Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [17,6,Mod(16,17)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(17, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("17.16");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 17 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 17.b (of order \(2\), degree \(1\), minimal) |
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: | no |
| Analytic conductor: | \(2.72652493682\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} + \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} + 167x^{4} + 9076x^{2} + 159156 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{6} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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
\( x^{6} + 167x^{4} + 9076x^{2} + 159156 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 2\nu^{3} + 116\nu ) / 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{5} + 105\nu^{3} + 2606\nu ) / 6 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{4} + 109\nu^{2} + 2826 ) / 12 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -5\nu^{4} - 497\nu^{2} - 11442 ) / 12 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{5} + 5\beta_{4} - 224 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 3\beta_{2} - 58\beta_1 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -109\beta_{5} - 497\beta_{4} + 13112 ) / 4 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 12\beta_{3} - 315\beta_{2} + 3484\beta_1 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/17\mathbb{Z}\right)^\times\).
| \(n\) | \(3\) |
| \(\chi(n)\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 16.1 |
|
−8.50095 | − | 12.4591i | 40.2661 | 92.1671i | 105.914i | 52.2138i | −70.2699 | 87.7709 | − | 783.508i | ||||||||||||||||||||||||||||||||||
| 16.2 | −8.50095 | 12.4591i | 40.2661 | − | 92.1671i | − | 105.914i | − | 52.2138i | −70.2699 | 87.7709 | 783.508i | ||||||||||||||||||||||||||||||||||
| 16.3 | −0.527478 | − | 16.3427i | −31.7218 | − | 31.4384i | 8.62043i | − | 127.899i | 33.6119 | −24.0844 | 16.5831i | ||||||||||||||||||||||||||||||||||
| 16.4 | −0.527478 | 16.3427i | −31.7218 | 31.4384i | − | 8.62043i | 127.899i | 33.6119 | −24.0844 | − | 16.5831i | |||||||||||||||||||||||||||||||||||
| 16.5 | 8.02843 | − | 15.6744i | 32.4556 | − | 2.20290i | − | 125.841i | 229.398i | 3.65807 | −2.68650 | − | 17.6858i | |||||||||||||||||||||||||||||||||
| 16.6 | 8.02843 | 15.6744i | 32.4556 | 2.20290i | 125.841i | − | 229.398i | 3.65807 | −2.68650 | 17.6858i | ||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 17.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 17.6.b.a | ✓ | 6 |
| 3.b | odd | 2 | 1 | 153.6.d.b | 6 | ||
| 4.b | odd | 2 | 1 | 272.6.b.c | 6 | ||
| 17.b | even | 2 | 1 | inner | 17.6.b.a | ✓ | 6 |
| 17.c | even | 4 | 2 | 289.6.a.f | 6 | ||
| 51.c | odd | 2 | 1 | 153.6.d.b | 6 | ||
| 68.d | odd | 2 | 1 | 272.6.b.c | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 17.6.b.a | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 17.6.b.a | ✓ | 6 | 17.b | even | 2 | 1 | inner |
| 153.6.d.b | 6 | 3.b | odd | 2 | 1 | ||
| 153.6.d.b | 6 | 51.c | odd | 2 | 1 | ||
| 272.6.b.c | 6 | 4.b | odd | 2 | 1 | ||
| 272.6.b.c | 6 | 68.d | odd | 2 | 1 | ||
| 289.6.a.f | 6 | 17.c | even | 4 | 2 | ||
Hecke kernels
This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(17, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{3} + T^{2} - 68 T - 36)^{2} \)
|
| $3$ |
\( T^{6} + 668 T^{4} + \cdots + 10185984 \)
|
| $5$ |
\( T^{6} + 9488 T^{4} + \cdots + 40743936 \)
|
| $7$ |
\( T^{6} + \cdots + 2346850713600 \)
|
| $11$ |
\( T^{6} + \cdots + 14\!\cdots\!84 \)
|
| $13$ |
\( (T^{3} + 606 T^{2} + \cdots - 43234040)^{2} \)
|
| $17$ |
\( T^{6} + \cdots + 28\!\cdots\!93 \)
|
| $19$ |
\( (T^{3} - 1980 T^{2} + \cdots + 3105566656)^{2} \)
|
| $23$ |
\( T^{6} + \cdots + 10\!\cdots\!36 \)
|
| $29$ |
\( T^{6} + \cdots + 14\!\cdots\!96 \)
|
| $31$ |
\( T^{6} + \cdots + 79\!\cdots\!64 \)
|
| $37$ |
\( T^{6} + \cdots + 50\!\cdots\!00 \)
|
| $41$ |
\( T^{6} + \cdots + 85\!\cdots\!04 \)
|
| $43$ |
\( (T^{3} + \cdots + 1416274833472)^{2} \)
|
| $47$ |
\( (T^{3} + \cdots - 3746462957568)^{2} \)
|
| $53$ |
\( (T^{3} + \cdots - 4147907218776)^{2} \)
|
| $59$ |
\( (T^{3} + \cdots - 2433323484864)^{2} \)
|
| $61$ |
\( T^{6} + \cdots + 59\!\cdots\!76 \)
|
| $67$ |
\( (T^{3} + \cdots - 37896271752000)^{2} \)
|
| $71$ |
\( T^{6} + \cdots + 89\!\cdots\!36 \)
|
| $73$ |
\( T^{6} + \cdots + 42\!\cdots\!96 \)
|
| $79$ |
\( T^{6} + \cdots + 13\!\cdots\!16 \)
|
| $83$ |
\( (T^{3} + \cdots + 64504720806336)^{2} \)
|
| $89$ |
\( (T^{3} + \cdots - 195681426220584)^{2} \)
|
| $97$ |
\( T^{6} + \cdots + 23\!\cdots\!76 \)
|
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