Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1368,4,Mod(1,1368)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1368, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("1368.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1368 = 2^{3} \cdot 3^{2} \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1368.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: | \(80.7146128879\) |
| Analytic rank: | \(0\) |
| Dimension: | \(5\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{5} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - x^{4} - 429x^{3} + 1657x^{2} + 46980x - 289104 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{4} \) |
| Twist minimal: | no (minimal twist has level 456) |
| 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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) 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
\( x^{5} - x^{4} - 429x^{3} + 1657x^{2} + 46980x - 289104 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{4} + 9\nu^{3} - 296\nu^{2} - 1475\nu + 22727 ) / 43 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -13\nu^{4} - 160\nu^{3} + 3633\nu^{2} + 27044\nu - 291624 ) / 344 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -13\nu^{4} - 160\nu^{3} + 3977\nu^{2} + 28420\nu - 351136 ) / 344 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{4} - \beta_{3} - 4\beta _1 + 173 \)
|
| \(\nu^{3}\) | \(=\) |
\( -5\beta_{4} - 3\beta_{3} - 13\beta_{2} + 203\beta _1 - 776 \)
|
| \(\nu^{4}\) | \(=\) |
\( 341\beta_{4} - 269\beta_{3} + 160\beta_{2} - 1536\beta _1 + 35465 \)
|
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 | 0 | 0 | −13.5437 | 0 | −1.98997 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||
| 1.2 | 0 | 0 | 0 | −13.0276 | 0 | 31.0001 | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||
| 1.3 | 0 | 0 | 0 | −8.82361 | 0 | −24.1803 | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 0 | 0 | 13.4806 | 0 | 33.1709 | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 0 | 0 | 15.9144 | 0 | −28.0007 | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(3\) | \(-1\) |
| \(19\) | \(1\) |
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 | 1368.4.a.l | 5 | |
| 3.b | odd | 2 | 1 | 456.4.a.h | ✓ | 5 | |
| 12.b | even | 2 | 1 | 912.4.a.w | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 456.4.a.h | ✓ | 5 | 3.b | odd | 2 | 1 | |
| 912.4.a.w | 5 | 12.b | even | 2 | 1 | ||
| 1368.4.a.l | 5 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{5} + 6T_{5}^{4} - 415T_{5}^{3} - 2928T_{5}^{2} + 42388T_{5} + 334000 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1368))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} \)
|
| $3$ |
\( T^{5} \)
|
| $5$ |
\( T^{5} + 6 T^{4} + \cdots + 334000 \)
|
| $7$ |
\( T^{5} - 10 T^{4} + \cdots + 1385472 \)
|
| $11$ |
\( T^{5} - 32 T^{4} + \cdots - 43946688 \)
|
| $13$ |
\( T^{5} - 36 T^{4} + \cdots - 24342016 \)
|
| $17$ |
\( T^{5} + \cdots + 1435501672 \)
|
| $19$ |
\( (T + 19)^{5} \)
|
| $23$ |
\( T^{5} + \cdots + 1231762176 \)
|
| $29$ |
\( T^{5} + \cdots + 308507748544 \)
|
| $31$ |
\( T^{5} + \cdots - 1317185280 \)
|
| $37$ |
\( T^{5} + \cdots - 226592112640 \)
|
| $41$ |
\( T^{5} + \cdots - 333683103744 \)
|
| $43$ |
\( T^{5} + \cdots - 984172080256 \)
|
| $47$ |
\( T^{5} + \cdots + 185759984592 \)
|
| $53$ |
\( T^{5} + \cdots + 22244430931904 \)
|
| $59$ |
\( T^{5} + \cdots + 6353174011904 \)
|
| $61$ |
\( T^{5} + \cdots - 234568720936 \)
|
| $67$ |
\( T^{5} + \cdots - 3881819148288 \)
|
| $71$ |
\( T^{5} + \cdots - 265155551232 \)
|
| $73$ |
\( T^{5} + \cdots - 24486705649224 \)
|
| $79$ |
\( T^{5} + \cdots - 126343242840064 \)
|
| $83$ |
\( T^{5} + \cdots + 30052573276672 \)
|
| $89$ |
\( T^{5} + \cdots + 799511662401792 \)
|
| $97$ |
\( T^{5} + \cdots + 11\!\cdots\!80 \)
|
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