Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [61,6,Mod(14,61)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(61, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([5]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("61.14");
S:= CuspForms(chi, 6);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 61 \) |
Weight: | \( k \) | \(=\) | \( 6 \) |
Character orbit: | \([\chi]\) | \(=\) | 61.f (of order \(6\), degree \(2\), 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: | \(9.78341300859\) |
Analytic rank: | \(0\) |
Dimension: | \(50\) |
Relative dimension: | \(25\) over \(\Q(\zeta_{6})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-expansion
The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
14.1 | −8.82463 | + | 5.09490i | 7.45682 | 35.9160 | − | 62.2084i | −44.2216 | − | 76.5941i | −65.8036 | + | 37.9917i | 52.4002 | − | 30.2533i | 405.881i | −187.396 | 780.479 | + | 450.610i | ||||||
14.2 | −8.66618 | + | 5.00342i | −4.30325 | 34.0684 | − | 59.0082i | 29.5085 | + | 51.1102i | 37.2927 | − | 21.5310i | −79.2178 | + | 45.7364i | 361.615i | −224.482 | −511.452 | − | 295.287i | ||||||
14.3 | −8.53217 | + | 4.92605i | −29.1038 | 32.5319 | − | 56.3470i | −22.4307 | − | 38.8511i | 248.319 | − | 143.367i | −49.6296 | + | 28.6537i | 325.749i | 604.033 | 382.765 | + | 220.990i | ||||||
14.4 | −7.84297 | + | 4.52814i | 17.1199 | 25.0081 | − | 43.3154i | 19.5582 | + | 33.8758i | −134.271 | + | 77.5214i | 156.523 | − | 90.3684i | 163.161i | 50.0917 | −306.789 | − | 177.125i | ||||||
14.5 | −7.27118 | + | 4.19802i | 29.5204 | 19.2467 | − | 33.3363i | 3.62206 | + | 6.27359i | −214.648 | + | 123.927i | −183.284 | + | 105.819i | 54.5194i | 628.451 | −52.6733 | − | 30.4110i | ||||||
14.6 | −6.13315 | + | 3.54098i | −20.4110 | 9.07704 | − | 15.7219i | 33.6596 | + | 58.3001i | 125.184 | − | 72.2750i | 182.780 | − | 105.528i | − | 98.0562i | 173.610 | −412.879 | − | 238.376i | |||||
14.7 | −5.30528 | + | 3.06301i | −9.50617 | 2.76401 | − | 4.78741i | −9.04682 | − | 15.6696i | 50.4329 | − | 29.1175i | −139.867 | + | 80.7525i | − | 162.168i | −152.633 | 95.9919 | + | 55.4210i | |||||
14.8 | −4.85971 | + | 2.80576i | −12.6609 | −0.255465 | + | 0.442478i | −31.8526 | − | 55.1703i | 61.5281 | − | 35.5233i | 102.473 | − | 59.1627i | − | 182.435i | −82.7026 | 309.589 | + | 178.741i | |||||
14.9 | −4.17180 | + | 2.40859i | 11.7686 | −4.39740 | + | 7.61653i | −15.4722 | − | 26.7986i | −49.0963 | + | 28.3458i | −6.23439 | + | 3.59943i | − | 196.516i | −104.499 | 129.094 | + | 74.5322i | |||||
14.10 | −2.56603 | + | 1.48150i | 13.1716 | −11.6103 | + | 20.1097i | 44.2550 | + | 76.6519i | −33.7986 | + | 19.5136i | −9.52914 | + | 5.50165i | − | 163.618i | −69.5101 | −227.119 | − | 131.127i | |||||
14.11 | −1.65849 | + | 0.957530i | −25.2946 | −14.1663 | + | 24.5367i | 37.7930 | + | 65.4593i | 41.9509 | − | 24.2203i | −194.301 | + | 112.180i | − | 115.540i | 396.817 | −125.358 | − | 72.3758i | |||||
14.12 | −1.33428 | + | 0.770348i | 28.3419 | −14.8131 | + | 25.6571i | −34.6172 | − | 59.9588i | −37.8161 | + | 21.8331i | 169.921 | − | 98.1041i | − | 94.9473i | 560.265 | 92.3783 | + | 53.3346i | |||||
14.13 | 0.532088 | − | 0.307201i | −12.3461 | −15.8113 | + | 27.3859i | 11.6874 | + | 20.2431i | −6.56920 | + | 3.79273i | 70.4812 | − | 40.6924i | 39.0898i | −90.5741 | 12.4374 | + | 7.18075i | ||||||
14.14 | 0.541363 | − | 0.312556i | −24.3819 | −15.8046 | + | 27.3744i | −46.8737 | − | 81.1877i | −13.1994 | + | 7.62070i | −21.4453 | + | 12.3814i | 39.7629i | 351.477 | −50.7514 | − | 29.3013i | ||||||
14.15 | 1.52349 | − | 0.879585i | 11.5259 | −14.4527 | + | 25.0327i | −32.6357 | − | 56.5268i | 17.5596 | − | 10.1380i | −167.355 | + | 96.6222i | 107.143i | −110.153 | −99.4402 | − | 57.4118i | ||||||
14.16 | 2.37669 | − | 1.37218i | 5.01039 | −12.2342 | + | 21.1903i | 7.66611 | + | 13.2781i | 11.9081 | − | 6.87517i | 137.587 | − | 79.4359i | 154.970i | −217.896 | 36.4400 | + | 21.0386i | ||||||
14.17 | 3.69873 | − | 2.13546i | 26.4579 | −6.87958 | + | 11.9158i | 23.9372 | + | 41.4604i | 97.8607 | − | 56.4999i | −33.9660 | + | 19.6103i | 195.434i | 457.020 | 177.074 | + | 102.234i | ||||||
14.18 | 5.36111 | − | 3.09524i | −2.54764 | 3.16098 | − | 5.47497i | 39.7841 | + | 68.9082i | −13.6582 | + | 7.88554i | −162.430 | + | 93.7789i | 158.959i | −236.510 | 426.574 | + | 246.283i | ||||||
14.19 | 5.97052 | − | 3.44708i | −13.9645 | 7.76473 | − | 13.4489i | −7.99258 | − | 13.8436i | −83.3754 | + | 48.1368i | −72.5498 | + | 41.8866i | 113.551i | −47.9923 | −95.4397 | − | 55.1022i | ||||||
14.20 | 6.07673 | − | 3.50840i | −30.1249 | 8.61775 | − | 14.9264i | 16.7440 | + | 29.0014i | −183.061 | + | 105.690i | 118.902 | − | 68.6479i | 103.600i | 664.511 | 203.497 | + | 117.489i | ||||||
See all 50 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
61.f | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 61.6.f.a | ✓ | 50 |
61.f | even | 6 | 1 | inner | 61.6.f.a | ✓ | 50 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
61.6.f.a | ✓ | 50 | 1.a | even | 1 | 1 | trivial |
61.6.f.a | ✓ | 50 | 61.f | even | 6 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(61, [\chi])\).