Properties

Label 54.9.f.a
Level $54$
Weight $9$
Character orbit 54.f
Analytic conductor $21.998$
Analytic rank $0$
Dimension $144$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [54,9,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, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("54.5");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 54 = 2 \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 9 \)
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: \(21.9984449433\)
Analytic rank: \(0\)
Dimension: \(144\)
Relative dimension: \(24\) over \(\Q(\zeta_{18})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 144 q + 882 q^{5} + 768 q^{6} + 25908 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 144 q + 882 q^{5} + 768 q^{6} + 25908 q^{9} - 45756 q^{11} - 21504 q^{12} + 94464 q^{14} - 70002 q^{15} - 182784 q^{18} - 225792 q^{20} - 2071668 q^{21} + 185472 q^{22} + 552996 q^{23} - 1015182 q^{25} - 4137588 q^{27} - 2856132 q^{29} - 861696 q^{30} - 1841490 q^{31} + 10547334 q^{33} - 661248 q^{34} - 12450834 q^{35} - 5104128 q^{36} + 14063616 q^{38} + 4239102 q^{39} - 15673536 q^{41} - 11028480 q^{42} - 4412520 q^{43} - 12558942 q^{45} + 15173838 q^{47} + 3440640 q^{48} - 11347488 q^{49} - 27744768 q^{50} - 3281850 q^{51} + 16379136 q^{54} + 12091392 q^{56} - 13553658 q^{57} - 86547168 q^{59} - 26125056 q^{60} - 34059456 q^{61} + 108329070 q^{63} + 150994944 q^{64} + 22190346 q^{65} + 152091648 q^{66} + 167100948 q^{67} + 13411584 q^{68} - 40190130 q^{69} - 118080000 q^{70} - 251437608 q^{71} - 60948480 q^{72} + 15265278 q^{73} + 41902848 q^{74} + 138868662 q^{75} + 67995648 q^{76} + 564911586 q^{77} + 201203712 q^{78} + 137535768 q^{79} - 23838948 q^{81} - 482857578 q^{83} - 128300544 q^{84} - 343215000 q^{85} - 46237824 q^{86} + 34353810 q^{87} + 47480832 q^{88} + 375969762 q^{89} + 452712960 q^{90} - 155093598 q^{91} + 97042176 q^{92} + 285095622 q^{93} - 70995456 q^{94} - 966996432 q^{95} - 88080384 q^{96} - 484917300 q^{97} - 234233856 q^{98} + 26756370 q^{99}+O(q^{100}) \) 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.

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 −7.27231 8.66680i −80.6183 + 7.85385i −22.2270 + 126.055i 208.139 + 571.857i 654.349 + 641.588i −808.155 4583.27i 1254.14 724.077i 6437.63 1266.33i 3442.52 5962.62i
5.2 −7.27231 8.66680i −74.4381 + 31.9369i −22.2270 + 126.055i −303.240 833.145i 818.128 + 412.886i −11.0691 62.7758i 1254.14 724.077i 4521.07 4754.64i −5015.45 + 8687.01i
5.3 −7.27231 8.66680i −71.9819 + 37.1430i −22.2270 + 126.055i 242.781 + 667.036i 845.386 + 353.737i 702.244 + 3982.62i 1254.14 724.077i 3801.79 5347.25i 4015.49 6955.03i
5.4 −7.27231 8.66680i −56.2872 58.2473i −22.2270 + 126.055i −19.4013 53.3047i −95.4793 + 911.423i −121.493 689.020i 1254.14 724.077i −224.491 + 6557.16i −320.889 + 555.796i
5.5 −7.27231 8.66680i −21.3841 + 78.1263i −22.2270 + 126.055i −192.252 528.207i 832.617 382.827i 240.528 + 1364.10i 1254.14 724.077i −5646.44 3341.33i −3179.75 + 5507.49i
5.6 −7.27231 8.66680i −14.8240 79.6320i −22.2270 + 126.055i 332.225 + 912.782i −582.350 + 707.585i 238.226 + 1351.05i 1254.14 724.077i −6121.50 + 2360.92i 5494.85 9517.37i
5.7 −7.27231 8.66680i 7.33820 + 80.6669i −22.2270 + 126.055i 58.6289 + 161.081i 645.759 650.234i −371.637 2107.66i 1254.14 724.077i −6453.30 + 1183.90i 969.694 1679.56i
5.8 −7.27231 8.66680i 24.6435 77.1602i −22.2270 + 126.055i −216.760 595.543i −847.948 + 347.553i −99.0312 561.634i 1254.14 724.077i −5346.40 3802.99i −3585.11 + 6209.59i
5.9 −7.27231 8.66680i 61.3803 + 52.8531i −22.2270 + 126.055i 209.630 + 575.954i 11.6907 916.336i 358.863 + 2035.21i 1254.14 724.077i 974.093 + 6488.29i 3467.19 6005.34i
5.10 −7.27231 8.66680i 66.5510 46.1732i −22.2270 + 126.055i 270.422 + 742.979i −884.154 240.999i −396.574 2249.09i 1254.14 724.077i 2297.08 6145.74i 4472.66 7746.87i
5.11 −7.27231 8.66680i 73.1569 + 34.7717i −22.2270 + 126.055i −324.095 890.442i −230.660 886.907i −575.632 3264.57i 1254.14 724.077i 4142.86 + 5087.58i −5360.37 + 9284.44i
5.12 −7.27231 8.66680i 76.2898 27.2188i −22.2270 + 126.055i −167.053 458.975i −790.704 463.245i 678.435 + 3847.60i 1254.14 724.077i 5079.27 4153.04i −2762.98 + 4785.62i
5.13 7.27231 + 8.66680i −80.9968 + 0.714754i −22.2270 + 126.055i −52.9870 145.581i −595.229 696.786i 710.104 + 4027.20i −1254.14 + 724.077i 6559.98 115.786i 876.380 1517.93i
5.14 7.27231 + 8.66680i −76.9844 25.1872i −22.2270 + 126.055i 421.201 + 1157.24i −341.562 850.378i −165.220 937.012i −1254.14 + 724.077i 5292.21 + 3878.04i −6966.47 + 12066.3i
5.15 7.27231 + 8.66680i −64.9711 48.3711i −22.2270 + 126.055i −403.520 1108.66i −53.2671 914.861i −619.994 3516.16i −1254.14 + 724.077i 1881.48 + 6285.44i 6674.03 11559.8i
5.16 7.27231 + 8.66680i −62.6465 + 51.3460i −22.2270 + 126.055i −16.8419 46.2726i −900.591 169.541i −338.197 1918.01i −1254.14 + 724.077i 1288.18 6433.30i 278.556 482.474i
5.17 7.27231 + 8.66680i −36.0536 72.5337i −22.2270 + 126.055i 0.862658 + 2.37013i 366.442 839.957i 117.968 + 669.031i −1254.14 + 724.077i −3961.28 + 5230.20i −14.2680 + 24.7128i
5.18 7.27231 + 8.66680i 13.3090 + 79.8991i −22.2270 + 126.055i 259.000 + 711.596i −595.682 + 696.398i 690.888 + 3918.22i −1254.14 + 724.077i −6206.74 + 2126.76i −4283.73 + 7419.64i
5.19 7.27231 + 8.66680i 24.9903 + 77.0486i −22.2270 + 126.055i 56.2230 + 154.472i −486.028 + 776.907i −208.662 1183.38i −1254.14 + 724.077i −5311.97 + 3850.93i −929.903 + 1610.64i
5.20 7.27231 + 8.66680i 30.8906 74.8784i −22.2270 + 126.055i 187.067 + 513.962i 873.602 276.817i −775.415 4397.60i −1254.14 + 724.077i −4652.55 4626.07i −3094.00 + 5358.96i
See next 80 embeddings (of 144 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.24
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 54.9.f.a 144
27.f odd 18 1 inner 54.9.f.a 144
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
54.9.f.a 144 1.a even 1 1 trivial
54.9.f.a 144 27.f odd 18 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{9}^{\mathrm{new}}(54, [\chi])\).