Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [168,4,Mod(5,168)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(168, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 3, 3, 5]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("168.5");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 168 = 2^{3} \cdot 3 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 168.ba (of order \(6\), 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: | \(9.91232088096\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{6})\) |
| Coefficient field: | \(\Q(\sqrt{2}, \sqrt{-3})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 2x^{2} + 4 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{U}(1)[D_{6}]$ |
$q$-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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} + 2x^{2} + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 2\beta_{2} \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{3} \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/168\mathbb{Z}\right)^\times\).
| \(n\) | \(73\) | \(85\) | \(113\) | \(127\) |
| \(\chi(n)\) | \(1 + \beta_{2}\) | \(-1\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 5.1 |
|
−1.41421 | + | 2.44949i | −4.50000 | + | 2.59808i | −4.00000 | − | 6.92820i | 10.3492 | + | 5.97514i | − | 14.6969i | 2.13604 | + | 18.3967i | 22.6274 | 13.5000 | − | 23.3827i | −29.2721 | + | 16.9002i | |||||||||||||||
| 5.2 | 1.41421 | − | 2.44949i | −4.50000 | + | 2.59808i | −4.00000 | − | 6.92820i | −19.3492 | − | 11.1713i | 14.6969i | 14.8640 | + | 11.0482i | −22.6274 | 13.5000 | − | 23.3827i | −54.7279 | + | 31.5972i | |||||||||||||||||
| 101.1 | −1.41421 | − | 2.44949i | −4.50000 | − | 2.59808i | −4.00000 | + | 6.92820i | 10.3492 | − | 5.97514i | 14.6969i | 2.13604 | − | 18.3967i | 22.6274 | 13.5000 | + | 23.3827i | −29.2721 | − | 16.9002i | |||||||||||||||||
| 101.2 | 1.41421 | + | 2.44949i | −4.50000 | − | 2.59808i | −4.00000 | + | 6.92820i | −19.3492 | + | 11.1713i | − | 14.6969i | 14.8640 | − | 11.0482i | −22.6274 | 13.5000 | + | 23.3827i | −54.7279 | − | 31.5972i | ||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 24.h | odd | 2 | 1 | CM by \(\Q(\sqrt{-6}) \) |
| 7.d | odd | 6 | 1 | inner |
| 168.ba | even | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 168.4.ba.a | ✓ | 4 |
| 3.b | odd | 2 | 1 | 168.4.ba.b | yes | 4 | |
| 7.d | odd | 6 | 1 | inner | 168.4.ba.a | ✓ | 4 |
| 8.b | even | 2 | 1 | 168.4.ba.b | yes | 4 | |
| 21.g | even | 6 | 1 | 168.4.ba.b | yes | 4 | |
| 24.h | odd | 2 | 1 | CM | 168.4.ba.a | ✓ | 4 |
| 56.j | odd | 6 | 1 | 168.4.ba.b | yes | 4 | |
| 168.ba | even | 6 | 1 | inner | 168.4.ba.a | ✓ | 4 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 168.4.ba.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 168.4.ba.a | ✓ | 4 | 7.d | odd | 6 | 1 | inner |
| 168.4.ba.a | ✓ | 4 | 24.h | odd | 2 | 1 | CM |
| 168.4.ba.a | ✓ | 4 | 168.ba | even | 6 | 1 | inner |
| 168.4.ba.b | yes | 4 | 3.b | odd | 2 | 1 | |
| 168.4.ba.b | yes | 4 | 8.b | even | 2 | 1 | |
| 168.4.ba.b | yes | 4 | 21.g | even | 6 | 1 | |
| 168.4.ba.b | yes | 4 | 56.j | odd | 6 | 1 | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} + 18T_{5}^{3} - 159T_{5}^{2} - 4806T_{5} + 71289 \)
acting on \(S_{4}^{\mathrm{new}}(168, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + 8T^{2} + 64 \)
|
| $3$ |
\( (T^{2} + 9 T + 27)^{2} \)
|
| $5$ |
\( T^{4} + 18 T^{3} + \cdots + 71289 \)
|
| $7$ |
\( T^{4} - 34 T^{3} + \cdots + 117649 \)
|
| $11$ |
\( T^{4} - 126 T^{3} + \cdots + 15689521 \)
|
| $13$ |
\( T^{4} \)
|
| $17$ |
\( T^{4} \)
|
| $19$ |
\( T^{4} \)
|
| $23$ |
\( T^{4} \)
|
| $29$ |
\( (T^{2} + 378 T + 23239)^{2} \)
|
| $31$ |
\( T^{4} + 210 T^{3} + \cdots + 619561881 \)
|
| $37$ |
\( T^{4} \)
|
| $41$ |
\( T^{4} \)
|
| $43$ |
\( T^{4} \)
|
| $47$ |
\( T^{4} \)
|
| $53$ |
\( T^{4} + \cdots + 12195005761 \)
|
| $59$ |
\( T^{4} + \cdots + 10393598601 \)
|
| $61$ |
\( T^{4} \)
|
| $67$ |
\( T^{4} \)
|
| $71$ |
\( T^{4} \)
|
| $73$ |
\( T^{4} + \cdots + 2109419283456 \)
|
| $79$ |
\( T^{4} + \cdots + 158231315089 \)
|
| $83$ |
\( T^{4} + \cdots + 874339073721 \)
|
| $89$ |
\( T^{4} \)
|
| $97$ |
\( T^{4} + \cdots + 340118740809 \)
|
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