Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [162,4,Mod(19,162)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(162, base_ring=CyclotomicField(18))
chi = DirichletCharacter(H, H._module([16]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("162.19");
S:= CuspForms(chi, 4);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 162 = 2 \cdot 3^{4} \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 162.e (of order \(9\), degree \(6\), 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: | \(9.55830942093\) |
Analytic rank: | \(0\) |
Dimension: | \(24\) |
Relative dimension: | \(4\) over \(\Q(\zeta_{9})\) |
Twist minimal: | no (minimal twist has level 54) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{9}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
19.1 | 1.87939 | − | 0.684040i | 0 | 3.06418 | − | 2.57115i | −1.31542 | − | 7.46011i | 0 | −2.90083 | − | 2.43409i | 4.00000 | − | 6.92820i | 0 | −7.57519 | − | 13.1206i | ||||||
19.2 | 1.87939 | − | 0.684040i | 0 | 3.06418 | − | 2.57115i | −1.18374 | − | 6.71331i | 0 | −25.7220 | − | 21.5833i | 4.00000 | − | 6.92820i | 0 | −6.81687 | − | 11.8072i | ||||||
19.3 | 1.87939 | − | 0.684040i | 0 | 3.06418 | − | 2.57115i | −0.203469 | − | 1.15393i | 0 | 19.9691 | + | 16.7561i | 4.00000 | − | 6.92820i | 0 | −1.17173 | − | 2.02949i | ||||||
19.4 | 1.87939 | − | 0.684040i | 0 | 3.06418 | − | 2.57115i | 2.50669 | + | 14.2161i | 0 | 0.855575 | + | 0.717913i | 4.00000 | − | 6.92820i | 0 | 14.4354 | + | 25.0029i | ||||||
37.1 | −1.53209 | + | 1.28558i | 0 | 0.694593 | − | 3.93923i | −18.7075 | + | 6.80898i | 0 | −3.26433 | − | 18.5129i | 4.00000 | + | 6.92820i | 0 | 19.9081 | − | 34.4819i | ||||||
37.2 | −1.53209 | + | 1.28558i | 0 | 0.694593 | − | 3.93923i | −3.45007 | + | 1.25572i | 0 | 0.157049 | + | 0.890672i | 4.00000 | + | 6.92820i | 0 | 3.67149 | − | 6.35921i | ||||||
37.3 | −1.53209 | + | 1.28558i | 0 | 0.694593 | − | 3.93923i | −1.07676 | + | 0.391909i | 0 | 0.470326 | + | 2.66735i | 4.00000 | + | 6.92820i | 0 | 1.14586 | − | 1.98470i | ||||||
37.4 | −1.53209 | + | 1.28558i | 0 | 0.694593 | − | 3.93923i | 19.8413 | − | 7.22164i | 0 | −3.38399 | − | 19.1916i | 4.00000 | + | 6.92820i | 0 | −21.1147 | + | 36.5717i | ||||||
73.1 | −0.347296 | + | 1.96962i | 0 | −3.75877 | − | 1.36808i | −5.41266 | + | 4.54176i | 0 | −4.71753 | + | 1.71704i | 4.00000 | − | 6.92820i | 0 | −7.06572 | − | 12.2382i | ||||||
73.2 | −0.347296 | + | 1.96962i | 0 | −3.75877 | − | 1.36808i | −3.46830 | + | 2.91025i | 0 | 22.9979 | − | 8.37056i | 4.00000 | − | 6.92820i | 0 | −4.52754 | − | 7.84193i | ||||||
73.3 | −0.347296 | + | 1.96962i | 0 | −3.75877 | − | 1.36808i | 5.66862 | − | 4.75653i | 0 | −11.1935 | + | 4.07412i | 4.00000 | − | 6.92820i | 0 | 7.39985 | + | 12.8169i | ||||||
73.4 | −0.347296 | + | 1.96962i | 0 | −3.75877 | − | 1.36808i | 12.8013 | − | 10.7416i | 0 | −9.76778 | + | 3.55518i | 4.00000 | − | 6.92820i | 0 | 16.7110 | + | 28.9442i | ||||||
91.1 | −0.347296 | − | 1.96962i | 0 | −3.75877 | + | 1.36808i | −5.41266 | − | 4.54176i | 0 | −4.71753 | − | 1.71704i | 4.00000 | + | 6.92820i | 0 | −7.06572 | + | 12.2382i | ||||||
91.2 | −0.347296 | − | 1.96962i | 0 | −3.75877 | + | 1.36808i | −3.46830 | − | 2.91025i | 0 | 22.9979 | + | 8.37056i | 4.00000 | + | 6.92820i | 0 | −4.52754 | + | 7.84193i | ||||||
91.3 | −0.347296 | − | 1.96962i | 0 | −3.75877 | + | 1.36808i | 5.66862 | + | 4.75653i | 0 | −11.1935 | − | 4.07412i | 4.00000 | + | 6.92820i | 0 | 7.39985 | − | 12.8169i | ||||||
91.4 | −0.347296 | − | 1.96962i | 0 | −3.75877 | + | 1.36808i | 12.8013 | + | 10.7416i | 0 | −9.76778 | − | 3.55518i | 4.00000 | + | 6.92820i | 0 | 16.7110 | − | 28.9442i | ||||||
127.1 | −1.53209 | − | 1.28558i | 0 | 0.694593 | + | 3.93923i | −18.7075 | − | 6.80898i | 0 | −3.26433 | + | 18.5129i | 4.00000 | − | 6.92820i | 0 | 19.9081 | + | 34.4819i | ||||||
127.2 | −1.53209 | − | 1.28558i | 0 | 0.694593 | + | 3.93923i | −3.45007 | − | 1.25572i | 0 | 0.157049 | − | 0.890672i | 4.00000 | − | 6.92820i | 0 | 3.67149 | + | 6.35921i | ||||||
127.3 | −1.53209 | − | 1.28558i | 0 | 0.694593 | + | 3.93923i | −1.07676 | − | 0.391909i | 0 | 0.470326 | − | 2.66735i | 4.00000 | − | 6.92820i | 0 | 1.14586 | + | 1.98470i | ||||||
127.4 | −1.53209 | − | 1.28558i | 0 | 0.694593 | + | 3.93923i | 19.8413 | + | 7.22164i | 0 | −3.38399 | + | 19.1916i | 4.00000 | − | 6.92820i | 0 | −21.1147 | − | 36.5717i | ||||||
See all 24 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
27.e | even | 9 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 162.4.e.a | 24 | |
3.b | odd | 2 | 1 | 54.4.e.a | ✓ | 24 | |
27.e | even | 9 | 1 | inner | 162.4.e.a | 24 | |
27.e | even | 9 | 1 | 1458.4.a.e | 12 | ||
27.f | odd | 18 | 1 | 54.4.e.a | ✓ | 24 | |
27.f | odd | 18 | 1 | 1458.4.a.h | 12 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
54.4.e.a | ✓ | 24 | 3.b | odd | 2 | 1 | |
54.4.e.a | ✓ | 24 | 27.f | odd | 18 | 1 | |
162.4.e.a | 24 | 1.a | even | 1 | 1 | trivial | |
162.4.e.a | 24 | 27.e | even | 9 | 1 | inner | |
1458.4.a.e | 12 | 27.e | even | 9 | 1 | ||
1458.4.a.h | 12 | 27.f | odd | 18 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{24} - 12 T_{5}^{23} - 360 T_{5}^{22} + 6699 T_{5}^{21} + 31221 T_{5}^{20} - 1138725 T_{5}^{19} + 32801526 T_{5}^{18} - 187511760 T_{5}^{17} + 1422469026 T_{5}^{16} + 40147364502 T_{5}^{15} + \cdots + 37\!\cdots\!01 \)
acting on \(S_{4}^{\mathrm{new}}(162, [\chi])\).