Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [107,2,Mod(1,107)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(107, 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("107.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 107 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 107.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: | \(0.854399301628\) |
| Analytic rank: | \(0\) |
| Dimension: | \(7\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{7} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{7} - x^{6} - 10x^{5} + 7x^{4} + 29x^{3} - 12x^{2} - 20x + 8 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| 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 \(1,\beta_1,\ldots,\beta_{6}\) 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
\( x^{7} - x^{6} - 10x^{5} + 7x^{4} + 29x^{3} - 12x^{2} - 20x + 8 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{6} - \nu^{5} - 10\nu^{4} + 7\nu^{3} + 25\nu^{2} - 8\nu - 4 ) / 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{6} + \nu^{5} + 10\nu^{4} - 3\nu^{3} - 29\nu^{2} - 8\nu + 16 ) / 4 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{6} - \nu^{5} - 8\nu^{4} + 5\nu^{3} + 15\nu^{2} - 2\nu - 2 ) / 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{6} - \nu^{5} - 10\nu^{4} + 7\nu^{3} + 29\nu^{2} - 8\nu - 16 ) / 4 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( \nu^{6} + \nu^{5} - 12\nu^{4} - 9\nu^{3} + 39\nu^{2} + 18\nu - 24 ) / 4 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{5} - \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{5} + \beta_{3} + 4\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 6\beta_{5} + \beta_{4} + \beta_{3} - 7\beta_{2} + \beta _1 + 14 \)
|
| \(\nu^{5}\) | \(=\) |
\( 2\beta_{6} + 7\beta_{5} + \beta_{4} + 9\beta_{3} - 2\beta_{2} + 20\beta _1 + 3 \)
|
| \(\nu^{6}\) | \(=\) |
\( 2\beta_{6} + 35\beta_{5} + 11\beta_{4} + 12\beta_{3} - 43\beta_{2} + 10\beta _1 + 72 \)
|
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 |
|
−2.51050 | 3.06691 | 4.30263 | 0.741172 | −7.69949 | −2.01143 | −5.78076 | 6.40594 | −1.86071 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.18501 | −0.577909 | 2.77426 | −1.96367 | 1.26274 | 4.45691 | −1.69177 | −2.66602 | 4.29064 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | −0.896638 | −2.53177 | −1.19604 | 4.31985 | 2.27008 | 1.44247 | 2.86569 | 3.40988 | −3.87334 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | −0.400681 | 2.05242 | −1.83945 | 0.858034 | −0.822367 | 3.08104 | 1.53840 | 1.21244 | −0.343798 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 1.07948 | 1.28677 | −0.834718 | 2.76688 | 1.38905 | −4.90639 | −3.06003 | −1.34422 | 2.98680 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.69574 | 1.42310 | 0.875548 | −3.10374 | 2.41322 | 1.50121 | −1.90678 | −0.974777 | −5.26315 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 2.21761 | −1.71952 | 2.91777 | 1.38148 | −3.81323 | 0.436175 | 2.03526 | −0.0432353 | 3.06357 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(107\) | \(-1\) |
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 | 107.2.a.b | ✓ | 7 |
| 3.b | odd | 2 | 1 | 963.2.a.f | 7 | ||
| 4.b | odd | 2 | 1 | 1712.2.a.t | 7 | ||
| 5.b | even | 2 | 1 | 2675.2.a.g | 7 | ||
| 7.b | odd | 2 | 1 | 5243.2.a.g | 7 | ||
| 8.b | even | 2 | 1 | 6848.2.a.bu | 7 | ||
| 8.d | odd | 2 | 1 | 6848.2.a.bv | 7 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 107.2.a.b | ✓ | 7 | 1.a | even | 1 | 1 | trivial |
| 963.2.a.f | 7 | 3.b | odd | 2 | 1 | ||
| 1712.2.a.t | 7 | 4.b | odd | 2 | 1 | ||
| 2675.2.a.g | 7 | 5.b | even | 2 | 1 | ||
| 5243.2.a.g | 7 | 7.b | odd | 2 | 1 | ||
| 6848.2.a.bu | 7 | 8.b | even | 2 | 1 | ||
| 6848.2.a.bv | 7 | 8.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{7} + T_{2}^{6} - 10T_{2}^{5} - 7T_{2}^{4} + 29T_{2}^{3} + 12T_{2}^{2} - 20T_{2} - 8 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(107))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} + T^{6} - 10 T^{5} + \cdots - 8 \)
|
| $3$ |
\( T^{7} - 3 T^{6} + \cdots + 29 \)
|
| $5$ |
\( T^{7} - 5 T^{6} + \cdots - 64 \)
|
| $7$ |
\( T^{7} - 4 T^{6} + \cdots - 128 \)
|
| $11$ |
\( T^{7} + 2 T^{6} + \cdots + 47 \)
|
| $13$ |
\( T^{7} - 18 T^{6} + \cdots - 1244 \)
|
| $17$ |
\( T^{7} + T^{6} + \cdots - 512 \)
|
| $19$ |
\( T^{7} + 4 T^{6} + \cdots - 1636 \)
|
| $23$ |
\( T^{7} - 123 T^{5} + \cdots + 21431 \)
|
| $29$ |
\( T^{7} + 3 T^{6} + \cdots - 11828 \)
|
| $31$ |
\( T^{7} - 4 T^{6} + \cdots + 256 \)
|
| $37$ |
\( T^{7} - 25 T^{6} + \cdots + 12113 \)
|
| $41$ |
\( T^{7} - 82 T^{5} + \cdots + 724 \)
|
| $43$ |
\( T^{7} - 11 T^{6} + \cdots - 21856 \)
|
| $47$ |
\( T^{7} + 9 T^{6} + \cdots - 30848 \)
|
| $53$ |
\( T^{7} - 8 T^{6} + \cdots - 143149 \)
|
| $59$ |
\( T^{7} + 19 T^{6} + \cdots - 16736 \)
|
| $61$ |
\( T^{7} - 25 T^{6} + \cdots - 123049 \)
|
| $67$ |
\( T^{7} + 24 T^{6} + \cdots + 333056 \)
|
| $71$ |
\( T^{7} + 19 T^{6} + \cdots + 1370816 \)
|
| $73$ |
\( T^{7} - 30 T^{6} + \cdots + 79712 \)
|
| $79$ |
\( T^{7} + 21 T^{6} + \cdots + 19859 \)
|
| $83$ |
\( T^{7} - 12 T^{6} + \cdots + 2420672 \)
|
| $89$ |
\( T^{7} + 22 T^{6} + \cdots - 14123 \)
|
| $97$ |
\( T^{7} + 4 T^{6} + \cdots + 139424 \)
|
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