Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [399,2,Mod(1,399)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(399, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("399.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 399 = 3 \cdot 7 \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 399.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: | \(3.18603104065\) |
| Analytic rank: | \(0\) |
| Dimension: | \(5\) |
| Coefficient field: | 5.5.368464.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - 2x^{4} - 6x^{3} + 6x^{2} + 6x - 4 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 2 \) |
| 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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) 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
\( x^{5} - 2x^{4} - 6x^{3} + 6x^{2} + 6x - 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 2\nu^{2} - 4\nu + 3 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - 2\nu^{3} - 5\nu^{2} + 4\nu + 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 2\beta_{2} + 6\beta _1 + 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{4} + 2\beta_{3} + 9\beta_{2} + 13\beta _1 + 19 \)
|
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 |
|
−2.14884 | −1.00000 | 2.61752 | 3.68348 | 2.14884 | 1.00000 | −1.32696 | 1.00000 | −7.91521 | |||||||||||||||||||||||||||||||||
| 1.2 | −0.178368 | −1.00000 | −1.96818 | −3.42801 | 0.178368 | 1.00000 | 0.707798 | 1.00000 | 0.611448 | ||||||||||||||||||||||||||||||||||
| 1.3 | 0.447457 | −1.00000 | −1.79978 | 3.78739 | −0.447457 | 1.00000 | −1.70024 | 1.00000 | 1.69470 | ||||||||||||||||||||||||||||||||||
| 1.4 | 2.09027 | −1.00000 | 2.36924 | 0.388134 | −2.09027 | 1.00000 | 0.771813 | 1.00000 | 0.811305 | ||||||||||||||||||||||||||||||||||
| 1.5 | 2.78948 | −1.00000 | 5.78120 | −0.430991 | −2.78948 | 1.00000 | 10.5476 | 1.00000 | −1.20224 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(1\) |
| \(7\) | \(-1\) |
| \(19\) | \(1\) |
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 | 399.2.a.g | ✓ | 5 |
| 3.b | odd | 2 | 1 | 1197.2.a.o | 5 | ||
| 4.b | odd | 2 | 1 | 6384.2.a.cf | 5 | ||
| 5.b | even | 2 | 1 | 9975.2.a.bp | 5 | ||
| 7.b | odd | 2 | 1 | 2793.2.a.bg | 5 | ||
| 19.b | odd | 2 | 1 | 7581.2.a.w | 5 | ||
| 21.c | even | 2 | 1 | 8379.2.a.cb | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 399.2.a.g | ✓ | 5 | 1.a | even | 1 | 1 | trivial |
| 1197.2.a.o | 5 | 3.b | odd | 2 | 1 | ||
| 2793.2.a.bg | 5 | 7.b | odd | 2 | 1 | ||
| 6384.2.a.cf | 5 | 4.b | odd | 2 | 1 | ||
| 7581.2.a.w | 5 | 19.b | odd | 2 | 1 | ||
| 8379.2.a.cb | 5 | 21.c | even | 2 | 1 | ||
| 9975.2.a.bp | 5 | 5.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(399))\):
|
\( T_{2}^{5} - 3T_{2}^{4} - 4T_{2}^{3} + 14T_{2}^{2} - 3T_{2} - 1 \)
|
|
\( T_{5}^{5} - 4T_{5}^{4} - 12T_{5}^{3} + 48T_{5}^{2} + 4T_{5} - 8 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} - 3 T^{4} + \cdots - 1 \)
|
| $3$ |
\( (T + 1)^{5} \)
|
| $5$ |
\( T^{5} - 4 T^{4} + \cdots - 8 \)
|
| $7$ |
\( (T - 1)^{5} \)
|
| $11$ |
\( T^{5} - 8 T^{4} + \cdots - 2816 \)
|
| $13$ |
\( T^{5} + 6 T^{4} + \cdots + 1984 \)
|
| $17$ |
\( T^{5} - 12 T^{4} + \cdots + 1256 \)
|
| $19$ |
\( (T + 1)^{5} \)
|
| $23$ |
\( T^{5} - 12 T^{4} + \cdots - 2432 \)
|
| $29$ |
\( T^{5} - 4 T^{4} + \cdots - 8 \)
|
| $31$ |
\( T^{5} + 8 T^{4} + \cdots - 1408 \)
|
| $37$ |
\( T^{5} - 2 T^{4} + \cdots - 416 \)
|
| $41$ |
\( T^{5} - 10 T^{4} + \cdots - 416 \)
|
| $43$ |
\( T^{5} + 16 T^{4} + \cdots - 1984 \)
|
| $47$ |
\( T^{5} + 2 T^{4} + \cdots + 32 \)
|
| $53$ |
\( T^{5} - 104 T^{3} + \cdots - 5416 \)
|
| $59$ |
\( T^{5} + 4 T^{4} + \cdots + 2048 \)
|
| $61$ |
\( T^{5} + 14 T^{4} + \cdots + 16736 \)
|
| $67$ |
\( T^{5} + 20 T^{4} + \cdots + 512 \)
|
| $71$ |
\( T^{5} + 6 T^{4} + \cdots - 1696 \)
|
| $73$ |
\( T^{5} - 10 T^{4} + \cdots - 11552 \)
|
| $79$ |
\( T^{5} - 192 T^{3} + \cdots + 512 \)
|
| $83$ |
\( T^{5} - 6 T^{4} + \cdots - 8912 \)
|
| $89$ |
\( T^{5} - 26 T^{4} + \cdots - 168352 \)
|
| $97$ |
\( T^{5} + 18 T^{4} + \cdots - 4448 \)
|
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