Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [38,3,Mod(3,38)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(38, base_ring=CyclotomicField(18))
chi = DirichletCharacter(H, H._module([13]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("38.3");
S:= CuspForms(chi, 3);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 38 = 2 \cdot 19 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 38.f (of order \(18\), degree \(6\), 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: | \(1.03542500457\) |
Analytic rank: | \(0\) |
Dimension: | \(24\) |
Relative dimension: | \(4\) over \(\Q(\zeta_{18})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{18}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
3.1 | −0.909039 | + | 1.08335i | −1.73025 | − | 4.75383i | −0.347296 | − | 1.96962i | 0.678914 | − | 3.85031i | 6.72293 | + | 2.44695i | 1.77574 | + | 3.07567i | 2.44949 | + | 1.41421i | −12.7107 | + | 10.6656i | 3.55408 | + | 4.23559i |
3.2 | −0.909039 | + | 1.08335i | 0.836494 | + | 2.29825i | −0.347296 | − | 1.96962i | −0.852562 | + | 4.83512i | −3.25021 | − | 1.18298i | 0.583107 | + | 1.00997i | 2.44949 | + | 1.41421i | 2.31218 | − | 1.94015i | −4.46312 | − | 5.31894i |
3.3 | 0.909039 | − | 1.08335i | −0.830875 | − | 2.28281i | −0.347296 | − | 1.96962i | 0.0233547 | − | 0.132451i | −3.22838 | − | 1.17503i | 4.79341 | + | 8.30244i | −2.44949 | − | 1.41421i | 2.37354 | − | 1.99163i | −0.122261 | − | 0.145705i |
3.4 | 0.909039 | − | 1.08335i | 1.41923 | + | 3.89929i | −0.347296 | − | 1.96962i | −0.197003 | + | 1.11726i | 5.51443 | + | 2.00709i | −5.55599 | − | 9.62326i | −2.44949 | − | 1.41421i | −6.29588 | + | 5.28287i | 1.03130 | + | 1.22906i |
13.1 | −0.909039 | − | 1.08335i | −1.73025 | + | 4.75383i | −0.347296 | + | 1.96962i | 0.678914 | + | 3.85031i | 6.72293 | − | 2.44695i | 1.77574 | − | 3.07567i | 2.44949 | − | 1.41421i | −12.7107 | − | 10.6656i | 3.55408 | − | 4.23559i |
13.2 | −0.909039 | − | 1.08335i | 0.836494 | − | 2.29825i | −0.347296 | + | 1.96962i | −0.852562 | − | 4.83512i | −3.25021 | + | 1.18298i | 0.583107 | − | 1.00997i | 2.44949 | − | 1.41421i | 2.31218 | + | 1.94015i | −4.46312 | + | 5.31894i |
13.3 | 0.909039 | + | 1.08335i | −0.830875 | + | 2.28281i | −0.347296 | + | 1.96962i | 0.0233547 | + | 0.132451i | −3.22838 | + | 1.17503i | 4.79341 | − | 8.30244i | −2.44949 | + | 1.41421i | 2.37354 | + | 1.99163i | −0.122261 | + | 0.145705i |
13.4 | 0.909039 | + | 1.08335i | 1.41923 | − | 3.89929i | −0.347296 | + | 1.96962i | −0.197003 | − | 1.11726i | 5.51443 | − | 2.00709i | −5.55599 | + | 9.62326i | −2.44949 | + | 1.41421i | −6.29588 | − | 5.28287i | 1.03130 | − | 1.22906i |
15.1 | −0.483690 | + | 1.32893i | −5.23364 | + | 0.922832i | −1.53209 | − | 1.28558i | −1.06384 | + | 0.892670i | 1.30508 | − | 7.40149i | −4.39759 | + | 7.61684i | 2.44949 | − | 1.41421i | 18.0821 | − | 6.58136i | −0.671723 | − | 1.84554i |
15.2 | −0.483690 | + | 1.32893i | 3.64826 | − | 0.643286i | −1.53209 | − | 1.28558i | 0.297798 | − | 0.249882i | −0.909744 | + | 5.15941i | −2.96431 | + | 5.13434i | 2.44949 | − | 1.41421i | 4.43872 | − | 1.61556i | 0.188033 | + | 0.516617i |
15.3 | 0.483690 | − | 1.32893i | −0.173101 | + | 0.0305223i | −1.53209 | − | 1.28558i | 6.04513 | − | 5.07246i | −0.0431650 | + | 0.244801i | −3.82954 | + | 6.63297i | −2.44949 | + | 1.41421i | −8.42820 | + | 3.06761i | −3.81697 | − | 10.4870i |
15.4 | 0.483690 | − | 1.32893i | 3.82266 | − | 0.674039i | −1.53209 | − | 1.28558i | −6.81117 | + | 5.71525i | 0.953235 | − | 5.40606i | 2.55329 | − | 4.42243i | −2.44949 | + | 1.41421i | 5.70119 | − | 2.07506i | 4.30065 | + | 11.8160i |
21.1 | −1.39273 | − | 0.245576i | −2.33610 | + | 2.78405i | 1.87939 | + | 0.684040i | −7.04907 | + | 2.56565i | 3.93724 | − | 3.30374i | 3.79852 | + | 6.57923i | −2.44949 | − | 1.41421i | −0.730761 | − | 4.14435i | 10.4475 | − | 1.84218i |
21.2 | −1.39273 | − | 0.245576i | 0.272417 | − | 0.324654i | 1.87939 | + | 0.684040i | 7.98876 | − | 2.90767i | −0.459130 | + | 0.385256i | −2.58545 | − | 4.47813i | −2.44949 | − | 1.41421i | 1.53164 | + | 8.68639i | −11.8402 | + | 2.08775i |
21.3 | 1.39273 | + | 0.245576i | −3.75536 | + | 4.47547i | 1.87939 | + | 0.684040i | 5.30025 | − | 1.92913i | −6.32926 | + | 5.31088i | −0.990297 | − | 1.71524i | 2.44949 | + | 1.41421i | −4.36422 | − | 24.7507i | 7.85555 | − | 1.38515i |
21.4 | 1.39273 | + | 0.245576i | 1.06027 | − | 1.26358i | 1.87939 | + | 0.684040i | −4.36056 | + | 1.58711i | 1.78697 | − | 1.49945i | −2.18088 | − | 3.77740i | 2.44949 | + | 1.41421i | 1.09037 | + | 6.18380i | −6.46283 | + | 1.13957i |
29.1 | −1.39273 | + | 0.245576i | −2.33610 | − | 2.78405i | 1.87939 | − | 0.684040i | −7.04907 | − | 2.56565i | 3.93724 | + | 3.30374i | 3.79852 | − | 6.57923i | −2.44949 | + | 1.41421i | −0.730761 | + | 4.14435i | 10.4475 | + | 1.84218i |
29.2 | −1.39273 | + | 0.245576i | 0.272417 | + | 0.324654i | 1.87939 | − | 0.684040i | 7.98876 | + | 2.90767i | −0.459130 | − | 0.385256i | −2.58545 | + | 4.47813i | −2.44949 | + | 1.41421i | 1.53164 | − | 8.68639i | −11.8402 | − | 2.08775i |
29.3 | 1.39273 | − | 0.245576i | −3.75536 | − | 4.47547i | 1.87939 | − | 0.684040i | 5.30025 | + | 1.92913i | −6.32926 | − | 5.31088i | −0.990297 | + | 1.71524i | 2.44949 | − | 1.41421i | −4.36422 | + | 24.7507i | 7.85555 | + | 1.38515i |
29.4 | 1.39273 | − | 0.245576i | 1.06027 | + | 1.26358i | 1.87939 | − | 0.684040i | −4.36056 | − | 1.58711i | 1.78697 | + | 1.49945i | −2.18088 | + | 3.77740i | 2.44949 | − | 1.41421i | 1.09037 | − | 6.18380i | −6.46283 | − | 1.13957i |
See all 24 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
19.f | odd | 18 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 38.3.f.a | ✓ | 24 |
3.b | odd | 2 | 1 | 342.3.z.b | 24 | ||
4.b | odd | 2 | 1 | 304.3.z.c | 24 | ||
19.e | even | 9 | 1 | 722.3.b.f | 24 | ||
19.f | odd | 18 | 1 | inner | 38.3.f.a | ✓ | 24 |
19.f | odd | 18 | 1 | 722.3.b.f | 24 | ||
57.j | even | 18 | 1 | 342.3.z.b | 24 | ||
76.k | even | 18 | 1 | 304.3.z.c | 24 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
38.3.f.a | ✓ | 24 | 1.a | even | 1 | 1 | trivial |
38.3.f.a | ✓ | 24 | 19.f | odd | 18 | 1 | inner |
304.3.z.c | 24 | 4.b | odd | 2 | 1 | ||
304.3.z.c | 24 | 76.k | even | 18 | 1 | ||
342.3.z.b | 24 | 3.b | odd | 2 | 1 | ||
342.3.z.b | 24 | 57.j | even | 18 | 1 | ||
722.3.b.f | 24 | 19.e | even | 9 | 1 | ||
722.3.b.f | 24 | 19.f | odd | 18 | 1 |
Hecke kernels
This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(38, [\chi])\).