Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [17,4,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, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("17.16");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 17 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| 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: | \(1.00303247010\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.0.4669632.2 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 74x^{2} + 1072 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2 \) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} + 74x^{2} + 1072 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} + 40 ) / 6 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} + 46\nu ) / 6 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 6\beta_{2} - 40 \)
|
| \(\nu^{3}\) | \(=\) |
\( 6\beta_{3} - 46\beta_1 \)
|
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 |
|
−3.37228 | − | 7.36435i | 3.37228 | − | 10.1060i | 24.8347i | 17.4703i | 15.6060 | −27.2337 | 34.0802i | ||||||||||||||||||||||||||||
| 16.2 | −3.37228 | 7.36435i | 3.37228 | 10.1060i | − | 24.8347i | − | 17.4703i | 15.6060 | −27.2337 | − | 34.0802i | ||||||||||||||||||||||||||||
| 16.3 | 2.37228 | − | 4.44593i | −2.37228 | 19.4389i | − | 10.5470i | − | 14.9929i | −24.6060 | 7.23369 | 46.1145i | ||||||||||||||||||||||||||||
| 16.4 | 2.37228 | 4.44593i | −2.37228 | − | 19.4389i | 10.5470i | 14.9929i | −24.6060 | 7.23369 | − | 46.1145i | |||||||||||||||||||||||||||||
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 | 17.4.b.a | ✓ | 4 |
| 3.b | odd | 2 | 1 | 153.4.d.b | 4 | ||
| 4.b | odd | 2 | 1 | 272.4.b.d | 4 | ||
| 5.b | even | 2 | 1 | 425.4.d.c | 4 | ||
| 5.c | odd | 4 | 2 | 425.4.c.c | 8 | ||
| 17.b | even | 2 | 1 | inner | 17.4.b.a | ✓ | 4 |
| 17.c | even | 4 | 2 | 289.4.a.e | 4 | ||
| 51.c | odd | 2 | 1 | 153.4.d.b | 4 | ||
| 68.d | odd | 2 | 1 | 272.4.b.d | 4 | ||
| 85.c | even | 2 | 1 | 425.4.d.c | 4 | ||
| 85.g | odd | 4 | 2 | 425.4.c.c | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 17.4.b.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 17.4.b.a | ✓ | 4 | 17.b | even | 2 | 1 | inner |
| 153.4.d.b | 4 | 3.b | odd | 2 | 1 | ||
| 153.4.d.b | 4 | 51.c | odd | 2 | 1 | ||
| 272.4.b.d | 4 | 4.b | odd | 2 | 1 | ||
| 272.4.b.d | 4 | 68.d | odd | 2 | 1 | ||
| 289.4.a.e | 4 | 17.c | even | 4 | 2 | ||
| 425.4.c.c | 8 | 5.c | odd | 4 | 2 | ||
| 425.4.c.c | 8 | 85.g | odd | 4 | 2 | ||
| 425.4.d.c | 4 | 5.b | even | 2 | 1 | ||
| 425.4.d.c | 4 | 85.c | even | 2 | 1 | ||
Hecke kernels
This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(17, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + T - 8)^{2} \)
|
| $3$ |
\( T^{4} + 74T^{2} + 1072 \)
|
| $5$ |
\( T^{4} + 480 T^{2} + 38592 \)
|
| $7$ |
\( T^{4} + 530 T^{2} + 68608 \)
|
| $11$ |
\( T^{4} + 3626 T^{2} + 2573872 \)
|
| $13$ |
\( (T^{2} - 70 T - 392)^{2} \)
|
| $17$ |
\( T^{4} - 112 T^{3} + \cdots + 24137569 \)
|
| $19$ |
\( (T + 28)^{4} \)
|
| $23$ |
\( T^{4} + 28322 T^{2} + 10295488 \)
|
| $29$ |
\( T^{4} + 23520 T^{2} + 92659392 \)
|
| $31$ |
\( T^{4} + 4274 T^{2} + 4390912 \)
|
| $37$ |
\( T^{4} + \cdots + 1245754048 \)
|
| $41$ |
\( T^{4} + \cdots + 2799480832 \)
|
| $43$ |
\( (T^{2} + 260 T - 30752)^{2} \)
|
| $47$ |
\( (T^{2} - 476 T + 50176)^{2} \)
|
| $53$ |
\( (T^{2} + 288 T - 49092)^{2} \)
|
| $59$ |
\( (T^{2} + 588 T - 75264)^{2} \)
|
| $61$ |
\( T^{4} + \cdots + 7146728128 \)
|
| $67$ |
\( (T^{2} + 120 T - 30192)^{2} \)
|
| $71$ |
\( T^{4} + 52626 T^{2} + 92659392 \)
|
| $73$ |
\( T^{4} + \cdots + 647466319872 \)
|
| $79$ |
\( T^{4} + \cdots + 70925616832 \)
|
| $83$ |
\( (T^{2} + 1428 T + 451584)^{2} \)
|
| $89$ |
\( (T^{2} - 994 T + 232456)^{2} \)
|
| $97$ |
\( T^{4} + \cdots + 16878665728 \)
|
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