# Properties

 Label 162.3.h.a Level $162$ Weight $3$ Character orbit 162.h Analytic conductor $4.414$ Analytic rank $0$ Dimension $324$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [162,3,Mod(5,162)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(162, base_ring=CyclotomicField(54))

chi = DirichletCharacter(H, H._module([23]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("162.5");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$162 = 2 \cdot 3^{4}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 162.h (of order $$54$$, 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: $$4.41418028264$$ Analytic rank: $$0$$ Dimension: $$324$$ Relative dimension: $$18$$ over $$\Q(\zeta_{54})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{54}]$

## $q$-expansion

The algebraic $$q$$-expansion of this newform has not been computed, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$324 q+O(q^{10})$$ 324 * q $$\operatorname{Tr}(f)(q) =$$ $$324 q + 72 q^{18} + 108 q^{20} + 270 q^{21} + 162 q^{23} - 54 q^{27} - 162 q^{29} - 216 q^{30} - 378 q^{33} - 486 q^{35} - 180 q^{36} - 108 q^{38} + 216 q^{41} + 432 q^{45} + 324 q^{47} + 126 q^{51} - 216 q^{57} - 378 q^{59} - 540 q^{63} - 1728 q^{65} - 1008 q^{66} + 702 q^{67} - 216 q^{68} - 1872 q^{69} + 540 q^{70} - 1296 q^{71} - 576 q^{72} - 864 q^{74} - 900 q^{75} + 108 q^{76} - 864 q^{77} - 288 q^{78} + 108 q^{79} + 144 q^{81} + 432 q^{83} + 144 q^{84} - 540 q^{85} + 864 q^{86} + 2016 q^{87} - 216 q^{88} + 1782 q^{89} + 1440 q^{90} + 864 q^{92} + 2124 q^{93} - 756 q^{94} + 2808 q^{95} + 288 q^{96} - 918 q^{97} + 1296 q^{98} + 1800 q^{99}+O(q^{100})$$ 324 * q + 72 * q^18 + 108 * q^20 + 270 * q^21 + 162 * q^23 - 54 * q^27 - 162 * q^29 - 216 * q^30 - 378 * q^33 - 486 * q^35 - 180 * q^36 - 108 * q^38 + 216 * q^41 + 432 * q^45 + 324 * q^47 + 126 * q^51 - 216 * q^57 - 378 * q^59 - 540 * q^63 - 1728 * q^65 - 1008 * q^66 + 702 * q^67 - 216 * q^68 - 1872 * q^69 + 540 * q^70 - 1296 * q^71 - 576 * q^72 - 864 * q^74 - 900 * q^75 + 108 * q^76 - 864 * q^77 - 288 * q^78 + 108 * q^79 + 144 * q^81 + 432 * q^83 + 144 * q^84 - 540 * q^85 + 864 * q^86 + 2016 * q^87 - 216 * q^88 + 1782 * q^89 + 1440 * q^90 + 864 * q^92 + 2124 * q^93 - 756 * q^94 + 2808 * q^95 + 288 * q^96 - 918 * q^97 + 1296 * q^98 + 1800 * 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   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
5.1 −0.326140 1.37609i −2.76034 + 1.17495i −1.78727 + 0.897598i 6.70564 4.99216i 2.51710 + 3.41529i −6.54938 + 4.30759i 1.81808 + 2.16670i 6.23898 6.48654i −9.05666 7.59944i
5.2 −0.326140 1.37609i −2.75282 1.19248i −1.78727 + 0.897598i −1.51997 + 1.13158i −0.743160 + 4.17705i −2.05983 + 1.35477i 1.81808 + 2.16670i 6.15598 + 6.56535i 2.05288 + 1.72257i
5.3 −0.326140 1.37609i −2.45330 + 1.72665i −1.78727 + 0.897598i −4.47588 + 3.33217i 3.17615 + 2.81284i 7.95611 5.23282i 1.81808 + 2.16670i 3.03734 8.47199i 6.04514 + 5.07248i
5.4 −0.326140 1.37609i −1.15412 2.76912i −1.78727 + 0.897598i 5.90063 4.39286i −3.43416 + 2.49129i 11.1239 7.31631i 1.81808 + 2.16670i −6.33603 + 6.39177i −7.96942 6.68713i
5.5 −0.326140 1.37609i −0.663139 + 2.92579i −1.78727 + 0.897598i 0.605521 0.450794i 4.24244 0.0416761i 1.59852 1.05136i 1.81808 + 2.16670i −8.12049 3.88041i −0.817819 0.686231i
5.6 −0.326140 1.37609i 0.243053 2.99014i −1.78727 + 0.897598i −2.26822 + 1.68862i −4.19398 + 0.640740i −8.30694 + 5.46356i 1.81808 + 2.16670i −8.88185 1.45353i 3.06346 + 2.57055i
5.7 −0.326140 1.37609i 2.26424 + 1.96805i −1.78727 + 0.897598i 5.11004 3.80428i 1.96976 3.75766i 1.09207 0.718268i 1.81808 + 2.16670i 1.25355 + 8.91227i −6.90163 5.79116i
5.8 −0.326140 1.37609i 2.56706 1.55248i −1.78727 + 0.897598i −0.00733898 + 0.00546367i −2.97357 3.02620i 2.72886 1.79480i 1.81808 + 2.16670i 4.17964 7.97061i 0.00991205 + 0.00831720i
5.9 −0.326140 1.37609i 2.89210 + 0.797352i −1.78727 + 0.897598i −7.40578 + 5.51340i 0.154002 4.23984i −1.07376 + 0.706223i 1.81808 + 2.16670i 7.72846 + 4.61204i 10.0023 + 8.39290i
5.10 0.326140 + 1.37609i −2.96504 0.456664i −1.78727 + 0.897598i −1.81547 + 1.35157i −0.338606 4.22911i −1.49360 + 0.982359i −1.81808 2.16670i 8.58292 + 2.70805i −2.45198 2.05746i
5.11 0.326140 + 1.37609i −2.78435 + 1.11687i −1.78727 + 0.897598i 5.92375 4.41007i −2.44501 3.46727i 6.83506 4.49549i −1.81808 2.16670i 6.50519 6.21953i 8.00064 + 6.71334i
5.12 0.326140 + 1.37609i −2.21854 2.01943i −1.78727 + 0.897598i 2.97271 2.21310i 2.05537 3.71153i −8.93104 + 5.87404i −1.81808 2.16670i 0.843823 + 8.96036i 4.01495 + 3.36895i
5.13 0.326140 + 1.37609i −1.18720 + 2.75509i −1.78727 + 0.897598i −3.75904 + 2.79850i −4.17846 0.735156i 1.03380 0.679940i −1.81808 2.16670i −6.18109 6.54172i −5.07698 4.26009i
5.14 0.326140 + 1.37609i −0.0281193 2.99987i −1.78727 + 0.897598i 1.73126 1.28887i 4.11893 1.01707i 3.67681 2.41828i −1.81808 2.16670i −8.99842 + 0.168708i 2.33824 + 1.96202i
5.15 0.326140 + 1.37609i 1.73974 2.44404i −1.78727 + 0.897598i −6.68826 + 4.97922i 3.93062 + 1.59694i −10.2458 + 6.73874i −1.81808 2.16670i −2.94663 8.50396i −9.03318 7.57974i
5.16 0.326140 + 1.37609i 2.18959 + 2.05078i −1.78727 + 0.897598i −1.03753 + 0.772415i −2.10794 + 3.68193i −4.52918 + 2.97889i −1.81808 2.16670i 0.588642 + 8.98073i −1.40130 1.17583i
5.17 0.326140 + 1.37609i 2.81870 1.02711i −1.78727 + 0.897598i −1.67736 + 1.24875i 2.33268 + 3.54381i 11.2134 7.37517i −1.81808 2.16670i 6.89011 5.79020i −2.26544 1.90093i
5.18 0.326140 + 1.37609i 2.88671 0.816639i −1.78727 + 0.897598i 6.99459 5.20728i 2.06524 + 3.70604i −4.06903 + 2.67625i −1.81808 2.16670i 7.66620 4.71480i 9.44691 + 7.92690i
11.1 −1.02866 0.970492i −2.92008 0.687859i 0.116290 + 1.99662i 0.533779 4.56677i 2.33621 + 3.54149i 7.85421 3.94453i 1.81808 2.16670i 8.05370 + 4.01720i −4.98110 + 4.17964i
11.2 −1.02866 0.970492i −2.72678 + 1.25087i 0.116290 + 1.99662i 0.295075 2.52452i 4.01889 + 1.35959i −6.23888 + 3.13328i 1.81808 2.16670i 5.87063 6.82170i −2.75356 + 2.31051i
See next 80 embeddings (of 324 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 5.18 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.h odd 54 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 162.3.h.a 324
81.h odd 54 1 inner 162.3.h.a 324

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
162.3.h.a 324 1.a even 1 1 trivial
162.3.h.a 324 81.h odd 54 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{3}^{\mathrm{new}}(162, [\chi])$$.