# Properties

 Label 162.2.g.b Level $162$ Weight $2$ Character orbit 162.g Analytic conductor $1.294$ Analytic rank $0$ Dimension $90$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [162,2,Mod(7,162)]

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([16]))

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

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

chi := DirichletCharacter("162.7");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$162 = 2 \cdot 3^{4}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 162.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: $$1.29357651274$$ Analytic rank: $$0$$ Dimension: $$90$$ Relative dimension: $$5$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$90 q - 9 q^{6}+O(q^{10})$$ 90 * q - 9 * q^6 $$\operatorname{Tr}(f)(q) =$$ $$90 q - 9 q^{6} - 18 q^{13} - 9 q^{18} - 9 q^{20} - 54 q^{21} + 27 q^{23} - 18 q^{25} - 27 q^{26} - 27 q^{27} - 18 q^{28} - 27 q^{29} + 9 q^{30} + 54 q^{31} - 63 q^{33} - 27 q^{35} - 9 q^{36} - 18 q^{38} - 9 q^{41} - 9 q^{42} - 36 q^{43} + 63 q^{45} + 18 q^{46} - 27 q^{47} - 9 q^{48} - 36 q^{51} + 36 q^{52} - 27 q^{53} - 54 q^{55} - 81 q^{57} - 9 q^{58} - 45 q^{59} - 63 q^{63} + 9 q^{65} + 36 q^{66} + 81 q^{67} + 36 q^{68} + 18 q^{69} - 72 q^{70} + 72 q^{71} + 18 q^{72} - 36 q^{73} + 45 q^{74} + 216 q^{75} - 18 q^{76} + 144 q^{77} + 54 q^{78} - 99 q^{79} + 18 q^{80} + 144 q^{81} + 72 q^{82} + 45 q^{83} + 18 q^{84} - 117 q^{85} + 72 q^{86} + 81 q^{87} - 18 q^{88} + 45 q^{89} + 162 q^{90} - 63 q^{91} + 36 q^{92} + 45 q^{93} - 72 q^{94} + 45 q^{95} + 18 q^{96} + 117 q^{97} + 36 q^{98} - 81 q^{99}+O(q^{100})$$ 90 * q - 9 * q^6 - 18 * q^13 - 9 * q^18 - 9 * q^20 - 54 * q^21 + 27 * q^23 - 18 * q^25 - 27 * q^26 - 27 * q^27 - 18 * q^28 - 27 * q^29 + 9 * q^30 + 54 * q^31 - 63 * q^33 - 27 * q^35 - 9 * q^36 - 18 * q^38 - 9 * q^41 - 9 * q^42 - 36 * q^43 + 63 * q^45 + 18 * q^46 - 27 * q^47 - 9 * q^48 - 36 * q^51 + 36 * q^52 - 27 * q^53 - 54 * q^55 - 81 * q^57 - 9 * q^58 - 45 * q^59 - 63 * q^63 + 9 * q^65 + 36 * q^66 + 81 * q^67 + 36 * q^68 + 18 * q^69 - 72 * q^70 + 72 * q^71 + 18 * q^72 - 36 * q^73 + 45 * q^74 + 216 * q^75 - 18 * q^76 + 144 * q^77 + 54 * q^78 - 99 * q^79 + 18 * q^80 + 144 * q^81 + 72 * q^82 + 45 * q^83 + 18 * q^84 - 117 * q^85 + 72 * q^86 + 81 * q^87 - 18 * q^88 + 45 * q^89 + 162 * q^90 - 63 * q^91 + 36 * q^92 + 45 * q^93 - 72 * q^94 + 45 * q^95 + 18 * q^96 + 117 * q^97 + 36 * q^98 - 81 * 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   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
7.1 0.597159 + 0.802123i −1.58049 + 0.708545i −0.286803 + 0.957990i −0.763144 + 0.501928i −1.51215 0.844638i −2.48772 + 2.63683i −0.939693 + 0.342020i 1.99593 2.23970i −0.858326 0.312405i
7.2 0.597159 + 0.802123i 0.156275 + 1.72499i −0.286803 + 0.957990i 3.59342 2.36343i −1.29033 + 1.15544i −0.0885835 + 0.0938930i −0.939693 + 0.342020i −2.95116 + 0.539145i 4.04160 + 1.47102i
7.3 0.597159 + 0.802123i 0.427379 1.67850i −0.286803 + 0.957990i 1.23282 0.810835i 1.60157 0.659517i 0.694984 0.736640i −0.939693 + 0.342020i −2.63469 1.43471i 1.38658 + 0.504672i
7.4 0.597159 + 0.802123i 0.497048 + 1.65920i −0.286803 + 0.957990i −3.09160 + 2.03338i −1.03407 + 1.38950i 2.90797 3.08226i −0.939693 + 0.342020i −2.50589 + 1.64940i −3.47720 1.26560i
7.5 0.597159 + 0.802123i 1.73136 0.0489119i −0.286803 + 0.957990i −0.660279 + 0.434273i 1.07313 + 1.35956i −2.31933 + 2.45835i −0.939693 + 0.342020i 2.99522 0.169368i −0.742632 0.270296i
13.1 0.893633 + 0.448799i −1.57770 0.714739i 0.597159 + 0.802123i 0.891212 + 2.97686i −1.08911 1.34679i 0.275210 + 0.638009i 0.173648 + 0.984808i 1.97830 + 2.25529i −0.539594 + 3.06019i
13.2 0.893633 + 0.448799i −0.977769 1.42967i 0.597159 + 0.802123i −1.15296 3.85116i −0.232129 1.71643i −0.663579 1.53835i 0.173648 + 0.984808i −1.08794 + 2.79578i 0.698073 3.95897i
13.3 0.893633 + 0.448799i −0.530234 + 1.64889i 0.597159 + 0.802123i −0.536865 1.79325i −1.21386 + 1.23554i 1.99514 + 4.62524i 0.173648 + 0.984808i −2.43770 1.74860i 0.325051 1.84346i
13.4 0.893633 + 0.448799i 1.02523 1.39603i 0.597159 + 0.802123i 0.664151 + 2.21842i 1.54272 0.787412i −0.590051 1.36789i 0.173648 + 0.984808i −0.897788 2.86251i −0.402118 + 2.28052i
13.5 0.893633 + 0.448799i 1.51179 + 0.845280i 0.597159 + 0.802123i −0.379535 1.26773i 0.971622 + 1.43386i −0.768169 1.78082i 0.173648 + 0.984808i 1.57100 + 2.55577i 0.229793 1.30322i
25.1 0.893633 0.448799i −1.57770 + 0.714739i 0.597159 0.802123i 0.891212 2.97686i −1.08911 + 1.34679i 0.275210 0.638009i 0.173648 0.984808i 1.97830 2.25529i −0.539594 3.06019i
25.2 0.893633 0.448799i −0.977769 + 1.42967i 0.597159 0.802123i −1.15296 + 3.85116i −0.232129 + 1.71643i −0.663579 + 1.53835i 0.173648 0.984808i −1.08794 2.79578i 0.698073 + 3.95897i
25.3 0.893633 0.448799i −0.530234 1.64889i 0.597159 0.802123i −0.536865 + 1.79325i −1.21386 1.23554i 1.99514 4.62524i 0.173648 0.984808i −2.43770 + 1.74860i 0.325051 + 1.84346i
25.4 0.893633 0.448799i 1.02523 + 1.39603i 0.597159 0.802123i 0.664151 2.21842i 1.54272 + 0.787412i −0.590051 + 1.36789i 0.173648 0.984808i −0.897788 + 2.86251i −0.402118 2.28052i
25.5 0.893633 0.448799i 1.51179 0.845280i 0.597159 0.802123i −0.379535 + 1.26773i 0.971622 1.43386i −0.768169 + 1.78082i 0.173648 0.984808i 1.57100 2.55577i 0.229793 + 1.30322i
31.1 0.396080 + 0.918216i −1.62304 + 0.604777i −0.686242 + 0.727374i −0.102641 + 1.76229i −1.19817 1.25076i −1.24262 + 0.294506i −0.939693 0.342020i 2.26849 1.96315i −1.65881 + 0.603759i
31.2 0.396080 + 0.918216i −1.57515 0.720348i −0.686242 + 0.727374i 0.193287 3.31862i 0.0375499 1.73164i 2.86227 0.678371i −0.939693 0.342020i 1.96220 + 2.26931i 3.12376 1.13696i
31.3 0.396080 + 0.918216i 0.178274 1.72285i −0.686242 + 0.727374i −0.253242 + 4.34800i 1.65256 0.518693i 3.96081 0.938728i −0.939693 0.342020i −2.93644 0.614278i −4.09271 + 1.48962i
31.4 0.396080 + 0.918216i 0.360138 + 1.69420i −0.686242 + 0.727374i −0.0437737 + 0.751566i −1.41299 + 1.00172i −2.40536 + 0.570080i −0.939693 0.342020i −2.74060 + 1.22029i −0.707437 + 0.257486i
31.5 0.396080 + 0.918216i 1.72468 + 0.159609i −0.686242 + 0.727374i 0.0350699 0.602127i 0.536555 + 1.64685i 0.510080 0.120891i −0.939693 0.342020i 2.94905 + 0.550550i 0.566774 0.206289i
See all 90 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 7.5 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 162.2.g.b 90
3.b odd 2 1 486.2.g.b 90
81.g even 27 1 inner 162.2.g.b 90
81.h odd 54 1 486.2.g.b 90

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
162.2.g.b 90 1.a even 1 1 trivial
162.2.g.b 90 81.g even 27 1 inner
486.2.g.b 90 3.b odd 2 1
486.2.g.b 90 81.h odd 54 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{90} + 9 T_{5}^{88} + 30 T_{5}^{87} + 45 T_{5}^{86} + 1377 T_{5}^{85} + 2673 T_{5}^{84} + \cdots + 21\!\cdots\!84$$ acting on $$S_{2}^{\mathrm{new}}(162, [\chi])$$.