# Properties

 Label 560.2.q.d Level $560$ Weight $2$ Character orbit 560.q Analytic conductor $4.472$ Analytic rank $0$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 70) 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 primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (2 \zeta_{6} - 2) q^{3} + \zeta_{6} q^{5} + (2 \zeta_{6} + 1) q^{7} - \zeta_{6} q^{9} +O(q^{10})$$ q + (2*z - 2) * q^3 + z * q^5 + (2*z + 1) * q^7 - z * q^9 $$q + (2 \zeta_{6} - 2) q^{3} + \zeta_{6} q^{5} + (2 \zeta_{6} + 1) q^{7} - \zeta_{6} q^{9} + ( - 3 \zeta_{6} + 3) q^{11} + 5 q^{13} - 2 q^{15} + (6 \zeta_{6} - 6) q^{17} - \zeta_{6} q^{19} + (2 \zeta_{6} - 6) q^{21} + 3 \zeta_{6} q^{23} + (\zeta_{6} - 1) q^{25} - 4 q^{27} - 6 q^{29} + (4 \zeta_{6} - 4) q^{31} + 6 \zeta_{6} q^{33} + (3 \zeta_{6} - 2) q^{35} - 11 \zeta_{6} q^{37} + (10 \zeta_{6} - 10) q^{39} + 3 q^{41} + 10 q^{43} + ( - \zeta_{6} + 1) q^{45} + 3 \zeta_{6} q^{47} + (8 \zeta_{6} - 3) q^{49} - 12 \zeta_{6} q^{51} + (3 \zeta_{6} - 3) q^{53} + 3 q^{55} + 2 q^{57} + 4 \zeta_{6} q^{61} + ( - 3 \zeta_{6} + 2) q^{63} + 5 \zeta_{6} q^{65} + (4 \zeta_{6} - 4) q^{67} - 6 q^{69} - 12 q^{71} + ( - 4 \zeta_{6} + 4) q^{73} - 2 \zeta_{6} q^{75} + ( - 3 \zeta_{6} + 9) q^{77} - 10 \zeta_{6} q^{79} + ( - 11 \zeta_{6} + 11) q^{81} + 12 q^{83} - 6 q^{85} + ( - 12 \zeta_{6} + 12) q^{87} - 6 \zeta_{6} q^{89} + (10 \zeta_{6} + 5) q^{91} - 8 \zeta_{6} q^{93} + ( - \zeta_{6} + 1) q^{95} + 14 q^{97} - 3 q^{99} +O(q^{100})$$ q + (2*z - 2) * q^3 + z * q^5 + (2*z + 1) * q^7 - z * q^9 + (-3*z + 3) * q^11 + 5 * q^13 - 2 * q^15 + (6*z - 6) * q^17 - z * q^19 + (2*z - 6) * q^21 + 3*z * q^23 + (z - 1) * q^25 - 4 * q^27 - 6 * q^29 + (4*z - 4) * q^31 + 6*z * q^33 + (3*z - 2) * q^35 - 11*z * q^37 + (10*z - 10) * q^39 + 3 * q^41 + 10 * q^43 + (-z + 1) * q^45 + 3*z * q^47 + (8*z - 3) * q^49 - 12*z * q^51 + (3*z - 3) * q^53 + 3 * q^55 + 2 * q^57 + 4*z * q^61 + (-3*z + 2) * q^63 + 5*z * q^65 + (4*z - 4) * q^67 - 6 * q^69 - 12 * q^71 + (-4*z + 4) * q^73 - 2*z * q^75 + (-3*z + 9) * q^77 - 10*z * q^79 + (-11*z + 11) * q^81 + 12 * q^83 - 6 * q^85 + (-12*z + 12) * q^87 - 6*z * q^89 + (10*z + 5) * q^91 - 8*z * q^93 + (-z + 1) * q^95 + 14 * q^97 - 3 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{3} + q^{5} + 4 q^{7} - q^{9}+O(q^{10})$$ 2 * q - 2 * q^3 + q^5 + 4 * q^7 - q^9 $$2 q - 2 q^{3} + q^{5} + 4 q^{7} - q^{9} + 3 q^{11} + 10 q^{13} - 4 q^{15} - 6 q^{17} - q^{19} - 10 q^{21} + 3 q^{23} - q^{25} - 8 q^{27} - 12 q^{29} - 4 q^{31} + 6 q^{33} - q^{35} - 11 q^{37} - 10 q^{39} + 6 q^{41} + 20 q^{43} + q^{45} + 3 q^{47} + 2 q^{49} - 12 q^{51} - 3 q^{53} + 6 q^{55} + 4 q^{57} + 4 q^{61} + q^{63} + 5 q^{65} - 4 q^{67} - 12 q^{69} - 24 q^{71} + 4 q^{73} - 2 q^{75} + 15 q^{77} - 10 q^{79} + 11 q^{81} + 24 q^{83} - 12 q^{85} + 12 q^{87} - 6 q^{89} + 20 q^{91} - 8 q^{93} + q^{95} + 28 q^{97} - 6 q^{99}+O(q^{100})$$ 2 * q - 2 * q^3 + q^5 + 4 * q^7 - q^9 + 3 * q^11 + 10 * q^13 - 4 * q^15 - 6 * q^17 - q^19 - 10 * q^21 + 3 * q^23 - q^25 - 8 * q^27 - 12 * q^29 - 4 * q^31 + 6 * q^33 - q^35 - 11 * q^37 - 10 * q^39 + 6 * q^41 + 20 * q^43 + q^45 + 3 * q^47 + 2 * q^49 - 12 * q^51 - 3 * q^53 + 6 * q^55 + 4 * q^57 + 4 * q^61 + q^63 + 5 * q^65 - 4 * q^67 - 12 * q^69 - 24 * q^71 + 4 * q^73 - 2 * q^75 + 15 * q^77 - 10 * q^79 + 11 * q^81 + 24 * q^83 - 12 * q^85 + 12 * q^87 - 6 * q^89 + 20 * q^91 - 8 * q^93 + q^95 + 28 * q^97 - 6 * q^99

## Character values

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

 $$n$$ $$241$$ $$337$$ $$351$$ $$421$$ $$\chi(n)$$ $$-\zeta_{6}$$ $$1$$ $$1$$ $$1$$

## 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
 0.5 + 0.866025i 0.5 − 0.866025i
0 −1.00000 + 1.73205i 0 0.500000 + 0.866025i 0 2.00000 + 1.73205i 0 −0.500000 0.866025i 0
401.1 0 −1.00000 1.73205i 0 0.500000 0.866025i 0 2.00000 1.73205i 0 −0.500000 + 0.866025i 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.d 2
4.b odd 2 1 70.2.e.b 2
7.c even 3 1 inner 560.2.q.d 2
7.c even 3 1 3920.2.a.be 1
7.d odd 6 1 3920.2.a.g 1
12.b even 2 1 630.2.k.e 2
20.d odd 2 1 350.2.e.h 2
20.e even 4 2 350.2.j.a 4
28.d even 2 1 490.2.e.a 2
28.f even 6 1 490.2.a.j 1
28.f even 6 1 490.2.e.a 2
28.g odd 6 1 70.2.e.b 2
28.g odd 6 1 490.2.a.g 1
84.j odd 6 1 4410.2.a.c 1
84.n even 6 1 630.2.k.e 2
84.n even 6 1 4410.2.a.m 1
140.p odd 6 1 350.2.e.h 2
140.p odd 6 1 2450.2.a.p 1
140.s even 6 1 2450.2.a.f 1
140.w even 12 2 350.2.j.a 4
140.w even 12 2 2450.2.c.f 2
140.x odd 12 2 2450.2.c.p 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.2.e.b 2 4.b odd 2 1
70.2.e.b 2 28.g odd 6 1
350.2.e.h 2 20.d odd 2 1
350.2.e.h 2 140.p odd 6 1
350.2.j.a 4 20.e even 4 2
350.2.j.a 4 140.w even 12 2
490.2.a.g 1 28.g odd 6 1
490.2.a.j 1 28.f even 6 1
490.2.e.a 2 28.d even 2 1
490.2.e.a 2 28.f even 6 1
560.2.q.d 2 1.a even 1 1 trivial
560.2.q.d 2 7.c even 3 1 inner
630.2.k.e 2 12.b even 2 1
630.2.k.e 2 84.n even 6 1
2450.2.a.f 1 140.s even 6 1
2450.2.a.p 1 140.p odd 6 1
2450.2.c.f 2 140.w even 12 2
2450.2.c.p 2 140.x odd 12 2
3920.2.a.g 1 7.d odd 6 1
3920.2.a.be 1 7.c even 3 1
4410.2.a.c 1 84.j odd 6 1
4410.2.a.m 1 84.n even 6 1

## Hecke kernels

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

 $$T_{3}^{2} + 2T_{3} + 4$$ T3^2 + 2*T3 + 4 $$T_{11}^{2} - 3T_{11} + 9$$ T11^2 - 3*T11 + 9 $$T_{13} - 5$$ T13 - 5

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 2T + 4$$
$5$ $$T^{2} - T + 1$$
$7$ $$T^{2} - 4T + 7$$
$11$ $$T^{2} - 3T + 9$$
$13$ $$(T - 5)^{2}$$
$17$ $$T^{2} + 6T + 36$$
$19$ $$T^{2} + T + 1$$
$23$ $$T^{2} - 3T + 9$$
$29$ $$(T + 6)^{2}$$
$31$ $$T^{2} + 4T + 16$$
$37$ $$T^{2} + 11T + 121$$
$41$ $$(T - 3)^{2}$$
$43$ $$(T - 10)^{2}$$
$47$ $$T^{2} - 3T + 9$$
$53$ $$T^{2} + 3T + 9$$
$59$ $$T^{2}$$
$61$ $$T^{2} - 4T + 16$$
$67$ $$T^{2} + 4T + 16$$
$71$ $$(T + 12)^{2}$$
$73$ $$T^{2} - 4T + 16$$
$79$ $$T^{2} + 10T + 100$$
$83$ $$(T - 12)^{2}$$
$89$ $$T^{2} + 6T + 36$$
$97$ $$(T - 14)^{2}$$