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 $2$

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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.

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

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:

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 \(S_{5}^{\mathrm{new}}(162, [\chi])\).