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 | −2.17725 | − | 7.41503i | 0 | 6.85488 | + | 7.91095i | 2.79964 | − | 0.402527i | 0 | −7.15707 | + | 8.25969i | ||||||
35.2 | −0.764582 | − | 1.18971i | 0 | −0.830830 | + | 1.81926i | −0.172109 | − | 0.586151i | 0 | −4.94341 | − | 5.70500i | 2.79964 | − | 0.402527i | 0 | −0.565759 | + | 0.652921i | ||||||
35.3 | −0.764582 | − | 1.18971i | 0 | −0.830830 | + | 1.81926i | 0.404235 | + | 1.37670i | 0 | −1.64257 | − | 1.89563i | 2.79964 | − | 0.402527i | 0 | 1.32880 | − | 1.53352i | ||||||
35.4 | −0.764582 | − | 1.18971i | 0 | −0.830830 | + | 1.81926i | 2.05853 | + | 7.01070i | 0 | 4.13834 | + | 4.77590i | 2.79964 | − | 0.402527i | 0 | 6.76681 | − | 7.80931i | ||||||
35.5 | 0.764582 | + | 1.18971i | 0 | −0.830830 | + | 1.81926i | −2.05853 | − | 7.01070i | 0 | 4.13834 | + | 4.77590i | −2.79964 | + | 0.402527i | 0 | 6.76681 | − | 7.80931i | ||||||
35.6 | 0.764582 | + | 1.18971i | 0 | −0.830830 | + | 1.81926i | −0.404235 | − | 1.37670i | 0 | −1.64257 | − | 1.89563i | −2.79964 | + | 0.402527i | 0 | 1.32880 | − | 1.53352i | ||||||
35.7 | 0.764582 | + | 1.18971i | 0 | −0.830830 | + | 1.81926i | 0.172109 | + | 0.586151i | 0 | −4.94341 | − | 5.70500i | −2.79964 | + | 0.402527i | 0 | −0.565759 | + | 0.652921i | ||||||
35.8 | 0.764582 | + | 1.18971i | 0 | −0.830830 | + | 1.81926i | 2.17725 | + | 7.41503i | 0 | 6.85488 | + | 7.91095i | −2.79964 | + | 0.402527i | 0 | −7.15707 | + | 8.25969i | ||||||
71.1 | −0.764582 | + | 1.18971i | 0 | −0.830830 | − | 1.81926i | −2.17725 | + | 7.41503i | 0 | 6.85488 | − | 7.91095i | 2.79964 | + | 0.402527i | 0 | −7.15707 | − | 8.25969i | ||||||
71.2 | −0.764582 | + | 1.18971i | 0 | −0.830830 | − | 1.81926i | −0.172109 | + | 0.586151i | 0 | −4.94341 | + | 5.70500i | 2.79964 | + | 0.402527i | 0 | −0.565759 | − | 0.652921i | ||||||
71.3 | −0.764582 | + | 1.18971i | 0 | −0.830830 | − | 1.81926i | 0.404235 | − | 1.37670i | 0 | −1.64257 | + | 1.89563i | 2.79964 | + | 0.402527i | 0 | 1.32880 | + | 1.53352i | ||||||
71.4 | −0.764582 | + | 1.18971i | 0 | −0.830830 | − | 1.81926i | 2.05853 | − | 7.01070i | 0 | 4.13834 | − | 4.77590i | 2.79964 | + | 0.402527i | 0 | 6.76681 | + | 7.80931i | ||||||
71.5 | 0.764582 | − | 1.18971i | 0 | −0.830830 | − | 1.81926i | −2.05853 | + | 7.01070i | 0 | 4.13834 | − | 4.77590i | −2.79964 | − | 0.402527i | 0 | 6.76681 | + | 7.80931i | ||||||
71.6 | 0.764582 | − | 1.18971i | 0 | −0.830830 | − | 1.81926i | −0.404235 | + | 1.37670i | 0 | −1.64257 | + | 1.89563i | −2.79964 | − | 0.402527i | 0 | 1.32880 | + | 1.53352i | ||||||
71.7 | 0.764582 | − | 1.18971i | 0 | −0.830830 | − | 1.81926i | 0.172109 | − | 0.586151i | 0 | −4.94341 | + | 5.70500i | −2.79964 | − | 0.402527i | 0 | −0.565759 | − | 0.652921i | ||||||
71.8 | 0.764582 | − | 1.18971i | 0 | −0.830830 | − | 1.81926i | 2.17725 | − | 7.41503i | 0 | 6.85488 | − | 7.91095i | −2.79964 | − | 0.402527i | 0 | −7.15707 | − | 8.25969i | ||||||
179.1 | −0.398430 | + | 1.35693i | 0 | −1.68251 | − | 1.08128i | −8.69286 | − | 1.24984i | 0 | −0.246783 | − | 0.540379i | 2.13758 | − | 1.85223i | 0 | 5.15945 | − | 11.2976i | ||||||
179.2 | −0.398430 | + | 1.35693i | 0 | −1.68251 | − | 1.08128i | 0.248013 | + | 0.0356589i | 0 | −2.43143 | − | 5.32409i | 2.13758 | − | 1.85223i | 0 | −0.147203 | + | 0.322329i | ||||||
179.3 | −0.398430 | + | 1.35693i | 0 | −1.68251 | − | 1.08128i | 0.751975 | + | 0.108118i | 0 | 0.750840 | + | 1.64411i | 2.13758 | − | 1.85223i | 0 | −0.446317 | + | 0.977298i | ||||||
179.4 | −0.398430 | + | 1.35693i | 0 | −1.68251 | − | 1.08128i | 9.52625 | + | 1.36967i | 0 | 5.11608 | + | 11.2026i | 2.13758 | − | 1.85223i | 0 | −5.65408 | + | 12.3807i | ||||||
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.b | ✓ | 80 |
3.b | odd | 2 | 1 | inner | 414.3.k.b | ✓ | 80 |
23.c | even | 11 | 1 | inner | 414.3.k.b | ✓ | 80 |
69.h | odd | 22 | 1 | inner | 414.3.k.b | ✓ | 80 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
414.3.k.b | ✓ | 80 | 1.a | even | 1 | 1 | trivial |
414.3.k.b | ✓ | 80 | 3.b | odd | 2 | 1 | inner |
414.3.k.b | ✓ | 80 | 23.c | even | 11 | 1 | inner |
414.3.k.b | ✓ | 80 | 69.h | odd | 22 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{80} - 228 T_{5}^{78} + 22970 T_{5}^{76} - 1637494 T_{5}^{74} + 104428799 T_{5}^{72} + \cdots + 15\!\cdots\!21 \) acting on \(S_{3}^{\mathrm{new}}(414, [\chi])\).