Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [41,2,Mod(9,41)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(41, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([3]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("41.9");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 41 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 41.c (of order \(4\), 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: | \(0.327386648287\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Relative dimension: | \(3\) over \(\Q(i)\) |
| Coefficient field: | 6.0.5089536.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 16x^{2} - 24x + 18 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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
\( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 16x^{2} - 24x + 18 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{5} + 24\nu^{4} - 6\nu^{3} - \nu^{2} + 6\nu + 285 ) / 131 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 6\nu^{5} - 13\nu^{4} + 36\nu^{3} + 6\nu^{2} + 95\nu - 138 ) / 131 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 23\nu^{5} - 28\nu^{4} + 7\nu^{3} + 154\nu^{2} + 386\nu - 267 ) / 393 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -23\nu^{5} + 28\nu^{4} - 7\nu^{3} - 23\nu^{2} - 386\nu + 267 ) / 131 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{5} + 3\beta_{4} \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{5} + 4\beta_{3} + \beta_{2} \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{3} + 6\beta_{2} - \beta _1 - 12 \)
|
| \(\nu^{5}\) | \(=\) |
\( -7\beta_{5} - 3\beta_{4} + 7\beta_{2} - 18\beta _1 - 3 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/41\mathbb{Z}\right)^\times\).
| \(n\) | \(6\) |
| \(\chi(n)\) | \(-\beta_{4}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 9.1 |
|
− | 2.51414i | 0.660442 | + | 0.660442i | −4.32088 | 3.51414i | 1.66044 | − | 1.66044i | −1.66044 | − | 1.66044i | 5.83502i | − | 2.12763i | 8.83502 | ||||||||||||||||||||||||||||
| 9.2 | − | 0.571993i | −2.33641 | − | 2.33641i | 1.67282 | 1.57199i | −1.33641 | + | 1.33641i | 1.33641 | + | 1.33641i | − | 2.10083i | 7.91764i | 0.899170 | |||||||||||||||||||||||||||||
| 9.3 | 2.08613i | −0.324030 | − | 0.324030i | −2.35194 | − | 1.08613i | 0.675970 | − | 0.675970i | −0.675970 | − | 0.675970i | − | 0.734191i | − | 2.79001i | 2.26581 | ||||||||||||||||||||||||||||
| 32.1 | − | 2.08613i | −0.324030 | + | 0.324030i | −2.35194 | 1.08613i | 0.675970 | + | 0.675970i | −0.675970 | + | 0.675970i | 0.734191i | 2.79001i | 2.26581 | ||||||||||||||||||||||||||||||
| 32.2 | 0.571993i | −2.33641 | + | 2.33641i | 1.67282 | − | 1.57199i | −1.33641 | − | 1.33641i | 1.33641 | − | 1.33641i | 2.10083i | − | 7.91764i | 0.899170 | |||||||||||||||||||||||||||||
| 32.3 | 2.51414i | 0.660442 | − | 0.660442i | −4.32088 | − | 3.51414i | 1.66044 | + | 1.66044i | −1.66044 | + | 1.66044i | − | 5.83502i | 2.12763i | 8.83502 | |||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 41.c | even | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 41.2.c.a | ✓ | 6 |
| 3.b | odd | 2 | 1 | 369.2.f.c | 6 | ||
| 4.b | odd | 2 | 1 | 656.2.l.g | 6 | ||
| 41.c | even | 4 | 1 | inner | 41.2.c.a | ✓ | 6 |
| 41.e | odd | 8 | 2 | 1681.2.a.g | 6 | ||
| 123.f | odd | 4 | 1 | 369.2.f.c | 6 | ||
| 164.e | odd | 4 | 1 | 656.2.l.g | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 41.2.c.a | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 41.2.c.a | ✓ | 6 | 41.c | even | 4 | 1 | inner |
| 369.2.f.c | 6 | 3.b | odd | 2 | 1 | ||
| 369.2.f.c | 6 | 123.f | odd | 4 | 1 | ||
| 656.2.l.g | 6 | 4.b | odd | 2 | 1 | ||
| 656.2.l.g | 6 | 164.e | odd | 4 | 1 | ||
| 1681.2.a.g | 6 | 41.e | odd | 8 | 2 | ||
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(41, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + 11 T^{4} + \cdots + 9 \)
|
| $3$ |
\( T^{6} + 4 T^{5} + \cdots + 2 \)
|
| $5$ |
\( T^{6} + 16 T^{4} + \cdots + 36 \)
|
| $7$ |
\( T^{6} + 2 T^{5} + \cdots + 18 \)
|
| $11$ |
\( T^{6} - 6 T^{5} + \cdots + 2 \)
|
| $13$ |
\( T^{6} + 10 T^{5} + \cdots + 72 \)
|
| $17$ |
\( (T^{2} + 6 T + 18)^{3} \)
|
| $19$ |
\( T^{6} - 4 T^{5} + \cdots + 162 \)
|
| $23$ |
\( (T + 2)^{6} \)
|
| $29$ |
\( T^{6} - 10 T^{5} + \cdots + 2312 \)
|
| $31$ |
\( (T^{3} + 4 T^{2} - 40 T - 16)^{2} \)
|
| $37$ |
\( (T^{3} - 48 T - 34)^{2} \)
|
| $41$ |
\( T^{6} - 14 T^{5} + \cdots + 68921 \)
|
| $43$ |
\( T^{6} + 96 T^{4} + \cdots + 26896 \)
|
| $47$ |
\( T^{6} - 12 T^{5} + \cdots + 578 \)
|
| $53$ |
\( (T^{2} - 2 T + 2)^{3} \)
|
| $59$ |
\( (T^{3} - 2 T^{2} - 84 T - 24)^{2} \)
|
| $61$ |
\( T^{6} + 168 T^{4} + \cdots + 26896 \)
|
| $67$ |
\( T^{6} + 18 T^{5} + \cdots + 1458 \)
|
| $71$ |
\( T^{6} + 24 T^{5} + \cdots + 816642 \)
|
| $73$ |
\( T^{6} + 12 T^{4} + \cdots + 4 \)
|
| $79$ |
\( T^{6} + 8 T^{5} + \cdots + 4269042 \)
|
| $83$ |
\( (T^{3} + 10 T^{2} + \cdots - 136)^{2} \)
|
| $89$ |
\( T^{6} - 42 T^{5} + \cdots + 291848 \)
|
| $97$ |
\( T^{6} - 2 T^{5} + \cdots + 668168 \)
|
| show more | |
| show less |