Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2169,2,Mod(1,2169)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2169, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2169.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2169 = 3^{2} \cdot 241 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2169.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: | \(17.3195521984\) |
| Analytic rank: | \(1\) |
| Dimension: | \(9\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{9} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{9} - 3x^{8} - 8x^{7} + 27x^{6} + 15x^{5} - 71x^{4} + 7x^{3} + 46x^{2} - 12x - 4 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 723) |
| 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 \(1,\beta_1,\ldots,\beta_{8}\) 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
\( x^{9} - 3x^{8} - 8x^{7} + 27x^{6} + 15x^{5} - 71x^{4} + 7x^{3} + 46x^{2} - 12x - 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 4\nu + 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - \nu^{3} - 5\nu^{2} + 3\nu + 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{5} - 2\nu^{4} - 5\nu^{3} + 9\nu^{2} + 3\nu - 4 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( \nu^{8} - 3\nu^{7} - 8\nu^{6} + 25\nu^{5} + 19\nu^{4} - 59\nu^{3} - 13\nu^{2} + 30\nu + 2 ) / 2 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -\nu^{8} + 3\nu^{7} + 10\nu^{6} - 29\nu^{5} - 31\nu^{4} + 79\nu^{3} + 29\nu^{2} - 44\nu - 6 ) / 2 \)
|
| \(\beta_{8}\) | \(=\) |
\( \nu^{8} - 2\nu^{7} - 10\nu^{6} + 18\nu^{5} + 31\nu^{4} - 46\nu^{3} - 29\nu^{2} + 25\nu + 5 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 4\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{4} + \beta_{3} + 6\beta_{2} + \beta _1 + 14 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{5} + 2\beta_{4} + 7\beta_{3} + 8\beta_{2} + 19\beta _1 + 10 \)
|
| \(\nu^{6}\) | \(=\) |
\( \beta_{7} + \beta_{6} + 2\beta_{5} + 10\beta_{4} + 10\beta_{3} + 34\beta_{2} + 11\beta _1 + 72 \)
|
| \(\nu^{7}\) | \(=\) |
\( \beta_{8} + 2\beta_{7} + 11\beta_{5} + 22\beta_{4} + 44\beta_{3} + 55\beta_{2} + 96\beta _1 + 78 \)
|
| \(\nu^{8}\) | \(=\) |
\( 3\beta_{8} + 14\beta_{7} + 10\beta_{6} + 24\beta_{5} + 77\beta_{4} + 77\beta_{3} + 195\beta_{2} + 88\beta _1 + 390 \)
|
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 |
|
−2.49543 | 0 | 4.22717 | −1.31835 | 0 | 2.06446 | −5.55775 | 0 | 3.28986 | |||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.31297 | 0 | 3.34985 | −4.02443 | 0 | −4.48318 | −3.12216 | 0 | 9.30841 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.84187 | 0 | 1.39248 | 0.763153 | 0 | 1.32171 | 1.11897 | 0 | −1.40563 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −0.845068 | 0 | −1.28586 | −3.77424 | 0 | 1.63086 | 2.77678 | 0 | 3.18949 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −0.620852 | 0 | −1.61454 | 2.97924 | 0 | −1.95766 | 2.24410 | 0 | −1.84967 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0.197518 | 0 | −1.96099 | −1.67366 | 0 | −4.59705 | −0.782365 | 0 | −0.330578 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0.900190 | 0 | −1.18966 | −2.66512 | 0 | 3.59926 | −2.87130 | 0 | −2.39912 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 1.94826 | 0 | 1.79574 | 0.566165 | 0 | −3.15848 | −0.397959 | 0 | 1.10304 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 2.07022 | 0 | 2.28581 | −2.85275 | 0 | 0.580097 | 0.591692 | 0 | −5.90581 | ||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(241\) | \(-1\) |
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 | 2169.2.a.f | 9 | |
| 3.b | odd | 2 | 1 | 723.2.a.e | ✓ | 9 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 723.2.a.e | ✓ | 9 | 3.b | odd | 2 | 1 | |
| 2169.2.a.f | 9 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{9} + 3T_{2}^{8} - 8T_{2}^{7} - 27T_{2}^{6} + 15T_{2}^{5} + 71T_{2}^{4} + 7T_{2}^{3} - 46T_{2}^{2} - 12T_{2} + 4 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2169))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{9} + 3 T^{8} + \cdots + 4 \)
|
| $3$ |
\( T^{9} \)
|
| $5$ |
\( T^{9} + 12 T^{8} + \cdots - 328 \)
|
| $7$ |
\( T^{9} + 5 T^{8} + \cdots - 1184 \)
|
| $11$ |
\( T^{9} + 3 T^{8} + \cdots - 448 \)
|
| $13$ |
\( T^{9} - 18 T^{8} + \cdots - 296 \)
|
| $17$ |
\( T^{9} + 16 T^{8} + \cdots - 392 \)
|
| $19$ |
\( T^{9} - 3 T^{8} + \cdots - 832 \)
|
| $23$ |
\( T^{9} + 9 T^{8} + \cdots + 14588 \)
|
| $29$ |
\( T^{9} + 15 T^{8} + \cdots - 2344 \)
|
| $31$ |
\( T^{9} + 13 T^{8} + \cdots - 12688 \)
|
| $37$ |
\( T^{9} - 19 T^{8} + \cdots - 19560944 \)
|
| $41$ |
\( T^{9} + 18 T^{8} + \cdots + 696416 \)
|
| $43$ |
\( T^{9} + 11 T^{8} + \cdots - 224 \)
|
| $47$ |
\( T^{9} + 19 T^{8} + \cdots - 263392 \)
|
| $53$ |
\( T^{9} + 31 T^{8} + \cdots + 1665712 \)
|
| $59$ |
\( T^{9} + 11 T^{8} + \cdots + 2391952 \)
|
| $61$ |
\( T^{9} - 2 T^{8} + \cdots + 497498 \)
|
| $67$ |
\( T^{9} + 12 T^{8} + \cdots - 29548 \)
|
| $71$ |
\( T^{9} + 6 T^{8} + \cdots + 1312144 \)
|
| $73$ |
\( T^{9} - 18 T^{8} + \cdots - 3824968 \)
|
| $79$ |
\( T^{9} + 32 T^{8} + \cdots - 6116744 \)
|
| $83$ |
\( T^{9} + 13 T^{8} + \cdots - 403904 \)
|
| $89$ |
\( T^{9} + 24 T^{8} + \cdots - 7531586 \)
|
| $97$ |
\( T^{9} - 23 T^{8} + \cdots - 87773474 \)
|
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