Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [43,7,Mod(7,43)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(43, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([5]))
N = Newforms(chi, 7, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("43.7");
S:= CuspForms(chi, 7);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 43 \) |
Weight: | \( k \) | \(=\) | \( 7 \) |
Character orbit: | \([\chi]\) | \(=\) | 43.d (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: | \(9.89232559565\) |
Analytic rank: | \(0\) |
Dimension: | \(42\) |
Relative dimension: | \(21\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
7.1 | − | 15.1507i | 15.3614 | − | 8.86889i | −165.545 | 63.9718 | − | 36.9341i | −134.370 | − | 232.736i | −298.109 | − | 172.114i | 1538.48i | −207.186 | + | 358.856i | −559.579 | − | 969.219i | |||||
7.2 | − | 13.4327i | −34.2137 | + | 19.7533i | −116.436 | −28.7813 | + | 16.6169i | 265.339 | + | 459.580i | −186.356 | − | 107.593i | 704.357i | 415.884 | − | 720.332i | 223.209 | + | 386.609i | |||||
7.3 | − | 12.4749i | 3.88212 | − | 2.24134i | −91.6224 | −151.098 | + | 87.2364i | −27.9605 | − | 48.4289i | 378.610 | + | 218.590i | 344.585i | −354.453 | + | 613.930i | 1088.26 | + | 1884.93i | |||||
7.4 | − | 10.6207i | −24.0449 | + | 13.8823i | −48.8003 | 200.632 | − | 115.835i | 147.441 | + | 255.375i | 396.423 | + | 228.875i | − | 161.432i | 20.9373 | − | 36.2644i | −1230.26 | − | 2130.86i | ||||
7.5 | − | 10.1559i | 30.5813 | − | 17.6561i | −39.1423 | 101.658 | − | 58.6923i | −179.314 | − | 310.581i | 191.878 | + | 110.781i | − | 252.452i | 258.977 | − | 448.562i | −596.073 | − | 1032.43i | ||||
7.6 | − | 7.84004i | 36.4833 | − | 21.0637i | 2.53376 | −116.673 | + | 67.3609i | −165.140 | − | 286.031i | −385.031 | − | 222.298i | − | 521.627i | 522.856 | − | 905.613i | 528.113 | + | 914.718i | ||||
7.7 | − | 6.95456i | −14.3583 | + | 8.28978i | 15.6341 | 9.86801 | − | 5.69730i | 57.6517 | + | 99.8557i | −310.875 | − | 179.484i | − | 553.820i | −227.059 | + | 393.278i | −39.6222 | − | 68.6277i | ||||
7.8 | − | 4.28699i | −41.9299 | + | 24.2082i | 45.6217 | −171.014 | + | 98.7351i | 103.780 | + | 179.753i | 223.732 | + | 129.172i | − | 469.947i | 807.577 | − | 1398.76i | 423.276 | + | 733.136i | ||||
7.9 | − | 2.64495i | −0.554504 | + | 0.320143i | 57.0043 | 14.6929 | − | 8.48295i | 0.846761 | + | 1.46663i | 131.768 | + | 76.0760i | − | 320.050i | −364.295 | + | 630.977i | −22.4370 | − | 38.8619i | ||||
7.10 | − | 0.178867i | 33.4310 | − | 19.3014i | 63.9680 | 21.1050 | − | 12.1850i | −3.45239 | − | 5.97972i | 409.144 | + | 236.219i | − | 22.8893i | 380.590 | − | 659.200i | −2.17950 | − | 3.77500i | ||||
7.11 | 1.51738i | 20.0424 | − | 11.5715i | 61.6975 | 211.318 | − | 122.005i | 17.5584 | + | 30.4120i | −447.414 | − | 258.315i | 190.731i | −96.7022 | + | 167.493i | 185.128 | + | 320.651i | ||||||
7.12 | 1.95704i | 10.4424 | − | 6.02890i | 60.1700 | −193.167 | + | 111.525i | 11.7988 | + | 20.4361i | −30.2778 | − | 17.4809i | 243.006i | −291.805 | + | 505.421i | −218.259 | − | 378.036i | ||||||
7.13 | 2.61500i | −35.7187 | + | 20.6222i | 57.1618 | 121.716 | − | 70.2726i | −53.9270 | − | 93.4043i | 140.639 | + | 81.1978i | 316.838i | 486.049 | − | 841.861i | 183.763 | + | 318.286i | ||||||
7.14 | 6.20332i | −29.5720 | + | 17.0734i | 25.5189 | −46.5574 | + | 26.8799i | −105.912 | − | 183.445i | −448.906 | − | 259.176i | 555.314i | 218.503 | − | 378.459i | −166.745 | − | 288.810i | ||||||
7.15 | 7.38637i | −8.75832 | + | 5.05662i | 9.44147 | 10.2848 | − | 5.93793i | −37.3501 | − | 64.6923i | 367.060 | + | 211.922i | 542.466i | −313.361 | + | 542.757i | 43.8598 | + | 75.9673i | ||||||
7.16 | 8.15234i | 41.5767 | − | 24.0043i | −2.46071 | −26.4676 | + | 15.2811i | 195.692 | + | 338.948i | 6.27128 | + | 3.62073i | 501.689i | 787.917 | − | 1364.71i | −124.577 | − | 215.773i | ||||||
7.17 | 9.90547i | 12.0709 | − | 6.96915i | −34.1183 | −41.4735 | + | 23.9447i | 69.0327 | + | 119.568i | −415.762 | − | 240.040i | 295.992i | −267.362 | + | 463.084i | −237.184 | − | 410.814i | ||||||
7.18 | 12.2048i | 14.4234 | − | 8.32735i | −84.9566 | 131.217 | − | 75.7580i | 101.633 | + | 176.034i | 221.781 | + | 128.045i | − | 255.771i | −225.811 | + | 391.115i | 924.609 | + | 1601.47i | |||||
7.19 | 12.8196i | −27.1856 | + | 15.6956i | −100.343 | −154.521 | + | 89.2129i | −201.212 | − | 348.509i | 237.672 | + | 137.220i | − | 465.907i | 128.203 | − | 222.054i | −1143.68 | − | 1980.91i | |||||
7.20 | 14.0830i | −28.7233 | + | 16.5834i | −134.330 | 147.495 | − | 85.1564i | −233.544 | − | 404.510i | −245.226 | − | 141.581i | − | 990.457i | 185.520 | − | 321.330i | 1199.26 | + | 2077.17i | |||||
See all 42 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
43.d | odd | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 43.7.d.a | ✓ | 42 |
43.d | odd | 6 | 1 | inner | 43.7.d.a | ✓ | 42 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
43.7.d.a | ✓ | 42 | 1.a | even | 1 | 1 | trivial |
43.7.d.a | ✓ | 42 | 43.d | odd | 6 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{7}^{\mathrm{new}}(43, [\chi])\).