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: | \(30\) |
Relative dimension: | \(5\) 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 | −3.05422 | − | 17.3214i | 0 | −19.9833 | − | 16.7680i | −4.00000 | + | 6.92820i | 0 | 17.5886 | + | 30.4643i | ||||||
19.2 | −1.87939 | + | 0.684040i | 0 | 3.06418 | − | 2.57115i | −2.39299 | − | 13.5713i | 0 | 20.6081 | + | 17.2922i | −4.00000 | + | 6.92820i | 0 | 13.7807 | + | 23.8688i | ||||||
19.3 | −1.87939 | + | 0.684040i | 0 | 3.06418 | − | 2.57115i | −0.127307 | − | 0.721992i | 0 | 1.18538 | + | 0.994655i | −4.00000 | + | 6.92820i | 0 | 0.733130 | + | 1.26982i | ||||||
19.4 | −1.87939 | + | 0.684040i | 0 | 3.06418 | − | 2.57115i | 2.36712 | + | 13.4246i | 0 | −10.7136 | − | 8.98975i | −4.00000 | + | 6.92820i | 0 | −13.6317 | − | 23.6108i | ||||||
19.5 | −1.87939 | + | 0.684040i | 0 | 3.06418 | − | 2.57115i | 3.01146 | + | 17.0788i | 0 | 16.7015 | + | 14.0142i | −4.00000 | + | 6.92820i | 0 | −17.3423 | − | 30.0378i | ||||||
37.1 | 1.53209 | − | 1.28558i | 0 | 0.694593 | − | 3.93923i | −13.2521 | + | 4.82337i | 0 | 3.00195 | + | 17.0249i | −4.00000 | − | 6.92820i | 0 | −14.1026 | + | 24.4264i | ||||||
37.2 | 1.53209 | − | 1.28558i | 0 | 0.694593 | − | 3.93923i | −11.4993 | + | 4.18541i | 0 | 3.15689 | + | 17.9036i | −4.00000 | − | 6.92820i | 0 | −12.2373 | + | 21.1956i | ||||||
37.3 | 1.53209 | − | 1.28558i | 0 | 0.694593 | − | 3.93923i | −3.40278 | + | 1.23851i | 0 | −3.51598 | − | 19.9401i | −4.00000 | − | 6.92820i | 0 | −3.62116 | + | 6.27204i | ||||||
37.4 | 1.53209 | − | 1.28558i | 0 | 0.694593 | − | 3.93923i | 7.46345 | − | 2.71647i | 0 | −2.66046 | − | 15.0882i | −4.00000 | − | 6.92820i | 0 | 7.94244 | − | 13.7567i | ||||||
37.5 | 1.53209 | − | 1.28558i | 0 | 0.694593 | − | 3.93923i | 17.2977 | − | 6.29584i | 0 | 6.03855 | + | 34.2463i | −4.00000 | − | 6.92820i | 0 | 18.4078 | − | 31.8832i | ||||||
73.1 | 0.347296 | − | 1.96962i | 0 | −3.75877 | − | 1.36808i | −7.73816 | + | 6.49309i | 0 | 7.54967 | − | 2.74785i | −4.00000 | + | 6.92820i | 0 | 10.1015 | + | 17.4962i | ||||||
73.2 | 0.347296 | − | 1.96962i | 0 | −3.75877 | − | 1.36808i | −6.77242 | + | 5.68273i | 0 | 30.7336 | − | 11.1861i | −4.00000 | + | 6.92820i | 0 | 8.84076 | + | 15.3127i | ||||||
73.3 | 0.347296 | − | 1.96962i | 0 | −3.75877 | − | 1.36808i | −0.657259 | + | 0.551505i | 0 | −20.5288 | + | 7.47187i | −4.00000 | + | 6.92820i | 0 | 0.857990 | + | 1.48608i | ||||||
73.4 | 0.347296 | − | 1.96962i | 0 | −3.75877 | − | 1.36808i | 9.84993 | − | 8.26508i | 0 | −26.8516 | + | 9.77320i | −4.00000 | + | 6.92820i | 0 | −12.8582 | − | 22.2710i | ||||||
73.5 | 0.347296 | − | 1.96962i | 0 | −3.75877 | − | 1.36808i | 14.9069 | − | 12.5084i | 0 | 11.7781 | − | 4.28689i | −4.00000 | + | 6.92820i | 0 | −19.4596 | − | 33.7050i | ||||||
91.1 | 0.347296 | + | 1.96962i | 0 | −3.75877 | + | 1.36808i | −7.73816 | − | 6.49309i | 0 | 7.54967 | + | 2.74785i | −4.00000 | − | 6.92820i | 0 | 10.1015 | − | 17.4962i | ||||||
91.2 | 0.347296 | + | 1.96962i | 0 | −3.75877 | + | 1.36808i | −6.77242 | − | 5.68273i | 0 | 30.7336 | + | 11.1861i | −4.00000 | − | 6.92820i | 0 | 8.84076 | − | 15.3127i | ||||||
91.3 | 0.347296 | + | 1.96962i | 0 | −3.75877 | + | 1.36808i | −0.657259 | − | 0.551505i | 0 | −20.5288 | − | 7.47187i | −4.00000 | − | 6.92820i | 0 | 0.857990 | − | 1.48608i | ||||||
91.4 | 0.347296 | + | 1.96962i | 0 | −3.75877 | + | 1.36808i | 9.84993 | + | 8.26508i | 0 | −26.8516 | − | 9.77320i | −4.00000 | − | 6.92820i | 0 | −12.8582 | + | 22.2710i | ||||||
91.5 | 0.347296 | + | 1.96962i | 0 | −3.75877 | + | 1.36808i | 14.9069 | + | 12.5084i | 0 | 11.7781 | + | 4.28689i | −4.00000 | − | 6.92820i | 0 | −19.4596 | + | 33.7050i | ||||||
See all 30 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.b | 30 | |
3.b | odd | 2 | 1 | 54.4.e.b | ✓ | 30 | |
27.e | even | 9 | 1 | inner | 162.4.e.b | 30 | |
27.e | even | 9 | 1 | 1458.4.a.j | 15 | ||
27.f | odd | 18 | 1 | 54.4.e.b | ✓ | 30 | |
27.f | odd | 18 | 1 | 1458.4.a.i | 15 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
54.4.e.b | ✓ | 30 | 3.b | odd | 2 | 1 | |
54.4.e.b | ✓ | 30 | 27.f | odd | 18 | 1 | |
162.4.e.b | 30 | 1.a | even | 1 | 1 | trivial | |
162.4.e.b | 30 | 27.e | even | 9 | 1 | inner | |
1458.4.a.i | 15 | 27.f | odd | 18 | 1 | ||
1458.4.a.j | 15 | 27.e | even | 9 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{30} - 12 T_{5}^{29} + 288 T_{5}^{28} - 1077 T_{5}^{27} - 7965 T_{5}^{26} + 1743363 T_{5}^{25} + \cdots + 54\!\cdots\!96 \) acting on \(S_{4}^{\mathrm{new}}(162, [\chi])\).