Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [42,4,Mod(25,42)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(42, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 4]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("42.25");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 42 = 2 \cdot 3 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 42.e (of order \(3\), 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: | \(2.47808022024\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\Q(\sqrt{-3}, \sqrt{1345})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} + 337x^{2} + 336x + 112896 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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
\( x^{4} - x^{3} + 337x^{2} + 336x + 112896 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{3} + 337\nu^{2} - 337\nu + 112896 ) / 113232 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} + 673 ) / 337 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + 336\beta_{2} + \beta _1 - 337 \)
|
| \(\nu^{3}\) | \(=\) |
\( 337\beta_{3} - 673 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/42\mathbb{Z}\right)^\times\).
| \(n\) | \(29\) | \(31\) |
| \(\chi(n)\) | \(1\) | \(-\beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 25.1 |
|
−1.00000 | − | 1.73205i | −1.50000 | + | 2.59808i | −2.00000 | + | 3.46410i | −10.4186 | − | 18.0455i | 6.00000 | −18.3371 | − | 2.59808i | 8.00000 | −4.50000 | − | 7.79423i | −20.8371 | + | 36.0910i | ||||||||||||||||
| 25.2 | −1.00000 | − | 1.73205i | −1.50000 | + | 2.59808i | −2.00000 | + | 3.46410i | 7.91856 | + | 13.7153i | 6.00000 | 18.3371 | − | 2.59808i | 8.00000 | −4.50000 | − | 7.79423i | 15.8371 | − | 27.4307i | |||||||||||||||||
| 37.1 | −1.00000 | + | 1.73205i | −1.50000 | − | 2.59808i | −2.00000 | − | 3.46410i | −10.4186 | + | 18.0455i | 6.00000 | −18.3371 | + | 2.59808i | 8.00000 | −4.50000 | + | 7.79423i | −20.8371 | − | 36.0910i | |||||||||||||||||
| 37.2 | −1.00000 | + | 1.73205i | −1.50000 | − | 2.59808i | −2.00000 | − | 3.46410i | 7.91856 | − | 13.7153i | 6.00000 | 18.3371 | + | 2.59808i | 8.00000 | −4.50000 | + | 7.79423i | 15.8371 | + | 27.4307i | |||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 42.4.e.c | ✓ | 4 |
| 3.b | odd | 2 | 1 | 126.4.g.g | 4 | ||
| 4.b | odd | 2 | 1 | 336.4.q.j | 4 | ||
| 7.b | odd | 2 | 1 | 294.4.e.l | 4 | ||
| 7.c | even | 3 | 1 | inner | 42.4.e.c | ✓ | 4 |
| 7.c | even | 3 | 1 | 294.4.a.n | 2 | ||
| 7.d | odd | 6 | 1 | 294.4.a.m | 2 | ||
| 7.d | odd | 6 | 1 | 294.4.e.l | 4 | ||
| 21.c | even | 2 | 1 | 882.4.g.bf | 4 | ||
| 21.g | even | 6 | 1 | 882.4.a.z | 2 | ||
| 21.g | even | 6 | 1 | 882.4.g.bf | 4 | ||
| 21.h | odd | 6 | 1 | 126.4.g.g | 4 | ||
| 21.h | odd | 6 | 1 | 882.4.a.v | 2 | ||
| 28.f | even | 6 | 1 | 2352.4.a.ca | 2 | ||
| 28.g | odd | 6 | 1 | 336.4.q.j | 4 | ||
| 28.g | odd | 6 | 1 | 2352.4.a.bq | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 42.4.e.c | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 42.4.e.c | ✓ | 4 | 7.c | even | 3 | 1 | inner |
| 126.4.g.g | 4 | 3.b | odd | 2 | 1 | ||
| 126.4.g.g | 4 | 21.h | odd | 6 | 1 | ||
| 294.4.a.m | 2 | 7.d | odd | 6 | 1 | ||
| 294.4.a.n | 2 | 7.c | even | 3 | 1 | ||
| 294.4.e.l | 4 | 7.b | odd | 2 | 1 | ||
| 294.4.e.l | 4 | 7.d | odd | 6 | 1 | ||
| 336.4.q.j | 4 | 4.b | odd | 2 | 1 | ||
| 336.4.q.j | 4 | 28.g | odd | 6 | 1 | ||
| 882.4.a.v | 2 | 21.h | odd | 6 | 1 | ||
| 882.4.a.z | 2 | 21.g | even | 6 | 1 | ||
| 882.4.g.bf | 4 | 21.c | even | 2 | 1 | ||
| 882.4.g.bf | 4 | 21.g | even | 6 | 1 | ||
| 2352.4.a.bq | 2 | 28.g | odd | 6 | 1 | ||
| 2352.4.a.ca | 2 | 28.f | even | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} + 5T_{5}^{3} + 355T_{5}^{2} - 1650T_{5} + 108900 \)
acting on \(S_{4}^{\mathrm{new}}(42, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 2 T + 4)^{2} \)
|
| $3$ |
\( (T^{2} + 3 T + 9)^{2} \)
|
| $5$ |
\( T^{4} + 5 T^{3} + \cdots + 108900 \)
|
| $7$ |
\( T^{4} - 659 T^{2} + 117649 \)
|
| $11$ |
\( T^{4} + 67 T^{3} + \cdots + 617796 \)
|
| $13$ |
\( (T^{2} - 41 T + 84)^{2} \)
|
| $17$ |
\( T^{4} - 92 T^{3} + \cdots + 10653696 \)
|
| $19$ |
\( T^{4} + 43 T^{3} + \cdots + 6574096 \)
|
| $23$ |
\( T^{4} + 148 T^{3} + \cdots + 9216 \)
|
| $29$ |
\( (T^{2} - 77 T - 39204)^{2} \)
|
| $31$ |
\( T^{4} + \cdots + 4389725025 \)
|
| $37$ |
\( T^{4} + 7 T^{3} + \cdots + 741146176 \)
|
| $41$ |
\( (T^{2} + 426 T + 33264)^{2} \)
|
| $43$ |
\( (T^{2} + 107 T - 72794)^{2} \)
|
| $47$ |
\( T^{4} + \cdots + 1191906576 \)
|
| $53$ |
\( T^{4} - 243 T^{3} + \cdots + 137733696 \)
|
| $59$ |
\( T^{4} + \cdots + 44160500736 \)
|
| $61$ |
\( T^{4} + 224 T^{3} + \cdots + 51322896 \)
|
| $67$ |
\( T^{4} + \cdots + 5976217636 \)
|
| $71$ |
\( (T^{2} - 472 T - 78804)^{2} \)
|
| $73$ |
\( T^{4} + \cdots + 39938423716 \)
|
| $79$ |
\( T^{4} + \cdots + 2270427201 \)
|
| $83$ |
\( (T^{2} + 221 T - 197946)^{2} \)
|
| $89$ |
\( T^{4} + \cdots + 196582277376 \)
|
| $97$ |
\( (T^{2} - 1953 T + 541646)^{2} \)
|
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