Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [806,2,Mod(439,806)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(806, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([5, 4]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("806.439");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 806 = 2 \cdot 13 \cdot 31 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 806.w (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: | \(6.43594240292\) |
Analytic rank: | \(0\) |
Dimension: | \(76\) |
Relative dimension: | \(38\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
439.1 | −0.866025 | + | 0.500000i | −3.09898 | 0.500000 | − | 0.866025i | 3.82547 | − | 2.20864i | 2.68380 | − | 1.54949i | −1.26741 | + | 0.731739i | 1.00000i | 6.60368 | −2.20864 | + | 3.82547i | ||||||
439.2 | −0.866025 | + | 0.500000i | −3.07789 | 0.500000 | − | 0.866025i | −0.798451 | + | 0.460986i | 2.66553 | − | 1.53895i | 4.50628 | − | 2.60170i | 1.00000i | 6.47342 | 0.460986 | − | 0.798451i | ||||||
439.3 | −0.866025 | + | 0.500000i | −3.05151 | 0.500000 | − | 0.866025i | −0.288325 | + | 0.166465i | 2.64269 | − | 1.52576i | −2.38358 | + | 1.37616i | 1.00000i | 6.31172 | 0.166465 | − | 0.288325i | ||||||
439.4 | −0.866025 | + | 0.500000i | −2.43132 | 0.500000 | − | 0.866025i | −3.48098 | + | 2.00974i | 2.10559 | − | 1.21566i | 0.459014 | − | 0.265012i | 1.00000i | 2.91134 | 2.00974 | − | 3.48098i | ||||||
439.5 | −0.866025 | + | 0.500000i | −1.63443 | 0.500000 | − | 0.866025i | 2.24632 | − | 1.29691i | 1.41546 | − | 0.817215i | 0.490644 | − | 0.283274i | 1.00000i | −0.328637 | −1.29691 | + | 2.24632i | ||||||
439.6 | −0.866025 | + | 0.500000i | −1.60467 | 0.500000 | − | 0.866025i | 2.03633 | − | 1.17568i | 1.38968 | − | 0.802334i | 3.91920 | − | 2.26275i | 1.00000i | −0.425039 | −1.17568 | + | 2.03633i | ||||||
439.7 | −0.866025 | + | 0.500000i | −1.52216 | 0.500000 | − | 0.866025i | −1.27474 | + | 0.735973i | 1.31823 | − | 0.761079i | −4.28830 | + | 2.47585i | 1.00000i | −0.683032 | 0.735973 | − | 1.27474i | ||||||
439.8 | −0.866025 | + | 0.500000i | −1.20830 | 0.500000 | − | 0.866025i | 0.897452 | − | 0.518144i | 1.04642 | − | 0.604149i | −0.236744 | + | 0.136684i | 1.00000i | −1.54002 | −0.518144 | + | 0.897452i | ||||||
439.9 | −0.866025 | + | 0.500000i | −0.339158 | 0.500000 | − | 0.866025i | 1.66872 | − | 0.963438i | 0.293719 | − | 0.169579i | −0.304644 | + | 0.175886i | 1.00000i | −2.88497 | −0.963438 | + | 1.66872i | ||||||
439.10 | −0.866025 | + | 0.500000i | −0.0977685 | 0.500000 | − | 0.866025i | −2.80209 | + | 1.61779i | 0.0846700 | − | 0.0488843i | 0.245298 | − | 0.141623i | 1.00000i | −2.99044 | 1.61779 | − | 2.80209i | ||||||
439.11 | −0.866025 | + | 0.500000i | 0.597655 | 0.500000 | − | 0.866025i | −2.25230 | + | 1.30037i | −0.517585 | + | 0.298828i | 4.23575 | − | 2.44551i | 1.00000i | −2.64281 | 1.30037 | − | 2.25230i | ||||||
439.12 | −0.866025 | + | 0.500000i | 0.616372 | 0.500000 | − | 0.866025i | −0.916907 | + | 0.529377i | −0.533794 | + | 0.308186i | 0.932513 | − | 0.538387i | 1.00000i | −2.62009 | 0.529377 | − | 0.916907i | ||||||
439.13 | −0.866025 | + | 0.500000i | 1.00798 | 0.500000 | − | 0.866025i | −0.178395 | + | 0.102997i | −0.872939 | + | 0.503992i | −1.66018 | + | 0.958507i | 1.00000i | −1.98397 | 0.102997 | − | 0.178395i | ||||||
439.14 | −0.866025 | + | 0.500000i | 1.85261 | 0.500000 | − | 0.866025i | 3.66076 | − | 2.11354i | −1.60441 | + | 0.926305i | 0.822470 | − | 0.474853i | 1.00000i | 0.432160 | −2.11354 | + | 3.66076i | ||||||
439.15 | −0.866025 | + | 0.500000i | 1.88976 | 0.500000 | − | 0.866025i | −0.545912 | + | 0.315182i | −1.63658 | + | 0.944882i | −3.21000 | + | 1.85330i | 1.00000i | 0.571208 | 0.315182 | − | 0.545912i | ||||||
439.16 | −0.866025 | + | 0.500000i | 2.16919 | 0.500000 | − | 0.866025i | −3.78268 | + | 2.18393i | −1.87857 | + | 1.08459i | −2.57189 | + | 1.48488i | 1.00000i | 1.70537 | 2.18393 | − | 3.78268i | ||||||
439.17 | −0.866025 | + | 0.500000i | 2.51146 | 0.500000 | − | 0.866025i | 0.600246 | − | 0.346552i | −2.17499 | + | 1.25573i | 2.39883 | − | 1.38497i | 1.00000i | 3.30746 | −0.346552 | + | 0.600246i | ||||||
439.18 | −0.866025 | + | 0.500000i | 2.99236 | 0.500000 | − | 0.866025i | −0.887634 | + | 0.512476i | −2.59146 | + | 1.49618i | 1.52289 | − | 0.879243i | 1.00000i | 5.95423 | 0.512476 | − | 0.887634i | ||||||
439.19 | −0.866025 | + | 0.500000i | 3.42879 | 0.500000 | − | 0.866025i | 2.27311 | − | 1.31238i | −2.96942 | + | 1.71440i | −2.97616 | + | 1.71829i | 1.00000i | 8.75662 | −1.31238 | + | 2.27311i | ||||||
439.20 | 0.866025 | − | 0.500000i | −3.16068 | 0.500000 | − | 0.866025i | −1.09564 | + | 0.632568i | −2.73723 | + | 1.58034i | 1.09699 | − | 0.633348i | − | 1.00000i | 6.98987 | −0.632568 | + | 1.09564i | |||||
See all 76 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
403.v | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 806.2.w.a | yes | 76 |
13.e | even | 6 | 1 | 806.2.o.a | ✓ | 76 | |
31.c | even | 3 | 1 | 806.2.o.a | ✓ | 76 | |
403.v | even | 6 | 1 | inner | 806.2.w.a | yes | 76 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
806.2.o.a | ✓ | 76 | 13.e | even | 6 | 1 | |
806.2.o.a | ✓ | 76 | 31.c | even | 3 | 1 | |
806.2.w.a | yes | 76 | 1.a | even | 1 | 1 | trivial |
806.2.w.a | yes | 76 | 403.v | even | 6 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(806, [\chi])\).