Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [17,8,Mod(1,17)]
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([0]))
N = Newforms(chi, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("17.1");
S:= CuspForms(chi, 8);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 17 \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 17.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: | \(5.31054543323\) |
| Analytic rank: | \(1\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.694349.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - 206x - 187 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2 \) |
| 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 \(1,\beta_1,\beta_2\) 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
\( x^{3} - 206x - 187 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} + \nu - 135 ) / 7 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 14\beta_{2} - \beta _1 + 270 ) / 2 \)
|
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 |
|
−19.2972 | 12.4174 | 244.384 | 154.984 | −239.621 | −730.360 | −2245.88 | −2032.81 | −2990.76 | |||||||||||||||||||||||||||
| 1.2 | 6.23536 | −12.7202 | −89.1203 | −358.828 | −79.3150 | 534.563 | −1353.82 | −2025.20 | −2237.42 | ||||||||||||||||||||||||||||
| 1.3 | 14.0619 | −85.6972 | 69.7366 | 5.84457 | −1205.06 | −1362.20 | −819.293 | 5157.01 | 82.1857 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(17\) | \(-1\) |
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 | 17.8.a.b | ✓ | 3 |
| 3.b | odd | 2 | 1 | 153.8.a.c | 3 | ||
| 4.b | odd | 2 | 1 | 272.8.a.g | 3 | ||
| 5.b | even | 2 | 1 | 425.8.a.b | 3 | ||
| 17.b | even | 2 | 1 | 289.8.a.b | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 17.8.a.b | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
| 153.8.a.c | 3 | 3.b | odd | 2 | 1 | ||
| 272.8.a.g | 3 | 4.b | odd | 2 | 1 | ||
| 289.8.a.b | 3 | 17.b | even | 2 | 1 | ||
| 425.8.a.b | 3 | 5.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{3} - T_{2}^{2} - 304T_{2} + 1692 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(17))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} - T^{2} + \cdots + 1692 \)
|
| $3$ |
\( T^{3} + 86 T^{2} + \cdots - 13536 \)
|
| $5$ |
\( T^{3} + 198 T^{2} + \cdots + 325032 \)
|
| $7$ |
\( T^{3} + 1558 T^{2} + \cdots - 531836208 \)
|
| $11$ |
\( T^{3} + \cdots + 1398581088 \)
|
| $13$ |
\( T^{3} + \cdots - 970059396232 \)
|
| $17$ |
\( (T - 4913)^{3} \)
|
| $19$ |
\( T^{3} + \cdots - 21199506858432 \)
|
| $23$ |
\( T^{3} + \cdots + 251907998984784 \)
|
| $29$ |
\( T^{3} + \cdots - 88\!\cdots\!36 \)
|
| $31$ |
\( T^{3} + \cdots + 74\!\cdots\!52 \)
|
| $37$ |
\( T^{3} + \cdots + 15\!\cdots\!92 \)
|
| $41$ |
\( T^{3} + \cdots - 16\!\cdots\!64 \)
|
| $43$ |
\( T^{3} + \cdots - 15\!\cdots\!08 \)
|
| $47$ |
\( T^{3} + \cdots - 15\!\cdots\!48 \)
|
| $53$ |
\( T^{3} + \cdots - 74\!\cdots\!68 \)
|
| $59$ |
\( T^{3} + \cdots + 60\!\cdots\!32 \)
|
| $61$ |
\( T^{3} + \cdots - 954978952597144 \)
|
| $67$ |
\( T^{3} + \cdots + 11\!\cdots\!68 \)
|
| $71$ |
\( T^{3} + \cdots - 13\!\cdots\!96 \)
|
| $73$ |
\( T^{3} + \cdots - 51\!\cdots\!92 \)
|
| $79$ |
\( T^{3} + \cdots + 16\!\cdots\!76 \)
|
| $83$ |
\( T^{3} + \cdots + 79\!\cdots\!44 \)
|
| $89$ |
\( T^{3} + \cdots + 13\!\cdots\!44 \)
|
| $97$ |
\( T^{3} + \cdots - 23\!\cdots\!00 \)
|
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