * q^91 - 416 * q^93 - 1040 * q^95 + 1220 * q^97 + 3136 * 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
 −2.44949 2.44949
0 −0.898979 0 −5.00000 0 −28.4949 0 −26.1918 0
1.2 0 8.89898 0 −5.00000 0 20.4949 0 52.1918 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 320.4.a.s 2
4.b odd 2 1 320.4.a.o 2
5.b even 2 1 1600.4.a.cd 2
8.b even 2 1 160.4.a.c 2
8.d odd 2 1 160.4.a.g yes 2
16.e even 4 2 1280.4.d.x 4
16.f odd 4 2 1280.4.d.q 4
20.d odd 2 1 1600.4.a.cn 2
24.f even 2 1 1440.4.a.x 2
24.h odd 2 1 1440.4.a.t 2
40.e odd 2 1 800.4.a.m 2
40.f even 2 1 800.4.a.s 2
40.i odd 4 2 800.4.c.i 4
40.k even 4 2 800.4.c.k 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.4.a.c 2 8.b even 2 1
160.4.a.g yes 2 8.d odd 2 1
320.4.a.o 2 4.b odd 2 1
320.4.a.s 2 1.a even 1 1 trivial
800.4.a.m 2 40.e odd 2 1
800.4.a.s 2 40.f even 2 1
800.4.c.i 4 40.i odd 4 2
800.4.c.k 4 40.k even 4 2
1280.4.d.q 4 16.f odd 4 2
1280.4.d.x 4 16.e even 4 2
1440.4.a.t 2 24.h odd 2 1
1440.4.a.x 2 24.f even 2 1
1600.4.a.cd 2 5.b even 2 1
1600.4.a.cn 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(320))$$:

 $$T_{3}^{2} - 8T_{3} - 8$$ T3^2 - 8*T3 - 8 $$T_{7}^{2} + 8T_{7} - 584$$ T7^2 + 8*T7 - 584

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - 8T - 8$$
$5$ $$(T + 5)^{2}$$
$7$ $$T^{2} + 8T - 584$$
$11$ $$T^{2} - 64T + 160$$
$13$ $$T^{2} + 12T - 1500$$
$17$ $$T^{2} - 4T - 13820$$
$19$ $$T^{2} - 208T + 9280$$
$23$ $$T^{2} + 120T + 696$$
$29$ $$T^{2} - 292T + 15172$$
$31$ $$T^{2} + 176T + 6880$$
$37$ $$T^{2} - 356T + 25540$$
$41$ $$T^{2} - 100T - 35900$$
$43$ $$T^{2} + 376T + 2488$$
$47$ $$T^{2} - 280T + 15544$$
$53$ $$T^{2} + 316T + 11140$$
$59$ $$T^{2} - 720T + 120000$$
$61$ $$T^{2} - 1268 T + 377380$$
$67$ $$T^{2} + 744T + 85368$$
$71$ $$T^{2} - 48T - 1140000$$
$73$ $$T^{2} + 940T - 124700$$
$79$ $$T^{2} + 32T - 709760$$
$83$ $$T^{2} + 1592 T + 580600$$
$89$ $$T^{2} + 780T + 96804$$
$97$ $$T^{2} - 1220 T + 333700$$