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: | 5.5.205225.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - x^{4} - 6x^{3} + 3x^{2} + 7x - 3 \)
|
| 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
\( x^{5} - x^{4} - 6x^{3} + 3x^{2} + 7x - 3 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 4\nu + 1 \)
|
| \(\beta_{3}\) | \(=\) |
\( -\nu^{3} + 2\nu^{2} + 3\nu - 4 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - \nu^{3} - 5\nu^{2} + \nu + 3 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + \beta_{2} + \beta _1 + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 2\beta_{2} + 5\beta _1 + 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{4} + 6\beta_{3} + 7\beta_{2} + 9\beta _1 + 14 \)
|
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 |
|
−1.54180 | −0.124960 | 0.377151 | 0 | 0.192664 | 4.01817 | 2.50211 | −2.98438 | 0 | |||||||||||||||||||||||||||||||||
| 1.2 | −0.170728 | −0.648789 | −1.97085 | 0 | 0.110767 | −2.03774 | 0.677937 | −2.57907 | 0 | ||||||||||||||||||||||||||||||||||
| 1.3 | 0.581067 | 2.46572 | −1.66236 | 0 | 1.43275 | 1.67419 | −2.12808 | 3.07975 | 0 | ||||||||||||||||||||||||||||||||||
| 1.4 | 2.35347 | 1.91723 | 3.53883 | 0 | 4.51216 | −0.845802 | 3.62160 | 0.675783 | 0 | ||||||||||||||||||||||||||||||||||
| 1.5 | 2.77799 | −2.60920 | 5.71723 | 0 | −7.24833 | 3.19118 | 10.3264 | 3.80792 | 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.j | yes | 5 |
| 3.b | odd | 2 | 1 | 6975.2.a.bq | 5 | ||
| 5.b | even | 2 | 1 | 775.2.a.i | ✓ | 5 | |
| 5.c | odd | 4 | 2 | 775.2.b.h | 10 | ||
| 15.d | odd | 2 | 1 | 6975.2.a.bx | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 775.2.a.i | ✓ | 5 | 5.b | even | 2 | 1 | |
| 775.2.a.j | yes | 5 | 1.a | even | 1 | 1 | trivial |
| 775.2.b.h | 10 | 5.c | odd | 4 | 2 | ||
| 6975.2.a.bq | 5 | 3.b | odd | 2 | 1 | ||
| 6975.2.a.bx | 5 | 15.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{5} - 4T_{2}^{4} + 11T_{2}^{2} - 4T_{2} - 1 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(775))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} - 4 T^{4} + \cdots - 1 \)
|
| $3$ |
\( T^{5} - T^{4} - 8 T^{3} + \cdots + 1 \)
|
| $5$ |
\( T^{5} \)
|
| $7$ |
\( T^{5} - 6 T^{4} + \cdots - 37 \)
|
| $11$ |
\( T^{5} - 23 T^{3} + \cdots + 37 \)
|
| $13$ |
\( T^{5} - 4 T^{4} + \cdots - 9 \)
|
| $17$ |
\( T^{5} - 11 T^{4} + \cdots + 59 \)
|
| $19$ |
\( T^{5} + 4 T^{4} + \cdots + 135 \)
|
| $23$ |
\( T^{5} - 12 T^{4} + \cdots - 167 \)
|
| $29$ |
\( T^{5} + 6 T^{4} + \cdots + 135 \)
|
| $31$ |
\( (T - 1)^{5} \)
|
| $37$ |
\( T^{5} + 2 T^{4} + \cdots - 27539 \)
|
| $41$ |
\( T^{5} + 2 T^{4} + \cdots - 27 \)
|
| $43$ |
\( T^{5} - 7 T^{4} + \cdots + 11727 \)
|
| $47$ |
\( T^{5} - 8 T^{4} + \cdots - 50279 \)
|
| $53$ |
\( T^{5} - 25 T^{4} + \cdots + 1719 \)
|
| $59$ |
\( T^{5} - 4 T^{4} + \cdots + 4595 \)
|
| $61$ |
\( T^{5} + 17 T^{4} + \cdots - 20609 \)
|
| $67$ |
\( T^{5} + 13 T^{4} + \cdots - 337 \)
|
| $71$ |
\( T^{5} + 6 T^{4} + \cdots - 549 \)
|
| $73$ |
\( T^{5} - 7 T^{4} + \cdots - 5507 \)
|
| $79$ |
\( T^{5} - 12 T^{4} + \cdots - 14985 \)
|
| $83$ |
\( T^{5} + 4 T^{4} + \cdots + 22459 \)
|
| $89$ |
\( T^{5} + 3 T^{4} + \cdots - 39965 \)
|
| $97$ |
\( T^{5} - 25 T^{4} + \cdots - 169317 \)
|
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