Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [368,3,Mod(17,368)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(368, base_ring=CyclotomicField(22))
chi = DirichletCharacter(H, H._module([0, 0, 7]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("368.17");
S:= CuspForms(chi, 3);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 368 = 2^{4} \cdot 23 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 368.p (of order \(22\), degree \(10\), not 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: | \(10.0272737285\) |
Analytic rank: | \(0\) |
Dimension: | \(30\) |
Relative dimension: | \(3\) over \(\Q(\zeta_{22})\) |
Twist minimal: | no (minimal twist has level 23) |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
17.1 | 0 | −0.844537 | − | 0.247978i | 0 | −3.24760 | − | 5.05336i | 0 | 3.20136 | + | 0.460286i | 0 | −6.91953 | − | 4.44691i | 0 | ||||||||||
17.2 | 0 | −0.201951 | − | 0.0592983i | 0 | 2.90325 | + | 4.51755i | 0 | −7.13192 | − | 1.02541i | 0 | −7.53401 | − | 4.84182i | 0 | ||||||||||
17.3 | 0 | 2.61464 | + | 0.767728i | 0 | 3.01510 | + | 4.69158i | 0 | 2.67922 | + | 0.385214i | 0 | −1.32434 | − | 0.851101i | 0 | ||||||||||
33.1 | 0 | −2.64748 | − | 1.70143i | 0 | −2.08252 | − | 0.951056i | 0 | 2.90589 | − | 9.89656i | 0 | 0.375553 | + | 0.822346i | 0 | ||||||||||
33.2 | 0 | 1.87410 | + | 1.20441i | 0 | −2.69128 | − | 1.22907i | 0 | 1.33225 | − | 4.53722i | 0 | −1.67709 | − | 3.67232i | 0 | ||||||||||
33.3 | 0 | 3.80315 | + | 2.44414i | 0 | 6.26010 | + | 2.85889i | 0 | −2.54176 | + | 8.65643i | 0 | 4.75142 | + | 10.4042i | 0 | ||||||||||
65.1 | 0 | −0.844537 | + | 0.247978i | 0 | −3.24760 | + | 5.05336i | 0 | 3.20136 | − | 0.460286i | 0 | −6.91953 | + | 4.44691i | 0 | ||||||||||
65.2 | 0 | −0.201951 | + | 0.0592983i | 0 | 2.90325 | − | 4.51755i | 0 | −7.13192 | + | 1.02541i | 0 | −7.53401 | + | 4.84182i | 0 | ||||||||||
65.3 | 0 | 2.61464 | − | 0.767728i | 0 | 3.01510 | − | 4.69158i | 0 | 2.67922 | − | 0.385214i | 0 | −1.32434 | + | 0.851101i | 0 | ||||||||||
97.1 | 0 | −2.35762 | + | 2.72084i | 0 | 5.05070 | + | 0.726181i | 0 | 8.85488 | − | 4.04389i | 0 | −0.563759 | − | 3.92103i | 0 | ||||||||||
97.2 | 0 | 0.590042 | − | 0.680945i | 0 | −1.77862 | − | 0.255727i | 0 | −2.68734 | + | 1.22727i | 0 | 1.16530 | + | 8.10482i | 0 | ||||||||||
97.3 | 0 | 3.16934 | − | 3.65762i | 0 | −5.40865 | − | 0.777647i | 0 | 0.889564 | − | 0.406250i | 0 | −2.05259 | − | 14.2761i | 0 | ||||||||||
113.1 | 0 | −0.749667 | − | 5.21405i | 0 | 1.45779 | − | 4.96477i | 0 | 1.27914 | + | 1.10838i | 0 | −17.9889 | + | 5.28201i | 0 | ||||||||||
113.2 | 0 | 0.238691 | + | 1.66013i | 0 | −2.65558 | + | 9.04406i | 0 | 5.31379 | + | 4.60443i | 0 | 5.93637 | − | 1.74307i | 0 | ||||||||||
113.3 | 0 | 0.365090 | + | 2.53926i | 0 | 0.682875 | − | 2.32566i | 0 | −6.72814 | − | 5.82996i | 0 | 2.32089 | − | 0.681474i | 0 | ||||||||||
129.1 | 0 | −2.35762 | − | 2.72084i | 0 | 5.05070 | − | 0.726181i | 0 | 8.85488 | + | 4.04389i | 0 | −0.563759 | + | 3.92103i | 0 | ||||||||||
129.2 | 0 | 0.590042 | + | 0.680945i | 0 | −1.77862 | + | 0.255727i | 0 | −2.68734 | − | 1.22727i | 0 | 1.16530 | − | 8.10482i | 0 | ||||||||||
129.3 | 0 | 3.16934 | + | 3.65762i | 0 | −5.40865 | + | 0.777647i | 0 | 0.889564 | + | 0.406250i | 0 | −2.05259 | + | 14.2761i | 0 | ||||||||||
145.1 | 0 | −2.64748 | + | 1.70143i | 0 | −2.08252 | + | 0.951056i | 0 | 2.90589 | + | 9.89656i | 0 | 0.375553 | − | 0.822346i | 0 | ||||||||||
145.2 | 0 | 1.87410 | − | 1.20441i | 0 | −2.69128 | + | 1.22907i | 0 | 1.33225 | + | 4.53722i | 0 | −1.67709 | + | 3.67232i | 0 | ||||||||||
See all 30 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
23.d | odd | 22 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 368.3.p.a | 30 | |
4.b | odd | 2 | 1 | 23.3.d.a | ✓ | 30 | |
12.b | even | 2 | 1 | 207.3.j.a | 30 | ||
23.d | odd | 22 | 1 | inner | 368.3.p.a | 30 | |
92.g | odd | 22 | 1 | 529.3.b.b | 30 | ||
92.h | even | 22 | 1 | 23.3.d.a | ✓ | 30 | |
92.h | even | 22 | 1 | 529.3.b.b | 30 | ||
276.j | odd | 22 | 1 | 207.3.j.a | 30 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
23.3.d.a | ✓ | 30 | 4.b | odd | 2 | 1 | |
23.3.d.a | ✓ | 30 | 92.h | even | 22 | 1 | |
207.3.j.a | 30 | 12.b | even | 2 | 1 | ||
207.3.j.a | 30 | 276.j | odd | 22 | 1 | ||
368.3.p.a | 30 | 1.a | even | 1 | 1 | trivial | |
368.3.p.a | 30 | 23.d | odd | 22 | 1 | inner | |
529.3.b.b | 30 | 92.g | odd | 22 | 1 | ||
529.3.b.b | 30 | 92.h | even | 22 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{30} - 11 T_{3}^{29} + 93 T_{3}^{28} - 600 T_{3}^{27} + 3512 T_{3}^{26} - 16076 T_{3}^{25} + 69780 T_{3}^{24} - 288419 T_{3}^{23} + 1153932 T_{3}^{22} - 3865349 T_{3}^{21} + 13169090 T_{3}^{20} + \cdots + 324612289 \)
acting on \(S_{3}^{\mathrm{new}}(368, [\chi])\).