Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [32,8,Mod(1,32)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(32, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("32.1");
S:= CuspForms(chi, 8);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 32 = 2^{5} \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 32.a (trivial) |
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: | yes |
| Analytic conductor: | \(9.99632081549\) |
| Analytic rank: | \(1\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{10}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - 10 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{4} \) |
| Twist minimal: | yes |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-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 \(\beta = 16\sqrt{10}\). We also show the integral \(q\)-expansion of the trace form.
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
0 | −42.5964 | 0 | 314.772 | 0 | 84.3502 | 0 | −372.543 | 0 | ||||||||||||||||||||||||
| 1.2 | 0 | 58.5964 | 0 | −494.772 | 0 | −1332.35 | 0 | 1246.54 | 0 | |||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 32.8.a.d | yes | 2 |
| 3.b | odd | 2 | 1 | 288.8.a.n | 2 | ||
| 4.b | odd | 2 | 1 | 32.8.a.b | ✓ | 2 | |
| 8.b | even | 2 | 1 | 64.8.a.h | 2 | ||
| 8.d | odd | 2 | 1 | 64.8.a.j | 2 | ||
| 12.b | even | 2 | 1 | 288.8.a.o | 2 | ||
| 16.e | even | 4 | 2 | 256.8.b.j | 4 | ||
| 16.f | odd | 4 | 2 | 256.8.b.h | 4 | ||
| 24.f | even | 2 | 1 | 576.8.a.bf | 2 | ||
| 24.h | odd | 2 | 1 | 576.8.a.be | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 32.8.a.b | ✓ | 2 | 4.b | odd | 2 | 1 | |
| 32.8.a.d | yes | 2 | 1.a | even | 1 | 1 | trivial |
| 64.8.a.h | 2 | 8.b | even | 2 | 1 | ||
| 64.8.a.j | 2 | 8.d | odd | 2 | 1 | ||
| 256.8.b.h | 4 | 16.f | odd | 4 | 2 | ||
| 256.8.b.j | 4 | 16.e | even | 4 | 2 | ||
| 288.8.a.n | 2 | 3.b | odd | 2 | 1 | ||
| 288.8.a.o | 2 | 12.b | even | 2 | 1 | ||
| 576.8.a.be | 2 | 24.h | odd | 2 | 1 | ||
| 576.8.a.bf | 2 | 24.f | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} - 16T_{3} - 2496 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(32))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} \)
|
| $3$ |
\( T^{2} - 16T - 2496 \)
|
| $5$ |
\( T^{2} + 180T - 155740 \)
|
| $7$ |
\( T^{2} + 1248 T - 112384 \)
|
| $11$ |
\( T^{2} + 9040 T + 6030400 \)
|
| $13$ |
\( T^{2} + 2500 T - 35301500 \)
|
| $17$ |
\( T^{2} - 17220 T - 73323900 \)
|
| $19$ |
\( T^{2} + \cdots + 1313422400 \)
|
| $23$ |
\( T^{2} + \cdots - 4248481536 \)
|
| $29$ |
\( T^{2} + \cdots + 2619280196 \)
|
| $31$ |
\( T^{2} + \cdots + 3425177600 \)
|
| $37$ |
\( T^{2} + \cdots + 57113332900 \)
|
| $41$ |
\( T^{2} + \cdots + 124732277540 \)
|
| $43$ |
\( T^{2} + \cdots - 95043921856 \)
|
| $47$ |
\( T^{2} + \cdots - 38062767104 \)
|
| $53$ |
\( T^{2} + \cdots - 412809891100 \)
|
| $59$ |
\( T^{2} + \cdots - 285681944000 \)
|
| $61$ |
\( T^{2} + \cdots - 775792836540 \)
|
| $67$ |
\( T^{2} + \cdots + 1403322476096 \)
|
| $71$ |
\( T^{2} + \cdots + 759262624000 \)
|
| $73$ |
\( T^{2} + \cdots - 16148101167900 \)
|
| $79$ |
\( T^{2} + \cdots + 1551858713600 \)
|
| $83$ |
\( T^{2} + \cdots + 10945727913024 \)
|
| $89$ |
\( T^{2} + \cdots - 14087847786844 \)
|
| $97$ |
\( T^{2} + \cdots - 20193384103100 \)
|
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