# Properties

 Label 980.6.a.h Level $980$ Weight $6$ Character orbit 980.a Self dual yes Analytic conductor $157.176$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$980 = 2^{2} \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 980.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$157.176143417$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: $$\mathbb{Q}[x]/(x^{3} - \cdots)$$ Defining polynomial: $$x^{3} - 499x - 210$$ x^3 - 499*x - 210 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 140) 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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_1 - 2) q^{3} - 25 q^{5} + (\beta_{2} + 5 \beta_1 + 94) q^{9}+O(q^{10})$$ q + (-b1 - 2) * q^3 - 25 * q^5 + (b2 + 5*b1 + 94) * q^9 $$q + ( - \beta_1 - 2) q^{3} - 25 q^{5} + (\beta_{2} + 5 \beta_1 + 94) q^{9} + ( - \beta_{2} + 11 \beta_1 - 5) q^{11} + (2 \beta_{2} - 13 \beta_1 + 2) q^{13} + (25 \beta_1 + 50) q^{15} + (2 \beta_{2} - 31 \beta_1 - 14) q^{17} + ( - 9 \beta_{2} - 6 \beta_1 - 779) q^{19} + ( - \beta_{2} - 70 \beta_1 + 1225) q^{23} + 625 q^{25} + ( - 6 \beta_{2} - 31 \beta_1 - 1244) q^{27} + ( - 15 \beta_{2} - 99 \beta_1 + 1359) q^{29} + (17 \beta_{2} + 248 \beta_1 - 1957) q^{31} + ( - 10 \beta_{2} + 137 \beta_1 - 3776) q^{33} + (\beta_{2} + 64 \beta_1 + 3793) q^{37} + (11 \beta_{2} - 293 \beta_1 + 4571) q^{39} + ( - 31 \beta_{2} - 142 \beta_1 - 3827) q^{41} + ( - 31 \beta_{2} - 322 \beta_1 + 6171) q^{43} + ( - 25 \beta_{2} - 125 \beta_1 - 2350) q^{45} + (16 \beta_{2} - 113 \beta_1 - 7246) q^{47} + (29 \beta_{2} - 223 \beta_1 + 10597) q^{51} + ( - 6 \beta_{2} + 594 \beta_1 + 2496) q^{53} + (25 \beta_{2} - 275 \beta_1 + 125) q^{55} + (15 \beta_{2} + 2282 \beta_1 + 2449) q^{57} + (64 \beta_{2} + 1840 \beta_1 - 4108) q^{59} + (71 \beta_{2} + 110 \beta_1 - 9037) q^{61} + ( - 50 \beta_{2} + 325 \beta_1 - 50) q^{65} + (8 \beta_{2} + 2228 \beta_1 - 2556) q^{67} + (71 \beta_{2} - 850 \beta_1 + 20737) q^{69} + (184 \beta_{2} + 136 \beta_1 + 27392) q^{71} + ( - 250 \beta_{2} + 1772 \beta_1 - 160) q^{73} + ( - 625 \beta_1 - 1250) q^{75} + (95 \beta_{2} - 3919 \beta_1 - 5277) q^{79} + ( - 206 \beta_{2} + 1112 \beta_1 - 10769) q^{81} + (138 \beta_{2} + 960 \beta_1 + 28746) q^{83} + ( - 50 \beta_{2} + 775 \beta_1 + 350) q^{85} + (114 \beta_{2} + 1413 \beta_1 + 28404) q^{87} + ( - 127 \beta_{2} + 2870 \beta_1 + 31861) q^{89} + ( - 265 \beta_{2} - 1592 \beta_1 - 76579) q^{93} + (225 \beta_{2} + 150 \beta_1 + 19475) q^{95} + ( - 322 \beta_{2} - 931 \beta_1 + 58622) q^{97} + (116 \beta_{2} + 2342 \beta_1 - 38084) q^{99}+O(q^{100})$$ q + (-b1 - 2) * q^3 - 25 * q^5 + (b2 + 5*b1 + 94) * q^9 + (-b2 + 11*b1 - 5) * q^11 + (2*b2 - 13*b1 + 2) * q^13 + (25*b1 + 50) * q^15 + (2*b2 - 31*b1 - 14) * q^17 + (-9*b2 - 6*b1 - 779) * q^19 + (-b2 - 70*b1 + 1225) * q^23 + 625 * q^25 + (-6*b2 - 31*b1 - 1244) * q^27 + (-15*b2 - 99*b1 + 1359) * q^29 + (17*b2 + 248*b1 - 1957) * q^31 + (-10*b2 + 137*b1 - 3776) * q^33 + (b2 + 64*b1 + 3793) * q^37 + (11*b2 - 293*b1 + 4571) * q^39 + (-31*b2 - 142*b1 - 3827) * q^41 + (-31*b2 - 322*b1 + 6171) * q^43 + (-25*b2 - 125*b1 - 2350) * q^45 + (16*b2 - 113*b1 - 7246) * q^47 + (29*b2 - 223*b1 + 10597) * q^51 + (-6*b2 + 594*b1 + 2496) * q^53 + (25*b2 - 275*b1 + 125) * q^55 + (15*b2 + 2282*b1 + 2449) * q^57 + (64*b2 + 1840*b1 - 4108) * q^59 + (71*b2 + 110*b1 - 9037) * q^61 + (-50*b2 + 325*b1 - 50) * q^65 + (8*b2 + 2228*b1 - 2556) * q^67 + (71*b2 - 850*b1 + 20737) * q^69 + (184*b2 + 136*b1 + 27392) * q^71 + (-250*b2 + 1772*b1 - 160) * q^73 + (-625*b1 - 1250) * q^75 + (95*b2 - 3919*b1 - 5277) * q^79 + (-206*b2 + 1112*b1 - 10769) * q^81 + (138*b2 + 960*b1 + 28746) * q^83 + (-50*b2 + 775*b1 + 350) * q^85 + (114*b2 + 1413*b1 + 28404) * q^87 + (-127*b2 + 2870*b1 + 31861) * q^89 + (-265*b2 - 1592*b1 - 76579) * q^93 + (225*b2 + 150*b1 + 19475) * q^95 + (-322*b2 - 931*b1 + 58622) * q^97 + (116*b2 + 2342*b1 - 38084) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 6 q^{3} - 75 q^{5} + 281 q^{9}+O(q^{10})$$ 3 * q - 6 * q^3 - 75 * q^5 + 281 * q^9 $$3 q - 6 q^{3} - 75 q^{5} + 281 q^{9} - 14 q^{11} + 4 q^{13} + 150 q^{15} - 44 q^{17} - 2328 q^{19} + 3676 q^{23} + 1875 q^{25} - 3726 q^{27} + 4092 q^{29} - 5888 q^{31} - 11318 q^{33} + 11378 q^{37} + 13702 q^{39} - 11450 q^{41} + 18544 q^{43} - 7025 q^{45} - 21754 q^{47} + 31762 q^{51} + 7494 q^{53} + 350 q^{55} + 7332 q^{57} - 12388 q^{59} - 27182 q^{61} - 100 q^{65} - 7676 q^{67} + 62140 q^{69} + 81992 q^{71} - 230 q^{73} - 3750 q^{75} - 15926 q^{79} - 32101 q^{81} + 86100 q^{83} + 1100 q^{85} + 85098 q^{87} + 95710 q^{89} - 229472 q^{93} + 58200 q^{95} + 176188 q^{97} - 114368 q^{99}+O(q^{100})$$ 3 * q - 6 * q^3 - 75 * q^5 + 281 * q^9 - 14 * q^11 + 4 * q^13 + 150 * q^15 - 44 * q^17 - 2328 * q^19 + 3676 * q^23 + 1875 * q^25 - 3726 * q^27 + 4092 * q^29 - 5888 * q^31 - 11318 * q^33 + 11378 * q^37 + 13702 * q^39 - 11450 * q^41 + 18544 * q^43 - 7025 * q^45 - 21754 * q^47 + 31762 * q^51 + 7494 * q^53 + 350 * q^55 + 7332 * q^57 - 12388 * q^59 - 27182 * q^61 - 100 * q^65 - 7676 * q^67 + 62140 * q^69 + 81992 * q^71 - 230 * q^73 - 3750 * q^75 - 15926 * q^79 - 32101 * q^81 + 86100 * q^83 + 1100 * q^85 + 85098 * q^87 + 95710 * q^89 - 229472 * q^93 + 58200 * q^95 + 176188 * q^97 - 114368 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - 499x - 210$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 333$$ v^2 - v - 333
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta _1 + 333$$ b2 + b1 + 333

## 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
 22.5458 −0.420991 −22.1248
0 −24.5458 0 −25.0000 0 0 0 359.498 0
1.2 0 −1.57901 0 −25.0000 0 0 0 −240.507 0
1.3 0 20.1248 0 −25.0000 0 0 0 162.009 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
$$2$$ $$-1$$
$$5$$ $$1$$
$$7$$ $$-1$$

## 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 980.6.a.h 3
7.b odd 2 1 140.6.a.d 3
28.d even 2 1 560.6.a.t 3
35.c odd 2 1 700.6.a.i 3
35.f even 4 2 700.6.e.g 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.6.a.d 3 7.b odd 2 1
560.6.a.t 3 28.d even 2 1
700.6.a.i 3 35.c odd 2 1
700.6.e.g 6 35.f even 4 2
980.6.a.h 3 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{3} + 6T_{3}^{2} - 487T_{3} - 780$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(980))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$T^{3} + 6 T^{2} - 487 T - 780$$
$5$ $$(T + 25)^{3}$$
$7$ $$T^{3}$$
$11$ $$T^{3} + 14 T^{2} - 147231 T + 12436740$$
$13$ $$T^{3} - 4 T^{2} - 425371 T + 6140734$$
$17$ $$T^{3} + 44 T^{2} + \cdots - 279119070$$
$19$ $$T^{3} + 2328 T^{2} + \cdots - 11429521264$$
$23$ $$T^{3} - 3676 T^{2} + \cdots + 2083660704$$
$29$ $$T^{3} - 4092 T^{2} + \cdots + 17580868722$$
$31$ $$T^{3} + 5888 T^{2} + \cdots - 211802104832$$
$37$ $$T^{3} - 11378 T^{2} + \cdots - 47286923800$$
$41$ $$T^{3} + 11450 T^{2} + \cdots - 478579953600$$
$43$ $$T^{3} - 18544 T^{2} + \cdots + 750561676176$$
$47$ $$T^{3} + 21754 T^{2} + \cdots + 173657376144$$
$53$ $$T^{3} - 7494 T^{2} + \cdots + 743911257600$$
$59$ $$T^{3} + 12388 T^{2} + \cdots - 41176040028480$$
$61$ $$T^{3} + 27182 T^{2} + \cdots + 169955356480$$
$67$ $$T^{3} + 7676 T^{2} + \cdots - 15170685707520$$
$71$ $$T^{3} + \cdots + 113425504819200$$
$73$ $$T^{3} + 230 T^{2} + \cdots + 11043630664360$$
$79$ $$T^{3} + \cdots - 274092525845520$$
$83$ $$T^{3} - 86100 T^{2} + \cdots + 40289422939200$$
$89$ $$T^{3} + \cdots + 305457269205600$$
$97$ $$T^{3} - 176188 T^{2} + \cdots + 41652594334882$$