Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [41,2,Mod(10,41)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(41, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([2]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("41.10");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 41 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 41.d (of order \(5\), degree \(4\), 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: | \(0.327386648287\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{5})\) |
| Coefficient field: | 8.0.13140625.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 3x^{7} + 5x^{6} - 3x^{5} + 4x^{4} + 3x^{3} + 5x^{2} + 3x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{5}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - 3x^{7} + 5x^{6} - 3x^{5} + 4x^{4} + 3x^{3} + 5x^{2} + 3x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{7} + 2\nu^{6} - 3\nu^{5} - 4\nu^{3} - 7\nu^{2} - 12\nu - 7 ) / 8 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{7} - 7\nu^{5} + 20\nu^{4} - 16\nu^{3} + 19\nu^{2} + 6\nu + 9 ) / 8 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{7} + 4\nu^{6} - 9\nu^{5} + 12\nu^{4} - 16\nu^{3} + 13\nu^{2} - 10\nu - 1 ) / 8 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -3\nu^{7} + 10\nu^{6} - 17\nu^{5} + 8\nu^{4} - 4\nu^{3} - 13\nu^{2} - 8\nu - 5 ) / 8 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 3\nu^{7} - 12\nu^{6} + 23\nu^{5} - 20\nu^{4} + 16\nu^{3} + \nu^{2} + 6\nu - 1 ) / 8 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -5\nu^{7} + 18\nu^{6} - 35\nu^{5} + 32\nu^{4} - 28\nu^{3} - 11\nu^{2} - 12\nu - 7 ) / 8 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{6} + \beta_{5} + \beta_{4} - \beta_{2} \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{7} + 3\beta_{6} + 2\beta_{5} + \beta_{4} - 3\beta_{2} - 2\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 3\beta_{7} + 4\beta_{6} + \beta_{4} - \beta_{3} - 5\beta_{2} - 4\beta_1 \)
|
| \(\nu^{5}\) | \(=\) |
\( 4\beta_{7} - 6\beta_{5} - 4\beta_{3} - 6\beta_{2} - 6\beta _1 - 1 \)
|
| \(\nu^{6}\) | \(=\) |
\( -16\beta_{6} - 16\beta_{5} - 6\beta_{4} - 6\beta_{3} - 7\beta _1 - 6 \)
|
| \(\nu^{7}\) | \(=\) |
\( -16\beta_{7} - 51\beta_{6} - 29\beta_{5} - 23\beta_{4} + 29\beta_{2} - 16 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/41\mathbb{Z}\right)^\times\).
| \(n\) | \(6\) |
| \(\chi(n)\) | \(-1 + \beta_{3} - \beta_{4} - \beta_{7}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 10.1 |
|
−0.886111 | − | 0.643798i | 1.09529 | −0.247316 | − | 0.761160i | 0.247316 | + | 0.761160i | −0.970553 | − | 0.705148i | 0.386111 | − | 0.280526i | −0.947813 | + | 2.91707i | −1.80033 | 0.270884 | − | 0.833694i | ||||||||||||||||||||||||||||
| 10.2 | 1.19513 | + | 0.868312i | −1.47726 | 0.0563329 | + | 0.173375i | −0.0563329 | − | 0.173375i | −1.76552 | − | 1.28272i | −1.69513 | + | 1.23158i | 0.829779 | − | 2.55380i | −0.817703 | 0.0832183 | − | 0.256120i | |||||||||||||||||||||||||||||
| 16.1 | −0.727943 | + | 2.24038i | −2.35567 | −2.87136 | − | 2.08617i | 2.87136 | + | 2.08617i | 1.71480 | − | 5.27760i | 0.227943 | + | 0.701538i | 2.95244 | − | 2.14507i | 2.54920 | −6.76400 | + | 4.91433i | |||||||||||||||||||||||||||||
| 16.2 | −0.0810736 | + | 0.249519i | −0.262360 | 1.56235 | + | 1.13511i | −1.56235 | − | 1.13511i | 0.0212704 | − | 0.0654637i | −0.418926 | − | 1.28932i | −0.834404 | + | 0.606230i | −2.93117 | 0.409897 | − | 0.297808i | |||||||||||||||||||||||||||||
| 18.1 | −0.727943 | − | 2.24038i | −2.35567 | −2.87136 | + | 2.08617i | 2.87136 | − | 2.08617i | 1.71480 | + | 5.27760i | 0.227943 | − | 0.701538i | 2.95244 | + | 2.14507i | 2.54920 | −6.76400 | − | 4.91433i | |||||||||||||||||||||||||||||
| 18.2 | −0.0810736 | − | 0.249519i | −0.262360 | 1.56235 | − | 1.13511i | −1.56235 | + | 1.13511i | 0.0212704 | + | 0.0654637i | −0.418926 | + | 1.28932i | −0.834404 | − | 0.606230i | −2.93117 | 0.409897 | + | 0.297808i | |||||||||||||||||||||||||||||
| 37.1 | −0.886111 | + | 0.643798i | 1.09529 | −0.247316 | + | 0.761160i | 0.247316 | − | 0.761160i | −0.970553 | + | 0.705148i | 0.386111 | + | 0.280526i | −0.947813 | − | 2.91707i | −1.80033 | 0.270884 | + | 0.833694i | |||||||||||||||||||||||||||||
| 37.2 | 1.19513 | − | 0.868312i | −1.47726 | 0.0563329 | − | 0.173375i | −0.0563329 | + | 0.173375i | −1.76552 | + | 1.28272i | −1.69513 | − | 1.23158i | 0.829779 | + | 2.55380i | −0.817703 | 0.0832183 | + | 0.256120i | |||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 41.d | even | 5 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 41.2.d.a | ✓ | 8 |
| 3.b | odd | 2 | 1 | 369.2.h.b | 8 | ||
| 4.b | odd | 2 | 1 | 656.2.u.d | 8 | ||
| 41.d | even | 5 | 1 | inner | 41.2.d.a | ✓ | 8 |
| 41.d | even | 5 | 1 | 1681.2.a.e | 4 | ||
| 41.f | even | 10 | 1 | 1681.2.a.f | 4 | ||
| 123.k | odd | 10 | 1 | 369.2.h.b | 8 | ||
| 164.j | odd | 10 | 1 | 656.2.u.d | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 41.2.d.a | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 41.2.d.a | ✓ | 8 | 41.d | even | 5 | 1 | inner |
| 369.2.h.b | 8 | 3.b | odd | 2 | 1 | ||
| 369.2.h.b | 8 | 123.k | odd | 10 | 1 | ||
| 656.2.u.d | 8 | 4.b | odd | 2 | 1 | ||
| 656.2.u.d | 8 | 164.j | odd | 10 | 1 | ||
| 1681.2.a.e | 4 | 41.d | even | 5 | 1 | ||
| 1681.2.a.f | 4 | 41.f | even | 10 | 1 | ||
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(41, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} + T^{7} + 4 T^{6} + \cdots + 1 \)
|
| $3$ |
\( (T^{4} + 3 T^{3} - 4 T - 1)^{2} \)
|
| $5$ |
\( T^{8} - 3 T^{7} + \cdots + 1 \)
|
| $7$ |
\( T^{8} + 3 T^{7} + \cdots + 1 \)
|
| $11$ |
\( T^{8} + 15 T^{6} + \cdots + 625 \)
|
| $13$ |
\( T^{8} - 11 T^{7} + \cdots + 121 \)
|
| $17$ |
\( T^{8} - 14 T^{7} + \cdots + 841 \)
|
| $19$ |
\( T^{8} + 4 T^{7} + \cdots + 271441 \)
|
| $23$ |
\( T^{8} + 15 T^{7} + \cdots + 19321 \)
|
| $29$ |
\( T^{8} + 9 T^{7} + \cdots + 10201 \)
|
| $31$ |
\( T^{8} - 6 T^{7} + \cdots + 92416 \)
|
| $37$ |
\( T^{8} + 36 T^{7} + \cdots + 8755681 \)
|
| $41$ |
\( T^{8} + 7 T^{7} + \cdots + 2825761 \)
|
| $43$ |
\( T^{8} - 13 T^{7} + \cdots + 358801 \)
|
| $47$ |
\( T^{8} + 12 T^{7} + \cdots + 674041 \)
|
| $53$ |
\( T^{8} - 8 T^{7} + \cdots + 6195121 \)
|
| $59$ |
\( T^{8} - 23 T^{7} + \cdots + 9740641 \)
|
| $61$ |
\( T^{8} + 15 T^{7} + \cdots + 203401 \)
|
| $67$ |
\( (T^{4} + 5 T^{3} + \cdots + 3025)^{2} \)
|
| $71$ |
\( T^{8} - 2 T^{7} + \cdots + 15768841 \)
|
| $73$ |
\( (T^{4} + 2 T^{3} + \cdots - 359)^{2} \)
|
| $79$ |
\( (T^{4} + 21 T^{3} + \cdots - 571)^{2} \)
|
| $83$ |
\( (T^{4} + 8 T^{3} + \cdots - 211)^{2} \)
|
| $89$ |
\( T^{8} + 9 T^{7} + \cdots + 6345361 \)
|
| $97$ |
\( T^{8} + 8 T^{7} + \cdots + 78961 \)
|
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