Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1338,2,Mod(1,1338)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1338, 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("1338.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1338 = 2 \cdot 3 \cdot 223 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1338.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: | \(10.6839837904\) |
| Analytic rank: | \(0\) |
| Dimension: | \(7\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{7} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{7} - 18x^{5} - 8x^{4} + 51x^{3} + 47x^{2} - 2x - 8 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| 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 \(1,\beta_1,\ldots,\beta_{6}\) 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
\( x^{7} - 18x^{5} - 8x^{4} + 51x^{3} + 47x^{2} - 2x - 8 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{6} - \nu^{5} - 17\nu^{4} + 9\nu^{3} + 42\nu^{2} + 5\nu - 8 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -9\nu^{6} + 7\nu^{5} + 156\nu^{4} - 49\nu^{3} - 413\nu^{2} - 104\nu + 88 ) / 3 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -21\nu^{6} + 16\nu^{5} + 366\nu^{4} - 112\nu^{3} - 989\nu^{2} - 221\nu + 226 ) / 6 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 11\nu^{6} - 9\nu^{5} - 191\nu^{4} + 68\nu^{3} + 512\nu^{2} + 104\nu - 123 ) / 3 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -14\nu^{6} + 10\nu^{5} + 245\nu^{4} - 63\nu^{3} - 670\nu^{2} - 181\nu + 148 ) / 3 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{6} + \beta_{5} - \beta_{3} - 2\beta_{2} + \beta _1 + 5 \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{6} - \beta_{5} - 4\beta_{4} - \beta_{2} + 13\beta _1 + 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( 17\beta_{6} + 11\beta_{5} - 4\beta_{4} - 18\beta_{3} - 29\beta_{2} + 18\beta _1 + 59 \)
|
| \(\nu^{5}\) | \(=\) |
\( 40\beta_{6} - 17\beta_{5} - 70\beta_{4} - 11\beta_{3} - 29\beta_{2} + 188\beta _1 + 57 \)
|
| \(\nu^{6}\) | \(=\) |
\( 269\beta_{6} + 137\beta_{5} - 102\beta_{4} - 275\beta_{3} - 428\beta_{2} + 330\beta _1 + 831 \)
|
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 |
|
1.00000 | −1.00000 | 1.00000 | −3.73564 | −1.00000 | 2.27800 | 1.00000 | 1.00000 | −3.73564 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | 1.00000 | −1.00000 | 1.00000 | −1.60399 | −1.00000 | −4.86544 | 1.00000 | 1.00000 | −1.60399 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | 1.00000 | −1.00000 | 1.00000 | −0.603045 | −1.00000 | 4.55050 | 1.00000 | 1.00000 | −0.603045 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | 1.00000 | −1.00000 | 1.00000 | 2.18886 | −1.00000 | −2.20076 | 1.00000 | 1.00000 | 2.18886 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 1.00000 | −1.00000 | 1.00000 | 2.69148 | −1.00000 | 2.17419 | 1.00000 | 1.00000 | 2.69148 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.00000 | −1.00000 | 1.00000 | 3.27125 | −1.00000 | 3.11004 | 1.00000 | 1.00000 | 3.27125 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 1.00000 | −1.00000 | 1.00000 | 3.79108 | −1.00000 | −2.04653 | 1.00000 | 1.00000 | 3.79108 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(3\) | \(1\) |
| \(223\) | \(1\) |
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 | 1338.2.a.j | ✓ | 7 |
| 3.b | odd | 2 | 1 | 4014.2.a.v | 7 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1338.2.a.j | ✓ | 7 | 1.a | even | 1 | 1 | trivial |
| 4014.2.a.v | 7 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{7} - 6T_{5}^{6} - 9T_{5}^{5} + 105T_{5}^{4} - 91T_{5}^{3} - 316T_{5}^{2} + 304T_{5} + 264 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1338))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1)^{7} \)
|
| $3$ |
\( (T + 1)^{7} \)
|
| $5$ |
\( T^{7} - 6 T^{6} + \cdots + 264 \)
|
| $7$ |
\( T^{7} - 3 T^{6} + \cdots + 1536 \)
|
| $11$ |
\( T^{7} + T^{6} + \cdots + 2512 \)
|
| $13$ |
\( T^{7} - 8 T^{6} + \cdots + 16 \)
|
| $17$ |
\( T^{7} - 16 T^{6} + \cdots - 6336 \)
|
| $19$ |
\( T^{7} - 2 T^{6} + \cdots - 6064 \)
|
| $23$ |
\( T^{7} - 8 T^{6} + \cdots + 18112 \)
|
| $29$ |
\( T^{7} - 4 T^{6} + \cdots + 1008 \)
|
| $31$ |
\( T^{7} - 11 T^{6} + \cdots - 15488 \)
|
| $37$ |
\( T^{7} - 17 T^{6} + \cdots - 124128 \)
|
| $41$ |
\( T^{7} - 18 T^{6} + \cdots - 412192 \)
|
| $43$ |
\( T^{7} + T^{6} + \cdots + 537552 \)
|
| $47$ |
\( T^{7} - 5 T^{6} + \cdots + 1909248 \)
|
| $53$ |
\( T^{7} - 6 T^{6} + \cdots + 97956 \)
|
| $59$ |
\( T^{7} + 7 T^{6} + \cdots + 59056 \)
|
| $61$ |
\( T^{7} - 24 T^{6} + \cdots + 5464 \)
|
| $67$ |
\( T^{7} + 4 T^{6} + \cdots - 13872 \)
|
| $71$ |
\( T^{7} + 7 T^{6} + \cdots + 512 \)
|
| $73$ |
\( T^{7} - 28 T^{6} + \cdots - 8586 \)
|
| $79$ |
\( T^{7} - 275 T^{5} + \cdots + 1214688 \)
|
| $83$ |
\( T^{7} + 2 T^{6} + \cdots - 2585664 \)
|
| $89$ |
\( T^{7} - 5 T^{6} + \cdots - 119008 \)
|
| $97$ |
\( T^{7} - 23 T^{6} + \cdots + 1877728 \)
|
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