Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [261,4,Mod(1,261)]
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, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("261.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 261 = 3^{2} \cdot 29 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 261.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: | \(15.3994985115\) |
| Analytic rank: | \(0\) |
| Dimension: | \(5\) |
| Coefficient field: | 5.5.13458092.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - x^{4} - 14x^{3} + 18x^{2} + 20x - 8 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | no (minimal twist has level 29) |
| 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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) 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
\( x^{5} - x^{4} - 14x^{3} + 18x^{2} + 20x - 8 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} + \nu^{2} - 8\nu - 2 ) / 2 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{3} - \nu^{2} + 12\nu + 2 ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{2} - 6 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{4} - 13\nu^{2} + 6\nu + 14 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + 6 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{3} + 4\beta_{2} + 6\beta _1 - 4 \)
|
| \(\nu^{4}\) | \(=\) |
\( 2\beta_{4} + 13\beta_{3} - 3\beta_{2} - 3\beta _1 + 64 \)
|
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 |
|
−4.47236 | 0 | 12.0020 | 6.52855 | 0 | 5.22706 | −17.8986 | 0 | −29.1981 | |||||||||||||||||||||||||||||||||
| 1.2 | −2.24125 | 0 | −2.97681 | −18.3339 | 0 | −16.8583 | 24.6017 | 0 | 41.0908 | ||||||||||||||||||||||||||||||||||
| 1.3 | −1.63099 | 0 | −5.33986 | 16.8209 | 0 | 5.21997 | 21.7572 | 0 | −27.4348 | ||||||||||||||||||||||||||||||||||
| 1.4 | 2.84972 | 0 | 0.120922 | −12.8729 | 0 | 26.0540 | −22.4532 | 0 | −36.6841 | ||||||||||||||||||||||||||||||||||
| 1.5 | 5.49488 | 0 | 22.1937 | −2.14270 | 0 | 20.3573 | 77.9928 | 0 | −11.7739 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(29\) | \(-1\) |
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 | 261.4.a.f | 5 | |
| 3.b | odd | 2 | 1 | 29.4.a.b | ✓ | 5 | |
| 12.b | even | 2 | 1 | 464.4.a.l | 5 | ||
| 15.d | odd | 2 | 1 | 725.4.a.c | 5 | ||
| 21.c | even | 2 | 1 | 1421.4.a.e | 5 | ||
| 24.f | even | 2 | 1 | 1856.4.a.bb | 5 | ||
| 24.h | odd | 2 | 1 | 1856.4.a.y | 5 | ||
| 87.d | odd | 2 | 1 | 841.4.a.b | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 29.4.a.b | ✓ | 5 | 3.b | odd | 2 | 1 | |
| 261.4.a.f | 5 | 1.a | even | 1 | 1 | trivial | |
| 464.4.a.l | 5 | 12.b | even | 2 | 1 | ||
| 725.4.a.c | 5 | 15.d | odd | 2 | 1 | ||
| 841.4.a.b | 5 | 87.d | odd | 2 | 1 | ||
| 1421.4.a.e | 5 | 21.c | even | 2 | 1 | ||
| 1856.4.a.y | 5 | 24.h | odd | 2 | 1 | ||
| 1856.4.a.bb | 5 | 24.f | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{5} - 33T_{2}^{3} - 28T_{2}^{2} + 192T_{2} + 256 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(261))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} - 33 T^{3} + \cdots + 256 \)
|
| $3$ |
\( T^{5} \)
|
| $5$ |
\( T^{5} + 10 T^{4} + \cdots + 55534 \)
|
| $7$ |
\( T^{5} - 40 T^{4} + \cdots + 243968 \)
|
| $11$ |
\( T^{5} + 12 T^{4} + \cdots - 30997958 \)
|
| $13$ |
\( T^{5} - 14 T^{4} + \cdots - 13078418 \)
|
| $17$ |
\( T^{5} + 66 T^{4} + \cdots - 19935872 \)
|
| $19$ |
\( T^{5} - 214 T^{4} + \cdots + 19441152 \)
|
| $23$ |
\( T^{5} + \cdots + 7938109184 \)
|
| $29$ |
\( (T - 29)^{5} \)
|
| $31$ |
\( T^{5} - 420 T^{4} + \cdots - 2094346 \)
|
| $37$ |
\( T^{5} + \cdots + 23564115968 \)
|
| $41$ |
\( T^{5} + \cdots - 59613728000 \)
|
| $43$ |
\( T^{5} + \cdots + 198643410886 \)
|
| $47$ |
\( T^{5} + \cdots + 203435244846 \)
|
| $53$ |
\( T^{5} + \cdots + 786854101018 \)
|
| $59$ |
\( T^{5} + \cdots - 109032704000 \)
|
| $61$ |
\( T^{5} + \cdots + 2140697762176 \)
|
| $67$ |
\( T^{5} + \cdots - 39308070146048 \)
|
| $71$ |
\( T^{5} + \cdots - 98341318953856 \)
|
| $73$ |
\( T^{5} + \cdots + 7201878016 \)
|
| $79$ |
\( T^{5} + \cdots - 240961986300538 \)
|
| $83$ |
\( T^{5} + \cdots - 6057622580224 \)
|
| $89$ |
\( T^{5} + \cdots - 21549994365568 \)
|
| $97$ |
\( T^{5} + \cdots - 20480102175488 \)
|
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