Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [241,2,Mod(3,241)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(241, base_ring=CyclotomicField(120))
chi = DirichletCharacter(H, H._module([91]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("241.3");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 241 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 241.s (of order \(120\), degree \(32\), 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: | \(1.92439468871\) |
Analytic rank: | \(0\) |
Dimension: | \(640\) |
Relative dimension: | \(20\) over \(\Q(\zeta_{120})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{120}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
3.1 | −2.63760 | − | 0.706742i | 0.0388054 | + | 0.00203370i | 4.72539 | + | 2.72821i | −1.46248 | + | 2.87029i | −0.100916 | − | 0.0327895i | −2.01129 | + | 1.90865i | −6.67384 | − | 6.67384i | −2.98206 | − | 0.313428i | 5.88600 | − | 6.53707i |
3.2 | −2.45155 | − | 0.656890i | 1.72096 | + | 0.0901919i | 3.84653 | + | 2.22079i | 1.44750 | − | 2.84088i | −4.15978 | − | 1.35159i | 1.62669 | − | 1.54367i | −4.38182 | − | 4.38182i | −0.0299826 | − | 0.00315130i | −5.41476 | + | 6.01370i |
3.3 | −2.20393 | − | 0.590541i | −3.02856 | − | 0.158720i | 2.77652 | + | 1.60302i | −0.162719 | + | 0.319354i | 6.58101 | + | 2.13830i | 1.66562 | − | 1.58062i | −1.94583 | − | 1.94583i | 6.16342 | + | 0.647802i | 0.547213 | − | 0.607742i |
3.4 | −1.85977 | − | 0.498323i | −1.14995 | − | 0.0602663i | 1.47836 | + | 0.853531i | 0.593949 | − | 1.16569i | 2.10861 | + | 0.685128i | −2.05081 | + | 1.94614i | 0.398819 | + | 0.398819i | −1.66481 | − | 0.174979i | −1.68550 | + | 1.87193i |
3.5 | −1.83755 | − | 0.492369i | 2.81865 | + | 0.147719i | 1.40210 | + | 0.809501i | −0.634709 | + | 1.24569i | −5.10666 | − | 1.65926i | −1.18204 | + | 1.12171i | 0.512511 | + | 0.512511i | 4.93940 | + | 0.519152i | 1.77964 | − | 1.97649i |
3.6 | −1.66779 | − | 0.446883i | 0.776835 | + | 0.0407122i | 0.849764 | + | 0.490612i | −1.75457 | + | 3.44354i | −1.27740 | − | 0.415054i | 2.74176 | − | 2.60184i | 1.24383 | + | 1.24383i | −2.38175 | − | 0.250332i | 4.46511 | − | 4.95901i |
3.7 | −1.21604 | − | 0.325838i | −0.393654 | − | 0.0206305i | −0.359457 | − | 0.207533i | 1.13796 | − | 2.23338i | 0.471979 | + | 0.153355i | 0.846934 | − | 0.803710i | 2.14991 | + | 2.14991i | −2.82903 | − | 0.297343i | −2.11153 | + | 2.34509i |
3.8 | −0.452936 | − | 0.121364i | −2.00149 | − | 0.104894i | −1.54163 | − | 0.890060i | −1.24052 | + | 2.43467i | 0.893818 | + | 0.290419i | 1.64261 | − | 1.55878i | 1.25338 | + | 1.25338i | 1.01141 | + | 0.106303i | 0.857359 | − | 0.952194i |
3.9 | −0.438186 | − | 0.117412i | 3.29106 | + | 0.172477i | −1.55383 | − | 0.897104i | 1.47341 | − | 2.89172i | −1.42185 | − | 0.461986i | −1.24246 | + | 1.17905i | 1.21708 | + | 1.21708i | 7.81779 | + | 0.821683i | −0.985147 | + | 1.09412i |
3.10 | −0.383790 | − | 0.102836i | 0.677022 | + | 0.0354812i | −1.59533 | − | 0.921065i | −0.689366 | + | 1.35296i | −0.256186 | − | 0.0832398i | −2.93731 | + | 2.78740i | 1.07946 | + | 1.07946i | −2.52647 | − | 0.265542i | 0.403705 | − | 0.448359i |
3.11 | −0.0899442 | − | 0.0241005i | −3.33752 | − | 0.174912i | −1.72454 | − | 0.995665i | 1.23222 | − | 2.41837i | 0.295975 | + | 0.0961681i | −2.76536 | + | 2.62422i | 0.262804 | + | 0.262804i | 8.12487 | + | 0.853959i | −0.169115 | + | 0.187821i |
3.12 | 0.544032 | + | 0.145773i | 1.49190 | + | 0.0781872i | −1.45733 | − | 0.841390i | 0.659390 | − | 1.29413i | 0.800244 | + | 0.260015i | 2.62383 | − | 2.48992i | −1.46670 | − | 1.46670i | −0.763912 | − | 0.0802904i | 0.547378 | − | 0.607925i |
3.13 | 0.945508 | + | 0.253348i | −0.900870 | − | 0.0472126i | −0.902251 | − | 0.520915i | 1.19603 | − | 2.34733i | −0.839819 | − | 0.272874i | 0.242852 | − | 0.230458i | −2.10543 | − | 2.10543i | −2.17423 | − | 0.228520i | 1.72555 | − | 1.91641i |
3.14 | 0.946629 | + | 0.253649i | 3.10227 | + | 0.162583i | −0.900282 | − | 0.519778i | −1.96330 | + | 3.85320i | 2.89546 | + | 0.940793i | 0.937006 | − | 0.889186i | −2.10635 | − | 2.10635i | 6.61410 | + | 0.695170i | −2.83588 | + | 3.14956i |
3.15 | 0.986895 | + | 0.264438i | −1.75155 | − | 0.0917949i | −0.828016 | − | 0.478055i | −0.804333 | + | 1.57859i | −1.70432 | − | 0.553768i | −0.238777 | + | 0.226591i | −2.13566 | − | 2.13566i | 0.0759399 | + | 0.00798161i | −1.21123 | + | 1.34521i |
3.16 | 1.64096 | + | 0.439693i | 2.06237 | + | 0.108084i | 0.767353 | + | 0.443032i | 0.619663 | − | 1.21616i | 3.33674 | + | 1.08417i | −3.03296 | + | 2.87817i | −1.33813 | − | 1.33813i | 1.25814 | + | 0.132236i | 1.55157 | − | 1.72320i |
3.17 | 2.12211 | + | 0.568619i | 1.06388 | + | 0.0557554i | 2.44799 | + | 1.41335i | −0.373752 | + | 0.733529i | 2.22596 | + | 0.723259i | −0.199145 | + | 0.188981i | 1.28427 | + | 1.28427i | −1.85484 | − | 0.194952i | −1.21024 | + | 1.34411i |
3.18 | 2.23659 | + | 0.599293i | −0.255295 | − | 0.0133794i | 2.91114 | + | 1.68075i | −1.11515 | + | 2.18860i | −0.562973 | − | 0.182921i | 2.86193 | − | 2.71587i | 2.22917 | + | 2.22917i | −2.91857 | − | 0.306754i | −3.80573 | + | 4.22670i |
3.19 | 2.45678 | + | 0.658292i | −1.06146 | − | 0.0556289i | 3.87036 | + | 2.23455i | 1.85617 | − | 3.64294i | −2.57116 | − | 0.835420i | −0.417872 | + | 0.396546i | 4.44065 | + | 4.44065i | −1.85996 | − | 0.195489i | 6.95831 | − | 7.72798i |
3.20 | 2.45948 | + | 0.659015i | −2.98534 | − | 0.156455i | 3.88268 | + | 2.24166i | −1.24067 | + | 2.43494i | −7.23927 | − | 2.35218i | −2.92640 | + | 2.77705i | 4.47114 | + | 4.47114i | 5.90421 | + | 0.620558i | −4.65605 | + | 5.17107i |
See next 80 embeddings (of 640 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
241.s | even | 120 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 241.2.s.a | ✓ | 640 |
241.s | even | 120 | 1 | inner | 241.2.s.a | ✓ | 640 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
241.2.s.a | ✓ | 640 | 1.a | even | 1 | 1 | trivial |
241.2.s.a | ✓ | 640 | 241.s | even | 120 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(241, [\chi])\).