Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [169,4,Mod(168,169)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(169, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("169.168");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 169 = 13^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 169.b (of order \(2\), degree \(1\), not 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: | \(9.97132279097\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(i, \sqrt{17})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 9x^{2} + 16 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 13) |
| 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
\( x^{4} + 9x^{2} + 16 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} + 5\nu ) / 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{2} + 5 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} - 5 \)
|
| \(\nu^{3}\) | \(=\) |
\( 4\beta_{2} - 5\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/169\mathbb{Z}\right)^\times\).
| \(n\) | \(2\) |
| \(\chi(n)\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 168.1 |
|
− | 4.56155i | 8.68466 | −12.8078 | 2.80776i | − | 39.6155i | − | 9.56155i | 21.9309i | 48.4233 | 12.8078 | |||||||||||||||||||||||||||
| 168.2 | − | 0.438447i | −3.68466 | 7.80776 | − | 17.8078i | 1.61553i | − | 5.43845i | − | 6.93087i | −13.4233 | −7.80776 | |||||||||||||||||||||||||||
| 168.3 | 0.438447i | −3.68466 | 7.80776 | 17.8078i | − | 1.61553i | 5.43845i | 6.93087i | −13.4233 | −7.80776 | ||||||||||||||||||||||||||||||
| 168.4 | 4.56155i | 8.68466 | −12.8078 | − | 2.80776i | 39.6155i | 9.56155i | − | 21.9309i | 48.4233 | 12.8078 | |||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 13.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 169.4.b.e | 4 | |
| 13.b | even | 2 | 1 | inner | 169.4.b.e | 4 | |
| 13.c | even | 3 | 2 | 169.4.e.g | 8 | ||
| 13.d | odd | 4 | 1 | 169.4.a.f | 2 | ||
| 13.d | odd | 4 | 1 | 169.4.a.j | 2 | ||
| 13.e | even | 6 | 2 | 169.4.e.g | 8 | ||
| 13.f | odd | 12 | 2 | 13.4.c.b | ✓ | 4 | |
| 13.f | odd | 12 | 2 | 169.4.c.f | 4 | ||
| 39.f | even | 4 | 1 | 1521.4.a.l | 2 | ||
| 39.f | even | 4 | 1 | 1521.4.a.t | 2 | ||
| 39.k | even | 12 | 2 | 117.4.g.d | 4 | ||
| 52.l | even | 12 | 2 | 208.4.i.e | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 13.4.c.b | ✓ | 4 | 13.f | odd | 12 | 2 | |
| 117.4.g.d | 4 | 39.k | even | 12 | 2 | ||
| 169.4.a.f | 2 | 13.d | odd | 4 | 1 | ||
| 169.4.a.j | 2 | 13.d | odd | 4 | 1 | ||
| 169.4.b.e | 4 | 1.a | even | 1 | 1 | trivial | |
| 169.4.b.e | 4 | 13.b | even | 2 | 1 | inner | |
| 169.4.c.f | 4 | 13.f | odd | 12 | 2 | ||
| 169.4.e.g | 8 | 13.c | even | 3 | 2 | ||
| 169.4.e.g | 8 | 13.e | even | 6 | 2 | ||
| 208.4.i.e | 4 | 52.l | even | 12 | 2 | ||
| 1521.4.a.l | 2 | 39.f | even | 4 | 1 | ||
| 1521.4.a.t | 2 | 39.f | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} + 21T_{2}^{2} + 4 \)
acting on \(S_{4}^{\mathrm{new}}(169, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + 21T^{2} + 4 \)
|
| $3$ |
\( (T^{2} - 5 T - 32)^{2} \)
|
| $5$ |
\( T^{4} + 325T^{2} + 2500 \)
|
| $7$ |
\( T^{4} + 121T^{2} + 2704 \)
|
| $11$ |
\( T^{4} + 2057 T^{2} + 781456 \)
|
| $13$ |
\( T^{4} \)
|
| $17$ |
\( (T^{2} + 70 T + 137)^{2} \)
|
| $19$ |
\( T^{4} + 10153 T^{2} + 23658496 \)
|
| $23$ |
\( (T^{2} + 145 T + 628)^{2} \)
|
| $29$ |
\( (T^{2} - 34 T - 15011)^{2} \)
|
| $31$ |
\( T^{4} + \cdots + 1413760000 \)
|
| $37$ |
\( T^{4} + 34506 T^{2} + 635209 \)
|
| $41$ |
\( T^{4} + \cdots + 4992976921 \)
|
| $43$ |
\( (T^{2} + 455 T + 11768)^{2} \)
|
| $47$ |
\( T^{4} + \cdots + 6789760000 \)
|
| $53$ |
\( (T^{2} - 545 T - 41450)^{2} \)
|
| $59$ |
\( T^{4} + \cdots + 22729783696 \)
|
| $61$ |
\( (T^{2} - 502 T - 106999)^{2} \)
|
| $67$ |
\( T^{4} + 183201 T^{2} + 449948944 \)
|
| $71$ |
\( T^{4} + \cdots + 1833894976 \)
|
| $73$ |
\( T^{4} + \cdots + 3008522500 \)
|
| $79$ |
\( (T^{2} - 240 T + 7600)^{2} \)
|
| $83$ |
\( T^{4} + 118800 T^{2} + 655360000 \)
|
| $89$ |
\( T^{4} + \cdots + 21215087716 \)
|
| $97$ |
\( T^{4} + \cdots + 795268001284 \)
|
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