# Properties

 Label 729.2.g.a Level $729$ Weight $2$ Character orbit 729.g Analytic conductor $5.821$ Analytic rank $0$ Dimension $144$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [729,2,Mod(28,729)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(729, base_ring=CyclotomicField(54))

chi = DirichletCharacter(H, H._module([44]))

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

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

chi := DirichletCharacter("729.28");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$729 = 3^{6}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 729.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: $$5.82109430735$$ Analytic rank: $$0$$ Dimension: $$144$$ Relative dimension: $$8$$ over $$\Q(\zeta_{27})$$ Twist minimal: no (minimal twist has level 81) 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.

 $$\operatorname{Tr}(f)(q) =$$ $$144 q - 9 q^{2} + 9 q^{4} - 9 q^{5} + 9 q^{7} + 18 q^{8}+O(q^{10})$$ 144 * q - 9 * q^2 + 9 * q^4 - 9 * q^5 + 9 * q^7 + 18 * q^8 $$\operatorname{Tr}(f)(q) =$$ $$144 q - 9 q^{2} + 9 q^{4} - 9 q^{5} + 9 q^{7} + 18 q^{8} - 18 q^{10} - 9 q^{11} + 9 q^{13} - 9 q^{14} + 9 q^{16} + 18 q^{17} - 18 q^{19} - 45 q^{20} + 9 q^{22} + 45 q^{23} + 9 q^{25} - 45 q^{26} - 9 q^{28} - 36 q^{29} + 9 q^{31} + 99 q^{32} + 9 q^{34} - 9 q^{35} - 18 q^{37} - 18 q^{38} + 9 q^{40} - 27 q^{41} + 9 q^{43} - 54 q^{44} - 18 q^{46} + 99 q^{47} + 9 q^{49} - 126 q^{50} - 27 q^{52} - 45 q^{53} - 9 q^{55} + 225 q^{56} + 9 q^{58} - 72 q^{59} + 9 q^{61} - 81 q^{62} - 18 q^{64} + 81 q^{65} - 45 q^{67} - 117 q^{68} - 99 q^{70} + 90 q^{71} - 18 q^{73} - 81 q^{74} - 153 q^{76} - 81 q^{77} - 99 q^{79} + 288 q^{80} - 36 q^{82} - 45 q^{83} - 99 q^{85} - 81 q^{86} - 153 q^{88} + 81 q^{89} - 18 q^{91} - 207 q^{92} - 99 q^{94} + 171 q^{95} - 45 q^{97} - 81 q^{98}+O(q^{100})$$ 144 * q - 9 * q^2 + 9 * q^4 - 9 * q^5 + 9 * q^7 + 18 * q^8 - 18 * q^10 - 9 * q^11 + 9 * q^13 - 9 * q^14 + 9 * q^16 + 18 * q^17 - 18 * q^19 - 45 * q^20 + 9 * q^22 + 45 * q^23 + 9 * q^25 - 45 * q^26 - 9 * q^28 - 36 * q^29 + 9 * q^31 + 99 * q^32 + 9 * q^34 - 9 * q^35 - 18 * q^37 - 18 * q^38 + 9 * q^40 - 27 * q^41 + 9 * q^43 - 54 * q^44 - 18 * q^46 + 99 * q^47 + 9 * q^49 - 126 * q^50 - 27 * q^52 - 45 * q^53 - 9 * q^55 + 225 * q^56 + 9 * q^58 - 72 * q^59 + 9 * q^61 - 81 * q^62 - 18 * q^64 + 81 * q^65 - 45 * q^67 - 117 * q^68 - 99 * q^70 + 90 * q^71 - 18 * q^73 - 81 * q^74 - 153 * q^76 - 81 * q^77 - 99 * q^79 + 288 * q^80 - 36 * q^82 - 45 * q^83 - 99 * q^85 - 81 * q^86 - 153 * q^88 + 81 * q^89 - 18 * q^91 - 207 * q^92 - 99 * q^94 + 171 * q^95 - 45 * q^97 - 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
28.1 −2.23012 + 1.46677i 0 2.02985 4.70573i 1.24595 1.32063i 0 2.56492 + 0.299797i 1.44840 + 8.21430i 0 −0.841551 + 4.77267i
28.2 −1.97349 + 1.29799i 0 1.41775 3.28671i −1.82524 + 1.93464i 0 −4.56284 0.533319i 0.647845 + 3.67411i 0 1.09096 6.18715i
28.3 −0.890910 + 0.585961i 0 −0.341789 + 0.792356i −0.585850 + 0.620965i 0 0.284505 + 0.0332539i −0.530120 3.00646i 0 0.158079 0.896509i
28.4 −0.463223 + 0.304667i 0 −0.670406 + 1.55417i 0.355980 0.377317i 0 −2.24586 0.262503i −0.355511 2.01620i 0 −0.0499424 + 0.283238i
28.5 0.103498 0.0680719i 0 −0.786081 + 1.82234i −2.31283 + 2.45146i 0 3.71541 + 0.434269i 0.0857144 + 0.486111i 0 −0.0724987 + 0.411161i
28.6 1.26928 0.834819i 0 0.121992 0.282809i −0.0546290 + 0.0579033i 0 0.848729 + 0.0992022i 0.446364 + 2.53145i 0 −0.0210007 + 0.119101i
28.7 1.68031 1.10516i 0 0.809915 1.87759i −2.40279 + 2.54681i 0 −2.90760 0.339850i −0.0156557 0.0887876i 0 −1.22281 + 6.93490i
28.8 1.76633 1.16173i 0 0.978129 2.26756i 2.05186 2.17484i 0 3.48897 + 0.407803i −0.172370 0.977557i 0 1.09767 6.22518i
55.1 −0.900807 + 2.08831i 0 −2.17708 2.30757i −0.0341238 0.585883i 0 3.70692 + 0.878555i 2.50575 0.912019i 0 1.25424 + 0.456507i
55.2 −0.837555 + 1.94167i 0 −1.69610 1.79777i −0.0261173 0.448416i 0 −2.37407 0.562666i 0.937080 0.341069i 0 0.892552 + 0.324862i
55.3 −0.361975 + 0.839152i 0 0.799333 + 0.847243i −0.221432 3.80183i 0 −0.706680 0.167486i −2.71786 + 0.989221i 0 3.27047 + 1.19035i
55.4 −0.280212 + 0.649604i 0 1.02902 + 1.09069i 0.199739 + 3.42939i 0 2.63236 + 0.623881i −2.32646 + 0.846761i 0 −2.28372 0.831205i
55.5 −0.0742143 + 0.172048i 0 1.34839 + 1.42921i 0.0921050 + 1.58138i 0 −3.93765 0.933240i −0.698107 + 0.254090i 0 −0.278909 0.101515i
55.6 0.415272 0.962709i 0 0.618125 + 0.655174i −0.0443725 0.761846i 0 2.82023 + 0.668406i 2.85789 1.04019i 0 −0.751863 0.273656i
55.7 0.677333 1.57023i 0 −0.634371 0.672394i 0.0798566 + 1.37108i 0 −0.301861 0.0715423i 1.72842 0.629095i 0 2.20701 + 0.803287i
55.8 0.971435 2.25204i 0 −2.75551 2.92067i −0.232513 3.99210i 0 −1.28109 0.303625i −4.64483 + 1.69058i 0 −9.21624 3.35444i
109.1 −1.80640 1.91467i 0 −0.286600 + 4.92073i 1.22242 0.142880i 0 3.38572 + 1.70037i 5.90638 4.95604i 0 −2.48175 2.08243i
109.2 −1.38579 1.46885i 0 −0.120822 + 2.07444i −2.74943 + 0.321362i 0 −0.546257 0.274341i 0.120591 0.101188i 0 4.28217 + 3.59317i
109.3 −1.31250 1.39117i 0 −0.0964037 + 1.65519i 2.46793 0.288460i 0 −4.06877 2.04341i −0.501084 + 0.420459i 0 −3.64045 3.05470i
109.4 −0.453861 0.481064i 0 0.0908564 1.55994i −2.49650 + 0.291799i 0 3.67465 + 1.84548i −1.80495 + 1.51453i 0 1.27344 + 1.06854i
See next 80 embeddings (of 144 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 703.8 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 729.2.g.a 144
3.b odd 2 1 729.2.g.d 144
9.c even 3 1 243.2.g.a 144
9.c even 3 1 729.2.g.b 144
9.d odd 6 1 81.2.g.a 144
9.d odd 6 1 729.2.g.c 144
81.g even 27 1 243.2.g.a 144
81.g even 27 1 inner 729.2.g.a 144
81.g even 27 1 729.2.g.b 144
81.g even 27 1 6561.2.a.d 72
81.h odd 54 1 81.2.g.a 144
81.h odd 54 1 729.2.g.c 144
81.h odd 54 1 729.2.g.d 144
81.h odd 54 1 6561.2.a.c 72

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
81.2.g.a 144 9.d odd 6 1
81.2.g.a 144 81.h odd 54 1
243.2.g.a 144 9.c even 3 1
243.2.g.a 144 81.g even 27 1
729.2.g.a 144 1.a even 1 1 trivial
729.2.g.a 144 81.g even 27 1 inner
729.2.g.b 144 9.c even 3 1
729.2.g.b 144 81.g even 27 1
729.2.g.c 144 9.d odd 6 1
729.2.g.c 144 81.h odd 54 1
729.2.g.d 144 3.b odd 2 1
729.2.g.d 144 81.h odd 54 1
6561.2.a.c 72 81.h odd 54 1
6561.2.a.d 72 81.g even 27 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{144} + 9 T_{2}^{143} + 36 T_{2}^{142} + 75 T_{2}^{141} + 45 T_{2}^{140} - 306 T_{2}^{139} - 1599 T_{2}^{138} - 4932 T_{2}^{137} - 10926 T_{2}^{136} - 13287 T_{2}^{135} + 27108 T_{2}^{134} + 227907 T_{2}^{133} + 766611 T_{2}^{132} + \cdots + 13966276041$$ acting on $$S_{2}^{\mathrm{new}}(729, [\chi])$$.