Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [422,2,Mod(1,422)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(422, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("422.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 422 = 2 \cdot 211 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 422.a (trivial) |
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: | yes |
| Analytic conductor: | \(3.36968696530\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.43983893.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 2x^{5} - 9x^{4} + 8x^{3} + 20x^{2} + 9x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-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
\( x^{6} - 2x^{5} - 9x^{4} + 8x^{3} + 20x^{2} + 9x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{5} - 2\nu^{4} - 9\nu^{3} + 9\nu^{2} + 19\nu + 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( -\nu^{5} + 2\nu^{4} + 9\nu^{3} - 8\nu^{2} - 20\nu - 7 \)
|
| \(\beta_{4}\) | \(=\) |
\( 2\nu^{5} - 5\nu^{4} - 15\nu^{3} + 23\nu^{2} + 25\nu + 5 \)
|
| \(\beta_{5}\) | \(=\) |
\( -4\nu^{5} + 10\nu^{4} + 31\nu^{3} - 47\nu^{2} - 57\nu - 10 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + \beta_{2} + \beta _1 + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{5} + 2\beta_{4} + \beta_{3} + \beta_{2} + 8\beta _1 + 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( 3\beta_{5} + 5\beta_{4} + 8\beta_{3} + 10\beta_{2} + 16\beta _1 + 21 \)
|
| \(\nu^{5}\) | \(=\) |
\( 15\beta_{5} + 28\beta_{4} + 16\beta_{3} + 21\beta_{2} + 76\beta _1 + 38 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
1.00000 | −2.37190 | 1.00000 | −1.04706 | −2.37190 | 3.12239 | 1.00000 | 2.62589 | −1.04706 | ||||||||||||||||||||||||||||||||||||
| 1.2 | 1.00000 | −1.22346 | 1.00000 | 1.29775 | −1.22346 | −0.524040 | 1.00000 | −1.50314 | 1.29775 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 1.00000 | 1.17049 | 1.00000 | 1.79293 | 1.17049 | 2.90214 | 1.00000 | −1.62996 | 1.79293 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 1.00000 | 1.48829 | 1.00000 | −3.69690 | 1.48829 | 4.25659 | 1.00000 | −0.784998 | −3.69690 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 1.00000 | 1.72422 | 1.00000 | 1.59832 | 1.72422 | −1.94176 | 1.00000 | −0.0270528 | 1.59832 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 1.00000 | 3.21236 | 1.00000 | −1.94504 | 3.21236 | −0.815314 | 1.00000 | 7.31926 | −1.94504 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(211\) | \(1\) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 422.2.a.f | ✓ | 6 |
| 3.b | odd | 2 | 1 | 3798.2.a.x | 6 | ||
| 4.b | odd | 2 | 1 | 3376.2.a.q | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 422.2.a.f | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 3376.2.a.q | 6 | 4.b | odd | 2 | 1 | ||
| 3798.2.a.x | 6 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{6} - 4T_{3}^{5} - 4T_{3}^{4} + 28T_{3}^{3} - 15T_{3}^{2} - 33T_{3} + 28 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(422))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1)^{6} \)
|
| $3$ |
\( T^{6} - 4 T^{5} + \cdots + 28 \)
|
| $5$ |
\( T^{6} + 2 T^{5} + \cdots - 28 \)
|
| $7$ |
\( T^{6} - 7 T^{5} + \cdots - 32 \)
|
| $11$ |
\( T^{6} - 4 T^{5} + \cdots - 68 \)
|
| $13$ |
\( T^{6} - 8 T^{5} + \cdots - 64 \)
|
| $17$ |
\( T^{6} - T^{5} + \cdots - 1616 \)
|
| $19$ |
\( T^{6} - 11 T^{5} + \cdots + 2996 \)
|
| $23$ |
\( T^{6} + 3 T^{5} + \cdots - 8 \)
|
| $29$ |
\( T^{6} + 9 T^{5} + \cdots + 14 \)
|
| $31$ |
\( T^{6} - T^{5} + \cdots - 424 \)
|
| $37$ |
\( T^{6} + 5 T^{5} + \cdots - 14684 \)
|
| $41$ |
\( T^{6} + 9 T^{5} + \cdots + 872 \)
|
| $43$ |
\( T^{6} - 2 T^{5} + \cdots - 23428 \)
|
| $47$ |
\( T^{6} + 12 T^{5} + \cdots - 6653 \)
|
| $53$ |
\( T^{6} + 17 T^{5} + \cdots - 1336 \)
|
| $59$ |
\( T^{6} + 7 T^{5} + \cdots - 4016 \)
|
| $61$ |
\( T^{6} - 2 T^{5} + \cdots - 56908 \)
|
| $67$ |
\( T^{6} - 2 T^{5} + \cdots - 47072 \)
|
| $71$ |
\( T^{6} + 9 T^{5} + \cdots + 30517 \)
|
| $73$ |
\( T^{6} - 26 T^{5} + \cdots + 120782 \)
|
| $79$ |
\( T^{6} - 8 T^{5} + \cdots - 436736 \)
|
| $83$ |
\( T^{6} - 14 T^{5} + \cdots - 137152 \)
|
| $89$ |
\( T^{6} - T^{5} + \cdots + 418816 \)
|
| $97$ |
\( T^{6} - 14 T^{5} + \cdots + 458816 \)
|
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