Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [381,2,Mod(1,381)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(381, 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("381.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 381 = 3 \cdot 127 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 381.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: | \(3.04230031701\) |
| Analytic rank: | \(0\) |
| 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: |
\( x^{9} - 2x^{8} - 14x^{7} + 26x^{6} + 59x^{5} - 99x^{4} - 66x^{3} + 102x^{2} - 24x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| 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
\( x^{9} - 2x^{8} - 14x^{7} + 26x^{6} + 59x^{5} - 99x^{4} - 66x^{3} + 102x^{2} - 24x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 6\nu \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{8} + \nu^{7} + 15\nu^{6} - 11\nu^{5} - 70\nu^{4} + 29\nu^{3} + 99\nu^{2} - 3\nu - 7 ) / 4 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -\nu^{8} + 3\nu^{7} + 14\nu^{6} - 40\nu^{5} - 58\nu^{4} + 157\nu^{3} + 59\nu^{2} - 170\nu + 32 ) / 6 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -5\nu^{8} + 9\nu^{7} + 73\nu^{6} - 113\nu^{5} - 338\nu^{4} + 401\nu^{3} + 499\nu^{2} - 337\nu - 5 ) / 12 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( \nu^{8} - 2\nu^{7} - 14\nu^{6} + 25\nu^{5} + 60\nu^{4} - 89\nu^{3} - 74\nu^{2} + 80\nu - 11 ) / 2 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 3\nu^{8} - 5\nu^{7} - 43\nu^{6} + 65\nu^{5} + 190\nu^{4} - 247\nu^{3} - 247\nu^{2} + 251\nu - 27 ) / 4 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 6\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -\beta_{6} + \beta_{5} + \beta_{4} + 7\beta_{2} + \beta _1 + 24 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{8} - \beta_{7} + \beta_{4} + 10\beta_{3} + 38\beta _1 + 3 \)
|
| \(\nu^{6}\) | \(=\) |
\( \beta_{8} + 3\beta_{7} - 8\beta_{6} + 14\beta_{5} + 13\beta_{4} + 2\beta_{3} + 46\beta_{2} + 11\beta _1 + 152 \)
|
| \(\nu^{7}\) | \(=\) |
\( 15\beta_{8} - 13\beta_{7} + 2\beta_{6} + 4\beta_{5} + 13\beta_{4} + 82\beta_{3} + \beta_{2} + 250\beta _1 + 36 \)
|
| \(\nu^{8}\) | \(=\) |
\( 19\beta_{8} + 43\beta_{7} - 48\beta_{6} + 144\beta_{5} + 123\beta_{4} + 31\beta_{3} + 300\beta_{2} + 98\beta _1 + 992 \)
|
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 |
|
−2.75841 | 1.00000 | 5.60882 | −3.68284 | −2.75841 | −3.98729 | −9.95461 | 1.00000 | 10.1588 | |||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.55353 | 1.00000 | 4.52051 | 2.44899 | −2.55353 | 3.89705 | −6.43620 | 1.00000 | −6.25358 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.92789 | 1.00000 | 1.71676 | −1.15841 | −1.92789 | 2.35752 | 0.546061 | 1.00000 | 2.23329 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −0.682747 | 1.00000 | −1.53386 | 3.92611 | −0.682747 | 1.75230 | 2.41273 | 1.00000 | −2.68054 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −0.247918 | 1.00000 | −1.93854 | 0.730098 | −0.247918 | −3.26267 | 0.976436 | 1.00000 | −0.181005 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −0.0532857 | 1.00000 | −1.99716 | −3.85537 | −0.0532857 | 4.59060 | 0.212992 | 1.00000 | 0.205436 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 1.44437 | 1.00000 | 0.0862103 | 0.766977 | 1.44437 | 3.56571 | −2.76422 | 1.00000 | 1.10780 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 2.14900 | 1.00000 | 2.61819 | 0.492551 | 2.14900 | −0.251160 | 1.32848 | 1.00000 | 1.05849 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 2.63041 | 1.00000 | 4.91907 | −3.66812 | 2.63041 | 1.33794 | 7.67834 | 1.00000 | −9.64866 | ||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(127\) | \(1\) |
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 | 381.2.a.e | ✓ | 9 |
| 3.b | odd | 2 | 1 | 1143.2.a.j | 9 | ||
| 4.b | odd | 2 | 1 | 6096.2.a.bk | 9 | ||
| 5.b | even | 2 | 1 | 9525.2.a.p | 9 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 381.2.a.e | ✓ | 9 | 1.a | even | 1 | 1 | trivial |
| 1143.2.a.j | 9 | 3.b | odd | 2 | 1 | ||
| 6096.2.a.bk | 9 | 4.b | odd | 2 | 1 | ||
| 9525.2.a.p | 9 | 5.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{9} + 2T_{2}^{8} - 14T_{2}^{7} - 26T_{2}^{6} + 59T_{2}^{5} + 99T_{2}^{4} - 66T_{2}^{3} - 102T_{2}^{2} - 24T_{2} - 1 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(381))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{9} + 2 T^{8} + \cdots - 1 \)
|
| $3$ |
\( (T - 1)^{9} \)
|
| $5$ |
\( T^{9} + 4 T^{8} + \cdots - 160 \)
|
| $7$ |
\( T^{9} - 10 T^{8} + \cdots + 1152 \)
|
| $11$ |
\( T^{9} - 8 T^{8} + \cdots - 5644 \)
|
| $13$ |
\( T^{9} - 14 T^{8} + \cdots + 418 \)
|
| $17$ |
\( T^{9} + 6 T^{8} + \cdots + 4768 \)
|
| $19$ |
\( T^{9} - 12 T^{8} + \cdots + 2816 \)
|
| $23$ |
\( T^{9} + 4 T^{8} + \cdots - 169984 \)
|
| $29$ |
\( T^{9} + 8 T^{8} + \cdots + 288 \)
|
| $31$ |
\( T^{9} - 4 T^{8} + \cdots + 123392 \)
|
| $37$ |
\( T^{9} - 22 T^{8} + \cdots + 136082 \)
|
| $41$ |
\( T^{9} + 2 T^{8} + \cdots - 30880 \)
|
| $43$ |
\( T^{9} - 6 T^{8} + \cdots + 292288 \)
|
| $47$ |
\( T^{9} + 2 T^{8} + \cdots + 1356304 \)
|
| $53$ |
\( T^{9} + 12 T^{8} + \cdots + 633760 \)
|
| $59$ |
\( T^{9} + 6 T^{8} + \cdots + 3599360 \)
|
| $61$ |
\( T^{9} - 2 T^{8} + \cdots - 6116062 \)
|
| $67$ |
\( T^{9} - 18 T^{8} + \cdots - 1696960 \)
|
| $71$ |
\( T^{9} - 24 T^{8} + \cdots - 165399656 \)
|
| $73$ |
\( T^{9} - 14 T^{8} + \cdots + 271190 \)
|
| $79$ |
\( T^{9} - 12 T^{8} + \cdots - 104192 \)
|
| $83$ |
\( T^{9} + 20 T^{8} + \cdots + 226643968 \)
|
| $89$ |
\( T^{9} + 30 T^{8} + \cdots - 825248 \)
|
| $97$ |
\( T^{9} - 12 T^{8} + \cdots - 13712896 \)
|
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