Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [164,3,Mod(163,164)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(164, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 1]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("164.163");
S:= CuspForms(chi, 3);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 164 = 2^{2} \cdot 41 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 164.d (of order \(2\), degree \(1\), 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: | \(4.46867633551\) |
Analytic rank: | \(0\) |
Dimension: | \(32\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
163.1 | −1.83775 | − | 0.789089i | −0.994532 | 2.75468 | + | 2.90030i | −1.69735 | 1.82770 | + | 0.784775i | 6.08835 | −2.77382 | − | 7.50373i | −8.01091 | 3.11931 | + | 1.33936i | ||||||||
163.2 | −1.83775 | − | 0.789089i | 0.994532 | 2.75468 | + | 2.90030i | −1.69735 | −1.82770 | − | 0.784775i | −6.08835 | −2.77382 | − | 7.50373i | −8.01091 | 3.11931 | + | 1.33936i | ||||||||
163.3 | −1.83775 | + | 0.789089i | −0.994532 | 2.75468 | − | 2.90030i | −1.69735 | 1.82770 | − | 0.784775i | 6.08835 | −2.77382 | + | 7.50373i | −8.01091 | 3.11931 | − | 1.33936i | ||||||||
163.4 | −1.83775 | + | 0.789089i | 0.994532 | 2.75468 | − | 2.90030i | −1.69735 | −1.82770 | + | 0.784775i | −6.08835 | −2.77382 | + | 7.50373i | −8.01091 | 3.11931 | − | 1.33936i | ||||||||
163.5 | −1.43715 | − | 1.39090i | −4.50897 | 0.130779 | + | 3.99786i | 3.66935 | 6.48005 | + | 6.27154i | 2.75944 | 5.37269 | − | 5.92741i | 11.3308 | −5.27339 | − | 5.10371i | ||||||||
163.6 | −1.43715 | − | 1.39090i | 4.50897 | 0.130779 | + | 3.99786i | 3.66935 | −6.48005 | − | 6.27154i | −2.75944 | 5.37269 | − | 5.92741i | 11.3308 | −5.27339 | − | 5.10371i | ||||||||
163.7 | −1.43715 | + | 1.39090i | −4.50897 | 0.130779 | − | 3.99786i | 3.66935 | 6.48005 | − | 6.27154i | 2.75944 | 5.37269 | + | 5.92741i | 11.3308 | −5.27339 | + | 5.10371i | ||||||||
163.8 | −1.43715 | + | 1.39090i | 4.50897 | 0.130779 | − | 3.99786i | 3.66935 | −6.48005 | + | 6.27154i | −2.75944 | 5.37269 | + | 5.92741i | 11.3308 | −5.27339 | + | 5.10371i | ||||||||
163.9 | −0.920996 | − | 1.77532i | −3.01133 | −2.30353 | + | 3.27013i | −7.77019 | 2.77343 | + | 5.34609i | 2.28597 | 7.92707 | + | 1.07774i | 0.0681311 | 7.15632 | + | 13.7946i | ||||||||
163.10 | −0.920996 | − | 1.77532i | 3.01133 | −2.30353 | + | 3.27013i | −7.77019 | −2.77343 | − | 5.34609i | −2.28597 | 7.92707 | + | 1.07774i | 0.0681311 | 7.15632 | + | 13.7946i | ||||||||
163.11 | −0.920996 | + | 1.77532i | −3.01133 | −2.30353 | − | 3.27013i | −7.77019 | 2.77343 | − | 5.34609i | 2.28597 | 7.92707 | − | 1.07774i | 0.0681311 | 7.15632 | − | 13.7946i | ||||||||
163.12 | −0.920996 | + | 1.77532i | 3.01133 | −2.30353 | − | 3.27013i | −7.77019 | −2.77343 | + | 5.34609i | −2.28597 | 7.92707 | − | 1.07774i | 0.0681311 | 7.15632 | − | 13.7946i | ||||||||
163.13 | −0.417115 | − | 1.95602i | −0.771966 | −3.65203 | + | 1.63177i | 4.30571 | 0.321999 | + | 1.50998i | −9.44612 | 4.71509 | + | 6.46281i | −8.40407 | −1.79598 | − | 8.42205i | ||||||||
163.14 | −0.417115 | − | 1.95602i | 0.771966 | −3.65203 | + | 1.63177i | 4.30571 | −0.321999 | − | 1.50998i | 9.44612 | 4.71509 | + | 6.46281i | −8.40407 | −1.79598 | − | 8.42205i | ||||||||
163.15 | −0.417115 | + | 1.95602i | −0.771966 | −3.65203 | − | 1.63177i | 4.30571 | 0.321999 | − | 1.50998i | −9.44612 | 4.71509 | − | 6.46281i | −8.40407 | −1.79598 | + | 8.42205i | ||||||||
163.16 | −0.417115 | + | 1.95602i | 0.771966 | −3.65203 | − | 1.63177i | 4.30571 | −0.321999 | + | 1.50998i | 9.44612 | 4.71509 | − | 6.46281i | −8.40407 | −1.79598 | + | 8.42205i | ||||||||
163.17 | 0.392436 | − | 1.96112i | −5.16771 | −3.69199 | − | 1.53923i | 0.943183 | −2.02800 | + | 10.1345i | −4.65695 | −4.46748 | + | 6.63638i | 17.7052 | 0.370139 | − | 1.84970i | ||||||||
163.18 | 0.392436 | − | 1.96112i | 5.16771 | −3.69199 | − | 1.53923i | 0.943183 | 2.02800 | − | 10.1345i | 4.65695 | −4.46748 | + | 6.63638i | 17.7052 | 0.370139 | − | 1.84970i | ||||||||
163.19 | 0.392436 | + | 1.96112i | −5.16771 | −3.69199 | + | 1.53923i | 0.943183 | −2.02800 | − | 10.1345i | −4.65695 | −4.46748 | − | 6.63638i | 17.7052 | 0.370139 | + | 1.84970i | ||||||||
163.20 | 0.392436 | + | 1.96112i | 5.16771 | −3.69199 | + | 1.53923i | 0.943183 | 2.02800 | + | 10.1345i | 4.65695 | −4.46748 | − | 6.63638i | 17.7052 | 0.370139 | + | 1.84970i | ||||||||
See all 32 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
4.b | odd | 2 | 1 | inner |
41.b | even | 2 | 1 | inner |
164.d | odd | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 164.3.d.c | ✓ | 32 |
4.b | odd | 2 | 1 | inner | 164.3.d.c | ✓ | 32 |
41.b | even | 2 | 1 | inner | 164.3.d.c | ✓ | 32 |
164.d | odd | 2 | 1 | inner | 164.3.d.c | ✓ | 32 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
164.3.d.c | ✓ | 32 | 1.a | even | 1 | 1 | trivial |
164.3.d.c | ✓ | 32 | 4.b | odd | 2 | 1 | inner |
164.3.d.c | ✓ | 32 | 41.b | even | 2 | 1 | inner |
164.3.d.c | ✓ | 32 | 164.d | odd | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{16} - 78 T_{3}^{14} + 2356 T_{3}^{12} - 35444 T_{3}^{10} + 285264 T_{3}^{8} - 1212680 T_{3}^{6} + \cdots + 619520 \) acting on \(S_{3}^{\mathrm{new}}(164, [\chi])\).