Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [184,4,Mod(1,184)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(184, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("184.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 184 = 2^{3} \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 184.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: | \(10.8563514411\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.2822449.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 27x^{2} - 24x - 5 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{3} \) |
| 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 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
\( x^{4} - x^{3} - 27x^{2} - 24x - 5 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu^{3} - 2\nu^{2} - 25\nu - 5 \)
|
| \(\beta_{2}\) | \(=\) |
\( 3\nu^{3} - 4\nu^{2} - 77\nu - 41 \)
|
| \(\beta_{3}\) | \(=\) |
\( 3\nu^{3} - 4\nu^{2} - 81\nu - 40 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{3} + \beta_{2} + 1 ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{3} + 3\beta_{2} - 6\beta _1 + 53 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -27\beta_{3} + 31\beta_{2} - 8\beta _1 + 151 ) / 4 \)
|
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 | −8.14810 | 0 | −3.35092 | 0 | −27.3609 | 0 | 39.3916 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | −1.90114 | 0 | −18.0297 | 0 | 0.923710 | 0 | −23.3857 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | 6.15452 | 0 | 1.44721 | 0 | 23.2480 | 0 | 10.8781 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | 8.89472 | 0 | 17.9334 | 0 | −28.8107 | 0 | 52.1161 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(23\) | \(-1\) |
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 | 184.4.a.f | ✓ | 4 |
| 3.b | odd | 2 | 1 | 1656.4.a.l | 4 | ||
| 4.b | odd | 2 | 1 | 368.4.a.m | 4 | ||
| 8.b | even | 2 | 1 | 1472.4.a.z | 4 | ||
| 8.d | odd | 2 | 1 | 1472.4.a.be | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 184.4.a.f | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 368.4.a.m | 4 | 4.b | odd | 2 | 1 | ||
| 1472.4.a.z | 4 | 8.b | even | 2 | 1 | ||
| 1472.4.a.be | 4 | 8.d | odd | 2 | 1 | ||
| 1656.4.a.l | 4 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} - 5T_{3}^{3} - 81T_{3}^{2} + 317T_{3} + 848 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(184))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} - 5 T^{3} + \cdots + 848 \)
|
| $5$ |
\( T^{4} + 2 T^{3} + \cdots + 1568 \)
|
| $7$ |
\( T^{4} + 32 T^{3} + \cdots + 16928 \)
|
| $11$ |
\( T^{4} - 56 T^{3} + \cdots - 480080 \)
|
| $13$ |
\( T^{4} - 139 T^{3} + \cdots - 608662 \)
|
| $17$ |
\( T^{4} + 6 T^{3} + \cdots - 2214128 \)
|
| $19$ |
\( T^{4} - 17368 T^{2} + \cdots + 69460176 \)
|
| $23$ |
\( (T - 23)^{4} \)
|
| $29$ |
\( T^{4} - 489 T^{3} + \cdots - 90222382 \)
|
| $31$ |
\( T^{4} + 127 T^{3} + \cdots + 514656 \)
|
| $37$ |
\( T^{4} + \cdots + 1125380768 \)
|
| $41$ |
\( T^{4} - 91 T^{3} + \cdots + 60003738 \)
|
| $43$ |
\( T^{4} + 762 T^{3} + \cdots - 325454848 \)
|
| $47$ |
\( T^{4} + \cdots + 2767397056 \)
|
| $53$ |
\( T^{4} + \cdots + 45208067808 \)
|
| $59$ |
\( T^{4} + \cdots + 12410250496 \)
|
| $61$ |
\( T^{4} + \cdots + 16900673344 \)
|
| $67$ |
\( T^{4} + \cdots - 191389824464 \)
|
| $71$ |
\( T^{4} + \cdots - 6719410208 \)
|
| $73$ |
\( T^{4} + 27 T^{3} + \cdots - 855134082 \)
|
| $79$ |
\( T^{4} + \cdots - 4467544416 \)
|
| $83$ |
\( T^{4} + \cdots - 175526395152 \)
|
| $89$ |
\( T^{4} + \cdots - 82915204480 \)
|
| $97$ |
\( T^{4} + \cdots - 734582347856 \)
|
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