# Properties

 Label 4560.2.a.bf Level $4560$ Weight $2$ Character orbit 4560.a Self dual yes Analytic conductor $36.412$ 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] = [4560,2,Mod(1,4560)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4560, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4560.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$4560 = 2^{4} \cdot 3 \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4560.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: $$36.4117833217$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 2$$ x^2 - 2 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 285) 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 $$\beta = \sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{3} - q^{5} + \beta q^{7} + q^{9}+O(q^{10})$$ q - q^3 - q^5 + b * q^7 + q^9 $$q - q^{3} - q^{5} + \beta q^{7} + q^{9} + (3 \beta - 2) q^{11} + ( - \beta - 2) q^{13} + q^{15} + ( - 2 \beta + 4) q^{17} + q^{19} - \beta q^{21} + ( - 4 \beta - 2) q^{23} + q^{25} - q^{27} + \beta q^{29} + ( - 2 \beta + 6) q^{31} + ( - 3 \beta + 2) q^{33} - \beta q^{35} + ( - \beta - 2) q^{37} + (\beta + 2) q^{39} + ( - 3 \beta + 4) q^{41} + ( - 3 \beta - 8) q^{43} - q^{45} + (4 \beta + 2) q^{47} - 5 q^{49} + (2 \beta - 4) q^{51} + 8 q^{53} + ( - 3 \beta + 2) q^{55} - q^{57} + ( - 6 \beta - 4) q^{59} + (8 \beta - 4) q^{61} + \beta q^{63} + (\beta + 2) q^{65} + (4 \beta + 4) q^{67} + (4 \beta + 2) q^{69} + (2 \beta + 8) q^{71} + ( - 4 \beta - 2) q^{73} - q^{75} + ( - 2 \beta + 6) q^{77} + q^{81} + ( - 2 \beta - 10) q^{83} + (2 \beta - 4) q^{85} - \beta q^{87} + (7 \beta - 4) q^{89} + ( - 2 \beta - 2) q^{91} + (2 \beta - 6) q^{93} - q^{95} + (3 \beta - 14) q^{97} + (3 \beta - 2) q^{99} +O(q^{100})$$ q - q^3 - q^5 + b * q^7 + q^9 + (3*b - 2) * q^11 + (-b - 2) * q^13 + q^15 + (-2*b + 4) * q^17 + q^19 - b * q^21 + (-4*b - 2) * q^23 + q^25 - q^27 + b * q^29 + (-2*b + 6) * q^31 + (-3*b + 2) * q^33 - b * q^35 + (-b - 2) * q^37 + (b + 2) * q^39 + (-3*b + 4) * q^41 + (-3*b - 8) * q^43 - q^45 + (4*b + 2) * q^47 - 5 * q^49 + (2*b - 4) * q^51 + 8 * q^53 + (-3*b + 2) * q^55 - q^57 + (-6*b - 4) * q^59 + (8*b - 4) * q^61 + b * q^63 + (b + 2) * q^65 + (4*b + 4) * q^67 + (4*b + 2) * q^69 + (2*b + 8) * q^71 + (-4*b - 2) * q^73 - q^75 + (-2*b + 6) * q^77 + q^81 + (-2*b - 10) * q^83 + (2*b - 4) * q^85 - b * q^87 + (7*b - 4) * q^89 + (-2*b - 2) * q^91 + (2*b - 6) * q^93 - q^95 + (3*b - 14) * q^97 + (3*b - 2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{3} - 2 q^{5} + 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^3 - 2 * q^5 + 2 * q^9 $$2 q - 2 q^{3} - 2 q^{5} + 2 q^{9} - 4 q^{11} - 4 q^{13} + 2 q^{15} + 8 q^{17} + 2 q^{19} - 4 q^{23} + 2 q^{25} - 2 q^{27} + 12 q^{31} + 4 q^{33} - 4 q^{37} + 4 q^{39} + 8 q^{41} - 16 q^{43} - 2 q^{45} + 4 q^{47} - 10 q^{49} - 8 q^{51} + 16 q^{53} + 4 q^{55} - 2 q^{57} - 8 q^{59} - 8 q^{61} + 4 q^{65} + 8 q^{67} + 4 q^{69} + 16 q^{71} - 4 q^{73} - 2 q^{75} + 12 q^{77} + 2 q^{81} - 20 q^{83} - 8 q^{85} - 8 q^{89} - 4 q^{91} - 12 q^{93} - 2 q^{95} - 28 q^{97} - 4 q^{99}+O(q^{100})$$ 2 * q - 2 * q^3 - 2 * q^5 + 2 * q^9 - 4 * q^11 - 4 * q^13 + 2 * q^15 + 8 * q^17 + 2 * q^19 - 4 * q^23 + 2 * q^25 - 2 * q^27 + 12 * q^31 + 4 * q^33 - 4 * q^37 + 4 * q^39 + 8 * q^41 - 16 * q^43 - 2 * q^45 + 4 * q^47 - 10 * q^49 - 8 * q^51 + 16 * q^53 + 4 * q^55 - 2 * q^57 - 8 * q^59 - 8 * q^61 + 4 * q^65 + 8 * q^67 + 4 * q^69 + 16 * q^71 - 4 * q^73 - 2 * q^75 + 12 * q^77 + 2 * q^81 - 20 * q^83 - 8 * q^85 - 8 * q^89 - 4 * q^91 - 12 * q^93 - 2 * q^95 - 28 * q^97 - 4 * 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
 −1.41421 1.41421
0 −1.00000 0 −1.00000 0 −1.41421 0 1.00000 0
1.2 0 −1.00000 0 −1.00000 0 1.41421 0 1.00000 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
$$2$$ $$-1$$
$$3$$ $$1$$
$$5$$ $$1$$
$$19$$ $$-1$$

## 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 4560.2.a.bf 2
4.b odd 2 1 285.2.a.g 2
12.b even 2 1 855.2.a.d 2
20.d odd 2 1 1425.2.a.k 2
20.e even 4 2 1425.2.c.l 4
60.h even 2 1 4275.2.a.y 2
76.d even 2 1 5415.2.a.n 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
285.2.a.g 2 4.b odd 2 1
855.2.a.d 2 12.b even 2 1
1425.2.a.k 2 20.d odd 2 1
1425.2.c.l 4 20.e even 4 2
4275.2.a.y 2 60.h even 2 1
4560.2.a.bf 2 1.a even 1 1 trivial
5415.2.a.n 2 76.d even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(4560))$$:

 $$T_{7}^{2} - 2$$ T7^2 - 2 $$T_{11}^{2} + 4T_{11} - 14$$ T11^2 + 4*T11 - 14 $$T_{13}^{2} + 4T_{13} + 2$$ T13^2 + 4*T13 + 2 $$T_{17}^{2} - 8T_{17} + 8$$ T17^2 - 8*T17 + 8

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$(T + 1)^{2}$$
$5$ $$(T + 1)^{2}$$
$7$ $$T^{2} - 2$$
$11$ $$T^{2} + 4T - 14$$
$13$ $$T^{2} + 4T + 2$$
$17$ $$T^{2} - 8T + 8$$
$19$ $$(T - 1)^{2}$$
$23$ $$T^{2} + 4T - 28$$
$29$ $$T^{2} - 2$$
$31$ $$T^{2} - 12T + 28$$
$37$ $$T^{2} + 4T + 2$$
$41$ $$T^{2} - 8T - 2$$
$43$ $$T^{2} + 16T + 46$$
$47$ $$T^{2} - 4T - 28$$
$53$ $$(T - 8)^{2}$$
$59$ $$T^{2} + 8T - 56$$
$61$ $$T^{2} + 8T - 112$$
$67$ $$T^{2} - 8T - 16$$
$71$ $$T^{2} - 16T + 56$$
$73$ $$T^{2} + 4T - 28$$
$79$ $$T^{2}$$
$83$ $$T^{2} + 20T + 92$$
$89$ $$T^{2} + 8T - 82$$
$97$ $$T^{2} + 28T + 178$$