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: | \(10\) |
| Relative dimension: | \(5\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{10} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{10} - x^{9} + 453 x^{8} - 3112 x^{7} + 159431 x^{6} - 994379 x^{5} + 23981359 x^{4} + \cdots + 79650950625 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| 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 \(1,\beta_1,\ldots,\beta_{9}\) 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
\( x^{10} - x^{9} + 453 x^{8} - 3112 x^{7} + 159431 x^{6} - 994379 x^{5} + 23981359 x^{4} + \cdots + 79650950625 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 38\!\cdots\!42 \nu^{9} + \cdots - 67\!\cdots\!25 ) / 47\!\cdots\!25 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 231243922881619 \nu^{9} + \cdots - 16\!\cdots\!50 ) / 25\!\cdots\!75 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 27\!\cdots\!91 \nu^{9} + \cdots - 19\!\cdots\!05 ) / 14\!\cdots\!20 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 10\!\cdots\!37 \nu^{9} + \cdots - 33\!\cdots\!75 ) / 35\!\cdots\!00 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 73\!\cdots\!33 \nu^{9} + \cdots - 13\!\cdots\!75 ) / 10\!\cdots\!00 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 12\!\cdots\!03 \nu^{9} + \cdots + 14\!\cdots\!50 ) / 14\!\cdots\!00 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 28\!\cdots\!18 \nu^{9} + \cdots - 14\!\cdots\!25 ) / 28\!\cdots\!00 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 20\!\cdots\!39 \nu^{9} + \cdots + 12\!\cdots\!50 ) / 18\!\cdots\!00 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 3\beta_{9} - \beta_{8} + \beta_{6} - 5\beta_{3} - 181\beta_{2} - 5\beta _1 - 181 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{6} + 23\beta_{5} - 22\beta_{4} + 246\beta_{3} + 838 \)
|
| \(\nu^{4}\) | \(=\) |
\( -1228\beta_{9} + 436\beta_{8} + 208\beta_{7} - 1228\beta_{5} + 208\beta_{4} + 43379\beta_{2} + 2246\beta_1 \)
|
| \(\nu^{5}\) | \(=\) |
\( 14514 \beta_{9} - 1798 \beta_{8} - 9736 \beta_{7} + 1798 \beta_{6} - 71201 \beta_{3} - 375714 \beta_{2} + \cdots - 375714 \)
|
| \(\nu^{6}\) | \(=\) |
\( -150401\beta_{6} + 441563\beta_{5} - 123484\beta_{4} + 866863\beta_{3} + 12280355 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 6387321 \beta_{9} + 1110367 \beta_{8} + 3621434 \beta_{7} - 6387321 \beta_{5} + 3621434 \beta_{4} + \cdots + 22430106 \beta_1 \)
|
| \(\nu^{8}\) | \(=\) |
\( 155279588 \beta_{9} - 49451116 \beta_{8} - 53629696 \beta_{7} + 49451116 \beta_{6} - 322057672 \beta_{3} + \cdots - 3815226877 \)
|
| \(\nu^{9}\) | \(=\) |
\( -513513560\beta_{6} + 2494931800\beta_{5} - 1290371080\beta_{4} + 7405340101\beta_{3} + 53684980260 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/186\mathbb{Z}\right)^\times\).
| \(n\) | \(125\) | \(127\) |
| \(\chi(n)\) | \(1\) | \(-1 - \beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 25.1 |
|
2.00000 | −1.50000 | − | 2.59808i | 4.00000 | −6.97346 | + | 12.0784i | −3.00000 | − | 5.19615i | −10.5378 | − | 18.2519i | 8.00000 | −4.50000 | + | 7.79423i | −13.9469 | + | 24.1568i | ||||||||||||||||||||||||||||||||||||
| 25.2 | 2.00000 | −1.50000 | − | 2.59808i | 4.00000 | −4.53514 | + | 7.85510i | −3.00000 | − | 5.19615i | −8.15214 | − | 14.1199i | 8.00000 | −4.50000 | + | 7.79423i | −9.07028 | + | 15.7102i | |||||||||||||||||||||||||||||||||||||
| 25.3 | 2.00000 | −1.50000 | − | 2.59808i | 4.00000 | −3.35677 | + | 5.81409i | −3.00000 | − | 5.19615i | 12.3457 | + | 21.3834i | 8.00000 | −4.50000 | + | 7.79423i | −6.71353 | + | 11.6282i | |||||||||||||||||||||||||||||||||||||
| 25.4 | 2.00000 | −1.50000 | − | 2.59808i | 4.00000 | 6.96382 | − | 12.0617i | −3.00000 | − | 5.19615i | 11.2066 | + | 19.4105i | 8.00000 | −4.50000 | + | 7.79423i | 13.9276 | − | 24.1234i | |||||||||||||||||||||||||||||||||||||
| 25.5 | 2.00000 | −1.50000 | − | 2.59808i | 4.00000 | 9.90155 | − | 17.1500i | −3.00000 | − | 5.19615i | −10.8624 | − | 18.8143i | 8.00000 | −4.50000 | + | 7.79423i | 19.8031 | − | 34.3000i | |||||||||||||||||||||||||||||||||||||
| 67.1 | 2.00000 | −1.50000 | + | 2.59808i | 4.00000 | −6.97346 | − | 12.0784i | −3.00000 | + | 5.19615i | −10.5378 | + | 18.2519i | 8.00000 | −4.50000 | − | 7.79423i | −13.9469 | − | 24.1568i | |||||||||||||||||||||||||||||||||||||
| 67.2 | 2.00000 | −1.50000 | + | 2.59808i | 4.00000 | −4.53514 | − | 7.85510i | −3.00000 | + | 5.19615i | −8.15214 | + | 14.1199i | 8.00000 | −4.50000 | − | 7.79423i | −9.07028 | − | 15.7102i | |||||||||||||||||||||||||||||||||||||
| 67.3 | 2.00000 | −1.50000 | + | 2.59808i | 4.00000 | −3.35677 | − | 5.81409i | −3.00000 | + | 5.19615i | 12.3457 | − | 21.3834i | 8.00000 | −4.50000 | − | 7.79423i | −6.71353 | − | 11.6282i | |||||||||||||||||||||||||||||||||||||
| 67.4 | 2.00000 | −1.50000 | + | 2.59808i | 4.00000 | 6.96382 | + | 12.0617i | −3.00000 | + | 5.19615i | 11.2066 | − | 19.4105i | 8.00000 | −4.50000 | − | 7.79423i | 13.9276 | + | 24.1234i | |||||||||||||||||||||||||||||||||||||
| 67.5 | 2.00000 | −1.50000 | + | 2.59808i | 4.00000 | 9.90155 | + | 17.1500i | −3.00000 | + | 5.19615i | −10.8624 | + | 18.8143i | 8.00000 | −4.50000 | − | 7.79423i | 19.8031 | + | 34.3000i | |||||||||||||||||||||||||||||||||||||
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.e | ✓ | 10 |
| 3.b | odd | 2 | 1 | 558.4.e.g | 10 | ||
| 31.c | even | 3 | 1 | inner | 186.4.e.e | ✓ | 10 |
| 93.h | odd | 6 | 1 | 558.4.e.g | 10 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 186.4.e.e | ✓ | 10 | 1.a | even | 1 | 1 | trivial |
| 186.4.e.e | ✓ | 10 | 31.c | even | 3 | 1 | inner |
| 558.4.e.g | 10 | 3.b | odd | 2 | 1 | ||
| 558.4.e.g | 10 | 93.h | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{10} - 4 T_{5}^{9} + 462 T_{5}^{8} + 2644 T_{5}^{7} + 148332 T_{5}^{6} + 816932 T_{5}^{5} + \cdots + 54868377600 \)
acting on \(S_{4}^{\mathrm{new}}(186, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 2)^{10} \)
|
| $3$ |
\( (T^{2} + 3 T + 9)^{5} \)
|
| $5$ |
\( T^{10} + \cdots + 54868377600 \)
|
| $7$ |
\( T^{10} + \cdots + 17067837992976 \)
|
| $11$ |
\( T^{10} + \cdots + 674451562500 \)
|
| $13$ |
\( T^{10} + \cdots + 41\!\cdots\!00 \)
|
| $17$ |
\( T^{10} + \cdots + 13\!\cdots\!36 \)
|
| $19$ |
\( T^{10} + \cdots + 44\!\cdots\!00 \)
|
| $23$ |
\( (T^{5} - 50 T^{4} + \cdots + 22470156288)^{2} \)
|
| $29$ |
\( (T^{5} - 250 T^{4} + \cdots + 6775050834)^{2} \)
|
| $31$ |
\( T^{10} + \cdots + 23\!\cdots\!51 \)
|
| $37$ |
\( T^{10} + \cdots + 82\!\cdots\!04 \)
|
| $41$ |
\( T^{10} + \cdots + 17\!\cdots\!36 \)
|
| $43$ |
\( T^{10} + \cdots + 83\!\cdots\!96 \)
|
| $47$ |
\( (T^{5} + \cdots - 1612247447412)^{2} \)
|
| $53$ |
\( T^{10} + \cdots + 18\!\cdots\!00 \)
|
| $59$ |
\( T^{10} + \cdots + 23\!\cdots\!96 \)
|
| $61$ |
\( (T^{5} + \cdots - 9316024080600)^{2} \)
|
| $67$ |
\( T^{10} + \cdots + 45\!\cdots\!96 \)
|
| $71$ |
\( T^{10} + \cdots + 23\!\cdots\!00 \)
|
| $73$ |
\( T^{10} + \cdots + 49\!\cdots\!36 \)
|
| $79$ |
\( T^{10} + \cdots + 75\!\cdots\!56 \)
|
| $83$ |
\( T^{10} + \cdots + 85\!\cdots\!64 \)
|
| $89$ |
\( (T^{5} + \cdots - 41901153595416)^{2} \)
|
| $97$ |
\( (T^{5} + \cdots + 499495118529031)^{2} \)
|
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