Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [164,3,Mod(83,164)]
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, 0]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("164.83");
S:= CuspForms(chi, 3);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 164 = 2^{2} \cdot 41 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 164.c (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: | \(40\) |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
83.1 | −1.99946 | − | 0.0466052i | 2.58485i | 3.99566 | + | 0.186370i | 6.26907 | 0.120467 | − | 5.16829i | − | 11.4160i | −7.98046 | − | 0.558857i | 2.31858 | −12.5347 | − | 0.292171i | |||||||
83.2 | −1.99946 | + | 0.0466052i | − | 2.58485i | 3.99566 | − | 0.186370i | 6.26907 | 0.120467 | + | 5.16829i | 11.4160i | −7.98046 | + | 0.558857i | 2.31858 | −12.5347 | + | 0.292171i | |||||||
83.3 | −1.99123 | − | 0.187081i | − | 3.90867i | 3.93000 | + | 0.745044i | −3.21306 | −0.731240 | + | 7.78307i | − | 7.36065i | −7.68616 | − | 2.21879i | −6.27772 | 6.39794 | + | 0.601103i | ||||||
83.4 | −1.99123 | + | 0.187081i | 3.90867i | 3.93000 | − | 0.745044i | −3.21306 | −0.731240 | − | 7.78307i | 7.36065i | −7.68616 | + | 2.21879i | −6.27772 | 6.39794 | − | 0.601103i | ||||||||
83.5 | −1.74626 | − | 0.974974i | 0.487501i | 2.09885 | + | 3.40512i | −6.86061 | 0.475301 | − | 0.851304i | 1.80145i | −0.345239 | − | 7.99255i | 8.76234 | 11.9804 | + | 6.68892i | ||||||||
83.6 | −1.74626 | + | 0.974974i | − | 0.487501i | 2.09885 | − | 3.40512i | −6.86061 | 0.475301 | + | 0.851304i | − | 1.80145i | −0.345239 | + | 7.99255i | 8.76234 | 11.9804 | − | 6.68892i | ||||||
83.7 | −1.54213 | − | 1.27351i | 2.47677i | 0.756327 | + | 3.92785i | −1.28625 | 3.15420 | − | 3.81950i | − | 3.58567i | 3.83581 | − | 7.02044i | 2.86563 | 1.98356 | + | 1.63805i | |||||||
83.8 | −1.54213 | + | 1.27351i | − | 2.47677i | 0.756327 | − | 3.92785i | −1.28625 | 3.15420 | + | 3.81950i | 3.58567i | 3.83581 | + | 7.02044i | 2.86563 | 1.98356 | − | 1.63805i | |||||||
83.9 | −1.50188 | − | 1.32074i | − | 2.23825i | 0.511316 | + | 3.96718i | 4.89966 | −2.95614 | + | 3.36160i | − | 0.843158i | 4.47166 | − | 6.63357i | 3.99022 | −7.35873 | − | 6.47116i | ||||||
83.10 | −1.50188 | + | 1.32074i | 2.23825i | 0.511316 | − | 3.96718i | 4.89966 | −2.95614 | − | 3.36160i | 0.843158i | 4.47166 | + | 6.63357i | 3.99022 | −7.35873 | + | 6.47116i | ||||||||
83.11 | −1.40870 | − | 1.41971i | 5.22032i | −0.0311413 | + | 3.99988i | 8.34602 | 7.41133 | − | 7.35385i | 10.3463i | 5.72253 | − | 5.59041i | −18.2517 | −11.7570 | − | 11.8489i | ||||||||
83.12 | −1.40870 | + | 1.41971i | − | 5.22032i | −0.0311413 | − | 3.99988i | 8.34602 | 7.41133 | + | 7.35385i | − | 10.3463i | 5.72253 | + | 5.59041i | −18.2517 | −11.7570 | + | 11.8489i | ||||||
83.13 | −1.29148 | − | 1.52711i | − | 5.51459i | −0.664159 | + | 3.94448i | −3.66998 | −8.42141 | + | 7.12198i | 8.18794i | 6.88142 | − | 4.07997i | −21.4107 | 4.73971 | + | 5.60448i | |||||||
83.14 | −1.29148 | + | 1.52711i | 5.51459i | −0.664159 | − | 3.94448i | −3.66998 | −8.42141 | − | 7.12198i | − | 8.18794i | 6.88142 | + | 4.07997i | −21.4107 | 4.73971 | − | 5.60448i | |||||||
83.15 | −0.688190 | − | 1.87787i | − | 1.75883i | −3.05279 | + | 2.58466i | 5.36632 | −3.30284 | + | 1.21041i | − | 7.30850i | 6.95456 | + | 3.95400i | 5.90653 | −3.69305 | − | 10.0773i | ||||||
83.16 | −0.688190 | + | 1.87787i | 1.75883i | −3.05279 | − | 2.58466i | 5.36632 | −3.30284 | − | 1.21041i | 7.30850i | 6.95456 | − | 3.95400i | 5.90653 | −3.69305 | + | 10.0773i | ||||||||
83.17 | −0.687445 | − | 1.87814i | 4.43708i | −3.05484 | + | 2.58224i | −5.62408 | 8.33347 | − | 3.05025i | − | 6.78235i | 6.94985 | + | 3.96227i | −10.6877 | 3.86624 | + | 10.5628i | |||||||
83.18 | −0.687445 | + | 1.87814i | − | 4.43708i | −3.05484 | − | 2.58224i | −5.62408 | 8.33347 | + | 3.05025i | 6.78235i | 6.94985 | − | 3.96227i | −10.6877 | 3.86624 | − | 10.5628i | |||||||
83.19 | −0.387677 | − | 1.96207i | − | 0.338353i | −3.69941 | + | 1.52129i | −3.21167 | −0.663871 | + | 0.131171i | 12.6392i | 4.41906 | + | 6.66873i | 8.88552 | 1.24509 | + | 6.30151i | |||||||
83.20 | −0.387677 | + | 1.96207i | 0.338353i | −3.69941 | − | 1.52129i | −3.21167 | −0.663871 | − | 0.131171i | − | 12.6392i | 4.41906 | − | 6.66873i | 8.88552 | 1.24509 | − | 6.30151i | |||||||
See all 40 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
4.b | 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.c.a | ✓ | 40 |
4.b | odd | 2 | 1 | inner | 164.3.c.a | ✓ | 40 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
164.3.c.a | ✓ | 40 | 1.a | even | 1 | 1 | trivial |
164.3.c.a | ✓ | 40 | 4.b | odd | 2 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(164, [\chi])\).