Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [7,7,Mod(3,7)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([1]))
N = Newforms(chi, 7, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("7.3");
S:= CuspForms(chi, 7);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 7 \) |
| Weight: | \( k \) | \(=\) | \( 7 \) |
| Character orbit: | \([\chi]\) | \(=\) | 7.d (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: | \(1.61037858534\) |
| 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: | \(\Z[a_1, \ldots, a_{4}]\) |
| Coefficient ring index: | \( 3 \) |
| 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
\( x^{4} + 2x^{2} + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{2} ) / 2 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} + 4\nu ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{3} + 2\nu ) / 2 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + \beta_{2} ) / 3 \)
|
| \(\nu^{2}\) | \(=\) |
\( 2\beta_1 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -4\beta_{3} + 2\beta_{2} ) / 3 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/7\mathbb{Z}\right)^\times\).
| \(n\) | \(3\) |
| \(\chi(n)\) | \(1 + \beta_{1}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 3.1 |
|
−4.12132 | − | 7.13834i | −23.0772 | − | 13.3236i | −1.97056 | + | 3.41311i | 68.5660 | − | 39.5866i | 219.643i | 337.286 | + | 62.3451i | −495.044 | −9.46299 | − | 16.3904i | −565.165 | − | 326.298i | ||||||||||||||||
| 3.2 | 0.121320 | + | 0.210133i | 32.0772 | + | 18.5198i | 31.9706 | − | 55.3746i | −143.566 | + | 82.8879i | 8.98729i | −197.286 | − | 280.583i | 31.0437 | 321.463 | + | 556.790i | −34.8350 | − | 20.1120i | |||||||||||||||||
| 5.1 | −4.12132 | + | 7.13834i | −23.0772 | + | 13.3236i | −1.97056 | − | 3.41311i | 68.5660 | + | 39.5866i | − | 219.643i | 337.286 | − | 62.3451i | −495.044 | −9.46299 | + | 16.3904i | −565.165 | + | 326.298i | ||||||||||||||||
| 5.2 | 0.121320 | − | 0.210133i | 32.0772 | − | 18.5198i | 31.9706 | + | 55.3746i | −143.566 | − | 82.8879i | − | 8.98729i | −197.286 | + | 280.583i | 31.0437 | 321.463 | − | 556.790i | −34.8350 | + | 20.1120i | ||||||||||||||||
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 | 7.7.d.b | ✓ | 4 |
| 3.b | odd | 2 | 1 | 63.7.m.b | 4 | ||
| 4.b | odd | 2 | 1 | 112.7.s.b | 4 | ||
| 7.b | odd | 2 | 1 | 49.7.d.c | 4 | ||
| 7.c | even | 3 | 1 | 49.7.b.b | 4 | ||
| 7.c | even | 3 | 1 | 49.7.d.c | 4 | ||
| 7.d | odd | 6 | 1 | inner | 7.7.d.b | ✓ | 4 |
| 7.d | odd | 6 | 1 | 49.7.b.b | 4 | ||
| 21.g | even | 6 | 1 | 63.7.m.b | 4 | ||
| 21.g | even | 6 | 1 | 441.7.d.b | 4 | ||
| 21.h | odd | 6 | 1 | 441.7.d.b | 4 | ||
| 28.f | even | 6 | 1 | 112.7.s.b | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 7.7.d.b | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 7.7.d.b | ✓ | 4 | 7.d | odd | 6 | 1 | inner |
| 49.7.b.b | 4 | 7.c | even | 3 | 1 | ||
| 49.7.b.b | 4 | 7.d | odd | 6 | 1 | ||
| 49.7.d.c | 4 | 7.b | odd | 2 | 1 | ||
| 49.7.d.c | 4 | 7.c | even | 3 | 1 | ||
| 63.7.m.b | 4 | 3.b | odd | 2 | 1 | ||
| 63.7.m.b | 4 | 21.g | even | 6 | 1 | ||
| 112.7.s.b | 4 | 4.b | odd | 2 | 1 | ||
| 112.7.s.b | 4 | 28.f | even | 6 | 1 | ||
| 441.7.d.b | 4 | 21.g | even | 6 | 1 | ||
| 441.7.d.b | 4 | 21.h | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} + 8T_{2}^{3} + 66T_{2}^{2} - 16T_{2} + 4 \)
acting on \(S_{7}^{\mathrm{new}}(7, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + 8 T^{3} + \cdots + 4 \)
|
| $3$ |
\( T^{4} - 18 T^{3} + \cdots + 974169 \)
|
| $5$ |
\( T^{4} + 150 T^{3} + \cdots + 172265625 \)
|
| $7$ |
\( T^{4} + \cdots + 13841287201 \)
|
| $11$ |
\( T^{4} + \cdots + 87487583089 \)
|
| $13$ |
\( T^{4} + \cdots + 37734434122896 \)
|
| $17$ |
\( T^{4} + \cdots + 226702498429449 \)
|
| $19$ |
\( T^{4} + \cdots + 718603078673529 \)
|
| $23$ |
\( T^{4} + \cdots + 44\!\cdots\!29 \)
|
| $29$ |
\( (T^{2} + 17272 T - 154827704)^{2} \)
|
| $31$ |
\( T^{4} + \cdots + 611468988954201 \)
|
| $37$ |
\( T^{4} + \cdots + 50\!\cdots\!09 \)
|
| $41$ |
\( T^{4} + \cdots + 26\!\cdots\!84 \)
|
| $43$ |
\( (T^{2} - 160108 T + 6274490716)^{2} \)
|
| $47$ |
\( T^{4} + \cdots + 81\!\cdots\!29 \)
|
| $53$ |
\( T^{4} + \cdots + 14\!\cdots\!25 \)
|
| $59$ |
\( T^{4} + \cdots + 35\!\cdots\!69 \)
|
| $61$ |
\( T^{4} + \cdots + 29\!\cdots\!29 \)
|
| $67$ |
\( T^{4} + \cdots + 14\!\cdots\!81 \)
|
| $71$ |
\( (T^{2} + 740764 T + 136232520316)^{2} \)
|
| $73$ |
\( T^{4} + \cdots + 56\!\cdots\!41 \)
|
| $79$ |
\( T^{4} + \cdots + 38\!\cdots\!69 \)
|
| $83$ |
\( T^{4} + \cdots + 10\!\cdots\!76 \)
|
| $89$ |
\( T^{4} + \cdots + 85\!\cdots\!21 \)
|
| $97$ |
\( T^{4} + \cdots + 95\!\cdots\!84 \)
|
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