Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [26,4,Mod(25,26)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(26, 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("26.25");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 26 = 2 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 26.b (of order \(2\), degree \(1\), 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: | \(1.53404966015\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(i, \sqrt{217})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 109x^{2} + 2916 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | yes |
| 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
\( x^{4} + 109x^{2} + 2916 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{3} - 109\nu ) / 54 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} + 55\nu ) / 27 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{2} + 55 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{2} - 2\beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} - 55 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 109\beta_{2} + 110\beta_1 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/26\mathbb{Z}\right)^\times\).
| \(n\) | \(15\) |
| \(\chi(n)\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 25.1 |
|
− | 2.00000i | −8.86546 | −4.00000 | − | 16.8655i | 17.7309i | 10.8655i | 8.00000i | 51.5964 | −33.7309 | ||||||||||||||||||||||||||||
| 25.2 | − | 2.00000i | 5.86546 | −4.00000 | − | 2.13454i | − | 11.7309i | − | 3.86546i | 8.00000i | 7.40362 | −4.26908 | |||||||||||||||||||||||||||
| 25.3 | 2.00000i | −8.86546 | −4.00000 | 16.8655i | − | 17.7309i | − | 10.8655i | − | 8.00000i | 51.5964 | −33.7309 | ||||||||||||||||||||||||||||
| 25.4 | 2.00000i | 5.86546 | −4.00000 | 2.13454i | 11.7309i | 3.86546i | − | 8.00000i | 7.40362 | −4.26908 | ||||||||||||||||||||||||||||||
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 | 26.4.b.a | ✓ | 4 |
| 3.b | odd | 2 | 1 | 234.4.b.b | 4 | ||
| 4.b | odd | 2 | 1 | 208.4.f.d | 4 | ||
| 5.b | even | 2 | 1 | 650.4.d.d | 4 | ||
| 5.c | odd | 4 | 1 | 650.4.c.e | 4 | ||
| 5.c | odd | 4 | 1 | 650.4.c.f | 4 | ||
| 8.b | even | 2 | 1 | 832.4.f.j | 4 | ||
| 8.d | odd | 2 | 1 | 832.4.f.h | 4 | ||
| 13.b | even | 2 | 1 | inner | 26.4.b.a | ✓ | 4 |
| 13.c | even | 3 | 2 | 338.4.e.g | 8 | ||
| 13.d | odd | 4 | 1 | 338.4.a.f | 2 | ||
| 13.d | odd | 4 | 1 | 338.4.a.i | 2 | ||
| 13.e | even | 6 | 2 | 338.4.e.g | 8 | ||
| 13.f | odd | 12 | 2 | 338.4.c.h | 4 | ||
| 13.f | odd | 12 | 2 | 338.4.c.i | 4 | ||
| 39.d | odd | 2 | 1 | 234.4.b.b | 4 | ||
| 52.b | odd | 2 | 1 | 208.4.f.d | 4 | ||
| 65.d | even | 2 | 1 | 650.4.d.d | 4 | ||
| 65.h | odd | 4 | 1 | 650.4.c.e | 4 | ||
| 65.h | odd | 4 | 1 | 650.4.c.f | 4 | ||
| 104.e | even | 2 | 1 | 832.4.f.j | 4 | ||
| 104.h | odd | 2 | 1 | 832.4.f.h | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 26.4.b.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 26.4.b.a | ✓ | 4 | 13.b | even | 2 | 1 | inner |
| 208.4.f.d | 4 | 4.b | odd | 2 | 1 | ||
| 208.4.f.d | 4 | 52.b | odd | 2 | 1 | ||
| 234.4.b.b | 4 | 3.b | odd | 2 | 1 | ||
| 234.4.b.b | 4 | 39.d | odd | 2 | 1 | ||
| 338.4.a.f | 2 | 13.d | odd | 4 | 1 | ||
| 338.4.a.i | 2 | 13.d | odd | 4 | 1 | ||
| 338.4.c.h | 4 | 13.f | odd | 12 | 2 | ||
| 338.4.c.i | 4 | 13.f | odd | 12 | 2 | ||
| 338.4.e.g | 8 | 13.c | even | 3 | 2 | ||
| 338.4.e.g | 8 | 13.e | even | 6 | 2 | ||
| 650.4.c.e | 4 | 5.c | odd | 4 | 1 | ||
| 650.4.c.e | 4 | 65.h | odd | 4 | 1 | ||
| 650.4.c.f | 4 | 5.c | odd | 4 | 1 | ||
| 650.4.c.f | 4 | 65.h | odd | 4 | 1 | ||
| 650.4.d.d | 4 | 5.b | even | 2 | 1 | ||
| 650.4.d.d | 4 | 65.d | even | 2 | 1 | ||
| 832.4.f.h | 4 | 8.d | odd | 2 | 1 | ||
| 832.4.f.h | 4 | 104.h | odd | 2 | 1 | ||
| 832.4.f.j | 4 | 8.b | even | 2 | 1 | ||
| 832.4.f.j | 4 | 104.e | even | 2 | 1 | ||
Hecke kernels
This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(26, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 4)^{2} \)
|
| $3$ |
\( (T^{2} + 3 T - 52)^{2} \)
|
| $5$ |
\( T^{4} + 289T^{2} + 1296 \)
|
| $7$ |
\( T^{4} + 133T^{2} + 1764 \)
|
| $11$ |
\( T^{4} + 4068 T^{2} + 3504384 \)
|
| $13$ |
\( T^{4} + 110 T^{3} + \cdots + 4826809 \)
|
| $17$ |
\( (T^{2} - 13 T - 1314)^{2} \)
|
| $19$ |
\( T^{4} + 22824 T^{2} + 17740944 \)
|
| $23$ |
\( (T^{2} + 98 T - 3024)^{2} \)
|
| $29$ |
\( (T^{2} - 374 T + 24336)^{2} \)
|
| $31$ |
\( T^{4} + \cdots + 1747908864 \)
|
| $37$ |
\( T^{4} + 31441 T^{2} + 244484496 \)
|
| $41$ |
\( T^{4} + \cdots + 1052872704 \)
|
| $43$ |
\( (T^{2} + 219 T - 21916)^{2} \)
|
| $47$ |
\( T^{4} + \cdots + 2466314244 \)
|
| $53$ |
\( (T^{2} - 24 T - 7668)^{2} \)
|
| $59$ |
\( T^{4} + \cdots + 7836498576 \)
|
| $61$ |
\( (T^{2} + 282 T - 94912)^{2} \)
|
| $67$ |
\( T^{4} + \cdots + 17799829056 \)
|
| $71$ |
\( T^{4} + \cdots + 11761402500 \)
|
| $73$ |
\( T^{4} + \cdots + 8100000000 \)
|
| $79$ |
\( (T^{2} + 722 T - 166752)^{2} \)
|
| $83$ |
\( T^{4} + 1623976 T^{2} + 797271696 \)
|
| $89$ |
\( T^{4} + \cdots + 827983524096 \)
|
| $97$ |
\( T^{4} + \cdots + 749484969984 \)
|
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