Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [798,2,Mod(493,798)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(798, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 1, 3]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("798.493");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 798 = 2 \cdot 3 \cdot 7 \cdot 19 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 798.be (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: | \(6.37206208130\) |
Analytic rank: | \(0\) |
Dimension: | \(28\) |
Relative dimension: | \(14\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
493.1 | −0.866025 | + | 0.500000i | −0.500000 | + | 0.866025i | 0.500000 | − | 0.866025i | −3.04056 | + | 1.75547i | − | 1.00000i | −0.775540 | − | 2.52953i | 1.00000i | −0.500000 | − | 0.866025i | 1.75547 | − | 3.04056i | |||
493.2 | −0.866025 | + | 0.500000i | −0.500000 | + | 0.866025i | 0.500000 | − | 0.866025i | −2.81074 | + | 1.62278i | − | 1.00000i | 2.61522 | − | 0.400774i | 1.00000i | −0.500000 | − | 0.866025i | 1.62278 | − | 2.81074i | |||
493.3 | −0.866025 | + | 0.500000i | −0.500000 | + | 0.866025i | 0.500000 | − | 0.866025i | −1.36428 | + | 0.787665i | − | 1.00000i | −2.58854 | + | 0.547225i | 1.00000i | −0.500000 | − | 0.866025i | 0.787665 | − | 1.36428i | |||
493.4 | −0.866025 | + | 0.500000i | −0.500000 | + | 0.866025i | 0.500000 | − | 0.866025i | 0.612231 | − | 0.353472i | − | 1.00000i | 0.810021 | + | 2.51870i | 1.00000i | −0.500000 | − | 0.866025i | −0.353472 | + | 0.612231i | |||
493.5 | −0.866025 | + | 0.500000i | −0.500000 | + | 0.866025i | 0.500000 | − | 0.866025i | 1.02150 | − | 0.589763i | − | 1.00000i | −0.486091 | − | 2.60071i | 1.00000i | −0.500000 | − | 0.866025i | −0.589763 | + | 1.02150i | |||
493.6 | −0.866025 | + | 0.500000i | −0.500000 | + | 0.866025i | 0.500000 | − | 0.866025i | 2.00675 | − | 1.15860i | − | 1.00000i | 2.55528 | − | 0.685945i | 1.00000i | −0.500000 | − | 0.866025i | −1.15860 | + | 2.00675i | |||
493.7 | −0.866025 | + | 0.500000i | −0.500000 | + | 0.866025i | 0.500000 | − | 0.866025i | 3.80714 | − | 2.19805i | − | 1.00000i | −2.63035 | + | 0.285014i | 1.00000i | −0.500000 | − | 0.866025i | −2.19805 | + | 3.80714i | |||
493.8 | 0.866025 | − | 0.500000i | −0.500000 | + | 0.866025i | 0.500000 | − | 0.866025i | −3.40054 | + | 1.96330i | 1.00000i | 0.223574 | − | 2.63629i | − | 1.00000i | −0.500000 | − | 0.866025i | −1.96330 | + | 3.40054i | |||
493.9 | 0.866025 | − | 0.500000i | −0.500000 | + | 0.866025i | 0.500000 | − | 0.866025i | −2.72411 | + | 1.57277i | 1.00000i | −0.413227 | + | 2.61328i | − | 1.00000i | −0.500000 | − | 0.866025i | −1.57277 | + | 2.72411i | |||
493.10 | 0.866025 | − | 0.500000i | −0.500000 | + | 0.866025i | 0.500000 | − | 0.866025i | −1.23273 | + | 0.711717i | 1.00000i | 2.49274 | + | 0.886713i | − | 1.00000i | −0.500000 | − | 0.866025i | −0.711717 | + | 1.23273i | |||
493.11 | 0.866025 | − | 0.500000i | −0.500000 | + | 0.866025i | 0.500000 | − | 0.866025i | −0.114418 | + | 0.0660594i | 1.00000i | −2.62389 | − | 0.339393i | − | 1.00000i | −0.500000 | − | 0.866025i | −0.0660594 | + | 0.114418i | |||
493.12 | 0.866025 | − | 0.500000i | −0.500000 | + | 0.866025i | 0.500000 | − | 0.866025i | 0.247308 | − | 0.142783i | 1.00000i | −0.779658 | + | 2.52827i | − | 1.00000i | −0.500000 | − | 0.866025i | 0.142783 | − | 0.247308i | |||
493.13 | 0.866025 | − | 0.500000i | −0.500000 | + | 0.866025i | 0.500000 | − | 0.866025i | 1.94282 | − | 1.12169i | 1.00000i | −2.03542 | − | 1.69029i | − | 1.00000i | −0.500000 | − | 0.866025i | 1.12169 | − | 1.94282i | |||
493.14 | 0.866025 | − | 0.500000i | −0.500000 | + | 0.866025i | 0.500000 | − | 0.866025i | 2.04963 | − | 1.18335i | 1.00000i | 2.63588 | − | 0.228315i | − | 1.00000i | −0.500000 | − | 0.866025i | 1.18335 | − | 2.04963i | |||
607.1 | −0.866025 | − | 0.500000i | −0.500000 | − | 0.866025i | 0.500000 | + | 0.866025i | −3.04056 | − | 1.75547i | 1.00000i | −0.775540 | + | 2.52953i | − | 1.00000i | −0.500000 | + | 0.866025i | 1.75547 | + | 3.04056i | |||
607.2 | −0.866025 | − | 0.500000i | −0.500000 | − | 0.866025i | 0.500000 | + | 0.866025i | −2.81074 | − | 1.62278i | 1.00000i | 2.61522 | + | 0.400774i | − | 1.00000i | −0.500000 | + | 0.866025i | 1.62278 | + | 2.81074i | |||
607.3 | −0.866025 | − | 0.500000i | −0.500000 | − | 0.866025i | 0.500000 | + | 0.866025i | −1.36428 | − | 0.787665i | 1.00000i | −2.58854 | − | 0.547225i | − | 1.00000i | −0.500000 | + | 0.866025i | 0.787665 | + | 1.36428i | |||
607.4 | −0.866025 | − | 0.500000i | −0.500000 | − | 0.866025i | 0.500000 | + | 0.866025i | 0.612231 | + | 0.353472i | 1.00000i | 0.810021 | − | 2.51870i | − | 1.00000i | −0.500000 | + | 0.866025i | −0.353472 | − | 0.612231i | |||
607.5 | −0.866025 | − | 0.500000i | −0.500000 | − | 0.866025i | 0.500000 | + | 0.866025i | 1.02150 | + | 0.589763i | 1.00000i | −0.486091 | + | 2.60071i | − | 1.00000i | −0.500000 | + | 0.866025i | −0.589763 | − | 1.02150i | |||
607.6 | −0.866025 | − | 0.500000i | −0.500000 | − | 0.866025i | 0.500000 | + | 0.866025i | 2.00675 | + | 1.15860i | 1.00000i | 2.55528 | + | 0.685945i | − | 1.00000i | −0.500000 | + | 0.866025i | −1.15860 | − | 2.00675i | |||
See all 28 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
133.o | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 798.2.be.a | ✓ | 28 |
7.d | odd | 6 | 1 | 798.2.be.b | yes | 28 | |
19.b | odd | 2 | 1 | 798.2.be.b | yes | 28 | |
133.o | even | 6 | 1 | inner | 798.2.be.a | ✓ | 28 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
798.2.be.a | ✓ | 28 | 1.a | even | 1 | 1 | trivial |
798.2.be.a | ✓ | 28 | 133.o | even | 6 | 1 | inner |
798.2.be.b | yes | 28 | 7.d | odd | 6 | 1 | |
798.2.be.b | yes | 28 | 19.b | odd | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{13}^{14} - 4 T_{13}^{13} - 96 T_{13}^{12} + 352 T_{13}^{11} + 3228 T_{13}^{10} - 10304 T_{13}^{9} + \cdots + 1075456 \) acting on \(S_{2}^{\mathrm{new}}(798, [\chi])\).