# Properties

 Label 6561.2.a.d Level $6561$ Weight $2$ Character orbit 6561.a Self dual yes Analytic conductor $52.390$ Analytic rank $0$ 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: $$0$$ 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 −2.63957 0 4.96734 1.13561 0 −0.795312 −7.83251 0 −2.99751
1.2 −2.60744 0 4.79872 −0.324106 0 −3.85500 −7.29749 0 0.845087
1.3 −2.54050 0 4.45416 3.18683 0 −1.66066 −6.23481 0 −8.09616
1.4 −2.52847 0 4.39319 2.84004 0 0.550190 −6.05111 0 −7.18097
1.5 −2.27431 0 3.17247 0.586876 0 3.80961 −2.66656 0 −1.33474
1.6 −2.22007 0 2.92869 0.455962 0 −3.73033 −2.06175 0 −1.01226
1.7 −2.19283 0 2.80849 3.59965 0 1.46244 −1.77287 0 −7.89340
1.8 −2.17581 0 2.73413 −0.646494 0 −1.82934 −1.59733 0 1.40664
1.9 −2.16061 0 2.66822 −1.96432 0 −0.358362 −1.44375 0 4.24413
1.10 −2.11461 0 2.47159 0.449176 0 −2.43984 −0.997220 0 −0.949834
1.11 −2.11413 0 2.46953 −2.98999 0 −3.51273 −0.992637 0 6.32121
1.12 −2.01118 0 2.04483 3.50138 0 2.92740 −0.0901573 0 −7.04188
1.13 −1.92033 0 1.68766 −0.788458 0 1.19178 0.599803 0 1.51410
1.14 −1.90646 0 1.63461 −1.42545 0 −0.181465 0.696609 0 2.71756
1.15 −1.57217 0 0.471721 −0.858316 0 3.32977 2.40272 0 1.34942
1.16 −1.55056 0 0.404245 −3.17771 0 0.292854 2.47432 0 4.92724
1.17 −1.54468 0 0.386029 0.508923 0 3.98845 2.49307 0 −0.786122
1.18 −1.51921 0 0.307998 0.0796060 0 −0.854507 2.57051 0 −0.120938
1.19 −1.38039 0 −0.0945210 −4.03747 0 −3.55546 2.89126 0 5.57329
1.20 −1.36534 0 −0.135854 3.81012 0 1.74698 2.91616 0 −5.20210
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.d 72
3.b odd 2 1 6561.2.a.c 72
81.g even 27 2 243.2.g.a 144
81.g even 27 2 729.2.g.a 144
81.g even 27 2 729.2.g.b 144
81.h odd 54 2 81.2.g.a 144
81.h odd 54 2 729.2.g.c 144
81.h odd 54 2 729.2.g.d 144

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