Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [441,3,Mod(55,441)]
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, 9]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("441.55");
S:= CuspForms(chi, 3);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 441 = 3^{2} \cdot 7^{2} \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 441.v (of order \(14\), 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: | \(12.0163796583\) |
Analytic rank: | \(0\) |
Dimension: | \(108\) |
Relative dimension: | \(18\) over \(\Q(\zeta_{14})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{14}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
55.1 | −2.22440 | + | 2.78931i | 0 | −1.94221 | − | 8.50938i | 0.587230 | − | 1.21939i | 0 | −7.00000 | + | 0.00817851i | 15.1981 | + | 7.31903i | 0 | 2.09504 | + | 4.35039i | ||||||
55.2 | −2.17139 | + | 2.72284i | 0 | −1.80883 | − | 7.92501i | −2.38642 | + | 4.95545i | 0 | 6.04459 | − | 3.53029i | 12.9552 | + | 6.23890i | 0 | −8.31106 | − | 17.2581i | ||||||
55.3 | −1.87534 | + | 2.35160i | 0 | −1.12304 | − | 4.92035i | 4.01706 | − | 8.34151i | 0 | 1.24401 | − | 6.88857i | 2.83699 | + | 1.36622i | 0 | 12.0825 | + | 25.0896i | ||||||
55.4 | −1.74663 | + | 2.19021i | 0 | −0.856201 | − | 3.75126i | 1.43978 | − | 2.98974i | 0 | 2.79088 | + | 6.41958i | −0.384316 | − | 0.185077i | 0 | 4.03338 | + | 8.37541i | ||||||
55.5 | −1.42918 | + | 1.79213i | 0 | −0.279105 | − | 1.22284i | −2.65777 | + | 5.51892i | 0 | −2.44049 | + | 6.56079i | −5.67051 | − | 2.73077i | 0 | −6.09221 | − | 12.6506i | ||||||
55.6 | −0.954441 | + | 1.19683i | 0 | 0.368636 | + | 1.61510i | 0.177110 | − | 0.367773i | 0 | −5.81764 | − | 3.89296i | −7.80168 | − | 3.75709i | 0 | 0.271121 | + | 0.562989i | ||||||
55.7 | −0.814787 | + | 1.02171i | 0 | 0.510069 | + | 2.23476i | −1.99918 | + | 4.15135i | 0 | 1.89189 | − | 6.73949i | −7.40848 | − | 3.56773i | 0 | −2.61256 | − | 5.42505i | ||||||
55.8 | −0.478613 | + | 0.600162i | 0 | 0.758960 | + | 3.32522i | 1.62579 | − | 3.37598i | 0 | 6.32334 | + | 3.00256i | −5.12539 | − | 2.46826i | 0 | 1.24801 | + | 2.59153i | ||||||
55.9 | −0.370791 | + | 0.464957i | 0 | 0.811384 | + | 3.55491i | 3.67587 | − | 7.63302i | 0 | −6.29446 | + | 3.06265i | −4.09697 | − | 1.97300i | 0 | 2.18605 | + | 4.53938i | ||||||
55.10 | 0.370791 | − | 0.464957i | 0 | 0.811384 | + | 3.55491i | −3.67587 | + | 7.63302i | 0 | −6.29446 | + | 3.06265i | 4.09697 | + | 1.97300i | 0 | 2.18605 | + | 4.53938i | ||||||
55.11 | 0.478613 | − | 0.600162i | 0 | 0.758960 | + | 3.32522i | −1.62579 | + | 3.37598i | 0 | 6.32334 | + | 3.00256i | 5.12539 | + | 2.46826i | 0 | 1.24801 | + | 2.59153i | ||||||
55.12 | 0.814787 | − | 1.02171i | 0 | 0.510069 | + | 2.23476i | 1.99918 | − | 4.15135i | 0 | 1.89189 | − | 6.73949i | 7.40848 | + | 3.56773i | 0 | −2.61256 | − | 5.42505i | ||||||
55.13 | 0.954441 | − | 1.19683i | 0 | 0.368636 | + | 1.61510i | −0.177110 | + | 0.367773i | 0 | −5.81764 | − | 3.89296i | 7.80168 | + | 3.75709i | 0 | 0.271121 | + | 0.562989i | ||||||
55.14 | 1.42918 | − | 1.79213i | 0 | −0.279105 | − | 1.22284i | 2.65777 | − | 5.51892i | 0 | −2.44049 | + | 6.56079i | 5.67051 | + | 2.73077i | 0 | −6.09221 | − | 12.6506i | ||||||
55.15 | 1.74663 | − | 2.19021i | 0 | −0.856201 | − | 3.75126i | −1.43978 | + | 2.98974i | 0 | 2.79088 | + | 6.41958i | 0.384316 | + | 0.185077i | 0 | 4.03338 | + | 8.37541i | ||||||
55.16 | 1.87534 | − | 2.35160i | 0 | −1.12304 | − | 4.92035i | −4.01706 | + | 8.34151i | 0 | 1.24401 | − | 6.88857i | −2.83699 | − | 1.36622i | 0 | 12.0825 | + | 25.0896i | ||||||
55.17 | 2.17139 | − | 2.72284i | 0 | −1.80883 | − | 7.92501i | 2.38642 | − | 4.95545i | 0 | 6.04459 | − | 3.53029i | −12.9552 | − | 6.23890i | 0 | −8.31106 | − | 17.2581i | ||||||
55.18 | 2.22440 | − | 2.78931i | 0 | −1.94221 | − | 8.50938i | −0.587230 | + | 1.21939i | 0 | −7.00000 | + | 0.00817851i | −15.1981 | − | 7.31903i | 0 | 2.09504 | + | 4.35039i | ||||||
118.1 | −3.46741 | − | 1.66982i | 0 | 6.74071 | + | 8.45258i | 6.54256 | − | 1.49330i | 0 | −6.67672 | − | 2.10271i | −5.83303 | − | 25.5562i | 0 | −25.1793 | − | 5.74701i | ||||||
118.2 | −3.15349 | − | 1.51864i | 0 | 5.14427 | + | 6.45071i | −3.39364 | + | 0.774577i | 0 | 3.67959 | + | 5.95488i | −3.31070 | − | 14.5051i | 0 | 11.8781 | + | 2.71110i | ||||||
See next 80 embeddings (of 108 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
49.f | odd | 14 | 1 | inner |
147.k | even | 14 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 441.3.v.b | ✓ | 108 |
3.b | odd | 2 | 1 | inner | 441.3.v.b | ✓ | 108 |
49.f | odd | 14 | 1 | inner | 441.3.v.b | ✓ | 108 |
147.k | even | 14 | 1 | inner | 441.3.v.b | ✓ | 108 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
441.3.v.b | ✓ | 108 | 1.a | even | 1 | 1 | trivial |
441.3.v.b | ✓ | 108 | 3.b | odd | 2 | 1 | inner |
441.3.v.b | ✓ | 108 | 49.f | odd | 14 | 1 | inner |
441.3.v.b | ✓ | 108 | 147.k | even | 14 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{108} + 52 T_{2}^{106} + 1592 T_{2}^{104} + 38340 T_{2}^{102} + 809333 T_{2}^{100} + \cdots + 94\!\cdots\!61 \) acting on \(S_{3}^{\mathrm{new}}(441, [\chi])\).