Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [432,5,Mod(271,432)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(432, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0, 0]))
N = Newforms(chi, 5, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("432.271");
S:= CuspForms(chi, 5);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 432 = 2^{4} \cdot 3^{3} \) |
| Weight: | \( k \) | \(=\) | \( 5 \) |
| Character orbit: | \([\chi]\) | \(=\) | 432.g (of order \(2\), degree \(1\), 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: | \(44.6558240522\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.0.280120707.3 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - x^{5} + 10x^{4} + 5x^{3} + 83x^{2} - 18x + 4 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{43}]\) |
| Coefficient ring index: | \( 2^{9}\cdot 3^{5} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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
\( x^{6} - x^{5} + 10x^{4} + 5x^{3} + 83x^{2} - 18x + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 45\nu^{5} - 44\nu^{4} + 440\nu^{3} + 325\nu^{2} + 3652\nu - 386 ) / 406 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 27\nu^{5} - 270\nu^{4} + 264\nu^{3} - 2241\nu^{2} + 486\nu - 15416 ) / 406 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -45\nu^{5} + 450\nu^{4} - 2064\nu^{3} + 3735\nu^{2} - 810\nu + 13784 ) / 406 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -3\nu^{5} + 2\nu^{4} - 20\nu^{3} - 31\nu^{2} - 82\nu + 8 ) / 7 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -1683\nu^{5} + 1402\nu^{4} - 16456\nu^{3} - 9719\nu^{2} - 138290\nu + 14680 ) / 406 \)
|
| \(\nu\) | \(=\) |
\( ( 3\beta_{4} + \beta_{3} + \beta_{2} + 12\beta _1 + 12 ) / 72 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{5} - \beta_{2} + 38\beta _1 - 38 ) / 12 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -3\beta_{3} - 5\beta_{2} - 88 ) / 12 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -60\beta_{5} - 21\beta_{4} - 7\beta_{3} - 67\beta_{2} - 2292\beta _1 - 2292 ) / 72 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -51\beta_{5} - 132\beta_{4} + 44\beta_{3} + 95\beta_{2} - 2106\beta _1 + 2106 ) / 36 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/432\mathbb{Z}\right)^\times\).
| \(n\) | \(271\) | \(325\) | \(353\) |
| \(\chi(n)\) | \(-1\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 271.1 |
|
0 | 0 | 0 | −26.1823 | 0 | − | 22.7768i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||
| 271.2 | 0 | 0 | 0 | −26.1823 | 0 | 22.7768i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||
| 271.3 | 0 | 0 | 0 | 2.46769 | 0 | − | 54.5282i | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||
| 271.4 | 0 | 0 | 0 | 2.46769 | 0 | 54.5282i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||
| 271.5 | 0 | 0 | 0 | 44.7146 | 0 | − | 78.5168i | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||
| 271.6 | 0 | 0 | 0 | 44.7146 | 0 | 78.5168i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 4.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 432.5.g.h | yes | 6 |
| 3.b | odd | 2 | 1 | 432.5.g.g | ✓ | 6 | |
| 4.b | odd | 2 | 1 | inner | 432.5.g.h | yes | 6 |
| 12.b | even | 2 | 1 | 432.5.g.g | ✓ | 6 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 432.5.g.g | ✓ | 6 | 3.b | odd | 2 | 1 | |
| 432.5.g.g | ✓ | 6 | 12.b | even | 2 | 1 | |
| 432.5.g.h | yes | 6 | 1.a | even | 1 | 1 | trivial |
| 432.5.g.h | yes | 6 | 4.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 \(S_{5}^{\mathrm{new}}(432, [\chi])\):
|
\( T_{5}^{3} - 21T_{5}^{2} - 1125T_{5} + 2889 \)
|
|
\( T_{7}^{6} + 9657T_{7}^{4} + 23070987T_{7}^{2} + 9509407803 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( T^{6} \)
|
| $5$ |
\( (T^{3} - 21 T^{2} + \cdots + 2889)^{2} \)
|
| $7$ |
\( T^{6} + \cdots + 9509407803 \)
|
| $11$ |
\( T^{6} + \cdots + 8240337570843 \)
|
| $13$ |
\( (T^{3} - 84 T^{2} + \cdots + 7949312)^{2} \)
|
| $17$ |
\( (T^{3} - 312 T^{2} + \cdots + 14262912)^{2} \)
|
| $19$ |
\( T^{6} + \cdots + 21\!\cdots\!88 \)
|
| $23$ |
\( T^{6} + \cdots + 178597746065088 \)
|
| $29$ |
\( (T^{3} - 798 T^{2} + \cdots - 82919592)^{2} \)
|
| $31$ |
\( T^{6} + \cdots + 78939896589603 \)
|
| $37$ |
\( (T^{3} + 1326 T^{2} + \cdots - 1133089208)^{2} \)
|
| $41$ |
\( (T^{3} - 4914 T^{2} + \cdots - 2677045464)^{2} \)
|
| $43$ |
\( T^{6} + \cdots + 23\!\cdots\!12 \)
|
| $47$ |
\( T^{6} + \cdots + 16\!\cdots\!08 \)
|
| $53$ |
\( (T^{3} - 6579 T^{2} + \cdots + 34915425111)^{2} \)
|
| $59$ |
\( T^{6} + \cdots + 75\!\cdots\!92 \)
|
| $61$ |
\( (T^{3} - 1968 T^{2} + \cdots + 793483264)^{2} \)
|
| $67$ |
\( T^{6} + \cdots + 18\!\cdots\!88 \)
|
| $71$ |
\( T^{6} + \cdots + 77\!\cdots\!92 \)
|
| $73$ |
\( (T^{3} - 1803 T^{2} + \cdots + 884366567)^{2} \)
|
| $79$ |
\( T^{6} + \cdots + 24\!\cdots\!88 \)
|
| $83$ |
\( T^{6} + \cdots + 24\!\cdots\!67 \)
|
| $89$ |
\( (T^{3} - 26310 T^{2} + \cdots - 572582283912)^{2} \)
|
| $97$ |
\( (T^{3} - 6645 T^{2} + \cdots + 265174138313)^{2} \)
|
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