# Properties

 Label 162.2.g.a Level 162 Weight 2 Character orbit 162.g Analytic conductor 1.294 Analytic rank 0 Dimension 72 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$162 = 2 \cdot 3^{4}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 162.g (of order $$27$$, degree $$18$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.29357651274$$ Analytic rank: $$0$$ Dimension: $$72$$ Relative dimension: $$4$$ over $$\Q(\zeta_{27})$$ Twist minimal: yes 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.

 $$\operatorname{Tr}(f)(q) =$$ $$72q + 9q^{6} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$72q + 9q^{6} + 18q^{13} - 9q^{20} - 81q^{23} + 18q^{25} - 27q^{26} - 27q^{27} + 18q^{28} - 27q^{29} - 63q^{30} - 54q^{31} + 9q^{33} - 27q^{35} - 9q^{36} + 9q^{38} - 9q^{41} + 9q^{42} + 36q^{43} - 117q^{45} - 18q^{46} - 27q^{47} + 9q^{48} - 27q^{51} - 36q^{52} - 27q^{53} + 54q^{55} + 27q^{57} + 9q^{58} - 18q^{59} + 9q^{63} + 9q^{65} + 36q^{66} - 135q^{67} - 18q^{68} + 108q^{69} + 18q^{70} + 72q^{71} + 54q^{72} + 36q^{73} + 99q^{74} - 36q^{75} - 9q^{76} + 144q^{77} + 90q^{78} - 9q^{79} + 18q^{80} - 72q^{82} + 99q^{83} + 18q^{84} + 9q^{85} + 72q^{86} + 207q^{87} - 9q^{88} + 126q^{89} - 18q^{90} + 63q^{91} + 36q^{92} + 81q^{93} + 18q^{94} + 45q^{95} - 171q^{97} + 36q^{98} + 99q^{99} + O(q^{100})$$

## 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.

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 −1.65406 + 0.513880i −0.286803 + 0.957990i 2.13228 1.40243i 1.39993 + 1.01990i −0.606844 + 0.643217i 0.939693 0.342020i 2.47185 1.69998i −2.39823 0.872884i
7.2 −0.597159 0.802123i −1.45578 0.938458i −0.286803 + 0.957990i −1.84048 + 1.21050i 0.116573 + 1.72812i −0.500394 + 0.530386i 0.939693 0.342020i 1.23859 + 2.73238i 2.07003 + 0.753431i
7.3 −0.597159 0.802123i 0.967452 + 1.43668i −0.286803 + 0.957990i 1.46985 0.966735i 0.574669 1.63394i −0.262959 + 0.278720i 0.939693 0.342020i −1.12808 + 2.77983i −1.65317 0.601706i
7.4 −0.597159 0.802123i 1.64484 0.542681i −0.286803 + 0.957990i −1.45045 + 0.953973i −1.41753 0.995298i 2.66288 2.82248i 0.939693 0.342020i 2.41100 1.78525i 1.63135 + 0.593763i
13.1 −0.893633 0.448799i −1.71649 + 0.231626i 0.597159 + 0.802123i −0.267154 0.892356i 1.63787 + 0.563372i 1.11188 + 2.57764i −0.173648 0.984808i 2.89270 0.795169i −0.161751 + 0.917337i
13.2 −0.893633 0.448799i −1.41362 1.00084i 0.597159 + 0.802123i 0.308002 + 1.02880i 0.814085 + 1.52881i −2.06499 4.78718i −0.173648 0.984808i 0.996657 + 2.82961i 0.186483 1.05760i
13.3 −0.893633 0.448799i 0.647569 1.60644i 0.597159 + 0.802123i −0.750365 2.50640i −1.29966 + 1.14494i 0.318489 + 0.738341i −0.173648 0.984808i −2.16131 2.08056i −0.454317 + 2.57656i
13.4 −0.893633 0.448799i 0.674015 + 1.59553i 0.597159 + 0.802123i 0.195520 + 0.653084i 0.113749 1.72831i 0.386069 + 0.895009i −0.173648 0.984808i −2.09141 + 2.15082i 0.118380 0.671366i
25.1 −0.893633 + 0.448799i −1.71649 0.231626i 0.597159 0.802123i −0.267154 + 0.892356i 1.63787 0.563372i 1.11188 2.57764i −0.173648 + 0.984808i 2.89270 + 0.795169i −0.161751 0.917337i
25.2 −0.893633 + 0.448799i −1.41362 + 1.00084i 0.597159 0.802123i 0.308002 1.02880i 0.814085 1.52881i −2.06499 + 4.78718i −0.173648 + 0.984808i 0.996657 2.82961i 0.186483 + 1.05760i
25.3 −0.893633 + 0.448799i 0.647569 + 1.60644i 0.597159 0.802123i −0.750365 + 2.50640i −1.29966 1.14494i 0.318489 0.738341i −0.173648 + 0.984808i −2.16131 + 2.08056i −0.454317 2.57656i
25.4 −0.893633 + 0.448799i 0.674015 1.59553i 0.597159 0.802123i 0.195520 0.653084i 0.113749 + 1.72831i 0.386069 0.895009i −0.173648 + 0.984808i −2.09141 2.15082i 0.118380 + 0.671366i
31.1 −0.396080 0.918216i −1.62371 + 0.602974i −0.686242 + 0.727374i −0.0263678 + 0.452718i 1.19678 + 1.25209i 4.03057 0.955263i 0.939693 + 0.342020i 2.27284 1.95811i 0.426137 0.155101i
31.2 −0.396080 0.918216i −1.31402 1.12843i −0.686242 + 0.727374i −0.205044 + 3.52046i −0.515687 + 1.65350i −4.52050 + 1.07138i 0.939693 + 0.342020i 0.453287 + 2.96556i 3.31376 1.20611i
31.3 −0.396080 0.918216i 1.05738 + 1.37184i −0.686242 + 0.727374i −0.141847 + 2.43542i 0.840836 1.51426i −0.675743 + 0.160154i 0.939693 + 0.342020i −0.763881 + 2.90112i 2.29243 0.834375i
31.4 −0.396080 0.918216i 1.72259 0.180780i −0.686242 + 0.727374i 0.201959 3.46750i −0.848278 1.51011i −2.51951 + 0.597135i 0.939693 + 0.342020i 2.93464 0.622818i −3.26390 + 1.18796i
43.1 0.286803 + 0.957990i −1.47699 0.904708i −0.835488 + 0.549509i 0.423867 + 0.982634i 0.443094 1.67442i −0.264903 + 4.54820i −0.766044 0.642788i 1.36301 + 2.67249i −0.819786 + 0.687882i
43.2 0.286803 + 0.957990i −0.671790 + 1.59646i −0.835488 + 0.549509i 0.417669 + 0.968265i −1.72207 0.185697i −0.0919975 + 1.57954i −0.766044 0.642788i −2.09740 2.14498i −0.807799 + 0.677824i
43.3 0.286803 + 0.957990i −0.524336 1.65078i −0.835488 + 0.549509i −0.757906 1.75702i 1.43105 0.975757i 0.278553 4.78256i −0.766044 0.642788i −2.45014 + 1.73113i 1.46584 1.22999i
43.4 0.286803 + 0.957990i 1.38972 + 1.03377i −0.835488 + 0.549509i 0.266390 + 0.617563i −0.591767 + 1.62782i 0.0791338 1.35867i −0.766044 0.642788i 0.862631 + 2.87330i −0.515217 + 0.432318i
See all 72 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 157.4 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.a 72
3.b odd 2 1 486.2.g.a 72
81.g even 27 1 inner 162.2.g.a 72
81.h odd 54 1 486.2.g.a 72

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
162.2.g.a 72 1.a even 1 1 trivial
162.2.g.a 72 81.g even 27 1 inner
486.2.g.a 72 3.b odd 2 1
486.2.g.a 72 81.h odd 54 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{72} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(162, [\chi])$$.

## Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database