Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [7,14,Mod(1,7)]
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([0]))
N = Newforms(chi, 14, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("7.1");
S:= CuspForms(chi, 14);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 7 \) |
| Weight: | \( k \) | \(=\) | \( 14 \) |
| Character orbit: | \([\chi]\) | \(=\) | 7.a (trivial) |
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: | yes |
| Analytic conductor: | \(7.50616502663\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 15567x^{2} - 590047x + 9242158 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{2}\cdot 3^{2}\cdot 7 \) |
| Twist minimal: | yes |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-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
\( x^{4} - x^{3} - 15567x^{2} - 590047x + 9242158 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} - 54\nu^{2} - 11277\nu - 31090 ) / 84 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{3} + 138\nu^{2} + 6405\nu - 621506 ) / 84 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 58\beta _1 + 7769 \)
|
| \(\nu^{3}\) | \(=\) |
\( 54\beta_{3} + 138\beta_{2} + 14409\beta _1 + 450616 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−92.5943 | −1055.94 | 381.706 | −56427.2 | 97773.6 | 117649. | 723189. | −479323. | 5.22483e6 | ||||||||||||||||||||||||||||||
| 1.2 | −71.8453 | 2405.30 | −3030.25 | 458.010 | −172809. | 117649. | 806266. | 4.19113e6 | −32905.9 | |||||||||||||||||||||||||||||||
| 1.3 | 4.93616 | −1947.96 | −8167.63 | 65245.5 | −9615.43 | 117649. | −80753.8 | 2.20021e6 | 322062. | |||||||||||||||||||||||||||||||
| 1.4 | 132.503 | 934.596 | 9365.18 | 14915.6 | 123837. | 117649. | 155450. | −720854. | 1.97637e6 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(7\) | \(-1\) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 7.14.a.b | ✓ | 4 |
| 3.b | odd | 2 | 1 | 63.14.a.f | 4 | ||
| 4.b | odd | 2 | 1 | 112.14.a.h | 4 | ||
| 5.b | even | 2 | 1 | 175.14.a.b | 4 | ||
| 7.b | odd | 2 | 1 | 49.14.a.c | 4 | ||
| 7.c | even | 3 | 2 | 49.14.c.d | 8 | ||
| 7.d | odd | 6 | 2 | 49.14.c.e | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 7.14.a.b | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 49.14.a.c | 4 | 7.b | odd | 2 | 1 | ||
| 49.14.c.d | 8 | 7.c | even | 3 | 2 | ||
| 49.14.c.e | 8 | 7.d | odd | 6 | 2 | ||
| 63.14.a.f | 4 | 3.b | odd | 2 | 1 | ||
| 112.14.a.h | 4 | 4.b | odd | 2 | 1 | ||
| 175.14.a.b | 4 | 5.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} + 27T_{2}^{3} - 15294T_{2}^{2} - 806760T_{2} + 4351104 \)
acting on \(S_{14}^{\mathrm{new}}(\Gamma_0(7))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + 27 T^{3} + \cdots + 4351104 \)
|
| $3$ |
\( T^{4} + \cdots + 4623906845376 \)
|
| $5$ |
\( T^{4} + \cdots - 25\!\cdots\!00 \)
|
| $7$ |
\( (T - 117649)^{4} \)
|
| $11$ |
\( T^{4} + \cdots - 16\!\cdots\!36 \)
|
| $13$ |
\( T^{4} + \cdots - 36\!\cdots\!88 \)
|
| $17$ |
\( T^{4} + \cdots - 10\!\cdots\!52 \)
|
| $19$ |
\( T^{4} + \cdots - 50\!\cdots\!60 \)
|
| $23$ |
\( T^{4} + \cdots - 79\!\cdots\!16 \)
|
| $29$ |
\( T^{4} + \cdots - 20\!\cdots\!00 \)
|
| $31$ |
\( T^{4} + \cdots + 40\!\cdots\!00 \)
|
| $37$ |
\( T^{4} + \cdots + 78\!\cdots\!04 \)
|
| $41$ |
\( T^{4} + \cdots - 11\!\cdots\!44 \)
|
| $43$ |
\( T^{4} + \cdots + 23\!\cdots\!48 \)
|
| $47$ |
\( T^{4} + \cdots + 15\!\cdots\!52 \)
|
| $53$ |
\( T^{4} + \cdots - 33\!\cdots\!76 \)
|
| $59$ |
\( T^{4} + \cdots + 50\!\cdots\!20 \)
|
| $61$ |
\( T^{4} + \cdots + 20\!\cdots\!64 \)
|
| $67$ |
\( T^{4} + \cdots - 14\!\cdots\!56 \)
|
| $71$ |
\( T^{4} + \cdots - 10\!\cdots\!64 \)
|
| $73$ |
\( T^{4} + \cdots - 98\!\cdots\!32 \)
|
| $79$ |
\( T^{4} + \cdots + 14\!\cdots\!00 \)
|
| $83$ |
\( T^{4} + \cdots + 52\!\cdots\!96 \)
|
| $89$ |
\( T^{4} + \cdots + 73\!\cdots\!60 \)
|
| $97$ |
\( T^{4} + \cdots + 50\!\cdots\!16 \)
|
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