Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1859,4,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, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1859.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1859 = 11 \cdot 13^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| 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: | \(109.684550701\) |
| Analytic rank: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.297133.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 19x^{2} - 2x + 52 \)
|
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - 19x^{2} - 2x + 52 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} - \nu - 10 ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} - \nu^{2} - 12\nu + 8 ) / 4 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 2\beta_{2} + \beta _1 + 10 \)
|
| \(\nu^{3}\) | \(=\) |
\( 4\beta_{3} + 2\beta_{2} + 13\beta _1 + 2 \)
|
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 |
|
−4.03811 | −3.27078 | 8.30634 | 16.3826 | 13.2078 | 1.85898 | −1.23702 | −16.3020 | −66.1546 | ||||||||||||||||||||||||||||||
| 1.2 | −1.74344 | 1.66539 | −4.96041 | −1.47352 | −2.90351 | 22.5004 | 22.5957 | −24.2265 | 2.56900 | |||||||||||||||||||||||||||||||
| 1.3 | 1.90566 | −6.07897 | −4.36845 | −8.17977 | −11.5845 | −17.8958 | −23.5701 | 9.95391 | −15.5879 | |||||||||||||||||||||||||||||||
| 1.4 | 3.87589 | 3.68437 | 7.02252 | −0.729260 | 14.2802 | 10.5365 | −3.78861 | −13.4254 | −2.82653 | |||||||||||||||||||||||||||||||
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 | 1859.4.a.b | 4 | |
| 13.b | even | 2 | 1 | 143.4.a.a | ✓ | 4 | |
| 39.d | odd | 2 | 1 | 1287.4.a.b | 4 | ||
| 52.b | odd | 2 | 1 | 2288.4.a.i | 4 | ||
| 143.d | odd | 2 | 1 | 1573.4.a.c | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 143.4.a.a | ✓ | 4 | 13.b | even | 2 | 1 | |
| 1287.4.a.b | 4 | 39.d | odd | 2 | 1 | ||
| 1573.4.a.c | 4 | 143.d | odd | 2 | 1 | ||
| 1859.4.a.b | 4 | 1.a | even | 1 | 1 | trivial | |
| 2288.4.a.i | 4 | 52.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} - 19T_{2}^{2} + 2T_{2} + 52 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1859))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - 19 T^{2} + \cdots + 52 \)
|
| $3$ |
\( T^{4} + 4 T^{3} + \cdots + 122 \)
|
| $5$ |
\( T^{4} - 6 T^{3} + \cdots - 144 \)
|
| $7$ |
\( T^{4} - 17 T^{3} + \cdots - 7887 \)
|
| $11$ |
\( (T - 11)^{4} \)
|
| $13$ |
\( T^{4} \)
|
| $17$ |
\( T^{4} + 41 T^{3} + \cdots + 32696 \)
|
| $19$ |
\( T^{4} - 41 T^{3} + \cdots + 1792256 \)
|
| $23$ |
\( T^{4} + 136 T^{3} + \cdots - 186204 \)
|
| $29$ |
\( T^{4} + 207 T^{3} + \cdots - 98922752 \)
|
| $31$ |
\( T^{4} - 348 T^{3} + \cdots + 432658176 \)
|
| $37$ |
\( T^{4} - 333 T^{3} + \cdots - 122208096 \)
|
| $41$ |
\( T^{4} - 198 T^{3} + \cdots + 21770232 \)
|
| $43$ |
\( T^{4} + 252 T^{3} + \cdots + 845198188 \)
|
| $47$ |
\( T^{4} + \cdots + 3989237104 \)
|
| $53$ |
\( T^{4} - 343 T^{3} + \cdots + 895612308 \)
|
| $59$ |
\( T^{4} - 541 T^{3} + \cdots + 50105064 \)
|
| $61$ |
\( T^{4} + 801 T^{3} + \cdots - 107700832 \)
|
| $67$ |
\( T^{4} + \cdots + 7976743328 \)
|
| $71$ |
\( T^{4} + \cdots - 24400890272 \)
|
| $73$ |
\( T^{4} + \cdots + 74727204864 \)
|
| $79$ |
\( T^{4} + \cdots - 98639841184 \)
|
| $83$ |
\( T^{4} + \cdots + 233243308272 \)
|
| $89$ |
\( T^{4} + \cdots - 5112256512 \)
|
| $97$ |
\( T^{4} + \cdots + 18716618752 \)
|
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