Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [42,5,Mod(19,42)]
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, 5]))
N = Newforms(chi, 5, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("42.19");
S:= CuspForms(chi, 5);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 42 = 2 \cdot 3 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 5 \) |
| Character orbit: | \([\chi]\) | \(=\) | 42.g (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: | \(4.34153844952\) |
| 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: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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
\( x^{4} + 2x^{2} + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( 2\beta_{2} \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/42\mathbb{Z}\right)^\times\).
| \(n\) | \(29\) | \(31\) |
| \(\chi(n)\) | \(1\) | \(1 + \beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 19.1 |
|
−1.41421 | + | 2.44949i | −4.50000 | + | 2.59808i | −4.00000 | − | 6.92820i | −12.2574 | − | 7.07679i | − | 14.6969i | 12.1985 | − | 47.4573i | 22.6274 | 13.5000 | − | 23.3827i | 34.6690 | − | 20.0162i | |||||||||||||||
| 19.2 | 1.41421 | − | 2.44949i | −4.50000 | + | 2.59808i | −4.00000 | − | 6.92820i | −20.7426 | − | 11.9758i | 14.6969i | −47.1985 | − | 13.1645i | −22.6274 | 13.5000 | − | 23.3827i | −58.6690 | + | 33.8726i | |||||||||||||||||
| 31.1 | −1.41421 | − | 2.44949i | −4.50000 | − | 2.59808i | −4.00000 | + | 6.92820i | −12.2574 | + | 7.07679i | 14.6969i | 12.1985 | + | 47.4573i | 22.6274 | 13.5000 | + | 23.3827i | 34.6690 | + | 20.0162i | |||||||||||||||||
| 31.2 | 1.41421 | + | 2.44949i | −4.50000 | − | 2.59808i | −4.00000 | + | 6.92820i | −20.7426 | + | 11.9758i | − | 14.6969i | −47.1985 | + | 13.1645i | −22.6274 | 13.5000 | + | 23.3827i | −58.6690 | − | 33.8726i | ||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.d | odd | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 42.5.g.a | ✓ | 4 |
| 3.b | odd | 2 | 1 | 126.5.n.b | 4 | ||
| 4.b | odd | 2 | 1 | 336.5.bh.d | 4 | ||
| 7.b | odd | 2 | 1 | 294.5.g.c | 4 | ||
| 7.c | even | 3 | 1 | 294.5.c.a | 4 | ||
| 7.c | even | 3 | 1 | 294.5.g.c | 4 | ||
| 7.d | odd | 6 | 1 | inner | 42.5.g.a | ✓ | 4 |
| 7.d | odd | 6 | 1 | 294.5.c.a | 4 | ||
| 21.g | even | 6 | 1 | 126.5.n.b | 4 | ||
| 21.g | even | 6 | 1 | 882.5.c.a | 4 | ||
| 21.h | odd | 6 | 1 | 882.5.c.a | 4 | ||
| 28.f | even | 6 | 1 | 336.5.bh.d | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 42.5.g.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 42.5.g.a | ✓ | 4 | 7.d | odd | 6 | 1 | inner |
| 126.5.n.b | 4 | 3.b | odd | 2 | 1 | ||
| 126.5.n.b | 4 | 21.g | even | 6 | 1 | ||
| 294.5.c.a | 4 | 7.c | even | 3 | 1 | ||
| 294.5.c.a | 4 | 7.d | odd | 6 | 1 | ||
| 294.5.g.c | 4 | 7.b | odd | 2 | 1 | ||
| 294.5.g.c | 4 | 7.c | even | 3 | 1 | ||
| 336.5.bh.d | 4 | 4.b | odd | 2 | 1 | ||
| 336.5.bh.d | 4 | 28.f | even | 6 | 1 | ||
| 882.5.c.a | 4 | 21.g | even | 6 | 1 | ||
| 882.5.c.a | 4 | 21.h | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} + 66T_{5}^{3} + 1791T_{5}^{2} + 22374T_{5} + 114921 \)
acting on \(S_{5}^{\mathrm{new}}(42, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + 8T^{2} + 64 \)
|
| $3$ |
\( (T^{2} + 9 T + 27)^{2} \)
|
| $5$ |
\( T^{4} + 66 T^{3} + \cdots + 114921 \)
|
| $7$ |
\( T^{4} + 70 T^{3} + \cdots + 5764801 \)
|
| $11$ |
\( T^{4} + 162 T^{3} + \cdots + 39350529 \)
|
| $13$ |
\( T^{4} + 63264 T^{2} + 569108736 \)
|
| $17$ |
\( T^{4} + 204 T^{3} + \cdots + 589324176 \)
|
| $19$ |
\( T^{4} + 444 T^{3} + \cdots + 128051856 \)
|
| $23$ |
\( T^{4} + \cdots + 525265461504 \)
|
| $29$ |
\( (T^{2} - 1362 T + 460233)^{2} \)
|
| $31$ |
\( T^{4} + \cdots + 1025818531929 \)
|
| $37$ |
\( T^{4} + \cdots + 219747187984 \)
|
| $41$ |
\( T^{4} + \cdots + 8311481425296 \)
|
| $43$ |
\( (T^{2} + 316 T - 10371836)^{2} \)
|
| $47$ |
\( T^{4} + \cdots + 5862054484224 \)
|
| $53$ |
\( T^{4} + \cdots + 6375054362769 \)
|
| $59$ |
\( T^{4} + \cdots + 105068670590601 \)
|
| $61$ |
\( T^{4} + \cdots + 2285563428864 \)
|
| $67$ |
\( T^{4} + \cdots + 26\!\cdots\!24 \)
|
| $71$ |
\( (T^{2} + 4848 T + 1589184)^{2} \)
|
| $73$ |
\( T^{4} + \cdots + 13\!\cdots\!44 \)
|
| $79$ |
\( T^{4} + \cdots + 45\!\cdots\!41 \)
|
| $83$ |
\( T^{4} + \cdots + 109011202550649 \)
|
| $89$ |
\( T^{4} + \cdots + 31\!\cdots\!04 \)
|
| $97$ |
\( T^{4} + \cdots + 9242250571449 \)
|
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