Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [390,2,Mod(239,390)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(390, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([2, 2, 3]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("390.239");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 390 = 2 \cdot 3 \cdot 5 \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 390.n (of order \(4\), 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: | \(3.11416567883\) |
Analytic rank: | \(0\) |
Dimension: | \(56\) |
Relative dimension: | \(28\) over \(\Q(i)\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
239.1 | −0.707107 | + | 0.707107i | −1.71366 | − | 0.251734i | − | 1.00000i | −1.77539 | − | 1.35941i | 1.38974 | − | 1.03374i | 0.242355 | − | 0.242355i | 0.707107 | + | 0.707107i | 2.87326 | + | 0.862772i | 2.21664 | − | 0.294144i | |
239.2 | −0.707107 | + | 0.707107i | −1.71366 | + | 0.251734i | − | 1.00000i | 1.35941 | + | 1.77539i | 1.03374 | − | 1.38974i | −0.242355 | + | 0.242355i | 0.707107 | + | 0.707107i | 2.87326 | − | 0.862772i | −2.21664 | − | 0.294144i | |
239.3 | −0.707107 | + | 0.707107i | −1.08285 | − | 1.35182i | − | 1.00000i | 2.23041 | + | 0.158919i | 1.72158 | + | 0.190189i | 2.29387 | − | 2.29387i | 0.707107 | + | 0.707107i | −0.654851 | + | 2.92766i | −1.68951 | + | 1.46477i | |
239.4 | −0.707107 | + | 0.707107i | −1.08285 | + | 1.35182i | − | 1.00000i | −0.158919 | − | 2.23041i | −0.190189 | − | 1.72158i | −2.29387 | + | 2.29387i | 0.707107 | + | 0.707107i | −0.654851 | − | 2.92766i | 1.68951 | + | 1.46477i | |
239.5 | −0.707107 | + | 0.707107i | −1.01321 | − | 1.40478i | − | 1.00000i | −1.62409 | + | 1.53698i | 1.70978 | + | 0.276879i | −0.656637 | + | 0.656637i | 0.707107 | + | 0.707107i | −0.946804 | + | 2.84668i | 0.0615952 | − | 2.23522i | |
239.6 | −0.707107 | + | 0.707107i | −1.01321 | + | 1.40478i | − | 1.00000i | −1.53698 | + | 1.62409i | −0.276879 | − | 1.70978i | 0.656637 | − | 0.656637i | 0.707107 | + | 0.707107i | −0.946804 | − | 2.84668i | −0.0615952 | − | 2.23522i | |
239.7 | −0.707107 | + | 0.707107i | 0.164052 | − | 1.72426i | − | 1.00000i | −0.365747 | − | 2.20595i | 1.10324 | + | 1.33524i | 2.08554 | − | 2.08554i | 0.707107 | + | 0.707107i | −2.94617 | − | 0.565738i | 1.81847 | + | 1.30122i | |
239.8 | −0.707107 | + | 0.707107i | 0.164052 | + | 1.72426i | − | 1.00000i | 2.20595 | + | 0.365747i | −1.33524 | − | 1.10324i | −2.08554 | + | 2.08554i | 0.707107 | + | 0.707107i | −2.94617 | + | 0.565738i | −1.81847 | + | 1.30122i | |
239.9 | −0.707107 | + | 0.707107i | 0.452930 | − | 1.67178i | − | 1.00000i | 0.934355 | + | 2.03150i | 0.861859 | + | 1.50240i | −3.48345 | + | 3.48345i | 0.707107 | + | 0.707107i | −2.58971 | − | 1.51440i | −2.09717 | − | 0.775797i | |
239.10 | −0.707107 | + | 0.707107i | 0.452930 | + | 1.67178i | − | 1.00000i | −2.03150 | − | 0.934355i | −1.50240 | − | 0.861859i | 3.48345 | − | 3.48345i | 0.707107 | + | 0.707107i | −2.58971 | + | 1.51440i | 2.09717 | − | 0.775797i | |
239.11 | −0.707107 | + | 0.707107i | 1.46346 | − | 0.926431i | − | 1.00000i | 0.989929 | − | 2.00500i | −0.379740 | + | 1.68991i | −1.27923 | + | 1.27923i | 0.707107 | + | 0.707107i | 1.28345 | − | 2.71160i | 0.717766 | + | 2.11774i | |
239.12 | −0.707107 | + | 0.707107i | 1.46346 | + | 0.926431i | − | 1.00000i | 2.00500 | − | 0.989929i | −1.68991 | + | 0.379740i | 1.27923 | − | 1.27923i | 0.707107 | + | 0.707107i | 1.28345 | + | 2.71160i | −0.717766 | + | 2.11774i | |
239.13 | −0.707107 | + | 0.707107i | 1.72928 | − | 0.0979139i | − | 1.00000i | 0.00363329 | + | 2.23607i | −1.15355 | + | 1.29202i | 1.76860 | − | 1.76860i | 0.707107 | + | 0.707107i | 2.98083 | − | 0.338641i | −1.58371 | − | 1.57857i | |
239.14 | −0.707107 | + | 0.707107i | 1.72928 | + | 0.0979139i | − | 1.00000i | −2.23607 | − | 0.00363329i | −1.29202 | + | 1.15355i | −1.76860 | + | 1.76860i | 0.707107 | + | 0.707107i | 2.98083 | + | 0.338641i | 1.58371 | − | 1.57857i | |
239.15 | 0.707107 | − | 0.707107i | −1.72928 | − | 0.0979139i | − | 1.00000i | −0.00363329 | − | 2.23607i | −1.29202 | + | 1.15355i | 1.76860 | − | 1.76860i | −0.707107 | − | 0.707107i | 2.98083 | + | 0.338641i | −1.58371 | − | 1.57857i | |
239.16 | 0.707107 | − | 0.707107i | −1.72928 | + | 0.0979139i | − | 1.00000i | 2.23607 | + | 0.00363329i | −1.15355 | + | 1.29202i | −1.76860 | + | 1.76860i | −0.707107 | − | 0.707107i | 2.98083 | − | 0.338641i | 1.58371 | − | 1.57857i | |
239.17 | 0.707107 | − | 0.707107i | −1.46346 | − | 0.926431i | − | 1.00000i | −0.989929 | + | 2.00500i | −1.68991 | + | 0.379740i | −1.27923 | + | 1.27923i | −0.707107 | − | 0.707107i | 1.28345 | + | 2.71160i | 0.717766 | + | 2.11774i | |
239.18 | 0.707107 | − | 0.707107i | −1.46346 | + | 0.926431i | − | 1.00000i | −2.00500 | + | 0.989929i | −0.379740 | + | 1.68991i | 1.27923 | − | 1.27923i | −0.707107 | − | 0.707107i | 1.28345 | − | 2.71160i | −0.717766 | + | 2.11774i | |
239.19 | 0.707107 | − | 0.707107i | −0.452930 | − | 1.67178i | − | 1.00000i | −0.934355 | − | 2.03150i | −1.50240 | − | 0.861859i | −3.48345 | + | 3.48345i | −0.707107 | − | 0.707107i | −2.58971 | + | 1.51440i | −2.09717 | − | 0.775797i | |
239.20 | 0.707107 | − | 0.707107i | −0.452930 | + | 1.67178i | − | 1.00000i | 2.03150 | + | 0.934355i | 0.861859 | + | 1.50240i | 3.48345 | − | 3.48345i | −0.707107 | − | 0.707107i | −2.58971 | − | 1.51440i | 2.09717 | − | 0.775797i | |
See all 56 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
5.b | even | 2 | 1 | inner |
13.d | odd | 4 | 1 | inner |
15.d | odd | 2 | 1 | inner |
39.f | even | 4 | 1 | inner |
65.g | odd | 4 | 1 | inner |
195.n | even | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 390.2.n.a | ✓ | 56 |
3.b | odd | 2 | 1 | inner | 390.2.n.a | ✓ | 56 |
5.b | even | 2 | 1 | inner | 390.2.n.a | ✓ | 56 |
13.d | odd | 4 | 1 | inner | 390.2.n.a | ✓ | 56 |
15.d | odd | 2 | 1 | inner | 390.2.n.a | ✓ | 56 |
39.f | even | 4 | 1 | inner | 390.2.n.a | ✓ | 56 |
65.g | odd | 4 | 1 | inner | 390.2.n.a | ✓ | 56 |
195.n | even | 4 | 1 | inner | 390.2.n.a | ✓ | 56 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
390.2.n.a | ✓ | 56 | 1.a | even | 1 | 1 | trivial |
390.2.n.a | ✓ | 56 | 3.b | odd | 2 | 1 | inner |
390.2.n.a | ✓ | 56 | 5.b | even | 2 | 1 | inner |
390.2.n.a | ✓ | 56 | 13.d | odd | 4 | 1 | inner |
390.2.n.a | ✓ | 56 | 15.d | odd | 2 | 1 | inner |
390.2.n.a | ✓ | 56 | 39.f | even | 4 | 1 | inner |
390.2.n.a | ✓ | 56 | 65.g | odd | 4 | 1 | inner |
390.2.n.a | ✓ | 56 | 195.n | even | 4 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(390, [\chi])\).