Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1805,2,Mod(1084,1805)]
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([1, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1805.1084");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1805 = 5 \cdot 19^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1805.b (of order \(2\), degree \(1\), minimal) |
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: | no |
| Analytic conductor: | \(14.4129975648\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.0.4227136.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} + 6x^{4} + 7x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 95) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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
\( x^{6} + 6x^{4} + 7x^{2} + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} + 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{4} + 5\nu^{2} + 3 \)
|
| \(\beta_{4}\) | \(=\) |
\( -\nu^{5} - 5\nu^{3} - 3\nu \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{5} + 6\nu^{3} + 6\nu \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} - 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{5} + \beta_{4} - 3\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{3} - 5\beta_{2} + 7 \)
|
| \(\nu^{5}\) | \(=\) |
\( -5\beta_{5} - 6\beta_{4} + 12\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1805\mathbb{Z}\right)^\times\).
| \(n\) | \(362\) | \(1446\) |
| \(\chi(n)\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1084.1 |
|
− | 2.45620i | 1.56104i | −4.03293 | 2.19869 | − | 0.407132i | 3.83424 | − | 4.50527i | 4.99330i | 0.563139 | −1.00000 | − | 5.40043i | ||||||||||||||||||||||||||||||
| 1084.2 | − | 0.862781i | 3.07914i | 1.25561 | −1.91223 | − | 1.15904i | 2.65662 | − | 0.566520i | − | 2.80888i | −6.48108 | −1.00000 | + | 1.64984i | ||||||||||||||||||||||||||||||
| 1084.3 | − | 0.471884i | − | 1.04022i | 1.77733 | 0.713538 | − | 2.11917i | −0.490864 | 1.17540i | − | 1.78246i | 1.91794 | −1.00000 | − | 0.336707i | ||||||||||||||||||||||||||||||
| 1084.4 | 0.471884i | 1.04022i | 1.77733 | 0.713538 | + | 2.11917i | −0.490864 | − | 1.17540i | 1.78246i | 1.91794 | −1.00000 | + | 0.336707i | ||||||||||||||||||||||||||||||||
| 1084.5 | 0.862781i | − | 3.07914i | 1.25561 | −1.91223 | + | 1.15904i | 2.65662 | 0.566520i | 2.80888i | −6.48108 | −1.00000 | − | 1.64984i | ||||||||||||||||||||||||||||||||
| 1084.6 | 2.45620i | − | 1.56104i | −4.03293 | 2.19869 | + | 0.407132i | 3.83424 | 4.50527i | − | 4.99330i | 0.563139 | −1.00000 | + | 5.40043i | |||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1805.2.b.f | 6 | |
| 5.b | even | 2 | 1 | inner | 1805.2.b.f | 6 | |
| 5.c | odd | 4 | 2 | 9025.2.a.bu | 6 | ||
| 19.b | odd | 2 | 1 | 1805.2.b.g | 6 | ||
| 19.c | even | 3 | 2 | 95.2.i.b | ✓ | 12 | |
| 57.h | odd | 6 | 2 | 855.2.be.d | 12 | ||
| 95.d | odd | 2 | 1 | 1805.2.b.g | 6 | ||
| 95.g | even | 4 | 2 | 9025.2.a.bt | 6 | ||
| 95.i | even | 6 | 2 | 95.2.i.b | ✓ | 12 | |
| 95.m | odd | 12 | 4 | 475.2.e.g | 12 | ||
| 285.n | odd | 6 | 2 | 855.2.be.d | 12 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 95.2.i.b | ✓ | 12 | 19.c | even | 3 | 2 | |
| 95.2.i.b | ✓ | 12 | 95.i | even | 6 | 2 | |
| 475.2.e.g | 12 | 95.m | odd | 12 | 4 | ||
| 855.2.be.d | 12 | 57.h | odd | 6 | 2 | ||
| 855.2.be.d | 12 | 285.n | odd | 6 | 2 | ||
| 1805.2.b.f | 6 | 1.a | even | 1 | 1 | trivial | |
| 1805.2.b.f | 6 | 5.b | even | 2 | 1 | inner | |
| 1805.2.b.g | 6 | 19.b | odd | 2 | 1 | ||
| 1805.2.b.g | 6 | 95.d | odd | 2 | 1 | ||
| 9025.2.a.bt | 6 | 95.g | even | 4 | 2 | ||
| 9025.2.a.bu | 6 | 5.c | odd | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1805, [\chi])\):
|
\( T_{2}^{6} + 7T_{2}^{4} + 6T_{2}^{2} + 1 \)
|
|
\( T_{29}^{3} - 6T_{29}^{2} - 27T_{29} + 27 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + 7 T^{4} + \cdots + 1 \)
|
| $3$ |
\( T^{6} + 13 T^{4} + \cdots + 25 \)
|
| $5$ |
\( T^{6} - 2 T^{5} + \cdots + 125 \)
|
| $7$ |
\( T^{6} + 22 T^{4} + \cdots + 9 \)
|
| $11$ |
\( (T^{3} - T^{2} - 4 T + 3)^{2} \)
|
| $13$ |
\( T^{6} + 31 T^{4} + \cdots + 9 \)
|
| $17$ |
\( T^{6} + 35 T^{4} + \cdots + 81 \)
|
| $19$ |
\( T^{6} \)
|
| $23$ |
\( T^{6} + 12 T^{4} + \cdots + 1 \)
|
| $29$ |
\( (T^{3} - 6 T^{2} - 27 T + 27)^{2} \)
|
| $31$ |
\( (T^{3} - 15 T^{2} + \cdots - 97)^{2} \)
|
| $37$ |
\( T^{6} + 98 T^{4} + \cdots + 729 \)
|
| $41$ |
\( (T^{3} - 6 T^{2} - 41 T + 3)^{2} \)
|
| $43$ |
\( T^{6} + 127 T^{4} + \cdots + 441 \)
|
| $47$ |
\( T^{6} + 214 T^{4} + \cdots + 218089 \)
|
| $53$ |
\( T^{6} + 99 T^{4} + \cdots + 11449 \)
|
| $59$ |
\( (T^{3} + 10 T^{2} + \cdots - 291)^{2} \)
|
| $61$ |
\( (T^{3} + T^{2} - 41 T - 113)^{2} \)
|
| $67$ |
\( T^{6} + 76 T^{4} + \cdots + 14400 \)
|
| $71$ |
\( (T^{3} + T^{2} - 124 T + 477)^{2} \)
|
| $73$ |
\( T^{6} + 236 T^{4} + \cdots + 99225 \)
|
| $79$ |
\( (T^{3} + 12 T^{2} + \cdots - 448)^{2} \)
|
| $83$ |
\( T^{6} + 459 T^{4} + \cdots + 966289 \)
|
| $89$ |
\( (T^{3} + 18 T^{2} + \cdots - 72)^{2} \)
|
| $97$ |
\( T^{6} + 529 T^{4} + \cdots + 1238769 \)
|
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