Properties

 Label 560.6.a.l Level $560$ Weight $6$ Character orbit 560.a Self dual yes Analytic conductor $89.815$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [560,6,Mod(1,560)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(560, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 6, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("560.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$89.8149390953$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{65})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 16$$ x^2 - x - 16 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 35) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{65})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - 3 \beta q^{3} - 25 q^{5} + 49 q^{7} + (9 \beta - 99) q^{9} +O(q^{10})$$ q - 3*b * q^3 - 25 * q^5 + 49 * q^7 + (9*b - 99) * q^9 $$q - 3 \beta q^{3} - 25 q^{5} + 49 q^{7} + (9 \beta - 99) q^{9} + (97 \beta + 252) q^{11} + ( - 53 \beta - 262) q^{13} + 75 \beta q^{15} + ( - 251 \beta + 146) q^{17} + ( - 86 \beta - 272) q^{19} - 147 \beta q^{21} + ( - 902 \beta + 672) q^{23} + 625 q^{25} + (999 \beta - 432) q^{27} + (945 \beta + 2470) q^{29} + (924 \beta - 264) q^{31} + ( - 1047 \beta - 4656) q^{33} - 1225 q^{35} + (1260 \beta - 5082) q^{37} + (945 \beta + 2544) q^{39} + (3818 \beta - 1022) q^{41} + (922 \beta + 13100) q^{43} + ( - 225 \beta + 2475) q^{45} + ( - 1575 \beta + 11432) q^{47} + 2401 q^{49} + (315 \beta + 12048) q^{51} + (454 \beta - 28018) q^{53} + ( - 2425 \beta - 6300) q^{55} + (1074 \beta + 4128) q^{57} + (5184 \beta - 32392) q^{59} + ( - 5706 \beta - 23070) q^{61} + (441 \beta - 4851) q^{63} + (1325 \beta + 6550) q^{65} + ( - 4568 \beta + 24956) q^{67} + (690 \beta + 43296) q^{69} + (5304 \beta - 43024) q^{71} + (4192 \beta - 8862) q^{73} - 1875 \beta q^{75} + (4753 \beta + 12348) q^{77} + (17635 \beta + 17080) q^{79} + ( - 3888 \beta - 23895) q^{81} + ( - 3924 \beta - 52952) q^{83} + (6275 \beta - 3650) q^{85} + ( - 10245 \beta - 45360) q^{87} + (5722 \beta - 21686) q^{89} + ( - 2597 \beta - 12838) q^{91} + ( - 1980 \beta - 44352) q^{93} + (2150 \beta + 6800) q^{95} + ( - 13943 \beta - 41198) q^{97} + ( - 6462 \beta - 10980) q^{99} +O(q^{100})$$ q - 3*b * q^3 - 25 * q^5 + 49 * q^7 + (9*b - 99) * q^9 + (97*b + 252) * q^11 + (-53*b - 262) * q^13 + 75*b * q^15 + (-251*b + 146) * q^17 + (-86*b - 272) * q^19 - 147*b * q^21 + (-902*b + 672) * q^23 + 625 * q^25 + (999*b - 432) * q^27 + (945*b + 2470) * q^29 + (924*b - 264) * q^31 + (-1047*b - 4656) * q^33 - 1225 * q^35 + (1260*b - 5082) * q^37 + (945*b + 2544) * q^39 + (3818*b - 1022) * q^41 + (922*b + 13100) * q^43 + (-225*b + 2475) * q^45 + (-1575*b + 11432) * q^47 + 2401 * q^49 + (315*b + 12048) * q^51 + (454*b - 28018) * q^53 + (-2425*b - 6300) * q^55 + (1074*b + 4128) * q^57 + (5184*b - 32392) * q^59 + (-5706*b - 23070) * q^61 + (441*b - 4851) * q^63 + (1325*b + 6550) * q^65 + (-4568*b + 24956) * q^67 + (690*b + 43296) * q^69 + (5304*b - 43024) * q^71 + (4192*b - 8862) * q^73 - 1875*b * q^75 + (4753*b + 12348) * q^77 + (17635*b + 17080) * q^79 + (-3888*b - 23895) * q^81 + (-3924*b - 52952) * q^83 + (6275*b - 3650) * q^85 + (-10245*b - 45360) * q^87 + (5722*b - 21686) * q^89 + (-2597*b - 12838) * q^91 + (-1980*b - 44352) * q^93 + (2150*b + 6800) * q^95 + (-13943*b - 41198) * q^97 + (-6462*b - 10980) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 3 q^{3} - 50 q^{5} + 98 q^{7} - 189 q^{9}+O(q^{10})$$ 2 * q - 3 * q^3 - 50 * q^5 + 98 * q^7 - 189 * q^9 $$2 q - 3 q^{3} - 50 q^{5} + 98 q^{7} - 189 q^{9} + 601 q^{11} - 577 q^{13} + 75 q^{15} + 41 q^{17} - 630 q^{19} - 147 q^{21} + 442 q^{23} + 1250 q^{25} + 135 q^{27} + 5885 q^{29} + 396 q^{31} - 10359 q^{33} - 2450 q^{35} - 8904 q^{37} + 6033 q^{39} + 1774 q^{41} + 27122 q^{43} + 4725 q^{45} + 21289 q^{47} + 4802 q^{49} + 24411 q^{51} - 55582 q^{53} - 15025 q^{55} + 9330 q^{57} - 59600 q^{59} - 51846 q^{61} - 9261 q^{63} + 14425 q^{65} + 45344 q^{67} + 87282 q^{69} - 80744 q^{71} - 13532 q^{73} - 1875 q^{75} + 29449 q^{77} + 51795 q^{79} - 51678 q^{81} - 109828 q^{83} - 1025 q^{85} - 100965 q^{87} - 37650 q^{89} - 28273 q^{91} - 90684 q^{93} + 15750 q^{95} - 96339 q^{97} - 28422 q^{99}+O(q^{100})$$ 2 * q - 3 * q^3 - 50 * q^5 + 98 * q^7 - 189 * q^9 + 601 * q^11 - 577 * q^13 + 75 * q^15 + 41 * q^17 - 630 * q^19 - 147 * q^21 + 442 * q^23 + 1250 * q^25 + 135 * q^27 + 5885 * q^29 + 396 * q^31 - 10359 * q^33 - 2450 * q^35 - 8904 * q^37 + 6033 * q^39 + 1774 * q^41 + 27122 * q^43 + 4725 * q^45 + 21289 * q^47 + 4802 * q^49 + 24411 * q^51 - 55582 * q^53 - 15025 * q^55 + 9330 * q^57 - 59600 * q^59 - 51846 * q^61 - 9261 * q^63 + 14425 * q^65 + 45344 * q^67 + 87282 * q^69 - 80744 * q^71 - 13532 * q^73 - 1875 * q^75 + 29449 * q^77 + 51795 * q^79 - 51678 * q^81 - 109828 * q^83 - 1025 * q^85 - 100965 * q^87 - 37650 * q^89 - 28273 * q^91 - 90684 * q^93 + 15750 * q^95 - 96339 * q^97 - 28422 * 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 4.53113 −3.53113
0 −13.5934 0 −25.0000 0 49.0000 0 −58.2198 0
1.2 0 10.5934 0 −25.0000 0 49.0000 0 −130.780 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.l 2
4.b odd 2 1 35.6.a.b 2
12.b even 2 1 315.6.a.c 2
20.d odd 2 1 175.6.a.d 2
20.e even 4 2 175.6.b.d 4
28.d even 2 1 245.6.a.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.6.a.b 2 4.b odd 2 1
175.6.a.d 2 20.d odd 2 1
175.6.b.d 4 20.e even 4 2
245.6.a.c 2 28.d even 2 1
315.6.a.c 2 12.b even 2 1
560.6.a.l 2 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} + 3T_{3} - 144$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(560))$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 3T - 144$$
$5$ $$(T + 25)^{2}$$
$7$ $$(T - 49)^{2}$$
$11$ $$T^{2} - 601T - 62596$$
$13$ $$T^{2} + 577T + 37586$$
$17$ $$T^{2} - 41T - 1023346$$
$19$ $$T^{2} + 630T - 20960$$
$23$ $$T^{2} - 442 T - 13172224$$
$29$ $$T^{2} - 5885 T - 5853350$$
$31$ $$T^{2} - 396 T - 13834656$$
$37$ $$T^{2} + 8904 T - 5978196$$
$41$ $$T^{2} - 1774 T - 236091496$$
$43$ $$T^{2} - 27122 T + 170086856$$
$47$ $$T^{2} - 21289 T + 72995224$$
$53$ $$T^{2} + 55582 T + 768990296$$
$59$ $$T^{2} + 59600 T + 451339840$$
$61$ $$T^{2} + 51846 T + 142927344$$
$67$ $$T^{2} - 45344 T + 174936944$$
$71$ $$T^{2} + 80744 T + 1172746624$$
$73$ $$T^{2} + 13532 T - 239780284$$
$79$ $$T^{2} - 51795 T - 4382959400$$
$83$ $$T^{2} + 109828 T + 2765333536$$
$89$ $$T^{2} + 37650 T - 177665240$$
$97$ $$T^{2} + 96339 T - 838817066$$