# Properties

 Label 486.2.g.b Level $486$ Weight $2$ Character orbit 486.g Analytic conductor $3.881$ Analytic rank $0$ Dimension $90$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [486,2,Mod(19,486)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(486, base_ring=CyclotomicField(54))

chi = DirichletCharacter(H, H._module([52]))

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

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

chi := DirichletCharacter("486.19");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$486 = 2 \cdot 3^{5}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 486.g (of order $$27$$, degree $$18$$, 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: $$3.88072953823$$ Analytic rank: $$0$$ Dimension: $$90$$ Relative dimension: $$5$$ over $$\Q(\zeta_{27})$$ Twist minimal: no (minimal twist has level 162) Sato-Tate group: $\mathrm{SU}(2)[C_{27}]$

## $q$-expansion

The dimension is sufficiently large that we do not compute an algebraic $$q$$-expansion, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$90 q+O(q^{10})$$ 90 * q $$\operatorname{Tr}(f)(q) =$$ $$90 q - 18 q^{13} + 9 q^{20} - 27 q^{23} - 18 q^{25} + 27 q^{26} - 18 q^{28} + 27 q^{29} + 54 q^{31} + 27 q^{35} + 18 q^{38} + 9 q^{41} - 36 q^{43} + 18 q^{46} + 27 q^{47} + 36 q^{52} + 27 q^{53} - 54 q^{55} - 9 q^{58} + 45 q^{59} - 9 q^{65} + 81 q^{67} - 36 q^{68} - 72 q^{70} - 72 q^{71} - 36 q^{73} - 45 q^{74} - 18 q^{76} - 144 q^{77} - 99 q^{79} - 18 q^{80} + 72 q^{82} - 45 q^{83} - 117 q^{85} - 72 q^{86} - 18 q^{88} - 45 q^{89} - 63 q^{91} - 36 q^{92} - 72 q^{94} - 45 q^{95} + 117 q^{97} - 36 q^{98}+O(q^{100})$$ 90 * q - 18 * q^13 + 9 * q^20 - 27 * q^23 - 18 * q^25 + 27 * q^26 - 18 * q^28 + 27 * q^29 + 54 * q^31 + 27 * q^35 + 18 * q^38 + 9 * q^41 - 36 * q^43 + 18 * q^46 + 27 * q^47 + 36 * q^52 + 27 * q^53 - 54 * q^55 - 9 * q^58 + 45 * q^59 - 9 * q^65 + 81 * q^67 - 36 * q^68 - 72 * q^70 - 72 * q^71 - 36 * q^73 - 45 * q^74 - 18 * q^76 - 144 * q^77 - 99 * q^79 - 18 * q^80 + 72 * q^82 - 45 * q^83 - 117 * q^85 - 72 * q^86 - 18 * q^88 - 45 * q^89 - 63 * q^91 - 36 * q^92 - 72 * q^94 - 45 * q^95 + 117 * q^97 - 36 * q^98

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
19.1 0.993238 0.116093i 0 0.973045 0.230616i −2.35524 1.18285i 0 −0.940572 3.14173i 0.939693 0.342020i 0 −2.47663 0.901421i
19.2 0.993238 0.116093i 0 0.973045 0.230616i −2.00838 1.00864i 0 −0.0359273 0.120006i 0.939693 0.342020i 0 −2.11189 0.768666i
19.3 0.993238 0.116093i 0 0.973045 0.230616i −0.917265 0.460668i 0 1.32329 + 4.42009i 0.939693 0.342020i 0 −0.964543 0.351065i
19.4 0.993238 0.116093i 0 0.973045 0.230616i 1.81065 + 0.909345i 0 0.911277 + 3.04388i 0.939693 0.342020i 0 1.90398 + 0.692992i
19.5 0.993238 0.116093i 0 0.973045 0.230616i 3.08919 + 1.55145i 0 −1.35243 4.51743i 0.939693 0.342020i 0 3.24841 + 1.18232i
37.1 0.286803 0.957990i 0 −0.835488 0.549509i −1.28359 + 2.97570i 0 −0.0354946 0.609419i −0.766044 + 0.642788i 0 2.48255 + 2.08311i
37.2 0.286803 0.957990i 0 −0.835488 0.549509i −1.03338 + 2.39565i 0 −0.170360 2.92496i −0.766044 + 0.642788i 0 1.99863 + 1.67705i
37.3 0.286803 0.957990i 0 −0.835488 0.549509i −0.307130 + 0.712007i 0 0.198033 + 3.40010i −0.766044 + 0.642788i 0 0.594009 + 0.498433i
37.4 0.286803 0.957990i 0 −0.835488 0.549509i 0.815630 1.89084i 0 0.163018 + 2.79890i −0.766044 + 0.642788i 0 −1.57748 1.32366i
37.5 0.286803 0.957990i 0 −0.835488 0.549509i 1.45845 3.38108i 0 −0.155983 2.67812i −0.766044 + 0.642788i 0 −2.82075 2.36689i
73.1 −0.893633 + 0.448799i 0 0.597159 0.802123i −0.891212 + 2.97686i 0 0.275210 0.638009i −0.173648 + 0.984808i 0 −0.539594 3.06019i
73.2 −0.893633 + 0.448799i 0 0.597159 0.802123i −0.664151 + 2.21842i 0 −0.590051 + 1.36789i −0.173648 + 0.984808i 0 −0.402118 2.28052i
73.3 −0.893633 + 0.448799i 0 0.597159 0.802123i 0.379535 1.26773i 0 −0.768169 + 1.78082i −0.173648 + 0.984808i 0 0.229793 + 1.30322i
73.4 −0.893633 + 0.448799i 0 0.597159 0.802123i 0.536865 1.79325i 0 1.99514 4.62524i −0.173648 + 0.984808i 0 0.325051 + 1.84346i
73.5 −0.893633 + 0.448799i 0 0.597159 0.802123i 1.15296 3.85116i 0 −0.663579 + 1.53835i −0.173648 + 0.984808i 0 0.698073 + 3.95897i
91.1 0.0581448 + 0.998308i 0 −0.993238 + 0.116093i −3.33598 0.790641i 0 1.21117 1.62688i −0.173648 0.984808i 0 0.595333 3.37630i
91.2 0.0581448 + 0.998308i 0 −0.993238 + 0.116093i −1.79878 0.426320i 0 −0.603713 + 0.810927i −0.173648 0.984808i 0 0.321008 1.82053i
91.3 0.0581448 + 0.998308i 0 −0.993238 + 0.116093i −0.597074 0.141509i 0 1.82935 2.45725i −0.173648 0.984808i 0 0.106553 0.604292i
91.4 0.0581448 + 0.998308i 0 −0.993238 + 0.116093i 1.61859 + 0.383612i 0 −2.67915 + 3.59872i −0.173648 0.984808i 0 −0.288850 + 1.63815i
91.5 0.0581448 + 0.998308i 0 −0.993238 + 0.116093i 1.97808 + 0.468813i 0 1.09412 1.46966i −0.173648 0.984808i 0 −0.353005 + 2.00199i
See all 90 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 19.5 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
81.g even 27 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 486.2.g.b 90
3.b odd 2 1 162.2.g.b 90
81.g even 27 1 inner 486.2.g.b 90
81.h odd 54 1 162.2.g.b 90

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
162.2.g.b 90 3.b odd 2 1
162.2.g.b 90 81.h odd 54 1
486.2.g.b 90 1.a even 1 1 trivial
486.2.g.b 90 81.g even 27 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{90} + 9 T_{5}^{88} - 30 T_{5}^{87} + 45 T_{5}^{86} - 1377 T_{5}^{85} + 2673 T_{5}^{84} - 10341 T_{5}^{83} + 18657 T_{5}^{82} - 199554 T_{5}^{81} + 401679 T_{5}^{80} - 1119420 T_{5}^{79} + \cdots + 21\!\cdots\!84$$ acting on $$S_{2}^{\mathrm{new}}(486, [\chi])$$.