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: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 2x^{5} - 38x^{4} + 42x^{3} + 393x^{2} - 72x - 432 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2\cdot 3^{3} \) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 2x^{5} - 38x^{4} + 42x^{3} + 393x^{2} - 72x - 432 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{5} - 2\nu^{4} - 26\nu^{3} + 30\nu^{2} + 141\nu - 36 ) / 24 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{2} - \nu - 13 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 19\nu + 1 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{4} - \nu^{3} - 21\nu^{2} + 3\nu + 30 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + \beta _1 + 13 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{4} + \beta_{3} + 20\beta _1 + 12 \)
|
| \(\nu^{4}\) | \(=\) |
\( 2\beta_{5} + \beta_{4} + 22\beta_{3} + 38\beta _1 + 255 \)
|
| \(\nu^{5}\) | \(=\) |
\( 4\beta_{5} + 28\beta_{4} + 40\beta_{3} + 24\beta_{2} + 425\beta _1 + 468 \)
|
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 |
|
−5.23336 | 0 | 19.3881 | −5.00000 | 0 | 33.0180 | −59.5981 | 0 | 26.1668 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −4.53444 | 0 | 12.5612 | −5.00000 | 0 | −2.63618 | −20.6823 | 0 | 22.6722 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −2.07326 | 0 | −3.70159 | −5.00000 | 0 | −4.66112 | 24.2604 | 0 | 10.3663 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0.149150 | 0 | −7.97775 | −5.00000 | 0 | 20.1424 | −2.38308 | 0 | −0.745751 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 3.57457 | 0 | 4.77759 | −5.00000 | 0 | 14.1597 | −11.5188 | 0 | −17.8729 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 4.11734 | 0 | 8.95250 | −5.00000 | 0 | −20.0229 | 3.92177 | 0 | −20.5867 | |||||||||||||||||||||||||||||||||||||
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.k | ✓ | 6 |
| 3.b | odd | 2 | 1 | 405.4.a.l | yes | 6 | |
| 5.b | even | 2 | 1 | 2025.4.a.z | 6 | ||
| 9.c | even | 3 | 2 | 405.4.e.x | 12 | ||
| 9.d | odd | 6 | 2 | 405.4.e.w | 12 | ||
| 15.d | odd | 2 | 1 | 2025.4.a.y | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 405.4.a.k | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 405.4.a.l | yes | 6 | 3.b | odd | 2 | 1 | |
| 405.4.e.w | 12 | 9.d | odd | 6 | 2 | ||
| 405.4.e.x | 12 | 9.c | even | 3 | 2 | ||
| 2025.4.a.y | 6 | 15.d | odd | 2 | 1 | ||
| 2025.4.a.z | 6 | 5.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} + 4T_{2}^{5} - 33T_{2}^{4} - 110T_{2}^{3} + 286T_{2}^{2} + 684T_{2} - 108 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(405))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + 4 T^{5} + \cdots - 108 \)
|
| $3$ |
\( T^{6} \)
|
| $5$ |
\( (T + 5)^{6} \)
|
| $7$ |
\( T^{6} - 40 T^{5} + \cdots - 2316924 \)
|
| $11$ |
\( T^{6} + 88 T^{5} + \cdots + 1197108 \)
|
| $13$ |
\( T^{6} - 20 T^{5} + \cdots - 185793728 \)
|
| $17$ |
\( T^{6} + 124 T^{5} + \cdots + 105966288 \)
|
| $19$ |
\( T^{6} + \cdots - 36821611175 \)
|
| $23$ |
\( T^{6} + \cdots + 332569842768 \)
|
| $29$ |
\( T^{6} + \cdots + 635447099088 \)
|
| $31$ |
\( T^{6} + \cdots - 69859854216 \)
|
| $37$ |
\( T^{6} + \cdots + 12008297128192 \)
|
| $41$ |
\( T^{6} + \cdots - 86213802377403 \)
|
| $43$ |
\( T^{6} + \cdots - 180465347194400 \)
|
| $47$ |
\( T^{6} + \cdots - 49451750433900 \)
|
| $53$ |
\( T^{6} + \cdots + 15\!\cdots\!00 \)
|
| $59$ |
\( T^{6} + \cdots - 168320389359483 \)
|
| $61$ |
\( T^{6} + \cdots - 20\!\cdots\!36 \)
|
| $67$ |
\( T^{6} + \cdots + 19\!\cdots\!00 \)
|
| $71$ |
\( T^{6} + \cdots - 11\!\cdots\!84 \)
|
| $73$ |
\( T^{6} + \cdots - 10\!\cdots\!36 \)
|
| $79$ |
\( T^{6} + \cdots - 15\!\cdots\!72 \)
|
| $83$ |
\( T^{6} + \cdots + 41\!\cdots\!32 \)
|
| $89$ |
\( T^{6} + \cdots + 16\!\cdots\!48 \)
|
| $97$ |
\( T^{6} + \cdots - 97\!\cdots\!24 \)
|
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