Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [368,4,Mod(1,368)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(368, 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("368.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 368 = 2^{4} \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 368.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: | \(21.7127028821\) |
| Analytic rank: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.167313.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 16x^{2} + 4x + 24 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{3} \) |
| Twist minimal: | no (minimal twist has level 184) |
| 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
\( x^{4} - x^{3} - 16x^{2} + 4x + 24 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} - \nu^{2} - 14\nu + 2 ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{2} - 8 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + 8 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 2\beta_{2} + 7\beta _1 + 6 \)
|
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 | −8.30570 | 0 | −19.3975 | 0 | 22.5880 | 0 | 41.9846 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | −0.911599 | 0 | −2.21395 | 0 | 0.592081 | 0 | −26.1690 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | 0.405785 | 0 | 5.36292 | 0 | 18.2150 | 0 | −26.8353 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | 7.81151 | 0 | −3.75144 | 0 | −31.3950 | 0 | 34.0197 | 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 | 368.4.a.n | 4 | |
| 4.b | odd | 2 | 1 | 184.4.a.e | ✓ | 4 | |
| 8.b | even | 2 | 1 | 1472.4.a.bd | 4 | ||
| 8.d | odd | 2 | 1 | 1472.4.a.ba | 4 | ||
| 12.b | even | 2 | 1 | 1656.4.a.n | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 184.4.a.e | ✓ | 4 | 4.b | odd | 2 | 1 | |
| 368.4.a.n | 4 | 1.a | even | 1 | 1 | trivial | |
| 1472.4.a.ba | 4 | 8.d | odd | 2 | 1 | ||
| 1472.4.a.bd | 4 | 8.b | even | 2 | 1 | ||
| 1656.4.a.n | 4 | 12.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} + T_{3}^{3} - 65T_{3}^{2} - 33T_{3} + 24 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(368))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} + T^{3} + \cdots + 24 \)
|
| $5$ |
\( T^{4} + 20 T^{3} + \cdots - 864 \)
|
| $7$ |
\( T^{4} - 10 T^{3} + \cdots - 7648 \)
|
| $11$ |
\( T^{4} - 30 T^{3} + \cdots + 984528 \)
|
| $13$ |
\( T^{4} + 153 T^{3} + \cdots - 9732878 \)
|
| $17$ |
\( T^{4} + 68 T^{3} + \cdots - 1525008 \)
|
| $19$ |
\( T^{4} - 120 T^{3} + \cdots - 1018224 \)
|
| $23$ |
\( (T - 23)^{4} \)
|
| $29$ |
\( T^{4} + 315 T^{3} + \cdots + 174665514 \)
|
| $31$ |
\( T^{4} + \cdots - 1710638448 \)
|
| $37$ |
\( T^{4} + 348 T^{3} + \cdots - 959571168 \)
|
| $41$ |
\( T^{4} + \cdots - 28721528046 \)
|
| $43$ |
\( T^{4} + \cdots + 1725576192 \)
|
| $47$ |
\( T^{4} + 205 T^{3} + \cdots + 884990496 \)
|
| $53$ |
\( T^{4} + \cdots + 2117560608 \)
|
| $59$ |
\( T^{4} + \cdots - 53186230528 \)
|
| $61$ |
\( T^{4} + 182 T^{3} + \cdots + 60956096 \)
|
| $67$ |
\( T^{4} + \cdots + 1683868176 \)
|
| $71$ |
\( T^{4} + \cdots - 48269410416 \)
|
| $73$ |
\( T^{4} + \cdots - 146762477034 \)
|
| $79$ |
\( T^{4} + \cdots - 25290376416 \)
|
| $83$ |
\( T^{4} + \cdots + 15216338064 \)
|
| $89$ |
\( T^{4} + \cdots + 272782530432 \)
|
| $97$ |
\( T^{4} + \cdots + 140583246192 \)
|
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