Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [663,2,Mod(8,663)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(663, base_ring=CyclotomicField(8))
chi = DirichletCharacter(H, H._module([4, 2, 5]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("663.8");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 663 = 3 \cdot 13 \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 663.bd (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: | \(5.29408165401\) |
| 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: | \(\Z[a_1, a_2, a_3]\) |
| 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 \(\left(\mathbb{Z}/663\mathbb{Z}\right)^\times\).
| \(n\) | \(443\) | \(547\) | \(613\) |
| \(\chi(n)\) | \(-1\) | \(\zeta_{8}\) | \(\zeta_{8}^{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 8.1 |
|
− | 2.41421i | −1.70711 | − | 0.292893i | −3.82843 | 1.70711 | − | 0.707107i | −0.707107 | + | 4.12132i | −1.70711 | + | 4.12132i | 4.41421i | 2.82843 | + | 1.00000i | −1.70711 | − | 4.12132i | |||||||||||||||||
| 83.1 | 2.41421i | −1.70711 | + | 0.292893i | −3.82843 | 1.70711 | + | 0.707107i | −0.707107 | − | 4.12132i | −1.70711 | − | 4.12132i | − | 4.41421i | 2.82843 | − | 1.00000i | −1.70711 | + | 4.12132i | ||||||||||||||||||
| 281.1 | 0.414214i | −0.292893 | − | 1.70711i | 1.82843 | 0.292893 | + | 0.707107i | 0.707107 | − | 0.121320i | −0.292893 | − | 0.121320i | 1.58579i | −2.82843 | + | 1.00000i | −0.292893 | + | 0.121320i | |||||||||||||||||||
| 512.1 | − | 0.414214i | −0.292893 | + | 1.70711i | 1.82843 | 0.292893 | − | 0.707107i | 0.707107 | + | 0.121320i | −0.292893 | + | 0.121320i | − | 1.58579i | −2.82843 | − | 1.00000i | −0.292893 | − | 0.121320i | |||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 663.bd | even | 8 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 663.2.bd.a | ✓ | 4 |
| 3.b | odd | 2 | 1 | 663.2.bd.b | yes | 4 | |
| 13.d | odd | 4 | 1 | 663.2.bi.a | yes | 4 | |
| 17.d | even | 8 | 1 | 663.2.bi.b | yes | 4 | |
| 39.f | even | 4 | 1 | 663.2.bi.b | yes | 4 | |
| 51.g | odd | 8 | 1 | 663.2.bi.a | yes | 4 | |
| 221.r | odd | 8 | 1 | 663.2.bd.b | yes | 4 | |
| 663.bd | even | 8 | 1 | inner | 663.2.bd.a | ✓ | 4 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 663.2.bd.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 663.2.bd.a | ✓ | 4 | 663.bd | even | 8 | 1 | inner |
| 663.2.bd.b | yes | 4 | 3.b | odd | 2 | 1 | |
| 663.2.bd.b | yes | 4 | 221.r | odd | 8 | 1 | |
| 663.2.bi.a | yes | 4 | 13.d | odd | 4 | 1 | |
| 663.2.bi.a | yes | 4 | 51.g | odd | 8 | 1 | |
| 663.2.bi.b | yes | 4 | 17.d | even | 8 | 1 | |
| 663.2.bi.b | yes | 4 | 39.f | even | 4 | 1 | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(663, [\chi])\):
|
\( T_{2}^{4} + 6T_{2}^{2} + 1 \)
|
|
\( T_{5}^{4} - 4T_{5}^{3} + 6T_{5}^{2} - 4T_{5} + 2 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + 6T^{2} + 1 \)
|
| $3$ |
\( T^{4} + 4 T^{3} + \cdots + 9 \)
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| $5$ |
\( T^{4} - 4 T^{3} + \cdots + 2 \)
|
| $7$ |
\( T^{4} + 4 T^{3} + \cdots + 2 \)
|
| $11$ |
\( T^{4} - 4 T^{3} + \cdots + 98 \)
|
| $13$ |
\( (T^{2} - 6 T + 13)^{2} \)
|
| $17$ |
\( (T^{2} - 6 T + 17)^{2} \)
|
| $19$ |
\( (T^{2} - 8)^{2} \)
|
| $23$ |
\( T^{4} - 4 T^{3} + \cdots + 2 \)
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| $29$ |
\( T^{4} + 4 T^{3} + \cdots + 2 \)
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| $31$ |
\( T^{4} - 12 T^{3} + \cdots + 1922 \)
|
| $37$ |
\( T^{4} - 20 T^{3} + \cdots + 3362 \)
|
| $41$ |
\( T^{4} + 4 T^{3} + \cdots + 98 \)
|
| $43$ |
\( T^{4} + 12 T^{3} + \cdots + 196 \)
|
| $47$ |
\( T^{4} + 4 T^{3} + \cdots + 4 \)
|
| $53$ |
\( T^{4} + 4 T^{3} + \cdots + 3844 \)
|
| $59$ |
\( T^{4} + 264 T^{2} + 15376 \)
|
| $61$ |
\( T^{4} + 12 T^{3} + \cdots + 578 \)
|
| $67$ |
\( T^{4} + 4 T^{3} + \cdots + 1156 \)
|
| $71$ |
\( T^{4} - 28 T^{3} + \cdots + 578 \)
|
| $73$ |
\( T^{4} - 36 T^{3} + \cdots + 25538 \)
|
| $79$ |
\( T^{4} - 20 T^{3} + \cdots + 4802 \)
|
| $83$ |
\( T^{4} + 136T^{2} + 16 \)
|
| $89$ |
\( (T^{2} + 2 T + 2)^{2} \)
|
| $97$ |
\( T^{4} + 20 T^{3} + \cdots + 114242 \)
|
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