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Properties

Label 54.5.f.a
Level $54$
Weight $5$
Character orbit 54.f
Analytic conductor $5.582$
Analytic rank $0$
Dimension $72$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [54,5,Mod(5,54)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(54, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([5]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("54.5");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 54 = 2 \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 54.f (of order \(18\), degree \(6\), 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: \(5.58197800653\)
Analytic rank: \(0\)
Dimension: \(72\)
Relative dimension: \(12\) over \(\Q(\zeta_{18})\)
Twist minimal: yes
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.

\(\operatorname{Tr}(f)(q) = \) \( 72 q - 18 q^{5} - 96 q^{6} + 204 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 72 q - 18 q^{5} - 96 q^{6} + 204 q^{9} + 720 q^{11} + 96 q^{12} - 288 q^{14} - 1422 q^{15} - 1344 q^{18} + 288 q^{20} + 4308 q^{21} - 1008 q^{22} + 4716 q^{23} - 882 q^{25} - 2808 q^{27} - 6084 q^{29} - 6336 q^{30} + 3330 q^{31} - 1026 q^{33} + 288 q^{34} + 5346 q^{35} + 2976 q^{36} + 576 q^{38} - 4974 q^{39} - 13356 q^{41} - 3840 q^{42} + 1260 q^{43} + 9558 q^{45} + 16578 q^{47} + 1920 q^{48} - 5904 q^{49} + 15552 q^{50} + 20898 q^{51} - 4896 q^{54} - 2304 q^{56} - 34218 q^{57} - 40104 q^{59} - 4176 q^{60} + 8352 q^{61} - 4110 q^{63} + 18432 q^{64} - 19674 q^{65} - 10368 q^{66} - 24192 q^{67} + 10224 q^{68} + 47106 q^{69} + 14400 q^{70} + 39528 q^{71} + 15360 q^{72} - 12222 q^{73} + 33120 q^{74} + 41682 q^{75} + 9792 q^{76} + 28206 q^{77} - 4608 q^{78} + 11304 q^{79} + 6804 q^{81} - 30078 q^{83} - 2592 q^{84} - 52200 q^{85} - 46224 q^{86} - 111690 q^{87} - 16128 q^{88} - 102222 q^{89} - 66240 q^{90} + 12078 q^{91} - 27504 q^{92} - 11958 q^{93} + 4032 q^{94} + 46728 q^{95} + 6144 q^{96} + 49680 q^{97} + 82944 q^{98} + 179622 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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 −1.81808 − 2.16670i −8.99966 − 0.0780629i −1.38919 + 7.87846i −15.0487 − 41.3459i 16.1929 + 19.6415i 12.0067 + 68.0936i 19.5959 − 11.3137i 80.9878 + 1.40508i −62.2245 + 107.776i
5.2 −1.81808 − 2.16670i −8.16410 − 3.78781i −1.38919 + 7.87846i 9.40666 + 25.8446i 6.63593 + 24.5757i −2.18926 − 12.4159i 19.5959 − 11.3137i 52.3050 + 61.8481i 38.8955 − 67.3689i
5.3 −1.81808 − 2.16670i −4.79972 + 7.61332i −1.38919 + 7.87846i 1.47744 + 4.05924i 25.2221 − 3.44207i −8.54442 − 48.4578i 19.5959 − 11.3137i −34.9254 − 73.0836i 6.10906 − 10.5812i
5.4 −1.81808 − 2.16670i −0.346638 − 8.99332i −1.38919 + 7.87846i −2.28016 − 6.26470i −18.8556 + 17.1016i −10.8277 − 61.4069i 19.5959 − 11.3137i −80.7597 + 6.23486i −9.42821 + 16.3301i
5.5 −1.81808 − 2.16670i 6.65069 − 6.06369i −1.38919 + 7.87846i 14.9830 + 41.1655i −25.2297 − 3.38581i 12.4750 + 70.7490i 19.5959 − 11.3137i 7.46343 − 80.6554i 61.9530 − 107.306i
5.6 −1.81808 − 2.16670i 8.98244 − 0.561982i −1.38919 + 7.87846i −10.5592 − 29.0112i −17.5484 − 18.4405i −0.904520 − 5.12979i 19.5959 − 11.3137i 80.3684 − 10.0959i −43.6611 + 75.6233i
5.7 1.81808 + 2.16670i −8.90566 + 1.29970i −1.38919 + 7.87846i −7.68730 − 21.1207i −19.0072 − 16.9330i −9.39505 − 53.2820i −19.5959 + 11.3137i 77.6216 − 23.1493i 31.7861 − 55.0551i
5.8 1.81808 + 2.16670i −6.22655 − 6.49847i −1.38919 + 7.87846i 6.13288 + 16.8500i 2.75990 − 25.3058i 8.35982 + 47.4109i −19.5959 + 11.3137i −3.46026 + 80.9261i −25.3588 + 43.9227i
5.9 1.81808 + 2.16670i −0.625816 + 8.97822i −1.38919 + 7.87846i 12.8057 + 35.1833i −20.5909 + 14.9671i −14.9416 − 84.7379i −19.5959 + 11.3137i −80.2167 − 11.2374i −52.9500 + 91.7121i
5.10 1.81808 + 2.16670i −0.457691 + 8.98835i −1.38919 + 7.87846i −5.13581 − 14.1105i −20.3072 + 15.3498i 13.6904 + 77.6422i −19.5959 + 11.3137i −80.5810 − 8.22778i 21.2360 − 36.7818i
5.11 1.81808 + 2.16670i 2.47357 − 8.65341i −1.38919 + 7.87846i −12.4033 − 34.0778i 23.2465 − 10.3731i −2.97171 − 16.8534i −19.5959 + 11.3137i −68.7629 − 42.8096i 51.2862 − 88.8303i
5.12 1.81808 + 2.16670i 8.99992 − 0.0388105i −1.38919 + 7.87846i 4.26691 + 11.7232i 16.4466 + 19.4296i 3.24231 + 18.3880i −19.5959 + 11.3137i 80.9970 − 0.698582i −17.6432 + 30.5589i
11.1 −1.81808 + 2.16670i −8.99966 + 0.0780629i −1.38919 − 7.87846i −15.0487 + 41.3459i 16.1929 − 19.6415i 12.0067 − 68.0936i 19.5959 + 11.3137i 80.9878 − 1.40508i −62.2245 − 107.776i
11.2 −1.81808 + 2.16670i −8.16410 + 3.78781i −1.38919 − 7.87846i 9.40666 − 25.8446i 6.63593 − 24.5757i −2.18926 + 12.4159i 19.5959 + 11.3137i 52.3050 − 61.8481i 38.8955 + 67.3689i
11.3 −1.81808 + 2.16670i −4.79972 − 7.61332i −1.38919 − 7.87846i 1.47744 − 4.05924i 25.2221 + 3.44207i −8.54442 + 48.4578i 19.5959 + 11.3137i −34.9254 + 73.0836i 6.10906 + 10.5812i
11.4 −1.81808 + 2.16670i −0.346638 + 8.99332i −1.38919 − 7.87846i −2.28016 + 6.26470i −18.8556 − 17.1016i −10.8277 + 61.4069i 19.5959 + 11.3137i −80.7597 − 6.23486i −9.42821 − 16.3301i
11.5 −1.81808 + 2.16670i 6.65069 + 6.06369i −1.38919 − 7.87846i 14.9830 − 41.1655i −25.2297 + 3.38581i 12.4750 − 70.7490i 19.5959 + 11.3137i 7.46343 + 80.6554i 61.9530 + 107.306i
11.6 −1.81808 + 2.16670i 8.98244 + 0.561982i −1.38919 − 7.87846i −10.5592 + 29.0112i −17.5484 + 18.4405i −0.904520 + 5.12979i 19.5959 + 11.3137i 80.3684 + 10.0959i −43.6611 − 75.6233i
11.7 1.81808 − 2.16670i −8.90566 − 1.29970i −1.38919 − 7.87846i −7.68730 + 21.1207i −19.0072 + 16.9330i −9.39505 + 53.2820i −19.5959 − 11.3137i 77.6216 + 23.1493i 31.7861 + 55.0551i
11.8 1.81808 − 2.16670i −6.22655 + 6.49847i −1.38919 − 7.87846i 6.13288 − 16.8500i 2.75990 + 25.3058i 8.35982 − 47.4109i −19.5959 − 11.3137i −3.46026 − 80.9261i −25.3588 − 43.9227i
See all 72 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.12
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 54.5.f.a ✓ 72
3.b odd 2 1 162.5.f.a 72
27.e even 9 1 162.5.f.a 72
27.f odd 18 1 inner 54.5.f.a ✓ 72
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
54.5.f.a ✓ 72 1.a even 1 1 trivial
54.5.f.a ✓ 72 27.f odd 18 1 inner
162.5.f.a 72 3.b odd 2 1
162.5.f.a 72 27.e even 9 1

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(54, [\chi])\).