Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [552,2,Mod(17,552)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(552, base_ring=CyclotomicField(22))
chi = DirichletCharacter(H, H._module([0, 0, 11, 7]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("552.17");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 552 = 2^{3} \cdot 3 \cdot 23 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 552.u (of order \(22\), 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: | \(4.40774219157\) |
Analytic rank: | \(0\) |
Dimension: | \(240\) |
Relative dimension: | \(24\) over \(\Q(\zeta_{22})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{22}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
17.1 | 0 | −1.68738 | − | 0.390832i | 0 | 3.08209 | − | 1.98074i | 0 | 0.266085 | + | 0.0382572i | 0 | 2.69450 | + | 1.31896i | 0 | ||||||||||
17.2 | 0 | −1.63887 | + | 0.560462i | 0 | 0.535830 | − | 0.344357i | 0 | −2.02223 | − | 0.290753i | 0 | 2.37176 | − | 1.83704i | 0 | ||||||||||
17.3 | 0 | −1.63081 | − | 0.583477i | 0 | −3.08209 | + | 1.98074i | 0 | 0.266085 | + | 0.0382572i | 0 | 2.31911 | + | 1.90309i | 0 | ||||||||||
17.4 | 0 | −1.52304 | + | 0.824833i | 0 | −1.21140 | + | 0.778520i | 0 | 4.71345 | + | 0.677692i | 0 | 1.63930 | − | 2.51251i | 0 | ||||||||||
17.5 | 0 | −1.49329 | + | 0.877542i | 0 | −2.99625 | + | 1.92558i | 0 | −3.72244 | − | 0.535207i | 0 | 1.45984 | − | 2.62085i | 0 | ||||||||||
17.6 | 0 | −1.21330 | + | 1.23609i | 0 | 1.30091 | − | 0.836045i | 0 | 2.87937 | + | 0.413991i | 0 | −0.0558237 | − | 2.99948i | 0 | ||||||||||
17.7 | 0 | −1.07569 | − | 1.35753i | 0 | −0.535830 | + | 0.344357i | 0 | −2.02223 | − | 0.290753i | 0 | −0.685767 | + | 2.92057i | 0 | ||||||||||
17.8 | 0 | −0.835324 | − | 1.51731i | 0 | 1.21140 | − | 0.778520i | 0 | 4.71345 | + | 0.677692i | 0 | −1.60447 | + | 2.53489i | 0 | ||||||||||
17.9 | 0 | −0.781802 | − | 1.54557i | 0 | 2.99625 | − | 1.92558i | 0 | −3.72244 | − | 0.535207i | 0 | −1.77757 | + | 2.41666i | 0 | ||||||||||
17.10 | 0 | −0.725957 | + | 1.57257i | 0 | 1.91532 | − | 1.23090i | 0 | −2.52083 | − | 0.362441i | 0 | −1.94597 | − | 2.28324i | 0 | ||||||||||
17.11 | 0 | −0.352411 | − | 1.69582i | 0 | −1.30091 | + | 0.836045i | 0 | 2.87937 | + | 0.413991i | 0 | −2.75161 | + | 1.19525i | 0 | ||||||||||
17.12 | 0 | 0.0283990 | + | 1.73182i | 0 | −2.72873 | + | 1.75365i | 0 | 3.60100 | + | 0.517745i | 0 | −2.99839 | + | 0.0983638i | 0 | ||||||||||
17.13 | 0 | 0.0794015 | + | 1.73023i | 0 | −0.260753 | + | 0.167576i | 0 | 0.155773 | + | 0.0223967i | 0 | −2.98739 | + | 0.274766i | 0 | ||||||||||
17.14 | 0 | 0.239483 | − | 1.71541i | 0 | −1.91532 | + | 1.23090i | 0 | −2.52083 | − | 0.362441i | 0 | −2.88530 | − | 0.821626i | 0 | ||||||||||
17.15 | 0 | 0.441043 | + | 1.67496i | 0 | 3.16079 | − | 2.03132i | 0 | −2.34525 | − | 0.337196i | 0 | −2.61096 | + | 1.47746i | 0 | ||||||||||
17.16 | 0 | 0.908372 | + | 1.47474i | 0 | −1.46204 | + | 0.939594i | 0 | −4.39604 | − | 0.632056i | 0 | −1.34972 | + | 2.67923i | 0 | ||||||||||
17.17 | 0 | 0.960182 | − | 1.44154i | 0 | 2.72873 | − | 1.75365i | 0 | 3.60100 | + | 0.517745i | 0 | −1.15610 | − | 2.76829i | 0 | ||||||||||
17.18 | 0 | 1.00223 | − | 1.41263i | 0 | 0.260753 | − | 0.167576i | 0 | 0.155773 | + | 0.0223967i | 0 | −0.991071 | − | 2.83157i | 0 | ||||||||||
17.19 | 0 | 1.27658 | − | 1.17062i | 0 | −3.16079 | + | 2.03132i | 0 | −2.34525 | − | 0.337196i | 0 | 0.259310 | − | 2.98877i | 0 | ||||||||||
17.20 | 0 | 1.48588 | + | 0.890042i | 0 | 1.88314 | − | 1.21022i | 0 | 0.355050 | + | 0.0510485i | 0 | 1.41565 | + | 2.64498i | 0 | ||||||||||
See next 80 embeddings (of 240 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
23.d | odd | 22 | 1 | inner |
69.g | even | 22 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 552.2.u.a | ✓ | 240 |
3.b | odd | 2 | 1 | inner | 552.2.u.a | ✓ | 240 |
23.d | odd | 22 | 1 | inner | 552.2.u.a | ✓ | 240 |
69.g | even | 22 | 1 | inner | 552.2.u.a | ✓ | 240 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
552.2.u.a | ✓ | 240 | 1.a | even | 1 | 1 | trivial |
552.2.u.a | ✓ | 240 | 3.b | odd | 2 | 1 | inner |
552.2.u.a | ✓ | 240 | 23.d | odd | 22 | 1 | inner |
552.2.u.a | ✓ | 240 | 69.g | even | 22 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(552, [\chi])\).