Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [261,6,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, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("261.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 261 = 3^{2} \cdot 29 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| 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: | \(41.8601769712\) |
| Analytic rank: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.3257317.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 34x^{2} - 27x + 10 \)
|
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - 34x^{2} - 27x + 10 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{2} + 3\nu - 17 ) / 3 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} - 3\nu - 17 ) / 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 2\nu^{3} - \nu^{2} - 67\nu - 25 ) / 3 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{2} + \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 3\beta_{2} + 3\beta _1 + 34 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 3\beta_{3} - 32\beta_{2} + 35\beta _1 + 42 ) / 2 \)
|
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 |
|
−5.91663 | 0 | 3.00648 | 97.7313 | 0 | 139.558 | 171.544 | 0 | −578.240 | ||||||||||||||||||||||||||||||
| 1.2 | −4.16235 | 0 | −14.6748 | −32.5670 | 0 | −220.793 | 194.277 | 0 | 135.555 | |||||||||||||||||||||||||||||||
| 1.3 | 0.863638 | 0 | −31.2541 | 44.4758 | 0 | −36.7447 | −54.6287 | 0 | 38.4110 | |||||||||||||||||||||||||||||||
| 1.4 | 9.21534 | 0 | 52.9225 | −41.6400 | 0 | −90.0205 | 192.808 | 0 | −383.727 | |||||||||||||||||||||||||||||||
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 | 261.6.a.a | 4 | |
| 3.b | odd | 2 | 1 | 29.6.a.a | ✓ | 4 | |
| 12.b | even | 2 | 1 | 464.6.a.i | 4 | ||
| 15.d | odd | 2 | 1 | 725.6.a.a | 4 | ||
| 87.d | odd | 2 | 1 | 841.6.a.a | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 29.6.a.a | ✓ | 4 | 3.b | odd | 2 | 1 | |
| 261.6.a.a | 4 | 1.a | even | 1 | 1 | trivial | |
| 464.6.a.i | 4 | 12.b | even | 2 | 1 | ||
| 725.6.a.a | 4 | 15.d | odd | 2 | 1 | ||
| 841.6.a.a | 4 | 87.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} - 69T_{2}^{2} - 168T_{2} + 196 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(261))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - 69 T^{2} + \cdots + 196 \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} - 68 T^{3} + \cdots + 5894489 \)
|
| $7$ |
\( T^{4} + 208 T^{3} + \cdots - 101924272 \)
|
| $11$ |
\( T^{4} + \cdots - 3717303119 \)
|
| $13$ |
\( T^{4} + \cdots - 116053863479 \)
|
| $17$ |
\( T^{4} + \cdots + 11464717824 \)
|
| $19$ |
\( T^{4} + \cdots + 2620094791680 \)
|
| $23$ |
\( T^{4} + \cdots + 2033080361984 \)
|
| $29$ |
\( (T - 841)^{4} \)
|
| $31$ |
\( T^{4} + \cdots - 421127233952247 \)
|
| $37$ |
\( T^{4} + \cdots + 16079593861120 \)
|
| $41$ |
\( T^{4} + \cdots + 10\!\cdots\!56 \)
|
| $43$ |
\( T^{4} + \cdots + 47\!\cdots\!37 \)
|
| $47$ |
\( T^{4} + \cdots - 35\!\cdots\!87 \)
|
| $53$ |
\( T^{4} + \cdots + 16\!\cdots\!53 \)
|
| $59$ |
\( T^{4} + \cdots - 43\!\cdots\!80 \)
|
| $61$ |
\( T^{4} + \cdots + 90\!\cdots\!24 \)
|
| $67$ |
\( T^{4} + \cdots + 20\!\cdots\!92 \)
|
| $71$ |
\( T^{4} + \cdots - 18\!\cdots\!16 \)
|
| $73$ |
\( T^{4} + \cdots - 40\!\cdots\!00 \)
|
| $79$ |
\( T^{4} + \cdots - 53\!\cdots\!75 \)
|
| $83$ |
\( T^{4} + \cdots - 18\!\cdots\!88 \)
|
| $89$ |
\( T^{4} + \cdots - 18\!\cdots\!60 \)
|
| $97$ |
\( T^{4} + \cdots - 12\!\cdots\!28 \)
|
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