Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [84,8,Mod(1,84)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(84, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("84.1");
S:= CuspForms(chi, 8);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 84 = 2^{2} \cdot 3 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 84.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: | \(26.2403421407\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{3649}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x - 912 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{2}\cdot 3 \) |
| 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 \(\beta = 6\sqrt{3649}\). 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 | −27.0000 | 0 | −230.442 | 0 | 343.000 | 0 | 729.000 | 0 | ||||||||||||||||||||||||
| 1.2 | 0 | −27.0000 | 0 | 494.442 | 0 | 343.000 | 0 | 729.000 | 0 | |||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(3\) | \( +1 \) |
| \(7\) | \( -1 \) |
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 | 84.8.a.c | ✓ | 2 |
| 3.b | odd | 2 | 1 | 252.8.a.c | 2 | ||
| 4.b | odd | 2 | 1 | 336.8.a.q | 2 | ||
| 7.b | odd | 2 | 1 | 588.8.a.f | 2 | ||
| 7.c | even | 3 | 2 | 588.8.i.k | 4 | ||
| 7.d | odd | 6 | 2 | 588.8.i.j | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 84.8.a.c | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
| 252.8.a.c | 2 | 3.b | odd | 2 | 1 | ||
| 336.8.a.q | 2 | 4.b | odd | 2 | 1 | ||
| 588.8.a.f | 2 | 7.b | odd | 2 | 1 | ||
| 588.8.i.j | 4 | 7.d | odd | 6 | 2 | ||
| 588.8.i.k | 4 | 7.c | even | 3 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{2} - 264T_{5} - 113940 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(84))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} \)
|
| $3$ |
\( (T + 27)^{2} \)
|
| $5$ |
\( T^{2} - 264T - 113940 \)
|
| $7$ |
\( (T - 343)^{2} \)
|
| $11$ |
\( T^{2} + 4980 T - 236736 \)
|
| $13$ |
\( T^{2} + 10148 T - 49920188 \)
|
| $17$ |
\( T^{2} - 17832 T - 475517844 \)
|
| $19$ |
\( T^{2} - 6256 T - 32777552 \)
|
| $23$ |
\( T^{2} - 14052 T - 576059328 \)
|
| $29$ |
\( T^{2} + \cdots + 14505368436 \)
|
| $31$ |
\( T^{2} + \cdots + 55376247808 \)
|
| $37$ |
\( T^{2} + \cdots - 55723566236 \)
|
| $41$ |
\( T^{2} + \cdots + 189493906140 \)
|
| $43$ |
\( T^{2} + \cdots - 246999010736 \)
|
| $47$ |
\( T^{2} + \cdots - 549432902016 \)
|
| $53$ |
\( T^{2} + \cdots + 1377408671172 \)
|
| $59$ |
\( T^{2} + \cdots - 6640635738384 \)
|
| $61$ |
\( T^{2} + \cdots - 3780042097868 \)
|
| $67$ |
\( T^{2} + \cdots - 9067890068432 \)
|
| $71$ |
\( T^{2} + \cdots - 3623141894400 \)
|
| $73$ |
\( T^{2} + \cdots - 10805280342380 \)
|
| $79$ |
\( T^{2} + \cdots - 795441820160 \)
|
| $83$ |
\( T^{2} + \cdots - 4261335046704 \)
|
| $89$ |
\( T^{2} + \cdots + 21509111254428 \)
|
| $97$ |
\( T^{2} + \cdots - 89368449155180 \)
|
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