Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [261,4,Mod(28,261)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(261, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("261.28");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 261 = 3^{2} \cdot 29 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 261.c (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: | \(15.3994985115\) |
| 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: |
\( x^{6} + 38x^{4} + 301x^{2} + 560 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{4} \) |
| Twist minimal: | no (minimal twist has level 29) |
| 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
\( x^{6} + 38x^{4} + 301x^{2} + 560 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{5} + 34\nu^{3} + 165\nu ) / 20 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{5} + 14\nu^{3} - 275\nu ) / 20 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{4} - 29\nu^{2} - 85 ) / 5 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{4} + 34\nu^{2} + 150 ) / 5 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{5} + \beta_{4} - 13 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{3} + \beta_{2} - 22\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -29\beta_{5} - 34\beta_{4} + 292 \)
|
| \(\nu^{5}\) | \(=\) |
\( 34\beta_{3} - 14\beta_{2} + 583\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/261\mathbb{Z}\right)^\times\).
| \(n\) | \(118\) | \(146\) |
| \(\chi(n)\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 28.1 |
|
− | 5.28644i | 0 | −19.9464 | −15.1112 | 0 | −16.1112 | 63.1540i | 0 | 79.8845i | |||||||||||||||||||||||||||||||||||
| 28.2 | − | 2.70440i | 0 | 0.686224 | 10.7216 | 0 | 9.72164 | − | 23.4910i | 0 | − | 28.9956i | ||||||||||||||||||||||||||||||||||
| 28.3 | − | 1.65524i | 0 | 5.26019 | −6.61042 | 0 | −7.61042 | − | 21.9488i | 0 | 10.9418i | |||||||||||||||||||||||||||||||||||
| 28.4 | 1.65524i | 0 | 5.26019 | −6.61042 | 0 | −7.61042 | 21.9488i | 0 | − | 10.9418i | ||||||||||||||||||||||||||||||||||||
| 28.5 | 2.70440i | 0 | 0.686224 | 10.7216 | 0 | 9.72164 | 23.4910i | 0 | 28.9956i | |||||||||||||||||||||||||||||||||||||
| 28.6 | 5.28644i | 0 | −19.9464 | −15.1112 | 0 | −16.1112 | − | 63.1540i | 0 | − | 79.8845i | |||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 29.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 261.4.c.c | 6 | |
| 3.b | odd | 2 | 1 | 29.4.b.a | ✓ | 6 | |
| 12.b | even | 2 | 1 | 464.4.e.a | 6 | ||
| 29.b | even | 2 | 1 | inner | 261.4.c.c | 6 | |
| 87.d | odd | 2 | 1 | 29.4.b.a | ✓ | 6 | |
| 87.f | even | 4 | 2 | 841.4.a.c | 6 | ||
| 348.b | even | 2 | 1 | 464.4.e.a | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 29.4.b.a | ✓ | 6 | 3.b | odd | 2 | 1 | |
| 29.4.b.a | ✓ | 6 | 87.d | odd | 2 | 1 | |
| 261.4.c.c | 6 | 1.a | even | 1 | 1 | trivial | |
| 261.4.c.c | 6 | 29.b | even | 2 | 1 | inner | |
| 464.4.e.a | 6 | 12.b | even | 2 | 1 | ||
| 464.4.e.a | 6 | 348.b | even | 2 | 1 | ||
| 841.4.a.c | 6 | 87.f | even | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} + 38T_{2}^{4} + 301T_{2}^{2} + 560 \)
acting on \(S_{4}^{\mathrm{new}}(261, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + 38 T^{4} + \cdots + 560 \)
|
| $3$ |
\( T^{6} \)
|
| $5$ |
\( (T^{3} + 11 T^{2} + \cdots - 1071)^{2} \)
|
| $7$ |
\( (T^{3} + 14 T^{2} + \cdots - 1192)^{2} \)
|
| $11$ |
\( T^{6} + 3909 T^{4} + \cdots + 193452035 \)
|
| $13$ |
\( (T^{3} - 15 T^{2} + \cdots + 119791)^{2} \)
|
| $17$ |
\( T^{6} + \cdots + 8077890240 \)
|
| $19$ |
\( T^{6} + \cdots + 5493277440 \)
|
| $23$ |
\( (T^{3} - 38 T^{2} + \cdots - 188856)^{2} \)
|
| $29$ |
\( T^{6} + \cdots + 14507145975869 \)
|
| $31$ |
\( T^{6} + \cdots + 1328705534835 \)
|
| $37$ |
\( T^{6} + \cdots + 844118984540160 \)
|
| $41$ |
\( T^{6} + \cdots + 16572819704000 \)
|
| $43$ |
\( T^{6} + \cdots + 207644764642875 \)
|
| $47$ |
\( T^{6} + \cdots + 328073602115 \)
|
| $53$ |
\( (T^{3} + 495 T^{2} + \cdots + 9329121)^{2} \)
|
| $59$ |
\( (T^{3} + 630 T^{2} + \cdots - 6664392)^{2} \)
|
| $61$ |
\( T^{6} + \cdots + 757905028251840 \)
|
| $67$ |
\( (T^{3} + 332 T^{2} + \cdots + 16765696)^{2} \)
|
| $71$ |
\( (T^{3} - 448 T^{2} + \cdots + 5011200)^{2} \)
|
| $73$ |
\( T^{6} + \cdots + 10\!\cdots\!40 \)
|
| $79$ |
\( T^{6} + \cdots + 424851885738315 \)
|
| $83$ |
\( (T^{3} - 1122 T^{2} + \cdots + 43495704)^{2} \)
|
| $89$ |
\( T^{6} + \cdots + 18\!\cdots\!40 \)
|
| $97$ |
\( T^{6} + \cdots + 18\!\cdots\!60 \)
|
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