Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [169,8,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, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("169.168");
S:= CuspForms(chi, 8);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 169 = 13^{2} \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| 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: | \(52.7930693068\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(i, \sqrt{337})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 169x^{2} + 7056 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2 \) |
| 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
\( x^{4} + 169x^{2} + 7056 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} + 85\nu ) / 42 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{2} + 85 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} - 85 \)
|
| \(\nu^{3}\) | \(=\) |
\( 42\beta_{2} - 85\beta_1 \)
|
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 |
|
− | 18.6788i | 50.0363 | −220.897 | − | 277.467i | − | 934.618i | 921.891i | 1735.20i | 316.635 | −5182.74 | |||||||||||||||||||||||||||
| 168.2 | − | 0.321220i | −5.03634 | 127.897 | − | 75.5334i | 1.61777i | 1087.11i | − | 82.1992i | −2161.64 | −24.2629 | ||||||||||||||||||||||||||||
| 168.3 | 0.321220i | −5.03634 | 127.897 | 75.5334i | − | 1.61777i | − | 1087.11i | 82.1992i | −2161.64 | −24.2629 | |||||||||||||||||||||||||||||
| 168.4 | 18.6788i | 50.0363 | −220.897 | 277.467i | 934.618i | − | 921.891i | − | 1735.20i | 316.635 | −5182.74 | |||||||||||||||||||||||||||||
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 | 169.8.b.b | 4 | |
| 13.b | even | 2 | 1 | inner | 169.8.b.b | 4 | |
| 13.d | odd | 4 | 1 | 13.8.a.b | ✓ | 2 | |
| 13.d | odd | 4 | 1 | 169.8.a.b | 2 | ||
| 39.f | even | 4 | 1 | 117.8.a.c | 2 | ||
| 52.f | even | 4 | 1 | 208.8.a.g | 2 | ||
| 65.g | odd | 4 | 1 | 325.8.a.b | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 13.8.a.b | ✓ | 2 | 13.d | odd | 4 | 1 | |
| 117.8.a.c | 2 | 39.f | even | 4 | 1 | ||
| 169.8.a.b | 2 | 13.d | odd | 4 | 1 | ||
| 169.8.b.b | 4 | 1.a | even | 1 | 1 | trivial | |
| 169.8.b.b | 4 | 13.b | even | 2 | 1 | inner | |
| 208.8.a.g | 2 | 52.f | even | 4 | 1 | ||
| 325.8.a.b | 2 | 65.g | odd | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} + 349T_{2}^{2} + 36 \)
acting on \(S_{8}^{\mathrm{new}}(169, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + 349T^{2} + 36 \)
|
| $3$ |
\( (T^{2} - 45 T - 252)^{2} \)
|
| $5$ |
\( T^{4} + 82693 T^{2} + 439237764 \)
|
| $7$ |
\( T^{4} + \cdots + 1004396822416 \)
|
| $11$ |
\( T^{4} + \cdots + 10\!\cdots\!04 \)
|
| $13$ |
\( T^{4} \)
|
| $17$ |
\( (T^{2} - 25361 T - 177216678)^{2} \)
|
| $19$ |
\( T^{4} + \cdots + 27\!\cdots\!44 \)
|
| $23$ |
\( (T^{2} - 26424 T - 3712002048)^{2} \)
|
| $29$ |
\( (T^{2} + 5804 T - 1049753004)^{2} \)
|
| $31$ |
\( T^{4} + \cdots + 40\!\cdots\!96 \)
|
| $37$ |
\( T^{4} + \cdots + 61\!\cdots\!36 \)
|
| $41$ |
\( T^{4} + \cdots + 13\!\cdots\!04 \)
|
| $43$ |
\( (T^{2} + 229307 T + 11546069484)^{2} \)
|
| $47$ |
\( T^{4} + \cdots + 44\!\cdots\!04 \)
|
| $53$ |
\( (T^{2} - 1046382 T - 648240166944)^{2} \)
|
| $59$ |
\( T^{4} + \cdots + 19\!\cdots\!84 \)
|
| $61$ |
\( (T^{2} + \cdots + 4947441275696)^{2} \)
|
| $67$ |
\( T^{4} + \cdots + 29\!\cdots\!36 \)
|
| $71$ |
\( T^{4} + \cdots + 61\!\cdots\!56 \)
|
| $73$ |
\( T^{4} + \cdots + 10\!\cdots\!96 \)
|
| $79$ |
\( (T^{2} + \cdots - 22472459585984)^{2} \)
|
| $83$ |
\( T^{4} + \cdots + 11\!\cdots\!16 \)
|
| $89$ |
\( T^{4} + \cdots + 10\!\cdots\!24 \)
|
| $97$ |
\( T^{4} + \cdots + 20\!\cdots\!16 \)
|
| show more | |
| show less |