# Properties

 Label 243.2.g.a Level $243$ Weight $2$ Character orbit 243.g Analytic conductor $1.940$ 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] = [243,2,Mod(10,243)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(243, base_ring=CyclotomicField(54))

chi = DirichletCharacter(H, H._module([8]))

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

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

chi := DirichletCharacter("243.10");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$243 = 3^{5}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 243.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: $$1.94036476912$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$144 q + 18 q^{2} - 18 q^{4} + 18 q^{5} - 18 q^{7} + 18 q^{8}+O(q^{10})$$ 144 * q + 18 * q^2 - 18 * q^4 + 18 * q^5 - 18 * q^7 + 18 * q^8 $$\operatorname{Tr}(f)(q) =$$ $$144 q + 18 q^{2} - 18 q^{4} + 18 q^{5} - 18 q^{7} + 18 q^{8} - 18 q^{10} + 18 q^{11} - 18 q^{13} + 18 q^{14} - 18 q^{16} + 18 q^{17} - 18 q^{19} - 18 q^{20} - 18 q^{22} - 9 q^{23} - 18 q^{25} - 45 q^{26} - 9 q^{28} - 9 q^{29} - 18 q^{31} - 36 q^{32} - 18 q^{34} - 9 q^{35} - 18 q^{37} + 9 q^{38} - 18 q^{40} - 18 q^{43} - 54 q^{44} - 18 q^{46} - 36 q^{47} - 18 q^{49} - 99 q^{50} - 45 q^{53} - 9 q^{55} - 126 q^{56} - 18 q^{58} - 45 q^{59} - 18 q^{61} - 81 q^{62} - 18 q^{64} + 9 q^{67} + 99 q^{68} + 36 q^{70} + 90 q^{71} - 18 q^{73} + 162 q^{74} + 63 q^{76} + 162 q^{77} + 36 q^{79} + 288 q^{80} - 36 q^{82} + 90 q^{83} + 36 q^{85} + 162 q^{86} + 63 q^{88} + 81 q^{89} - 18 q^{91} + 144 q^{92} + 36 q^{94} - 18 q^{95} + 9 q^{97} - 81 q^{98}+O(q^{100})$$ 144 * q + 18 * q^2 - 18 * q^4 + 18 * q^5 - 18 * q^7 + 18 * q^8 - 18 * q^10 + 18 * q^11 - 18 * q^13 + 18 * q^14 - 18 * q^16 + 18 * q^17 - 18 * q^19 - 18 * q^20 - 18 * q^22 - 9 * q^23 - 18 * q^25 - 45 * q^26 - 9 * q^28 - 9 * q^29 - 18 * q^31 - 36 * q^32 - 18 * q^34 - 9 * q^35 - 18 * q^37 + 9 * q^38 - 18 * q^40 - 18 * q^43 - 54 * q^44 - 18 * q^46 - 36 * q^47 - 18 * q^49 - 99 * q^50 - 45 * q^53 - 9 * q^55 - 126 * q^56 - 18 * q^58 - 45 * q^59 - 18 * q^61 - 81 * q^62 - 18 * q^64 + 9 * q^67 + 99 * q^68 + 36 * q^70 + 90 * q^71 - 18 * q^73 + 162 * q^74 + 63 * q^76 + 162 * q^77 + 36 * q^79 + 288 * q^80 - 36 * q^82 + 90 * q^83 + 36 * q^85 + 162 * q^86 + 63 * q^88 + 81 * q^89 - 18 * q^91 + 144 * q^92 + 36 * q^94 - 18 * q^95 + 9 * 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
10.1 −1.94437 0.976500i 0 1.63271 + 2.19311i 0.185416 + 0.619334i 0 −0.724564 1.67973i −0.277373 1.57306i 0 0.244261 1.38527i
10.2 −1.38037 0.693250i 0 0.230520 + 0.309643i −0.145961 0.487543i 0 1.57975 + 3.66226i 0.432916 + 2.45519i 0 −0.136509 + 0.774179i
10.3 −0.744407 0.373855i 0 −0.779943 1.04765i −0.345929 1.15548i 0 −0.520803 1.20736i 0.478230 + 2.71217i 0 −0.174471 + 0.989477i
10.4 0.507233 + 0.254742i 0 −1.00193 1.34582i 0.832798 + 2.78174i 0 1.23966 + 2.87385i −0.362501 2.05585i 0 −0.286203 + 1.62314i
10.5 0.698417 + 0.350758i 0 −0.829563 1.11430i −0.424677 1.41852i 0 −1.50560 3.49038i −0.459961 2.60857i 0 0.200956 1.13968i
10.6 1.07782 + 0.541304i 0 −0.325622 0.437386i −1.12732 3.76550i 0 0.736444 + 1.70727i −0.533084 3.02327i 0 0.823230 4.66877i
10.7 1.89071 + 0.949549i 0 1.47882 + 1.98639i 1.11651 + 3.72940i 0 −1.32100 3.06243i 0.175036 + 0.992677i 0 −1.43025 + 8.11138i
10.8 2.38576 + 1.19817i 0 3.06192 + 4.11287i −0.727126 2.42877i 0 0.202369 + 0.469143i 1.44988 + 8.22270i 0 1.17534 6.66569i
19.1 −2.43604 + 0.284732i 0 3.90713 0.926007i 3.57352 + 1.79469i 0 0.377600 + 1.26127i −4.64483 + 1.69058i 0 −9.21624 3.35444i
19.2 −1.69853 + 0.198530i 0 0.899496 0.213184i −1.22732 0.616384i 0 0.0889729 + 0.297190i 1.72842 0.629095i 0 2.20701 + 0.803287i
19.3 −1.04137 + 0.121718i 0 −0.876460 + 0.207725i 0.681964 + 0.342495i 0 −0.831257 2.77659i 2.85789 1.04019i 0 −0.751863 0.273656i
19.4 0.186105 0.0217526i 0 −1.91193 + 0.453135i −1.41557 0.710926i 0 1.16062 + 3.87673i −0.698107 + 0.254090i 0 −0.278909 0.101515i
19.5 0.702679 0.0821314i 0 −1.45908 + 0.345808i −3.06981 1.54172i 0 −0.775884 2.59163i −2.32646 + 0.846761i 0 −2.28372 0.831205i
19.6 0.907715 0.106097i 0 −1.13340 + 0.268621i 3.40320 + 1.70915i 0 0.208293 + 0.695746i −2.71786 + 0.989221i 0 3.27047 + 1.19035i
19.7 2.10031 0.245492i 0 2.40496 0.569987i 0.401399 + 0.201590i 0 0.699754 + 2.33734i 0.937080 0.341069i 0 0.892552 + 0.324862i
19.8 2.25893 0.264031i 0 3.08696 0.731623i 0.524452 + 0.263390i 0 −1.09261 3.64956i 2.50575 0.912019i 0 1.25424 + 0.456507i
37.1 −0.761699 + 2.54425i 0 −4.22205 2.77689i 1.12810 2.61524i 0 −0.111224 1.90965i 6.21207 5.21255i 0 5.79455 + 4.86220i
37.2 −0.587043 + 1.96086i 0 −1.82938 1.20320i −0.00823679 + 0.0190950i 0 0.0874756 + 1.50190i 0.297287 0.249453i 0 −0.0326074 0.0273608i
37.3 −0.423731 + 1.41536i 0 −0.152717 0.100444i −1.03062 + 2.38925i 0 0.0288073 + 0.494602i −2.05667 + 1.72575i 0 −2.94494 2.47110i
37.4 −0.0876154 + 0.292656i 0 1.59300 + 1.04774i 1.04144 2.41432i 0 −0.211661 3.63407i −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 235.8 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 243.2.g.a 144
3.b odd 2 1 81.2.g.a 144
9.c even 3 1 729.2.g.a 144
9.c even 3 1 729.2.g.b 144
9.d odd 6 1 729.2.g.c 144
9.d odd 6 1 729.2.g.d 144
81.g even 27 1 inner 243.2.g.a 144
81.g even 27 1 729.2.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 3.b odd 2 1
81.2.g.a 144 81.h odd 54 1
243.2.g.a 144 1.a even 1 1 trivial
243.2.g.a 144 81.g even 27 1 inner
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 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 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 is the entire newspace $$S_{2}^{\mathrm{new}}(243, [\chi])$$.