Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [157,2,Mod(1,157)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(157, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("157.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 157 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 157.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: | \(1.25365131173\) |
| Analytic rank: | \(1\) |
| Dimension: | \(5\) |
| Coefficient field: | 5.5.24217.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - 5x^{3} - x^{2} + 3x + 1 \)
|
| 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
\( x^{5} - 5x^{3} - x^{2} + 3x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu^{4} - 5\nu^{2} + 2 \)
|
| \(\beta_{2}\) | \(=\) |
\( -\nu^{4} + 5\nu^{2} + \nu - 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( -\nu^{4} + \nu^{3} + 5\nu^{2} - 3\nu - 3 \)
|
| \(\beta_{4}\) | \(=\) |
\( 2\nu^{4} - \nu^{3} - 9\nu^{2} + 3\nu + 3 \)
|
| \(\nu\) | \(=\) |
\( \beta_{2} + \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{4} + \beta_{3} - \beta _1 + 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 3\beta_{2} + 4\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 5\beta_{4} + 5\beta_{3} - 4\beta _1 + 8 \)
|
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 |
|
−2.69649 | −3.13883 | 5.27108 | 0.728280 | 8.46384 | 1.90023 | −8.82046 | 6.85225 | −1.96380 | |||||||||||||||||||||||||||||||||
| 1.2 | −2.27684 | 0.705039 | 3.18400 | −2.92186 | −1.60526 | −0.255206 | −2.69578 | −2.50292 | 6.65260 | ||||||||||||||||||||||||||||||||||
| 1.3 | −1.19993 | −1.48980 | −0.560160 | 3.49147 | 1.78766 | −4.39354 | 3.07202 | −0.780487 | −4.18953 | ||||||||||||||||||||||||||||||||||
| 1.4 | 0.130127 | −0.616516 | −1.98307 | −3.26834 | −0.0802254 | 2.70173 | −0.518305 | −2.61991 | −0.425300 | ||||||||||||||||||||||||||||||||||
| 1.5 | 1.04314 | −2.45989 | −0.911857 | −1.02955 | −2.56601 | −2.95321 | −3.03748 | 3.05107 | −1.07397 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(157\) | \(1\) |
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 | 157.2.a.a | ✓ | 5 |
| 3.b | odd | 2 | 1 | 1413.2.a.d | 5 | ||
| 4.b | odd | 2 | 1 | 2512.2.a.f | 5 | ||
| 5.b | even | 2 | 1 | 3925.2.a.f | 5 | ||
| 7.b | odd | 2 | 1 | 7693.2.a.b | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 157.2.a.a | ✓ | 5 | 1.a | even | 1 | 1 | trivial |
| 1413.2.a.d | 5 | 3.b | odd | 2 | 1 | ||
| 2512.2.a.f | 5 | 4.b | odd | 2 | 1 | ||
| 3925.2.a.f | 5 | 5.b | even | 2 | 1 | ||
| 7693.2.a.b | 5 | 7.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{5} + 5T_{2}^{4} + 5T_{2}^{3} - 6T_{2}^{2} - 7T_{2} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(157))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} + 5 T^{4} + \cdots + 1 \)
|
| $3$ |
\( T^{5} + 7 T^{4} + \cdots - 5 \)
|
| $5$ |
\( T^{5} + 3 T^{4} + \cdots + 25 \)
|
| $7$ |
\( T^{5} + 3 T^{4} + \cdots + 17 \)
|
| $11$ |
\( T^{5} + 14 T^{4} + \cdots + 1 \)
|
| $13$ |
\( T^{5} + 7 T^{4} + \cdots - 59 \)
|
| $17$ |
\( T^{5} + 9 T^{4} + \cdots - 139 \)
|
| $19$ |
\( T^{5} + 3 T^{4} + \cdots + 557 \)
|
| $23$ |
\( T^{5} + 13 T^{4} + \cdots - 631 \)
|
| $29$ |
\( T^{5} + 2 T^{4} + \cdots - 5 \)
|
| $31$ |
\( T^{5} - 3 T^{4} + \cdots + 2797 \)
|
| $37$ |
\( T^{5} - 3 T^{4} + \cdots - 641 \)
|
| $41$ |
\( T^{5} - 5 T^{4} + \cdots - 85 \)
|
| $43$ |
\( T^{5} + 23 T^{4} + \cdots + 53 \)
|
| $47$ |
\( T^{5} + 8 T^{4} + \cdots - 25 \)
|
| $53$ |
\( T^{5} + 25 T^{4} + \cdots - 36997 \)
|
| $59$ |
\( T^{5} - 7 T^{4} + \cdots - 45421 \)
|
| $61$ |
\( T^{5} - 4 T^{4} + \cdots - 145 \)
|
| $67$ |
\( T^{5} + 12 T^{4} + \cdots + 11339 \)
|
| $71$ |
\( T^{5} + 18 T^{4} + \cdots + 2105 \)
|
| $73$ |
\( T^{5} + 3 T^{4} + \cdots - 14337 \)
|
| $79$ |
\( T^{5} - 22 T^{4} + \cdots + 21877 \)
|
| $83$ |
\( T^{5} + 27 T^{4} + \cdots - 17375 \)
|
| $89$ |
\( T^{5} - 25 T^{4} + \cdots + 86269 \)
|
| $97$ |
\( T^{5} - 20 T^{4} + \cdots - 37157 \)
|
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