# Properties

 Label 560.2.q.k Level $560$ Weight $2$ Character orbit 560.q Analytic conductor $4.472$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [560,2,Mod(81,560)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(560, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 0, 4]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("560.81");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$560 = 2^{4} \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 560.q (of order $$3$$, degree $$2$$, not minimal)

## 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: no Analytic conductor: $$4.47162251319$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{2}, \sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 2x^{2} + 4$$ x^4 + 2*x^2 + 4 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 35) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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 a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_{2} + \beta_1 + 1) q^{3} + \beta_{2} q^{5} + ( - 2 \beta_{3} - \beta_{2} - \beta_1 - 1) q^{7} + (2 \beta_{3} + 2 \beta_1) q^{9}+O(q^{10})$$ q + (b2 + b1 + 1) * q^3 + b2 * q^5 + (-2*b3 - b2 - b1 - 1) * q^7 + (2*b3 + 2*b1) * q^9 $$q + (\beta_{2} + \beta_1 + 1) q^{3} + \beta_{2} q^{5} + ( - 2 \beta_{3} - \beta_{2} - \beta_1 - 1) q^{7} + (2 \beta_{3} + 2 \beta_1) q^{9} + (2 \beta_{2} + 2 \beta_1 + 2) q^{11} + ( - 2 \beta_{3} - 2) q^{13} + (\beta_{3} - 1) q^{15} + ( - 2 \beta_{2} + 2 \beta_1 - 2) q^{17} + (2 \beta_{3} + 2 \beta_1) q^{19} + ( - 2 \beta_{3} + \beta_{2} + 4) q^{21} + (\beta_{3} + \beta_{2} + \beta_1) q^{23} + ( - \beta_{2} - 1) q^{25} + ( - \beta_{3} - 1) q^{27} - q^{29} + ( - 6 \beta_{2} - 6) q^{31} + (4 \beta_{3} + 6 \beta_{2} + 4 \beta_1) q^{33} + (\beta_{3} + 2 \beta_1 + 1) q^{35} + (2 \beta_{2} + 2) q^{39} + ( - 2 \beta_{3} - 5) q^{41} + (\beta_{3} - 5) q^{43} - 2 \beta_1 q^{45} - 2 \beta_{2} q^{47} + (2 \beta_{3} - 5 \beta_{2} - 2 \beta_1) q^{49} + 2 \beta_{2} q^{51} + (4 \beta_{2} + 2 \beta_1 + 4) q^{53} + (2 \beta_{3} - 2) q^{55} + (2 \beta_{3} - 4) q^{57} + ( - 4 \beta_{2} - 6 \beta_1 - 4) q^{59} + ( - 6 \beta_{3} - 3 \beta_{2} - 6 \beta_1) q^{61} + ( - 2 \beta_{3} + 8 \beta_{2} + 4) q^{63} + (2 \beta_{3} - 2 \beta_{2} + 2 \beta_1) q^{65} + (11 \beta_{2} + \beta_1 + 11) q^{67} + (2 \beta_{3} - 3) q^{69} + ( - 6 \beta_{3} + 4) q^{71} + ( - 2 \beta_{2} - 2 \beta_1 - 2) q^{73} + ( - \beta_{3} - \beta_{2} - \beta_1) q^{75} + ( - 4 \beta_{3} + 2 \beta_{2} + 8) q^{77} + (2 \beta_{3} - 12 \beta_{2} + 2 \beta_1) q^{79} + (\beta_{2} + 6 \beta_1 + 1) q^{81} + ( - 9 \beta_{3} - 1) q^{83} + (2 \beta_{3} + 2) q^{85} + ( - \beta_{2} - \beta_1 - 1) q^{87} + (4 \beta_{3} - 3 \beta_{2} + 4 \beta_1) q^{89} + (4 \beta_{3} - 2 \beta_{2} + 6) q^{91} + ( - 6 \beta_{3} - 6 \beta_{2} - 6 \beta_1) q^{93} - 2 \beta_1 q^{95} + (4 \beta_{3} + 6) q^{97} + (4 \beta_{3} - 8) q^{99}+O(q^{100})$$ q + (b2 + b1 + 1) * q^3 + b2 * q^5 + (-2*b3 - b2 - b1 - 1) * q^7 + (2*b3 + 2*b1) * q^9 + (2*b2 + 2*b1 + 2) * q^11 + (-2*b3 - 2) * q^13 + (b3 - 1) * q^15 + (-2*b2 + 2*b1 - 2) * q^17 + (2*b3 + 2*b1) * q^19 + (-2*b3 + b2 + 4) * q^21 + (b3 + b2 + b1) * q^23 + (-b2 - 1) * q^25 + (-b3 - 1) * q^27 - q^29 + (-6*b2 - 6) * q^31 + (4*b3 + 6*b2 + 4*b1) * q^33 + (b3 + 2*b1 + 1) * q^35 + (2*b2 + 2) * q^39 + (-2*b3 - 5) * q^41 + (b3 - 5) * q^43 - 2*b1 * q^45 - 2*b2 * q^47 + (2*b3 - 5*b2 - 2*b1) * q^49 + 2*b2 * q^51 + (4*b2 + 2*b1 + 4) * q^53 + (2*b3 - 2) * q^55 + (2*b3 - 4) * q^57 + (-4*b2 - 6*b1 - 4) * q^59 + (-6*b3 - 3*b2 - 6*b1) * q^61 + (-2*b3 + 8*b2 + 4) * q^63 + (2*b3 - 2*b2 + 2*b1) * q^65 + (11*b2 + b1 + 11) * q^67 + (2*b3 - 3) * q^69 + (-6*b3 + 4) * q^71 + (-2*b2 - 2*b1 - 2) * q^73 + (-b3 - b2 - b1) * q^75 + (-4*b3 + 2*b2 + 8) * q^77 + (2*b3 - 12*b2 + 2*b1) * q^79 + (b2 + 6*b1 + 1) * q^81 + (-9*b3 - 1) * q^83 + (2*b3 + 2) * q^85 + (-b2 - b1 - 1) * q^87 + (4*b3 - 3*b2 + 4*b1) * q^89 + (4*b3 - 2*b2 + 6) * q^91 + (-6*b3 - 6*b2 - 6*b1) * q^93 - 2*b1 * q^95 + (4*b3 + 6) * q^97 + (4*b3 - 8) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{3} - 2 q^{5} - 2 q^{7}+O(q^{10})$$ 4 * q + 2 * q^3 - 2 * q^5 - 2 * q^7 $$4 q + 2 q^{3} - 2 q^{5} - 2 q^{7} + 4 q^{11} - 8 q^{13} - 4 q^{15} - 4 q^{17} + 14 q^{21} - 2 q^{23} - 2 q^{25} - 4 q^{27} - 4 q^{29} - 12 q^{31} - 12 q^{33} + 4 q^{35} + 4 q^{39} - 20 q^{41} - 20 q^{43} + 4 q^{47} + 10 q^{49} - 4 q^{51} + 8 q^{53} - 8 q^{55} - 16 q^{57} - 8 q^{59} + 6 q^{61} + 4 q^{65} + 22 q^{67} - 12 q^{69} + 16 q^{71} - 4 q^{73} + 2 q^{75} + 28 q^{77} + 24 q^{79} + 2 q^{81} - 4 q^{83} + 8 q^{85} - 2 q^{87} + 6 q^{89} + 28 q^{91} + 12 q^{93} + 24 q^{97} - 32 q^{99}+O(q^{100})$$ 4 * q + 2 * q^3 - 2 * q^5 - 2 * q^7 + 4 * q^11 - 8 * q^13 - 4 * q^15 - 4 * q^17 + 14 * q^21 - 2 * q^23 - 2 * q^25 - 4 * q^27 - 4 * q^29 - 12 * q^31 - 12 * q^33 + 4 * q^35 + 4 * q^39 - 20 * q^41 - 20 * q^43 + 4 * q^47 + 10 * q^49 - 4 * q^51 + 8 * q^53 - 8 * q^55 - 16 * q^57 - 8 * q^59 + 6 * q^61 + 4 * q^65 + 22 * q^67 - 12 * q^69 + 16 * q^71 - 4 * q^73 + 2 * q^75 + 28 * q^77 + 24 * q^79 + 2 * q^81 - 4 * q^83 + 8 * q^85 - 2 * q^87 + 6 * q^89 + 28 * q^91 + 12 * q^93 + 24 * q^97 - 32 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 2x^{2} + 4$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{2} ) / 2$$ (v^2) / 2 $$\beta_{3}$$ $$=$$ $$( \nu^{3} ) / 2$$ (v^3) / 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$2\beta_{2}$$ 2*b2 $$\nu^{3}$$ $$=$$ $$2\beta_{3}$$ 2*b3

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/560\mathbb{Z}\right)^\times$$.

 $$n$$ $$241$$ $$337$$ $$351$$ $$421$$ $$\chi(n)$$ $$\beta_{2}$$ $$1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
81.1
 −0.707107 + 1.22474i 0.707107 − 1.22474i −0.707107 − 1.22474i 0.707107 + 1.22474i
0 −0.207107 + 0.358719i 0 −0.500000 0.866025i 0 −2.62132 0.358719i 0 1.41421 + 2.44949i 0
81.2 0 1.20711 2.09077i 0 −0.500000 0.866025i 0 1.62132 + 2.09077i 0 −1.41421 2.44949i 0
401.1 0 −0.207107 0.358719i 0 −0.500000 + 0.866025i 0 −2.62132 + 0.358719i 0 1.41421 2.44949i 0
401.2 0 1.20711 + 2.09077i 0 −0.500000 + 0.866025i 0 1.62132 2.09077i 0 −1.41421 + 2.44949i 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 560.2.q.k 4
4.b odd 2 1 35.2.e.a 4
7.c even 3 1 inner 560.2.q.k 4
7.c even 3 1 3920.2.a.bq 2
7.d odd 6 1 3920.2.a.bv 2
12.b even 2 1 315.2.j.e 4
20.d odd 2 1 175.2.e.c 4
20.e even 4 2 175.2.k.a 8
28.d even 2 1 245.2.e.e 4
28.f even 6 1 245.2.a.g 2
28.f even 6 1 245.2.e.e 4
28.g odd 6 1 35.2.e.a 4
28.g odd 6 1 245.2.a.h 2
84.j odd 6 1 2205.2.a.q 2
84.n even 6 1 315.2.j.e 4
84.n even 6 1 2205.2.a.n 2
140.p odd 6 1 175.2.e.c 4
140.p odd 6 1 1225.2.a.k 2
140.s even 6 1 1225.2.a.m 2
140.w even 12 2 175.2.k.a 8
140.w even 12 2 1225.2.b.g 4
140.x odd 12 2 1225.2.b.h 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.2.e.a 4 4.b odd 2 1
35.2.e.a 4 28.g odd 6 1
175.2.e.c 4 20.d odd 2 1
175.2.e.c 4 140.p odd 6 1
175.2.k.a 8 20.e even 4 2
175.2.k.a 8 140.w even 12 2
245.2.a.g 2 28.f even 6 1
245.2.a.h 2 28.g odd 6 1
245.2.e.e 4 28.d even 2 1
245.2.e.e 4 28.f even 6 1
315.2.j.e 4 12.b even 2 1
315.2.j.e 4 84.n even 6 1
560.2.q.k 4 1.a even 1 1 trivial
560.2.q.k 4 7.c even 3 1 inner
1225.2.a.k 2 140.p odd 6 1
1225.2.a.m 2 140.s even 6 1
1225.2.b.g 4 140.w even 12 2
1225.2.b.h 4 140.x odd 12 2
2205.2.a.n 2 84.n even 6 1
2205.2.a.q 2 84.j odd 6 1
3920.2.a.bq 2 7.c even 3 1
3920.2.a.bv 2 7.d odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(560, [\chi])$$:

 $$T_{3}^{4} - 2T_{3}^{3} + 5T_{3}^{2} + 2T_{3} + 1$$ T3^4 - 2*T3^3 + 5*T3^2 + 2*T3 + 1 $$T_{11}^{4} - 4T_{11}^{3} + 20T_{11}^{2} + 16T_{11} + 16$$ T11^4 - 4*T11^3 + 20*T11^2 + 16*T11 + 16 $$T_{13}^{2} + 4T_{13} - 4$$ T13^2 + 4*T13 - 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} - 2 T^{3} + 5 T^{2} + 2 T + 1$$
$5$ $$(T^{2} + T + 1)^{2}$$
$7$ $$T^{4} + 2 T^{3} - 3 T^{2} + 14 T + 49$$
$11$ $$T^{4} - 4 T^{3} + 20 T^{2} + 16 T + 16$$
$13$ $$(T^{2} + 4 T - 4)^{2}$$
$17$ $$T^{4} + 4 T^{3} + 20 T^{2} - 16 T + 16$$
$19$ $$T^{4} + 8T^{2} + 64$$
$23$ $$T^{4} + 2 T^{3} + 5 T^{2} - 2 T + 1$$
$29$ $$(T + 1)^{4}$$
$31$ $$(T^{2} + 6 T + 36)^{2}$$
$37$ $$T^{4}$$
$41$ $$(T^{2} + 10 T + 17)^{2}$$
$43$ $$(T^{2} + 10 T + 23)^{2}$$
$47$ $$(T^{2} - 2 T + 4)^{2}$$
$53$ $$T^{4} - 8 T^{3} + 56 T^{2} - 64 T + 64$$
$59$ $$T^{4} + 8 T^{3} + 120 T^{2} + \cdots + 3136$$
$61$ $$T^{4} - 6 T^{3} + 99 T^{2} + \cdots + 3969$$
$67$ $$T^{4} - 22 T^{3} + 365 T^{2} + \cdots + 14161$$
$71$ $$(T^{2} - 8 T - 56)^{2}$$
$73$ $$T^{4} + 4 T^{3} + 20 T^{2} - 16 T + 16$$
$79$ $$T^{4} - 24 T^{3} + 440 T^{2} + \cdots + 18496$$
$83$ $$(T^{2} + 2 T - 161)^{2}$$
$89$ $$T^{4} - 6 T^{3} + 59 T^{2} + 138 T + 529$$
$97$ $$(T^{2} - 12 T + 4)^{2}$$