Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [103,2,Mod(1,103)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(103, 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("103.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 103 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 103.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: | \(0.822459140819\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.6999257.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 2x^{5} - 6x^{4} + 7x^{3} + 11x^{2} + x - 1 \)
|
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 2x^{5} - 6x^{4} + 7x^{3} + 11x^{2} + x - 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{5} - 3\nu^{4} - 4\nu^{3} + 11\nu^{2} + 5\nu - 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( -\nu^{5} + 2\nu^{4} + 6\nu^{3} - 7\nu^{2} - 11\nu - 1 \)
|
| \(\beta_{5}\) | \(=\) |
\( 2\nu^{5} - 5\nu^{4} - 9\nu^{3} + 18\nu^{2} + 11\nu - 3 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{5} + \beta_{4} - \beta_{3} + 5\beta _1 + 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( 2\beta_{5} + \beta_{4} - 3\beta_{3} + 4\beta_{2} + 8\beta _1 + 9 \)
|
| \(\nu^{5}\) | \(=\) |
\( 10\beta_{5} + 7\beta_{4} - 12\beta_{3} + \beta_{2} + 28\beta _1 + 15 \)
|
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 |
|
−1.68129 | 1.73515 | 0.826745 | 1.52866 | −2.91730 | −0.782217 | 1.97258 | 0.0107559 | −2.57012 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −1.15811 | −2.77524 | −0.658781 | 2.87877 | 3.21404 | 3.25000 | 3.07916 | 4.70198 | −3.33393 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 0.752951 | 3.01362 | −1.43307 | −0.416604 | 2.26911 | −3.96468 | −2.58493 | 6.08193 | −0.313682 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 1.47106 | −0.565073 | 0.164010 | 3.06831 | −0.831255 | −0.0154195 | −2.70085 | −2.68069 | 4.51366 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 1.83327 | 0.860083 | 1.36089 | −3.15863 | 1.57677 | 2.30447 | −1.17166 | −2.26026 | −5.79063 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 2.78212 | −2.26854 | 5.74020 | −0.900498 | −6.31136 | −2.79215 | 10.4057 | 2.14628 | −2.50530 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(103\) | \(-1\) |
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 | 103.2.a.b | ✓ | 6 |
| 3.b | odd | 2 | 1 | 927.2.a.f | 6 | ||
| 4.b | odd | 2 | 1 | 1648.2.a.m | 6 | ||
| 5.b | even | 2 | 1 | 2575.2.a.k | 6 | ||
| 7.b | odd | 2 | 1 | 5047.2.a.d | 6 | ||
| 8.b | even | 2 | 1 | 6592.2.a.bd | 6 | ||
| 8.d | odd | 2 | 1 | 6592.2.a.be | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 103.2.a.b | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 927.2.a.f | 6 | 3.b | odd | 2 | 1 | ||
| 1648.2.a.m | 6 | 4.b | odd | 2 | 1 | ||
| 2575.2.a.k | 6 | 5.b | even | 2 | 1 | ||
| 5047.2.a.d | 6 | 7.b | odd | 2 | 1 | ||
| 6592.2.a.bd | 6 | 8.b | even | 2 | 1 | ||
| 6592.2.a.be | 6 | 8.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} - 4T_{2}^{5} - T_{2}^{4} + 17T_{2}^{3} - 9T_{2}^{2} - 16T_{2} + 11 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(103))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} - 4 T^{5} + \cdots + 11 \)
|
| $3$ |
\( T^{6} - 13 T^{4} + \cdots - 16 \)
|
| $5$ |
\( T^{6} - 3 T^{5} + \cdots - 16 \)
|
| $7$ |
\( T^{6} + 2 T^{5} + \cdots + 1 \)
|
| $11$ |
\( T^{6} + T^{5} + \cdots + 272 \)
|
| $13$ |
\( T^{6} + T^{5} + \cdots + 55 \)
|
| $17$ |
\( T^{6} - 21 T^{5} + \cdots - 1745 \)
|
| $19$ |
\( T^{6} + 7 T^{5} + \cdots - 241 \)
|
| $23$ |
\( T^{6} - 12 T^{5} + \cdots + 12268 \)
|
| $29$ |
\( T^{6} - 12 T^{5} + \cdots + 4 \)
|
| $31$ |
\( T^{6} + 16 T^{5} + \cdots - 400 \)
|
| $37$ |
\( T^{6} - 83 T^{4} + \cdots + 176 \)
|
| $41$ |
\( T^{6} - 14 T^{5} + \cdots - 15152 \)
|
| $43$ |
\( T^{6} + 6 T^{5} + \cdots - 23984 \)
|
| $47$ |
\( T^{6} - T^{5} + \cdots - 22384 \)
|
| $53$ |
\( T^{6} - 19 T^{5} + \cdots - 80 \)
|
| $59$ |
\( T^{6} - 3 T^{5} + \cdots - 78173 \)
|
| $61$ |
\( T^{6} - T^{5} + \cdots - 2495 \)
|
| $67$ |
\( T^{6} + 12 T^{5} + \cdots + 22576 \)
|
| $71$ |
\( T^{6} + 27 T^{5} + \cdots + 83632 \)
|
| $73$ |
\( T^{6} + 7 T^{5} + \cdots - 4624 \)
|
| $79$ |
\( T^{6} + 21 T^{5} + \cdots + 5779 \)
|
| $83$ |
\( T^{6} + 9 T^{5} + \cdots + 9637 \)
|
| $89$ |
\( T^{6} + 14 T^{5} + \cdots + 1667776 \)
|
| $97$ |
\( T^{6} + 8 T^{5} + \cdots - 560468 \)
|
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