Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [165,8,Mod(1,165)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(165, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("165.1");
S:= CuspForms(chi, 8);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 165 = 3 \cdot 5 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 165.a (trivial) |
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: | yes |
| Analytic conductor: | \(51.5435292049\) |
| Analytic rank: | \(1\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{2}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - 2 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | yes |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-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 \(\beta = 2\sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−3.65685 | 27.0000 | −114.627 | 125.000 | −98.7351 | −1185.94 | 887.253 | 729.000 | −457.107 | ||||||||||||||||||||||||
| 1.2 | 7.65685 | 27.0000 | −69.3726 | 125.000 | 206.735 | 233.935 | −1511.25 | 729.000 | 957.107 | |||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(5\) | \(-1\) |
| \(11\) | \(-1\) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 165.8.a.b | ✓ | 2 |
| 3.b | odd | 2 | 1 | 495.8.a.b | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 165.8.a.b | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
| 495.8.a.b | 2 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} - 4T_{2} - 28 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(165))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} - 4T - 28 \)
|
| $3$ |
\( (T - 27)^{2} \)
|
| $5$ |
\( (T - 125)^{2} \)
|
| $7$ |
\( T^{2} + 952T - 277432 \)
|
| $11$ |
\( (T - 1331)^{2} \)
|
| $13$ |
\( T^{2} + 1044 T - 74098724 \)
|
| $17$ |
\( T^{2} + 10964 T - 40277476 \)
|
| $19$ |
\( T^{2} + 8144 T + 15450176 \)
|
| $23$ |
\( T^{2} + 16304 T - 236124896 \)
|
| $29$ |
\( T^{2} + \cdots + 5555001284 \)
|
| $31$ |
\( T^{2} + \cdots - 3336802272 \)
|
| $37$ |
\( T^{2} + \cdots - 10944833756 \)
|
| $41$ |
\( T^{2} + \cdots + 41852544836 \)
|
| $43$ |
\( T^{2} + \cdots - 185313529928 \)
|
| $47$ |
\( T^{2} + \cdots + 154633278496 \)
|
| $53$ |
\( T^{2} + \cdots - 301887067324 \)
|
| $59$ |
\( T^{2} + \cdots + 192826164976 \)
|
| $61$ |
\( T^{2} + \cdots + 347552012836 \)
|
| $67$ |
\( T^{2} + \cdots - 6751556100752 \)
|
| $71$ |
\( T^{2} + \cdots + 460543680032 \)
|
| $73$ |
\( T^{2} + \cdots - 467627279972 \)
|
| $79$ |
\( T^{2} + \cdots - 18383201340912 \)
|
| $83$ |
\( T^{2} + \cdots - 7962488696 \)
|
| $89$ |
\( T^{2} + \cdots - 9701814770812 \)
|
| $97$ |
\( T^{2} + \cdots + 161706093888484 \)
|
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