Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [186,4,Mod(25,186)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(186, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 2]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("186.25");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 186 = 2 \cdot 3 \cdot 31 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 186.e (of order \(3\), degree \(2\), 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: | \(10.9743552611\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Relative dimension: | \(3\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | 6.0.4956535827.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - x^{5} + 32x^{4} + 23x^{3} + 965x^{2} - 124x + 16 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{4} \) |
| 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
\( x^{6} - x^{5} + 32x^{4} + 23x^{3} + 965x^{2} - 124x + 16 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -7\nu^{5} + 224\nu^{4} + 521\nu^{3} + 6755\nu^{2} - 868\nu + 166178 ) / 23067 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 17\nu^{5} - 544\nu^{4} + 2030\nu^{3} - 16405\nu^{2} + 2108\nu - 321193 ) / 23067 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -248\nu^{5} + 247\nu^{4} - 7904\nu^{3} - 6728\nu^{2} - 238355\nu + 30628 ) / 30756 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 5453\nu^{5} - 5338\nu^{4} + 170816\nu^{3} + 120155\nu^{2} + 5151170\nu - 661912 ) / 92268 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 10106\nu^{5} - 8143\nu^{4} + 322088\nu^{3} + 274166\nu^{2} + 9949403\nu + 5216 ) / 92268 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{5} - 2\beta_{4} - \beta_{3} - 2\beta _1 + 1 ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -3\beta_{5} - 6\beta_{4} - 85\beta_{3} - 3\beta_{2} ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( 7\beta_{2} + 17\beta _1 - 25 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 69\beta_{5} + 246\beta_{4} + 2731\beta_{3} + 246\beta _1 - 2731 ) / 4 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -811\beta_{5} + 2330\beta_{4} + 5491\beta_{3} - 811\beta_{2} ) / 4 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/186\mathbb{Z}\right)^\times\).
| \(n\) | \(125\) | \(127\) |
| \(\chi(n)\) | \(1\) | \(-\beta_{3}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 25.1 |
|
2.00000 | 1.50000 | + | 2.59808i | 4.00000 | −6.45639 | + | 11.1828i | 3.00000 | + | 5.19615i | 0.900372 | + | 1.55949i | 8.00000 | −4.50000 | + | 7.79423i | −12.9128 | + | 22.3656i | ||||||||||||||||||||||||
| 25.2 | 2.00000 | 1.50000 | + | 2.59808i | 4.00000 | −0.241447 | + | 0.418199i | 3.00000 | + | 5.19615i | 9.90980 | + | 17.1643i | 8.00000 | −4.50000 | + | 7.79423i | −0.482895 | + | 0.836399i | |||||||||||||||||||||||||
| 25.3 | 2.00000 | 1.50000 | + | 2.59808i | 4.00000 | 7.69784 | − | 13.3330i | 3.00000 | + | 5.19615i | 2.68983 | + | 4.65892i | 8.00000 | −4.50000 | + | 7.79423i | 15.3957 | − | 26.6661i | |||||||||||||||||||||||||
| 67.1 | 2.00000 | 1.50000 | − | 2.59808i | 4.00000 | −6.45639 | − | 11.1828i | 3.00000 | − | 5.19615i | 0.900372 | − | 1.55949i | 8.00000 | −4.50000 | − | 7.79423i | −12.9128 | − | 22.3656i | |||||||||||||||||||||||||
| 67.2 | 2.00000 | 1.50000 | − | 2.59808i | 4.00000 | −0.241447 | − | 0.418199i | 3.00000 | − | 5.19615i | 9.90980 | − | 17.1643i | 8.00000 | −4.50000 | − | 7.79423i | −0.482895 | − | 0.836399i | |||||||||||||||||||||||||
| 67.3 | 2.00000 | 1.50000 | − | 2.59808i | 4.00000 | 7.69784 | + | 13.3330i | 3.00000 | − | 5.19615i | 2.68983 | − | 4.65892i | 8.00000 | −4.50000 | − | 7.79423i | 15.3957 | + | 26.6661i | |||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 31.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 186.4.e.c | ✓ | 6 |
| 3.b | odd | 2 | 1 | 558.4.e.b | 6 | ||
| 31.c | even | 3 | 1 | inner | 186.4.e.c | ✓ | 6 |
| 93.h | odd | 6 | 1 | 558.4.e.b | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 186.4.e.c | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 186.4.e.c | ✓ | 6 | 31.c | even | 3 | 1 | inner |
| 558.4.e.b | 6 | 3.b | odd | 2 | 1 | ||
| 558.4.e.b | 6 | 93.h | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{6} - 2T_{5}^{5} + 204T_{5}^{4} + 592T_{5}^{3} + 39808T_{5}^{2} + 19200T_{5} + 9216 \)
acting on \(S_{4}^{\mathrm{new}}(186, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 2)^{6} \)
|
| $3$ |
\( (T^{2} - 3 T + 9)^{3} \)
|
| $5$ |
\( T^{6} - 2 T^{5} + \cdots + 9216 \)
|
| $7$ |
\( T^{6} - 27 T^{5} + \cdots + 36864 \)
|
| $11$ |
\( T^{6} - 15 T^{5} + \cdots + 688747536 \)
|
| $13$ |
\( T^{6} + 42 T^{5} + \cdots + 266864896 \)
|
| $17$ |
\( T^{6} + \cdots + 1719926784 \)
|
| $19$ |
\( T^{6} + \cdots + 58444996516 \)
|
| $23$ |
\( (T^{3} + 12 T^{2} + \cdots + 882144)^{2} \)
|
| $29$ |
\( (T^{3} - 85 T^{2} + \cdots + 4409424)^{2} \)
|
| $31$ |
\( T^{6} + \cdots + 26439622160671 \)
|
| $37$ |
\( T^{6} + \cdots + 745498805022976 \)
|
| $41$ |
\( T^{6} + \cdots + 42535231785216 \)
|
| $43$ |
\( T^{6} + \cdots + 4486856256 \)
|
| $47$ |
\( (T^{3} + 186 T^{2} + \cdots + 1159584)^{2} \)
|
| $53$ |
\( T^{6} + \cdots + 11\!\cdots\!96 \)
|
| $59$ |
\( T^{6} + \cdots + 13\!\cdots\!84 \)
|
| $61$ |
\( (T^{3} + 132 T^{2} + \cdots + 20001792)^{2} \)
|
| $67$ |
\( T^{6} + \cdots + 10\!\cdots\!76 \)
|
| $71$ |
\( T^{6} + \cdots + 61473628422144 \)
|
| $73$ |
\( T^{6} + \cdots + 408499800658884 \)
|
| $79$ |
\( T^{6} + \cdots + 44\!\cdots\!04 \)
|
| $83$ |
\( T^{6} + \cdots + 11\!\cdots\!64 \)
|
| $89$ |
\( (T^{3} + 1666 T^{2} + \cdots - 30094056)^{2} \)
|
| $97$ |
\( (T^{3} - 29 T^{2} + \cdots - 45009451)^{2} \)
|
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