Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [420,2,Mod(139,420)]
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, 0, 1, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("420.139");
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.i (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: | \(48\) |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
139.1 | −1.36843 | − | 0.356944i | − | 1.00000i | 1.74518 | + | 0.976904i | −1.65234 | + | 1.50657i | −0.356944 | + | 1.36843i | −2.07310 | + | 1.64386i | −2.03945 | − | 1.95975i | −1.00000 | 2.79887 | − | 1.47184i | |||
139.2 | −1.36843 | − | 0.356944i | 1.00000i | 1.74518 | + | 0.976904i | 1.65234 | − | 1.50657i | 0.356944 | − | 1.36843i | −2.07310 | − | 1.64386i | −2.03945 | − | 1.95975i | −1.00000 | −2.79887 | + | 1.47184i | ||||
139.3 | −1.36843 | + | 0.356944i | − | 1.00000i | 1.74518 | − | 0.976904i | 1.65234 | + | 1.50657i | 0.356944 | + | 1.36843i | −2.07310 | + | 1.64386i | −2.03945 | + | 1.95975i | −1.00000 | −2.79887 | − | 1.47184i | |||
139.4 | −1.36843 | + | 0.356944i | 1.00000i | 1.74518 | − | 0.976904i | −1.65234 | − | 1.50657i | −0.356944 | − | 1.36843i | −2.07310 | − | 1.64386i | −2.03945 | + | 1.95975i | −1.00000 | 2.79887 | + | 1.47184i | ||||
139.5 | −1.33186 | − | 0.475542i | − | 1.00000i | 1.54772 | + | 1.26671i | 2.22087 | + | 0.260295i | −0.475542 | + | 1.33186i | 1.17807 | − | 2.36900i | −1.45898 | − | 2.42309i | −1.00000 | −2.83411 | − | 1.40279i | |||
139.6 | −1.33186 | − | 0.475542i | 1.00000i | 1.54772 | + | 1.26671i | −2.22087 | − | 0.260295i | 0.475542 | − | 1.33186i | 1.17807 | + | 2.36900i | −1.45898 | − | 2.42309i | −1.00000 | 2.83411 | + | 1.40279i | ||||
139.7 | −1.33186 | + | 0.475542i | − | 1.00000i | 1.54772 | − | 1.26671i | −2.22087 | + | 0.260295i | 0.475542 | + | 1.33186i | 1.17807 | − | 2.36900i | −1.45898 | + | 2.42309i | −1.00000 | 2.83411 | − | 1.40279i | |||
139.8 | −1.33186 | + | 0.475542i | 1.00000i | 1.54772 | − | 1.26671i | 2.22087 | − | 0.260295i | −0.475542 | − | 1.33186i | 1.17807 | + | 2.36900i | −1.45898 | + | 2.42309i | −1.00000 | −2.83411 | + | 1.40279i | ||||
139.9 | −1.08769 | − | 0.903842i | − | 1.00000i | 0.366139 | + | 1.96620i | −0.660150 | − | 2.13640i | −0.903842 | + | 1.08769i | −2.64346 | − | 0.110159i | 1.37889 | − | 2.46955i | −1.00000 | −1.21293 | + | 2.92041i | |||
139.10 | −1.08769 | − | 0.903842i | 1.00000i | 0.366139 | + | 1.96620i | 0.660150 | + | 2.13640i | 0.903842 | − | 1.08769i | −2.64346 | + | 0.110159i | 1.37889 | − | 2.46955i | −1.00000 | 1.21293 | − | 2.92041i | ||||
139.11 | −1.08769 | + | 0.903842i | − | 1.00000i | 0.366139 | − | 1.96620i | 0.660150 | − | 2.13640i | 0.903842 | + | 1.08769i | −2.64346 | − | 0.110159i | 1.37889 | + | 2.46955i | −1.00000 | 1.21293 | + | 2.92041i | |||
139.12 | −1.08769 | + | 0.903842i | 1.00000i | 0.366139 | − | 1.96620i | −0.660150 | + | 2.13640i | −0.903842 | − | 1.08769i | −2.64346 | + | 0.110159i | 1.37889 | + | 2.46955i | −1.00000 | −1.21293 | − | 2.92041i | ||||
139.13 | −0.846354 | − | 1.13300i | − | 1.00000i | −0.567368 | + | 1.91784i | −2.22361 | + | 0.235752i | −1.13300 | + | 0.846354i | 2.54904 | + | 0.708793i | 2.65310 | − | 0.980342i | −1.00000 | 2.14906 | + | 2.31981i | |||
139.14 | −0.846354 | − | 1.13300i | 1.00000i | −0.567368 | + | 1.91784i | 2.22361 | − | 0.235752i | 1.13300 | − | 0.846354i | 2.54904 | − | 0.708793i | 2.65310 | − | 0.980342i | −1.00000 | −2.14906 | − | 2.31981i | ||||
139.15 | −0.846354 | + | 1.13300i | − | 1.00000i | −0.567368 | − | 1.91784i | 2.22361 | + | 0.235752i | 1.13300 | + | 0.846354i | 2.54904 | + | 0.708793i | 2.65310 | + | 0.980342i | −1.00000 | −2.14906 | + | 2.31981i | |||
139.16 | −0.846354 | + | 1.13300i | 1.00000i | −0.567368 | − | 1.91784i | −2.22361 | − | 0.235752i | −1.13300 | − | 0.846354i | 2.54904 | − | 0.708793i | 2.65310 | + | 0.980342i | −1.00000 | 2.14906 | − | 2.31981i | ||||
139.17 | −0.654401 | − | 1.25370i | − | 1.00000i | −1.14352 | + | 1.64084i | −0.206931 | + | 2.22647i | −1.25370 | + | 0.654401i | −0.859661 | − | 2.50220i | 2.80544 | + | 0.359858i | −1.00000 | 2.92674 | − | 1.19758i | |||
139.18 | −0.654401 | − | 1.25370i | 1.00000i | −1.14352 | + | 1.64084i | 0.206931 | − | 2.22647i | 1.25370 | − | 0.654401i | −0.859661 | + | 2.50220i | 2.80544 | + | 0.359858i | −1.00000 | −2.92674 | + | 1.19758i | ||||
139.19 | −0.654401 | + | 1.25370i | − | 1.00000i | −1.14352 | − | 1.64084i | 0.206931 | + | 2.22647i | 1.25370 | + | 0.654401i | −0.859661 | − | 2.50220i | 2.80544 | − | 0.359858i | −1.00000 | −2.92674 | − | 1.19758i | |||
139.20 | −0.654401 | + | 1.25370i | 1.00000i | −1.14352 | − | 1.64084i | −0.206931 | − | 2.22647i | −1.25370 | − | 0.654401i | −0.859661 | + | 2.50220i | 2.80544 | − | 0.359858i | −1.00000 | 2.92674 | + | 1.19758i | ||||
See all 48 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
4.b | odd | 2 | 1 | inner |
5.b | even | 2 | 1 | inner |
7.b | odd | 2 | 1 | inner |
20.d | odd | 2 | 1 | inner |
28.d | even | 2 | 1 | inner |
35.c | odd | 2 | 1 | inner |
140.c | 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.i.a | ✓ | 48 |
4.b | odd | 2 | 1 | inner | 420.2.i.a | ✓ | 48 |
5.b | even | 2 | 1 | inner | 420.2.i.a | ✓ | 48 |
7.b | odd | 2 | 1 | inner | 420.2.i.a | ✓ | 48 |
20.d | odd | 2 | 1 | inner | 420.2.i.a | ✓ | 48 |
28.d | even | 2 | 1 | inner | 420.2.i.a | ✓ | 48 |
35.c | odd | 2 | 1 | inner | 420.2.i.a | ✓ | 48 |
140.c | even | 2 | 1 | inner | 420.2.i.a | ✓ | 48 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
420.2.i.a | ✓ | 48 | 1.a | even | 1 | 1 | trivial |
420.2.i.a | ✓ | 48 | 4.b | odd | 2 | 1 | inner |
420.2.i.a | ✓ | 48 | 5.b | even | 2 | 1 | inner |
420.2.i.a | ✓ | 48 | 7.b | odd | 2 | 1 | inner |
420.2.i.a | ✓ | 48 | 20.d | odd | 2 | 1 | inner |
420.2.i.a | ✓ | 48 | 28.d | even | 2 | 1 | inner |
420.2.i.a | ✓ | 48 | 35.c | odd | 2 | 1 | inner |
420.2.i.a | ✓ | 48 | 140.c | even | 2 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(420, [\chi])\).