Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [405,4,Mod(136,405)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(405, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([4, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("405.136");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 405 = 3^{4} \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 405.e (of order \(3\), degree \(2\), not minimal) |
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: | no |
| Analytic conductor: | \(23.8957735523\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Relative dimension: | \(3\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | 6.0.148347072.2 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} + 29x^{4} + 223x^{2} + 243 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 3 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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
\( x^{6} + 29x^{4} + 223x^{2} + 243 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} + \nu^{2} + 13\nu + 9 ) / 4 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{5} + 20\nu^{3} + 79\nu - 36 ) / 72 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{2} + 9 ) / 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{5} - 9\nu^{4} - 20\nu^{3} - 144\nu^{2} + 29\nu - 207 ) / 72 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -\nu^{5} + 9\nu^{4} - 20\nu^{3} + 144\nu^{2} + 29\nu + 207 ) / 72 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{5} + \beta_{4} + 2\beta_{2} + 1 ) / 3 \)
|
| \(\nu^{2}\) | \(=\) |
\( 2\beta_{3} - 9 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -13\beta_{5} - 13\beta_{4} - 6\beta_{3} - 26\beta_{2} + 12\beta _1 - 13 ) / 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( 4\beta_{5} - 4\beta_{4} - 32\beta_{3} + 121 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 181\beta_{5} + 181\beta_{4} + 120\beta_{3} + 578\beta_{2} - 240\beta _1 + 289 ) / 3 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/405\mathbb{Z}\right)^\times\).
| \(n\) | \(82\) | \(326\) |
| \(\chi(n)\) | \(1\) | \(-1 - \beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 136.1 |
|
−1.76174 | + | 3.05142i | 0 | −2.20744 | − | 3.82340i | 2.50000 | + | 4.33013i | 0 | −12.7162 | + | 22.0252i | −12.6321 | 0 | −17.6174 | ||||||||||||||||||||||||||||
| 136.2 | −0.663404 | + | 1.14905i | 0 | 3.11979 | + | 5.40363i | 2.50000 | + | 4.33013i | 0 | 12.0521 | − | 20.8749i | −18.8932 | 0 | −6.63404 | |||||||||||||||||||||||||||||
| 136.3 | 1.92514 | − | 3.33444i | 0 | −3.41235 | − | 5.91036i | 2.50000 | + | 4.33013i | 0 | 13.1641 | − | 22.8009i | 4.52526 | 0 | 19.2514 | |||||||||||||||||||||||||||||
| 271.1 | −1.76174 | − | 3.05142i | 0 | −2.20744 | + | 3.82340i | 2.50000 | − | 4.33013i | 0 | −12.7162 | − | 22.0252i | −12.6321 | 0 | −17.6174 | |||||||||||||||||||||||||||||
| 271.2 | −0.663404 | − | 1.14905i | 0 | 3.11979 | − | 5.40363i | 2.50000 | − | 4.33013i | 0 | 12.0521 | + | 20.8749i | −18.8932 | 0 | −6.63404 | |||||||||||||||||||||||||||||
| 271.3 | 1.92514 | + | 3.33444i | 0 | −3.41235 | + | 5.91036i | 2.50000 | − | 4.33013i | 0 | 13.1641 | + | 22.8009i | 4.52526 | 0 | 19.2514 | |||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 9.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 405.4.e.s | 6 | |
| 3.b | odd | 2 | 1 | 405.4.e.u | 6 | ||
| 9.c | even | 3 | 1 | 405.4.a.i | yes | 3 | |
| 9.c | even | 3 | 1 | inner | 405.4.e.s | 6 | |
| 9.d | odd | 6 | 1 | 405.4.a.g | ✓ | 3 | |
| 9.d | odd | 6 | 1 | 405.4.e.u | 6 | ||
| 45.h | odd | 6 | 1 | 2025.4.a.r | 3 | ||
| 45.j | even | 6 | 1 | 2025.4.a.p | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 405.4.a.g | ✓ | 3 | 9.d | odd | 6 | 1 | |
| 405.4.a.i | yes | 3 | 9.c | even | 3 | 1 | |
| 405.4.e.s | 6 | 1.a | even | 1 | 1 | trivial | |
| 405.4.e.s | 6 | 9.c | even | 3 | 1 | inner | |
| 405.4.e.u | 6 | 3.b | odd | 2 | 1 | ||
| 405.4.e.u | 6 | 9.d | odd | 6 | 1 | ||
| 2025.4.a.p | 3 | 45.j | even | 6 | 1 | ||
| 2025.4.a.r | 3 | 45.h | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(405, [\chi])\):
|
\( T_{2}^{6} + T_{2}^{5} + 15T_{2}^{4} + 22T_{2}^{3} + 214T_{2}^{2} + 252T_{2} + 324 \)
|
|
\( T_{7}^{6} - 25T_{7}^{5} + 1273T_{7}^{4} - 16080T_{7}^{3} + 823404T_{7}^{2} - 10458720T_{7} + 260499600 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + T^{5} + \cdots + 324 \)
|
| $3$ |
\( T^{6} \)
|
| $5$ |
\( (T^{2} - 5 T + 25)^{3} \)
|
| $7$ |
\( T^{6} - 25 T^{5} + \cdots + 260499600 \)
|
| $11$ |
\( T^{6} + 58 T^{5} + \cdots + 9000000 \)
|
| $13$ |
\( T^{6} + \cdots + 137160603904 \)
|
| $17$ |
\( (T^{3} - 34 T^{2} + \cdots + 90984)^{2} \)
|
| $19$ |
\( (T^{3} + 5 T^{2} + \cdots - 299645)^{2} \)
|
| $23$ |
\( T^{6} + \cdots + 1084005816336 \)
|
| $29$ |
\( T^{6} + \cdots + 126287249817600 \)
|
| $31$ |
\( T^{6} + \cdots + 90993741996096 \)
|
| $37$ |
\( (T^{3} + 414 T^{2} + \cdots - 577760)^{2} \)
|
| $41$ |
\( T^{6} + \cdots + 316826721339129 \)
|
| $43$ |
\( T^{6} + \cdots + 349427949912064 \)
|
| $47$ |
\( T^{6} + \cdots + 81096796901376 \)
|
| $53$ |
\( (T^{3} - 505 T^{2} + \cdots + 1500684)^{2} \)
|
| $59$ |
\( T^{6} + \cdots + 498077813969649 \)
|
| $61$ |
\( T^{6} + \cdots + 554264938580224 \)
|
| $67$ |
\( T^{6} + \cdots + 246999193899264 \)
|
| $71$ |
\( (T^{3} - 452 T^{2} + \cdots + 116183454)^{2} \)
|
| $73$ |
\( (T^{3} + 710 T^{2} + \cdots + 8707528)^{2} \)
|
| $79$ |
\( T^{6} + \cdots + 25533374939136 \)
|
| $83$ |
\( T^{6} + \cdots + 49\!\cdots\!04 \)
|
| $89$ |
\( (T^{3} + 852 T^{2} + \cdots + 17926434)^{2} \)
|
| $97$ |
\( T^{6} + \cdots + 49\!\cdots\!64 \)
|
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