Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1694,4,Mod(1,1694)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1694, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1694.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1694 = 2 \cdot 7 \cdot 11^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1694.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: | \(99.9492355497\) |
| Analytic rank: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.4388525.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 2x^{3} - 208x^{2} + 209x + 10919 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 154) |
| 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
\( x^{4} - 2x^{3} - 208x^{2} + 209x + 10919 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 105 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 105\nu \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 105 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 106\beta _1 + 105 \)
|
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 |
|
2.00000 | −5.85410 | 4.00000 | −14.1521 | −11.7082 | 7.00000 | 8.00000 | 7.27051 | −28.3042 | ||||||||||||||||||||||||||||||
| 1.2 | 2.00000 | −5.85410 | 4.00000 | 6.20784 | −11.7082 | 7.00000 | 8.00000 | 7.27051 | 12.4157 | |||||||||||||||||||||||||||||||
| 1.3 | 2.00000 | 0.854102 | 4.00000 | −5.31708 | 1.70820 | 7.00000 | 8.00000 | −26.2705 | −10.6342 | |||||||||||||||||||||||||||||||
| 1.4 | 2.00000 | 0.854102 | 4.00000 | 15.2614 | 1.70820 | 7.00000 | 8.00000 | −26.2705 | 30.5227 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(7\) | \(-1\) |
| \(11\) | \(-1\) |
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 | 1694.4.a.x | 4 | |
| 11.b | odd | 2 | 1 | 1694.4.a.v | 4 | ||
| 11.d | odd | 10 | 2 | 154.4.f.c | ✓ | 8 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 154.4.f.c | ✓ | 8 | 11.d | odd | 10 | 2 | |
| 1694.4.a.v | 4 | 11.b | odd | 2 | 1 | ||
| 1694.4.a.x | 4 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1694))\):
|
\( T_{3}^{2} + 5T_{3} - 5 \)
|
|
\( T_{5}^{4} - 2T_{5}^{3} - 248T_{5}^{2} + 229T_{5} + 7129 \)
|
|
\( T_{13}^{4} + 46T_{13}^{3} - 6342T_{13}^{2} - 412333T_{13} - 5340851 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 2)^{4} \)
|
| $3$ |
\( (T^{2} + 5 T - 5)^{2} \)
|
| $5$ |
\( T^{4} - 2 T^{3} + \cdots + 7129 \)
|
| $7$ |
\( (T - 7)^{4} \)
|
| $11$ |
\( T^{4} \)
|
| $13$ |
\( T^{4} + 46 T^{3} + \cdots - 5340851 \)
|
| $17$ |
\( T^{4} + 97 T^{3} + \cdots - 41685061 \)
|
| $19$ |
\( T^{4} - 73 T^{3} + \cdots + 2882779 \)
|
| $23$ |
\( T^{4} + 40 T^{3} + \cdots - 163259 \)
|
| $29$ |
\( T^{4} + 343 T^{3} + \cdots + 11513725 \)
|
| $31$ |
\( T^{4} + \cdots - 1504915219 \)
|
| $37$ |
\( T^{4} + \cdots + 2699334311 \)
|
| $41$ |
\( T^{4} + \cdots + 1294447625 \)
|
| $43$ |
\( T^{4} - 678 T^{3} + \cdots - 950575601 \)
|
| $47$ |
\( T^{4} - 282 T^{3} + \cdots + 63570695 \)
|
| $53$ |
\( T^{4} + \cdots + 28796531059 \)
|
| $59$ |
\( T^{4} + \cdots - 9716759305 \)
|
| $61$ |
\( T^{4} + \cdots + 7779459391 \)
|
| $67$ |
\( T^{4} + \cdots - 7487660169 \)
|
| $71$ |
\( T^{4} + \cdots + 7019148619 \)
|
| $73$ |
\( T^{4} + \cdots + 285179060069 \)
|
| $79$ |
\( T^{4} + \cdots + 204415644895 \)
|
| $83$ |
\( T^{4} + \cdots + 651982614061 \)
|
| $89$ |
\( T^{4} + \cdots - 1863561605 \)
|
| $97$ |
\( T^{4} + \cdots + 491066725369 \)
|
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