Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [239,4,Mod(1,239)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(239, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("239.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 239 \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 239.a (trivial) |
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: | yes |
Analytic conductor: | \(14.1014564914\) |
Analytic rank: | \(0\) |
Dimension: | \(37\) |
Twist minimal: | yes |
Fricke sign: | \(1\) |
Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1.1 | −5.56532 | 9.56504 | 22.9727 | 8.90443 | −53.2325 | −11.2801 | −83.3280 | 64.4899 | −49.5560 | ||||||||||||||||||
1.2 | −5.30310 | −10.0457 | 20.1229 | −9.35421 | 53.2736 | 16.9697 | −64.2888 | 73.9171 | 49.6063 | ||||||||||||||||||
1.3 | −5.19331 | −0.852071 | 18.9705 | −14.4938 | 4.42507 | −33.8516 | −56.9730 | −26.2740 | 75.2706 | ||||||||||||||||||
1.4 | −5.02414 | 4.31973 | 17.2420 | −11.3995 | −21.7029 | 16.2461 | −46.4330 | −8.33993 | 57.2725 | ||||||||||||||||||
1.5 | −4.59397 | −3.54103 | 13.1045 | 4.72841 | 16.2674 | 31.8111 | −23.4500 | −14.4611 | −21.7221 | ||||||||||||||||||
1.6 | −4.20811 | −8.89126 | 9.70818 | 14.3148 | 37.4154 | −14.1622 | −7.18821 | 52.0545 | −60.2384 | ||||||||||||||||||
1.7 | −3.82561 | 4.58903 | 6.63530 | 13.2443 | −17.5559 | −25.5656 | 5.22079 | −5.94079 | −50.6676 | ||||||||||||||||||
1.8 | −3.50254 | −2.72284 | 4.26778 | −11.8178 | 9.53686 | 22.4937 | 13.0723 | −19.5861 | 41.3925 | ||||||||||||||||||
1.9 | −3.37354 | 7.98211 | 3.38078 | 13.8556 | −26.9280 | 24.5625 | 15.5831 | 36.7140 | −46.7423 | ||||||||||||||||||
1.10 | −3.12953 | −6.58952 | 1.79393 | 20.5928 | 20.6221 | 28.8946 | 19.4220 | 16.4217 | −64.4456 | ||||||||||||||||||
1.11 | −2.91546 | −4.96191 | 0.499907 | 3.11303 | 14.4663 | −17.8402 | 21.8662 | −2.37941 | −9.07590 | ||||||||||||||||||
1.12 | −2.86154 | 3.04040 | 0.188430 | −5.35419 | −8.70025 | −31.0877 | 22.3531 | −17.7559 | 15.3212 | ||||||||||||||||||
1.13 | −1.94502 | 7.25713 | −4.21689 | 3.86550 | −14.1153 | 19.7745 | 23.7621 | 25.6659 | −7.51848 | ||||||||||||||||||
1.14 | −1.82758 | 2.53373 | −4.65994 | −21.6433 | −4.63060 | −18.0702 | 23.1371 | −20.5802 | 39.5549 | ||||||||||||||||||
1.15 | −1.50268 | −0.901740 | −5.74194 | 13.8014 | 1.35503 | 16.9859 | 20.6498 | −26.1869 | −20.7392 | ||||||||||||||||||
1.16 | −0.483342 | 8.62567 | −7.76638 | −20.7756 | −4.16915 | 33.2708 | 7.62055 | 47.4022 | 10.0417 | ||||||||||||||||||
1.17 | −0.104490 | −3.03341 | −7.98908 | −1.78609 | 0.316959 | −23.0255 | 1.67069 | −17.7985 | 0.186628 | ||||||||||||||||||
1.18 | 0.123948 | 9.22900 | −7.98464 | 3.85596 | 1.14391 | −18.4037 | −1.98126 | 58.1744 | 0.477937 | ||||||||||||||||||
1.19 | 0.271729 | −5.48441 | −7.92616 | −11.3325 | −1.49027 | −27.2237 | −4.32760 | 3.07872 | −3.07937 | ||||||||||||||||||
1.20 | 0.412743 | −3.80494 | −7.82964 | −15.2997 | −1.57046 | 4.25811 | −6.53358 | −12.5225 | −6.31484 | ||||||||||||||||||
See all 37 embeddings |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(239\) | \(1\) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 239.4.a.b | ✓ | 37 |
3.b | odd | 2 | 1 | 2151.4.a.f | 37 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
239.4.a.b | ✓ | 37 | 1.a | even | 1 | 1 | trivial |
2151.4.a.f | 37 | 3.b | odd | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{37} - 4 T_{2}^{36} - 225 T_{2}^{35} + 891 T_{2}^{34} + 22979 T_{2}^{33} - 89996 T_{2}^{32} + \cdots + 8610703477760 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(239))\).