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: | \(1\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.761.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 6x - 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| 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 \(1,\beta_1,\beta_2\) 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
\( x^{3} - x^{2} - 6x - 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( 4\nu - 1 \)
|
| \(\beta_{2}\) | \(=\) |
\( 2\nu^{2} - 2\nu - 8 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 1 ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 2\beta_{2} + \beta _1 + 17 ) / 4 \)
|
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 | −8.59554 | 0 | 10.2855 | 0 | −5.78430 | 0 | 46.8833 | 0 | |||||||||||||||||||||||||||
| 1.2 | 0 | 1.94289 | 0 | 6.62493 | 0 | −30.0785 | 0 | −23.2252 | 0 | ||||||||||||||||||||||||||||
| 1.3 | 0 | 3.65265 | 0 | −14.9104 | 0 | 7.86283 | 0 | −13.6582 | 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.c | ✓ | 3 |
| 3.b | odd | 2 | 1 | 1656.4.a.j | 3 | ||
| 4.b | odd | 2 | 1 | 368.4.a.j | 3 | ||
| 8.b | even | 2 | 1 | 1472.4.a.v | 3 | ||
| 8.d | odd | 2 | 1 | 1472.4.a.q | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 184.4.a.c | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
| 368.4.a.j | 3 | 4.b | odd | 2 | 1 | ||
| 1472.4.a.q | 3 | 8.d | odd | 2 | 1 | ||
| 1472.4.a.v | 3 | 8.b | even | 2 | 1 | ||
| 1656.4.a.j | 3 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{3} + 3T_{3}^{2} - 41T_{3} + 61 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(184))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} \)
|
| $3$ |
\( T^{3} + 3 T^{2} + \cdots + 61 \)
|
| $5$ |
\( T^{3} - 2 T^{2} + \cdots + 1016 \)
|
| $7$ |
\( T^{3} + 28 T^{2} + \cdots - 1368 \)
|
| $11$ |
\( T^{3} + 22 T^{2} + \cdots + 216 \)
|
| $13$ |
\( T^{3} + 27 T^{2} + \cdots - 12263 \)
|
| $17$ |
\( T^{3} + 108 T^{2} + \cdots - 364328 \)
|
| $19$ |
\( T^{3} + 218 T^{2} + \cdots + 232184 \)
|
| $23$ |
\( (T - 23)^{3} \)
|
| $29$ |
\( T^{3} + 209 T^{2} + \cdots - 42541 \)
|
| $31$ |
\( T^{3} + 359 T^{2} + \cdots + 142937 \)
|
| $37$ |
\( T^{3} - 230 T^{2} + \cdots - 1435928 \)
|
| $41$ |
\( T^{3} - 53 T^{2} + \cdots - 1259303 \)
|
| $43$ |
\( T^{3} + 248 T^{2} + \cdots + 7906816 \)
|
| $47$ |
\( T^{3} + 667 T^{2} + \cdots + 4301949 \)
|
| $53$ |
\( T^{3} - 742 T^{2} + \cdots - 11295528 \)
|
| $59$ |
\( T^{3} - 400 T^{2} + \cdots - 37423296 \)
|
| $61$ |
\( T^{3} - 562 T^{2} + \cdots + 120065576 \)
|
| $67$ |
\( T^{3} - 190 T^{2} + \cdots + 5217864 \)
|
| $71$ |
\( T^{3} + 575 T^{2} + \cdots - 183391207 \)
|
| $73$ |
\( T^{3} - 671 T^{2} + \cdots + 197534723 \)
|
| $79$ |
\( T^{3} - 138 T^{2} + \cdots - 338243128 \)
|
| $83$ |
\( T^{3} + 408 T^{2} + \cdots - 945084168 \)
|
| $89$ |
\( T^{3} + 1316 T^{2} + \cdots - 470184768 \)
|
| $97$ |
\( T^{3} - 944 T^{2} + \cdots + 37901864 \)
|
| show more | |
| show less |