# Properties

 Label 320.6.a.y Level $320$ Weight $6$ Character orbit 320.a Self dual yes Analytic conductor $51.323$ Analytic rank $0$ Dimension $3$ 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: $$3$$ Coefficient field: 3.3.39180.1 Defining polynomial: $$x^{3} - x^{2} - 36x - 24$$ x^3 - x^2 - 36*x - 24 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{6}$$ 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 a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_1 + 3) q^{3} + 25 q^{5} + ( - \beta_{2} + \beta_1 + 2) q^{7} + (2 \beta_{2} + 12 \beta_1 + 151) q^{9}+O(q^{10})$$ q + (b1 + 3) * q^3 + 25 * q^5 + (-b2 + b1 + 2) * q^7 + (2*b2 + 12*b1 + 151) * q^9 $$q + (\beta_1 + 3) q^{3} + 25 q^{5} + ( - \beta_{2} + \beta_1 + 2) q^{7} + (2 \beta_{2} + 12 \beta_1 + 151) q^{9} + ( - 3 \beta_{2} - 131) q^{11} + (2 \beta_{2} - 20 \beta_1 + 124) q^{13} + (25 \beta_1 + 75) q^{15} + ( - 6 \beta_{2} + 60 \beta_1 + 368) q^{17} + (\beta_{2} + 50 \beta_1 + 1047) q^{19} + (4 \beta_{2} - 72 \beta_1 + 296) q^{21} + ( - 7 \beta_{2} + 43 \beta_1 - 2054) q^{23} + 625 q^{25} + (20 \beta_{2} + 182 \beta_1 + 4534) q^{27} + ( - 52 \beta_{2} + 40 \beta_1 - 138) q^{29} + (42 \beta_{2} + 270 \beta_1 - 1196) q^{31} + (6 \beta_{2} - 380 \beta_1 - 678) q^{33} + ( - 25 \beta_{2} + 25 \beta_1 + 50) q^{35} + (76 \beta_{2} + 200 \beta_1 + 3762) q^{37} + ( - 44 \beta_{2} + 110 \beta_1 - 7138) q^{39} + (22 \beta_{2} + 68 \beta_1 + 4120) q^{41} + ( - 14 \beta_{2} + 221 \beta_1 + 8713) q^{43} + (50 \beta_{2} + 300 \beta_1 + 3775) q^{45} + (25 \beta_{2} + 77 \beta_1 - 12288) q^{47} + ( - 74 \beta_{2} - 316 \beta_1 - 1153) q^{49} + (132 \beta_{2} + 410 \beta_1 + 23634) q^{51} + ( - 114 \beta_{2} + 1140 \beta_1 + 6712) q^{53} + ( - 75 \beta_{2} - 3275) q^{55} + (98 \beta_{2} + 1580 \beta_1 + 22486) q^{57} + ( - 185 \beta_{2} - 70 \beta_1 + 11765) q^{59} + ( - 32 \beta_{2} - 1984 \beta_1 + 8718) q^{61} + (91 \beta_{2} - 263 \beta_1 - 26938) q^{63} + (50 \beta_{2} - 500 \beta_1 + 3100) q^{65} + ( - 82 \beta_{2} - 2345 \beta_1 - 2381) q^{67} + (100 \beta_{2} - 2248 \beta_1 + 9728) q^{69} + ( - 56 \beta_{2} - 1030 \beta_1 - 29002) q^{71} + ( - 150 \beta_{2} - 2340 \beta_1 + 23080) q^{73} + (625 \beta_1 + 1875) q^{75} + ( - 106 \beta_{2} - 860 \beta_1 + 45818) q^{77} + ( - 164 \beta_{2} - 700 \beta_1 + 31272) q^{79} + ( - 162 \beta_{2} + 4916 \beta_1 + 48879) q^{81} + (376 \beta_{2} - 2923 \beta_1 + 10935) q^{83} + ( - 150 \beta_{2} + 1500 \beta_1 + 9200) q^{85} + (184 \beta_{2} - 4094 \beta_1 + 10046) q^{87} + ( - 36 \beta_{2} - 600 \beta_1 + 57774) q^{89} + ( - 106 \beta_{2} + 2110 \beta_1 - 36272) q^{91} + (456 \beta_{2} + 4720 \beta_1 + 104352) q^{93} + (25 \beta_{2} + 1250 \beta_1 + 26175) q^{95} + (430 \beta_{2} - 1420 \beta_1 + 57300) q^{97} + ( - 43 \beta_{2} - 3600 \beta_1 - 115931) q^{99}+O(q^{100})$$ q + (b1 + 3) * q^3 + 25 * q^5 + (-b2 + b1 + 2) * q^7 + (2*b2 + 12*b1 + 151) * q^9 + (-3*b2 - 131) * q^11 + (2*b2 - 20*b1 + 124) * q^13 + (25*b1 + 75) * q^15 + (-6*b2 + 60*b1 + 368) * q^17 + (b2 + 50*b1 + 1047) * q^19 + (4*b2 - 72*b1 + 296) * q^21 + (-7*b2 + 43*b1 - 2054) * q^23 + 625 * q^25 + (20*b2 + 182*b1 + 4534) * q^27 + (-52*b2 + 40*b1 - 138) * q^29 + (42*b2 + 270*b1 - 1196) * q^31 + (6*b2 - 380*b1 - 678) * q^33 + (-25*b2 + 25*b1 + 50) * q^35 + (76*b2 + 200*b1 + 3762) * q^37 + (-44*b2 + 110*b1 - 7138) * q^39 + (22*b2 + 68*b1 + 4120) * q^41 + (-14*b2 + 221*b1 + 8713) * q^43 + (50*b2 + 300*b1 + 3775) * q^45 + (25*b2 + 77*b1 - 12288) * q^47 + (-74*b2 - 316*b1 - 1153) * q^49 + (132*b2 + 410*b1 + 23634) * q^51 + (-114*b2 + 1140*b1 + 6712) * q^53 + (-75*b2 - 3275) * q^55 + (98*b2 + 1580*b1 + 22486) * q^57 + (-185*b2 - 70*b1 + 11765) * q^59 + (-32*b2 - 1984*b1 + 8718) * q^61 + (91*b2 - 263*b1 - 26938) * q^63 + (50*b2 - 500*b1 + 3100) * q^65 + (-82*b2 - 2345*b1 - 2381) * q^67 + (100*b2 - 2248*b1 + 9728) * q^69 + (-56*b2 - 1030*b1 - 29002) * q^71 + (-150*b2 - 2340*b1 + 23080) * q^73 + (625*b1 + 1875) * q^75 + (-106*b2 - 860*b1 + 45818) * q^77 + (-164*b2 - 700*b1 + 31272) * q^79 + (-162*b2 + 4916*b1 + 48879) * q^81 + (376*b2 - 2923*b1 + 10935) * q^83 + (-150*b2 + 1500*b1 + 9200) * q^85 + (184*b2 - 4094*b1 + 10046) * q^87 + (-36*b2 - 600*b1 + 57774) * q^89 + (-106*b2 + 2110*b1 - 36272) * q^91 + (456*b2 + 4720*b1 + 104352) * q^93 + (25*b2 + 1250*b1 + 26175) * q^95 + (430*b2 - 1420*b1 + 57300) * q^97 + (-43*b2 - 3600*b1 - 115931) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 10 q^{3} + 75 q^{5} + 6 q^{7} + 467 q^{9}+O(q^{10})$$ 3 * q + 10 * q^3 + 75 * q^5 + 6 * q^7 + 467 * q^9 $$3 q + 10 q^{3} + 75 q^{5} + 6 q^{7} + 467 q^{9} - 396 q^{11} + 354 q^{13} + 250 q^{15} + 1158 q^{17} + 3192 q^{19} + 820 q^{21} - 6126 q^{23} + 1875 q^{25} + 13804 q^{27} - 426 q^{29} - 3276 q^{31} - 2408 q^{33} + 150 q^{35} + 11562 q^{37} - 21348 q^{39} + 12450 q^{41} + 26346 q^{43} + 11675 q^{45} - 36762 q^{47} - 3849 q^{49} + 71444 q^{51} + 21162 q^{53} - 9900 q^{55} + 69136 q^{57} + 35040 q^{59} + 24138 q^{61} - 80986 q^{63} + 8850 q^{65} - 9570 q^{67} + 27036 q^{69} - 88092 q^{71} + 66750 q^{73} + 6250 q^{75} + 136488 q^{77} + 92952 q^{79} + 151391 q^{81} + 30258 q^{83} + 28950 q^{85} + 26228 q^{87} + 172686 q^{89} - 106812 q^{91} + 318232 q^{93} + 79800 q^{95} + 170910 q^{97} - 351436 q^{99}+O(q^{100})$$ 3 * q + 10 * q^3 + 75 * q^5 + 6 * q^7 + 467 * q^9 - 396 * q^11 + 354 * q^13 + 250 * q^15 + 1158 * q^17 + 3192 * q^19 + 820 * q^21 - 6126 * q^23 + 1875 * q^25 + 13804 * q^27 - 426 * q^29 - 3276 * q^31 - 2408 * q^33 + 150 * q^35 + 11562 * q^37 - 21348 * q^39 + 12450 * q^41 + 26346 * q^43 + 11675 * q^45 - 36762 * q^47 - 3849 * q^49 + 71444 * q^51 + 21162 * q^53 - 9900 * q^55 + 69136 * q^57 + 35040 * q^59 + 24138 * q^61 - 80986 * q^63 + 8850 * q^65 - 9570 * q^67 + 27036 * q^69 - 88092 * q^71 + 66750 * q^73 + 6250 * q^75 + 136488 * q^77 + 92952 * q^79 + 151391 * q^81 + 30258 * q^83 + 28950 * q^85 + 26228 * q^87 + 172686 * q^89 - 106812 * q^91 + 318232 * q^93 + 79800 * q^95 + 170910 * q^97 - 351436 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 36x - 24$$ :

 $$\beta_{1}$$ $$=$$ $$4\nu - 1$$ 4*v - 1 $$\beta_{2}$$ $$=$$ $$8\nu^{2} - 16\nu - 189$$ 8*v^2 - 16*v - 189
 $$\nu$$ $$=$$ $$( \beta _1 + 1 ) / 4$$ (b1 + 1) / 4 $$\nu^{2}$$ $$=$$ $$( \beta_{2} + 4\beta _1 + 193 ) / 8$$ (b2 + 4*b1 + 193) / 8

## 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
 −5.11788 −0.688934 6.80681
0 −18.4715 0 25.0000 0 −121.899 0 98.1968 0
1.2 0 −0.755735 0 25.0000 0 172.424 0 −242.429 0
1.3 0 29.2272 0 25.0000 0 −44.5253 0 611.232 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.y 3
4.b odd 2 1 320.6.a.x 3
8.b even 2 1 160.6.a.f 3
8.d odd 2 1 160.6.a.g yes 3
40.e odd 2 1 800.6.a.n 3
40.f even 2 1 800.6.a.o 3
40.i odd 4 2 800.6.c.k 6
40.k even 4 2 800.6.c.j 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.6.a.f 3 8.b even 2 1
160.6.a.g yes 3 8.d odd 2 1
320.6.a.x 3 4.b odd 2 1
320.6.a.y 3 1.a even 1 1 trivial
800.6.a.n 3 40.e odd 2 1
800.6.a.o 3 40.f even 2 1
800.6.c.j 6 40.k even 4 2
800.6.c.k 6 40.i odd 4 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{3} - 10T_{3}^{2} - 548T_{3} - 408$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(320))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$T^{3} - 10 T^{2} - 548 T - 408$$
$5$ $$(T - 25)^{3}$$
$7$ $$T^{3} - 6 T^{2} - 23268 T - 935848$$
$11$ $$T^{3} + 396 T^{2} + \cdots - 59934400$$
$13$ $$T^{3} - 354 T^{2} + \cdots - 28863000$$
$17$ $$T^{3} - 1158 T^{2} + \cdots + 2743838200$$
$19$ $$T^{3} - 3192 T^{2} + \cdots - 126323200$$
$23$ $$T^{3} + 6126 T^{2} + \cdots + 5283148104$$
$29$ $$T^{3} + 426 T^{2} + \cdots - 159249002312$$
$31$ $$T^{3} + 3276 T^{2} + \cdots - 229217617600$$
$37$ $$T^{3} - 11562 T^{2} + \cdots + 1078137168200$$
$41$ $$T^{3} - 12450 T^{2} + \cdots - 1203781400$$
$43$ $$T^{3} - 26346 T^{2} + \cdots - 352818354968$$
$47$ $$T^{3} + 36762 T^{2} + \cdots + 1628197032152$$
$53$ $$T^{3} - 21162 T^{2} + \cdots + 18581204112200$$
$59$ $$T^{3} - 35040 T^{2} + \cdots - 887454720000$$
$61$ $$T^{3} - 24138 T^{2} + \cdots + 47677792189640$$
$67$ $$T^{3} + 9570 T^{2} + \cdots + 57239443872504$$
$71$ $$T^{3} + 88092 T^{2} + \cdots + 11664981864000$$
$73$ $$T^{3} + \cdots + 165394983407000$$
$79$ $$T^{3} - 92952 T^{2} + \cdots - 1808931596800$$
$83$ $$T^{3} + \cdots - 187637358980920$$
$89$ $$T^{3} + \cdots - 175016035497384$$
$97$ $$T^{3} - 170910 T^{2} + \cdots + 83009979455000$$