Properties

Label 27.6.e.a
Level $27$
Weight $6$
Character orbit 27.e
Analytic conductor $4.330$
Analytic rank $0$
Dimension $84$
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] = [27,6,Mod(4,27)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(27, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([2]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("27.4");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 27 = 3^{3} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 27.e (of order \(9\), 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: \(4.33036313495\)
Analytic rank: \(0\)
Dimension: \(84\)
Relative dimension: \(14\) over \(\Q(\zeta_{9})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 84 q - 6 q^{2} - 6 q^{3} - 6 q^{4} - 93 q^{5} - 126 q^{6} - 6 q^{7} + 573 q^{8} + 324 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 84 q - 6 q^{2} - 6 q^{3} - 6 q^{4} - 93 q^{5} - 126 q^{6} - 6 q^{7} + 573 q^{8} + 324 q^{9} - 3 q^{10} + 111 q^{11} - 2769 q^{12} - 6 q^{13} - 1641 q^{14} + 1989 q^{15} + 90 q^{16} + 3465 q^{17} - 99 q^{18} - 3 q^{19} + 9987 q^{20} + 5424 q^{21} - 2850 q^{22} - 7716 q^{23} - 18486 q^{24} + 4953 q^{25} - 7806 q^{26} - 8109 q^{27} - 12 q^{28} - 20418 q^{29} - 6453 q^{30} - 6657 q^{31} + 51192 q^{32} + 50634 q^{33} - 11394 q^{34} + 35868 q^{35} - 3600 q^{36} - 3 q^{37} - 44076 q^{38} - 13971 q^{39} + 12441 q^{40} - 79077 q^{41} - 64422 q^{42} - 9465 q^{43} + 110757 q^{44} + 92493 q^{45} - 3 q^{46} + 103557 q^{47} + 134049 q^{48} + 5484 q^{49} - 105513 q^{50} - 124866 q^{51} + 68625 q^{52} - 206406 q^{53} - 350946 q^{54} - 12 q^{55} - 147237 q^{56} + 67767 q^{57} - 9753 q^{58} + 116484 q^{59} + 241290 q^{60} - 70116 q^{61} + 246066 q^{62} + 135981 q^{63} - 86019 q^{64} - 19815 q^{65} - 32931 q^{66} + 48117 q^{67} - 48105 q^{68} + 25263 q^{69} + 270111 q^{70} + 279531 q^{71} + 441684 q^{72} - 27012 q^{73} + 233691 q^{74} + 18399 q^{75} - 125670 q^{76} - 345135 q^{77} - 520974 q^{78} - 216186 q^{79} - 924114 q^{80} - 428616 q^{81} - 12 q^{82} - 370401 q^{83} - 522390 q^{84} - 43731 q^{85} + 116682 q^{86} + 304173 q^{87} + 371418 q^{88} + 154827 q^{89} + 327366 q^{90} - 91002 q^{91} + 1279059 q^{92} + 890079 q^{93} + 11667 q^{94} + 1087671 q^{95} + 1165698 q^{96} - 420621 q^{97} + 463410 q^{98} - 28323 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1 −9.69504 3.52870i −5.72592 + 14.4988i 57.0285 + 47.8526i 2.39953 13.6084i 106.675 120.361i −83.8831 + 70.3863i −218.960 379.250i −177.428 166.037i −71.2837 + 123.467i
4.2 −8.79489 3.20108i 13.9939 6.86812i 42.5898 + 35.7371i −11.0981 + 62.9404i −145.060 + 15.6088i 158.627 133.104i −110.426 191.263i 148.658 192.223i 299.084 518.028i
4.3 −7.30121 2.65742i −0.728957 15.5714i 21.7323 + 18.2356i 15.2097 86.2586i −36.0575 + 115.627i −90.4944 + 75.9338i 14.1040 + 24.4288i −241.937 + 22.7018i −340.275 + 589.373i
4.4 −5.55790 2.02291i −15.5209 1.44989i 2.28464 + 1.91704i −3.90711 + 22.1583i 83.3305 + 39.4557i 69.5798 58.3844i 85.8137 + 148.634i 238.796 + 45.0070i 66.5396 115.250i
4.5 −4.77365 1.73746i 11.5896 + 10.4250i −4.74451 3.98112i −2.77896 + 15.7603i −37.2117 69.9018i −95.3445 + 80.0035i 97.0117 + 168.029i 25.6389 + 241.644i 40.6487 70.4057i
4.6 −1.14127 0.415390i −3.98171 + 15.0714i −23.3835 19.6211i 10.2948 58.3849i 10.8047 15.5466i 144.334 121.111i 37.9689 + 65.7640i −211.292 120.020i −36.0017 + 62.3568i
4.7 −0.752106 0.273744i −2.15220 15.4392i −24.0227 20.1574i −15.6402 + 88.6998i −2.60770 + 12.2010i −70.2017 + 58.9062i 25.3556 + 43.9172i −233.736 + 66.4565i 36.0441 62.4302i
4.8 0.224938 + 0.0818707i 14.5788 5.51901i −24.4695 20.5324i 7.42390 42.1031i 3.73116 0.0478599i 7.97190 6.68922i −7.65311 13.2556i 182.081 160.921i 5.11693 8.86277i
4.9 3.25396 + 1.18435i −12.0549 + 9.88326i −15.3278 12.8616i −5.36783 + 30.4425i −50.9315 + 17.8826i −155.790 + 130.724i −90.0483 155.968i 47.6424 238.284i −53.5211 + 92.7013i
4.10 4.64663 + 1.69124i −11.1566 10.8872i −5.78253 4.85212i 9.66519 54.8140i −33.4278 69.4570i 60.6928 50.9273i −97.7806 169.361i 5.93943 + 242.927i 137.614 238.354i
4.11 5.36004 + 1.95090i 9.76327 + 12.1523i 0.410645 + 0.344572i −14.4659 + 82.0402i 28.6236 + 84.1840i 131.197 110.088i −89.7358 155.427i −52.3572 + 237.292i −237.590 + 411.517i
4.12 8.01235 + 2.91626i 10.1819 11.8038i 31.1798 + 26.1630i −3.30454 + 18.7410i 116.004 64.8829i −6.74337 + 5.65836i 37.1007 + 64.2603i −35.6577 240.370i −81.1307 + 140.523i
4.13 8.47173 + 3.08346i 8.38757 + 13.1396i 37.7491 + 31.6753i 17.5895 99.7549i 30.5419 + 137.178i −143.917 + 120.761i 77.8841 + 134.899i −102.297 + 220.418i 456.604 790.861i
4.14 10.0391 + 3.65394i −15.3240 + 2.85912i 62.9193 + 52.7956i −7.04748 + 39.9683i −164.287 27.2900i 72.7973 61.0842i 267.808 + 463.857i 226.651 87.6263i −216.792 + 375.495i
7.1 −9.69504 + 3.52870i −5.72592 14.4988i 57.0285 47.8526i 2.39953 + 13.6084i 106.675 + 120.361i −83.8831 70.3863i −218.960 + 379.250i −177.428 + 166.037i −71.2837 123.467i
7.2 −8.79489 + 3.20108i 13.9939 + 6.86812i 42.5898 35.7371i −11.0981 62.9404i −145.060 15.6088i 158.627 + 133.104i −110.426 + 191.263i 148.658 + 192.223i 299.084 + 518.028i
7.3 −7.30121 + 2.65742i −0.728957 + 15.5714i 21.7323 18.2356i 15.2097 + 86.2586i −36.0575 115.627i −90.4944 75.9338i 14.1040 24.4288i −241.937 22.7018i −340.275 589.373i
7.4 −5.55790 + 2.02291i −15.5209 + 1.44989i 2.28464 1.91704i −3.90711 22.1583i 83.3305 39.4557i 69.5798 + 58.3844i 85.8137 148.634i 238.796 45.0070i 66.5396 + 115.250i
7.5 −4.77365 + 1.73746i 11.5896 10.4250i −4.74451 + 3.98112i −2.77896 15.7603i −37.2117 + 69.9018i −95.3445 80.0035i 97.0117 168.029i 25.6389 241.644i 40.6487 + 70.4057i
7.6 −1.14127 + 0.415390i −3.98171 15.0714i −23.3835 + 19.6211i 10.2948 + 58.3849i 10.8047 + 15.5466i 144.334 + 121.111i 37.9689 65.7640i −211.292 + 120.020i −36.0017 62.3568i
See all 84 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.14
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
27.e even 9 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 27.6.e.a 84
3.b odd 2 1 81.6.e.a 84
27.e even 9 1 inner 27.6.e.a 84
27.e even 9 1 729.6.a.c 42
27.f odd 18 1 81.6.e.a 84
27.f odd 18 1 729.6.a.e 42
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
27.6.e.a 84 1.a even 1 1 trivial
27.6.e.a 84 27.e even 9 1 inner
81.6.e.a 84 3.b odd 2 1
81.6.e.a 84 27.f odd 18 1
729.6.a.c 42 27.e even 9 1
729.6.a.e 42 27.f odd 18 1

Hecke kernels

This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(27, [\chi])\).