Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [63,4,Mod(5,63)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(63, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([5, 5]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("63.5");
S:= CuspForms(chi, 4);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 63 = 3^{2} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 63.i (of order \(6\), degree \(2\), 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: | \(3.71712033036\) |
Analytic rank: | \(0\) |
Dimension: | \(44\) |
Relative dimension: | \(22\) over \(\Q(\zeta_{6})\) |
Twist minimal: | yes |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
5.1 | − | 5.38106i | −0.968958 | + | 5.10501i | −20.9559 | −5.57991 | − | 9.66469i | 27.4704 | + | 5.21402i | −8.46770 | + | 16.4711i | 69.7163i | −25.1222 | − | 9.89308i | −52.0063 | + | 30.0259i | |||||
5.2 | − | 5.07706i | −4.42649 | − | 2.72143i | −17.7766 | 4.21335 | + | 7.29774i | −13.8169 | + | 22.4736i | −4.29909 | − | 18.0144i | 49.6363i | 12.1877 | + | 24.0928i | 37.0511 | − | 21.3915i | |||||
5.3 | − | 4.25176i | 5.13732 | − | 0.779690i | −10.0774 | −1.48754 | − | 2.57649i | −3.31505 | − | 21.8426i | −13.6257 | − | 12.5435i | 8.83278i | 25.7842 | − | 8.01104i | −10.9546 | + | 6.32464i | |||||
5.4 | − | 4.10714i | 1.00877 | − | 5.09729i | −8.86861 | −3.80591 | − | 6.59204i | −20.9353 | − | 4.14318i | 13.1152 | + | 13.0764i | 3.56749i | −24.9647 | − | 10.2840i | −27.0744 | + | 15.6314i | |||||
5.5 | − | 3.72062i | 3.26996 | + | 4.03824i | −5.84303 | 6.83336 | + | 11.8357i | 15.0248 | − | 12.1663i | 18.4941 | − | 0.984293i | − | 8.02526i | −5.61469 | + | 26.4098i | 44.0363 | − | 25.4243i | ||||
5.6 | − | 2.91468i | −4.68054 | + | 2.25667i | −0.495360 | 8.60567 | + | 14.9055i | 6.57748 | + | 13.6423i | −4.20673 | + | 18.0362i | − | 21.8736i | 16.8149 | − | 21.1249i | 43.4446 | − | 25.0828i | ||||
5.7 | − | 2.41653i | −3.31226 | + | 4.00362i | 2.16037 | −5.35965 | − | 9.28319i | 9.67487 | + | 8.00418i | 3.86176 | − | 18.1132i | − | 24.5529i | −5.05789 | − | 26.5220i | −22.4331 | + | 12.9518i | ||||
5.8 | − | 1.81805i | −3.99842 | − | 3.31853i | 4.69469 | −5.16236 | − | 8.94146i | −6.03325 | + | 7.26934i | −16.8039 | + | 7.78656i | − | 23.0796i | 4.97478 | + | 26.5377i | −16.2560 | + | 9.38543i | ||||
5.9 | − | 1.10690i | 0.171156 | − | 5.19333i | 6.77476 | 6.23688 | + | 10.8026i | −5.74852 | − | 0.189454i | 11.7098 | − | 14.3485i | − | 16.3542i | −26.9414 | − | 1.77774i | 11.9574 | − | 6.90363i | ||||
5.10 | − | 0.837567i | 4.56931 | + | 2.47415i | 7.29848 | −10.9584 | − | 18.9806i | 2.07226 | − | 3.82711i | 14.8276 | + | 11.0969i | − | 12.8135i | 14.7572 | + | 22.6103i | −15.8975 | + | 9.17842i | ||||
5.11 | − | 0.747815i | 4.77381 | − | 2.05201i | 7.44077 | 4.35110 | + | 7.53632i | −1.53452 | − | 3.56993i | −11.6200 | + | 14.4213i | − | 11.5468i | 18.5785 | − | 19.5918i | 5.63577 | − | 3.25382i | ||||
5.12 | 0.257625i | 1.70621 | + | 4.90804i | 7.93363 | 3.19386 | + | 5.53193i | −1.26443 | + | 0.439563i | −15.2112 | − | 10.5650i | 4.10490i | −21.1777 | + | 16.7483i | −1.42516 | + | 0.822818i | ||||||
5.13 | 1.15257i | −5.00666 | − | 1.39045i | 6.67158 | 0.137359 | + | 0.237913i | 1.60260 | − | 5.77053i | 18.5199 | + | 0.122959i | 16.9100i | 23.1333 | + | 13.9231i | −0.274212 | + | 0.158316i | ||||||
5.14 | 1.78786i | 2.28100 | − | 4.66873i | 4.80356 | −8.47168 | − | 14.6734i | 8.34703 | + | 4.07812i | −8.67298 | − | 16.3640i | 22.8910i | −16.5940 | − | 21.2988i | 26.2340 | − | 15.1462i | ||||||
5.15 | 1.83815i | −1.68860 | + | 4.91412i | 4.62119 | 0.207277 | + | 0.359014i | −9.03291 | − | 3.10391i | 5.26194 | + | 17.7570i | 23.1997i | −21.2972 | − | 16.5960i | −0.659923 | + | 0.381007i | ||||||
5.16 | 3.06129i | 4.98958 | + | 1.45054i | −1.37153 | 2.75111 | + | 4.76507i | −4.44054 | + | 15.2746i | 8.35389 | − | 16.5291i | 20.2917i | 22.7918 | + | 14.4752i | −14.5873 | + | 8.42197i | ||||||
5.17 | 3.46625i | −2.57609 | − | 4.51263i | −4.01488 | 9.02701 | + | 15.6352i | 15.6419 | − | 8.92935i | −18.1650 | + | 3.61017i | 13.8134i | −13.7276 | + | 23.2498i | −54.1956 | + | 31.2898i | ||||||
5.18 | 3.64983i | −4.97539 | + | 1.49850i | −5.32127 | −6.11568 | − | 10.5927i | −5.46929 | − | 18.1593i | −17.8879 | − | 4.79841i | 9.77690i | 22.5090 | − | 14.9113i | 38.6615 | − | 22.3212i | ||||||
5.19 | 3.72101i | 3.89612 | − | 3.43806i | −5.84592 | 1.33006 | + | 2.30373i | 12.7931 | + | 14.4975i | 8.98650 | + | 16.1939i | 8.01534i | 3.35949 | − | 26.7902i | −8.57221 | + | 4.94917i | ||||||
5.20 | 4.92859i | 3.60211 | + | 3.74497i | −16.2910 | −4.62434 | − | 8.00960i | −18.4575 | + | 17.7533i | −13.0699 | + | 13.1217i | − | 40.8632i | −1.04964 | + | 26.9796i | 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 | 63.4.i.a | ✓ | 44 |
3.b | odd | 2 | 1 | 189.4.i.a | 44 | ||
7.d | odd | 6 | 1 | 63.4.s.a | yes | 44 | |
9.c | even | 3 | 1 | 189.4.s.a | 44 | ||
9.d | odd | 6 | 1 | 63.4.s.a | yes | 44 | |
21.g | even | 6 | 1 | 189.4.s.a | 44 | ||
63.i | even | 6 | 1 | inner | 63.4.i.a | ✓ | 44 |
63.t | odd | 6 | 1 | 189.4.i.a | 44 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
63.4.i.a | ✓ | 44 | 1.a | even | 1 | 1 | trivial |
63.4.i.a | ✓ | 44 | 63.i | even | 6 | 1 | inner |
63.4.s.a | yes | 44 | 7.d | odd | 6 | 1 | |
63.4.s.a | yes | 44 | 9.d | odd | 6 | 1 | |
189.4.i.a | 44 | 3.b | odd | 2 | 1 | ||
189.4.i.a | 44 | 63.t | odd | 6 | 1 | ||
189.4.s.a | 44 | 9.c | even | 3 | 1 | ||
189.4.s.a | 44 | 21.g | even | 6 | 1 |
Hecke kernels
This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(63, [\chi])\).