# Properties

 Label 162.5.f.a Level $162$ Weight $5$ Character orbit 162.f Analytic conductor $16.746$ Analytic rank $0$ Dimension $72$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$162 = 2 \cdot 3^{4}$$ Weight: $$k$$ $$=$$ $$5$$ Character orbit: $$[\chi]$$ $$=$$ 162.f (of order $$18$$, degree $$6$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$16.7459340196$$ Analytic rank: $$0$$ Dimension: $$72$$ Relative dimension: $$12$$ over $$\Q(\zeta_{18})$$ Twist minimal: no (minimal twist has level 54) Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

## $q$-expansion

The dimension is sufficiently large that we do not compute an algebraic $$q$$-expansion, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$72 q + 18 q^{5}+O(q^{10})$$ 72 * q + 18 * q^5 $$\operatorname{Tr}(f)(q) =$$ $$72 q + 18 q^{5} - 720 q^{11} + 288 q^{14} - 288 q^{20} - 1008 q^{22} - 4716 q^{23} - 882 q^{25} + 6084 q^{29} + 3330 q^{31} + 288 q^{34} - 5346 q^{35} - 576 q^{38} + 13356 q^{41} + 1260 q^{43} - 16578 q^{47} - 5904 q^{49} - 15552 q^{50} + 2304 q^{56} + 40104 q^{59} + 8352 q^{61} + 18432 q^{64} + 19674 q^{65} - 24192 q^{67} - 10224 q^{68} + 14400 q^{70} - 39528 q^{71} - 12222 q^{73} - 33120 q^{74} + 9792 q^{76} - 28206 q^{77} + 11304 q^{79} + 30078 q^{83} - 52200 q^{85} + 46224 q^{86} - 16128 q^{88} + 102222 q^{89} + 12078 q^{91} + 27504 q^{92} + 4032 q^{94} - 46728 q^{95} + 49680 q^{97} - 82944 q^{98}+O(q^{100})$$ 72 * q + 18 * q^5 - 720 * q^11 + 288 * q^14 - 288 * q^20 - 1008 * q^22 - 4716 * q^23 - 882 * q^25 + 6084 * q^29 + 3330 * q^31 + 288 * q^34 - 5346 * q^35 - 576 * q^38 + 13356 * q^41 + 1260 * q^43 - 16578 * q^47 - 5904 * q^49 - 15552 * q^50 + 2304 * q^56 + 40104 * q^59 + 8352 * q^61 + 18432 * q^64 + 19674 * q^65 - 24192 * q^67 - 10224 * q^68 + 14400 * q^70 - 39528 * q^71 - 12222 * q^73 - 33120 * q^74 + 9792 * q^76 - 28206 * q^77 + 11304 * q^79 + 30078 * q^83 - 52200 * q^85 + 46224 * q^86 - 16128 * q^88 + 102222 * q^89 + 12078 * q^91 + 27504 * q^92 + 4032 * q^94 - 46728 * q^95 + 49680 * q^97 - 82944 * 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.

Label $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
17.1 −0.967379 + 2.65785i 0 −6.12836 5.14230i 45.7302 + 8.06347i 0 −14.1135 + 11.8427i 19.5959 11.3137i 0 −65.6700 + 113.744i
17.2 −0.967379 + 2.65785i 0 −6.12836 5.14230i 5.14300 + 0.906850i 0 −21.7252 + 18.2296i 19.5959 11.3137i 0 −7.38550 + 12.7921i
17.3 −0.967379 + 2.65785i 0 −6.12836 5.14230i 3.66245 + 0.645790i 0 67.2236 56.4073i 19.5959 11.3137i 0 −5.25940 + 9.10954i
17.4 −0.967379 + 2.65785i 0 −6.12836 5.14230i 20.4117 + 3.59913i 0 −31.5466 + 26.4708i 19.5959 11.3137i 0 −29.3118 + 50.7695i
17.5 −0.967379 + 2.65785i 0 −6.12836 5.14230i −31.9962 5.64180i 0 29.5375 24.7849i 19.5959 11.3137i 0 45.9475 79.5835i
17.6 −0.967379 + 2.65785i 0 −6.12836 5.14230i −39.1530 6.90373i 0 −3.77039 + 3.16373i 19.5959 11.3137i 0 56.2249 97.3844i
17.7 0.967379 2.65785i 0 −6.12836 5.14230i −41.7224 7.35678i 0 19.0320 15.9698i −19.5959 + 11.3137i 0 −59.9146 + 103.775i
17.8 0.967379 2.65785i 0 −6.12836 5.14230i 33.5989 + 5.92440i 0 −63.3453 + 53.1530i −19.5959 + 11.3137i 0 48.2491 83.5699i
17.9 0.967379 2.65785i 0 −6.12836 5.14230i −5.05113 0.890650i 0 14.8841 12.4892i −19.5959 + 11.3137i 0 −7.25357 + 12.5636i
17.10 0.967379 2.65785i 0 −6.12836 5.14230i −6.60957 1.16545i 0 −57.7034 + 48.4189i −19.5959 + 11.3137i 0 −9.49154 + 16.4398i
17.11 0.967379 2.65785i 0 −6.12836 5.14230i −0.622849 0.109825i 0 42.1999 35.4099i −19.5959 + 11.3137i 0 −0.894430 + 1.54920i
17.12 0.967379 2.65785i 0 −6.12836 5.14230i 24.2051 + 4.26801i 0 19.3273 16.2175i −19.5959 + 11.3137i 0 34.7593 60.2048i
35.1 −1.81808 + 2.16670i 0 −1.38919 7.87846i −4.26691 + 11.7232i 0 3.24231 18.3880i 19.5959 + 11.3137i 0 −17.6432 30.5589i
35.2 −1.81808 + 2.16670i 0 −1.38919 7.87846i 5.13581 14.1105i 0 13.6904 77.6422i 19.5959 + 11.3137i 0 21.2360 + 36.7818i
35.3 −1.81808 + 2.16670i 0 −1.38919 7.87846i −6.13288 + 16.8500i 0 8.35982 47.4109i 19.5959 + 11.3137i 0 −25.3588 43.9227i
35.4 −1.81808 + 2.16670i 0 −1.38919 7.87846i 7.68730 21.1207i 0 −9.39505 + 53.2820i 19.5959 + 11.3137i 0 31.7861 + 55.0551i
35.5 −1.81808 + 2.16670i 0 −1.38919 7.87846i −12.8057 + 35.1833i 0 −14.9416 + 84.7379i 19.5959 + 11.3137i 0 −52.9500 91.7121i
35.6 −1.81808 + 2.16670i 0 −1.38919 7.87846i 12.4033 34.0778i 0 −2.97171 + 16.8534i 19.5959 + 11.3137i 0 51.2862 + 88.8303i
35.7 1.81808 2.16670i 0 −1.38919 7.87846i −14.9830 + 41.1655i 0 12.4750 70.7490i −19.5959 11.3137i 0 61.9530 + 107.306i
35.8 1.81808 2.16670i 0 −1.38919 7.87846i 2.28016 6.26470i 0 −10.8277 + 61.4069i −19.5959 11.3137i 0 −9.42821 16.3301i
See all 72 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 143.12 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
27.f odd 18 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 162.5.f.a 72
3.b odd 2 1 54.5.f.a 72
27.e even 9 1 54.5.f.a 72
27.f odd 18 1 inner 162.5.f.a 72

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
54.5.f.a 72 3.b odd 2 1
54.5.f.a 72 27.e even 9 1
162.5.f.a 72 1.a even 1 1 trivial
162.5.f.a 72 27.f odd 18 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{5}^{\mathrm{new}}(162, [\chi])$$.