Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [163,2,Mod(1,163)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(163, 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("163.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 163 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 163.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: | \(1.30156155295\) |
| 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: |
\( x^{7} - 3x^{6} - 5x^{5} + 19x^{4} - 23x^{2} + 4x + 6 \)
|
| 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 \(1,\beta_1,\ldots,\beta_{6}\) 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
\( x^{7} - 3x^{6} - 5x^{5} + 19x^{4} - 23x^{2} + 4x + 6 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{4} - 6\nu^{2} + 4 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{5} - \nu^{4} - 6\nu^{3} + 5\nu^{2} + 5\nu - 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( -\nu^{6} + 2\nu^{5} + 6\nu^{4} - 11\nu^{3} - 6\nu^{2} + 7\nu + 2 \)
|
| \(\beta_{6}\) | \(=\) |
\( \nu^{6} - \nu^{5} - 7\nu^{4} + 6\nu^{3} + 11\nu^{2} - 6\nu - 4 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{6} + \beta_{5} - \beta_{4} + 4\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{3} + 6\beta_{2} + 14 \)
|
| \(\nu^{5}\) | \(=\) |
\( 6\beta_{6} + 6\beta_{5} - 5\beta_{4} + \beta_{3} + \beta_{2} + 19\beta _1 + 1 \)
|
| \(\nu^{6}\) | \(=\) |
\( \beta_{6} + \beta_{4} + 8\beta_{3} + 32\beta_{2} + \beta _1 + 70 \)
|
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 |
|
−2.19784 | 0.245778 | 2.83050 | 2.71621 | −0.540180 | −2.14321 | −1.82530 | −2.93959 | −5.96979 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.03490 | 2.49677 | −0.928983 | 0.273679 | −2.58391 | 1.43756 | 3.03120 | 3.23386 | −0.283230 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | −0.472573 | −2.68646 | −1.77667 | 1.65562 | 1.26955 | 2.09759 | 1.78476 | 4.21705 | −0.782401 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0.805826 | 2.05443 | −1.35064 | 3.57006 | 1.65551 | −2.79937 | −2.70004 | 1.22067 | 2.87684 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 1.32510 | 0.446910 | −0.244113 | 0.929287 | 0.592200 | 4.32010 | −2.97367 | −2.80027 | 1.23140 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 2.22634 | −2.16715 | 2.95661 | 3.52340 | −4.82482 | −1.14118 | 2.12974 | 1.69653 | 7.84429 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 2.34804 | 0.609719 | 3.51331 | −1.66825 | 1.43165 | −1.77149 | 3.55331 | −2.62824 | −3.91711 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(163\) | \(-1\) |
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 | 163.2.a.c | ✓ | 7 |
| 3.b | odd | 2 | 1 | 1467.2.a.f | 7 | ||
| 4.b | odd | 2 | 1 | 2608.2.a.n | 7 | ||
| 5.b | even | 2 | 1 | 4075.2.a.f | 7 | ||
| 7.b | odd | 2 | 1 | 7987.2.a.h | 7 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 163.2.a.c | ✓ | 7 | 1.a | even | 1 | 1 | trivial |
| 1467.2.a.f | 7 | 3.b | odd | 2 | 1 | ||
| 2608.2.a.n | 7 | 4.b | odd | 2 | 1 | ||
| 4075.2.a.f | 7 | 5.b | even | 2 | 1 | ||
| 7987.2.a.h | 7 | 7.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{7} - 3T_{2}^{6} - 5T_{2}^{5} + 19T_{2}^{4} - 23T_{2}^{2} + 4T_{2} + 6 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(163))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} - 3 T^{6} + \cdots + 6 \)
|
| $3$ |
\( T^{7} - T^{6} - 11 T^{5} + \cdots - 2 \)
|
| $5$ |
\( T^{7} - 11 T^{6} + \cdots + 24 \)
|
| $7$ |
\( T^{7} - 21 T^{5} + \cdots - 158 \)
|
| $11$ |
\( T^{7} - 2 T^{6} + \cdots + 12 \)
|
| $13$ |
\( T^{7} - 10 T^{6} + \cdots - 334 \)
|
| $17$ |
\( T^{7} - 13 T^{6} + \cdots - 90 \)
|
| $19$ |
\( T^{7} + 5 T^{6} + \cdots - 962 \)
|
| $23$ |
\( T^{7} - 2 T^{6} + \cdots - 564 \)
|
| $29$ |
\( T^{7} - 7 T^{6} + \cdots + 83922 \)
|
| $31$ |
\( T^{7} + 11 T^{6} + \cdots - 16738 \)
|
| $37$ |
\( T^{7} - 3 T^{6} + \cdots + 1286 \)
|
| $41$ |
\( T^{7} - 17 T^{6} + \cdots + 30237 \)
|
| $43$ |
\( T^{7} + 10 T^{6} + \cdots - 31793 \)
|
| $47$ |
\( T^{7} - 11 T^{6} + \cdots + 2048493 \)
|
| $53$ |
\( T^{7} - 18 T^{6} + \cdots - 93987 \)
|
| $59$ |
\( T^{7} - 11 T^{6} + \cdots - 269034 \)
|
| $61$ |
\( T^{7} - 4 T^{6} + \cdots + 12119 \)
|
| $67$ |
\( T^{7} + 18 T^{6} + \cdots - 839836 \)
|
| $71$ |
\( T^{7} + 3 T^{6} + \cdots - 13023 \)
|
| $73$ |
\( T^{7} - 2 T^{6} + \cdots - 2554 \)
|
| $79$ |
\( T^{7} - 353 T^{5} + \cdots + 1197688 \)
|
| $83$ |
\( T^{7} - 18 T^{6} + \cdots + 62745 \)
|
| $89$ |
\( T^{7} - 18 T^{6} + \cdots + 2340 \)
|
| $97$ |
\( T^{7} - 21 T^{6} + \cdots - 8371133 \)
|
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