Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1859,2,Mod(1,1859)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1859, 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("1859.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1859 = 11 \cdot 13^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1859.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: | \(14.8441897358\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.28561300.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 9x^{4} - x^{3} + 22x^{2} + 4x - 12 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 143) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 9x^{4} - x^{3} + 22x^{2} + 4x - 12 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 4\nu \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{5} - 7\nu^{3} - \nu^{2} + 10\nu + 2 ) / 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -\nu^{5} + 2\nu^{4} + 7\nu^{3} - 13\nu^{2} - 10\nu + 16 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 4\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{5} + \beta_{4} + 7\beta_{2} + 12 \)
|
| \(\nu^{5}\) | \(=\) |
\( 2\beta_{4} + 7\beta_{3} + \beta_{2} + 18\beta _1 + 1 \)
|
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 |
|
−2.53464 | −2.99024 | 4.42441 | 3.35258 | 7.57919 | −1.91841 | −6.14501 | 5.94153 | −8.49760 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −1.97462 | −0.571492 | 1.89914 | −1.23039 | 1.12848 | 2.55803 | 0.199164 | −2.67340 | 2.42955 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −0.268396 | 2.35117 | −1.92796 | 2.80787 | −0.631044 | −2.38467 | 1.05425 | 2.52800 | −0.753621 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0.498953 | −0.240597 | −1.75105 | −0.581470 | −0.120047 | −1.41017 | −1.87160 | −2.94211 | −0.290126 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 1.82172 | 3.11527 | 1.31865 | −0.0854874 | 5.67515 | 4.51942 | −1.24122 | 6.70494 | −0.155734 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 2.45699 | −0.664116 | 4.03681 | 1.73689 | −1.63173 | 1.63580 | 5.00442 | −2.55895 | 4.26753 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(11\) | \(-1\) |
| \(13\) | \(1\) |
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 | 1859.2.a.l | 6 | |
| 13.b | even | 2 | 1 | 1859.2.a.k | 6 | ||
| 13.e | even | 6 | 2 | 143.2.e.c | ✓ | 12 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 143.2.e.c | ✓ | 12 | 13.e | even | 6 | 2 | |
| 1859.2.a.k | 6 | 13.b | even | 2 | 1 | ||
| 1859.2.a.l | 6 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1859))\):
|
\( T_{2}^{6} - 10T_{2}^{4} + T_{2}^{3} + 24T_{2}^{2} - 5T_{2} - 3 \)
|
|
\( T_{7}^{6} - 3T_{7}^{5} - 16T_{7}^{4} + 27T_{7}^{3} + 82T_{7}^{2} - 52T_{7} - 122 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} - 10 T^{4} + \cdots - 3 \)
|
| $3$ |
\( T^{6} - T^{5} - 12 T^{4} + \cdots + 2 \)
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| $5$ |
\( T^{6} - 6 T^{5} + \cdots - 1 \)
|
| $7$ |
\( T^{6} - 3 T^{5} + \cdots - 122 \)
|
| $11$ |
\( (T - 1)^{6} \)
|
| $13$ |
\( T^{6} \)
|
| $17$ |
\( T^{6} - 2 T^{5} + \cdots - 139 \)
|
| $19$ |
\( T^{6} - 10 T^{5} + \cdots - 302 \)
|
| $23$ |
\( T^{6} - 3 T^{5} + \cdots + 592 \)
|
| $29$ |
\( T^{6} - 3 T^{5} + \cdots - 19891 \)
|
| $31$ |
\( T^{6} - 5 T^{5} + \cdots + 120 \)
|
| $37$ |
\( T^{6} - 25 T^{5} + \cdots - 5171 \)
|
| $41$ |
\( T^{6} - 24 T^{5} + \cdots - 6427 \)
|
| $43$ |
\( T^{6} + 8 T^{5} + \cdots + 1448 \)
|
| $47$ |
\( T^{6} - 10 T^{5} + \cdots - 2 \)
|
| $53$ |
\( T^{6} - 10 T^{5} + \cdots + 121 \)
|
| $59$ |
\( T^{6} + 4 T^{5} + \cdots - 18398 \)
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| $61$ |
\( T^{6} + 21 T^{5} + \cdots - 655 \)
|
| $67$ |
\( T^{6} - 21 T^{5} + \cdots + 2978 \)
|
| $71$ |
\( T^{6} + 3 T^{5} + \cdots - 101256 \)
|
| $73$ |
\( T^{6} - 13 T^{5} + \cdots + 296 \)
|
| $79$ |
\( T^{6} + 4 T^{5} + \cdots - 1572016 \)
|
| $83$ |
\( T^{6} - 8 T^{5} + \cdots - 12114 \)
|
| $89$ |
\( T^{6} + 9 T^{5} + \cdots + 648364 \)
|
| $97$ |
\( T^{6} - 15 T^{5} + \cdots - 7628 \)
|
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