Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [803,2,Mod(1,803)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(803, 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("803.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 803 = 11 \cdot 73 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 803.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: | \(6.41198728231\) |
| Analytic rank: | \(1\) |
| Dimension: | \(10\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{10} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{10} - 5x^{9} + 29x^{7} - 21x^{6} - 53x^{5} + 47x^{4} + 32x^{3} - 27x^{2} - 6x + 4 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| 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 \(1,\beta_1,\ldots,\beta_{9}\) 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
\( x^{10} - 5x^{9} + 29x^{7} - 21x^{6} - 53x^{5} + 47x^{4} + 32x^{3} - 27x^{2} - 6x + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{9} + 5\nu^{8} - 27\nu^{6} + 13\nu^{5} + 51\nu^{4} - 17\nu^{3} - 38\nu^{2} + \nu + 6 ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{9} - 5\nu^{8} + 2\nu^{7} + 21\nu^{6} - 27\nu^{5} - 9\nu^{4} + 45\nu^{3} - 36\nu^{2} - 13\nu + 16 ) / 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{9} + 5\nu^{8} - 29\nu^{6} + 23\nu^{5} + 45\nu^{4} - 47\nu^{3} - 8\nu^{2} + 15\nu - 4 ) / 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{9} - 5\nu^{8} + 29\nu^{6} - 21\nu^{5} - 51\nu^{4} + 43\nu^{3} + 24\nu^{2} - 17\nu ) / 2 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( \nu^{9} - 5\nu^{8} + 29\nu^{6} - 21\nu^{5} - 51\nu^{4} + 41\nu^{3} + 28\nu^{2} - 11\nu - 6 ) / 2 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -\nu^{9} + 5\nu^{8} - 29\nu^{6} + 21\nu^{5} + 53\nu^{4} - 45\nu^{3} - 34\nu^{2} + 17\nu + 8 ) / 2 \)
|
| \(\beta_{8}\) | \(=\) |
\( \nu^{9} - 4\nu^{8} - 4\nu^{7} + 25\nu^{6} + 4\nu^{5} - 49\nu^{4} - 2\nu^{3} + 30\nu^{2} + 3\nu - 3 \)
|
| \(\beta_{9}\) | \(=\) |
\( -\nu^{9} + 4\nu^{8} + 4\nu^{7} - 25\nu^{6} - 4\nu^{5} + 50\nu^{4} - 34\nu^{2} + \nu + 6 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{9} - \beta_{8} + \beta_{7} + \beta_{6} + \beta _1 + 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( -2\beta_{9} - 2\beta_{8} + 2\beta_{7} + \beta_{6} + \beta_{5} + 5\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -7\beta_{9} - 7\beta_{8} + 8\beta_{7} + 6\beta_{6} + 2\beta_{5} + 10\beta _1 + 7 \)
|
| \(\nu^{5}\) | \(=\) |
\( -17\beta_{9} - 17\beta_{8} + 20\beta_{7} + 12\beta_{6} + 9\beta_{5} + \beta_{4} + 33\beta _1 + 9 \)
|
| \(\nu^{6}\) | \(=\) |
\( -49\beta_{9} - 49\beta_{8} + 61\beta_{7} + 42\beta_{6} + 24\beta_{5} + 4\beta_{4} + \beta_{2} + 82\beta _1 + 34 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 128 \beta_{9} - 128 \beta_{8} + 164 \beta_{7} + 107 \beta_{6} + 79 \beta_{5} + 19 \beta_{4} + \cdots + 67 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 353 \beta_{9} - 352 \beta_{8} + 468 \beta_{7} + 323 \beta_{6} + 226 \beta_{5} + 67 \beta_{4} + \cdots + 201 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 948 \beta_{9} - 943 \beta_{8} + 1289 \beta_{7} + 888 \beta_{6} + 684 \beta_{5} + 240 \beta_{4} + \cdots + 474 \)
|
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 |
|
−2.60085 | −2.67544 | 4.76443 | 1.80992 | 6.95841 | 0.443469 | −7.18986 | 4.15795 | −4.70734 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.49667 | 1.94349 | 4.23334 | 0.743105 | −4.85224 | −3.71529 | −5.57591 | 0.777142 | −1.85529 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.74377 | 1.96022 | 1.04073 | −3.14000 | −3.41818 | 1.53678 | 1.67274 | 0.842476 | 5.47544 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.52695 | −2.65074 | 0.331565 | −3.22427 | 4.04754 | −1.92698 | 2.54761 | 4.02642 | 4.92329 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −0.606728 | −0.504418 | −1.63188 | 0.0765050 | 0.306044 | 0.287484 | 2.20356 | −2.74556 | −0.0464177 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −0.235626 | −1.06305 | −1.94448 | 4.25907 | 0.250483 | −1.84823 | 0.929422 | −1.86992 | −1.00355 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0.256977 | 1.16155 | −1.93396 | −1.94649 | 0.298491 | 2.92970 | −1.01094 | −1.65081 | −0.500205 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0.545416 | 1.68037 | −1.70252 | 0.888470 | 0.916498 | −5.05491 | −2.01941 | −0.176368 | 0.484585 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 1.57202 | −2.76236 | 0.471260 | 0.295833 | −4.34250 | 1.68379 | −2.40322 | 4.63063 | 0.465057 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 1.83617 | −0.0896154 | 1.37152 | −1.76214 | −0.164549 | −2.33580 | −1.15400 | −2.99197 | −3.23558 | |||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(11\) | \(-1\) |
| \(73\) | \(-1\) |
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 | 803.2.a.d | ✓ | 10 |
| 3.b | odd | 2 | 1 | 7227.2.a.q | 10 | ||
| 11.b | odd | 2 | 1 | 8833.2.a.i | 10 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 803.2.a.d | ✓ | 10 | 1.a | even | 1 | 1 | trivial |
| 7227.2.a.q | 10 | 3.b | odd | 2 | 1 | ||
| 8833.2.a.i | 10 | 11.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{10} + 5T_{2}^{9} - 31T_{2}^{7} - 28T_{2}^{6} + 52T_{2}^{5} + 62T_{2}^{4} - 15T_{2}^{3} - 20T_{2}^{2} + T_{2} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(803))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{10} + 5 T^{9} + \cdots + 1 \)
|
| $3$ |
\( T^{10} + 3 T^{9} + \cdots + 7 \)
|
| $5$ |
\( T^{10} + 2 T^{9} + \cdots + 4 \)
|
| $7$ |
\( T^{10} + 8 T^{9} + \cdots - 151 \)
|
| $11$ |
\( (T - 1)^{10} \)
|
| $13$ |
\( T^{10} + 24 T^{9} + \cdots + 397 \)
|
| $17$ |
\( T^{10} + 9 T^{9} + \cdots - 676675 \)
|
| $19$ |
\( T^{10} + 7 T^{9} + \cdots - 700 \)
|
| $23$ |
\( T^{10} + 8 T^{9} + \cdots - 1759 \)
|
| $29$ |
\( T^{10} + 22 T^{9} + \cdots - 329 \)
|
| $31$ |
\( T^{10} + T^{9} + \cdots + 196 \)
|
| $37$ |
\( T^{10} + 34 T^{9} + \cdots + 4748941 \)
|
| $41$ |
\( T^{10} - 7 T^{9} + \cdots - 26429708 \)
|
| $43$ |
\( T^{10} + 26 T^{9} + \cdots - 3630784 \)
|
| $47$ |
\( T^{10} + 6 T^{9} + \cdots - 3364 \)
|
| $53$ |
\( T^{10} + 19 T^{9} + \cdots + 8379100 \)
|
| $59$ |
\( T^{10} - 12 T^{9} + \cdots - 324484 \)
|
| $61$ |
\( T^{10} + 32 T^{9} + \cdots + 249536 \)
|
| $67$ |
\( T^{10} + 25 T^{9} + \cdots - 10129 \)
|
| $71$ |
\( T^{10} + \cdots + 438323557 \)
|
| $73$ |
\( (T - 1)^{10} \)
|
| $79$ |
\( T^{10} + \cdots + 4016685668 \)
|
| $83$ |
\( T^{10} + 13 T^{9} + \cdots - 9023 \)
|
| $89$ |
\( T^{10} + \cdots + 2399072519 \)
|
| $97$ |
\( T^{10} + 16 T^{9} + \cdots + 39591979 \)
|
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