Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [143,2,Mod(1,143)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(143, 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("143.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 143 = 11 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 143.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: | \(1.14186074890\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.194616205.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 10x^{4} - 2x^{3} + 24x^{2} + 7x - 12 \)
|
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 10x^{4} - 2x^{3} + 24x^{2} + 7x - 12 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{5} - \nu^{4} - 8\nu^{3} + 6\nu^{2} + 11\nu - 6 \)
|
| \(\beta_{4}\) | \(=\) |
\( -\nu^{5} + \nu^{4} + 9\nu^{3} - 7\nu^{2} - 16\nu + 8 \)
|
| \(\beta_{5}\) | \(=\) |
\( -\nu^{5} + 2\nu^{4} + 8\nu^{3} - 14\nu^{2} - 12\nu + 15 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{4} + \beta_{3} + \beta_{2} + 5\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{5} + \beta_{3} + 8\beta_{2} + \beta _1 + 15 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{5} + 8\beta_{4} + 10\beta_{3} + 10\beta_{2} + 30\beta _1 + 11 \)
|
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 |
|
−2.70899 | 1.82693 | 5.33863 | 0.610508 | −4.94913 | 2.05182 | −9.04431 | 0.337672 | −1.65386 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −1.70126 | −2.43752 | 0.894288 | −4.04114 | 4.14685 | 4.09965 | 1.88110 | 2.94150 | 6.87504 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −0.633036 | 2.27944 | −1.59927 | 4.04223 | −1.44297 | −3.23808 | 2.27847 | 2.19585 | −2.55888 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 1.23127 | 0.357107 | −0.483971 | 1.04428 | 0.439695 | 4.82820 | −3.05844 | −2.87247 | 1.28579 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 1.36536 | 3.30864 | −0.135780 | −3.38172 | 4.51750 | −1.56538 | −2.91612 | 7.94711 | −4.61728 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 2.44665 | −2.33460 | 3.98610 | 2.72585 | −5.71195 | −2.17620 | 4.85930 | 2.45035 | 6.66920 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(11\) | \(1\) |
| \(13\) | \(-1\) |
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 | 143.2.a.c | ✓ | 6 |
| 3.b | odd | 2 | 1 | 1287.2.a.q | 6 | ||
| 4.b | odd | 2 | 1 | 2288.2.a.z | 6 | ||
| 5.b | even | 2 | 1 | 3575.2.a.p | 6 | ||
| 7.b | odd | 2 | 1 | 7007.2.a.r | 6 | ||
| 8.b | even | 2 | 1 | 9152.2.a.cm | 6 | ||
| 8.d | odd | 2 | 1 | 9152.2.a.cs | 6 | ||
| 11.b | odd | 2 | 1 | 1573.2.a.m | 6 | ||
| 13.b | even | 2 | 1 | 1859.2.a.m | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 143.2.a.c | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 1287.2.a.q | 6 | 3.b | odd | 2 | 1 | ||
| 1573.2.a.m | 6 | 11.b | odd | 2 | 1 | ||
| 1859.2.a.m | 6 | 13.b | even | 2 | 1 | ||
| 2288.2.a.z | 6 | 4.b | odd | 2 | 1 | ||
| 3575.2.a.p | 6 | 5.b | even | 2 | 1 | ||
| 7007.2.a.r | 6 | 7.b | odd | 2 | 1 | ||
| 9152.2.a.cm | 6 | 8.b | even | 2 | 1 | ||
| 9152.2.a.cs | 6 | 8.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} - 10T_{2}^{4} + 2T_{2}^{3} + 24T_{2}^{2} - 7T_{2} - 12 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(143))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} - 10 T^{4} + \cdots - 12 \)
|
| $3$ |
\( T^{6} - 3 T^{5} + \cdots + 28 \)
|
| $5$ |
\( T^{6} - T^{5} + \cdots + 96 \)
|
| $7$ |
\( T^{6} - 4 T^{5} + \cdots - 448 \)
|
| $11$ |
\( (T + 1)^{6} \)
|
| $13$ |
\( (T - 1)^{6} \)
|
| $17$ |
\( T^{6} - 40 T^{4} + \cdots - 768 \)
|
| $19$ |
\( T^{6} + 10 T^{5} + \cdots - 104 \)
|
| $23$ |
\( T^{6} - 11 T^{5} + \cdots + 13176 \)
|
| $29$ |
\( T^{6} - 2 T^{5} + \cdots + 1344 \)
|
| $31$ |
\( T^{6} + 9 T^{5} + \cdots - 1664 \)
|
| $37$ |
\( T^{6} - 15 T^{5} + \cdots - 2560 \)
|
| $41$ |
\( T^{6} + 4 T^{5} + \cdots - 252 \)
|
| $43$ |
\( T^{6} + 2 T^{5} + \cdots - 1024 \)
|
| $47$ |
\( T^{6} - 6 T^{5} + \cdots - 7680 \)
|
| $53$ |
\( T^{6} - 2 T^{5} + \cdots - 10116 \)
|
| $59$ |
\( T^{6} - 11 T^{5} + \cdots - 57792 \)
|
| $61$ |
\( T^{6} - 16 T^{5} + \cdots + 19648 \)
|
| $67$ |
\( T^{6} - 9 T^{5} + \cdots - 832 \)
|
| $71$ |
\( T^{6} + 15 T^{5} + \cdots + 33024 \)
|
| $73$ |
\( T^{6} - 32 T^{5} + \cdots + 17456 \)
|
| $79$ |
\( T^{6} - 14 T^{5} + \cdots + 2048 \)
|
| $83$ |
\( T^{6} + 26 T^{5} + \cdots + 584400 \)
|
| $89$ |
\( T^{6} + 23 T^{5} + \cdots + 61152 \)
|
| $97$ |
\( T^{6} - 27 T^{5} + \cdots - 65312 \)
|
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