Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [946,2,Mod(1,946)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(946, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("946.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 946 = 2 \cdot 11 \cdot 43 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 946.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: | \(7.55384803121\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.262888000.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 2x^{5} - 13x^{4} + 24x^{3} + 33x^{2} - 56x + 16 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| 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
\( x^{6} - 2x^{5} - 13x^{4} + 24x^{3} + 33x^{2} - 56x + 16 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{5} + 6\nu^{4} + 21\nu^{3} - 76\nu^{2} - 113\nu + 156 ) / 32 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 7\nu^{5} - 10\nu^{4} - 83\nu^{3} + 116\nu^{2} + 183\nu - 228 ) / 32 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{5} - 2\nu^{4} - 13\nu^{3} + 20\nu^{2} + 37\nu - 28 ) / 4 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -13\nu^{5} + 14\nu^{4} + 177\nu^{3} - 156\nu^{2} - 509\nu + 300 ) / 32 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{4} + \beta_{3} - \beta_{2} + 5 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{5} + 2\beta_{3} + \beta_{2} + 8\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -2\beta_{5} - 13\beta_{4} + 10\beta_{3} - 8\beta_{2} + 3\beta _1 + 38 \)
|
| \(\nu^{5}\) | \(=\) |
\( 9\beta_{5} - 2\beta_{4} + 26\beta_{3} + 17\beta_{2} + 73\beta _1 + 4 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−1.00000 | −3.01044 | 1.00000 | −0.447830 | 3.01044 | 3.74445 | −1.00000 | 6.06272 | 0.447830 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −1.00000 | −1.92579 | 1.00000 | 0.200631 | 1.92579 | −3.44261 | −1.00000 | 0.708667 | −0.200631 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −1.00000 | 0.403115 | 1.00000 | 3.54032 | −0.403115 | 2.68212 | −1.00000 | −2.83750 | −3.54032 | |||||||||||||||||||||||||||||||||||||
| 1.4 | −1.00000 | 0.867300 | 1.00000 | −3.43079 | −0.867300 | −2.50308 | −1.00000 | −2.24779 | 3.43079 | |||||||||||||||||||||||||||||||||||||
| 1.5 | −1.00000 | 2.47014 | 1.00000 | 3.63325 | −2.47014 | −1.74921 | −1.00000 | 3.10159 | −3.63325 | |||||||||||||||||||||||||||||||||||||
| 1.6 | −1.00000 | 3.19567 | 1.00000 | 0.504412 | −3.19567 | 1.26833 | −1.00000 | 7.21231 | −0.504412 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(11\) | \(-1\) |
| \(43\) | \(1\) |
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 | 946.2.a.j | ✓ | 6 |
| 3.b | odd | 2 | 1 | 8514.2.a.bk | 6 | ||
| 4.b | odd | 2 | 1 | 7568.2.a.bf | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 946.2.a.j | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 7568.2.a.bf | 6 | 4.b | odd | 2 | 1 | ||
| 8514.2.a.bk | 6 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(946))\):
|
\( T_{3}^{6} - 2T_{3}^{5} - 13T_{3}^{4} + 24T_{3}^{3} + 33T_{3}^{2} - 56T_{3} + 16 \)
|
|
\( T_{5}^{6} - 4T_{5}^{5} - 11T_{5}^{4} + 48T_{5}^{3} - 9T_{5}^{2} - 10T_{5} + 2 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 1)^{6} \)
|
| $3$ |
\( T^{6} - 2 T^{5} + \cdots + 16 \)
|
| $5$ |
\( T^{6} - 4 T^{5} + \cdots + 2 \)
|
| $7$ |
\( T^{6} - 22 T^{4} + \cdots - 192 \)
|
| $11$ |
\( (T - 1)^{6} \)
|
| $13$ |
\( T^{6} + 4 T^{5} + \cdots - 40 \)
|
| $17$ |
\( T^{6} - 8 T^{5} + \cdots - 1668 \)
|
| $19$ |
\( T^{6} + 4 T^{5} + \cdots + 1532 \)
|
| $23$ |
\( T^{6} - 16 T^{5} + \cdots + 288 \)
|
| $29$ |
\( T^{6} + 4 T^{5} + \cdots + 9788 \)
|
| $31$ |
\( T^{6} - 22 T^{5} + \cdots - 14232 \)
|
| $37$ |
\( T^{6} + 12 T^{5} + \cdots + 21002 \)
|
| $41$ |
\( T^{6} - 10 T^{5} + \cdots + 15836 \)
|
| $43$ |
\( (T + 1)^{6} \)
|
| $47$ |
\( T^{6} - 18 T^{5} + \cdots + 33064 \)
|
| $53$ |
\( T^{6} - 12 T^{5} + \cdots + 18768 \)
|
| $59$ |
\( T^{6} - 18 T^{5} + \cdots + 34640 \)
|
| $61$ |
\( T^{6} + 12 T^{5} + \cdots - 3392 \)
|
| $67$ |
\( T^{6} - 4 T^{5} + \cdots + 141104 \)
|
| $71$ |
\( T^{6} - 24 T^{5} + \cdots + 220520 \)
|
| $73$ |
\( T^{6} + 34 T^{5} + \cdots - 110496 \)
|
| $79$ |
\( T^{6} - 10 T^{5} + \cdots - 203986 \)
|
| $83$ |
\( T^{6} - 2 T^{5} + \cdots - 47360 \)
|
| $89$ |
\( T^{6} - 26 T^{5} + \cdots + 184720 \)
|
| $97$ |
\( T^{6} + 6 T^{5} + \cdots + 101908 \)
|
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