Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [68,3,Mod(15,68)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(68, base_ring=CyclotomicField(8))
chi = DirichletCharacter(H, H._module([4, 3]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("68.15");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 68 = 2^{2} \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 68.g (of order \(8\), degree \(4\), 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: | \(1.85286579765\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\zeta_{8})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{8}]$ |
$q$-expansion
comment: q-expansion
sage: f.q_expansion() # note that sage often uses an isomorphic number field
gp: mfcoefs(f, 20)
Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{8}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/68\mathbb{Z}\right)^\times\).
| \(n\) | \(35\) | \(37\) |
| \(\chi(n)\) | \(-1\) | \(\zeta_{8}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 15.1 |
|
2.00000 | 1.41421 | − | 0.585786i | 4.00000 | −4.12132 | + | 1.70711i | 2.82843 | − | 1.17157i | 1.41421 | − | 3.41421i | 8.00000 | −4.70711 | + | 4.70711i | −8.24264 | + | 3.41421i | ||||||||||||||||||
| 19.1 | 2.00000 | −1.41421 | − | 3.41421i | 4.00000 | 0.121320 | + | 0.292893i | −2.82843 | − | 6.82843i | −1.41421 | − | 0.585786i | 8.00000 | −3.29289 | + | 3.29289i | 0.242641 | + | 0.585786i | |||||||||||||||||||
| 43.1 | 2.00000 | −1.41421 | + | 3.41421i | 4.00000 | 0.121320 | − | 0.292893i | −2.82843 | + | 6.82843i | −1.41421 | + | 0.585786i | 8.00000 | −3.29289 | − | 3.29289i | 0.242641 | − | 0.585786i | |||||||||||||||||||
| 59.1 | 2.00000 | 1.41421 | + | 0.585786i | 4.00000 | −4.12132 | − | 1.70711i | 2.82843 | + | 1.17157i | 1.41421 | + | 3.41421i | 8.00000 | −4.70711 | − | 4.70711i | −8.24264 | − | 3.41421i | |||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 68.g | odd | 8 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 68.3.g.d | yes | 4 |
| 4.b | odd | 2 | 1 | 68.3.g.a | ✓ | 4 | |
| 17.d | even | 8 | 1 | 68.3.g.a | ✓ | 4 | |
| 68.g | odd | 8 | 1 | inner | 68.3.g.d | yes | 4 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 68.3.g.a | ✓ | 4 | 4.b | odd | 2 | 1 | |
| 68.3.g.a | ✓ | 4 | 17.d | even | 8 | 1 | |
| 68.3.g.d | yes | 4 | 1.a | even | 1 | 1 | trivial |
| 68.3.g.d | yes | 4 | 68.g | odd | 8 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(68, [\chi])\):
|
\( T_{3}^{4} + 8T_{3}^{2} - 32T_{3} + 32 \)
|
|
\( T_{5}^{4} + 8T_{5}^{3} + 18T_{5}^{2} - 4T_{5} + 2 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 2)^{4} \)
|
| $3$ |
\( T^{4} + 8 T^{2} + \cdots + 32 \)
|
| $5$ |
\( T^{4} + 8 T^{3} + \cdots + 2 \)
|
| $7$ |
\( T^{4} + 8 T^{2} + \cdots + 32 \)
|
| $11$ |
\( T^{4} + 24 T^{3} + \cdots + 2592 \)
|
| $13$ |
\( T^{4} + 132T^{2} + 1156 \)
|
| $17$ |
\( (T^{2} + 289)^{2} \)
|
| $19$ |
\( T^{4} + 24 T^{3} + \cdots + 107584 \)
|
| $23$ |
\( T^{4} + 40 T^{3} + \cdots + 161312 \)
|
| $29$ |
\( T^{4} - 4 T^{3} + \cdots + 1839362 \)
|
| $31$ |
\( T^{4} - 56 T^{3} + \cdots + 76832 \)
|
| $37$ |
\( T^{4} + 40 T^{3} + \cdots + 1922 \)
|
| $41$ |
\( T^{4} - 44 T^{3} + \cdots + 1033922 \)
|
| $43$ |
\( T^{4} - 72 T^{3} + \cdots + 61504 \)
|
| $47$ |
\( (T - 60)^{4} \)
|
| $53$ |
\( T^{4} + 92 T^{3} + \cdots + 432964 \)
|
| $59$ |
\( (T^{2} + 52 T + 1352)^{2} \)
|
| $61$ |
\( T^{4} - 104 T^{3} + \cdots + 1659842 \)
|
| $67$ |
\( T^{4} + 25632 T^{2} + 163430656 \)
|
| $71$ |
\( T^{4} + 304 T^{3} + \cdots + 178076192 \)
|
| $73$ |
\( T^{4} + 60 T^{3} + \cdots + 6188162 \)
|
| $79$ |
\( T^{4} + 264 T^{3} + \cdots + 129154592 \)
|
| $83$ |
\( T^{4} + 152 T^{3} + \cdots + 12334144 \)
|
| $89$ |
\( (T^{2} + 14450)^{2} \)
|
| $97$ |
\( T^{4} - 160 T^{3} + \cdots + 19857602 \)
|
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