Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [7,11,Mod(6,7)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1]))
N = Newforms(chi, 11, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("7.6");
S:= CuspForms(chi, 11);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 7 \) |
| Weight: | \( k \) | \(=\) | \( 11 \) |
| Character orbit: | \([\chi]\) | \(=\) | 7.b (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: | \(4.44750076872\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.0.373770240.2 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 368x^{2} + 2760 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{7}\cdot 3\cdot 5\cdot 7 \) |
| 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
\( x^{4} + 368x^{2} + 2760 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -3\nu^{3} - 1254\nu ) / 13 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 2\nu^{2} + 368 ) / 13 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -35\nu^{3} - 10990\nu ) / 13 \)
|
| \(\nu\) | \(=\) |
\( ( 3\beta_{3} - 35\beta_1 ) / 840 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 13\beta_{2} - 368 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -627\beta_{3} + 5495\beta_1 ) / 420 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/7\mathbb{Z}\right)^\times\).
| \(n\) | \(3\) |
| \(\chi(n)\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 6.1 |
|
−39.1293 | − | 252.583i | 507.104 | 2873.50i | 9883.42i | −2763.01 | + | 16578.3i | 20225.8 | −4749.36 | − | 112438.i | ||||||||||||||||||||||||||
| 6.2 | −39.1293 | 252.583i | 507.104 | − | 2873.50i | − | 9883.42i | −2763.01 | − | 16578.3i | 20225.8 | −4749.36 | 112438.i | |||||||||||||||||||||||||||
| 6.3 | 15.1293 | − | 262.072i | −795.104 | − | 1758.44i | − | 3964.97i | 5213.01 | − | 15978.1i | −27521.8 | −9632.64 | − | 26604.0i | |||||||||||||||||||||||||
| 6.4 | 15.1293 | 262.072i | −795.104 | 1758.44i | 3964.97i | 5213.01 | + | 15978.1i | −27521.8 | −9632.64 | 26604.0i | |||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 7.11.b.b | ✓ | 4 |
| 3.b | odd | 2 | 1 | 63.11.d.c | 4 | ||
| 4.b | odd | 2 | 1 | 112.11.c.b | 4 | ||
| 7.b | odd | 2 | 1 | inner | 7.11.b.b | ✓ | 4 |
| 7.c | even | 3 | 2 | 49.11.d.b | 8 | ||
| 7.d | odd | 6 | 2 | 49.11.d.b | 8 | ||
| 21.c | even | 2 | 1 | 63.11.d.c | 4 | ||
| 28.d | even | 2 | 1 | 112.11.c.b | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 7.11.b.b | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 7.11.b.b | ✓ | 4 | 7.b | odd | 2 | 1 | inner |
| 49.11.d.b | 8 | 7.c | even | 3 | 2 | ||
| 49.11.d.b | 8 | 7.d | odd | 6 | 2 | ||
| 63.11.d.c | 4 | 3.b | odd | 2 | 1 | ||
| 63.11.d.c | 4 | 21.c | even | 2 | 1 | ||
| 112.11.c.b | 4 | 4.b | odd | 2 | 1 | ||
| 112.11.c.b | 4 | 28.d | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} + 24T_{2} - 592 \)
acting on \(S_{11}^{\mathrm{new}}(7, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 24 T - 592)^{2} \)
|
| $3$ |
\( T^{4} + \cdots + 4381776000 \)
|
| $5$ |
\( T^{4} + \cdots + 25531635024000 \)
|
| $7$ |
\( T^{4} + \cdots + 79\!\cdots\!01 \)
|
| $11$ |
\( (T^{2} - 42996 T - 12545617372)^{2} \)
|
| $13$ |
\( T^{4} + \cdots + 48\!\cdots\!00 \)
|
| $17$ |
\( T^{4} + \cdots + 30\!\cdots\!00 \)
|
| $19$ |
\( T^{4} + \cdots + 16\!\cdots\!00 \)
|
| $23$ |
\( (T^{2} + \cdots - 28039440658492)^{2} \)
|
| $29$ |
\( (T^{2} + \cdots - 199771975916572)^{2} \)
|
| $31$ |
\( T^{4} + \cdots + 11\!\cdots\!00 \)
|
| $37$ |
\( (T^{2} + \cdots - 21\!\cdots\!32)^{2} \)
|
| $41$ |
\( T^{4} + \cdots + 62\!\cdots\!00 \)
|
| $43$ |
\( (T^{2} + \cdots - 22\!\cdots\!92)^{2} \)
|
| $47$ |
\( T^{4} + \cdots + 78\!\cdots\!00 \)
|
| $53$ |
\( (T^{2} + \cdots + 75\!\cdots\!64)^{2} \)
|
| $59$ |
\( T^{4} + \cdots + 42\!\cdots\!00 \)
|
| $61$ |
\( T^{4} + \cdots + 36\!\cdots\!00 \)
|
| $67$ |
\( (T^{2} + \cdots + 23\!\cdots\!88)^{2} \)
|
| $71$ |
\( (T^{2} + \cdots - 12\!\cdots\!32)^{2} \)
|
| $73$ |
\( T^{4} + \cdots + 16\!\cdots\!00 \)
|
| $79$ |
\( (T^{2} + \cdots + 59\!\cdots\!68)^{2} \)
|
| $83$ |
\( T^{4} + \cdots + 83\!\cdots\!00 \)
|
| $89$ |
\( T^{4} + \cdots + 19\!\cdots\!00 \)
|
| $97$ |
\( T^{4} + \cdots + 53\!\cdots\!00 \)
|
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