Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [121,2,Mod(12,121)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(121, base_ring=CyclotomicField(22))
chi = DirichletCharacter(H, H._module([18]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("121.12");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 121 = 11^{2} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 121.e (of order \(11\), degree \(10\), 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: | \(0.966189864457\) |
Analytic rank: | \(0\) |
Dimension: | \(100\) |
Relative dimension: | \(10\) over \(\Q(\zeta_{11})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{11}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
12.1 | −2.28929 | + | 1.47124i | −1.98337 | 2.24548 | − | 4.91691i | 0.00109106 | − | 0.00758846i | 4.54050 | − | 2.91800i | 1.07261 | − | 1.23786i | 1.31883 | + | 9.17268i | 0.933746 | 0.00866667 | + | 0.0189774i | ||||
12.2 | −1.90982 | + | 1.22737i | 2.89591 | 1.31016 | − | 2.86885i | −0.301049 | + | 2.09384i | −5.53069 | + | 3.55436i | 2.32393 | − | 2.68196i | 0.372795 | + | 2.59284i | 5.38632 | −1.99496 | − | 4.36836i | ||||
12.3 | −1.29895 | + | 0.834784i | −0.564640 | 0.159574 | − | 0.349419i | −0.144022 | + | 1.00170i | 0.733439 | − | 0.471353i | −1.26378 | + | 1.45848i | −0.355076 | − | 2.46961i | −2.68118 | −0.649122 | − | 1.42138i | ||||
12.4 | −0.800834 | + | 0.514665i | −2.23328 | −0.454375 | + | 0.994941i | 0.373665 | − | 2.59890i | 1.78848 | − | 1.14939i | 1.22249 | − | 1.41083i | −0.419137 | − | 2.91516i | 1.98753 | 1.03832 | + | 2.27360i | ||||
12.5 | −0.322674 | + | 0.207370i | 1.45814 | −0.769714 | + | 1.68544i | −0.307743 | + | 2.14040i | −0.470506 | + | 0.302376i | −0.298898 | + | 0.344946i | −0.210316 | − | 1.46278i | −0.873817 | −0.344555 | − | 0.754469i | ||||
12.6 | −0.0579015 | + | 0.0372110i | 2.65925 | −0.828862 | + | 1.81495i | 0.499158 | − | 3.47172i | −0.153975 | + | 0.0989535i | −1.42926 | + | 1.64945i | −0.0391344 | − | 0.272185i | 4.07163 | 0.100284 | + | 0.219592i | ||||
12.7 | 0.668610 | − | 0.429690i | −2.29704 | −0.568424 | + | 1.24467i | −0.598868 | + | 4.16522i | −1.53583 | + | 0.987017i | 2.14854 | − | 2.47954i | 0.380987 | + | 2.64983i | 2.27641 | 1.38934 | + | 3.04223i | ||||
12.8 | 1.05761 | − | 0.679684i | 0.322850 | −0.174263 | + | 0.381583i | 0.250017 | − | 1.73891i | 0.341450 | − | 0.219436i | 2.04681 | − | 2.36215i | 0.432885 | + | 3.01078i | −2.89577 | −0.917487 | − | 2.00902i | ||||
12.9 | 1.77043 | − | 1.13779i | 0.298538 | 1.00904 | − | 2.20948i | −0.0612837 | + | 0.426238i | 0.528540 | − | 0.339672i | −1.45441 | + | 1.67848i | −0.128482 | − | 0.893611i | −2.91088 | 0.376469 | + | 0.824352i | ||||
12.10 | 2.02796 | − | 1.30329i | −3.38720 | 1.58323 | − | 3.46678i | 0.366767 | − | 2.55092i | −6.86911 | + | 4.41451i | −0.194745 | + | 0.224747i | −0.621365 | − | 4.32169i | 8.47314 | −2.58080 | − | 5.65116i | ||||
23.1 | −1.14719 | + | 2.51200i | −1.28359 | −3.68436 | − | 4.25198i | −0.739579 | − | 0.217160i | 1.47253 | − | 3.22438i | −0.216919 | − | 1.50870i | 9.60824 | − | 2.82123i | −1.35239 | 1.39394 | − | 1.60870i | ||||
23.2 | −0.917263 | + | 2.00852i | 3.12379 | −1.88308 | − | 2.17319i | −1.00380 | − | 0.294742i | −2.86534 | + | 6.27421i | 0.184381 | + | 1.28240i | 1.85494 | − | 0.544660i | 6.75808 | 1.51274 | − | 1.74580i | ||||
23.3 | −0.671026 | + | 1.46934i | −1.09234 | −0.398970 | − | 0.460436i | 3.18729 | + | 0.935873i | 0.732990 | − | 1.60502i | 0.233950 | + | 1.62716i | −2.15551 | + | 0.632915i | −1.80679 | −3.51387 | + | 4.05523i | ||||
23.4 | −0.597130 | + | 1.30753i | −0.640035 | −0.0433545 | − | 0.0500337i | −3.97756 | − | 1.16792i | 0.382184 | − | 0.836867i | 0.623172 | + | 4.33426i | −2.66710 | + | 0.783131i | −2.59035 | 3.90221 | − | 4.50339i | ||||
23.5 | −0.389908 | + | 0.853778i | 1.43258 | 0.732812 | + | 0.845710i | 0.815711 | + | 0.239514i | −0.558572 | + | 1.22310i | −0.497109 | − | 3.45747i | −2.80893 | + | 0.824777i | −0.947723 | −0.522544 | + | 0.603048i | ||||
23.6 | 0.194417 | − | 0.425713i | −3.09292 | 1.16629 | + | 1.34597i | 1.44135 | + | 0.423218i | −0.601314 | + | 1.31669i | 0.384441 | + | 2.67384i | 1.69784 | − | 0.498530i | 6.56613 | 0.460391 | − | 0.531320i | ||||
23.7 | 0.333827 | − | 0.730979i | 0.0294246 | 0.886832 | + | 1.02346i | 0.779896 | + | 0.228998i | 0.00982271 | − | 0.0215087i | 0.0964221 | + | 0.670630i | 2.58627 | − | 0.759397i | −2.99913 | 0.427743 | − | 0.493642i | ||||
23.8 | 0.569265 | − | 1.24652i | 2.07869 | 0.0799801 | + | 0.0923019i | −3.60635 | − | 1.05892i | 1.18333 | − | 2.59113i | −0.0268425 | − | 0.186694i | 2.79027 | − | 0.819298i | 1.32097 | −3.37293 | + | 3.89257i | ||||
23.9 | 0.934014 | − | 2.04521i | −1.99480 | −2.00076 | − | 2.30900i | −1.29355 | − | 0.379821i | −1.86317 | + | 4.07977i | −0.411373 | − | 2.86116i | −2.27650 | + | 0.668441i | 0.979213 | −1.98501 | + | 2.29082i | ||||
23.10 | 1.04868 | − | 2.29629i | 0.748916 | −2.86349 | − | 3.30464i | 1.29867 | + | 0.381324i | 0.785372 | − | 1.71973i | 0.561499 | + | 3.90531i | −5.74697 | + | 1.68746i | −2.43913 | 2.23752 | − | 2.58224i | ||||
See all 100 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
121.e | even | 11 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 121.2.e.a | ✓ | 100 |
121.e | even | 11 | 1 | inner | 121.2.e.a | ✓ | 100 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
121.2.e.a | ✓ | 100 | 1.a | even | 1 | 1 | trivial |
121.2.e.a | ✓ | 100 | 121.e | even | 11 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(121, [\chi])\).