# Properties

 Label 486.2.g.a Level $486$ Weight $2$ Character orbit 486.g Analytic conductor $3.881$ Analytic rank $0$ Dimension $72$ 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: $$72$$ Relative dimension: $$4$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$72 q+O(q^{10})$$ 72 * q $$\operatorname{Tr}(f)(q) =$$ $$72 q + 18 q^{13} + 9 q^{20} + 81 q^{23} + 18 q^{25} + 27 q^{26} + 18 q^{28} + 27 q^{29} - 54 q^{31} + 27 q^{35} - 9 q^{38} + 9 q^{41} + 36 q^{43} - 18 q^{46} + 27 q^{47} - 36 q^{52} + 27 q^{53} + 54 q^{55} + 9 q^{58} + 18 q^{59} - 9 q^{65} - 135 q^{67} + 18 q^{68} + 18 q^{70} - 72 q^{71} + 36 q^{73} - 99 q^{74} - 9 q^{76} - 144 q^{77} - 9 q^{79} - 18 q^{80} - 72 q^{82} - 99 q^{83} + 9 q^{85} - 72 q^{86} - 9 q^{88} - 126 q^{89} + 63 q^{91} - 36 q^{92} + 18 q^{94} - 45 q^{95} - 171 q^{97} - 36 q^{98}+O(q^{100})$$ 72 * q + 18 * q^13 + 9 * q^20 + 81 * q^23 + 18 * q^25 + 27 * q^26 + 18 * q^28 + 27 * q^29 - 54 * q^31 + 27 * q^35 - 9 * 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 + 18 * q^59 - 9 * q^65 - 135 * q^67 + 18 * q^68 + 18 * q^70 - 72 * q^71 + 36 * q^73 - 99 * q^74 - 9 * q^76 - 144 * q^77 - 9 * q^79 - 18 * q^80 - 72 * q^82 - 99 * q^83 + 9 * q^85 - 72 * q^86 - 9 * q^88 - 126 * q^89 + 63 * q^91 - 36 * q^92 + 18 * q^94 - 45 * q^95 - 171 * 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 −2.13185 1.07065i 0 0.413572 + 1.38143i −0.939693 + 0.342020i 0 2.24173 + 0.815922i
19.2 −0.993238 + 0.116093i 0 0.973045 0.230616i −0.907364 0.455695i 0 −0.755228 2.52264i −0.939693 + 0.342020i 0 0.954132 + 0.347276i
19.3 −0.993238 + 0.116093i 0 0.973045 0.230616i 0.110030 + 0.0552591i 0 −0.0823094 0.274932i −0.939693 + 0.342020i 0 −0.115701 0.0421118i
19.4 −0.993238 + 0.116093i 0 0.973045 0.230616i 2.54814 + 1.27972i 0 0.518331 + 1.73135i −0.939693 + 0.342020i 0 −2.67948 0.975250i
37.1 −0.286803 + 0.957990i 0 −0.835488 0.549509i −0.423867 + 0.982634i 0 −0.264903 4.54820i 0.766044 0.642788i 0 −0.819786 0.687882i
37.2 −0.286803 + 0.957990i 0 −0.835488 0.549509i −0.417669 + 0.968265i 0 −0.0919975 1.57954i 0.766044 0.642788i 0 −0.807799 0.677824i
37.3 −0.286803 + 0.957990i 0 −0.835488 0.549509i −0.266390 + 0.617563i 0 0.0791338 + 1.35867i 0.766044 0.642788i 0 −0.515217 0.432318i
37.4 −0.286803 + 0.957990i 0 −0.835488 0.549509i 0.757906 1.75702i 0 0.278553 + 4.78256i 0.766044 0.642788i 0 1.46584 + 1.22999i
73.1 0.893633 0.448799i 0 0.597159 0.802123i −0.308002 + 1.02880i 0 −2.06499 + 4.78718i 0.173648 0.984808i 0 0.186483 + 1.05760i
73.2 0.893633 0.448799i 0 0.597159 0.802123i −0.195520 + 0.653084i 0 0.386069 0.895009i 0.173648 0.984808i 0 0.118380 + 0.671366i
73.3 0.893633 0.448799i 0 0.597159 0.802123i 0.267154 0.892356i 0 1.11188 2.57764i 0.173648 0.984808i 0 −0.161751 0.917337i
73.4 0.893633 0.448799i 0 0.597159 0.802123i 0.750365 2.50640i 0 0.318489 0.738341i 0.173648 0.984808i 0 −0.454317 2.57656i
91.1 −0.0581448 0.998308i 0 −0.993238 + 0.116093i −3.64873 0.864766i 0 1.03565 1.39113i 0.173648 + 0.984808i 0 −0.651148 + 3.69284i
91.2 −0.0581448 0.998308i 0 −0.993238 + 0.116093i −1.08715 0.257659i 0 1.38530 1.86078i 0.173648 + 0.984808i 0 −0.194011 + 1.10029i
91.3 −0.0581448 0.998308i 0 −0.993238 + 0.116093i −0.789983 0.187229i 0 −1.95256 + 2.62274i 0.173648 + 0.984808i 0 −0.140979 + 0.799532i
91.4 −0.0581448 0.998308i 0 −0.993238 + 0.116093i 3.39069 + 0.803609i 0 −1.32018 + 1.77332i 0.173648 + 0.984808i 0 0.605098 3.43168i
127.1 −0.686242 + 0.727374i 0 −0.0581448 0.998308i −1.83788 0.214818i 0 −0.0854199 + 0.0428995i 0.766044 + 0.642788i 0 1.41748 1.18941i
127.2 −0.686242 + 0.727374i 0 −0.0581448 0.998308i −1.08261 0.126540i 0 −0.387485 + 0.194602i 0.766044 + 0.642788i 0 0.834977 0.700629i
127.3 −0.686242 + 0.727374i 0 −0.0581448 0.998308i 1.97104 + 0.230382i 0 0.868864 0.436360i 0.766044 + 0.642788i 0 −1.52019 + 1.27559i
127.4 −0.686242 + 0.727374i 0 −0.0581448 0.998308i 3.71787 + 0.434557i 0 −3.25050 + 1.63246i 0.766044 + 0.642788i 0 −2.86744 + 2.40607i
See all 72 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 19.4 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.a 72
3.b odd 2 1 162.2.g.a 72
81.g even 27 1 inner 486.2.g.a 72
81.h odd 54 1 162.2.g.a 72

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
162.2.g.a 72 3.b odd 2 1
162.2.g.a 72 81.h odd 54 1
486.2.g.a 72 1.a even 1 1 trivial
486.2.g.a 72 81.g even 27 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{72} - 9 T_{5}^{70} - 30 T_{5}^{69} + 63 T_{5}^{68} + 81 T_{5}^{67} - 1665 T_{5}^{66} + 4023 T_{5}^{65} - 18009 T_{5}^{64} - 92904 T_{5}^{63} + 832275 T_{5}^{62} + 782244 T_{5}^{61} - 3669129 T_{5}^{60} + \cdots + 18487617876729$$ acting on $$S_{2}^{\mathrm{new}}(486, [\chi])$$.