# Properties

 Label 729.2.g.b Level $729$ Weight $2$ Character orbit 729.g Analytic conductor $5.821$ Analytic rank $0$ Dimension $144$ 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 algebraic $$q$$-expansion of this newform has not been computed, 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 $$\operatorname{Tr}(f)(q) =$$ $$144 q - 9 q^{2} + 9 q^{4} - 9 q^{5} + 9 q^{7} + 18 q^{8} - 18 q^{10} - 9 q^{11} + 9 q^{13} - 9 q^{14} + 9 q^{16} + 18 q^{17} - 18 q^{19} + 63 q^{20} + 9 q^{22} - 36 q^{23} + 9 q^{25} - 45 q^{26} - 9 q^{28} + 45 q^{29} + 9 q^{31} - 63 q^{32} + 9 q^{34} - 9 q^{35} - 18 q^{37} + 9 q^{38} + 9 q^{40} + 27 q^{41} + 9 q^{43} - 54 q^{44} - 18 q^{46} - 63 q^{47} + 9 q^{49} + 225 q^{50} + 27 q^{52} - 45 q^{53} - 9 q^{55} - 99 q^{56} + 9 q^{58} + 117 q^{59} + 9 q^{61} - 81 q^{62} - 18 q^{64} - 81 q^{65} + 36 q^{67} + 18 q^{68} + 63 q^{70} + 90 q^{71} - 18 q^{73} - 81 q^{74} + 90 q^{76} - 81 q^{77} + 63 q^{79} + 288 q^{80} - 36 q^{82} - 45 q^{83} + 63 q^{85} - 81 q^{86} + 90 q^{88} + 81 q^{89} - 18 q^{91} + 63 q^{92} + 63 q^{94} - 153 q^{95} + 36 q^{97} - 81 q^{98}+O(q^{100})$$ 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 + 63 * q^20 + 9 * q^22 - 36 * q^23 + 9 * q^25 - 45 * q^26 - 9 * q^28 + 45 * q^29 + 9 * q^31 - 63 * q^32 + 9 * q^34 - 9 * q^35 - 18 * q^37 + 9 * q^38 + 9 * q^40 + 27 * q^41 + 9 * q^43 - 54 * q^44 - 18 * q^46 - 63 * q^47 + 9 * q^49 + 225 * q^50 + 27 * q^52 - 45 * q^53 - 9 * q^55 - 99 * q^56 + 9 * q^58 + 117 * q^59 + 9 * q^61 - 81 * q^62 - 18 * q^64 - 81 * q^65 + 36 * q^67 + 18 * q^68 + 63 * q^70 + 90 * q^71 - 18 * q^73 - 81 * q^74 + 90 * q^76 - 81 * q^77 + 63 * q^79 + 288 * q^80 - 36 * q^82 - 45 * q^83 + 63 * q^85 - 81 * q^86 + 90 * q^88 + 81 * q^89 - 18 * q^91 + 63 * q^92 + 63 * q^94 - 153 * q^95 + 36 * 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 −2.23053 + 1.46704i 0 2.03089 4.70814i −1.73981 + 1.84410i 0 −0.507474 0.0593153i 1.44988 + 8.22270i 0 1.17534 6.66569i
28.2 −1.76769 + 1.16263i 0 0.980860 2.27389i 2.67150 2.83162i 0 3.31264 + 0.387192i 0.175036 + 0.992677i 0 −1.43025 + 8.11138i
28.3 −1.00769 + 0.662771i 0 −0.215977 + 0.500690i −2.69736 + 2.85904i 0 −1.84676 0.215855i −0.533084 3.02327i 0 0.823230 4.66877i
28.4 −0.652974 + 0.429468i 0 −0.550227 + 1.27557i −1.01614 + 1.07704i 0 3.77556 + 0.441300i −0.459961 2.60857i 0 0.200956 1.13968i
28.5 −0.474230 + 0.311906i 0 −0.664551 + 1.54060i 1.99266 2.11209i 0 −3.10865 0.363349i −0.362501 2.05585i 0 −0.286203 + 1.62314i
28.6 0.695972 0.457748i 0 −0.517316 + 1.19927i −0.827713 + 0.877324i 0 1.30600 + 0.152650i 0.478230 + 2.71217i 0 −0.174471 + 0.989477i
28.7 1.29056 0.848814i 0 0.152898 0.354458i −0.349244 + 0.370177i 0 −3.96148 0.463031i 0.432916 + 2.45519i 0 −0.136509 + 0.774179i
28.8 1.81786 1.19562i 0 1.08293 2.51052i 0.443651 0.470242i 0 1.81697 + 0.212373i −0.277373 1.57306i 0 0.244261 1.38527i
55.1 −1.03275 + 2.39419i 0 −3.29308 3.49047i 0.0188451 + 0.323558i 0 −3.75109 0.889024i 6.85740 2.49589i 0 −0.794122 0.289037i
55.2 −0.614147 + 1.42375i 0 −0.277410 0.294037i 0.184768 + 3.17234i 0 0.284960 + 0.0675368i −2.32510 + 0.846267i 0 −4.63010 1.68522i
55.3 −0.311913 + 0.723096i 0 0.946905 + 1.00366i −0.161980 2.78108i 0 4.84803 + 1.14900i −2.50111 + 0.910331i 0 2.06152 + 0.750330i
55.4 0.0800459 0.185567i 0 1.34446 + 1.42504i 0.0529885 + 0.909778i 0 0.159621 + 0.0378310i 0.751874 0.273660i 0 0.173067 + 0.0629911i
55.5 0.314515 0.729128i 0 0.939775 + 0.996103i −0.127484 2.18881i 0 −3.45959 0.819939i 2.51422 0.915103i 0 −1.63602 0.595462i
55.6 0.588281 1.36379i 0 −0.141360 0.149832i 0.0932044 + 1.60026i 0 −1.85061 0.438602i 2.50387 0.911335i 0 2.23724 + 0.814290i
55.7 0.880274 2.04070i 0 −2.01711 2.13801i −0.171269 2.94057i 0 2.87602 + 0.681629i −1.96178 + 0.714030i 0 −6.15160 2.23900i
55.8 0.964822 2.23671i 0 −2.69950 2.86130i 0.171598 + 2.94623i 0 2.22815 + 0.528081i −4.42639 + 1.61107i 0 6.75541 + 2.45877i
109.1 −1.82254 1.93178i 0 −0.293829 + 5.04485i −2.82892 + 0.330653i 0 1.70942 + 0.858501i 6.21207 5.21255i 0 5.79455 + 4.86220i
109.2 −1.40463 1.48883i 0 −0.127314 + 2.18589i 0.0206552 0.00241424i 0 −1.34442 0.675193i 0.297287 0.249453i 0 −0.0326074 0.0273608i
109.3 −1.01387 1.07464i 0 −0.0106282 + 0.182479i 2.58446 0.302080i 0 −0.442742 0.222353i −2.05667 + 1.72575i 0 −2.94494 2.47110i
109.4 −0.209640 0.222205i 0 0.110863 1.90345i −2.61158 + 0.305250i 0 3.25303 + 1.63373i −0.914234 + 0.767134i 0 0.615319 + 0.516314i
See next 80 embeddings (of 144 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 28.8 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.b 144
3.b odd 2 1 729.2.g.c 144
9.c even 3 1 243.2.g.a 144
9.c even 3 1 729.2.g.a 144
9.d odd 6 1 81.2.g.a 144
9.d odd 6 1 729.2.g.d 144
81.g even 27 1 243.2.g.a 144
81.g even 27 1 729.2.g.a 144
81.g even 27 1 inner 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 9.c even 3 1
729.2.g.a 144 81.g even 27 1
729.2.g.b 144 1.a even 1 1 trivial
729.2.g.b 144 81.g even 27 1 inner
729.2.g.c 144 3.b odd 2 1
729.2.g.c 144 81.h odd 54 1
729.2.g.d 144 9.d odd 6 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 $$T_{2}^{144} + 9 T_{2}^{143} + 36 T_{2}^{142} + 75 T_{2}^{141} + 45 T_{2}^{140} - 144 T_{2}^{139} + \cdots + 13966276041$$ acting on $$S_{2}^{\mathrm{new}}(729, [\chi])$$.