Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1805,2,Mod(1,1805)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1805, 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("1805.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1805 = 5 \cdot 19^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1805.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: | \(14.4129975648\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.5822000.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - x^{5} - 8x^{4} + 5x^{3} + 14x^{2} + x - 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - x^{5} - 8x^{4} + 5x^{3} + 14x^{2} + x - 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{4} - 5\nu^{2} - 1 ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{5} - \nu^{4} - 7\nu^{3} + 5\nu^{2} + 9\nu - 1 ) / 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{5} - 2\nu^{4} - 7\nu^{3} + 12\nu^{2} + 9\nu - 6 ) / 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -\nu^{5} + \nu^{4} + 9\nu^{3} - 5\nu^{2} - 19\nu - 1 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{4} - \beta_{3} + \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{5} + \beta_{3} + 5\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 5\beta_{4} - 5\beta_{3} + 7\beta_{2} + 16 \)
|
| \(\nu^{5}\) | \(=\) |
\( 7\beta_{5} + 9\beta_{3} + 2\beta_{2} + 26\beta _1 + 9 \)
|
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 |
|
−2.65583 | −2.41205 | 5.05344 | −1.00000 | 6.40599 | −2.14656 | −8.10942 | 2.81796 | 2.65583 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −0.860448 | 0.867394 | −1.25963 | −1.00000 | −0.746348 | −0.263921 | 2.80474 | −2.24763 | 0.860448 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −0.274673 | 3.09431 | −1.92455 | −1.00000 | −0.849925 | 3.63772 | 1.07797 | 6.57477 | 0.274673 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 1.38913 | 3.31798 | −0.0703236 | −1.00000 | 4.60910 | −1.97948 | −2.87594 | 8.00899 | −1.38913 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 1.69444 | −0.918335 | 0.871115 | −1.00000 | −1.55606 | 3.12687 | −1.91282 | −2.15666 | −1.69444 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 2.70739 | 0.0506943 | 5.32995 | −1.00000 | 0.137249 | 4.62536 | 9.01547 | −2.99743 | −2.70739 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \(1\) |
| \(19\) | \(-1\) |
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 | 1805.2.a.r | yes | 6 |
| 5.b | even | 2 | 1 | 9025.2.a.bq | 6 | ||
| 19.b | odd | 2 | 1 | 1805.2.a.q | ✓ | 6 | |
| 95.d | odd | 2 | 1 | 9025.2.a.ca | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1805.2.a.q | ✓ | 6 | 19.b | odd | 2 | 1 | |
| 1805.2.a.r | yes | 6 | 1.a | even | 1 | 1 | trivial |
| 9025.2.a.bq | 6 | 5.b | even | 2 | 1 | ||
| 9025.2.a.ca | 6 | 95.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1805))\):
|
\( T_{2}^{6} - 2T_{2}^{5} - 8T_{2}^{4} + 16T_{2}^{3} + 7T_{2}^{2} - 14T_{2} - 4 \)
|
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\( T_{3}^{6} - 4T_{3}^{5} - 6T_{3}^{4} + 28T_{3}^{3} + 4T_{3}^{2} - 20T_{3} + 1 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} - 2 T^{5} + \cdots - 4 \)
|
| $3$ |
\( T^{6} - 4 T^{5} + \cdots + 1 \)
|
| $5$ |
\( (T + 1)^{6} \)
|
| $7$ |
\( T^{6} - 7 T^{5} + \cdots - 59 \)
|
| $11$ |
\( T^{6} - 15 T^{5} + \cdots - 4496 \)
|
| $13$ |
\( T^{6} - 6 T^{5} + \cdots + 304 \)
|
| $17$ |
\( T^{6} - 5 T^{5} + \cdots - 16 \)
|
| $19$ |
\( T^{6} \)
|
| $23$ |
\( T^{6} - 68 T^{4} + \cdots + 61 \)
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| $29$ |
\( T^{6} + 20 T^{5} + \cdots - 17236 \)
|
| $31$ |
\( T^{6} - 12 T^{5} + \cdots - 16 \)
|
| $37$ |
\( T^{6} - 20 T^{5} + \cdots - 304 \)
|
| $41$ |
\( T^{6} - 8 T^{5} + \cdots + 149 \)
|
| $43$ |
\( T^{6} - 27 T^{5} + \cdots + 496 \)
|
| $47$ |
\( T^{6} + 8 T^{5} + \cdots + 29 \)
|
| $53$ |
\( T^{6} - 19 T^{5} + \cdots - 4544 \)
|
| $59$ |
\( T^{6} + 41 T^{5} + \cdots + 6416 \)
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| $61$ |
\( T^{6} - 6 T^{5} + \cdots + 11584 \)
|
| $67$ |
\( T^{6} + 15 T^{5} + \cdots - 961 \)
|
| $71$ |
\( T^{6} - 2 T^{5} + \cdots - 114224 \)
|
| $73$ |
\( T^{6} - 8 T^{5} + \cdots + 496 \)
|
| $79$ |
\( T^{6} - 18 T^{5} + \cdots + 64 \)
|
| $83$ |
\( T^{6} + 10 T^{5} + \cdots + 24304 \)
|
| $89$ |
\( T^{6} - 17 T^{5} + \cdots - 4139 \)
|
| $97$ |
\( T^{6} + 4 T^{5} + \cdots - 198704 \)
|
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