Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [50,20,Mod(49,50)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(50, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1]))
N = Newforms(chi, 20, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("50.49");
S:= CuspForms(chi, 20);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 50 = 2 \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 20 \) |
| Character orbit: | \([\chi]\) | \(=\) | 50.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: | \(114.408348278\) |
| 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: |
\( x^{4} + 5851705x^{2} + 8560609925904 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{6}\cdot 5^{2} \) |
| Twist minimal: | no (minimal twist has level 10) |
| 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
\( x^{4} + 5851705x^{2} + 8560609925904 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} + 2925853\nu ) / 1462926 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 5\nu^{3} + 43887785\nu ) / 1462926 \)
|
| \(\beta_{3}\) | \(=\) |
\( 40\nu^{2} + 117034100 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} - 5\beta_1 ) / 20 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} - 117034100 ) / 40 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -2925853\beta_{2} + 43887785\beta_1 ) / 20 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/50\mathbb{Z}\right)^\times\).
| \(n\) | \(27\) |
| \(\chi(n)\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 49.1 |
|
− | 512.000i | − | 17348.2i | −262144. | 0 | −8.88230e6 | − | 5.57963e7i | 1.34218e8i | 8.61300e8 | 0 | |||||||||||||||||||||||||||
| 49.2 | − | 512.000i | 51072.2i | −262144. | 0 | 2.61490e7 | − | 2.72650e7i | 1.34218e8i | −1.44611e9 | 0 | |||||||||||||||||||||||||||||
| 49.3 | 512.000i | − | 51072.2i | −262144. | 0 | 2.61490e7 | 2.72650e7i | − | 1.34218e8i | −1.44611e9 | 0 | |||||||||||||||||||||||||||||
| 49.4 | 512.000i | 17348.2i | −262144. | 0 | −8.88230e6 | 5.57963e7i | − | 1.34218e8i | 8.61300e8 | 0 | ||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 50.20.b.f | 4 | |
| 5.b | even | 2 | 1 | inner | 50.20.b.f | 4 | |
| 5.c | odd | 4 | 1 | 10.20.a.d | ✓ | 2 | |
| 5.c | odd | 4 | 1 | 50.20.a.f | 2 | ||
| 15.e | even | 4 | 1 | 90.20.a.g | 2 | ||
| 20.e | even | 4 | 1 | 80.20.a.d | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 10.20.a.d | ✓ | 2 | 5.c | odd | 4 | 1 | |
| 50.20.a.f | 2 | 5.c | odd | 4 | 1 | ||
| 50.20.b.f | 4 | 1.a | even | 1 | 1 | trivial | |
| 50.20.b.f | 4 | 5.b | even | 2 | 1 | inner | |
| 80.20.a.d | 2 | 20.e | even | 4 | 1 | ||
| 90.20.a.g | 2 | 15.e | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} + 2909335888T_{3}^{2} + 785020553023988736 \)
acting on \(S_{20}^{\mathrm{new}}(50, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 262144)^{2} \)
|
| $3$ |
\( T^{4} + \cdots + 78\!\cdots\!36 \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( T^{4} + \cdots + 23\!\cdots\!56 \)
|
| $11$ |
\( (T^{2} + \cdots - 47\!\cdots\!16)^{2} \)
|
| $13$ |
\( T^{4} + \cdots + 72\!\cdots\!76 \)
|
| $17$ |
\( T^{4} + \cdots + 16\!\cdots\!96 \)
|
| $19$ |
\( (T^{2} + \cdots - 22\!\cdots\!00)^{2} \)
|
| $23$ |
\( T^{4} + \cdots + 22\!\cdots\!96 \)
|
| $29$ |
\( (T^{2} + \cdots + 34\!\cdots\!00)^{2} \)
|
| $31$ |
\( (T^{2} + \cdots + 43\!\cdots\!24)^{2} \)
|
| $37$ |
\( T^{4} + \cdots + 84\!\cdots\!16 \)
|
| $41$ |
\( (T^{2} + \cdots - 39\!\cdots\!96)^{2} \)
|
| $43$ |
\( T^{4} + \cdots + 25\!\cdots\!76 \)
|
| $47$ |
\( T^{4} + \cdots + 40\!\cdots\!96 \)
|
| $53$ |
\( T^{4} + \cdots + 48\!\cdots\!96 \)
|
| $59$ |
\( (T^{2} + \cdots + 90\!\cdots\!00)^{2} \)
|
| $61$ |
\( (T^{2} + \cdots - 50\!\cdots\!56)^{2} \)
|
| $67$ |
\( T^{4} + \cdots + 22\!\cdots\!36 \)
|
| $71$ |
\( (T^{2} + \cdots - 14\!\cdots\!96)^{2} \)
|
| $73$ |
\( T^{4} + \cdots + 25\!\cdots\!56 \)
|
| $79$ |
\( (T^{2} + \cdots - 92\!\cdots\!00)^{2} \)
|
| $83$ |
\( T^{4} + \cdots + 27\!\cdots\!16 \)
|
| $89$ |
\( (T^{2} + \cdots - 17\!\cdots\!00)^{2} \)
|
| $97$ |
\( T^{4} + \cdots + 14\!\cdots\!36 \)
|
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