Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [189,4,Mod(143,189)]
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, 1]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("189.143");
S:= CuspForms(chi, 4);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 189 = 3^{3} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 189.i (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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
143.1 | − | 5.38106i | 0 | −20.9559 | 5.57991 | − | 9.66469i | 0 | −8.46770 | − | 16.4711i | 69.7163i | 0 | −52.0063 | − | 30.0259i | |||||||||||
143.2 | − | 5.07706i | 0 | −17.7766 | −4.21335 | + | 7.29774i | 0 | −4.29909 | + | 18.0144i | 49.6363i | 0 | 37.0511 | + | 21.3915i | |||||||||||
143.3 | − | 4.25176i | 0 | −10.0774 | 1.48754 | − | 2.57649i | 0 | −13.6257 | + | 12.5435i | 8.83278i | 0 | −10.9546 | − | 6.32464i | |||||||||||
143.4 | − | 4.10714i | 0 | −8.86861 | 3.80591 | − | 6.59204i | 0 | 13.1152 | − | 13.0764i | 3.56749i | 0 | −27.0744 | − | 15.6314i | |||||||||||
143.5 | − | 3.72062i | 0 | −5.84303 | −6.83336 | + | 11.8357i | 0 | 18.4941 | + | 0.984293i | − | 8.02526i | 0 | 44.0363 | + | 25.4243i | ||||||||||
143.6 | − | 2.91468i | 0 | −0.495360 | −8.60567 | + | 14.9055i | 0 | −4.20673 | − | 18.0362i | − | 21.8736i | 0 | 43.4446 | + | 25.0828i | ||||||||||
143.7 | − | 2.41653i | 0 | 2.16037 | 5.35965 | − | 9.28319i | 0 | 3.86176 | + | 18.1132i | − | 24.5529i | 0 | −22.4331 | − | 12.9518i | ||||||||||
143.8 | − | 1.81805i | 0 | 4.69469 | 5.16236 | − | 8.94146i | 0 | −16.8039 | − | 7.78656i | − | 23.0796i | 0 | −16.2560 | − | 9.38543i | ||||||||||
143.9 | − | 1.10690i | 0 | 6.77476 | −6.23688 | + | 10.8026i | 0 | 11.7098 | + | 14.3485i | − | 16.3542i | 0 | 11.9574 | + | 6.90363i | ||||||||||
143.10 | − | 0.837567i | 0 | 7.29848 | 10.9584 | − | 18.9806i | 0 | 14.8276 | − | 11.0969i | − | 12.8135i | 0 | −15.8975 | − | 9.17842i | ||||||||||
143.11 | − | 0.747815i | 0 | 7.44077 | −4.35110 | + | 7.53632i | 0 | −11.6200 | − | 14.4213i | − | 11.5468i | 0 | 5.63577 | + | 3.25382i | ||||||||||
143.12 | 0.257625i | 0 | 7.93363 | −3.19386 | + | 5.53193i | 0 | −15.2112 | + | 10.5650i | 4.10490i | 0 | −1.42516 | − | 0.822818i | ||||||||||||
143.13 | 1.15257i | 0 | 6.67158 | −0.137359 | + | 0.237913i | 0 | 18.5199 | − | 0.122959i | 16.9100i | 0 | −0.274212 | − | 0.158316i | ||||||||||||
143.14 | 1.78786i | 0 | 4.80356 | 8.47168 | − | 14.6734i | 0 | −8.67298 | + | 16.3640i | 22.8910i | 0 | 26.2340 | + | 15.1462i | ||||||||||||
143.15 | 1.83815i | 0 | 4.62119 | −0.207277 | + | 0.359014i | 0 | 5.26194 | − | 17.7570i | 23.1997i | 0 | −0.659923 | − | 0.381007i | ||||||||||||
143.16 | 3.06129i | 0 | −1.37153 | −2.75111 | + | 4.76507i | 0 | 8.35389 | + | 16.5291i | 20.2917i | 0 | −14.5873 | − | 8.42197i | ||||||||||||
143.17 | 3.46625i | 0 | −4.01488 | −9.02701 | + | 15.6352i | 0 | −18.1650 | − | 3.61017i | 13.8134i | 0 | −54.1956 | − | 31.2898i | ||||||||||||
143.18 | 3.64983i | 0 | −5.32127 | 6.11568 | − | 10.5927i | 0 | −17.8879 | + | 4.79841i | 9.77690i | 0 | 38.6615 | + | 22.3212i | ||||||||||||
143.19 | 3.72101i | 0 | −5.84592 | −1.33006 | + | 2.30373i | 0 | 8.98650 | − | 16.1939i | 8.01534i | 0 | −8.57221 | − | 4.94917i | ||||||||||||
143.20 | 4.92859i | 0 | −16.2910 | 4.62434 | − | 8.00960i | 0 | −13.0699 | − | 13.1217i | − | 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.i | 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.i.a | 44 | |
3.b | odd | 2 | 1 | 63.4.i.a | ✓ | 44 | |
7.d | odd | 6 | 1 | 189.4.s.a | 44 | ||
9.c | even | 3 | 1 | 63.4.s.a | yes | 44 | |
9.d | odd | 6 | 1 | 189.4.s.a | 44 | ||
21.g | even | 6 | 1 | 63.4.s.a | yes | 44 | |
63.i | even | 6 | 1 | inner | 189.4.i.a | 44 | |
63.t | odd | 6 | 1 | 63.4.i.a | ✓ | 44 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
63.4.i.a | ✓ | 44 | 3.b | odd | 2 | 1 | |
63.4.i.a | ✓ | 44 | 63.t | odd | 6 | 1 | |
63.4.s.a | yes | 44 | 9.c | even | 3 | 1 | |
63.4.s.a | yes | 44 | 21.g | even | 6 | 1 | |
189.4.i.a | 44 | 1.a | even | 1 | 1 | trivial | |
189.4.i.a | 44 | 63.i | even | 6 | 1 | inner | |
189.4.s.a | 44 | 7.d | odd | 6 | 1 | ||
189.4.s.a | 44 | 9.d | odd | 6 | 1 |
Hecke kernels
This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(189, [\chi])\).