Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [378,4,Mod(289,378)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(378, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([4, 2]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("378.289");
S:= CuspForms(chi, 4);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 378 = 2 \cdot 3^{3} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 378.h (of order \(3\), degree \(2\), not 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: | \(22.3027219822\) |
Analytic rank: | \(0\) |
Dimension: | \(24\) |
Relative dimension: | \(12\) over \(\Q(\zeta_{3})\) |
Twist minimal: | no (minimal twist has level 126) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
289.1 | −1.00000 | − | 1.73205i | 0 | −2.00000 | + | 3.46410i | −18.7213 | 0 | −15.3695 | − | 10.3334i | 8.00000 | 0 | 18.7213 | + | 32.4263i | ||||||||||
289.2 | −1.00000 | − | 1.73205i | 0 | −2.00000 | + | 3.46410i | −15.6350 | 0 | 17.1578 | + | 6.97217i | 8.00000 | 0 | 15.6350 | + | 27.0807i | ||||||||||
289.3 | −1.00000 | − | 1.73205i | 0 | −2.00000 | + | 3.46410i | −15.5553 | 0 | 6.99950 | − | 17.1466i | 8.00000 | 0 | 15.5553 | + | 26.9425i | ||||||||||
289.4 | −1.00000 | − | 1.73205i | 0 | −2.00000 | + | 3.46410i | −3.49343 | 0 | −11.9430 | + | 14.1550i | 8.00000 | 0 | 3.49343 | + | 6.05080i | ||||||||||
289.5 | −1.00000 | − | 1.73205i | 0 | −2.00000 | + | 3.46410i | −1.46289 | 0 | −3.71747 | + | 18.1433i | 8.00000 | 0 | 1.46289 | + | 2.53380i | ||||||||||
289.6 | −1.00000 | − | 1.73205i | 0 | −2.00000 | + | 3.46410i | 0.521453 | 0 | −3.56519 | − | 18.1739i | 8.00000 | 0 | −0.521453 | − | 0.903183i | ||||||||||
289.7 | −1.00000 | − | 1.73205i | 0 | −2.00000 | + | 3.46410i | 1.52253 | 0 | 18.1220 | + | 3.82009i | 8.00000 | 0 | −1.52253 | − | 2.63709i | ||||||||||
289.8 | −1.00000 | − | 1.73205i | 0 | −2.00000 | + | 3.46410i | 3.11229 | 0 | 8.38218 | + | 16.5148i | 8.00000 | 0 | −3.11229 | − | 5.39064i | ||||||||||
289.9 | −1.00000 | − | 1.73205i | 0 | −2.00000 | + | 3.46410i | 4.38272 | 0 | −16.3832 | − | 8.63665i | 8.00000 | 0 | −4.38272 | − | 7.59110i | ||||||||||
289.10 | −1.00000 | − | 1.73205i | 0 | −2.00000 | + | 3.46410i | 14.9231 | 0 | 14.0821 | − | 12.0289i | 8.00000 | 0 | −14.9231 | − | 25.8476i | ||||||||||
289.11 | −1.00000 | − | 1.73205i | 0 | −2.00000 | + | 3.46410i | 19.8520 | 0 | −18.1695 | + | 3.58722i | 8.00000 | 0 | −19.8520 | − | 34.3846i | ||||||||||
289.12 | −1.00000 | − | 1.73205i | 0 | −2.00000 | + | 3.46410i | 20.5539 | 0 | 18.4043 | − | 2.06929i | 8.00000 | 0 | −20.5539 | − | 35.6003i | ||||||||||
361.1 | −1.00000 | + | 1.73205i | 0 | −2.00000 | − | 3.46410i | −18.7213 | 0 | −15.3695 | + | 10.3334i | 8.00000 | 0 | 18.7213 | − | 32.4263i | ||||||||||
361.2 | −1.00000 | + | 1.73205i | 0 | −2.00000 | − | 3.46410i | −15.6350 | 0 | 17.1578 | − | 6.97217i | 8.00000 | 0 | 15.6350 | − | 27.0807i | ||||||||||
361.3 | −1.00000 | + | 1.73205i | 0 | −2.00000 | − | 3.46410i | −15.5553 | 0 | 6.99950 | + | 17.1466i | 8.00000 | 0 | 15.5553 | − | 26.9425i | ||||||||||
361.4 | −1.00000 | + | 1.73205i | 0 | −2.00000 | − | 3.46410i | −3.49343 | 0 | −11.9430 | − | 14.1550i | 8.00000 | 0 | 3.49343 | − | 6.05080i | ||||||||||
361.5 | −1.00000 | + | 1.73205i | 0 | −2.00000 | − | 3.46410i | −1.46289 | 0 | −3.71747 | − | 18.1433i | 8.00000 | 0 | 1.46289 | − | 2.53380i | ||||||||||
361.6 | −1.00000 | + | 1.73205i | 0 | −2.00000 | − | 3.46410i | 0.521453 | 0 | −3.56519 | + | 18.1739i | 8.00000 | 0 | −0.521453 | + | 0.903183i | ||||||||||
361.7 | −1.00000 | + | 1.73205i | 0 | −2.00000 | − | 3.46410i | 1.52253 | 0 | 18.1220 | − | 3.82009i | 8.00000 | 0 | −1.52253 | + | 2.63709i | ||||||||||
361.8 | −1.00000 | + | 1.73205i | 0 | −2.00000 | − | 3.46410i | 3.11229 | 0 | 8.38218 | − | 16.5148i | 8.00000 | 0 | −3.11229 | + | 5.39064i | ||||||||||
See all 24 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
63.g | even | 3 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 378.4.h.a | 24 | |
3.b | odd | 2 | 1 | 126.4.h.b | yes | 24 | |
7.c | even | 3 | 1 | 378.4.e.b | 24 | ||
9.c | even | 3 | 1 | 378.4.e.b | 24 | ||
9.d | odd | 6 | 1 | 126.4.e.a | ✓ | 24 | |
21.h | odd | 6 | 1 | 126.4.e.a | ✓ | 24 | |
63.g | even | 3 | 1 | inner | 378.4.h.a | 24 | |
63.n | odd | 6 | 1 | 126.4.h.b | yes | 24 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
126.4.e.a | ✓ | 24 | 9.d | odd | 6 | 1 | |
126.4.e.a | ✓ | 24 | 21.h | odd | 6 | 1 | |
126.4.h.b | yes | 24 | 3.b | odd | 2 | 1 | |
126.4.h.b | yes | 24 | 63.n | odd | 6 | 1 | |
378.4.e.b | 24 | 7.c | even | 3 | 1 | ||
378.4.e.b | 24 | 9.c | even | 3 | 1 | ||
378.4.h.a | 24 | 1.a | even | 1 | 1 | trivial | |
378.4.h.a | 24 | 63.g | even | 3 | 1 | inner |