Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [136,3,Mod(41,136)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(136, base_ring=CyclotomicField(16))
chi = DirichletCharacter(H, H._module([0, 0, 11]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("136.41");
S:= CuspForms(chi, 3);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 136 = 2^{3} \cdot 17 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 136.t (of order \(16\), degree \(8\), 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: | \(3.70573159530\) |
Analytic rank: | \(0\) |
Dimension: | \(40\) |
Relative dimension: | \(5\) over \(\Q(\zeta_{16})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{16}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
41.1 | 0 | −3.07751 | + | 4.60583i | 0 | 1.22413 | + | 6.15412i | 0 | 0.478359 | − | 2.40487i | 0 | −8.29839 | − | 20.0341i | 0 | ||||||||||
41.2 | 0 | −1.26339 | + | 1.89080i | 0 | −0.944701 | − | 4.74933i | 0 | −2.41180 | + | 12.1249i | 0 | 1.46519 | + | 3.53728i | 0 | ||||||||||
41.3 | 0 | −0.708293 | + | 1.06004i | 0 | −0.482121 | − | 2.42379i | 0 | 2.22606 | − | 11.1911i | 0 | 2.82215 | + | 6.81328i | 0 | ||||||||||
41.4 | 0 | 1.26753 | − | 1.89699i | 0 | 1.74040 | + | 8.74956i | 0 | −0.705386 | + | 3.54621i | 0 | 1.45221 | + | 3.50595i | 0 | ||||||||||
41.5 | 0 | 2.38532 | − | 3.56988i | 0 | −0.996509 | − | 5.00979i | 0 | −0.0460348 | + | 0.231433i | 0 | −3.61016 | − | 8.71570i | 0 | ||||||||||
57.1 | 0 | −3.88441 | + | 0.772657i | 0 | 2.37425 | − | 3.55331i | 0 | 2.98990 | + | 4.47471i | 0 | 6.17671 | − | 2.55848i | 0 | ||||||||||
57.2 | 0 | −1.73192 | + | 0.344500i | 0 | −0.638729 | + | 0.955926i | 0 | −5.52260 | − | 8.26516i | 0 | −5.43405 | + | 2.25086i | 0 | ||||||||||
57.3 | 0 | 1.63930 | − | 0.326077i | 0 | −4.61793 | + | 6.91122i | 0 | 2.96901 | + | 4.44344i | 0 | −5.73394 | + | 2.37508i | 0 | ||||||||||
57.4 | 0 | 1.67722 | − | 0.333620i | 0 | 3.22552 | − | 4.82733i | 0 | 5.51299 | + | 8.25077i | 0 | −5.61314 | + | 2.32504i | 0 | ||||||||||
57.5 | 0 | 5.47199 | − | 1.08845i | 0 | 0.963457 | − | 1.44192i | 0 | −5.64274 | − | 8.44495i | 0 | 20.4430 | − | 8.46777i | 0 | ||||||||||
65.1 | 0 | −1.10801 | + | 5.57033i | 0 | −3.32830 | + | 2.22390i | 0 | −0.387063 | − | 0.258627i | 0 | −21.4860 | − | 8.89979i | 0 | ||||||||||
65.2 | 0 | −0.262425 | + | 1.31930i | 0 | 6.57938 | − | 4.39620i | 0 | −6.01468 | − | 4.01888i | 0 | 6.64323 | + | 2.75172i | 0 | ||||||||||
65.3 | 0 | −0.0253220 | + | 0.127302i | 0 | −2.11962 | + | 1.41628i | 0 | 5.83745 | + | 3.90046i | 0 | 8.29935 | + | 3.43770i | 0 | ||||||||||
65.4 | 0 | 0.657237 | − | 3.30415i | 0 | −5.77013 | + | 3.85548i | 0 | −10.9767 | − | 7.33440i | 0 | −2.17054 | − | 0.899066i | 0 | ||||||||||
65.5 | 0 | 0.980550 | − | 4.92956i | 0 | 3.33211 | − | 2.22644i | 0 | 9.23443 | + | 6.17025i | 0 | −15.0241 | − | 6.22320i | 0 | ||||||||||
73.1 | 0 | −3.07751 | − | 4.60583i | 0 | 1.22413 | − | 6.15412i | 0 | 0.478359 | + | 2.40487i | 0 | −8.29839 | + | 20.0341i | 0 | ||||||||||
73.2 | 0 | −1.26339 | − | 1.89080i | 0 | −0.944701 | + | 4.74933i | 0 | −2.41180 | − | 12.1249i | 0 | 1.46519 | − | 3.53728i | 0 | ||||||||||
73.3 | 0 | −0.708293 | − | 1.06004i | 0 | −0.482121 | + | 2.42379i | 0 | 2.22606 | + | 11.1911i | 0 | 2.82215 | − | 6.81328i | 0 | ||||||||||
73.4 | 0 | 1.26753 | + | 1.89699i | 0 | 1.74040 | − | 8.74956i | 0 | −0.705386 | − | 3.54621i | 0 | 1.45221 | − | 3.50595i | 0 | ||||||||||
73.5 | 0 | 2.38532 | + | 3.56988i | 0 | −0.996509 | + | 5.00979i | 0 | −0.0460348 | − | 0.231433i | 0 | −3.61016 | + | 8.71570i | 0 | ||||||||||
See all 40 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
17.e | odd | 16 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 136.3.t.b | ✓ | 40 |
4.b | odd | 2 | 1 | 272.3.bh.g | 40 | ||
17.e | odd | 16 | 1 | inner | 136.3.t.b | ✓ | 40 |
68.i | even | 16 | 1 | 272.3.bh.g | 40 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
136.3.t.b | ✓ | 40 | 1.a | even | 1 | 1 | trivial |
136.3.t.b | ✓ | 40 | 17.e | odd | 16 | 1 | inner |
272.3.bh.g | 40 | 4.b | odd | 2 | 1 | ||
272.3.bh.g | 40 | 68.i | even | 16 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{40} - 8 T_{3}^{39} + 40 T_{3}^{38} - 272 T_{3}^{37} + 1542 T_{3}^{36} - 6216 T_{3}^{35} + 35492 T_{3}^{34} - 188488 T_{3}^{33} + 750578 T_{3}^{32} - 3515808 T_{3}^{31} + 12615864 T_{3}^{30} + \cdots + 558577609736192 \)
acting on \(S_{3}^{\mathrm{new}}(136, [\chi])\).