Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [420,2,Mod(71,420)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(420, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 1, 0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("420.71");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 420 = 2^{2} \cdot 3 \cdot 5 \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 420.n (of order \(2\), degree \(1\), 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: | \(3.35371688489\) |
Analytic rank: | \(0\) |
Dimension: | \(24\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
71.1 | −1.41258 | − | 0.0678578i | 1.66901 | − | 0.463024i | 1.99079 | + | 0.191710i | 1.00000i | −2.38904 | + | 0.540805i | 1.00000i | −2.79915 | − | 0.405897i | 2.57122 | − | 1.54559i | 0.0678578 | − | 1.41258i | ||||
71.2 | −1.41258 | + | 0.0678578i | 1.66901 | + | 0.463024i | 1.99079 | − | 0.191710i | − | 1.00000i | −2.38904 | − | 0.540805i | − | 1.00000i | −2.79915 | + | 0.405897i | 2.57122 | + | 1.54559i | 0.0678578 | + | 1.41258i | ||
71.3 | −1.38203 | − | 0.299986i | −0.491627 | + | 1.66081i | 1.82002 | + | 0.829179i | − | 1.00000i | 1.17766 | − | 2.14781i | − | 1.00000i | −2.26658 | − | 1.69193i | −2.51661 | − | 1.63300i | −0.299986 | + | 1.38203i | ||
71.4 | −1.38203 | + | 0.299986i | −0.491627 | − | 1.66081i | 1.82002 | − | 0.829179i | 1.00000i | 1.17766 | + | 2.14781i | 1.00000i | −2.26658 | + | 1.69193i | −2.51661 | + | 1.63300i | −0.299986 | − | 1.38203i | ||||
71.5 | −1.09313 | − | 0.897253i | −1.62224 | + | 0.606916i | 0.389874 | + | 1.96163i | 1.00000i | 2.31788 | + | 0.792118i | 1.00000i | 1.33390 | − | 2.49414i | 2.26331 | − | 1.96912i | 0.897253 | − | 1.09313i | ||||
71.6 | −1.09313 | + | 0.897253i | −1.62224 | − | 0.606916i | 0.389874 | − | 1.96163i | − | 1.00000i | 2.31788 | − | 0.792118i | − | 1.00000i | 1.33390 | + | 2.49414i | 2.26331 | + | 1.96912i | 0.897253 | + | 1.09313i | ||
71.7 | −0.859821 | − | 1.12281i | −1.72451 | − | 0.161413i | −0.521414 | + | 1.93084i | − | 1.00000i | 1.30154 | + | 2.07509i | − | 1.00000i | 2.61629 | − | 1.07472i | 2.94789 | + | 0.556719i | −1.12281 | + | 0.859821i | ||
71.8 | −0.859821 | + | 1.12281i | −1.72451 | + | 0.161413i | −0.521414 | − | 1.93084i | 1.00000i | 1.30154 | − | 2.07509i | 1.00000i | 2.61629 | + | 1.07472i | 2.94789 | − | 0.556719i | −1.12281 | − | 0.859821i | ||||
71.9 | −0.637711 | − | 1.26227i | 1.30156 | − | 1.14278i | −1.18665 | + | 1.60993i | 1.00000i | −2.27252 | − | 0.914147i | 1.00000i | 2.78890 | + | 0.471201i | 0.388091 | − | 2.97479i | 1.26227 | − | 0.637711i | ||||
71.10 | −0.637711 | + | 1.26227i | 1.30156 | + | 1.14278i | −1.18665 | − | 1.60993i | − | 1.00000i | −2.27252 | + | 0.914147i | − | 1.00000i | 2.78890 | − | 0.471201i | 0.388091 | + | 2.97479i | 1.26227 | + | 0.637711i | ||
71.11 | −0.0797590 | − | 1.41196i | −1.52443 | − | 0.822258i | −1.98728 | + | 0.225233i | 1.00000i | −1.03941 | + | 2.21802i | 1.00000i | 0.476524 | + | 2.78800i | 1.64778 | + | 2.50695i | 1.41196 | − | 0.0797590i | ||||
71.12 | −0.0797590 | + | 1.41196i | −1.52443 | + | 0.822258i | −1.98728 | − | 0.225233i | − | 1.00000i | −1.03941 | − | 2.21802i | − | 1.00000i | 0.476524 | − | 2.78800i | 1.64778 | − | 2.50695i | 1.41196 | + | 0.0797590i | ||
71.13 | 0.269719 | − | 1.38825i | −0.0343495 | + | 1.73171i | −1.85450 | − | 0.748876i | 1.00000i | 2.39479 | + | 0.514760i | 1.00000i | −1.53983 | + | 2.37254i | −2.99764 | − | 0.118967i | 1.38825 | + | 0.269719i | ||||
71.14 | 0.269719 | + | 1.38825i | −0.0343495 | − | 1.73171i | −1.85450 | + | 0.748876i | − | 1.00000i | 2.39479 | − | 0.514760i | − | 1.00000i | −1.53983 | − | 2.37254i | −2.99764 | + | 0.118967i | 1.38825 | − | 0.269719i | ||
71.15 | 0.275453 | − | 1.38713i | 1.30386 | − | 1.14015i | −1.84825 | − | 0.764177i | − | 1.00000i | −1.22239 | − | 2.12268i | − | 1.00000i | −1.56912 | + | 2.35327i | 0.400103 | − | 2.97320i | −1.38713 | − | 0.275453i | ||
71.16 | 0.275453 | + | 1.38713i | 1.30386 | + | 1.14015i | −1.84825 | + | 0.764177i | 1.00000i | −1.22239 | + | 2.12268i | 1.00000i | −1.56912 | − | 2.35327i | 0.400103 | + | 2.97320i | −1.38713 | + | 0.275453i | ||||
71.17 | 1.09123 | − | 0.899566i | −0.826488 | − | 1.52214i | 0.381563 | − | 1.96327i | − | 1.00000i | −2.27116 | − | 0.917526i | − | 1.00000i | −1.34971 | − | 2.48561i | −1.63383 | + | 2.51607i | −0.899566 | − | 1.09123i | ||
71.18 | 1.09123 | + | 0.899566i | −0.826488 | + | 1.52214i | 0.381563 | + | 1.96327i | 1.00000i | −2.27116 | + | 0.917526i | 1.00000i | −1.34971 | + | 2.48561i | −1.63383 | − | 2.51607i | −0.899566 | + | 1.09123i | ||||
71.19 | 1.19442 | − | 0.757207i | −0.667879 | + | 1.59810i | 0.853276 | − | 1.80884i | − | 1.00000i | 0.412368 | + | 2.41453i | − | 1.00000i | −0.350500 | − | 2.80663i | −2.10788 | − | 2.13468i | −0.757207 | − | 1.19442i | ||
71.20 | 1.19442 | + | 0.757207i | −0.667879 | − | 1.59810i | 0.853276 | + | 1.80884i | 1.00000i | 0.412368 | − | 2.41453i | 1.00000i | −0.350500 | + | 2.80663i | −2.10788 | + | 2.13468i | −0.757207 | + | 1.19442i | ||||
See all 24 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
12.b | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 420.2.n.b | yes | 24 |
3.b | odd | 2 | 1 | 420.2.n.a | ✓ | 24 | |
4.b | odd | 2 | 1 | 420.2.n.a | ✓ | 24 | |
12.b | even | 2 | 1 | inner | 420.2.n.b | yes | 24 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
420.2.n.a | ✓ | 24 | 3.b | odd | 2 | 1 | |
420.2.n.a | ✓ | 24 | 4.b | odd | 2 | 1 | |
420.2.n.b | yes | 24 | 1.a | even | 1 | 1 | trivial |
420.2.n.b | yes | 24 | 12.b | even | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{11}^{12} - 76 T_{11}^{10} - 100 T_{11}^{9} + 1829 T_{11}^{8} + 4844 T_{11}^{7} - 11202 T_{11}^{6} + \cdots + 7552 \) acting on \(S_{2}^{\mathrm{new}}(420, [\chi])\).