Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [43,7,Mod(2,43)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(43, base_ring=CyclotomicField(14))
chi = DirichletCharacter(H, H._module([9]))
N = Newforms(chi, 7, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("43.2");
S:= CuspForms(chi, 7);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 43 \) |
Weight: | \( k \) | \(=\) | \( 7 \) |
Character orbit: | \([\chi]\) | \(=\) | 43.f (of order \(14\), 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: | \(9.89232559565\) |
Analytic rank: | \(0\) |
Dimension: | \(126\) |
Relative dimension: | \(21\) over \(\Q(\zeta_{14})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{14}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
2.1 | −6.68851 | − | 13.8888i | 4.36630 | − | 9.06671i | −108.260 | + | 135.754i | 1.12402 | + | 0.256549i | −155.130 | 114.871i | 1647.71 | + | 376.078i | 391.383 | + | 490.779i | −3.95482 | − | 17.3272i | ||||
2.2 | −5.62328 | − | 11.6769i | −22.1545 | + | 46.0043i | −64.8243 | + | 81.2871i | 43.4728 | + | 9.92237i | 661.767 | 306.680i | 505.038 | + | 115.272i | −1171.05 | − | 1468.45i | −128.597 | − | 563.421i | ||||
2.3 | −5.10721 | − | 10.6052i | −7.80492 | + | 16.2071i | −46.4840 | + | 58.2891i | −223.373 | − | 50.9834i | 211.741 | − | 193.213i | 121.123 | + | 27.6454i | 252.771 | + | 316.965i | 600.121 | + | 2629.30i | |||
2.4 | −5.02991 | − | 10.4447i | 20.6257 | − | 42.8298i | −43.8887 | + | 55.0347i | 127.013 | + | 28.9898i | −551.091 | − | 619.099i | 72.2435 | + | 16.4891i | −954.446 | − | 1196.84i | −336.072 | − | 1472.43i | |||
2.5 | −4.79242 | − | 9.95156i | −6.97408 | + | 14.4818i | −36.1629 | + | 45.3469i | 117.045 | + | 26.7147i | 177.540 | − | 466.480i | −64.6015 | − | 14.7449i | 293.438 | + | 367.960i | −295.075 | − | 1292.81i | |||
2.6 | −4.35509 | − | 9.04344i | 15.2129 | − | 31.5900i | −22.9137 | + | 28.7328i | −93.5256 | − | 21.3466i | −351.936 | 345.453i | −266.657 | − | 60.8626i | −311.968 | − | 391.196i | 214.266 | + | 938.759i | ||||
2.7 | −3.64714 | − | 7.57336i | 2.80632 | − | 5.82738i | −4.15080 | + | 5.20494i | 190.781 | + | 43.5445i | −54.3678 | 593.422i | −469.925 | − | 107.257i | 428.441 | + | 537.248i | −366.026 | − | 1603.66i | ||||
2.8 | −2.72088 | − | 5.64997i | −8.85233 | + | 18.3821i | 15.3844 | − | 19.2915i | −52.8314 | − | 12.0584i | 127.944 | − | 8.48034i | −542.136 | − | 123.739i | 194.988 | + | 244.507i | 75.6182 | + | 331.305i | |||
2.9 | −1.12993 | − | 2.34633i | 12.1637 | − | 25.2581i | 35.6748 | − | 44.7348i | −135.516 | − | 30.9307i | −73.0081 | − | 261.823i | −307.765 | − | 70.2454i | −35.4943 | − | 44.5085i | 80.5506 | + | 352.915i | |||
2.10 | −0.716329 | − | 1.48747i | 3.40007 | − | 7.06031i | 38.2039 | − | 47.9062i | 69.5466 | + | 15.8736i | −12.9376 | − | 168.130i | −201.638 | − | 46.0227i | 416.237 | + | 521.944i | −26.2068 | − | 114.819i | |||
2.11 | −0.546272 | − | 1.13435i | −16.4415 | + | 34.1411i | 38.9150 | − | 48.7979i | −89.4697 | − | 20.4209i | 47.7093 | 356.575i | −155.169 | − | 35.4164i | −440.768 | − | 552.706i | 25.7105 | + | 112.645i | ||||
2.12 | −0.0322332 | − | 0.0669329i | −17.7426 | + | 36.8430i | 39.8999 | − | 50.0329i | 157.975 | + | 36.0568i | 3.03791 | − | 203.317i | −9.27029 | − | 2.11588i | −588.078 | − | 737.427i | −2.67865 | − | 11.7360i | |||
2.13 | 1.11483 | + | 2.31496i | 18.6979 | − | 38.8265i | 35.7871 | − | 44.8756i | 137.550 | + | 31.3949i | 110.727 | 167.161i | 304.102 | + | 69.4092i | −703.365 | − | 881.992i | 80.6666 | + | 353.424i | ||||
2.14 | 2.61426 | + | 5.42856i | 1.01637 | − | 2.11052i | 17.2684 | − | 21.6539i | −102.842 | − | 23.4731i | 14.1142 | 486.006i | 538.641 | + | 122.941i | 451.103 | + | 565.665i | −141.431 | − | 619.650i | ||||
2.15 | 2.79658 | + | 5.80715i | −16.1345 | + | 33.5037i | 14.0012 | − | 17.5569i | −177.830 | − | 40.5886i | −239.682 | − | 522.371i | 543.278 | + | 124.000i | −407.649 | − | 511.176i | −261.612 | − | 1146.20i | |||
2.16 | 3.53371 | + | 7.33783i | −9.42884 | + | 19.5792i | −1.45324 | + | 1.82231i | 197.653 | + | 45.1129i | −176.987 | 222.322i | 489.664 | + | 111.763i | 160.083 | + | 200.737i | 367.416 | + | 1609.76i | ||||
2.17 | 3.58164 | + | 7.43735i | 3.11543 | − | 6.46926i | −2.58264 | + | 3.23853i | 50.0201 | + | 11.4168i | 59.2725 | − | 578.950i | 481.727 | + | 109.951i | 422.379 | + | 529.646i | 94.2435 | + | 412.908i | |||
2.18 | 4.32423 | + | 8.97936i | 20.4277 | − | 42.4185i | −22.0266 | + | 27.6205i | −183.723 | − | 41.9336i | 469.225 | − | 28.9921i | 278.591 | + | 63.5867i | −927.514 | − | 1163.07i | −417.925 | − | 1831.05i | |||
2.19 | 5.62961 | + | 11.6900i | −16.3881 | + | 34.0303i | −65.0604 | + | 81.5832i | 15.2899 | + | 3.48982i | −490.074 | 124.671i | −510.398 | − | 116.495i | −434.969 | − | 545.434i | 45.2802 | + | 198.386i | ||||
2.20 | 5.84606 | + | 12.1395i | 12.7263 | − | 26.4263i | −73.2871 | + | 91.8991i | 119.457 | + | 27.2654i | 395.200 | 91.2949i | −703.346 | − | 160.534i | −81.8698 | − | 102.661i | 367.368 | + | 1609.55i | ||||
See next 80 embeddings (of 126 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
43.f | odd | 14 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 43.7.f.a | ✓ | 126 |
43.f | odd | 14 | 1 | inner | 43.7.f.a | ✓ | 126 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
43.7.f.a | ✓ | 126 | 1.a | even | 1 | 1 | trivial |
43.7.f.a | ✓ | 126 | 43.f | odd | 14 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{7}^{\mathrm{new}}(43, [\chi])\).