Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [405,4,Mod(1,405)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(405, 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("405.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 405 = 3^{4} \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 405.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: | \(23.8957735523\) |
| Analytic rank: | \(1\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.7032.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 14x + 18 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| 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 \(1,\beta_1,\beta_2\) 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
\( x^{3} - x^{2} - 14x + 18 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} + \nu - 10 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} - \beta _1 + 10 \)
|
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 |
|
−3.52348 | 0 | 4.41489 | 5.00000 | 0 | 25.4325 | 12.6321 | 0 | −17.6174 | |||||||||||||||||||||||||||
| 1.2 | −1.32681 | 0 | −6.23958 | 5.00000 | 0 | −24.1043 | 18.8932 | 0 | −6.63404 | ||||||||||||||||||||||||||||
| 1.3 | 3.85028 | 0 | 6.82469 | 5.00000 | 0 | −26.3282 | −4.52526 | 0 | 19.2514 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(1\) |
| \(5\) | \(-1\) |
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 | 405.4.a.g | ✓ | 3 |
| 3.b | odd | 2 | 1 | 405.4.a.i | yes | 3 | |
| 5.b | even | 2 | 1 | 2025.4.a.r | 3 | ||
| 9.c | even | 3 | 2 | 405.4.e.u | 6 | ||
| 9.d | odd | 6 | 2 | 405.4.e.s | 6 | ||
| 15.d | odd | 2 | 1 | 2025.4.a.p | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 405.4.a.g | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
| 405.4.a.i | yes | 3 | 3.b | odd | 2 | 1 | |
| 405.4.e.s | 6 | 9.d | odd | 6 | 2 | ||
| 405.4.e.u | 6 | 9.c | even | 3 | 2 | ||
| 2025.4.a.p | 3 | 15.d | odd | 2 | 1 | ||
| 2025.4.a.r | 3 | 5.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{3} + T_{2}^{2} - 14T_{2} - 18 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(405))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} + T^{2} + \cdots - 18 \)
|
| $3$ |
\( T^{3} \)
|
| $5$ |
\( (T - 5)^{3} \)
|
| $7$ |
\( T^{3} + 25 T^{2} + \cdots - 16140 \)
|
| $11$ |
\( T^{3} + 58 T^{2} + \cdots + 3000 \)
|
| $13$ |
\( T^{3} + 47 T^{2} + \cdots - 370352 \)
|
| $17$ |
\( T^{3} + 34 T^{2} + \cdots - 90984 \)
|
| $19$ |
\( T^{3} + 5 T^{2} + \cdots - 299645 \)
|
| $23$ |
\( T^{3} - 51 T^{2} + \cdots + 1041156 \)
|
| $29$ |
\( T^{3} + 350 T^{2} + \cdots - 11237760 \)
|
| $31$ |
\( T^{3} - 638 T^{2} + \cdots - 9539064 \)
|
| $37$ |
\( T^{3} + 414 T^{2} + \cdots - 577760 \)
|
| $41$ |
\( T^{3} + 179 T^{2} + \cdots - 17799627 \)
|
| $43$ |
\( T^{3} + 836 T^{2} + \cdots + 18692992 \)
|
| $47$ |
\( T^{3} + 235 T^{2} + \cdots - 9005376 \)
|
| $53$ |
\( T^{3} + 505 T^{2} + \cdots - 1500684 \)
|
| $59$ |
\( T^{3} + 535 T^{2} + \cdots + 22317657 \)
|
| $61$ |
\( T^{3} + 104 T^{2} + \cdots - 23542832 \)
|
| $67$ |
\( T^{3} + 40 T^{2} + \cdots - 15716208 \)
|
| $71$ |
\( T^{3} + 452 T^{2} + \cdots - 116183454 \)
|
| $73$ |
\( T^{3} + 710 T^{2} + \cdots + 8707528 \)
|
| $79$ |
\( T^{3} + 634 T^{2} + \cdots + 5053056 \)
|
| $83$ |
\( T^{3} + 1734 T^{2} + \cdots - 222334848 \)
|
| $89$ |
\( T^{3} - 852 T^{2} + \cdots - 17926434 \)
|
| $97$ |
\( T^{3} - 1575168 T - 703275008 \)
|
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