Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [61,7,Mod(8,61)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(61, base_ring=CyclotomicField(20))
chi = DirichletCharacter(H, H._module([1]))
N = Newforms(chi, 7, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("61.8");
S:= CuspForms(chi, 7);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 61 \) |
Weight: | \( k \) | \(=\) | \( 7 \) |
Character orbit: | \([\chi]\) | \(=\) | 61.j (of order \(20\), degree \(8\), 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: | \(14.0332991008\) |
Analytic rank: | \(0\) |
Dimension: | \(240\) |
Relative dimension: | \(30\) over \(\Q(\zeta_{20})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{20}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
8.1 | −14.9359 | − | 2.36562i | 24.3471 | + | 33.5109i | 156.618 | + | 50.8882i | 15.7864 | + | 5.12931i | −284.372 | − | 558.112i | −57.0555 | − | 360.234i | −1356.52 | − | 691.182i | −304.927 | + | 938.468i | −223.650 | − | 113.955i |
8.2 | −14.5200 | − | 2.29974i | 2.94531 | + | 4.05387i | 144.673 | + | 47.0071i | −140.596 | − | 45.6824i | −33.4430 | − | 65.6355i | 102.471 | + | 646.975i | −1154.23 | − | 588.108i | 217.514 | − | 669.440i | 1936.39 | + | 986.640i |
8.3 | −14.3157 | − | 2.26738i | −17.1536 | − | 23.6099i | 138.930 | + | 45.1412i | 124.471 | + | 40.4431i | 192.033 | + | 376.886i | −3.66476 | − | 23.1384i | −1060.01 | − | 540.102i | −37.9086 | + | 116.671i | −1690.19 | − | 861.194i |
8.4 | −12.0990 | − | 1.91629i | 10.2537 | + | 14.1131i | 81.8450 | + | 26.5930i | 10.5714 | + | 3.43485i | −97.0148 | − | 190.402i | −33.3080 | − | 210.298i | −240.743 | − | 122.664i | 131.234 | − | 403.897i | −121.321 | − | 61.8159i |
8.5 | −11.5786 | − | 1.83388i | −13.3265 | − | 18.3423i | 69.8344 | + | 22.6906i | −47.6522 | − | 15.4831i | 120.665 | + | 236.818i | −61.4872 | − | 388.215i | −98.4799 | − | 50.1780i | 66.4280 | − | 204.444i | 523.354 | + | 266.662i |
8.6 | −11.0664 | − | 1.75275i | −26.9919 | − | 37.1511i | 58.5256 | + | 19.0161i | −159.846 | − | 51.9370i | 233.587 | + | 458.439i | 12.0727 | + | 76.2243i | 24.5840 | + | 12.5261i | −426.372 | + | 1312.24i | 1677.89 | + | 854.925i |
8.7 | −10.4150 | − | 1.64958i | 11.0227 | + | 15.1715i | 44.8844 | + | 14.5838i | 234.100 | + | 76.0636i | −89.7755 | − | 176.194i | 60.1035 | + | 379.479i | 157.899 | + | 80.4537i | 116.600 | − | 358.859i | −2312.69 | − | 1178.37i |
8.8 | −8.70749 | − | 1.37913i | 11.9365 | + | 16.4292i | 13.0507 | + | 4.24042i | −207.954 | − | 67.5685i | −81.2790 | − | 159.519i | −31.8422 | − | 201.044i | 394.938 | + | 201.231i | 97.8350 | − | 301.105i | 1717.57 | + | 875.147i |
8.9 | −8.37352 | − | 1.32624i | 29.2938 | + | 40.3194i | 7.48937 | + | 2.43344i | −17.1595 | − | 5.57546i | −191.819 | − | 376.466i | 46.4125 | + | 293.037i | 423.962 | + | 216.020i | −542.257 | + | 1668.90i | 136.291 | + | 69.4437i |
8.10 | −6.48483 | − | 1.02710i | −4.26947 | − | 5.87641i | −19.8695 | − | 6.45601i | 59.1716 | + | 19.2260i | 21.6511 | + | 42.4927i | −22.3687 | − | 141.230i | 496.623 | + | 253.042i | 208.969 | − | 643.142i | −363.971 | − | 185.452i |
8.11 | −5.80061 | − | 0.918727i | −14.6267 | − | 20.1319i | −28.0646 | − | 9.11873i | −3.89918 | − | 1.26692i | 66.3479 | + | 130.215i | 92.8806 | + | 586.425i | 489.314 | + | 249.318i | 33.9202 | − | 104.396i | 21.4537 | + | 10.9312i |
8.12 | −5.34019 | − | 0.845804i | −30.9690 | − | 42.6251i | −33.0653 | − | 10.7436i | 186.226 | + | 60.5084i | 129.328 | + | 253.820i | −13.0661 | − | 82.4960i | 475.806 | + | 242.435i | −632.550 | + | 1946.79i | −943.303 | − | 480.637i |
8.13 | −2.46414 | − | 0.390282i | 19.7627 | + | 27.2011i | −54.9479 | − | 17.8537i | 152.454 | + | 49.5352i | −38.0821 | − | 74.7403i | −103.627 | − | 654.278i | 270.699 | + | 137.928i | −124.060 | + | 381.816i | −356.335 | − | 181.562i |
8.14 | −0.798753 | − | 0.126510i | 12.4887 | + | 17.1893i | −60.2456 | − | 19.5750i | −105.947 | − | 34.4242i | −7.80080 | − | 15.3099i | 36.6098 | + | 231.145i | 91.7612 | + | 46.7547i | 85.7711 | − | 263.976i | 80.2704 | + | 40.8998i |
8.15 | −0.794529 | − | 0.125841i | −12.5523 | − | 17.2768i | −60.2522 | − | 19.5771i | −219.917 | − | 71.4552i | 7.79905 | + | 15.3065i | −40.0746 | − | 253.021i | 91.2809 | + | 46.5099i | 84.3471 | − | 259.594i | 165.738 | + | 84.4478i |
8.16 | 0.120641 | + | 0.0191077i | 16.5235 | + | 22.7426i | −60.8534 | − | 19.7725i | 22.1092 | + | 7.18371i | 1.55885 | + | 3.05942i | 21.9570 | + | 138.631i | −13.9288 | − | 7.09710i | −18.9281 | + | 58.2546i | 2.53001 | + | 1.28910i |
8.17 | 0.244988 | + | 0.0388022i | −17.8282 | − | 24.5385i | −60.8091 | − | 19.7581i | −7.75245 | − | 2.51892i | −3.41555 | − | 6.70339i | −75.3402 | − | 475.679i | −28.2752 | − | 14.4070i | −59.0165 | + | 181.634i | −1.80151 | − | 0.917917i |
8.18 | 3.55204 | + | 0.562587i | 0.0740963 | + | 0.101985i | −48.5672 | − | 15.7804i | 141.256 | + | 45.8967i | 0.205817 | + | 0.403939i | 51.7148 | + | 326.514i | −368.712 | − | 187.868i | 225.268 | − | 693.305i | 475.924 | + | 242.495i |
8.19 | 4.37160 | + | 0.692394i | −24.1162 | − | 33.1931i | −42.2361 | − | 13.7233i | −99.3178 | − | 32.2703i | −82.4438 | − | 161.805i | 50.4996 | + | 318.842i | −427.533 | − | 217.839i | −294.918 | + | 907.664i | −411.834 | − | 209.840i |
8.20 | 4.61063 | + | 0.730252i | −13.3088 | − | 18.3180i | −40.1430 | − | 13.0432i | 200.376 | + | 65.1061i | −47.9851 | − | 94.1760i | 4.63036 | + | 29.2349i | −441.755 | − | 225.086i | 66.8494 | − | 205.741i | 876.316 | + | 446.505i |
See next 80 embeddings (of 240 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
61.j | odd | 20 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 61.7.j.a | ✓ | 240 |
61.j | odd | 20 | 1 | inner | 61.7.j.a | ✓ | 240 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
61.7.j.a | ✓ | 240 | 1.a | even | 1 | 1 | trivial |
61.7.j.a | ✓ | 240 | 61.j | odd | 20 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{7}^{\mathrm{new}}(61, [\chi])\).