# Properties

 Label 1080.2.d.d Level $1080$ Weight $2$ Character orbit 1080.d Analytic conductor $8.624$ Analytic rank $0$ Dimension $4$ CM discriminant -24 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1080,2,Mod(109,1080)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1080, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0, 1]))

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

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

chi := DirichletCharacter("1080.109");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1080 = 2^{3} \cdot 3^{3} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1080.d (of order $$2$$, degree $$1$$, 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: $$8.62384341830$$ Analytic rank: $$0$$ Dimension: $$4$$ 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: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q - \beta_{3} q^{2} + 2 q^{4} + ( - \beta_{3} - \beta_{2} - \beta_1 - 2) q^{5} + ( - \beta_{3} + 2 \beta_{2} - 2 \beta_1 + 1) q^{7} - 2 \beta_{3} q^{8}+O(q^{10})$$ q - b3 * q^2 + 2 * q^4 + (-b3 - b2 - b1 - 2) * q^5 + (-b3 + 2*b2 - 2*b1 + 1) * q^7 - 2*b3 * q^8 $$q - \beta_{3} q^{2} + 2 q^{4} + ( - \beta_{3} - \beta_{2} - \beta_1 - 2) q^{5} + ( - \beta_{3} + 2 \beta_{2} - 2 \beta_1 + 1) q^{7} - 2 \beta_{3} q^{8} + (\beta_{3} - 2 \beta_{2} - \beta_1) q^{10} + ( - 2 \beta_{3} + 2 \beta_{2} - 4 \beta_1 + 1) q^{11} + (\beta_{3} - 4 \beta_{2} + 2 \beta_1 - 2) q^{14} + 4 q^{16} + ( - 2 \beta_{3} - 2 \beta_{2} - 2 \beta_1 - 4) q^{20} + (\beta_{3} - 8 \beta_{2} + 2 \beta_1 - 4) q^{22} + (4 \beta_{3} + \beta_{2} + 2 \beta_1 + 1) q^{25} + ( - 2 \beta_{3} + 4 \beta_{2} - 4 \beta_1 + 2) q^{28} + ( - 12 \beta_{2} - 6) q^{29} + ( - 3 \beta_{3} - 5) q^{31} - 4 \beta_{3} q^{32} + (2 \beta_{3} - 5 \beta_{2} + 4 \beta_1 - 4) q^{35} + (2 \beta_{3} - 4 \beta_{2} - 2 \beta_1) q^{40} + ( - 4 \beta_{3} + 4 \beta_{2} - 8 \beta_1 + 2) q^{44} + ( - 6 \beta_{3} - 2) q^{49} + (4 \beta_{2} + \beta_1 - 4) q^{50} + ( - 5 \beta_{3} + 3) q^{53} + (5 \beta_{3} - 7 \beta_{2} + 7 \beta_1 - 8) q^{55} + (2 \beta_{3} - 8 \beta_{2} + 4 \beta_1 - 4) q^{56} + ( - 6 \beta_{3} - 12 \beta_1) q^{58} + (12 \beta_{2} + 6) q^{59} + (5 \beta_{3} + 6) q^{62} + 8 q^{64} + ( - \beta_{3} + 8 \beta_{2} - 5 \beta_1 + 4) q^{70} + (2 \beta_{3} + 14 \beta_{2} + 4 \beta_1 + 7) q^{73} + ( - 9 \beta_{3} - 15) q^{77} - 10 q^{79} + ( - 4 \beta_{3} - 4 \beta_{2} - 4 \beta_1 - 8) q^{80} + (2 \beta_{3} + 15) q^{83} + (2 \beta_{3} - 16 \beta_{2} + 4 \beta_1 - 8) q^{88} + (4 \beta_{3} - 2 \beta_{2} + 8 \beta_1 - 1) q^{97} + (2 \beta_{3} + 12) q^{98}+O(q^{100})$$ q - b3 * q^2 + 2 * q^4 + (-b3 - b2 - b1 - 2) * q^5 + (-b3 + 2*b2 - 2*b1 + 1) * q^7 - 2*b3 * q^8 + (b3 - 2*b2 - b1) * q^10 + (-2*b3 + 2*b2 - 4*b1 + 1) * q^11 + (b3 - 4*b2 + 2*b1 - 2) * q^14 + 4 * q^16 + (-2*b3 - 2*b2 - 2*b1 - 4) * q^20 + (b3 - 8*b2 + 2*b1 - 4) * q^22 + (4*b3 + b2 + 2*b1 + 1) * q^25 + (-2*b3 + 4*b2 - 4*b1 + 2) * q^28 + (-12*b2 - 6) * q^29 + (-3*b3 - 5) * q^31 - 4*b3 * q^32 + (2*b3 - 5*b2 + 4*b1 - 4) * q^35 + (2*b3 - 4*b2 - 2*b1) * q^40 + (-4*b3 + 4*b2 - 8*b1 + 2) * q^44 + (-6*b3 - 2) * q^49 + (4*b2 + b1 - 4) * q^50 + (-5*b3 + 3) * q^53 + (5*b3 - 7*b2 + 7*b1 - 8) * q^55 + (2*b3 - 8*b2 + 4*b1 - 4) * q^56 + (-6*b3 - 12*b1) * q^58 + (12*b2 + 6) * q^59 + (5*b3 + 6) * q^62 + 8 * q^64 + (-b3 + 8*b2 - 5*b1 + 4) * q^70 + (2*b3 + 14*b2 + 4*b1 + 7) * q^73 + (-9*b3 - 15) * q^77 - 10 * q^79 + (-4*b3 - 4*b2 - 4*b1 - 8) * q^80 + (2*b3 + 15) * q^83 + (2*b3 - 16*b2 + 4*b1 - 8) * q^88 + (4*b3 - 2*b2 + 8*b1 - 1) * q^97 + (2*b3 + 12) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 8 q^{4} - 6 q^{5}+O(q^{10})$$ 4 * q + 8 * q^4 - 6 * q^5 $$4 q + 8 q^{4} - 6 q^{5} + 4 q^{10} + 16 q^{16} - 12 q^{20} + 2 q^{25} - 20 q^{31} - 6 q^{35} + 8 q^{40} - 8 q^{49} - 24 q^{50} + 12 q^{53} - 18 q^{55} + 24 q^{62} + 32 q^{64} - 60 q^{77} - 40 q^{79} - 24 q^{80} + 60 q^{83} + 48 q^{98}+O(q^{100})$$ 4 * q + 8 * q^4 - 6 * q^5 + 4 * q^10 + 16 * q^16 - 12 * q^20 + 2 * q^25 - 20 * q^31 - 6 * q^35 + 8 * q^40 - 8 * q^49 - 24 * q^50 + 12 * q^53 - 18 * q^55 + 24 * q^62 + 32 * q^64 - 60 * q^77 - 40 * q^79 - 24 * q^80 + 60 * q^83 + 48 * q^98

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 $$\left(\mathbb{Z}/1080\mathbb{Z}\right)^\times$$.

 $$n$$ $$217$$ $$271$$ $$541$$ $$1001$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
109.1
 −0.707107 + 1.22474i −0.707107 − 1.22474i 0.707107 + 1.22474i 0.707107 − 1.22474i
−1.41421 0 2.00000 −2.20711 0.358719i 0 4.18154i −2.82843 0 3.12132 + 0.507306i
109.2 −1.41421 0 2.00000 −2.20711 + 0.358719i 0 4.18154i −2.82843 0 3.12132 0.507306i
109.3 1.41421 0 2.00000 −0.792893 2.09077i 0 0.717439i 2.82843 0 −1.12132 2.95680i
109.4 1.41421 0 2.00000 −0.792893 + 2.09077i 0 0.717439i 2.82843 0 −1.12132 + 2.95680i
 $$n$$: 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
24.h odd 2 1 CM by $$\Q(\sqrt{-6})$$
15.d odd 2 1 inner
40.f even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1080.2.d.d 4
3.b odd 2 1 1080.2.d.e yes 4
4.b odd 2 1 4320.2.d.a 4
5.b even 2 1 1080.2.d.e yes 4
8.b even 2 1 1080.2.d.e yes 4
8.d odd 2 1 4320.2.d.f 4
12.b even 2 1 4320.2.d.f 4
15.d odd 2 1 inner 1080.2.d.d 4
20.d odd 2 1 4320.2.d.f 4
24.f even 2 1 4320.2.d.a 4
24.h odd 2 1 CM 1080.2.d.d 4
40.e odd 2 1 4320.2.d.a 4
40.f even 2 1 inner 1080.2.d.d 4
60.h even 2 1 4320.2.d.a 4
120.i odd 2 1 1080.2.d.e yes 4
120.m even 2 1 4320.2.d.f 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1080.2.d.d 4 1.a even 1 1 trivial
1080.2.d.d 4 15.d odd 2 1 inner
1080.2.d.d 4 24.h odd 2 1 CM
1080.2.d.d 4 40.f even 2 1 inner
1080.2.d.e yes 4 3.b odd 2 1
1080.2.d.e yes 4 5.b even 2 1
1080.2.d.e yes 4 8.b even 2 1
1080.2.d.e yes 4 120.i odd 2 1
4320.2.d.a 4 4.b odd 2 1
4320.2.d.a 4 24.f even 2 1
4320.2.d.a 4 40.e odd 2 1
4320.2.d.a 4 60.h even 2 1
4320.2.d.f 4 8.d odd 2 1
4320.2.d.f 4 12.b even 2 1
4320.2.d.f 4 20.d odd 2 1
4320.2.d.f 4 120.m even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1080, [\chi])$$:

 $$T_{7}^{4} + 18T_{7}^{2} + 9$$ T7^4 + 18*T7^2 + 9 $$T_{11}^{4} + 54T_{11}^{2} + 441$$ T11^4 + 54*T11^2 + 441 $$T_{13}$$ T13 $$T_{53}^{2} - 6T_{53} - 41$$ T53^2 - 6*T53 - 41

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} - 2)^{2}$$
$3$ $$T^{4}$$
$5$ $$T^{4} + 6 T^{3} + 17 T^{2} + 30 T + 25$$
$7$ $$T^{4} + 18T^{2} + 9$$
$11$ $$T^{4} + 54T^{2} + 441$$
$13$ $$T^{4}$$
$17$ $$T^{4}$$
$19$ $$T^{4}$$
$23$ $$T^{4}$$
$29$ $$(T^{2} + 108)^{2}$$
$31$ $$(T^{2} + 10 T + 7)^{2}$$
$37$ $$T^{4}$$
$41$ $$T^{4}$$
$43$ $$T^{4}$$
$47$ $$T^{4}$$
$53$ $$(T^{2} - 6 T - 41)^{2}$$
$59$ $$(T^{2} + 108)^{2}$$
$61$ $$T^{4}$$
$67$ $$T^{4}$$
$71$ $$T^{4}$$
$73$ $$T^{4} + 342 T^{2} + 15129$$
$79$ $$(T + 10)^{4}$$
$83$ $$(T^{2} - 30 T + 217)^{2}$$
$89$ $$T^{4}$$
$97$ $$T^{4} + 198T^{2} + 8649$$