Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [684,2,Mod(113,684)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(684, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 5, 3]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("684.113");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 684 = 2^{2} \cdot 3^{2} \cdot 19 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 684.bd (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: | \(5.46176749826\) |
Analytic rank: | \(0\) |
Dimension: | \(40\) |
Relative dimension: | \(20\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
113.1 | 0 | −1.72128 | − | 0.192885i | 0 | 2.63936 | + | 1.52384i | 0 | 0.965020 | + | 1.67146i | 0 | 2.92559 | + | 0.664016i | 0 | ||||||||||
113.2 | 0 | −1.68427 | + | 0.404004i | 0 | −2.56778 | − | 1.48251i | 0 | 2.26083 | + | 3.91587i | 0 | 2.67356 | − | 1.36091i | 0 | ||||||||||
113.3 | 0 | −1.46881 | + | 0.917931i | 0 | 0.592959 | + | 0.342345i | 0 | −0.568624 | − | 0.984886i | 0 | 1.31480 | − | 2.69653i | 0 | ||||||||||
113.4 | 0 | −1.46081 | + | 0.930611i | 0 | −2.45164 | − | 1.41545i | 0 | −1.38460 | − | 2.39819i | 0 | 1.26792 | − | 2.71889i | 0 | ||||||||||
113.5 | 0 | −1.28485 | − | 1.16154i | 0 | −0.480485 | − | 0.277408i | 0 | 0.467121 | + | 0.809078i | 0 | 0.301655 | + | 2.98480i | 0 | ||||||||||
113.6 | 0 | −1.26241 | − | 1.18589i | 0 | −3.02392 | − | 1.74586i | 0 | −1.08684 | − | 1.88246i | 0 | 0.187345 | + | 2.99414i | 0 | ||||||||||
113.7 | 0 | −0.912030 | − | 1.47248i | 0 | 2.57094 | + | 1.48433i | 0 | −2.04328 | − | 3.53907i | 0 | −1.33640 | + | 2.68589i | 0 | ||||||||||
113.8 | 0 | −0.653696 | + | 1.60396i | 0 | 3.00821 | + | 1.73679i | 0 | 0.760917 | + | 1.31795i | 0 | −2.14536 | − | 2.09700i | 0 | ||||||||||
113.9 | 0 | −0.333490 | − | 1.69964i | 0 | −0.106486 | − | 0.0614798i | 0 | 1.69340 | + | 2.93305i | 0 | −2.77757 | + | 1.13363i | 0 | ||||||||||
113.10 | 0 | −0.210302 | + | 1.71924i | 0 | −0.181163 | − | 0.104594i | 0 | −1.56394 | − | 2.70883i | 0 | −2.91155 | − | 0.723118i | 0 | ||||||||||
113.11 | 0 | 0.210302 | − | 1.71924i | 0 | −0.181163 | − | 0.104594i | 0 | −1.56394 | − | 2.70883i | 0 | −2.91155 | − | 0.723118i | 0 | ||||||||||
113.12 | 0 | 0.333490 | + | 1.69964i | 0 | −0.106486 | − | 0.0614798i | 0 | 1.69340 | + | 2.93305i | 0 | −2.77757 | + | 1.13363i | 0 | ||||||||||
113.13 | 0 | 0.653696 | − | 1.60396i | 0 | 3.00821 | + | 1.73679i | 0 | 0.760917 | + | 1.31795i | 0 | −2.14536 | − | 2.09700i | 0 | ||||||||||
113.14 | 0 | 0.912030 | + | 1.47248i | 0 | 2.57094 | + | 1.48433i | 0 | −2.04328 | − | 3.53907i | 0 | −1.33640 | + | 2.68589i | 0 | ||||||||||
113.15 | 0 | 1.26241 | + | 1.18589i | 0 | −3.02392 | − | 1.74586i | 0 | −1.08684 | − | 1.88246i | 0 | 0.187345 | + | 2.99414i | 0 | ||||||||||
113.16 | 0 | 1.28485 | + | 1.16154i | 0 | −0.480485 | − | 0.277408i | 0 | 0.467121 | + | 0.809078i | 0 | 0.301655 | + | 2.98480i | 0 | ||||||||||
113.17 | 0 | 1.46081 | − | 0.930611i | 0 | −2.45164 | − | 1.41545i | 0 | −1.38460 | − | 2.39819i | 0 | 1.26792 | − | 2.71889i | 0 | ||||||||||
113.18 | 0 | 1.46881 | − | 0.917931i | 0 | 0.592959 | + | 0.342345i | 0 | −0.568624 | − | 0.984886i | 0 | 1.31480 | − | 2.69653i | 0 | ||||||||||
113.19 | 0 | 1.68427 | − | 0.404004i | 0 | −2.56778 | − | 1.48251i | 0 | 2.26083 | + | 3.91587i | 0 | 2.67356 | − | 1.36091i | 0 | ||||||||||
113.20 | 0 | 1.72128 | + | 0.192885i | 0 | 2.63936 | + | 1.52384i | 0 | 0.965020 | + | 1.67146i | 0 | 2.92559 | + | 0.664016i | 0 | ||||||||||
See all 40 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
9.d | odd | 6 | 1 | inner |
19.b | odd | 2 | 1 | inner |
171.l | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 684.2.bd.a | ✓ | 40 |
3.b | odd | 2 | 1 | 2052.2.bd.a | 40 | ||
9.c | even | 3 | 1 | 2052.2.bd.a | 40 | ||
9.c | even | 3 | 1 | 6156.2.d.f | 40 | ||
9.d | odd | 6 | 1 | inner | 684.2.bd.a | ✓ | 40 |
9.d | odd | 6 | 1 | 6156.2.d.f | 40 | ||
19.b | odd | 2 | 1 | inner | 684.2.bd.a | ✓ | 40 |
57.d | even | 2 | 1 | 2052.2.bd.a | 40 | ||
171.l | even | 6 | 1 | inner | 684.2.bd.a | ✓ | 40 |
171.l | even | 6 | 1 | 6156.2.d.f | 40 | ||
171.o | odd | 6 | 1 | 2052.2.bd.a | 40 | ||
171.o | odd | 6 | 1 | 6156.2.d.f | 40 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
684.2.bd.a | ✓ | 40 | 1.a | even | 1 | 1 | trivial |
684.2.bd.a | ✓ | 40 | 9.d | odd | 6 | 1 | inner |
684.2.bd.a | ✓ | 40 | 19.b | odd | 2 | 1 | inner |
684.2.bd.a | ✓ | 40 | 171.l | even | 6 | 1 | inner |
2052.2.bd.a | 40 | 3.b | odd | 2 | 1 | ||
2052.2.bd.a | 40 | 9.c | even | 3 | 1 | ||
2052.2.bd.a | 40 | 57.d | even | 2 | 1 | ||
2052.2.bd.a | 40 | 171.o | odd | 6 | 1 | ||
6156.2.d.f | 40 | 9.c | even | 3 | 1 | ||
6156.2.d.f | 40 | 9.d | odd | 6 | 1 | ||
6156.2.d.f | 40 | 171.l | even | 6 | 1 | ||
6156.2.d.f | 40 | 171.o | odd | 6 | 1 |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(684, [\chi])\).