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: | \(0\) |
| 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: |
\( x^{6} - 3x^{5} - 604x^{4} + 760x^{3} + 102128x^{2} - 41712x - 4749120 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{4}\cdot 3 \) |
| 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
\( x^{6} - 3x^{5} - 604x^{4} + 760x^{3} + 102128x^{2} - 41712x - 4749120 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 71\nu^{5} - 399\nu^{4} - 43730\nu^{3} + 272540\nu^{2} + 5573320\nu - 35238624 ) / 214848 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 11\nu^{5} + 411\nu^{4} - 5924\nu^{3} - 173380\nu^{2} + 394816\nu + 11741952 ) / 53712 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 119\nu^{5} - 1047\nu^{4} - 52490\nu^{3} + 307004\nu^{2} + 4688680\nu - 16189152 ) / 214848 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{5} - 573\nu^{4} + 3734\nu^{3} + 235708\nu^{2} - 777112\nu - 17829408 ) / 53712 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} + 3\beta _1 + 202 \)
|
| \(\nu^{3}\) | \(=\) |
\( 15\beta_{5} - 5\beta_{4} + 17\beta_{3} - 3\beta_{2} + 279\beta _1 + 394 \)
|
| \(\nu^{4}\) | \(=\) |
\( 429\beta_{5} - 447\beta_{4} + 539\beta_{3} + 391\beta_{2} + 1857\beta _1 + 55022 \)
|
| \(\nu^{5}\) | \(=\) |
\( 7811\beta_{5} - 1753\beta_{4} + 9661\beta_{3} - 463\beta_{2} + 92263\beta _1 + 272802 \)
|
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 |
|
−15.3402 | −70.5591 | 107.322 | −537.404 | 1082.39 | 362.843 | 317.198 | 2791.58 | 8243.89 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −11.2090 | 70.4777 | −2.35934 | −132.938 | −789.982 | 1295.63 | 1461.19 | 2780.11 | 1490.10 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −6.77791 | −52.1459 | −82.0600 | 356.107 | 353.440 | 248.464 | 1423.77 | 532.198 | −2413.66 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 10.8108 | 75.6036 | −11.1256 | 188.752 | 817.339 | −871.217 | −1504.07 | 3528.91 | 2040.57 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 15.1882 | −6.67205 | 102.682 | 344.873 | −101.337 | 1626.36 | −384.534 | −2142.48 | 5238.01 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 22.3280 | 23.2957 | 370.540 | −403.390 | 520.147 | −598.084 | 5415.44 | −1644.31 | −9006.90 | |||||||||||||||||||||||||||||||||||||
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.c | ✓ | 6 |
| 3.b | odd | 2 | 1 | 153.8.a.h | 6 | ||
| 4.b | odd | 2 | 1 | 272.8.a.i | 6 | ||
| 5.b | even | 2 | 1 | 425.8.a.c | 6 | ||
| 17.b | even | 2 | 1 | 289.8.a.c | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 17.8.a.c | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 153.8.a.h | 6 | 3.b | odd | 2 | 1 | ||
| 272.8.a.i | 6 | 4.b | odd | 2 | 1 | ||
| 289.8.a.c | 6 | 17.b | even | 2 | 1 | ||
| 425.8.a.c | 6 | 5.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} - 15T_{2}^{5} - 514T_{2}^{4} + 5312T_{2}^{3} + 83552T_{2}^{2} - 422208T_{2} - 4272768 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(17))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} - 15 T^{5} + \cdots - 4272768 \)
|
| $3$ |
\( T^{6} + \cdots - 3047209200 \)
|
| $5$ |
\( T^{6} + \cdots - 668046873600000 \)
|
| $7$ |
\( T^{6} + \cdots + 98\!\cdots\!52 \)
|
| $11$ |
\( T^{6} + \cdots + 25\!\cdots\!80 \)
|
| $13$ |
\( T^{6} + \cdots - 87\!\cdots\!92 \)
|
| $17$ |
\( (T + 4913)^{6} \)
|
| $19$ |
\( T^{6} + \cdots - 15\!\cdots\!40 \)
|
| $23$ |
\( T^{6} + \cdots - 26\!\cdots\!00 \)
|
| $29$ |
\( T^{6} + \cdots + 51\!\cdots\!00 \)
|
| $31$ |
\( T^{6} + \cdots + 43\!\cdots\!92 \)
|
| $37$ |
\( T^{6} + \cdots + 84\!\cdots\!24 \)
|
| $41$ |
\( T^{6} + \cdots + 75\!\cdots\!76 \)
|
| $43$ |
\( T^{6} + \cdots + 71\!\cdots\!36 \)
|
| $47$ |
\( T^{6} + \cdots - 75\!\cdots\!60 \)
|
| $53$ |
\( T^{6} + \cdots - 45\!\cdots\!64 \)
|
| $59$ |
\( T^{6} + \cdots - 39\!\cdots\!20 \)
|
| $61$ |
\( T^{6} + \cdots - 11\!\cdots\!00 \)
|
| $67$ |
\( T^{6} + \cdots + 37\!\cdots\!64 \)
|
| $71$ |
\( T^{6} + \cdots + 63\!\cdots\!64 \)
|
| $73$ |
\( T^{6} + \cdots + 11\!\cdots\!24 \)
|
| $79$ |
\( T^{6} + \cdots + 46\!\cdots\!60 \)
|
| $83$ |
\( T^{6} + \cdots + 39\!\cdots\!04 \)
|
| $89$ |
\( T^{6} + \cdots + 10\!\cdots\!00 \)
|
| $97$ |
\( T^{6} + \cdots - 49\!\cdots\!00 \)
|
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