Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [945,2,Mod(323,945)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(945, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([2, 3, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("945.323");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 945 = 3^{3} \cdot 5 \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 945.m (of order \(4\), degree \(2\), 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: | \(7.54586299101\) |
Analytic rank: | \(0\) |
Dimension: | \(48\) |
Relative dimension: | \(24\) over \(\Q(i)\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
323.1 | −1.92327 | − | 1.92327i | 0 | 5.39793i | 1.20974 | − | 1.88057i | 0 | −0.707107 | + | 0.707107i | 6.53512 | − | 6.53512i | 0 | −5.94349 | + | 1.29019i | ||||||||
323.2 | −1.80466 | − | 1.80466i | 0 | 4.51361i | −2.22504 | − | 0.221835i | 0 | −0.707107 | + | 0.707107i | 4.53621 | − | 4.53621i | 0 | 3.61510 | + | 4.41578i | ||||||||
323.3 | −1.71292 | − | 1.71292i | 0 | 3.86818i | 1.70013 | − | 1.45243i | 0 | 0.707107 | − | 0.707107i | 3.20004 | − | 3.20004i | 0 | −5.40008 | − | 0.424302i | ||||||||
323.4 | −1.66804 | − | 1.66804i | 0 | 3.56469i | 0.420676 | + | 2.19614i | 0 | 0.707107 | − | 0.707107i | 2.60996 | − | 2.60996i | 0 | 2.96154 | − | 4.36494i | ||||||||
323.5 | −1.36262 | − | 1.36262i | 0 | 1.71349i | −1.41633 | − | 1.73032i | 0 | 0.707107 | − | 0.707107i | −0.390409 | + | 0.390409i | 0 | −0.427853 | + | 4.28770i | ||||||||
323.6 | −1.20546 | − | 1.20546i | 0 | 0.906269i | 2.12218 | + | 0.704515i | 0 | −0.707107 | + | 0.707107i | −1.31845 | + | 1.31845i | 0 | −1.70894 | − | 3.40747i | ||||||||
323.7 | −1.05516 | − | 1.05516i | 0 | 0.226725i | 2.07957 | − | 0.821810i | 0 | 0.707107 | − | 0.707107i | −1.87109 | + | 1.87109i | 0 | −3.06142 | − | 1.32714i | ||||||||
323.8 | −0.885953 | − | 0.885953i | 0 | − | 0.430174i | −0.684062 | − | 2.12886i | 0 | −0.707107 | + | 0.707107i | −2.15302 | + | 2.15302i | 0 | −1.28003 | + | 2.49212i | |||||||
323.9 | −0.754986 | − | 0.754986i | 0 | − | 0.859992i | −2.23288 | + | 0.119338i | 0 | 0.707107 | − | 0.707107i | −2.15925 | + | 2.15925i | 0 | 1.77589 | + | 1.59570i | |||||||
323.10 | −0.493412 | − | 0.493412i | 0 | − | 1.51309i | 0.449429 | + | 2.19044i | 0 | 0.707107 | − | 0.707107i | −1.73340 | + | 1.73340i | 0 | 0.859034 | − | 1.30254i | |||||||
323.11 | −0.488896 | − | 0.488896i | 0 | − | 1.52196i | −2.07865 | + | 0.824144i | 0 | −0.707107 | + | 0.707107i | −1.72187 | + | 1.72187i | 0 | 1.41916 | + | 0.613323i | |||||||
323.12 | −0.259165 | − | 0.259165i | 0 | − | 1.86567i | 1.30674 | − | 1.81450i | 0 | −0.707107 | + | 0.707107i | −1.00185 | + | 1.00185i | 0 | −0.808919 | + | 0.131594i | |||||||
323.13 | 0.259165 | + | 0.259165i | 0 | − | 1.86567i | −1.30674 | + | 1.81450i | 0 | −0.707107 | + | 0.707107i | 1.00185 | − | 1.00185i | 0 | −0.808919 | + | 0.131594i | |||||||
323.14 | 0.488896 | + | 0.488896i | 0 | − | 1.52196i | 2.07865 | − | 0.824144i | 0 | −0.707107 | + | 0.707107i | 1.72187 | − | 1.72187i | 0 | 1.41916 | + | 0.613323i | |||||||
323.15 | 0.493412 | + | 0.493412i | 0 | − | 1.51309i | −0.449429 | − | 2.19044i | 0 | 0.707107 | − | 0.707107i | 1.73340 | − | 1.73340i | 0 | 0.859034 | − | 1.30254i | |||||||
323.16 | 0.754986 | + | 0.754986i | 0 | − | 0.859992i | 2.23288 | − | 0.119338i | 0 | 0.707107 | − | 0.707107i | 2.15925 | − | 2.15925i | 0 | 1.77589 | + | 1.59570i | |||||||
323.17 | 0.885953 | + | 0.885953i | 0 | − | 0.430174i | 0.684062 | + | 2.12886i | 0 | −0.707107 | + | 0.707107i | 2.15302 | − | 2.15302i | 0 | −1.28003 | + | 2.49212i | |||||||
323.18 | 1.05516 | + | 1.05516i | 0 | 0.226725i | −2.07957 | + | 0.821810i | 0 | 0.707107 | − | 0.707107i | 1.87109 | − | 1.87109i | 0 | −3.06142 | − | 1.32714i | ||||||||
323.19 | 1.20546 | + | 1.20546i | 0 | 0.906269i | −2.12218 | − | 0.704515i | 0 | −0.707107 | + | 0.707107i | 1.31845 | − | 1.31845i | 0 | −1.70894 | − | 3.40747i | ||||||||
323.20 | 1.36262 | + | 1.36262i | 0 | 1.71349i | 1.41633 | + | 1.73032i | 0 | 0.707107 | − | 0.707107i | 0.390409 | − | 0.390409i | 0 | −0.427853 | + | 4.28770i | ||||||||
See all 48 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
5.c | odd | 4 | 1 | inner |
15.e | even | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 945.2.m.a | ✓ | 48 |
3.b | odd | 2 | 1 | inner | 945.2.m.a | ✓ | 48 |
5.c | odd | 4 | 1 | inner | 945.2.m.a | ✓ | 48 |
15.e | even | 4 | 1 | inner | 945.2.m.a | ✓ | 48 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
945.2.m.a | ✓ | 48 | 1.a | even | 1 | 1 | trivial |
945.2.m.a | ✓ | 48 | 3.b | odd | 2 | 1 | inner |
945.2.m.a | ✓ | 48 | 5.c | odd | 4 | 1 | inner |
945.2.m.a | ✓ | 48 | 15.e | even | 4 | 1 | inner |