Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [414,3,Mod(35,414)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(414, base_ring=CyclotomicField(22))
chi = DirichletCharacter(H, H._module([11, 20]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("414.35");
S:= CuspForms(chi, 3);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 414 = 2 \cdot 3^{2} \cdot 23 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 414.k (of order \(22\), degree \(10\), 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: | \(11.2806829445\) |
Analytic rank: | \(0\) |
Dimension: | \(80\) |
Relative dimension: | \(8\) over \(\Q(\zeta_{22})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{22}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
35.1 | −0.764582 | − | 1.18971i | 0 | −0.830830 | + | 1.81926i | −1.90263 | − | 6.47977i | 0 | −7.74127 | − | 8.93390i | 2.79964 | − | 0.402527i | 0 | −6.25434 | + | 7.21790i | ||||||
35.2 | −0.764582 | − | 1.18971i | 0 | −0.830830 | + | 1.81926i | −1.27049 | − | 4.32689i | 0 | 6.32390 | + | 7.29817i | 2.79964 | − | 0.402527i | 0 | −4.17636 | + | 4.81978i | ||||||
35.3 | −0.764582 | − | 1.18971i | 0 | −0.830830 | + | 1.81926i | 0.466986 | + | 1.59041i | 0 | −2.89002 | − | 3.33526i | 2.79964 | − | 0.402527i | 0 | 1.53508 | − | 1.77157i | ||||||
35.4 | −0.764582 | − | 1.18971i | 0 | −0.830830 | + | 1.81926i | 2.59273 | + | 8.83003i | 0 | −0.0998440 | − | 0.115226i | 2.79964 | − | 0.402527i | 0 | 8.52284 | − | 9.83588i | ||||||
35.5 | 0.764582 | + | 1.18971i | 0 | −0.830830 | + | 1.81926i | −2.59273 | − | 8.83003i | 0 | −0.0998440 | − | 0.115226i | −2.79964 | + | 0.402527i | 0 | 8.52284 | − | 9.83588i | ||||||
35.6 | 0.764582 | + | 1.18971i | 0 | −0.830830 | + | 1.81926i | −0.466986 | − | 1.59041i | 0 | −2.89002 | − | 3.33526i | −2.79964 | + | 0.402527i | 0 | 1.53508 | − | 1.77157i | ||||||
35.7 | 0.764582 | + | 1.18971i | 0 | −0.830830 | + | 1.81926i | 1.27049 | + | 4.32689i | 0 | 6.32390 | + | 7.29817i | −2.79964 | + | 0.402527i | 0 | −4.17636 | + | 4.81978i | ||||||
35.8 | 0.764582 | + | 1.18971i | 0 | −0.830830 | + | 1.81926i | 1.90263 | + | 6.47977i | 0 | −7.74127 | − | 8.93390i | −2.79964 | + | 0.402527i | 0 | −6.25434 | + | 7.21790i | ||||||
71.1 | −0.764582 | + | 1.18971i | 0 | −0.830830 | − | 1.81926i | −1.90263 | + | 6.47977i | 0 | −7.74127 | + | 8.93390i | 2.79964 | + | 0.402527i | 0 | −6.25434 | − | 7.21790i | ||||||
71.2 | −0.764582 | + | 1.18971i | 0 | −0.830830 | − | 1.81926i | −1.27049 | + | 4.32689i | 0 | 6.32390 | − | 7.29817i | 2.79964 | + | 0.402527i | 0 | −4.17636 | − | 4.81978i | ||||||
71.3 | −0.764582 | + | 1.18971i | 0 | −0.830830 | − | 1.81926i | 0.466986 | − | 1.59041i | 0 | −2.89002 | + | 3.33526i | 2.79964 | + | 0.402527i | 0 | 1.53508 | + | 1.77157i | ||||||
71.4 | −0.764582 | + | 1.18971i | 0 | −0.830830 | − | 1.81926i | 2.59273 | − | 8.83003i | 0 | −0.0998440 | + | 0.115226i | 2.79964 | + | 0.402527i | 0 | 8.52284 | + | 9.83588i | ||||||
71.5 | 0.764582 | − | 1.18971i | 0 | −0.830830 | − | 1.81926i | −2.59273 | + | 8.83003i | 0 | −0.0998440 | + | 0.115226i | −2.79964 | − | 0.402527i | 0 | 8.52284 | + | 9.83588i | ||||||
71.6 | 0.764582 | − | 1.18971i | 0 | −0.830830 | − | 1.81926i | −0.466986 | + | 1.59041i | 0 | −2.89002 | + | 3.33526i | −2.79964 | − | 0.402527i | 0 | 1.53508 | + | 1.77157i | ||||||
71.7 | 0.764582 | − | 1.18971i | 0 | −0.830830 | − | 1.81926i | 1.27049 | − | 4.32689i | 0 | 6.32390 | − | 7.29817i | −2.79964 | − | 0.402527i | 0 | −4.17636 | − | 4.81978i | ||||||
71.8 | 0.764582 | − | 1.18971i | 0 | −0.830830 | − | 1.81926i | 1.90263 | − | 6.47977i | 0 | −7.74127 | + | 8.93390i | −2.79964 | − | 0.402527i | 0 | −6.25434 | − | 7.21790i | ||||||
179.1 | −0.398430 | + | 1.35693i | 0 | −1.68251 | − | 1.08128i | −5.13916 | − | 0.738899i | 0 | −5.58919 | − | 12.2386i | 2.13758 | − | 1.85223i | 0 | 3.05023 | − | 6.67907i | ||||||
179.2 | −0.398430 | + | 1.35693i | 0 | −1.68251 | − | 1.08128i | −4.66712 | − | 0.671030i | 0 | 3.18481 | + | 6.97376i | 2.13758 | − | 1.85223i | 0 | 2.77006 | − | 6.06558i | ||||||
179.3 | −0.398430 | + | 1.35693i | 0 | −1.68251 | − | 1.08128i | 1.62786 | + | 0.234051i | 0 | 2.08299 | + | 4.56111i | 2.13758 | − | 1.85223i | 0 | −0.966179 | + | 2.11564i | ||||||
179.4 | −0.398430 | + | 1.35693i | 0 | −1.68251 | − | 1.08128i | 6.34504 | + | 0.912279i | 0 | −2.86731 | − | 6.27853i | 2.13758 | − | 1.85223i | 0 | −3.76595 | + | 8.24628i | ||||||
See all 80 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
23.c | even | 11 | 1 | inner |
69.h | odd | 22 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 414.3.k.a | ✓ | 80 |
3.b | odd | 2 | 1 | inner | 414.3.k.a | ✓ | 80 |
23.c | even | 11 | 1 | inner | 414.3.k.a | ✓ | 80 |
69.h | odd | 22 | 1 | inner | 414.3.k.a | ✓ | 80 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
414.3.k.a | ✓ | 80 | 1.a | even | 1 | 1 | trivial |
414.3.k.a | ✓ | 80 | 3.b | odd | 2 | 1 | inner |
414.3.k.a | ✓ | 80 | 23.c | even | 11 | 1 | inner |
414.3.k.a | ✓ | 80 | 69.h | odd | 22 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{80} + 76 T_{5}^{78} + 7354 T_{5}^{76} - 308566 T_{5}^{74} - 6443281 T_{5}^{72} + \cdots + 26\!\cdots\!01 \) acting on \(S_{3}^{\mathrm{new}}(414, [\chi])\).