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Properties

Label 27.5.f.a
Level $27$
Weight $5$
Character orbit 27.f
Analytic conductor $2.791$
Analytic rank $0$
Dimension $66$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [27,5,Mod(2,27)]
 
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([1]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("27.2");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 27 = 3^{3} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 27.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: \(2.79098900326\)
Analytic rank: \(0\)
Dimension: \(66\)
Relative dimension: \(11\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 66 q - 6 q^{2} - 6 q^{3} - 6 q^{4} + 3 q^{5} + 90 q^{6} - 6 q^{7} - 9 q^{8} - 108 q^{9} - 3 q^{10} - 492 q^{11} - 339 q^{12} - 6 q^{13} + 1137 q^{14} + 1017 q^{15} - 54 q^{16} - 9 q^{17} + 603 q^{18} - 3 q^{19}+ \cdots - 162405 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1 −7.11993 − 1.25544i 6.30423 − 6.42314i 34.0822 + 12.4049i −16.3502 − 19.4854i −52.9495 + 37.8177i −60.7132 + 22.0978i −126.911 − 73.2722i −1.51338 − 80.9859i 91.9496 + 159.261i
2.2 −6.19782 − 1.09284i −7.35835 + 5.18216i 22.1836 + 8.07418i −7.06026 − 8.41408i 51.2690 − 24.0766i 62.3269 − 22.6852i −41.4621 − 23.9381i 27.2905 − 76.2642i 34.5629 + 59.8648i
2.3 −4.35935 − 0.768672i −5.98497 − 6.72162i 3.37802 + 1.22950i 27.2417 + 32.4654i 20.9239 + 33.9024i −49.1081 + 17.8739i 47.5559 + 27.4564i −9.36028 + 80.4573i −93.8009 − 162.468i
2.4 −3.58910 − 0.632855i 8.76651 + 2.03673i −2.55397 − 0.929568i 12.8585 + 15.3241i −30.1749 − 12.8580i 75.5885 − 27.5120i 59.0773 + 34.1083i 72.7034 + 35.7101i −36.4523 − 63.1373i
2.5 −2.38785 − 0.421042i 1.69434 + 8.83907i −9.51055 − 3.46156i −12.2061 − 14.5467i −0.324205 − 21.8197i −77.2633 + 28.1215i 54.8497 + 31.6675i −75.2584 + 29.9528i 23.0216 + 39.8746i
2.6 −0.185922 − 0.0327832i 3.20767 − 8.40898i −15.0016 − 5.46013i −10.5929 − 12.6241i −0.872050 + 1.45826i 26.3432 − 9.58813i 5.22609 + 3.01729i −60.4218 − 53.9464i 1.55560 + 2.69438i
2.7 1.18141 + 0.208314i −8.94516 − 0.991988i −13.6827 − 4.98011i −12.1061 − 14.4275i −10.3613 − 3.03535i 0.372391 − 0.135539i −31.7501 − 18.3309i 79.0319 + 17.7470i −11.2968 − 19.5667i
2.8 2.89555 + 0.510564i −3.30987 + 8.36927i −6.91152 − 2.51559i 31.8237 + 37.9260i −13.8570 + 22.5438i 28.2667 − 10.2882i −59.4692 − 34.3346i −59.0895 − 55.4025i 72.7837 + 126.065i
2.9 4.49504 + 0.792597i 8.84760 + 1.64926i 4.54211 + 1.65319i −3.40574 − 4.05880i 38.4631 + 14.4261i −28.7185 + 10.4527i −44.1393 − 25.4838i 75.5599 + 29.1840i −12.0919 − 20.9439i
2.10 5.98494 + 1.05531i −1.69482 − 8.83898i 19.6707 + 7.15955i 10.5942 + 12.6257i −0.815584 − 54.6893i 7.09675 − 2.58301i 25.9633 + 14.9899i −75.2551 + 29.9610i 50.0816 + 86.7438i
2.11 7.34334 + 1.29483i −4.72310 + 7.66109i 37.2130 + 13.5444i −22.9421 − 27.3414i −44.6032 + 50.1424i 14.0427 − 5.11111i 152.408 + 87.9929i −36.3846 − 72.3682i −133.070 − 230.483i
5.1 −5.07357 − 6.04645i 4.83746 + 7.58940i −8.04003 + 45.5973i 2.20171 + 6.04914i 21.3457 − 67.7548i 3.89095 + 22.0667i 207.124 − 119.583i −34.1980 + 73.4268i 25.4053 − 44.0033i
5.2 −3.69447 − 4.40289i −2.23464 − 8.71817i −2.95802 + 16.7758i −4.30375 − 11.8244i −30.1294 + 42.0478i 8.03917 + 45.5924i 5.14963 − 2.97314i −71.0128 + 38.9638i −36.1617 + 62.6340i
5.3 −2.66286 − 3.17348i −8.41268 + 3.19794i −0.201747 + 1.14416i 12.7013 + 34.8965i 32.5504 + 18.1818i 4.71194 + 26.7227i −53.2345 + 30.7349i 60.5463 − 53.8065i 76.9216 − 133.232i
5.4 −2.46212 − 2.93424i 8.40252 − 3.22453i 0.230635 − 1.30800i 3.23200 + 8.87984i −30.1496 − 16.7158i −13.4127 − 76.0673i −57.4812 + 33.1868i 60.2048 − 54.1884i 18.0980 − 31.3467i
5.5 −1.73231 − 2.06448i −2.55344 + 8.63017i 1.51717 − 8.60428i −14.6121 − 40.1465i 22.2402 − 9.67858i −2.26393 − 12.8394i −57.7345 + 33.3330i −67.9598 − 44.0734i −57.5690 + 99.7125i
5.6 0.712681 + 0.849340i 5.75088 + 6.92296i 2.56491 − 14.5463i 9.60722 + 26.3956i −1.78140 + 9.81832i 2.66962 + 15.1401i 29.5458 − 17.0583i −14.8546 + 79.6262i −15.5720 + 26.9714i
5.7 0.868224 + 1.03471i −7.41018 − 5.10776i 2.46156 − 13.9602i −3.20330 − 8.80100i −1.14866 − 12.1021i −7.41386 − 42.0461i 35.2980 − 20.3793i 28.8216 + 75.6988i 6.32529 − 10.9557i
5.8 1.32398 + 1.57786i 6.23064 − 6.49455i 2.04165 − 11.5788i −6.36554 − 17.4892i 18.4968 + 1.23241i 14.2733 + 80.9481i 49.5136 − 28.5867i −3.35832 − 80.9304i 19.1677 − 33.1993i
5.9 3.59971 + 4.28997i −7.31429 + 5.24416i −2.66754 + 15.1284i 2.69449 + 7.40305i −48.8266 − 12.5006i 7.02945 + 39.8660i 3.09545 − 1.78716i 25.9976 − 76.7146i −22.0595 + 38.2081i
See all 66 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2.11
Significant digits:
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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 27.5.f.a ✓ 66
3.b odd 2 1 81.5.f.a 66
27.e even 9 1 81.5.f.a 66
27.f odd 18 1 inner 27.5.f.a ✓ 66
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
27.5.f.a ✓ 66 1.a even 1 1 trivial
27.5.f.a ✓ 66 27.f odd 18 1 inner
81.5.f.a 66 3.b odd 2 1
81.5.f.a 66 27.e even 9 1

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(27, [\chi])\).