Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [387,2,Mod(26,387)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(387, base_ring=CyclotomicField(42))
chi = DirichletCharacter(H, H._module([21, 17]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("387.26");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 387 = 3^{2} \cdot 43 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 387.bc (of order \(42\), degree \(12\), 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: | \(3.09021055822\) |
Analytic rank: | \(0\) |
Dimension: | \(168\) |
Relative dimension: | \(14\) over \(\Q(\zeta_{42})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{42}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
26.1 | −0.527337 | − | 2.31042i | 0 | −3.25800 | + | 1.56897i | 0.512348 | − | 1.30544i | 0 | −2.21115 | − | 1.27661i | 2.38790 | + | 2.99433i | 0 | −3.28629 | − | 0.495329i | ||||||
26.2 | −0.520055 | − | 2.27851i | 0 | −3.11920 | + | 1.50213i | −0.631486 | + | 1.60900i | 0 | 3.26254 | + | 1.88363i | 2.13045 | + | 2.67150i | 0 | 3.99453 | + | 0.602078i | ||||||
26.3 | −0.449553 | − | 1.96962i | 0 | −1.87536 | + | 0.903128i | −0.914267 | + | 2.32952i | 0 | 2.03412 | + | 1.17440i | 0.102655 | + | 0.128725i | 0 | 4.99927 | + | 0.753519i | ||||||
26.4 | −0.341624 | − | 1.49675i | 0 | −0.321628 | + | 0.154888i | −1.26738 | + | 3.22924i | 0 | −3.78274 | − | 2.18397i | −1.57272 | − | 1.97212i | 0 | 5.26634 | + | 0.793773i | ||||||
26.5 | −0.284573 | − | 1.24679i | 0 | 0.328424 | − | 0.158161i | 1.02032 | − | 2.59974i | 0 | 0.443921 | + | 0.256298i | −1.88536 | − | 2.36417i | 0 | −3.53169 | − | 0.532317i | ||||||
26.6 | −0.192228 | − | 0.842204i | 0 | 1.12958 | − | 0.543978i | −0.389792 | + | 0.993173i | 0 | −0.428070 | − | 0.247146i | −1.75250 | − | 2.19756i | 0 | 0.911383 | + | 0.137369i | ||||||
26.7 | −0.0709389 | − | 0.310803i | 0 | 1.71037 | − | 0.823671i | 0.920597 | − | 2.34564i | 0 | 2.29370 | + | 1.32427i | −0.774864 | − | 0.971649i | 0 | −0.794340 | − | 0.119727i | ||||||
26.8 | 0.0709389 | + | 0.310803i | 0 | 1.71037 | − | 0.823671i | −0.920597 | + | 2.34564i | 0 | 2.29370 | + | 1.32427i | 0.774864 | + | 0.971649i | 0 | −0.794340 | − | 0.119727i | ||||||
26.9 | 0.192228 | + | 0.842204i | 0 | 1.12958 | − | 0.543978i | 0.389792 | − | 0.993173i | 0 | −0.428070 | − | 0.247146i | 1.75250 | + | 2.19756i | 0 | 0.911383 | + | 0.137369i | ||||||
26.10 | 0.284573 | + | 1.24679i | 0 | 0.328424 | − | 0.158161i | −1.02032 | + | 2.59974i | 0 | 0.443921 | + | 0.256298i | 1.88536 | + | 2.36417i | 0 | −3.53169 | − | 0.532317i | ||||||
26.11 | 0.341624 | + | 1.49675i | 0 | −0.321628 | + | 0.154888i | 1.26738 | − | 3.22924i | 0 | −3.78274 | − | 2.18397i | 1.57272 | + | 1.97212i | 0 | 5.26634 | + | 0.793773i | ||||||
26.12 | 0.449553 | + | 1.96962i | 0 | −1.87536 | + | 0.903128i | 0.914267 | − | 2.32952i | 0 | 2.03412 | + | 1.17440i | −0.102655 | − | 0.128725i | 0 | 4.99927 | + | 0.753519i | ||||||
26.13 | 0.520055 | + | 2.27851i | 0 | −3.11920 | + | 1.50213i | 0.631486 | − | 1.60900i | 0 | 3.26254 | + | 1.88363i | −2.13045 | − | 2.67150i | 0 | 3.99453 | + | 0.602078i | ||||||
26.14 | 0.527337 | + | 2.31042i | 0 | −3.25800 | + | 1.56897i | −0.512348 | + | 1.30544i | 0 | −2.21115 | − | 1.27661i | −2.38790 | − | 2.99433i | 0 | −3.28629 | − | 0.495329i | ||||||
62.1 | −1.61315 | + | 2.02283i | 0 | −1.04453 | − | 4.57640i | −3.04879 | + | 2.07863i | 0 | −2.30472 | + | 1.33063i | 6.28013 | + | 3.02435i | 0 | 0.713450 | − | 9.52033i | ||||||
62.2 | −1.44603 | + | 1.81327i | 0 | −0.751887 | − | 3.29423i | −0.852423 | + | 0.581172i | 0 | 1.78964 | − | 1.03325i | 2.88143 | + | 1.38762i | 0 | 0.178811 | − | 2.38607i | ||||||
62.3 | −1.22767 | + | 1.53944i | 0 | −0.417683 | − | 1.82999i | 0.566552 | − | 0.386269i | 0 | 2.46740 | − | 1.42456i | −0.218114 | − | 0.105038i | 0 | −0.100898 | + | 1.34638i | ||||||
62.4 | −0.895578 | + | 1.12302i | 0 | −0.0140704 | − | 0.0616464i | 2.30357 | − | 1.57055i | 0 | −3.82546 | + | 2.20863i | −2.50646 | − | 1.20705i | 0 | −0.299272 | + | 3.99351i | ||||||
62.5 | −0.722716 | + | 0.906257i | 0 | 0.146058 | + | 0.639922i | −1.18640 | + | 0.808874i | 0 | −1.10528 | + | 0.638135i | −2.77420 | − | 1.33599i | 0 | 0.124383 | − | 1.65977i | ||||||
62.6 | −0.435866 | + | 0.546559i | 0 | 0.336295 | + | 1.47340i | −1.07035 | + | 0.729752i | 0 | 2.64692 | − | 1.52820i | −2.21157 | − | 1.06504i | 0 | 0.0676768 | − | 0.903084i | ||||||
See next 80 embeddings (of 168 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
43.h | odd | 42 | 1 | inner |
129.n | even | 42 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 387.2.bc.a | ✓ | 168 |
3.b | odd | 2 | 1 | inner | 387.2.bc.a | ✓ | 168 |
43.h | odd | 42 | 1 | inner | 387.2.bc.a | ✓ | 168 |
129.n | even | 42 | 1 | inner | 387.2.bc.a | ✓ | 168 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
387.2.bc.a | ✓ | 168 | 1.a | even | 1 | 1 | trivial |
387.2.bc.a | ✓ | 168 | 3.b | odd | 2 | 1 | inner |
387.2.bc.a | ✓ | 168 | 43.h | odd | 42 | 1 | inner |
387.2.bc.a | ✓ | 168 | 129.n | even | 42 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(387, [\chi])\).