# Properties

 Label 81.2.g.a Level $81$ Weight $2$ Character orbit 81.g Analytic conductor $0.647$ 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] = [81,2,Mod(4,81)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(81, base_ring=CyclotomicField(54))

chi = DirichletCharacter(H, H._module([2]))

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

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

chi := DirichletCharacter("81.4");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$81 = 3^{4}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 81.g (of order $$27$$, degree $$18$$, 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: $$0.646788256372$$ Analytic rank: $$0$$ Dimension: $$144$$ Relative dimension: $$8$$ over $$\Q(\zeta_{27})$$ Twist minimal: yes 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.

 $$\operatorname{Tr}(f)(q) =$$ $$144 q - 18 q^{2} - 18 q^{3} - 18 q^{4} - 18 q^{5} - 18 q^{6} - 18 q^{7} - 18 q^{8} - 18 q^{9}+O(q^{10})$$ 144 * q - 18 * q^2 - 18 * q^3 - 18 * q^4 - 18 * q^5 - 18 * q^6 - 18 * q^7 - 18 * q^8 - 18 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$144 q - 18 q^{2} - 18 q^{3} - 18 q^{4} - 18 q^{5} - 18 q^{6} - 18 q^{7} - 18 q^{8} - 18 q^{9} - 18 q^{10} - 18 q^{11} - 18 q^{12} - 18 q^{13} - 18 q^{14} - 18 q^{15} - 18 q^{16} - 18 q^{17} - 9 q^{18} - 18 q^{19} + 18 q^{20} + 9 q^{21} - 18 q^{22} + 9 q^{23} + 36 q^{24} - 18 q^{25} + 45 q^{26} + 9 q^{27} - 9 q^{28} + 9 q^{29} + 36 q^{30} - 18 q^{31} + 36 q^{32} + 9 q^{33} - 18 q^{34} + 9 q^{35} + 18 q^{36} - 18 q^{37} - 9 q^{38} - 18 q^{39} - 18 q^{40} + 27 q^{42} - 18 q^{43} + 54 q^{44} + 36 q^{45} - 18 q^{46} + 36 q^{47} + 81 q^{48} - 18 q^{49} + 99 q^{50} + 45 q^{51} + 45 q^{53} + 108 q^{54} - 9 q^{55} + 126 q^{56} + 36 q^{57} - 18 q^{58} + 45 q^{59} + 99 q^{60} - 18 q^{61} + 81 q^{62} + 36 q^{63} - 18 q^{64} - 18 q^{66} + 9 q^{67} - 99 q^{68} - 72 q^{69} + 36 q^{70} - 90 q^{71} - 234 q^{72} - 18 q^{73} - 162 q^{74} - 108 q^{75} + 63 q^{76} - 162 q^{77} - 135 q^{78} + 36 q^{79} - 288 q^{80} - 90 q^{81} - 36 q^{82} - 90 q^{83} - 243 q^{84} + 36 q^{85} - 162 q^{86} - 162 q^{87} + 63 q^{88} - 81 q^{89} - 99 q^{90} - 18 q^{91} - 144 q^{92} + 36 q^{94} + 18 q^{95} - 27 q^{96} + 9 q^{97} + 81 q^{98} + 54 q^{99}+O(q^{100})$$ 144 * q - 18 * q^2 - 18 * q^3 - 18 * q^4 - 18 * q^5 - 18 * q^6 - 18 * q^7 - 18 * q^8 - 18 * q^9 - 18 * q^10 - 18 * q^11 - 18 * q^12 - 18 * q^13 - 18 * q^14 - 18 * q^15 - 18 * q^16 - 18 * q^17 - 9 * q^18 - 18 * q^19 + 18 * q^20 + 9 * q^21 - 18 * q^22 + 9 * q^23 + 36 * q^24 - 18 * q^25 + 45 * q^26 + 9 * q^27 - 9 * q^28 + 9 * q^29 + 36 * q^30 - 18 * q^31 + 36 * q^32 + 9 * q^33 - 18 * q^34 + 9 * q^35 + 18 * q^36 - 18 * q^37 - 9 * q^38 - 18 * q^39 - 18 * q^40 + 27 * q^42 - 18 * q^43 + 54 * q^44 + 36 * q^45 - 18 * q^46 + 36 * q^47 + 81 * q^48 - 18 * q^49 + 99 * q^50 + 45 * q^51 + 45 * q^53 + 108 * q^54 - 9 * q^55 + 126 * q^56 + 36 * q^57 - 18 * q^58 + 45 * q^59 + 99 * q^60 - 18 * q^61 + 81 * q^62 + 36 * q^63 - 18 * q^64 - 18 * q^66 + 9 * q^67 - 99 * q^68 - 72 * q^69 + 36 * q^70 - 90 * q^71 - 234 * q^72 - 18 * q^73 - 162 * q^74 - 108 * q^75 + 63 * q^76 - 162 * q^77 - 135 * q^78 + 36 * q^79 - 288 * q^80 - 90 * q^81 - 36 * q^82 - 90 * q^83 - 243 * q^84 + 36 * q^85 - 162 * q^86 - 162 * q^87 + 63 * q^88 - 81 * q^89 - 99 * q^90 - 18 * q^91 - 144 * q^92 + 36 * q^94 + 18 * q^95 - 27 * q^96 + 9 * q^97 + 81 * q^98 + 54 * q^99

## 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   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
4.1 −2.25893 0.264031i −1.57391 0.723054i 3.08696 + 0.731623i −0.524452 + 0.263390i 3.36444 + 2.04889i −1.09261 + 3.64956i −2.50575 0.912019i 1.95439 + 2.27604i 1.25424 0.456507i
4.2 −2.10031 0.245492i 1.49640 0.872226i 2.40496 + 0.569987i −0.401399 + 0.201590i −3.35704 + 1.46459i 0.699754 2.33734i −0.937080 0.341069i 1.47844 2.61040i 0.892552 0.324862i
4.3 −0.907715 0.106097i −0.350962 + 1.69612i −1.13340 0.268621i −3.40320 + 1.70915i 0.498526 1.50236i 0.208293 0.695746i 2.71786 + 0.989221i −2.75365 1.19055i 3.27047 1.19035i
4.4 −0.702679 0.0821314i 1.28612 + 1.16012i −1.45908 0.345808i 3.06981 1.54172i −0.808449 0.920826i −0.775884 + 2.59163i 2.32646 + 0.846761i 0.308226 + 2.98412i −2.28372 + 0.831205i
4.5 −0.186105 0.0217526i −0.680105 1.59294i −1.91193 0.453135i 1.41557 0.710926i 0.0919205 + 0.311248i 1.16062 3.87673i 0.698107 + 0.254090i −2.07491 + 2.16673i −0.278909 + 0.101515i
4.6 1.04137 + 0.121718i 1.32397 1.11673i −0.876460 0.207725i −0.681964 + 0.342495i 1.51467 1.00178i −0.831257 + 2.77659i −2.85789 1.04019i 0.505808 2.95705i −0.751863 + 0.273656i
4.7 1.69853 + 0.198530i −1.07544 + 1.35773i 0.899496 + 0.213184i 1.22732 0.616384i −2.09621 + 2.09264i 0.0889729 0.297190i −1.72842 0.629095i −0.686860 2.92031i 2.20701 0.803287i
4.8 2.43604 + 0.284732i −1.18775 1.26065i 3.90713 + 0.926007i −3.57352 + 1.79469i −2.53446 3.40919i 0.377600 1.26127i 4.64483 + 1.69058i −0.178488 + 2.99469i −9.21624 + 3.35444i
7.1 −1.45464 1.95391i −0.684440 + 1.59108i −1.12821 + 3.76849i −2.46571 + 1.62172i 4.10445 0.977107i −1.57141 + 1.66559i 4.42639 1.61107i −2.06308 2.17800i 6.75541 + 2.45877i
7.2 −1.32716 1.78269i 1.62274 0.605570i −0.843016 + 2.81587i 2.46097 1.61861i −3.23319 2.08915i −2.02832 + 2.14989i 1.96178 0.714030i 2.26657 1.96536i −6.15160 2.23900i
7.3 −0.886934 1.19136i −0.340526 1.69825i −0.0590788 + 0.197337i −1.33926 + 0.880846i −1.72120 + 1.91192i 1.30515 1.38337i −2.50387 + 0.911335i −2.76808 + 1.15659i 2.23724 + 0.814290i
7.4 −0.474186 0.636942i −1.48279 + 0.895173i 0.392763 1.31192i 1.83182 1.20481i 1.27329 + 0.519973i 2.43989 2.58613i −2.51422 + 0.915103i 1.39733 2.65471i −1.63602 0.595462i
7.5 −0.120683 0.162105i 1.25647 + 1.19218i 0.561893 1.87685i −0.761396 + 0.500778i 0.0416250 0.347556i −0.112573 + 0.119321i −0.751874 + 0.273660i 0.157410 + 2.99587i 0.173067 + 0.0629911i
7.6 0.470263 + 0.631673i −1.10956 1.32999i 0.395743 1.32187i 2.32750 1.53082i 0.318331 1.32633i −3.41908 + 3.62402i 2.50111 0.910331i −0.537739 + 2.95141i 2.06152 + 0.750330i
7.7 0.925932 + 1.24374i 1.03494 1.38884i −0.115939 + 0.387262i −2.65494 + 1.74618i 2.68565 + 0.00122935i −0.200969 + 0.213014i 2.32510 0.846267i −0.857778 2.87476i −4.63010 1.68522i
7.8 1.55705 + 2.09149i −1.63477 0.572310i −1.37629 + 4.59713i −0.270787 + 0.178099i −1.34844 4.31021i 2.64546 2.80403i −6.85740 + 2.49589i 2.34492 + 1.87119i −0.794122 0.289037i
13.1 −2.38576 1.19817i 1.71560 0.238151i 3.06192 + 4.11287i 0.727126 + 2.42877i −4.37836 1.48742i 0.202369 + 0.469143i −1.44988 8.22270i 2.88657 0.817145i 1.17534 6.66569i
13.2 −1.89071 0.949549i −0.540889 + 1.64543i 1.47882 + 1.98639i −1.11651 3.72940i 2.58508 2.59743i −1.32100 3.06243i −0.175036 0.992677i −2.41488 1.77999i −1.43025 + 8.11138i
13.3 −1.07782 0.541304i −1.45923 + 0.933081i −0.325622 0.437386i 1.12732 + 3.76550i 2.07788 0.215810i 0.736444 + 1.70727i 0.533084 + 3.02327i 1.25872 2.72317i 0.823230 4.66877i
13.4 −0.698417 0.350758i 0.295453 1.70667i −0.829563 1.11430i 0.424677 + 1.41852i −0.804976 + 1.08833i −1.50560 3.49038i 0.459961 + 2.60857i −2.82541 1.00848i 0.200956 1.13968i
See next 80 embeddings (of 144 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 4.8 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 81.2.g.a 144
3.b odd 2 1 243.2.g.a 144
9.c even 3 1 729.2.g.c 144
9.c even 3 1 729.2.g.d 144
9.d odd 6 1 729.2.g.a 144
9.d odd 6 1 729.2.g.b 144
81.g even 27 1 inner 81.2.g.a 144
81.g even 27 1 729.2.g.c 144
81.g even 27 1 729.2.g.d 144
81.g even 27 1 6561.2.a.c 72
81.h odd 54 1 243.2.g.a 144
81.h odd 54 1 729.2.g.a 144
81.h odd 54 1 729.2.g.b 144
81.h odd 54 1 6561.2.a.d 72

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

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(81, [\chi])$$.