Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [165,2,Mod(23,165)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(165, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([2, 3, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("165.23");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 165 = 3 \cdot 5 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 165.k (of order \(4\), 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: | \(1.31753163335\) |
| Analytic rank: | \(1\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(i)\) |
| 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: | \(\Z[a_1, \ldots, a_{4}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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 \(\left(\mathbb{Z}/165\mathbb{Z}\right)^\times\).
| \(n\) | \(46\) | \(56\) | \(67\) |
| \(\chi(n)\) | \(1\) | \(-1\) | \(\zeta_{8}^{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 23.1 |
|
−1.70711 | − | 1.70711i | −1.00000 | − | 1.41421i | 3.82843i | −2.00000 | − | 1.00000i | −0.707107 | + | 4.12132i | −0.585786 | + | 0.585786i | 3.12132 | − | 3.12132i | −1.00000 | + | 2.82843i | 1.70711 | + | 5.12132i | ||||||||||||||
| 23.2 | −0.292893 | − | 0.292893i | −1.00000 | + | 1.41421i | − | 1.82843i | −2.00000 | − | 1.00000i | 0.707107 | − | 0.121320i | −3.41421 | + | 3.41421i | −1.12132 | + | 1.12132i | −1.00000 | − | 2.82843i | 0.292893 | + | 0.878680i | ||||||||||||||
| 122.1 | −1.70711 | + | 1.70711i | −1.00000 | + | 1.41421i | − | 3.82843i | −2.00000 | + | 1.00000i | −0.707107 | − | 4.12132i | −0.585786 | − | 0.585786i | 3.12132 | + | 3.12132i | −1.00000 | − | 2.82843i | 1.70711 | − | 5.12132i | ||||||||||||||
| 122.2 | −0.292893 | + | 0.292893i | −1.00000 | − | 1.41421i | 1.82843i | −2.00000 | + | 1.00000i | 0.707107 | + | 0.121320i | −3.41421 | − | 3.41421i | −1.12132 | − | 1.12132i | −1.00000 | + | 2.82843i | 0.292893 | − | 0.878680i | |||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 15.e | even | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 165.2.k.a | ✓ | 4 |
| 3.b | odd | 2 | 1 | 165.2.k.b | yes | 4 | |
| 5.b | even | 2 | 1 | 825.2.k.f | 4 | ||
| 5.c | odd | 4 | 1 | 165.2.k.b | yes | 4 | |
| 5.c | odd | 4 | 1 | 825.2.k.c | 4 | ||
| 15.d | odd | 2 | 1 | 825.2.k.c | 4 | ||
| 15.e | even | 4 | 1 | inner | 165.2.k.a | ✓ | 4 |
| 15.e | even | 4 | 1 | 825.2.k.f | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 165.2.k.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 165.2.k.a | ✓ | 4 | 15.e | even | 4 | 1 | inner |
| 165.2.k.b | yes | 4 | 3.b | odd | 2 | 1 | |
| 165.2.k.b | yes | 4 | 5.c | odd | 4 | 1 | |
| 825.2.k.c | 4 | 5.c | odd | 4 | 1 | ||
| 825.2.k.c | 4 | 15.d | odd | 2 | 1 | ||
| 825.2.k.f | 4 | 5.b | even | 2 | 1 | ||
| 825.2.k.f | 4 | 15.e | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} + 4T_{2}^{3} + 8T_{2}^{2} + 4T_{2} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(165, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + 4 T^{3} + \cdots + 1 \)
|
| $3$ |
\( (T^{2} + 2 T + 3)^{2} \)
|
| $5$ |
\( (T^{2} + 4 T + 5)^{2} \)
|
| $7$ |
\( T^{4} + 8 T^{3} + \cdots + 16 \)
|
| $11$ |
\( (T^{2} + 1)^{2} \)
|
| $13$ |
\( (T^{2} + 4 T + 8)^{2} \)
|
| $17$ |
\( T^{4} + 256 \)
|
| $19$ |
\( (T^{2} + 8)^{2} \)
|
| $23$ |
\( T^{4} + 4 T^{3} + \cdots + 1156 \)
|
| $29$ |
\( (T^{2} + 4 T - 4)^{2} \)
|
| $31$ |
\( (T^{2} + 12 T + 28)^{2} \)
|
| $37$ |
\( T^{4} + 12 T^{3} + \cdots + 4 \)
|
| $41$ |
\( T^{4} + 72T^{2} + 784 \)
|
| $43$ |
\( T^{4} + 16 T^{3} + \cdots + 16 \)
|
| $47$ |
\( T^{4} + 12 T^{3} + \cdots + 324 \)
|
| $53$ |
\( (T^{2} - 2 T + 2)^{2} \)
|
| $59$ |
\( (T + 4)^{4} \)
|
| $61$ |
\( (T^{2} - 4 T - 68)^{2} \)
|
| $67$ |
\( T^{4} + 20 T^{3} + \cdots + 2116 \)
|
| $71$ |
\( T^{4} + 216T^{2} + 1296 \)
|
| $73$ |
\( T^{4} \)
|
| $79$ |
\( T^{4} + 144T^{2} + 3136 \)
|
| $83$ |
\( T^{4} + 8 T^{3} + \cdots + 8464 \)
|
| $89$ |
\( (T^{2} + 8 T - 16)^{2} \)
|
| $97$ |
\( T^{4} - 20 T^{3} + \cdots + 196 \)
|
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