Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [369,3,Mod(278,369)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(369, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([2, 1]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("369.278");
S:= CuspForms(chi, 3);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 369 = 3^{2} \cdot 41 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 369.g (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: | \(10.0545217549\) |
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 algebraic \(q\)-expansion of this newform has not been computed, 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
278.1 | −3.87511 | 0 | 11.0165 | −2.44085 | 0 | −6.91222 | + | 6.91222i | −27.1895 | 0 | 9.45854 | ||||||||||||||||
278.2 | −3.75560 | 0 | 10.1045 | −1.37623 | 0 | 1.88410 | − | 1.88410i | −22.9260 | 0 | 5.16856 | ||||||||||||||||
278.3 | −3.17750 | 0 | 6.09651 | 9.24351 | 0 | −8.22970 | + | 8.22970i | −6.66167 | 0 | −29.3713 | ||||||||||||||||
278.4 | −3.11478 | 0 | 5.70186 | 2.46109 | 0 | 5.80271 | − | 5.80271i | −5.30091 | 0 | −7.66577 | ||||||||||||||||
278.5 | −3.02801 | 0 | 5.16885 | −7.81511 | 0 | 7.82646 | − | 7.82646i | −3.53928 | 0 | 23.6642 | ||||||||||||||||
278.6 | −2.76859 | 0 | 3.66508 | −6.51777 | 0 | 0.248864 | − | 0.248864i | 0.927255 | 0 | 18.0450 | ||||||||||||||||
278.7 | −2.38729 | 0 | 1.69916 | 4.01428 | 0 | −3.72450 | + | 3.72450i | 5.49277 | 0 | −9.58325 | ||||||||||||||||
278.8 | −2.36522 | 0 | 1.59427 | 6.17585 | 0 | 8.22083 | − | 8.22083i | 5.69007 | 0 | −14.6073 | ||||||||||||||||
278.9 | −1.65305 | 0 | −1.26743 | −2.76858 | 0 | −7.08063 | + | 7.08063i | 8.70732 | 0 | 4.57659 | ||||||||||||||||
278.10 | −1.29986 | 0 | −2.31036 | −8.88333 | 0 | −5.37438 | + | 5.37438i | 8.20259 | 0 | 11.5471 | ||||||||||||||||
278.11 | −1.14055 | 0 | −2.69914 | −1.53165 | 0 | 0.674505 | − | 0.674505i | 7.64072 | 0 | 1.74693 | ||||||||||||||||
278.12 | −0.955263 | 0 | −3.08747 | 0.755334 | 0 | 3.41913 | − | 3.41913i | 6.77040 | 0 | −0.721543 | ||||||||||||||||
278.13 | −0.474550 | 0 | −3.77480 | 3.29396 | 0 | −2.12416 | + | 2.12416i | 3.68953 | 0 | −1.56315 | ||||||||||||||||
278.14 | −0.304166 | 0 | −3.90748 | 8.85950 | 0 | 5.36900 | − | 5.36900i | 2.40519 | 0 | −2.69476 | ||||||||||||||||
278.15 | 0.304166 | 0 | −3.90748 | −8.85950 | 0 | 5.36900 | − | 5.36900i | −2.40519 | 0 | −2.69476 | ||||||||||||||||
278.16 | 0.474550 | 0 | −3.77480 | −3.29396 | 0 | −2.12416 | + | 2.12416i | −3.68953 | 0 | −1.56315 | ||||||||||||||||
278.17 | 0.955263 | 0 | −3.08747 | −0.755334 | 0 | 3.41913 | − | 3.41913i | −6.77040 | 0 | −0.721543 | ||||||||||||||||
278.18 | 1.14055 | 0 | −2.69914 | 1.53165 | 0 | 0.674505 | − | 0.674505i | −7.64072 | 0 | 1.74693 | ||||||||||||||||
278.19 | 1.29986 | 0 | −2.31036 | 8.88333 | 0 | −5.37438 | + | 5.37438i | −8.20259 | 0 | 11.5471 | ||||||||||||||||
278.20 | 1.65305 | 0 | −1.26743 | 2.76858 | 0 | −7.08063 | + | 7.08063i | −8.70732 | 0 | 4.57659 | ||||||||||||||||
See all 56 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
41.c | even | 4 | 1 | inner |
123.f | odd | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 369.3.g.a | ✓ | 56 |
3.b | odd | 2 | 1 | inner | 369.3.g.a | ✓ | 56 |
41.c | even | 4 | 1 | inner | 369.3.g.a | ✓ | 56 |
123.f | odd | 4 | 1 | inner | 369.3.g.a | ✓ | 56 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
369.3.g.a | ✓ | 56 | 1.a | even | 1 | 1 | trivial |
369.3.g.a | ✓ | 56 | 3.b | odd | 2 | 1 | inner |
369.3.g.a | ✓ | 56 | 41.c | even | 4 | 1 | inner |
369.3.g.a | ✓ | 56 | 123.f | odd | 4 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(369, [\chi])\).