Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [197,2,Mod(1,197)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(197, 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("197.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 197 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 197.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: | \(1.57305291982\) |
| 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 |
|
−2.17442 | −3.26823 | 2.72812 | 2.11405 | 7.10652 | −0.641841 | −1.58325 | 7.68134 | −4.59684 | |||||||||||||||||||||||||||||||||
| 1.2 | −0.878095 | 1.36778 | −1.22895 | −2.05392 | −1.20104 | −4.16404 | 2.83532 | −1.12918 | 1.80354 | ||||||||||||||||||||||||||||||||||
| 1.3 | 0.369680 | −1.84170 | −1.86334 | 2.08422 | −0.680841 | −2.69034 | −1.42820 | 0.391870 | 0.770496 | ||||||||||||||||||||||||||||||||||
| 1.4 | 0.722813 | −0.905942 | −1.47754 | −3.28633 | −0.654827 | 1.73780 | −2.51361 | −2.17927 | −2.37540 | ||||||||||||||||||||||||||||||||||
| 1.5 | 1.96003 | −3.35190 | 1.84170 | −2.85802 | −6.56982 | −4.24157 | −0.310264 | 8.23524 | −5.60180 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(197\) | \( +1 \) |
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 | 197.2.a.b | ✓ | 5 |
| 3.b | odd | 2 | 1 | 1773.2.a.e | 5 | ||
| 4.b | odd | 2 | 1 | 3152.2.a.j | 5 | ||
| 5.b | even | 2 | 1 | 4925.2.a.h | 5 | ||
| 7.b | odd | 2 | 1 | 9653.2.a.h | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 197.2.a.b | ✓ | 5 | 1.a | even | 1 | 1 | trivial |
| 1773.2.a.e | 5 | 3.b | odd | 2 | 1 | ||
| 3152.2.a.j | 5 | 4.b | odd | 2 | 1 | ||
| 4925.2.a.h | 5 | 5.b | even | 2 | 1 | ||
| 9653.2.a.h | 5 | 7.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{5} - 5T_{2}^{3} + T_{2}^{2} + 3T_{2} - 1 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(197))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} - 5 T^{3} + \cdots - 1 \)
|
| $3$ |
\( T^{5} + 8 T^{4} + \cdots - 25 \)
|
| $5$ |
\( T^{5} + 4 T^{4} + \cdots + 85 \)
|
| $7$ |
\( T^{5} + 10 T^{4} + \cdots - 53 \)
|
| $11$ |
\( T^{5} + 8 T^{4} + \cdots + 59 \)
|
| $13$ |
\( T^{5} + 8 T^{4} + \cdots + 493 \)
|
| $17$ |
\( T^{5} - 9 T^{4} + \cdots - 289 \)
|
| $19$ |
\( T^{5} + 16 T^{4} + \cdots - 761 \)
|
| $23$ |
\( T^{5} + T^{4} - 27 T^{3} + \cdots - 1 \)
|
| $29$ |
\( T^{5} - 2 T^{4} + \cdots - 1 \)
|
| $31$ |
\( T^{5} - 2 T^{4} + \cdots + 235 \)
|
| $37$ |
\( T^{5} + 17 T^{4} + \cdots - 7121 \)
|
| $41$ |
\( T^{5} + 5 T^{4} + \cdots + 16859 \)
|
| $43$ |
\( T^{5} + 26 T^{4} + \cdots - 2027 \)
|
| $47$ |
\( T^{5} - 16 T^{4} + \cdots - 169 \)
|
| $53$ |
\( T^{5} - 2 T^{4} + \cdots - 5615 \)
|
| $59$ |
\( T^{5} + 13 T^{4} + \cdots + 7055 \)
|
| $61$ |
\( T^{5} + 3 T^{4} + \cdots - 4835 \)
|
| $67$ |
\( T^{5} + 40 T^{4} + \cdots + 745 \)
|
| $71$ |
\( T^{5} + 5 T^{4} + \cdots - 617 \)
|
| $73$ |
\( T^{5} + 9 T^{4} + \cdots + 10889 \)
|
| $79$ |
\( T^{5} - 13 T^{4} + \cdots + 20567 \)
|
| $83$ |
\( T^{5} + 17 T^{4} + \cdots + 4447 \)
|
| $89$ |
\( T^{5} - 10 T^{4} + \cdots - 9043 \)
|
| $97$ |
\( T^{5} + 42 T^{4} + \cdots + 16711 \)
|
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