Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [189,4,Mod(17,189)]
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([5, 1]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("189.17");
S:= CuspForms(chi, 4);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 189 = 3^{3} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 189.s (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: | \(11.1513609911\) |
Analytic rank: | \(0\) |
Dimension: | \(44\) |
Relative dimension: | \(22\) over \(\Q(\zeta_{6})\) |
Twist minimal: | no (minimal twist has level 63) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
17.1 | −4.66014 | − | 2.69053i | 0 | 10.4779 | + | 18.1483i | 11.1598 | 0 | 18.4983 | + | 0.902321i | − | 69.7163i | 0 | −52.0063 | − | 30.0259i | |||||||||
17.2 | −4.39687 | − | 2.53853i | 0 | 8.88829 | + | 15.3950i | −8.42670 | 0 | −13.4514 | − | 12.7303i | − | 49.6363i | 0 | 37.0511 | + | 21.3915i | |||||||||
17.3 | −3.68213 | − | 2.12588i | 0 | 5.03872 | + | 8.72732i | 2.97507 | 0 | −4.05014 | − | 18.0720i | − | 8.83278i | 0 | −10.9546 | − | 6.32464i | |||||||||
17.4 | −3.55689 | − | 2.05357i | 0 | 4.43430 | + | 7.68044i | 7.61183 | 0 | 4.76690 | + | 17.8963i | − | 3.56749i | 0 | −27.0744 | − | 15.6314i | |||||||||
17.5 | −3.22215 | − | 1.86031i | 0 | 2.92152 | + | 5.06021i | −13.6667 | 0 | −10.0995 | + | 15.5242i | 8.02526i | 0 | 44.0363 | + | 25.4243i | ||||||||||
17.6 | −2.52419 | − | 1.45734i | 0 | 0.247680 | + | 0.428994i | −17.2113 | 0 | 17.7231 | + | 5.37495i | 21.8736i | 0 | 43.4446 | + | 25.0828i | ||||||||||
17.7 | −2.09278 | − | 1.20827i | 0 | −1.08019 | − | 1.87094i | 10.7193 | 0 | −17.6173 | − | 5.71220i | 24.5529i | 0 | −22.4331 | − | 12.9518i | ||||||||||
17.8 | −1.57448 | − | 0.909026i | 0 | −2.34735 | − | 4.06572i | 10.3247 | 0 | 15.1453 | − | 10.6593i | 23.0796i | 0 | −16.2560 | − | 9.38543i | ||||||||||
17.9 | −0.958607 | − | 0.553452i | 0 | −3.38738 | − | 5.86712i | −12.4738 | 0 | −18.2811 | + | 2.96677i | 16.3542i | 0 | 11.9574 | + | 6.90363i | ||||||||||
17.10 | −0.725355 | − | 0.418784i | 0 | −3.64924 | − | 6.32067i | 21.9169 | 0 | 2.19637 | + | 18.3896i | 12.8135i | 0 | −15.8975 | − | 9.17842i | ||||||||||
17.11 | −0.647627 | − | 0.373907i | 0 | −3.72039 | − | 6.44390i | −8.70220 | 0 | 18.2993 | − | 2.85258i | 11.5468i | 0 | 5.63577 | + | 3.25382i | ||||||||||
17.12 | 0.223110 | + | 0.128812i | 0 | −3.96681 | − | 6.87072i | −6.38772 | 0 | −1.54394 | − | 18.4558i | − | 4.10490i | 0 | −1.42516 | − | 0.822818i | |||||||||
17.13 | 0.998155 | + | 0.576285i | 0 | −3.33579 | − | 5.77776i | −0.274718 | 0 | −9.15344 | + | 16.1001i | − | 16.9100i | 0 | −0.274212 | − | 0.158316i | |||||||||
17.14 | 1.54833 | + | 0.893930i | 0 | −2.40178 | − | 4.16000i | 16.9434 | 0 | −9.83512 | − | 15.6930i | − | 22.8910i | 0 | 26.2340 | + | 15.1462i | |||||||||
17.15 | 1.59189 | + | 0.919076i | 0 | −2.31060 | − | 4.00207i | −0.414554 | 0 | 12.7471 | + | 13.4355i | − | 23.1997i | 0 | −0.659923 | − | 0.381007i | |||||||||
17.16 | 2.65116 | + | 1.53065i | 0 | 0.685763 | + | 1.18778i | −5.50223 | 0 | −18.4916 | − | 1.02989i | − | 20.2917i | 0 | −14.5873 | − | 8.42197i | |||||||||
17.17 | 3.00186 | + | 1.73312i | 0 | 2.00744 | + | 3.47699i | −18.0540 | 0 | 12.2090 | − | 13.9263i | − | 13.8134i | 0 | −54.1956 | − | 31.2898i | |||||||||
17.18 | 3.16085 | + | 1.82492i | 0 | 2.66064 | + | 4.60836i | 12.2314 | 0 | 4.78838 | − | 17.8905i | − | 9.77690i | 0 | 38.6615 | + | 22.3212i | |||||||||
17.19 | 3.22249 | + | 1.86051i | 0 | 2.92296 | + | 5.06272i | −2.66012 | 0 | 9.53109 | + | 15.8795i | − | 8.01534i | 0 | −8.57221 | − | 4.94917i | |||||||||
17.20 | 4.26829 | + | 2.46430i | 0 | 8.14552 | + | 14.1085i | 9.24869 | 0 | 17.8986 | − | 4.75801i | 40.8632i | 0 | 39.4761 | + | 22.7915i | ||||||||||
See all 44 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
63.s | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 189.4.s.a | 44 | |
3.b | odd | 2 | 1 | 63.4.s.a | yes | 44 | |
7.d | odd | 6 | 1 | 189.4.i.a | 44 | ||
9.c | even | 3 | 1 | 63.4.i.a | ✓ | 44 | |
9.d | odd | 6 | 1 | 189.4.i.a | 44 | ||
21.g | even | 6 | 1 | 63.4.i.a | ✓ | 44 | |
63.k | odd | 6 | 1 | 63.4.s.a | yes | 44 | |
63.s | even | 6 | 1 | inner | 189.4.s.a | 44 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
63.4.i.a | ✓ | 44 | 9.c | even | 3 | 1 | |
63.4.i.a | ✓ | 44 | 21.g | even | 6 | 1 | |
63.4.s.a | yes | 44 | 3.b | odd | 2 | 1 | |
63.4.s.a | yes | 44 | 63.k | odd | 6 | 1 | |
189.4.i.a | 44 | 7.d | odd | 6 | 1 | ||
189.4.i.a | 44 | 9.d | odd | 6 | 1 | ||
189.4.s.a | 44 | 1.a | even | 1 | 1 | trivial | |
189.4.s.a | 44 | 63.s | even | 6 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(189, [\chi])\).