Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [775,2,Mod(1,775)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(775, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("775.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 775 = 5^{2} \cdot 31 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 775.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: | \(6.18840615665\) |
| Analytic rank: | \(0\) |
| Dimension: | \(5\) |
| Coefficient field: | 5.5.144209.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - 6x^{3} - x^{2} + 6x - 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| 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
\( x^{5} - 6x^{3} - x^{2} + 6x - 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 4\nu + 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - 5\nu^{2} - 2\nu + 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 4\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{4} + 5\beta_{2} + 2\beta _1 + 8 \)
|
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 |
|
−1.70816 | 0.930775 | 0.917797 | 0 | −1.58991 | 1.50771 | 1.84857 | −2.13366 | 0 | |||||||||||||||||||||||||||||||||
| 1.2 | −0.264183 | 2.96850 | −1.93021 | 0 | −0.784228 | −2.14598 | 1.03829 | 5.81200 | 0 | ||||||||||||||||||||||||||||||||||
| 1.3 | 1.14876 | −2.39873 | −0.680351 | 0 | −2.75556 | 1.07521 | −3.07908 | 2.75390 | 0 | ||||||||||||||||||||||||||||||||||
| 1.4 | 2.23959 | −0.740841 | 3.01578 | 0 | −1.65918 | 3.67497 | 2.27494 | −2.45115 | 0 | ||||||||||||||||||||||||||||||||||
| 1.5 | 2.58398 | 2.24029 | 4.67698 | 0 | 5.78888 | −2.11190 | 6.91727 | 2.01891 | 0 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \(-1\) |
| \(31\) | \(1\) |
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 | 775.2.a.k | yes | 5 |
| 3.b | odd | 2 | 1 | 6975.2.a.bp | 5 | ||
| 5.b | even | 2 | 1 | 775.2.a.h | ✓ | 5 | |
| 5.c | odd | 4 | 2 | 775.2.b.g | 10 | ||
| 15.d | odd | 2 | 1 | 6975.2.a.by | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 775.2.a.h | ✓ | 5 | 5.b | even | 2 | 1 | |
| 775.2.a.k | yes | 5 | 1.a | even | 1 | 1 | trivial |
| 775.2.b.g | 10 | 5.c | odd | 4 | 2 | ||
| 6975.2.a.bp | 5 | 3.b | odd | 2 | 1 | ||
| 6975.2.a.by | 5 | 15.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{5} - 4T_{2}^{4} + 13T_{2}^{2} - 8T_{2} - 3 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(775))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} - 4 T^{4} + \cdots - 3 \)
|
| $3$ |
\( T^{5} - 3 T^{4} + \cdots - 11 \)
|
| $5$ |
\( T^{5} \)
|
| $7$ |
\( T^{5} - 2 T^{4} + \cdots - 27 \)
|
| $11$ |
\( T^{5} + 2 T^{4} + \cdots - 233 \)
|
| $13$ |
\( T^{5} - 4 T^{4} + \cdots - 153 \)
|
| $17$ |
\( T^{5} - 19 T^{4} + \cdots + 59 \)
|
| $19$ |
\( T^{5} - 8 T^{4} + \cdots + 59 \)
|
| $23$ |
\( T^{5} - 12 T^{4} + \cdots - 503 \)
|
| $29$ |
\( T^{5} - 6 T^{4} + \cdots - 8711 \)
|
| $31$ |
\( (T + 1)^{5} \)
|
| $37$ |
\( T^{5} - 16 T^{4} + \cdots + 4369 \)
|
| $41$ |
\( T^{5} + 2 T^{4} + \cdots - 507 \)
|
| $43$ |
\( T^{5} - T^{4} + \cdots - 17 \)
|
| $47$ |
\( T^{5} - 8 T^{4} + \cdots + 291 \)
|
| $53$ |
\( T^{5} - 3 T^{4} + \cdots - 389 \)
|
| $59$ |
\( T^{5} + 4 T^{4} + \cdots + 15079 \)
|
| $61$ |
\( T^{5} + 5 T^{4} + \cdots - 3 \)
|
| $67$ |
\( T^{5} - 13 T^{4} + \cdots - 124659 \)
|
| $71$ |
\( T^{5} - 6 T^{4} + \cdots + 57279 \)
|
| $73$ |
\( T^{5} - 19 T^{4} + \cdots - 3291 \)
|
| $79$ |
\( T^{5} + 6 T^{4} + \cdots + 1873 \)
|
| $83$ |
\( T^{5} - 18 T^{4} + \cdots + 1983 \)
|
| $89$ |
\( T^{5} + 31 T^{4} + \cdots + 729 \)
|
| $97$ |
\( T^{5} + T^{4} + \cdots + 25961 \)
|
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