# Properties

 Label 6561.2.a.c Level $6561$ Weight $2$ Character orbit 6561.a Self dual yes Analytic conductor $52.390$ Analytic rank $1$ Dimension $72$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6561,2,Mod(1,6561)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6561, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("6561.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$6561 = 3^{8}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 6561.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: $$52.3898487662$$ Analytic rank: $$1$$ Dimension: $$72$$ Twist minimal: no (minimal twist has level 81) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$72 q - 9 q^{2} + 63 q^{4} - 18 q^{5} - 27 q^{8}+O(q^{10})$$ 72 * q - 9 * q^2 + 63 * q^4 - 18 * q^5 - 27 * q^8 $$\operatorname{Tr}(f)(q) =$$ $$72 q - 9 q^{2} + 63 q^{4} - 18 q^{5} - 27 q^{8} - 36 q^{11} - 36 q^{14} + 45 q^{16} - 36 q^{17} - 54 q^{20} - 54 q^{23} + 36 q^{25} - 45 q^{26} + 9 q^{28} - 54 q^{29} - 63 q^{32} - 72 q^{35} - 54 q^{38} - 72 q^{41} - 90 q^{44} - 90 q^{47} + 18 q^{49} - 45 q^{50} - 45 q^{53} + 9 q^{55} - 108 q^{56} + 18 q^{58} - 108 q^{59} - 72 q^{62} + 9 q^{64} - 72 q^{65} - 108 q^{68} - 126 q^{71} - 90 q^{74} - 72 q^{77} - 144 q^{80} - 18 q^{82} - 108 q^{83} - 90 q^{86} - 108 q^{89} - 72 q^{92} - 144 q^{95} - 81 q^{98}+O(q^{100})$$ 72 * q - 9 * q^2 + 63 * q^4 - 18 * q^5 - 27 * q^8 - 36 * q^11 - 36 * q^14 + 45 * q^16 - 36 * q^17 - 54 * q^20 - 54 * q^23 + 36 * q^25 - 45 * q^26 + 9 * q^28 - 54 * q^29 - 63 * q^32 - 72 * q^35 - 54 * q^38 - 72 * q^41 - 90 * q^44 - 90 * q^47 + 18 * q^49 - 45 * q^50 - 45 * q^53 + 9 * q^55 - 108 * q^56 + 18 * q^58 - 108 * q^59 - 72 * q^62 + 9 * q^64 - 72 * q^65 - 108 * q^68 - 126 * q^71 - 90 * q^74 - 72 * q^77 - 144 * q^80 - 18 * q^82 - 108 * q^83 - 90 * q^86 - 108 * q^89 - 72 * q^92 - 144 * q^95 - 81 * 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1 −2.76978 0 5.67166 −1.02650 0 1.57628 −10.1697 0 2.84319
1.2 −2.66973 0 5.12748 −2.53528 0 0.510929 −8.34954 0 6.76852
1.3 −2.66924 0 5.12486 1.81561 0 −2.58238 −8.34102 0 −4.84630
1.4 −2.65582 0 5.05340 −2.84817 0 1.91288 −8.10929 0 7.56425
1.5 −2.63231 0 4.92907 1.23074 0 3.78872 −7.71023 0 −3.23969
1.6 −2.49505 0 4.22529 −1.40756 0 −3.19016 −5.55222 0 3.51194
1.7 −2.45262 0 4.01536 −3.99887 0 −1.31658 −4.94293 0 9.80772
1.8 −2.43593 0 3.93374 2.95122 0 2.28987 −4.71046 0 −7.18896
1.9 −2.38302 0 3.67881 1.07976 0 3.12254 −4.00064 0 −2.57310
1.10 −2.36209 0 3.57945 −2.65977 0 4.59390 −3.73079 0 6.28260
1.11 −2.22247 0 2.93935 −2.94555 0 2.95569 −2.08768 0 6.54639
1.12 −2.11576 0 2.47642 3.89294 0 −3.33519 −1.00799 0 −8.23651
1.13 −2.04685 0 2.18960 0.0207958 0 −1.50444 −0.388080 0 −0.0425659
1.14 −2.01939 0 2.07795 −2.76814 0 −0.611277 −0.157420 0 5.58998
1.15 −1.91259 0 1.65799 2.48473 0 −4.55307 0.654119 0 −4.75227
1.16 −1.73329 0 1.00429 1.91493 0 4.47719 1.72585 0 −3.31913
1.17 −1.73145 0 0.997916 −1.37705 0 3.28825 1.73506 0 2.38429
1.18 −1.71009 0 0.924414 1.37341 0 −0.310223 1.83935 0 −2.34865
1.19 −1.55207 0 0.408920 3.09765 0 1.06604 2.46947 0 −4.80776
1.20 −1.50881 0 0.276511 −2.79694 0 −3.90283 2.60042 0 4.22006
See all 72 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.72 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$1$$

## 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 6561.2.a.c 72
3.b odd 2 1 6561.2.a.d 72
81.g even 27 2 81.2.g.a 144
81.g even 27 2 729.2.g.c 144
81.g even 27 2 729.2.g.d 144
81.h odd 54 2 243.2.g.a 144
81.h odd 54 2 729.2.g.a 144
81.h odd 54 2 729.2.g.b 144

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
81.2.g.a 144 81.g even 27 2
243.2.g.a 144 81.h odd 54 2
729.2.g.a 144 81.h odd 54 2
729.2.g.b 144 81.h odd 54 2
729.2.g.c 144 81.g even 27 2
729.2.g.d 144 81.g even 27 2
6561.2.a.c 72 1.a even 1 1 trivial
6561.2.a.d 72 3.b odd 2 1