Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1815,4,Mod(1,1815)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1815, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("1815.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1815 = 3 \cdot 5 \cdot 11^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1815.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: | \(107.088466660\) |
| Analytic rank: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.744012.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 21x^{2} + 48 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - 21x^{2} + 48 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} - 17\nu ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{2} - 11 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + 11 \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{2} + 17\beta_1 \)
|
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 |
|
−4.28834 | −3.00000 | 10.3899 | 5.00000 | 12.8650 | 2.98011 | −10.2486 | 9.00000 | −21.4417 | ||||||||||||||||||||||||||||||
| 1.2 | −1.61559 | −3.00000 | −5.38987 | 5.00000 | 4.84677 | −11.6241 | 21.6325 | 9.00000 | −8.07795 | |||||||||||||||||||||||||||||||
| 1.3 | 1.61559 | −3.00000 | −5.38987 | 5.00000 | −4.84677 | 11.6241 | −21.6325 | 9.00000 | 8.07795 | |||||||||||||||||||||||||||||||
| 1.4 | 4.28834 | −3.00000 | 10.3899 | 5.00000 | −12.8650 | −2.98011 | 10.2486 | 9.00000 | 21.4417 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(1\) |
| \(5\) | \(-1\) |
| \(11\) | \(1\) |
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 11.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1815.4.a.v | ✓ | 4 |
| 11.b | odd | 2 | 1 | inner | 1815.4.a.v | ✓ | 4 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1815.4.a.v | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 1815.4.a.v | ✓ | 4 | 11.b | odd | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1815))\):
|
\( T_{2}^{4} - 21T_{2}^{2} + 48 \)
|
|
\( T_{7}^{4} - 144T_{7}^{2} + 1200 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - 21T^{2} + 48 \)
|
| $3$ |
\( (T + 3)^{4} \)
|
| $5$ |
\( (T - 5)^{4} \)
|
| $7$ |
\( T^{4} - 144T^{2} + 1200 \)
|
| $11$ |
\( T^{4} \)
|
| $13$ |
\( T^{4} - 3564 T^{2} + 1660608 \)
|
| $17$ |
\( T^{4} - 7056 T^{2} + 2881200 \)
|
| $19$ |
\( T^{4} - 4116 T^{2} + 1843968 \)
|
| $23$ |
\( (T^{2} + 48 T - 15360)^{2} \)
|
| $29$ |
\( T^{4} - 14148 T^{2} + 4853952 \)
|
| $31$ |
\( (T + 8)^{4} \)
|
| $37$ |
\( (T^{2} + 88 T - 22964)^{2} \)
|
| $41$ |
\( T^{4} + \cdots + 4626869952 \)
|
| $43$ |
\( T^{4} - 58608 T^{2} + 843899952 \)
|
| $47$ |
\( (T^{2} - 132 T - 116160)^{2} \)
|
| $53$ |
\( (T^{2} + 60 T - 62844)^{2} \)
|
| $59$ |
\( (T^{2} + 204 T - 14496)^{2} \)
|
| $61$ |
\( T^{4} + \cdots + 1180876800 \)
|
| $67$ |
\( (T^{2} - 296 T + 5968)^{2} \)
|
| $71$ |
\( (T^{2} + 900 T + 34176)^{2} \)
|
| $73$ |
\( T^{4} + \cdots + 6744831168 \)
|
| $79$ |
\( T^{4} + \cdots + 27390554112 \)
|
| $83$ |
\( T^{4} + \cdots + 1093723320000 \)
|
| $89$ |
\( (T^{2} + 120 T - 116916)^{2} \)
|
| $97$ |
\( (T^{2} + 1016 T + 249100)^{2} \)
|
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