Properties

 Label 560.6.a.s Level $560$ Weight $6$ Character orbit 560.a Self dual yes Analytic conductor $89.815$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$560 = 2^{4} \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 560.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$89.8149390953$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: $$\mathbb{Q}[x]/(x^{3} - \cdots)$$ Defining polynomial: $$x^{3} - 463 x - 1890$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 280) 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.

 $$f(q)$$ $$=$$ $$q + ( -2 + \beta_{1} ) q^{3} -25 q^{5} -49 q^{7} + ( 70 + 2 \beta_{1} + \beta_{2} ) q^{9} +O(q^{10})$$ $$q + ( -2 + \beta_{1} ) q^{3} -25 q^{5} -49 q^{7} + ( 70 + 2 \beta_{1} + \beta_{2} ) q^{9} + ( 193 + 4 \beta_{1} + \beta_{2} ) q^{11} + ( 26 + 23 \beta_{1} - 2 \beta_{2} ) q^{13} + ( 50 - 25 \beta_{1} ) q^{15} + ( -416 - 13 \beta_{1} - 4 \beta_{2} ) q^{17} + ( -315 - 105 \beta_{1} - \beta_{2} ) q^{19} + ( 98 - 49 \beta_{1} ) q^{21} + ( -371 - 41 \beta_{1} - 17 \beta_{2} ) q^{23} + 625 q^{25} + ( 1000 - 47 \beta_{1} - 6 \beta_{2} ) q^{27} + ( 625 + 48 \beta_{1} + 7 \beta_{2} ) q^{29} + ( 2215 - 177 \beta_{1} + 25 \beta_{2} ) q^{31} + ( 886 + 327 \beta_{1} - 4 \beta_{2} ) q^{33} + 1225 q^{35} + ( -1351 + 115 \beta_{1} + 5 \beta_{2} ) q^{37} + ( 6983 - 118 \beta_{1} + 39 \beta_{2} ) q^{39} + ( -3195 - 319 \beta_{1} + 17 \beta_{2} ) q^{41} + ( 4099 + \beta_{1} - 43 \beta_{2} ) q^{43} + ( -1750 - 50 \beta_{1} - 25 \beta_{2} ) q^{45} + ( 13688 - 93 \beta_{1} + 38 \beta_{2} ) q^{47} + 2401 q^{49} + ( -3329 - 940 \beta_{1} + 19 \beta_{2} ) q^{51} + ( -12516 - 1060 \beta_{1} + 62 \beta_{2} ) q^{53} + ( -4825 - 100 \beta_{1} - 25 \beta_{2} ) q^{55} + ( -31851 - 853 \beta_{1} - 97 \beta_{2} ) q^{57} + ( 12564 + 644 \beta_{1} + 28 \beta_{2} ) q^{59} + ( -2769 - 633 \beta_{1} + 79 \beta_{2} ) q^{61} + ( -3430 - 98 \beta_{1} - 49 \beta_{2} ) q^{63} + ( -650 - 575 \beta_{1} + 50 \beta_{2} ) q^{65} + ( -23136 - 996 \beta_{1} - 68 \beta_{2} ) q^{67} + ( -12539 - 2541 \beta_{1} + 95 \beta_{2} ) q^{69} + ( 11352 - 152 \beta_{1} + 40 \beta_{2} ) q^{71} + ( -15144 + 926 \beta_{1} - 118 \beta_{2} ) q^{73} + ( -1250 + 625 \beta_{1} ) q^{75} + ( -9457 - 196 \beta_{1} - 49 \beta_{2} ) q^{77} + ( -537 + 2168 \beta_{1} - 229 \beta_{2} ) q^{79} + ( -33749 - 382 \beta_{1} - 242 \beta_{2} ) q^{81} + ( 3414 - 2174 \beta_{1} + 114 \beta_{2} ) q^{83} + ( 10400 + 325 \beta_{1} + 100 \beta_{2} ) q^{85} + ( 13834 + 1643 \beta_{1} - 8 \beta_{2} ) q^{87} + ( -74331 - 1743 \beta_{1} - 183 \beta_{2} ) q^{89} + ( -1274 - 1127 \beta_{1} + 98 \beta_{2} ) q^{91} + ( -58223 + 4457 \beta_{1} - 377 \beta_{2} ) q^{93} + ( 7875 + 2625 \beta_{1} + 25 \beta_{2} ) q^{95} + ( -53080 - 2889 \beta_{1} + 236 \beta_{2} ) q^{97} + ( 52228 + 750 \beta_{1} + 116 \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 6 q^{3} - 75 q^{5} - 147 q^{7} + 209 q^{9} + O(q^{10})$$ $$3 q - 6 q^{3} - 75 q^{5} - 147 q^{7} + 209 q^{9} + 578 q^{11} + 80 q^{13} + 150 q^{15} - 1244 q^{17} - 944 q^{19} + 294 q^{21} - 1096 q^{23} + 1875 q^{25} + 3006 q^{27} + 1868 q^{29} + 6620 q^{31} + 2662 q^{33} + 3675 q^{35} - 4058 q^{37} + 20910 q^{39} - 9602 q^{41} + 12340 q^{43} - 5225 q^{45} + 41026 q^{47} + 7203 q^{49} - 10006 q^{51} - 37610 q^{53} - 14450 q^{55} - 95456 q^{57} + 37664 q^{59} - 8386 q^{61} - 10241 q^{63} - 2000 q^{65} - 69340 q^{67} - 37712 q^{69} + 34016 q^{71} - 45314 q^{73} - 3750 q^{75} - 28322 q^{77} - 1382 q^{79} - 101005 q^{81} + 10128 q^{83} + 31100 q^{85} + 41510 q^{87} - 222810 q^{89} - 3920 q^{91} - 174292 q^{93} + 23600 q^{95} - 159476 q^{97} + 156568 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - 463 x - 1890$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 6 \nu - 309$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 6 \beta_{1} + 309$$

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
 −19.0769 −4.24759 23.3245
0 −21.0769 0 −25.0000 0 −49.0000 0 201.235 0
1.2 0 −6.24759 0 −25.0000 0 −49.0000 0 −203.968 0
1.3 0 21.3245 0 −25.0000 0 −49.0000 0 211.733 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 560.6.a.s 3
4.b odd 2 1 280.6.a.e 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.6.a.e 3 4.b odd 2 1
560.6.a.s 3 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{3} + 6 T_{3}^{2} - 451 T_{3} - 2808$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(560))$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$-2808 - 451 T + 6 T^{2} + T^{3}$$
$5$ $$( 25 + T )^{3}$$
$7$ $$( 49 + T )^{3}$$
$11$ $$9760932 + 49393 T - 578 T^{2} + T^{3}$$
$13$ $$128496494 - 453967 T - 80 T^{2} + T^{3}$$
$17$ $$-651843290 - 433999 T + 1244 T^{2} + T^{3}$$
$19$ $$1721529888 - 4873604 T + 944 T^{2} + T^{3}$$
$23$ $$-31363349376 - 16093508 T + 1096 T^{2} + T^{3}$$
$29$ $$3100258778 - 2593027 T - 1868 T^{2} + T^{3}$$
$31$ $$16721033600 - 33207952 T - 6620 T^{2} + T^{3}$$
$37$ $$-15436371144 - 2052212 T + 4058 T^{2} + T^{3}$$
$41$ $$-330620399712 - 31400472 T + 9602 T^{2} + T^{3}$$
$43$ $$1675236784 - 49275940 T - 12340 T^{2} + T^{3}$$
$47$ $$-1323866448632 + 479315957 T - 41026 T^{2} + T^{3}$$
$53$ $$-13966928161632 - 249207744 T + 37610 T^{2} + T^{3}$$
$59$ $$-363808567296 + 236362752 T - 37664 T^{2} + T^{3}$$
$61$ $$-4786876164960 - 494041688 T + 8386 T^{2} + T^{3}$$
$67$ $$705690174528 + 885475696 T + 69340 T^{2} + T^{3}$$
$71$ $$-336586973184 + 289125120 T - 34016 T^{2} + T^{3}$$
$73$ $$-3081736975464 - 453450708 T + 45314 T^{2} + T^{3}$$
$79$ $$116381435931432 - 4956254607 T + 1382 T^{2} + T^{3}$$
$83$ $$-41224716201984 - 2828960272 T - 10128 T^{2} + T^{3}$$
$89$ $$175059765854496 + 13293219528 T + 222810 T^{2} + T^{3}$$
$97$ $$-426037755101066 + 1677515609 T + 159476 T^{2} + T^{3}$$