Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [684,3,Mod(343,684)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(684, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0, 0]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("684.343");
S:= CuspForms(chi, 3);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 684 = 2^{2} \cdot 3^{2} \cdot 19 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 684.g (of order \(2\), degree \(1\), 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: | \(18.6376500822\) |
Analytic rank: | \(0\) |
Dimension: | \(36\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
343.1 | −1.98942 | − | 0.205433i | 0 | 3.91559 | + | 0.817384i | −0.0956487 | 0 | 9.56723i | −7.62185 | − | 2.43051i | 0 | 0.190286 | + | 0.0196494i | ||||||||||
343.2 | −1.98942 | + | 0.205433i | 0 | 3.91559 | − | 0.817384i | −0.0956487 | 0 | − | 9.56723i | −7.62185 | + | 2.43051i | 0 | 0.190286 | − | 0.0196494i | |||||||||
343.3 | −1.83969 | − | 0.784567i | 0 | 2.76891 | + | 2.88672i | −6.85656 | 0 | − | 3.48151i | −2.82911 | − | 7.48306i | 0 | 12.6139 | + | 5.37943i | |||||||||
343.4 | −1.83969 | + | 0.784567i | 0 | 2.76891 | − | 2.88672i | −6.85656 | 0 | 3.48151i | −2.82911 | + | 7.48306i | 0 | 12.6139 | − | 5.37943i | ||||||||||
343.5 | −1.58671 | − | 1.21751i | 0 | 1.03532 | + | 3.86369i | 1.46825 | 0 | 6.38949i | 3.06136 | − | 7.39108i | 0 | −2.32969 | − | 1.78762i | ||||||||||
343.6 | −1.58671 | + | 1.21751i | 0 | 1.03532 | − | 3.86369i | 1.46825 | 0 | − | 6.38949i | 3.06136 | + | 7.39108i | 0 | −2.32969 | + | 1.78762i | |||||||||
343.7 | −1.55174 | − | 1.26178i | 0 | 0.815807 | + | 3.91592i | 2.61138 | 0 | − | 10.9942i | 3.67512 | − | 7.10587i | 0 | −4.05218 | − | 3.29499i | |||||||||
343.8 | −1.55174 | + | 1.26178i | 0 | 0.815807 | − | 3.91592i | 2.61138 | 0 | 10.9942i | 3.67512 | + | 7.10587i | 0 | −4.05218 | + | 3.29499i | ||||||||||
343.9 | −1.42379 | − | 1.40457i | 0 | 0.0543607 | + | 3.99963i | 5.95758 | 0 | − | 3.23592i | 5.54037 | − | 5.77099i | 0 | −8.48235 | − | 8.36785i | |||||||||
343.10 | −1.42379 | + | 1.40457i | 0 | 0.0543607 | − | 3.99963i | 5.95758 | 0 | 3.23592i | 5.54037 | + | 5.77099i | 0 | −8.48235 | + | 8.36785i | ||||||||||
343.11 | −1.07034 | − | 1.68949i | 0 | −1.70873 | + | 3.61666i | −6.66582 | 0 | 12.1510i | 7.93923 | − | 0.984190i | 0 | 7.13471 | + | 11.2618i | ||||||||||
343.12 | −1.07034 | + | 1.68949i | 0 | −1.70873 | − | 3.61666i | −6.66582 | 0 | − | 12.1510i | 7.93923 | + | 0.984190i | 0 | 7.13471 | − | 11.2618i | |||||||||
343.13 | −0.690218 | − | 1.87713i | 0 | −3.04720 | + | 2.59125i | 8.91109 | 0 | 7.74115i | 6.96733 | + | 3.93145i | 0 | −6.15059 | − | 16.7272i | ||||||||||
343.14 | −0.690218 | + | 1.87713i | 0 | −3.04720 | − | 2.59125i | 8.91109 | 0 | − | 7.74115i | 6.96733 | − | 3.93145i | 0 | −6.15059 | + | 16.7272i | |||||||||
343.15 | −0.671560 | − | 1.88388i | 0 | −3.09801 | + | 2.53028i | −4.54598 | 0 | − | 5.76325i | 6.84725 | + | 4.13706i | 0 | 3.05290 | + | 8.56408i | |||||||||
343.16 | −0.671560 | + | 1.88388i | 0 | −3.09801 | − | 2.53028i | −4.54598 | 0 | 5.76325i | 6.84725 | − | 4.13706i | 0 | 3.05290 | − | 8.56408i | ||||||||||
343.17 | −0.363289 | − | 1.96673i | 0 | −3.73604 | + | 1.42898i | −0.0632854 | 0 | − | 2.71864i | 4.16769 | + | 6.82864i | 0 | 0.0229909 | + | 0.124465i | |||||||||
343.18 | −0.363289 | + | 1.96673i | 0 | −3.73604 | − | 1.42898i | −0.0632854 | 0 | 2.71864i | 4.16769 | − | 6.82864i | 0 | 0.0229909 | − | 0.124465i | ||||||||||
343.19 | 0.363289 | − | 1.96673i | 0 | −3.73604 | − | 1.42898i | 0.0632854 | 0 | 2.71864i | −4.16769 | + | 6.82864i | 0 | 0.0229909 | − | 0.124465i | ||||||||||
343.20 | 0.363289 | + | 1.96673i | 0 | −3.73604 | + | 1.42898i | 0.0632854 | 0 | − | 2.71864i | −4.16769 | − | 6.82864i | 0 | 0.0229909 | + | 0.124465i | |||||||||
See all 36 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
4.b | odd | 2 | 1 | inner |
12.b | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 684.3.g.d | ✓ | 36 |
3.b | odd | 2 | 1 | inner | 684.3.g.d | ✓ | 36 |
4.b | odd | 2 | 1 | inner | 684.3.g.d | ✓ | 36 |
12.b | even | 2 | 1 | inner | 684.3.g.d | ✓ | 36 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
684.3.g.d | ✓ | 36 | 1.a | even | 1 | 1 | trivial |
684.3.g.d | ✓ | 36 | 3.b | odd | 2 | 1 | inner |
684.3.g.d | ✓ | 36 | 4.b | odd | 2 | 1 | inner |
684.3.g.d | ✓ | 36 | 12.b | even | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{18} - 236 T_{5}^{16} + 21734 T_{5}^{14} - 996540 T_{5}^{12} + 23802465 T_{5}^{10} - 279141136 T_{5}^{8} + \cdots - 65536 \) acting on \(S_{3}^{\mathrm{new}}(684, [\chi])\).