Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1800,4,Mod(1,1800)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1800, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 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("1800.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1800 = 2^{3} \cdot 3^{2} \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1800.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: | \(106.203438010\) |
| Analytic rank: | \(1\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.121909.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 87x + 270 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{3} \) |
| 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 \(1,\beta_1,\beta_2\) 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
\( x^{3} - x^{2} - 87x + 270 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 2\nu^{2} + 10\nu - 120 ) / 3 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -2\nu^{2} + 2\nu + 117 ) / 3 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + \beta _1 + 1 ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -5\beta_{2} + \beta _1 + 235 ) / 4 \)
|
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 |
|
0 | 0 | 0 | 0 | 0 | −23.6993 | 0 | 0 | 0 | |||||||||||||||||||||||||||
| 1.2 | 0 | 0 | 0 | 0 | 0 | −7.96885 | 0 | 0 | 0 | ||||||||||||||||||||||||||||
| 1.3 | 0 | 0 | 0 | 0 | 0 | 22.6681 | 0 | 0 | 0 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(3\) | \(1\) |
| \(5\) | \(-1\) |
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 | 1800.4.a.br | ✓ | 3 |
| 3.b | odd | 2 | 1 | 1800.4.a.bs | yes | 3 | |
| 5.b | even | 2 | 1 | 1800.4.a.bt | yes | 3 | |
| 5.c | odd | 4 | 2 | 1800.4.f.ba | 6 | ||
| 15.d | odd | 2 | 1 | 1800.4.a.bu | yes | 3 | |
| 15.e | even | 4 | 2 | 1800.4.f.bb | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1800.4.a.br | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
| 1800.4.a.bs | yes | 3 | 3.b | odd | 2 | 1 | |
| 1800.4.a.bt | yes | 3 | 5.b | even | 2 | 1 | |
| 1800.4.a.bu | yes | 3 | 15.d | odd | 2 | 1 | |
| 1800.4.f.ba | 6 | 5.c | odd | 4 | 2 | ||
| 1800.4.f.bb | 6 | 15.e | even | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1800))\):
|
\( T_{7}^{3} + 9T_{7}^{2} - 529T_{7} - 4281 \)
|
|
\( T_{11}^{3} + 8T_{11}^{2} - 1376T_{11} + 11712 \)
|
|
\( T_{17}^{3} - 48T_{17}^{2} - 12160T_{17} + 26304 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} \)
|
| $3$ |
\( T^{3} \)
|
| $5$ |
\( T^{3} \)
|
| $7$ |
\( T^{3} + 9 T^{2} + \cdots - 4281 \)
|
| $11$ |
\( T^{3} + 8 T^{2} + \cdots + 11712 \)
|
| $13$ |
\( T^{3} + 17 T^{2} + \cdots - 66573 \)
|
| $17$ |
\( T^{3} - 48 T^{2} + \cdots + 26304 \)
|
| $19$ |
\( T^{3} + 11 T^{2} + \cdots - 137891 \)
|
| $23$ |
\( T^{3} - 40 T^{2} + \cdots + 344256 \)
|
| $29$ |
\( T^{3} - 30016 T + 1273920 \)
|
| $31$ |
\( T^{3} - 59 T^{2} + \cdots + 1145475 \)
|
| $37$ |
\( T^{3} - 10 T^{2} + \cdots - 2282040 \)
|
| $41$ |
\( T^{3} + 400 T^{2} + \cdots - 8631360 \)
|
| $43$ |
\( T^{3} - 159 T^{2} + \cdots - 5492625 \)
|
| $47$ |
\( T^{3} - 272 T^{2} + \cdots - 3495424 \)
|
| $53$ |
\( T^{3} + 256 T^{2} + \cdots + 1038528 \)
|
| $59$ |
\( T^{3} + 176 T^{2} + \cdots - 27431424 \)
|
| $61$ |
\( T^{3} + 375 T^{2} + \cdots - 151917099 \)
|
| $67$ |
\( T^{3} - 143 T^{2} + \cdots + 66765983 \)
|
| $71$ |
\( T^{3} + 1288 T^{2} + \cdots - 35439552 \)
|
| $73$ |
\( T^{3} + 398 T^{2} + \cdots - 146538072 \)
|
| $79$ |
\( T^{3} + 292 T^{2} + \cdots + 868032 \)
|
| $83$ |
\( T^{3} + 1424 T^{2} + \cdots + 4050432 \)
|
| $89$ |
\( T^{3} + 928 T^{2} + \cdots - 132710400 \)
|
| $97$ |
\( T^{3} + 697 T^{2} + \cdots - 10998517 \)
|
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