Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1368,3,Mod(809,1368)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1368, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 0, 3, 4]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1368.809");
S:= CuspForms(chi, 3);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 1368 = 2^{3} \cdot 3^{2} \cdot 19 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 1368.cs (of order \(6\), 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: | \(37.2753001645\) |
Analytic rank: | \(0\) |
Dimension: | \(40\) |
Relative dimension: | \(20\) over \(\Q(\zeta_{6})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
809.1 | 0 | 0 | 0 | −7.33109 | − | 4.23261i | 0 | 4.59226 | 0 | 0 | 0 | ||||||||||||||||
809.2 | 0 | 0 | 0 | −7.27843 | − | 4.20220i | 0 | −8.93653 | 0 | 0 | 0 | ||||||||||||||||
809.3 | 0 | 0 | 0 | −5.69866 | − | 3.29012i | 0 | −0.331562 | 0 | 0 | 0 | ||||||||||||||||
809.4 | 0 | 0 | 0 | −5.20909 | − | 3.00747i | 0 | −0.0305841 | 0 | 0 | 0 | ||||||||||||||||
809.5 | 0 | 0 | 0 | −4.59641 | − | 2.65374i | 0 | 0.408339 | 0 | 0 | 0 | ||||||||||||||||
809.6 | 0 | 0 | 0 | −4.04818 | − | 2.33722i | 0 | −12.0768 | 0 | 0 | 0 | ||||||||||||||||
809.7 | 0 | 0 | 0 | −2.45740 | − | 1.41878i | 0 | 13.4036 | 0 | 0 | 0 | ||||||||||||||||
809.8 | 0 | 0 | 0 | −1.59721 | − | 0.922150i | 0 | −7.57779 | 0 | 0 | 0 | ||||||||||||||||
809.9 | 0 | 0 | 0 | −0.741654 | − | 0.428194i | 0 | −3.22148 | 0 | 0 | 0 | ||||||||||||||||
809.10 | 0 | 0 | 0 | −0.124535 | − | 0.0719003i | 0 | 9.77058 | 0 | 0 | 0 | ||||||||||||||||
809.11 | 0 | 0 | 0 | 0.124535 | + | 0.0719003i | 0 | 9.77058 | 0 | 0 | 0 | ||||||||||||||||
809.12 | 0 | 0 | 0 | 0.741654 | + | 0.428194i | 0 | −3.22148 | 0 | 0 | 0 | ||||||||||||||||
809.13 | 0 | 0 | 0 | 1.59721 | + | 0.922150i | 0 | −7.57779 | 0 | 0 | 0 | ||||||||||||||||
809.14 | 0 | 0 | 0 | 2.45740 | + | 1.41878i | 0 | 13.4036 | 0 | 0 | 0 | ||||||||||||||||
809.15 | 0 | 0 | 0 | 4.04818 | + | 2.33722i | 0 | −12.0768 | 0 | 0 | 0 | ||||||||||||||||
809.16 | 0 | 0 | 0 | 4.59641 | + | 2.65374i | 0 | 0.408339 | 0 | 0 | 0 | ||||||||||||||||
809.17 | 0 | 0 | 0 | 5.20909 | + | 3.00747i | 0 | −0.0305841 | 0 | 0 | 0 | ||||||||||||||||
809.18 | 0 | 0 | 0 | 5.69866 | + | 3.29012i | 0 | −0.331562 | 0 | 0 | 0 | ||||||||||||||||
809.19 | 0 | 0 | 0 | 7.27843 | + | 4.20220i | 0 | −8.93653 | 0 | 0 | 0 | ||||||||||||||||
809.20 | 0 | 0 | 0 | 7.33109 | + | 4.23261i | 0 | 4.59226 | 0 | 0 | 0 | ||||||||||||||||
See all 40 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
19.c | even | 3 | 1 | inner |
57.h | odd | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 1368.3.cs.b | ✓ | 40 |
3.b | odd | 2 | 1 | inner | 1368.3.cs.b | ✓ | 40 |
19.c | even | 3 | 1 | inner | 1368.3.cs.b | ✓ | 40 |
57.h | odd | 6 | 1 | inner | 1368.3.cs.b | ✓ | 40 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
1368.3.cs.b | ✓ | 40 | 1.a | even | 1 | 1 | trivial |
1368.3.cs.b | ✓ | 40 | 3.b | odd | 2 | 1 | inner |
1368.3.cs.b | ✓ | 40 | 19.c | even | 3 | 1 | inner |
1368.3.cs.b | ✓ | 40 | 57.h | odd | 6 | 1 | inner |