Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [164,2,Mod(9,164)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(164, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([0, 3]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("164.9");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 164 = 2^{2} \cdot 41 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 164.f (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: | \(1.30954659315\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Relative dimension: | \(3\) over \(\Q(i)\) |
| Coefficient field: | 6.0.5089536.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 16x^{2} - 24x + 18 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| 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 basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of a root \(\nu\) of
\( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 16x^{2} - 24x + 18 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{5} + 24\nu^{4} - 6\nu^{3} - \nu^{2} + 6\nu + 285 ) / 131 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 6\nu^{5} - 13\nu^{4} + 36\nu^{3} + 6\nu^{2} + 95\nu - 138 ) / 131 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 23\nu^{5} - 28\nu^{4} + 7\nu^{3} + 154\nu^{2} + 386\nu - 267 ) / 393 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -23\nu^{5} + 28\nu^{4} - 7\nu^{3} - 23\nu^{2} - 386\nu + 267 ) / 131 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{5} + 3\beta_{4} \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{5} + 4\beta_{3} + \beta_{2} \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{3} + 6\beta_{2} - \beta _1 - 12 \)
|
| \(\nu^{5}\) | \(=\) |
\( -7\beta_{5} - 3\beta_{4} + 7\beta_{2} - 18\beta _1 - 3 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/164\mathbb{Z}\right)^\times\).
| \(n\) | \(83\) | \(129\) |
| \(\chi(n)\) | \(1\) | \(-\beta_{4}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 9.1 |
|
0 | −1.33641 | − | 1.33641i | 0 | − | 0.428007i | 0 | −1.90841 | − | 1.90841i | 0 | 0.571993i | 0 | |||||||||||||||||||||||||||||||
| 9.2 | 0 | 0.675970 | + | 0.675970i | 0 | − | 3.08613i | 0 | 2.76210 | + | 2.76210i | 0 | − | 2.08613i | 0 | |||||||||||||||||||||||||||||||
| 9.3 | 0 | 1.66044 | + | 1.66044i | 0 | 1.51414i | 0 | −0.853695 | − | 0.853695i | 0 | 2.51414i | 0 | |||||||||||||||||||||||||||||||||
| 73.1 | 0 | −1.33641 | + | 1.33641i | 0 | 0.428007i | 0 | −1.90841 | + | 1.90841i | 0 | − | 0.571993i | 0 | ||||||||||||||||||||||||||||||||
| 73.2 | 0 | 0.675970 | − | 0.675970i | 0 | 3.08613i | 0 | 2.76210 | − | 2.76210i | 0 | 2.08613i | 0 | |||||||||||||||||||||||||||||||||
| 73.3 | 0 | 1.66044 | − | 1.66044i | 0 | − | 1.51414i | 0 | −0.853695 | + | 0.853695i | 0 | − | 2.51414i | 0 | |||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 41.c | even | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 164.2.f.a | ✓ | 6 |
| 3.b | odd | 2 | 1 | 1476.2.k.a | 6 | ||
| 4.b | odd | 2 | 1 | 656.2.l.f | 6 | ||
| 41.c | even | 4 | 1 | inner | 164.2.f.a | ✓ | 6 |
| 41.e | odd | 8 | 2 | 6724.2.a.e | 6 | ||
| 123.f | odd | 4 | 1 | 1476.2.k.a | 6 | ||
| 164.e | odd | 4 | 1 | 656.2.l.f | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 164.2.f.a | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 164.2.f.a | ✓ | 6 | 41.c | even | 4 | 1 | inner |
| 656.2.l.f | 6 | 4.b | odd | 2 | 1 | ||
| 656.2.l.f | 6 | 164.e | odd | 4 | 1 | ||
| 1476.2.k.a | 6 | 3.b | odd | 2 | 1 | ||
| 1476.2.k.a | 6 | 123.f | odd | 4 | 1 | ||
| 6724.2.a.e | 6 | 41.e | odd | 8 | 2 | ||
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(164, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( T^{6} - 2 T^{5} + \cdots + 18 \)
|
| $5$ |
\( T^{6} + 12 T^{4} + \cdots + 4 \)
|
| $7$ |
\( T^{6} + 18 T^{3} + \cdots + 162 \)
|
| $11$ |
\( T^{6} + 4 T^{5} + \cdots + 2 \)
|
| $13$ |
\( T^{6} + 2 T^{5} + \cdots + 72 \)
|
| $17$ |
\( T^{6} + 2 T^{5} + \cdots + 72 \)
|
| $19$ |
\( T^{6} - 2 T^{5} + \cdots + 1458 \)
|
| $23$ |
\( (T^{3} - 2 T^{2} - 20 T + 24)^{2} \)
|
| $29$ |
\( T^{6} + 10 T^{5} + \cdots + 2312 \)
|
| $31$ |
\( (T^{3} - 4 T^{2} - 40 T + 16)^{2} \)
|
| $37$ |
\( (T^{3} + 18 T^{2} + \cdots - 82)^{2} \)
|
| $41$ |
\( T^{6} - 6 T^{5} + \cdots + 68921 \)
|
| $43$ |
\( T^{6} + 204 T^{4} + \cdots + 258064 \)
|
| $47$ |
\( T^{6} + 26 T^{5} + \cdots + 7442 \)
|
| $53$ |
\( T^{6} + 14 T^{5} + \cdots + 20808 \)
|
| $59$ |
\( (T^{3} + 2 T^{2} - 20 T - 24)^{2} \)
|
| $61$ |
\( T^{6} + 24 T^{4} + \cdots + 16 \)
|
| $67$ |
\( T^{6} - 16 T^{5} + \cdots + 2068578 \)
|
| $71$ |
\( T^{6} - 26 T^{5} + \cdots + 13122 \)
|
| $73$ |
\( T^{6} + 456 T^{4} + \cdots + 2910436 \)
|
| $79$ |
\( T^{6} + 2 T^{5} + \cdots + 8978 \)
|
| $83$ |
\( (T^{3} - 18 T^{2} + \cdots + 1304)^{2} \)
|
| $89$ |
\( T^{6} - 14 T^{5} + \cdots + 267912 \)
|
| $97$ |
\( T^{6} + 6 T^{5} + \cdots + 1352 \)
|
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