Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [112,10,Mod(31,112)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(112, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([3, 0, 1]))
N = Newforms(chi, 10, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("112.31");
S:= CuspForms(chi, 10);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 112 = 2^{4} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 10 \) |
Character orbit: | \([\chi]\) | \(=\) | 112.p (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: | \(57.6840136504\) |
Analytic rank: | \(0\) |
Dimension: | \(24\) |
Relative dimension: | \(12\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
31.1 | 0 | −128.906 | + | 223.272i | 0 | 125.209 | − | 72.2894i | 0 | 5420.95 | + | 3311.64i | 0 | −23392.1 | − | 40516.3i | 0 | ||||||||||
31.2 | 0 | −104.726 | + | 181.391i | 0 | 1126.36 | − | 650.304i | 0 | −5633.01 | − | 2936.47i | 0 | −12093.7 | − | 20946.9i | 0 | ||||||||||
31.3 | 0 | −84.0026 | + | 145.497i | 0 | −2357.52 | + | 1361.12i | 0 | −2187.92 | − | 5963.77i | 0 | −4271.38 | − | 7398.25i | 0 | ||||||||||
31.4 | 0 | −48.7996 | + | 84.5234i | 0 | −62.4476 | + | 36.0541i | 0 | 6259.38 | + | 1083.41i | 0 | 5078.69 | + | 8796.55i | 0 | ||||||||||
31.5 | 0 | −41.2793 | + | 71.4979i | 0 | −557.845 | + | 322.072i | 0 | −3078.62 | + | 5556.59i | 0 | 6433.53 | + | 11143.2i | 0 | ||||||||||
31.6 | 0 | −29.1853 | + | 50.5504i | 0 | 2152.25 | − | 1242.60i | 0 | 2216.90 | − | 5953.06i | 0 | 8137.94 | + | 14095.3i | 0 | ||||||||||
31.7 | 0 | 29.1853 | − | 50.5504i | 0 | 2152.25 | − | 1242.60i | 0 | −2216.90 | + | 5953.06i | 0 | 8137.94 | + | 14095.3i | 0 | ||||||||||
31.8 | 0 | 41.2793 | − | 71.4979i | 0 | −557.845 | + | 322.072i | 0 | 3078.62 | − | 5556.59i | 0 | 6433.53 | + | 11143.2i | 0 | ||||||||||
31.9 | 0 | 48.7996 | − | 84.5234i | 0 | −62.4476 | + | 36.0541i | 0 | −6259.38 | − | 1083.41i | 0 | 5078.69 | + | 8796.55i | 0 | ||||||||||
31.10 | 0 | 84.0026 | − | 145.497i | 0 | −2357.52 | + | 1361.12i | 0 | 2187.92 | + | 5963.77i | 0 | −4271.38 | − | 7398.25i | 0 | ||||||||||
31.11 | 0 | 104.726 | − | 181.391i | 0 | 1126.36 | − | 650.304i | 0 | 5633.01 | + | 2936.47i | 0 | −12093.7 | − | 20946.9i | 0 | ||||||||||
31.12 | 0 | 128.906 | − | 223.272i | 0 | 125.209 | − | 72.2894i | 0 | −5420.95 | − | 3311.64i | 0 | −23392.1 | − | 40516.3i | 0 | ||||||||||
47.1 | 0 | −128.906 | − | 223.272i | 0 | 125.209 | + | 72.2894i | 0 | 5420.95 | − | 3311.64i | 0 | −23392.1 | + | 40516.3i | 0 | ||||||||||
47.2 | 0 | −104.726 | − | 181.391i | 0 | 1126.36 | + | 650.304i | 0 | −5633.01 | + | 2936.47i | 0 | −12093.7 | + | 20946.9i | 0 | ||||||||||
47.3 | 0 | −84.0026 | − | 145.497i | 0 | −2357.52 | − | 1361.12i | 0 | −2187.92 | + | 5963.77i | 0 | −4271.38 | + | 7398.25i | 0 | ||||||||||
47.4 | 0 | −48.7996 | − | 84.5234i | 0 | −62.4476 | − | 36.0541i | 0 | 6259.38 | − | 1083.41i | 0 | 5078.69 | − | 8796.55i | 0 | ||||||||||
47.5 | 0 | −41.2793 | − | 71.4979i | 0 | −557.845 | − | 322.072i | 0 | −3078.62 | − | 5556.59i | 0 | 6433.53 | − | 11143.2i | 0 | ||||||||||
47.6 | 0 | −29.1853 | − | 50.5504i | 0 | 2152.25 | + | 1242.60i | 0 | 2216.90 | + | 5953.06i | 0 | 8137.94 | − | 14095.3i | 0 | ||||||||||
47.7 | 0 | 29.1853 | + | 50.5504i | 0 | 2152.25 | + | 1242.60i | 0 | −2216.90 | − | 5953.06i | 0 | 8137.94 | − | 14095.3i | 0 | ||||||||||
47.8 | 0 | 41.2793 | + | 71.4979i | 0 | −557.845 | − | 322.072i | 0 | 3078.62 | + | 5556.59i | 0 | 6433.53 | − | 11143.2i | 0 | ||||||||||
See all 24 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 | 112.10.p.b | ✓ | 24 |
4.b | odd | 2 | 1 | inner | 112.10.p.b | ✓ | 24 |
7.d | odd | 6 | 1 | inner | 112.10.p.b | ✓ | 24 |
28.f | even | 6 | 1 | inner | 112.10.p.b | ✓ | 24 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
112.10.p.b | ✓ | 24 | 1.a | even | 1 | 1 | trivial |
112.10.p.b | ✓ | 24 | 4.b | odd | 2 | 1 | inner |
112.10.p.b | ✓ | 24 | 7.d | odd | 6 | 1 | inner |
112.10.p.b | ✓ | 24 | 28.f | even | 6 | 1 | inner |