Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1456,2,Mod(495,1456)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1456, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([3, 0, 5, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1456.495");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 1456 = 2^{4} \cdot 7 \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 1456.cp (of order \(6\), 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: | \(11.6262185343\) |
Analytic rank: | \(0\) |
Dimension: | \(32\) |
Relative dimension: | \(16\) over \(\Q(\zeta_{6})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
495.1 | 0 | −1.68604 | − | 2.92030i | 0 | 3.35997 | + | 1.93988i | 0 | 2.60194 | − | 0.479515i | 0 | −4.18545 | + | 7.24942i | 0 | ||||||||||
495.2 | 0 | −1.36228 | − | 2.35953i | 0 | −2.81406 | − | 1.62470i | 0 | 1.05058 | − | 2.42822i | 0 | −2.21160 | + | 3.83060i | 0 | ||||||||||
495.3 | 0 | −1.14890 | − | 1.98995i | 0 | −1.39339 | − | 0.804472i | 0 | 1.87264 | + | 1.86901i | 0 | −1.13993 | + | 1.97442i | 0 | ||||||||||
495.4 | 0 | −1.01129 | − | 1.75161i | 0 | 0.650003 | + | 0.375279i | 0 | −0.669765 | + | 2.55957i | 0 | −0.545433 | + | 0.944717i | 0 | ||||||||||
495.5 | 0 | −0.696784 | − | 1.20687i | 0 | 1.38995 | + | 0.802488i | 0 | 0.300336 | − | 2.62865i | 0 | 0.528984 | − | 0.916227i | 0 | ||||||||||
495.6 | 0 | −0.669244 | − | 1.15916i | 0 | 3.22760 | + | 1.86346i | 0 | −2.10372 | − | 1.60448i | 0 | 0.604224 | − | 1.04655i | 0 | ||||||||||
495.7 | 0 | −0.120774 | − | 0.209186i | 0 | −2.59019 | − | 1.49545i | 0 | 1.09375 | − | 2.40909i | 0 | 1.47083 | − | 2.54755i | 0 | ||||||||||
495.8 | 0 | −0.103962 | − | 0.180067i | 0 | 1.17011 | + | 0.675562i | 0 | 2.63794 | − | 0.203192i | 0 | 1.47838 | − | 2.56064i | 0 | ||||||||||
495.9 | 0 | 0.103962 | + | 0.180067i | 0 | 1.17011 | + | 0.675562i | 0 | −2.63794 | + | 0.203192i | 0 | 1.47838 | − | 2.56064i | 0 | ||||||||||
495.10 | 0 | 0.120774 | + | 0.209186i | 0 | −2.59019 | − | 1.49545i | 0 | −1.09375 | + | 2.40909i | 0 | 1.47083 | − | 2.54755i | 0 | ||||||||||
495.11 | 0 | 0.669244 | + | 1.15916i | 0 | 3.22760 | + | 1.86346i | 0 | 2.10372 | + | 1.60448i | 0 | 0.604224 | − | 1.04655i | 0 | ||||||||||
495.12 | 0 | 0.696784 | + | 1.20687i | 0 | 1.38995 | + | 0.802488i | 0 | −0.300336 | + | 2.62865i | 0 | 0.528984 | − | 0.916227i | 0 | ||||||||||
495.13 | 0 | 1.01129 | + | 1.75161i | 0 | 0.650003 | + | 0.375279i | 0 | 0.669765 | − | 2.55957i | 0 | −0.545433 | + | 0.944717i | 0 | ||||||||||
495.14 | 0 | 1.14890 | + | 1.98995i | 0 | −1.39339 | − | 0.804472i | 0 | −1.87264 | − | 1.86901i | 0 | −1.13993 | + | 1.97442i | 0 | ||||||||||
495.15 | 0 | 1.36228 | + | 2.35953i | 0 | −2.81406 | − | 1.62470i | 0 | −1.05058 | + | 2.42822i | 0 | −2.21160 | + | 3.83060i | 0 | ||||||||||
495.16 | 0 | 1.68604 | + | 2.92030i | 0 | 3.35997 | + | 1.93988i | 0 | −2.60194 | + | 0.479515i | 0 | −4.18545 | + | 7.24942i | 0 | ||||||||||
703.1 | 0 | −1.68604 | + | 2.92030i | 0 | 3.35997 | − | 1.93988i | 0 | 2.60194 | + | 0.479515i | 0 | −4.18545 | − | 7.24942i | 0 | ||||||||||
703.2 | 0 | −1.36228 | + | 2.35953i | 0 | −2.81406 | + | 1.62470i | 0 | 1.05058 | + | 2.42822i | 0 | −2.21160 | − | 3.83060i | 0 | ||||||||||
703.3 | 0 | −1.14890 | + | 1.98995i | 0 | −1.39339 | + | 0.804472i | 0 | 1.87264 | − | 1.86901i | 0 | −1.13993 | − | 1.97442i | 0 | ||||||||||
703.4 | 0 | −1.01129 | + | 1.75161i | 0 | 0.650003 | − | 0.375279i | 0 | −0.669765 | − | 2.55957i | 0 | −0.545433 | − | 0.944717i | 0 | ||||||||||
See all 32 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
4.b | odd | 2 | 1 | inner |
7.d | odd | 6 | 1 | inner |
28.f | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 1456.2.cp.c | ✓ | 32 |
4.b | odd | 2 | 1 | inner | 1456.2.cp.c | ✓ | 32 |
7.d | odd | 6 | 1 | inner | 1456.2.cp.c | ✓ | 32 |
28.f | even | 6 | 1 | inner | 1456.2.cp.c | ✓ | 32 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
1456.2.cp.c | ✓ | 32 | 1.a | even | 1 | 1 | trivial |
1456.2.cp.c | ✓ | 32 | 4.b | odd | 2 | 1 | inner |
1456.2.cp.c | ✓ | 32 | 7.d | odd | 6 | 1 | inner |
1456.2.cp.c | ✓ | 32 | 28.f | even | 6 | 1 | inner |