Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [168,2,Mod(155,168)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(168, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 1, 1, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("168.155");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 168 = 2^{3} \cdot 3 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 168.j (of order \(2\), degree \(1\), 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: | \(1.34148675396\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(i, \sqrt{5})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 3x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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 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
\( x^{4} + 3x^{2} + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu^{2} + \nu + 1 \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{3} + 2\nu \)
|
| \(\beta_{3}\) | \(=\) |
\( -\nu^{2} + \nu - 1 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{3} + \beta _1 - 2 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{3} + \beta_{2} - \beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/168\mathbb{Z}\right)^\times\).
| \(n\) | \(73\) | \(85\) | \(113\) | \(127\) |
| \(\chi(n)\) | \(1\) | \(-1\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 155.1 |
|
−1.00000 | − | 1.00000i | −1.61803 | − | 0.618034i | 2.00000i | 3.23607 | 1.00000 | + | 2.23607i | 1.00000i | 2.00000 | − | 2.00000i | 2.23607 | + | 2.00000i | −3.23607 | − | 3.23607i | ||||||||||||||||||
| 155.2 | −1.00000 | − | 1.00000i | 0.618034 | + | 1.61803i | 2.00000i | −1.23607 | 1.00000 | − | 2.23607i | 1.00000i | 2.00000 | − | 2.00000i | −2.23607 | + | 2.00000i | 1.23607 | + | 1.23607i | |||||||||||||||||||
| 155.3 | −1.00000 | + | 1.00000i | −1.61803 | + | 0.618034i | − | 2.00000i | 3.23607 | 1.00000 | − | 2.23607i | − | 1.00000i | 2.00000 | + | 2.00000i | 2.23607 | − | 2.00000i | −3.23607 | + | 3.23607i | |||||||||||||||||
| 155.4 | −1.00000 | + | 1.00000i | 0.618034 | − | 1.61803i | − | 2.00000i | −1.23607 | 1.00000 | + | 2.23607i | − | 1.00000i | 2.00000 | + | 2.00000i | −2.23607 | − | 2.00000i | 1.23607 | − | 1.23607i | |||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 24.f | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 168.2.j.a | ✓ | 4 |
| 3.b | odd | 2 | 1 | 168.2.j.c | yes | 4 | |
| 4.b | odd | 2 | 1 | 672.2.j.c | 4 | ||
| 8.b | even | 2 | 1 | 672.2.j.b | 4 | ||
| 8.d | odd | 2 | 1 | 168.2.j.c | yes | 4 | |
| 12.b | even | 2 | 1 | 672.2.j.b | 4 | ||
| 24.f | even | 2 | 1 | inner | 168.2.j.a | ✓ | 4 |
| 24.h | odd | 2 | 1 | 672.2.j.c | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 168.2.j.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 168.2.j.a | ✓ | 4 | 24.f | even | 2 | 1 | inner |
| 168.2.j.c | yes | 4 | 3.b | odd | 2 | 1 | |
| 168.2.j.c | yes | 4 | 8.d | odd | 2 | 1 | |
| 672.2.j.b | 4 | 8.b | even | 2 | 1 | ||
| 672.2.j.b | 4 | 12.b | even | 2 | 1 | ||
| 672.2.j.c | 4 | 4.b | odd | 2 | 1 | ||
| 672.2.j.c | 4 | 24.h | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{2} - 2T_{5} - 4 \)
acting on \(S_{2}^{\mathrm{new}}(168, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 2 T + 2)^{2} \)
|
| $3$ |
\( T^{4} + 2 T^{3} + \cdots + 9 \)
|
| $5$ |
\( (T^{2} - 2 T - 4)^{2} \)
|
| $7$ |
\( (T^{2} + 1)^{2} \)
|
| $11$ |
\( (T^{2} + 20)^{2} \)
|
| $13$ |
\( T^{4} + 12T^{2} + 16 \)
|
| $17$ |
\( (T^{2} + 4)^{2} \)
|
| $19$ |
\( (T^{2} - 10 T + 20)^{2} \)
|
| $23$ |
\( (T^{2} - 8 T - 4)^{2} \)
|
| $29$ |
\( T^{4} \)
|
| $31$ |
\( (T^{2} + 80)^{2} \)
|
| $37$ |
\( T^{4} + 48T^{2} + 256 \)
|
| $41$ |
\( (T^{2} + 20)^{2} \)
|
| $43$ |
\( (T^{2} - 8 T - 4)^{2} \)
|
| $47$ |
\( (T^{2} + 4 T - 16)^{2} \)
|
| $53$ |
\( (T^{2} + 12 T + 16)^{2} \)
|
| $59$ |
\( T^{4} + 12T^{2} + 16 \)
|
| $61$ |
\( T^{4} + 140T^{2} + 400 \)
|
| $67$ |
\( (T^{2} + 4 T - 76)^{2} \)
|
| $71$ |
\( (T^{2} - 4 T - 76)^{2} \)
|
| $73$ |
\( (T^{2} - 8 T - 4)^{2} \)
|
| $79$ |
\( T^{4} + 192T^{2} + 4096 \)
|
| $83$ |
\( T^{4} + 252 T^{2} + 13456 \)
|
| $89$ |
\( (T^{2} + 36)^{2} \)
|
| $97$ |
\( (T^{2} - 16 T + 44)^{2} \)
|
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