Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [8,14,Mod(5,8)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1]))
N = Newforms(chi, 14, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("8.5");
S:= CuspForms(chi, 14);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 8 = 2^{3} \) |
| Weight: | \( k \) | \(=\) | \( 14 \) |
| Character orbit: | \([\chi]\) | \(=\) | 8.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: | \(8.57847431615\) |
| Analytic rank: | \(1\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{-79}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x + 20 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{2} \) |
| 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 \(\beta = 2\sqrt{-79}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/8\mathbb{Z}\right)^\times\).
| \(n\) | \(5\) | \(7\) |
| \(\chi(n)\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 5.1 |
|
−56.0000 | − | 71.1056i | 2293.15i | −1920.00 | + | 7963.82i | − | 22576.0i | 163056. | − | 128417.i | −175832. | 673792. | − | 309451.i | −3.66423e6 | −1.60528e6 | + | 1.26426e6i | |||||||||||||
| 5.2 | −56.0000 | + | 71.1056i | − | 2293.15i | −1920.00 | − | 7963.82i | 22576.0i | 163056. | + | 128417.i | −175832. | 673792. | + | 309451.i | −3.66423e6 | −1.60528e6 | − | 1.26426e6i | ||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 8.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 8.14.b.a | ✓ | 2 |
| 3.b | odd | 2 | 1 | 72.14.d.b | 2 | ||
| 4.b | odd | 2 | 1 | 32.14.b.a | 2 | ||
| 8.b | even | 2 | 1 | inner | 8.14.b.a | ✓ | 2 |
| 8.d | odd | 2 | 1 | 32.14.b.a | 2 | ||
| 24.h | odd | 2 | 1 | 72.14.d.b | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 8.14.b.a | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
| 8.14.b.a | ✓ | 2 | 8.b | even | 2 | 1 | inner |
| 32.14.b.a | 2 | 4.b | odd | 2 | 1 | ||
| 32.14.b.a | 2 | 8.d | odd | 2 | 1 | ||
| 72.14.d.b | 2 | 3.b | odd | 2 | 1 | ||
| 72.14.d.b | 2 | 24.h | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} + 5258556 \)
acting on \(S_{14}^{\mathrm{new}}(8, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} + 112T + 8192 \)
|
| $3$ |
\( T^{2} + 5258556 \)
|
| $5$ |
\( T^{2} + 509676400 \)
|
| $7$ |
\( (T + 175832)^{2} \)
|
| $11$ |
\( T^{2} + 6944599287900 \)
|
| $13$ |
\( T^{2} + 966142544060656 \)
|
| $17$ |
\( (T + 133520302)^{2} \)
|
| $19$ |
\( T^{2} + 12\!\cdots\!36 \)
|
| $23$ |
\( (T + 35585416)^{2} \)
|
| $29$ |
\( T^{2} + 25\!\cdots\!36 \)
|
| $31$ |
\( (T + 5765001568)^{2} \)
|
| $37$ |
\( T^{2} + 17\!\cdots\!24 \)
|
| $41$ |
\( (T + 23546348918)^{2} \)
|
| $43$ |
\( T^{2} + 21\!\cdots\!56 \)
|
| $47$ |
\( (T + 68107736592)^{2} \)
|
| $53$ |
\( T^{2} + 27\!\cdots\!16 \)
|
| $59$ |
\( T^{2} + 16\!\cdots\!36 \)
|
| $61$ |
\( T^{2} + 17\!\cdots\!00 \)
|
| $67$ |
\( T^{2} + 14\!\cdots\!44 \)
|
| $71$ |
\( (T + 1309471657368)^{2} \)
|
| $73$ |
\( (T - 478647871914)^{2} \)
|
| $79$ |
\( (T + 364547231600)^{2} \)
|
| $83$ |
\( T^{2} + 76\!\cdots\!56 \)
|
| $89$ |
\( (T + 102457641350)^{2} \)
|
| $97$ |
\( (T + 6157717373342)^{2} \)
|
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