# Properties

 Label 560.2.g.e Level $560$ Weight $2$ Character orbit 560.g Analytic conductor $4.472$ Analytic rank $0$ Dimension $4$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [560,2,Mod(449,560)]

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, 1, 0]))

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

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

chi := DirichletCharacter("560.449");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$560 = 2^{4} \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 560.g (of order $$2$$, degree $$1$$, 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: $$4$$ Coefficient field: $$\Q(i, \sqrt{6})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 9$$ x^4 + 9 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 70) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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 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.

 $$f(q)$$ $$=$$ $$q + (\beta_{3} + \beta_1) q^{3} + ( - \beta_{2} + \beta_1 + 1) q^{5} - \beta_{2} q^{7} - 3 q^{9}+O(q^{10})$$ q + (b3 + b1) * q^3 + (-b2 + b1 + 1) * q^5 - b2 * q^7 - 3 * q^9 $$q + (\beta_{3} + \beta_1) q^{3} + ( - \beta_{2} + \beta_1 + 1) q^{5} - \beta_{2} q^{7} - 3 q^{9} + ( - 2 \beta_{3} + 2 \beta_1) q^{11} + (\beta_{3} + 2 \beta_{2} + \beta_1) q^{13} + (3 \beta_{2} + 2 \beta_1 - 3) q^{15} - 2 \beta_{2} q^{17} + (\beta_{3} - \beta_1 + 4) q^{19} + ( - \beta_{3} + \beta_1) q^{21} + ( - 2 \beta_{3} + 2 \beta_{2} - 2 \beta_1) q^{23} + ( - 2 \beta_{3} + \beta_{2} + 2 \beta_1) q^{25} + (2 \beta_{3} - 2 \beta_1 - 2) q^{29} + (2 \beta_{3} - 2 \beta_1 - 4) q^{31} + 12 \beta_{2} q^{33} + ( - \beta_{3} - \beta_{2} - 1) q^{35} - 2 \beta_{2} q^{37} + (2 \beta_{3} - 2 \beta_1 - 6) q^{39} + ( - 2 \beta_{3} + 2 \beta_1 - 6) q^{41} + (2 \beta_{3} - 4 \beta_{2} + 2 \beta_1) q^{43} + (3 \beta_{2} - 3 \beta_1 - 3) q^{45} + (2 \beta_{3} + 4 \beta_{2} + 2 \beta_1) q^{47} - q^{49} + ( - 2 \beta_{3} + 2 \beta_1) q^{51} + ( - 2 \beta_{3} - 6 \beta_{2} - 2 \beta_1) q^{53} + ( - 4 \beta_{3} + 6 \beta_{2} + 6) q^{55} + (4 \beta_{3} - 6 \beta_{2} + 4 \beta_1) q^{57} + ( - \beta_{3} + \beta_1 - 4) q^{59} + (\beta_{3} - \beta_1 + 6) q^{61} + 3 \beta_{2} q^{63} + (2 \beta_{3} + 5 \beta_{2} + 2 \beta_1 - 1) q^{65} - 8 \beta_{2} q^{67} + (2 \beta_{3} - 2 \beta_1 + 12) q^{69} + (2 \beta_{3} - 2 \beta_1 + 6) q^{71} + (2 \beta_{3} - 2 \beta_{2} + 2 \beta_1) q^{73} + (\beta_{3} + 12 \beta_{2} - \beta_1) q^{75} + ( - 2 \beta_{3} - 2 \beta_1) q^{77} + ( - 2 \beta_{3} + 2 \beta_1 + 2) q^{79} - 9 q^{81} + (\beta_{3} + \beta_1) q^{83} + ( - 2 \beta_{3} - 2 \beta_{2} - 2) q^{85} + ( - 2 \beta_{3} - 12 \beta_{2} - 2 \beta_1) q^{87} + 10 q^{89} + ( - \beta_{3} + \beta_1 + 2) q^{91} + ( - 4 \beta_{3} - 12 \beta_{2} - 4 \beta_1) q^{93} + (2 \beta_{3} - 7 \beta_{2} + 4 \beta_1 + 1) q^{95} + ( - 4 \beta_{3} - 6 \beta_{2} - 4 \beta_1) q^{97} + (6 \beta_{3} - 6 \beta_1) q^{99}+O(q^{100})$$ q + (b3 + b1) * q^3 + (-b2 + b1 + 1) * q^5 - b2 * q^7 - 3 * q^9 + (-2*b3 + 2*b1) * q^11 + (b3 + 2*b2 + b1) * q^13 + (3*b2 + 2*b1 - 3) * q^15 - 2*b2 * q^17 + (b3 - b1 + 4) * q^19 + (-b3 + b1) * q^21 + (-2*b3 + 2*b2 - 2*b1) * q^23 + (-2*b3 + b2 + 2*b1) * q^25 + (2*b3 - 2*b1 - 2) * q^29 + (2*b3 - 2*b1 - 4) * q^31 + 12*b2 * q^33 + (-b3 - b2 - 1) * q^35 - 2*b2 * q^37 + (2*b3 - 2*b1 - 6) * q^39 + (-2*b3 + 2*b1 - 6) * q^41 + (2*b3 - 4*b2 + 2*b1) * q^43 + (3*b2 - 3*b1 - 3) * q^45 + (2*b3 + 4*b2 + 2*b1) * q^47 - q^49 + (-2*b3 + 2*b1) * q^51 + (-2*b3 - 6*b2 - 2*b1) * q^53 + (-4*b3 + 6*b2 + 6) * q^55 + (4*b3 - 6*b2 + 4*b1) * q^57 + (-b3 + b1 - 4) * q^59 + (b3 - b1 + 6) * q^61 + 3*b2 * q^63 + (2*b3 + 5*b2 + 2*b1 - 1) * q^65 - 8*b2 * q^67 + (2*b3 - 2*b1 + 12) * q^69 + (2*b3 - 2*b1 + 6) * q^71 + (2*b3 - 2*b2 + 2*b1) * q^73 + (b3 + 12*b2 - b1) * q^75 + (-2*b3 - 2*b1) * q^77 + (-2*b3 + 2*b1 + 2) * q^79 - 9 * q^81 + (b3 + b1) * q^83 + (-2*b3 - 2*b2 - 2) * q^85 + (-2*b3 - 12*b2 - 2*b1) * q^87 + 10 * q^89 + (-b3 + b1 + 2) * q^91 + (-4*b3 - 12*b2 - 4*b1) * q^93 + (2*b3 - 7*b2 + 4*b1 + 1) * q^95 + (-4*b3 - 6*b2 - 4*b1) * q^97 + (6*b3 - 6*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 4 q^{5} - 12 q^{9}+O(q^{10})$$ 4 * q + 4 * q^5 - 12 * q^9 $$4 q + 4 q^{5} - 12 q^{9} - 12 q^{15} + 16 q^{19} - 8 q^{29} - 16 q^{31} - 4 q^{35} - 24 q^{39} - 24 q^{41} - 12 q^{45} - 4 q^{49} + 24 q^{55} - 16 q^{59} + 24 q^{61} - 4 q^{65} + 48 q^{69} + 24 q^{71} + 8 q^{79} - 36 q^{81} - 8 q^{85} + 40 q^{89} + 8 q^{91} + 4 q^{95}+O(q^{100})$$ 4 * q + 4 * q^5 - 12 * q^9 - 12 * q^15 + 16 * q^19 - 8 * q^29 - 16 * q^31 - 4 * q^35 - 24 * q^39 - 24 * q^41 - 12 * q^45 - 4 * q^49 + 24 * q^55 - 16 * q^59 + 24 * q^61 - 4 * q^65 + 48 * q^69 + 24 * q^71 + 8 * q^79 - 36 * q^81 - 8 * q^85 + 40 * q^89 + 8 * q^91 + 4 * q^95

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 9$$ :

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

## Character values

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

 $$n$$ $$241$$ $$337$$ $$351$$ $$421$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
449.1
 −1.22474 − 1.22474i 1.22474 − 1.22474i −1.22474 + 1.22474i 1.22474 + 1.22474i
0 2.44949i 0 −0.224745 2.22474i 0 1.00000i 0 −3.00000 0
449.2 0 2.44949i 0 2.22474 0.224745i 0 1.00000i 0 −3.00000 0
449.3 0 2.44949i 0 −0.224745 + 2.22474i 0 1.00000i 0 −3.00000 0
449.4 0 2.44949i 0 2.22474 + 0.224745i 0 1.00000i 0 −3.00000 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
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 560.2.g.e 4
3.b odd 2 1 5040.2.t.t 4
4.b odd 2 1 70.2.c.a 4
5.b even 2 1 inner 560.2.g.e 4
5.c odd 4 1 2800.2.a.bl 2
5.c odd 4 1 2800.2.a.bm 2
8.b even 2 1 2240.2.g.i 4
8.d odd 2 1 2240.2.g.j 4
12.b even 2 1 630.2.g.g 4
15.d odd 2 1 5040.2.t.t 4
20.d odd 2 1 70.2.c.a 4
20.e even 4 1 350.2.a.g 2
20.e even 4 1 350.2.a.h 2
28.d even 2 1 490.2.c.e 4
28.f even 6 2 490.2.i.f 8
28.g odd 6 2 490.2.i.c 8
40.e odd 2 1 2240.2.g.j 4
40.f even 2 1 2240.2.g.i 4
60.h even 2 1 630.2.g.g 4
60.l odd 4 1 3150.2.a.bs 2
60.l odd 4 1 3150.2.a.bt 2
140.c even 2 1 490.2.c.e 4
140.j odd 4 1 2450.2.a.bl 2
140.j odd 4 1 2450.2.a.bq 2
140.p odd 6 2 490.2.i.c 8
140.s even 6 2 490.2.i.f 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.2.c.a 4 4.b odd 2 1
70.2.c.a 4 20.d odd 2 1
350.2.a.g 2 20.e even 4 1
350.2.a.h 2 20.e even 4 1
490.2.c.e 4 28.d even 2 1
490.2.c.e 4 140.c even 2 1
490.2.i.c 8 28.g odd 6 2
490.2.i.c 8 140.p odd 6 2
490.2.i.f 8 28.f even 6 2
490.2.i.f 8 140.s even 6 2
560.2.g.e 4 1.a even 1 1 trivial
560.2.g.e 4 5.b even 2 1 inner
630.2.g.g 4 12.b even 2 1
630.2.g.g 4 60.h even 2 1
2240.2.g.i 4 8.b even 2 1
2240.2.g.i 4 40.f even 2 1
2240.2.g.j 4 8.d odd 2 1
2240.2.g.j 4 40.e odd 2 1
2450.2.a.bl 2 140.j odd 4 1
2450.2.a.bq 2 140.j odd 4 1
2800.2.a.bl 2 5.c odd 4 1
2800.2.a.bm 2 5.c odd 4 1
3150.2.a.bs 2 60.l odd 4 1
3150.2.a.bt 2 60.l odd 4 1
5040.2.t.t 4 3.b odd 2 1
5040.2.t.t 4 15.d odd 2 1

## Hecke kernels

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

 $$T_{3}^{2} + 6$$ T3^2 + 6 $$T_{11}^{2} - 24$$ T11^2 - 24

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$(T^{2} + 6)^{2}$$
$5$ $$T^{4} - 4 T^{3} + \cdots + 25$$
$7$ $$(T^{2} + 1)^{2}$$
$11$ $$(T^{2} - 24)^{2}$$
$13$ $$T^{4} + 20T^{2} + 4$$
$17$ $$(T^{2} + 4)^{2}$$
$19$ $$(T^{2} - 8 T + 10)^{2}$$
$23$ $$T^{4} + 56T^{2} + 400$$
$29$ $$(T^{2} + 4 T - 20)^{2}$$
$31$ $$(T^{2} + 8 T - 8)^{2}$$
$37$ $$(T^{2} + 4)^{2}$$
$41$ $$(T^{2} + 12 T + 12)^{2}$$
$43$ $$T^{4} + 80T^{2} + 64$$
$47$ $$T^{4} + 80T^{2} + 64$$
$53$ $$T^{4} + 120T^{2} + 144$$
$59$ $$(T^{2} + 8 T + 10)^{2}$$
$61$ $$(T^{2} - 12 T + 30)^{2}$$
$67$ $$(T^{2} + 64)^{2}$$
$71$ $$(T^{2} - 12 T + 12)^{2}$$
$73$ $$T^{4} + 56T^{2} + 400$$
$79$ $$(T^{2} - 4 T - 20)^{2}$$
$83$ $$(T^{2} + 6)^{2}$$
$89$ $$(T - 10)^{4}$$
$97$ $$T^{4} + 264T^{2} + 3600$$