Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [189,3,Mod(170,189)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(189, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([1, 2]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("189.170");
S:= CuspForms(chi, 3);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 189 = 3^{3} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 189.n (of order \(6\), degree \(2\), 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: | \(5.14987699641\) |
Analytic rank: | \(0\) |
Dimension: | \(22\) |
Relative dimension: | \(11\) over \(\Q(\zeta_{6})\) |
Twist minimal: | no (minimal twist has level 63) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-expansion
The algebraic \(q\)-expansion of this newform has not been computed, 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
170.1 | −2.48702 | + | 1.43588i | 0 | 2.12350 | − | 3.67801i | − | 7.54889i | 0 | −6.82274 | + | 1.56534i | 0.709334i | 0 | 10.8393 | + | 18.7742i | |||||||||
170.2 | −2.37724 | + | 1.37250i | 0 | 1.76751 | − | 3.06142i | 2.68504i | 0 | 6.00002 | − | 3.60552i | − | 1.27635i | 0 | −3.68521 | − | 6.38297i | |||||||||
170.3 | −1.86624 | + | 1.07747i | 0 | 0.321900 | − | 0.557548i | 1.87862i | 0 | −3.79886 | − | 5.87951i | − | 7.23243i | 0 | −2.02416 | − | 3.50595i | |||||||||
170.4 | −1.11318 | + | 0.642694i | 0 | −1.17389 | + | 2.03324i | 7.87519i | 0 | 0.417718 | + | 6.98753i | − | 8.15936i | 0 | −5.06133 | − | 8.76649i | |||||||||
170.5 | −0.0664669 | + | 0.0383747i | 0 | −1.99705 | + | 3.45900i | − | 4.07697i | 0 | −6.97461 | + | 0.595686i | − | 0.613543i | 0 | 0.156452 | + | 0.270983i | ||||||||
170.6 | 0.444866 | − | 0.256844i | 0 | −1.86806 | + | 3.23558i | − | 7.02462i | 0 | 5.34652 | − | 4.51827i | 3.97395i | 0 | −1.80423 | − | 3.12502i | |||||||||
170.7 | 1.11793 | − | 0.645440i | 0 | −1.16681 | + | 2.02098i | 4.83636i | 0 | −1.74042 | − | 6.78019i | 8.17595i | 0 | 3.12158 | + | 5.40673i | ||||||||||
170.8 | 1.26920 | − | 0.732774i | 0 | −0.926086 | + | 1.60403i | 1.15270i | 0 | −2.90907 | + | 6.36689i | 8.57663i | 0 | 0.844669 | + | 1.46301i | ||||||||||
170.9 | 2.09020 | − | 1.20678i | 0 | 0.912615 | − | 1.58070i | 6.85138i | 0 | 5.41224 | + | 4.43934i | 5.24892i | 0 | 8.26808 | + | 14.3207i | ||||||||||
170.10 | 2.79169 | − | 1.61178i | 0 | 3.19568 | − | 5.53509i | − | 5.53294i | 0 | 3.31972 | + | 6.16275i | − | 7.70873i | 0 | −8.91790 | − | 15.4463i | ||||||||
170.11 | 3.19625 | − | 1.84536i | 0 | 4.81069 | − | 8.33236i | 5.83234i | 0 | 3.24949 | − | 6.20007i | − | 20.7469i | 0 | 10.7628 | + | 18.6416i | |||||||||
179.1 | −2.48702 | − | 1.43588i | 0 | 2.12350 | + | 3.67801i | 7.54889i | 0 | −6.82274 | − | 1.56534i | − | 0.709334i | 0 | 10.8393 | − | 18.7742i | |||||||||
179.2 | −2.37724 | − | 1.37250i | 0 | 1.76751 | + | 3.06142i | − | 2.68504i | 0 | 6.00002 | + | 3.60552i | 1.27635i | 0 | −3.68521 | + | 6.38297i | |||||||||
179.3 | −1.86624 | − | 1.07747i | 0 | 0.321900 | + | 0.557548i | − | 1.87862i | 0 | −3.79886 | + | 5.87951i | 7.23243i | 0 | −2.02416 | + | 3.50595i | |||||||||
179.4 | −1.11318 | − | 0.642694i | 0 | −1.17389 | − | 2.03324i | − | 7.87519i | 0 | 0.417718 | − | 6.98753i | 8.15936i | 0 | −5.06133 | + | 8.76649i | |||||||||
179.5 | −0.0664669 | − | 0.0383747i | 0 | −1.99705 | − | 3.45900i | 4.07697i | 0 | −6.97461 | − | 0.595686i | 0.613543i | 0 | 0.156452 | − | 0.270983i | ||||||||||
179.6 | 0.444866 | + | 0.256844i | 0 | −1.86806 | − | 3.23558i | 7.02462i | 0 | 5.34652 | + | 4.51827i | − | 3.97395i | 0 | −1.80423 | + | 3.12502i | |||||||||
179.7 | 1.11793 | + | 0.645440i | 0 | −1.16681 | − | 2.02098i | − | 4.83636i | 0 | −1.74042 | + | 6.78019i | − | 8.17595i | 0 | 3.12158 | − | 5.40673i | ||||||||
179.8 | 1.26920 | + | 0.732774i | 0 | −0.926086 | − | 1.60403i | − | 1.15270i | 0 | −2.90907 | − | 6.36689i | − | 8.57663i | 0 | 0.844669 | − | 1.46301i | ||||||||
179.9 | 2.09020 | + | 1.20678i | 0 | 0.912615 | + | 1.58070i | − | 6.85138i | 0 | 5.41224 | − | 4.43934i | − | 5.24892i | 0 | 8.26808 | − | 14.3207i | ||||||||
See all 22 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
63.n | odd | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 189.3.n.b | 22 | |
3.b | odd | 2 | 1 | 63.3.n.b | yes | 22 | |
7.c | even | 3 | 1 | 189.3.j.b | 22 | ||
9.c | even | 3 | 1 | 63.3.j.b | ✓ | 22 | |
9.d | odd | 6 | 1 | 189.3.j.b | 22 | ||
21.c | even | 2 | 1 | 441.3.n.f | 22 | ||
21.g | even | 6 | 1 | 441.3.j.f | 22 | ||
21.g | even | 6 | 1 | 441.3.r.f | 22 | ||
21.h | odd | 6 | 1 | 63.3.j.b | ✓ | 22 | |
21.h | odd | 6 | 1 | 441.3.r.g | 22 | ||
63.g | even | 3 | 1 | 63.3.n.b | yes | 22 | |
63.h | even | 3 | 1 | 441.3.r.g | 22 | ||
63.k | odd | 6 | 1 | 441.3.n.f | 22 | ||
63.l | odd | 6 | 1 | 441.3.j.f | 22 | ||
63.n | odd | 6 | 1 | inner | 189.3.n.b | 22 | |
63.t | odd | 6 | 1 | 441.3.r.f | 22 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
63.3.j.b | ✓ | 22 | 9.c | even | 3 | 1 | |
63.3.j.b | ✓ | 22 | 21.h | odd | 6 | 1 | |
63.3.n.b | yes | 22 | 3.b | odd | 2 | 1 | |
63.3.n.b | yes | 22 | 63.g | even | 3 | 1 | |
189.3.j.b | 22 | 7.c | even | 3 | 1 | ||
189.3.j.b | 22 | 9.d | odd | 6 | 1 | ||
189.3.n.b | 22 | 1.a | even | 1 | 1 | trivial | |
189.3.n.b | 22 | 63.n | odd | 6 | 1 | inner | |
441.3.j.f | 22 | 21.g | even | 6 | 1 | ||
441.3.j.f | 22 | 63.l | odd | 6 | 1 | ||
441.3.n.f | 22 | 21.c | even | 2 | 1 | ||
441.3.n.f | 22 | 63.k | odd | 6 | 1 | ||
441.3.r.f | 22 | 21.g | even | 6 | 1 | ||
441.3.r.f | 22 | 63.t | odd | 6 | 1 | ||
441.3.r.g | 22 | 21.h | odd | 6 | 1 | ||
441.3.r.g | 22 | 63.h | even | 3 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{22} - 6 T_{2}^{21} - 10 T_{2}^{20} + 132 T_{2}^{19} + 63 T_{2}^{18} - 1884 T_{2}^{17} + \cdots + 2187 \)
acting on \(S_{3}^{\mathrm{new}}(189, [\chi])\).