Properties

 Label 320.6.a.v Level $320$ Weight $6$ Character orbit 320.a Self dual yes Analytic conductor $51.323$ Analytic rank $1$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{70})$$ Defining polynomial: $$x^{2} - 70$$ x^2 - 70 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 $$\beta = 2\sqrt{70}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta + 4) q^{3} - 25 q^{5} + (\beta - 52) q^{7} + (8 \beta + 53) q^{9}+O(q^{10})$$ q + (b + 4) * q^3 - 25 * q^5 + (b - 52) * q^7 + (8*b + 53) * q^9 $$q + (\beta + 4) q^{3} - 25 q^{5} + (\beta - 52) q^{7} + (8 \beta + 53) q^{9} + ( - 10 \beta + 160) q^{11} + ( - 40 \beta + 50) q^{13} + ( - 25 \beta - 100) q^{15} + (40 \beta + 290) q^{17} + ( - 40 \beta + 360) q^{19} + ( - 48 \beta + 72) q^{21} + ( - 97 \beta - 844) q^{23} + 625 q^{25} + ( - 158 \beta + 1480) q^{27} + (80 \beta - 54) q^{29} + ( - 130 \beta - 4920) q^{31} + (120 \beta - 2160) q^{33} + ( - 25 \beta + 1300) q^{35} + (560 \beta - 3270) q^{37} + ( - 110 \beta - 11000) q^{39} + ( - 808 \beta - 5310) q^{41} + ( - 579 \beta + 12836) q^{43} + ( - 200 \beta - 1325) q^{45} + ( - 383 \beta - 14148) q^{47} + ( - 104 \beta - 13823) q^{49} + (450 \beta + 12360) q^{51} + ( - 1080 \beta - 15670) q^{53} + (250 \beta - 4000) q^{55} + (200 \beta - 9760) q^{57} + ( - 20 \beta + 15400) q^{59} + (1184 \beta - 12270) q^{61} + ( - 363 \beta - 516) q^{63} + (1000 \beta - 1250) q^{65} + (495 \beta + 17292) q^{67} + ( - 1232 \beta - 30536) q^{69} + (2990 \beta + 6200) q^{71} + (3720 \beta - 3590) q^{73} + (625 \beta + 2500) q^{75} + (680 \beta - 11120) q^{77} + (220 \beta - 35920) q^{79} + ( - 1096 \beta - 51199) q^{81} + (2817 \beta - 15964) q^{83} + ( - 1000 \beta - 7250) q^{85} + (266 \beta + 22184) q^{87} + ( - 3280 \beta - 20374) q^{89} + (2130 \beta - 13800) q^{91} + ( - 5440 \beta - 56080) q^{93} + (1000 \beta - 9000) q^{95} + ( - 4840 \beta - 95070) q^{97} + (750 \beta - 13920) q^{99}+O(q^{100})$$ q + (b + 4) * q^3 - 25 * q^5 + (b - 52) * q^7 + (8*b + 53) * q^9 + (-10*b + 160) * q^11 + (-40*b + 50) * q^13 + (-25*b - 100) * q^15 + (40*b + 290) * q^17 + (-40*b + 360) * q^19 + (-48*b + 72) * q^21 + (-97*b - 844) * q^23 + 625 * q^25 + (-158*b + 1480) * q^27 + (80*b - 54) * q^29 + (-130*b - 4920) * q^31 + (120*b - 2160) * q^33 + (-25*b + 1300) * q^35 + (560*b - 3270) * q^37 + (-110*b - 11000) * q^39 + (-808*b - 5310) * q^41 + (-579*b + 12836) * q^43 + (-200*b - 1325) * q^45 + (-383*b - 14148) * q^47 + (-104*b - 13823) * q^49 + (450*b + 12360) * q^51 + (-1080*b - 15670) * q^53 + (250*b - 4000) * q^55 + (200*b - 9760) * q^57 + (-20*b + 15400) * q^59 + (1184*b - 12270) * q^61 + (-363*b - 516) * q^63 + (1000*b - 1250) * q^65 + (495*b + 17292) * q^67 + (-1232*b - 30536) * q^69 + (2990*b + 6200) * q^71 + (3720*b - 3590) * q^73 + (625*b + 2500) * q^75 + (680*b - 11120) * q^77 + (220*b - 35920) * q^79 + (-1096*b - 51199) * q^81 + (2817*b - 15964) * q^83 + (-1000*b - 7250) * q^85 + (266*b + 22184) * q^87 + (-3280*b - 20374) * q^89 + (2130*b - 13800) * q^91 + (-5440*b - 56080) * q^93 + (1000*b - 9000) * q^95 + (-4840*b - 95070) * q^97 + (750*b - 13920) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 8 q^{3} - 50 q^{5} - 104 q^{7} + 106 q^{9}+O(q^{10})$$ 2 * q + 8 * q^3 - 50 * q^5 - 104 * q^7 + 106 * q^9 $$2 q + 8 q^{3} - 50 q^{5} - 104 q^{7} + 106 q^{9} + 320 q^{11} + 100 q^{13} - 200 q^{15} + 580 q^{17} + 720 q^{19} + 144 q^{21} - 1688 q^{23} + 1250 q^{25} + 2960 q^{27} - 108 q^{29} - 9840 q^{31} - 4320 q^{33} + 2600 q^{35} - 6540 q^{37} - 22000 q^{39} - 10620 q^{41} + 25672 q^{43} - 2650 q^{45} - 28296 q^{47} - 27646 q^{49} + 24720 q^{51} - 31340 q^{53} - 8000 q^{55} - 19520 q^{57} + 30800 q^{59} - 24540 q^{61} - 1032 q^{63} - 2500 q^{65} + 34584 q^{67} - 61072 q^{69} + 12400 q^{71} - 7180 q^{73} + 5000 q^{75} - 22240 q^{77} - 71840 q^{79} - 102398 q^{81} - 31928 q^{83} - 14500 q^{85} + 44368 q^{87} - 40748 q^{89} - 27600 q^{91} - 112160 q^{93} - 18000 q^{95} - 190140 q^{97} - 27840 q^{99}+O(q^{100})$$ 2 * q + 8 * q^3 - 50 * q^5 - 104 * q^7 + 106 * q^9 + 320 * q^11 + 100 * q^13 - 200 * q^15 + 580 * q^17 + 720 * q^19 + 144 * q^21 - 1688 * q^23 + 1250 * q^25 + 2960 * q^27 - 108 * q^29 - 9840 * q^31 - 4320 * q^33 + 2600 * q^35 - 6540 * q^37 - 22000 * q^39 - 10620 * q^41 + 25672 * q^43 - 2650 * q^45 - 28296 * q^47 - 27646 * q^49 + 24720 * q^51 - 31340 * q^53 - 8000 * q^55 - 19520 * q^57 + 30800 * q^59 - 24540 * q^61 - 1032 * q^63 - 2500 * q^65 + 34584 * q^67 - 61072 * q^69 + 12400 * q^71 - 7180 * q^73 + 5000 * q^75 - 22240 * q^77 - 71840 * q^79 - 102398 * q^81 - 31928 * q^83 - 14500 * q^85 + 44368 * q^87 - 40748 * q^89 - 27600 * q^91 - 112160 * q^93 - 18000 * q^95 - 190140 * q^97 - 27840 * 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
 −8.36660 8.36660
0 −12.7332 0 −25.0000 0 −68.7332 0 −80.8656 0
1.2 0 20.7332 0 −25.0000 0 −35.2668 0 186.866 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.v 2
4.b odd 2 1 320.6.a.r 2
8.b even 2 1 160.6.a.a 2
8.d odd 2 1 160.6.a.e yes 2
40.e odd 2 1 800.6.a.g 2
40.f even 2 1 800.6.a.l 2
40.i odd 4 2 800.6.c.f 4
40.k even 4 2 800.6.c.g 4

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} - 8T_{3} - 264$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(320))$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - 8T - 264$$
$5$ $$(T + 25)^{2}$$
$7$ $$T^{2} + 104T + 2424$$
$11$ $$T^{2} - 320T - 2400$$
$13$ $$T^{2} - 100T - 445500$$
$17$ $$T^{2} - 580T - 363900$$
$19$ $$T^{2} - 720T - 318400$$
$23$ $$T^{2} + 1688 T - 1922184$$
$29$ $$T^{2} + 108 T - 1789084$$
$31$ $$T^{2} + 9840 T + 19474400$$
$37$ $$T^{2} + 6540 T - 77115100$$
$41$ $$T^{2} + 10620 T - 154605820$$
$43$ $$T^{2} - 25672 T + 70895416$$
$47$ $$T^{2} + 28296 T + 159092984$$
$53$ $$T^{2} + 31340 T - 81043100$$
$59$ $$T^{2} - 30800 T + 237048000$$
$61$ $$T^{2} + 24540 T - 241966780$$
$67$ $$T^{2} - 34584 T + 230406264$$
$71$ $$T^{2} - 12400 T - 2464788000$$
$73$ $$T^{2} + 7180 T - 3861863900$$
$79$ $$T^{2} + 71840 T + 1276694400$$
$83$ $$T^{2} + 31928 T - 1967087624$$
$89$ $$T^{2} + 40748 T - 2597252124$$
$97$ $$T^{2} + 190140 T + 2479136900$$