Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [378,3,Mod(73,378)]
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, 1]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("378.73");
S:= CuspForms(chi, 3);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 378 = 2 \cdot 3^{3} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 378.p (of order \(6\), 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: | \(10.2997539928\) |
Analytic rank: | \(0\) |
Dimension: | \(32\) |
Relative dimension: | \(16\) over \(\Q(\zeta_{6})\) |
Twist minimal: | no (minimal twist has level 126) |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
73.1 | −1.41421 | 0 | 2.00000 | −6.25411 | − | 3.61081i | 0 | −6.40955 | + | 2.81382i | −2.82843 | 0 | 8.84464 | + | 5.10646i | ||||||||||||
73.2 | −1.41421 | 0 | 2.00000 | −3.75454 | − | 2.16769i | 0 | 5.02745 | − | 4.87080i | −2.82843 | 0 | 5.30973 | + | 3.06557i | ||||||||||||
73.3 | −1.41421 | 0 | 2.00000 | −2.65919 | − | 1.53529i | 0 | −3.27459 | + | 6.18685i | −2.82843 | 0 | 3.76067 | + | 2.17122i | ||||||||||||
73.4 | −1.41421 | 0 | 2.00000 | −1.56851 | − | 0.905579i | 0 | 6.71333 | + | 1.98274i | −2.82843 | 0 | 2.21821 | + | 1.28068i | ||||||||||||
73.5 | −1.41421 | 0 | 2.00000 | −1.56104 | − | 0.901265i | 0 | −0.490625 | − | 6.98279i | −2.82843 | 0 | 2.20764 | + | 1.27458i | ||||||||||||
73.6 | −1.41421 | 0 | 2.00000 | 3.25801 | + | 1.88101i | 0 | 4.45903 | + | 5.39603i | −2.82843 | 0 | −4.60752 | − | 2.66015i | ||||||||||||
73.7 | −1.41421 | 0 | 2.00000 | 6.01208 | + | 3.47108i | 0 | −6.97949 | − | 0.535409i | −2.82843 | 0 | −8.50237 | − | 4.90885i | ||||||||||||
73.8 | −1.41421 | 0 | 2.00000 | 6.52730 | + | 3.76854i | 0 | −1.66686 | − | 6.79864i | −2.82843 | 0 | −9.23099 | − | 5.32952i | ||||||||||||
73.9 | 1.41421 | 0 | 2.00000 | −6.30090 | − | 3.63782i | 0 | −5.27174 | + | 4.60530i | 2.82843 | 0 | −8.91081 | − | 5.14466i | ||||||||||||
73.10 | 1.41421 | 0 | 2.00000 | −4.87915 | − | 2.81698i | 0 | 5.20700 | + | 4.67837i | 2.82843 | 0 | −6.90016 | − | 3.98381i | ||||||||||||
73.11 | 1.41421 | 0 | 2.00000 | −4.82569 | − | 2.78612i | 0 | 4.90353 | − | 4.99554i | 2.82843 | 0 | −6.82456 | − | 3.94016i | ||||||||||||
73.12 | 1.41421 | 0 | 2.00000 | −1.59708 | − | 0.922074i | 0 | −6.32875 | − | 2.99115i | 2.82843 | 0 | −2.25861 | − | 1.30401i | ||||||||||||
73.13 | 1.41421 | 0 | 2.00000 | −0.214369 | − | 0.123766i | 0 | −1.01424 | + | 6.92613i | 2.82843 | 0 | −0.303164 | − | 0.175032i | ||||||||||||
73.14 | 1.41421 | 0 | 2.00000 | 3.45664 | + | 1.99569i | 0 | 4.98028 | − | 4.91903i | 2.82843 | 0 | 4.88842 | + | 2.82233i | ||||||||||||
73.15 | 1.41421 | 0 | 2.00000 | 6.47195 | + | 3.73658i | 0 | −6.15173 | − | 3.34009i | 2.82843 | 0 | 9.15272 | + | 5.28432i | ||||||||||||
73.16 | 1.41421 | 0 | 2.00000 | 7.88860 | + | 4.55449i | 0 | 5.29697 | + | 4.57626i | 2.82843 | 0 | 11.1562 | + | 6.44102i | ||||||||||||
145.1 | −1.41421 | 0 | 2.00000 | −6.25411 | + | 3.61081i | 0 | −6.40955 | − | 2.81382i | −2.82843 | 0 | 8.84464 | − | 5.10646i | ||||||||||||
145.2 | −1.41421 | 0 | 2.00000 | −3.75454 | + | 2.16769i | 0 | 5.02745 | + | 4.87080i | −2.82843 | 0 | 5.30973 | − | 3.06557i | ||||||||||||
145.3 | −1.41421 | 0 | 2.00000 | −2.65919 | + | 1.53529i | 0 | −3.27459 | − | 6.18685i | −2.82843 | 0 | 3.76067 | − | 2.17122i | ||||||||||||
145.4 | −1.41421 | 0 | 2.00000 | −1.56851 | + | 0.905579i | 0 | 6.71333 | − | 1.98274i | −2.82843 | 0 | 2.21821 | − | 1.28068i | ||||||||||||
See all 32 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
63.t | odd | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 378.3.p.a | 32 | |
3.b | odd | 2 | 1 | 126.3.p.a | yes | 32 | |
7.d | odd | 6 | 1 | 378.3.j.a | 32 | ||
9.c | even | 3 | 1 | 378.3.j.a | 32 | ||
9.d | odd | 6 | 1 | 126.3.j.a | ✓ | 32 | |
21.g | even | 6 | 1 | 126.3.j.a | ✓ | 32 | |
63.i | even | 6 | 1 | 126.3.p.a | yes | 32 | |
63.t | odd | 6 | 1 | inner | 378.3.p.a | 32 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
126.3.j.a | ✓ | 32 | 9.d | odd | 6 | 1 | |
126.3.j.a | ✓ | 32 | 21.g | even | 6 | 1 | |
126.3.p.a | yes | 32 | 3.b | odd | 2 | 1 | |
126.3.p.a | yes | 32 | 63.i | even | 6 | 1 | |
378.3.j.a | 32 | 7.d | odd | 6 | 1 | ||
378.3.j.a | 32 | 9.c | even | 3 | 1 | ||
378.3.p.a | 32 | 1.a | even | 1 | 1 | trivial | |
378.3.p.a | 32 | 63.t | odd | 6 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(378, [\chi])\).