Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [289,4,Mod(288,289)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(289, 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("289.288");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 289 = 17^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 289.b (of order \(2\), degree \(1\), not 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: | \(17.0515519917\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.0.27793984.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 2x^{3} + 49x^{2} - 14x + 2 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{5} \) |
| Twist minimal: | no (minimal twist has level 17) |
| 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
\( x^{6} - 2x^{3} + 49x^{2} - 14x + 2 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 4\nu^{5} + 25\nu^{4} + 28\nu^{3} - 4\nu^{2} + 799 ) / 171 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -7\nu^{5} - \nu^{4} - 49\nu^{3} + 7\nu^{2} - 684\nu + 98 ) / 342 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{5} + 8\nu^{4} - 7\nu^{3} + \nu^{2} + 299 ) / 57 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -49\nu^{5} - 7\nu^{4} - \nu^{3} + 49\nu^{2} - 2394\nu + 344 ) / 171 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -245\nu^{5} - 35\nu^{4} - 5\nu^{3} + 587\nu^{2} - 11970\nu + 1720 ) / 171 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} - 2\beta_{2} - \beta_1 ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{5} - 5\beta_{4} ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 2\beta_{4} - 7\beta_{3} - 14\beta_{2} + 7\beta _1 + 4 ) / 4 \)
|
| \(\nu^{4}\) | \(=\) |
\( 4\beta_{3} + 3\beta _1 - 35 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 2\beta_{5} - 24\beta_{4} - 51\beta_{3} + 98\beta_{2} + 47\beta _1 + 48 ) / 4 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/289\mathbb{Z}\right)^\times\).
| \(n\) | \(3\) |
| \(\chi(n)\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 288.1 |
|
−4.67129 | − | 7.62999i | 13.8209 | − | 11.9174i | 35.6419i | − | 26.1222i | −27.1912 | −31.2167 | 55.6696i | |||||||||||||||||||||||||||||||||
| 288.2 | −4.67129 | 7.62999i | 13.8209 | 11.9174i | − | 35.6419i | 26.1222i | −27.1912 | −31.2167 | − | 55.6696i | |||||||||||||||||||||||||||||||||||
| 288.3 | −1.36122 | − | 3.15463i | −6.14708 | − | 3.03171i | 4.29415i | − | 7.94049i | 19.2573 | 17.0483 | 4.12682i | ||||||||||||||||||||||||||||||||||
| 288.4 | −1.36122 | 3.15463i | −6.14708 | 3.03171i | − | 4.29415i | 7.94049i | 19.2573 | 17.0483 | − | 4.12682i | |||||||||||||||||||||||||||||||||||
| 288.5 | 5.03251 | − | 8.47535i | 17.3261 | − | 0.885690i | − | 42.6523i | 3.81828i | 46.9339 | −44.8316 | − | 4.45724i | |||||||||||||||||||||||||||||||||
| 288.6 | 5.03251 | 8.47535i | 17.3261 | 0.885690i | 42.6523i | − | 3.81828i | 46.9339 | −44.8316 | 4.45724i | ||||||||||||||||||||||||||||||||||||
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 | 289.4.b.b | 6 | |
| 17.b | even | 2 | 1 | inner | 289.4.b.b | 6 | |
| 17.c | even | 4 | 1 | 17.4.a.b | ✓ | 3 | |
| 17.c | even | 4 | 1 | 289.4.a.b | 3 | ||
| 51.f | odd | 4 | 1 | 153.4.a.g | 3 | ||
| 68.f | odd | 4 | 1 | 272.4.a.h | 3 | ||
| 85.f | odd | 4 | 1 | 425.4.b.f | 6 | ||
| 85.i | odd | 4 | 1 | 425.4.b.f | 6 | ||
| 85.j | even | 4 | 1 | 425.4.a.g | 3 | ||
| 119.f | odd | 4 | 1 | 833.4.a.d | 3 | ||
| 136.i | even | 4 | 1 | 1088.4.a.v | 3 | ||
| 136.j | odd | 4 | 1 | 1088.4.a.x | 3 | ||
| 187.f | odd | 4 | 1 | 2057.4.a.e | 3 | ||
| 204.l | even | 4 | 1 | 2448.4.a.bi | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 17.4.a.b | ✓ | 3 | 17.c | even | 4 | 1 | |
| 153.4.a.g | 3 | 51.f | odd | 4 | 1 | ||
| 272.4.a.h | 3 | 68.f | odd | 4 | 1 | ||
| 289.4.a.b | 3 | 17.c | even | 4 | 1 | ||
| 289.4.b.b | 6 | 1.a | even | 1 | 1 | trivial | |
| 289.4.b.b | 6 | 17.b | even | 2 | 1 | inner | |
| 425.4.a.g | 3 | 85.j | even | 4 | 1 | ||
| 425.4.b.f | 6 | 85.f | odd | 4 | 1 | ||
| 425.4.b.f | 6 | 85.i | odd | 4 | 1 | ||
| 833.4.a.d | 3 | 119.f | odd | 4 | 1 | ||
| 1088.4.a.v | 3 | 136.i | even | 4 | 1 | ||
| 1088.4.a.x | 3 | 136.j | odd | 4 | 1 | ||
| 2057.4.a.e | 3 | 187.f | odd | 4 | 1 | ||
| 2448.4.a.bi | 3 | 204.l | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{3} + T_{2}^{2} - 24T_{2} - 32 \)
acting on \(S_{4}^{\mathrm{new}}(289, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{3} + T^{2} - 24 T - 32)^{2} \)
|
| $3$ |
\( T^{6} + 140 T^{4} + \cdots + 41616 \)
|
| $5$ |
\( T^{6} + 152 T^{4} + \cdots + 1024 \)
|
| $7$ |
\( T^{6} + 760 T^{4} + \cdots + 627264 \)
|
| $11$ |
\( T^{6} + 3516 T^{4} + \cdots + 22014864 \)
|
| $13$ |
\( (T^{3} - 30 T^{2} + \cdots - 9392)^{2} \)
|
| $17$ |
\( T^{6} \)
|
| $19$ |
\( (T^{3} + 80 T^{2} + \cdots - 340128)^{2} \)
|
| $23$ |
\( T^{6} + \cdots + 2561741095936 \)
|
| $29$ |
\( T^{6} + \cdots + 2306218853376 \)
|
| $31$ |
\( T^{6} + \cdots + 6659865664 \)
|
| $37$ |
\( T^{6} + \cdots + 38152265269504 \)
|
| $41$ |
\( T^{6} + \cdots + 2685481897536 \)
|
| $43$ |
\( (T^{3} + 556 T^{2} + \cdots - 7270272)^{2} \)
|
| $47$ |
\( (T^{3} - 640 T^{2} + \cdots - 1671168)^{2} \)
|
| $53$ |
\( (T^{3} + 302 T^{2} + \cdots - 18162072)^{2} \)
|
| $59$ |
\( (T^{3} + 636 T^{2} + \cdots - 49419072)^{2} \)
|
| $61$ |
\( T^{6} + \cdots + 46141914470656 \)
|
| $67$ |
\( (T^{3} - 1008 T^{2} + \cdots - 765952)^{2} \)
|
| $71$ |
\( T^{6} + \cdots + 75\!\cdots\!56 \)
|
| $73$ |
\( T^{6} + \cdots + 398302285230144 \)
|
| $79$ |
\( T^{6} + \cdots + 55\!\cdots\!76 \)
|
| $83$ |
\( (T^{3} - 2396 T^{2} + \cdots - 142080704)^{2} \)
|
| $89$ |
\( (T^{3} + 170 T^{2} + \cdots - 446571376)^{2} \)
|
| $97$ |
\( T^{6} + \cdots + 42\!\cdots\!00 \)
|
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