Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [8,5,Mod(3,8)]
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([1, 1]))
N = Newforms(chi, 5, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("8.3");
S:= CuspForms(chi, 5);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 8 = 2^{3} \) |
| Weight: | \( k \) | \(=\) | \( 5 \) |
| Character orbit: | \([\chi]\) | \(=\) | 8.d (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: | \(0.826959704671\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{-15}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x + 4 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| 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 \(\beta = \sqrt{-15}\). 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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 3.1 |
|
−1.00000 | − | 3.87298i | 6.00000 | −14.0000 | + | 7.74597i | 30.9839i | −6.00000 | − | 23.2379i | − | 61.9677i | 44.0000 | + | 46.4758i | −45.0000 | 120.000 | − | 30.9839i | |||||||||||||
| 3.2 | −1.00000 | + | 3.87298i | 6.00000 | −14.0000 | − | 7.74597i | − | 30.9839i | −6.00000 | + | 23.2379i | 61.9677i | 44.0000 | − | 46.4758i | −45.0000 | 120.000 | + | 30.9839i | ||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 8.d | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 8.5.d.b | ✓ | 2 |
| 3.b | odd | 2 | 1 | 72.5.b.b | 2 | ||
| 4.b | odd | 2 | 1 | 32.5.d.b | 2 | ||
| 5.b | even | 2 | 1 | 200.5.g.d | 2 | ||
| 5.c | odd | 4 | 2 | 200.5.e.c | 4 | ||
| 8.b | even | 2 | 1 | 32.5.d.b | 2 | ||
| 8.d | odd | 2 | 1 | inner | 8.5.d.b | ✓ | 2 |
| 12.b | even | 2 | 1 | 288.5.b.b | 2 | ||
| 16.e | even | 4 | 2 | 256.5.c.i | 4 | ||
| 16.f | odd | 4 | 2 | 256.5.c.i | 4 | ||
| 20.d | odd | 2 | 1 | 800.5.g.d | 2 | ||
| 20.e | even | 4 | 2 | 800.5.e.c | 4 | ||
| 24.f | even | 2 | 1 | 72.5.b.b | 2 | ||
| 24.h | odd | 2 | 1 | 288.5.b.b | 2 | ||
| 40.e | odd | 2 | 1 | 200.5.g.d | 2 | ||
| 40.f | even | 2 | 1 | 800.5.g.d | 2 | ||
| 40.i | odd | 4 | 2 | 800.5.e.c | 4 | ||
| 40.k | even | 4 | 2 | 200.5.e.c | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 8.5.d.b | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
| 8.5.d.b | ✓ | 2 | 8.d | odd | 2 | 1 | inner |
| 32.5.d.b | 2 | 4.b | odd | 2 | 1 | ||
| 32.5.d.b | 2 | 8.b | even | 2 | 1 | ||
| 72.5.b.b | 2 | 3.b | odd | 2 | 1 | ||
| 72.5.b.b | 2 | 24.f | even | 2 | 1 | ||
| 200.5.e.c | 4 | 5.c | odd | 4 | 2 | ||
| 200.5.e.c | 4 | 40.k | even | 4 | 2 | ||
| 200.5.g.d | 2 | 5.b | even | 2 | 1 | ||
| 200.5.g.d | 2 | 40.e | odd | 2 | 1 | ||
| 256.5.c.i | 4 | 16.e | even | 4 | 2 | ||
| 256.5.c.i | 4 | 16.f | odd | 4 | 2 | ||
| 288.5.b.b | 2 | 12.b | even | 2 | 1 | ||
| 288.5.b.b | 2 | 24.h | odd | 2 | 1 | ||
| 800.5.e.c | 4 | 20.e | even | 4 | 2 | ||
| 800.5.e.c | 4 | 40.i | odd | 4 | 2 | ||
| 800.5.g.d | 2 | 20.d | odd | 2 | 1 | ||
| 800.5.g.d | 2 | 40.f | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3} - 6 \)
acting on \(S_{5}^{\mathrm{new}}(8, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} + 2T + 16 \)
|
| $3$ |
\( (T - 6)^{2} \)
|
| $5$ |
\( T^{2} + 960 \)
|
| $7$ |
\( T^{2} + 3840 \)
|
| $11$ |
\( (T + 26)^{2} \)
|
| $13$ |
\( T^{2} + 960 \)
|
| $17$ |
\( (T - 226)^{2} \)
|
| $19$ |
\( (T - 134)^{2} \)
|
| $23$ |
\( T^{2} + 96000 \)
|
| $29$ |
\( T^{2} + 116160 \)
|
| $31$ |
\( T^{2} + 1536000 \)
|
| $37$ |
\( T^{2} + 3119040 \)
|
| $41$ |
\( (T - 994)^{2} \)
|
| $43$ |
\( (T + 1882)^{2} \)
|
| $47$ |
\( T^{2} + 4439040 \)
|
| $53$ |
\( T^{2} + 14523840 \)
|
| $59$ |
\( (T + 5018)^{2} \)
|
| $61$ |
\( T^{2} + 4309440 \)
|
| $67$ |
\( (T - 8006)^{2} \)
|
| $71$ |
\( T^{2} + 311040 \)
|
| $73$ |
\( (T - 386)^{2} \)
|
| $79$ |
\( T^{2} + 121666560 \)
|
| $83$ |
\( (T + 2234)^{2} \)
|
| $89$ |
\( (T + 10046)^{2} \)
|
| $97$ |
\( (T - 8738)^{2} \)
|
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