Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [441,2,Mod(64,441)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(441, base_ring=CyclotomicField(14))
chi = DirichletCharacter(H, H._module([0, 10]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("441.64");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 441 = 3^{2} \cdot 7^{2} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 441.u (of order \(7\), degree \(6\), 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: | \(3.52140272914\) |
Analytic rank: | \(0\) |
Dimension: | \(24\) |
Relative dimension: | \(4\) over \(\Q(\zeta_{7})\) |
Twist minimal: | no (minimal twist has level 147) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{7}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
64.1 | −0.540253 | + | 2.36700i | 0 | −3.50889 | − | 1.68979i | −1.42001 | + | 1.78063i | 0 | −1.01369 | − | 2.44386i | 2.86791 | − | 3.59625i | 0 | −3.44760 | − | 4.32315i | ||||||
64.2 | −0.271791 | + | 1.19079i | 0 | 0.457820 | + | 0.220474i | 0.164209 | − | 0.205912i | 0 | 2.60345 | + | 0.471214i | −1.91005 | + | 2.39513i | 0 | 0.200568 | + | 0.251504i | ||||||
64.3 | 0.243969 | − | 1.06890i | 0 | 0.718913 | + | 0.346210i | 2.19062 | − | 2.74696i | 0 | 0.626501 | − | 2.57050i | 1.91263 | − | 2.39836i | 0 | −2.40177 | − | 3.01173i | ||||||
64.4 | 0.345553 | − | 1.51397i | 0 | −0.370753 | − | 0.178545i | −0.0888147 | + | 0.111370i | 0 | −0.524245 | + | 2.59329i | 1.53801 | − | 1.92860i | 0 | 0.137921 | + | 0.172947i | ||||||
127.1 | −1.72952 | − | 0.832892i | 0 | 1.05054 | + | 1.31734i | 0.379489 | + | 1.66265i | 0 | 0.548424 | + | 2.58829i | 0.134579 | + | 0.589630i | 0 | 0.728476 | − | 3.19166i | ||||||
127.2 | −1.28533 | − | 0.618981i | 0 | 0.0219496 | + | 0.0275239i | −0.813628 | − | 3.56474i | 0 | −0.766737 | − | 2.53222i | 0.623724 | + | 2.73271i | 0 | −1.16073 | + | 5.08547i | ||||||
127.3 | 0.0441061 | + | 0.0212404i | 0 | −1.24549 | − | 1.56179i | 0.763028 | + | 3.34304i | 0 | 1.92953 | − | 1.81022i | −0.0435471 | − | 0.190792i | 0 | −0.0373533 | + | 0.163656i | ||||||
127.4 | 2.06977 | + | 0.996749i | 0 | 2.04346 | + | 2.56242i | 0.349558 | + | 1.53152i | 0 | −0.354322 | + | 2.62192i | 0.653025 | + | 2.86109i | 0 | −0.803031 | + | 3.51831i | ||||||
190.1 | −1.18466 | + | 1.48552i | 0 | −0.358304 | − | 1.56983i | 0.295313 | + | 0.142215i | 0 | −2.01260 | − | 1.71740i | −0.667291 | − | 0.321350i | 0 | −0.561110 | + | 0.270216i | ||||||
190.2 | −0.159899 | + | 0.200507i | 0 | 0.430406 | + | 1.88573i | 1.50958 | + | 0.726973i | 0 | 2.63685 | − | 0.216906i | −0.909048 | − | 0.437774i | 0 | −0.387144 | + | 0.186439i | ||||||
190.3 | 0.309296 | − | 0.387845i | 0 | 0.390282 | + | 1.70994i | −0.734303 | − | 0.353621i | 0 | −2.35634 | + | 1.20320i | 1.67779 | + | 0.807983i | 0 | −0.364267 | + | 0.175422i | ||||||
190.4 | 1.65876 | − | 2.08002i | 0 | −1.12995 | − | 4.95062i | −2.59504 | − | 1.24971i | 0 | −1.31683 | + | 2.29477i | −7.37774 | − | 3.55293i | 0 | −6.90396 | + | 3.32477i | ||||||
253.1 | −1.18466 | − | 1.48552i | 0 | −0.358304 | + | 1.56983i | 0.295313 | − | 0.142215i | 0 | −2.01260 | + | 1.71740i | −0.667291 | + | 0.321350i | 0 | −0.561110 | − | 0.270216i | ||||||
253.2 | −0.159899 | − | 0.200507i | 0 | 0.430406 | − | 1.88573i | 1.50958 | − | 0.726973i | 0 | 2.63685 | + | 0.216906i | −0.909048 | + | 0.437774i | 0 | −0.387144 | − | 0.186439i | ||||||
253.3 | 0.309296 | + | 0.387845i | 0 | 0.390282 | − | 1.70994i | −0.734303 | + | 0.353621i | 0 | −2.35634 | − | 1.20320i | 1.67779 | − | 0.807983i | 0 | −0.364267 | − | 0.175422i | ||||||
253.4 | 1.65876 | + | 2.08002i | 0 | −1.12995 | + | 4.95062i | −2.59504 | + | 1.24971i | 0 | −1.31683 | − | 2.29477i | −7.37774 | + | 3.55293i | 0 | −6.90396 | − | 3.32477i | ||||||
316.1 | −1.72952 | + | 0.832892i | 0 | 1.05054 | − | 1.31734i | 0.379489 | − | 1.66265i | 0 | 0.548424 | − | 2.58829i | 0.134579 | − | 0.589630i | 0 | 0.728476 | + | 3.19166i | ||||||
316.2 | −1.28533 | + | 0.618981i | 0 | 0.0219496 | − | 0.0275239i | −0.813628 | + | 3.56474i | 0 | −0.766737 | + | 2.53222i | 0.623724 | − | 2.73271i | 0 | −1.16073 | − | 5.08547i | ||||||
316.3 | 0.0441061 | − | 0.0212404i | 0 | −1.24549 | + | 1.56179i | 0.763028 | − | 3.34304i | 0 | 1.92953 | + | 1.81022i | −0.0435471 | + | 0.190792i | 0 | −0.0373533 | − | 0.163656i | ||||||
316.4 | 2.06977 | − | 0.996749i | 0 | 2.04346 | − | 2.56242i | 0.349558 | − | 1.53152i | 0 | −0.354322 | − | 2.62192i | 0.653025 | − | 2.86109i | 0 | −0.803031 | − | 3.51831i | ||||||
See all 24 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
49.e | even | 7 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 441.2.u.c | 24 | |
3.b | odd | 2 | 1 | 147.2.i.a | ✓ | 24 | |
49.e | even | 7 | 1 | inner | 441.2.u.c | 24 | |
147.k | even | 14 | 1 | 7203.2.a.b | 12 | ||
147.l | odd | 14 | 1 | 147.2.i.a | ✓ | 24 | |
147.l | odd | 14 | 1 | 7203.2.a.a | 12 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
147.2.i.a | ✓ | 24 | 3.b | odd | 2 | 1 | |
147.2.i.a | ✓ | 24 | 147.l | odd | 14 | 1 | |
441.2.u.c | 24 | 1.a | even | 1 | 1 | trivial | |
441.2.u.c | 24 | 49.e | even | 7 | 1 | inner | |
7203.2.a.a | 12 | 147.l | odd | 14 | 1 | ||
7203.2.a.b | 12 | 147.k | even | 14 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{24} + T_{2}^{23} + 6 T_{2}^{22} + 8 T_{2}^{21} + 40 T_{2}^{20} + 95 T_{2}^{19} + 210 T_{2}^{18} + \cdots + 1 \) acting on \(S_{2}^{\mathrm{new}}(441, [\chi])\).