Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [6006,2,Mod(1,6006)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6006, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 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("6006.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 6006 = 2 \cdot 3 \cdot 7 \cdot 11 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 6006.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: | \(47.9581514540\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.72306708.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 11x^{4} - x^{3} + 10x^{2} - 2 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{23}]\) |
| Coefficient ring index: | \( 2^{4} \) |
| 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
\( x^{6} - 11x^{4} - x^{3} + 10x^{2} - 2 \)
:
| \(\beta_{1}\) | \(=\) |
\( -\nu^{5} + 11\nu^{3} + \nu^{2} - 8\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{5} - 11\nu^{3} - \nu^{2} + 10\nu \)
|
| \(\beta_{3}\) | \(=\) |
\( 4\nu^{5} - 42\nu^{3} - 4\nu^{2} + 20\nu - 1 \)
|
| \(\beta_{4}\) | \(=\) |
\( -4\nu^{5} - 2\nu^{4} + 42\nu^{3} + 26\nu^{2} - 18\nu - 12 \)
|
| \(\beta_{5}\) | \(=\) |
\( 5\nu^{5} + 4\nu^{4} - 53\nu^{3} - 47\nu^{2} + 24\nu + 18 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{5} + 2\beta_{4} + \beta_{3} + \beta _1 + 7 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( \beta_{3} + 6\beta_{2} + 10\beta _1 + 1 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 11\beta_{5} + 21\beta_{4} + 10\beta_{3} + \beta_{2} + 12\beta _1 + 64 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( \beta_{5} + 2\beta_{4} + 12\beta_{3} + 58\beta_{2} + 101\beta _1 + 18 ) / 2 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
1.00000 | −1.00000 | 1.00000 | −3.93175 | −1.00000 | 1.00000 | 1.00000 | 1.00000 | −3.93175 | ||||||||||||||||||||||||||||||||||||
| 1.2 | 1.00000 | −1.00000 | 1.00000 | −2.18192 | −1.00000 | 1.00000 | 1.00000 | 1.00000 | −2.18192 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 1.00000 | −1.00000 | 1.00000 | −0.643320 | −1.00000 | 1.00000 | 1.00000 | 1.00000 | −0.643320 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 1.00000 | −1.00000 | 1.00000 | 0.621238 | −1.00000 | 1.00000 | 1.00000 | 1.00000 | 0.621238 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 1.00000 | −1.00000 | 1.00000 | 2.78772 | −1.00000 | 1.00000 | 1.00000 | 1.00000 | 2.78772 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 1.00000 | −1.00000 | 1.00000 | 3.34804 | −1.00000 | 1.00000 | 1.00000 | 1.00000 | 3.34804 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(3\) | \(1\) |
| \(7\) | \(-1\) |
| \(11\) | \(-1\) |
| \(13\) | \(1\) |
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 | 6006.2.a.cf | ✓ | 6 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 6006.2.a.cf | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6006))\):
|
\( T_{5}^{6} - 20T_{5}^{4} + 4T_{5}^{3} + 88T_{5}^{2} - 32 \)
|
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\( T_{17}^{6} - 44T_{17}^{4} - 8T_{17}^{3} + 160T_{17}^{2} - 128 \)
|
|
\( T_{19}^{6} - 2T_{19}^{5} - 64T_{19}^{4} + 24T_{19}^{3} + 864T_{19}^{2} - 384T_{19} - 2432 \)
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\( T_{23}^{6} - 10T_{23}^{5} - 16T_{23}^{4} + 376T_{23}^{3} - 640T_{23}^{2} - 1024T_{23} - 256 \)
|
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\( T_{31}^{6} - 2T_{31}^{5} - 116T_{31}^{4} + 204T_{31}^{3} + 2728T_{31}^{2} + 576T_{31} - 5088 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1)^{6} \)
|
| $3$ |
\( (T + 1)^{6} \)
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| $5$ |
\( T^{6} - 20 T^{4} + \cdots - 32 \)
|
| $7$ |
\( (T - 1)^{6} \)
|
| $11$ |
\( (T - 1)^{6} \)
|
| $13$ |
\( (T + 1)^{6} \)
|
| $17$ |
\( T^{6} - 44 T^{4} + \cdots - 128 \)
|
| $19$ |
\( T^{6} - 2 T^{5} + \cdots - 2432 \)
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| $23$ |
\( T^{6} - 10 T^{5} + \cdots - 256 \)
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| $29$ |
\( T^{6} - 6 T^{5} + \cdots - 1152 \)
|
| $31$ |
\( T^{6} - 2 T^{5} + \cdots - 5088 \)
|
| $37$ |
\( T^{6} - 12 T^{5} + \cdots + 608 \)
|
| $41$ |
\( T^{6} - 4 T^{5} + \cdots + 15168 \)
|
| $43$ |
\( T^{6} - 4 T^{5} + \cdots - 25344 \)
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| $47$ |
\( T^{6} - 4 T^{5} + \cdots + 346368 \)
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| $53$ |
\( T^{6} - 18 T^{5} + \cdots - 576 \)
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| $59$ |
\( T^{6} - 14 T^{5} + \cdots + 20736 \)
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| $61$ |
\( T^{6} - 252 T^{4} + \cdots - 53056 \)
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| $67$ |
\( T^{6} - 4 T^{5} + \cdots + 26368 \)
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| $71$ |
\( T^{6} - 14 T^{5} + \cdots + 96256 \)
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| $73$ |
\( T^{6} - 2 T^{5} + \cdots - 152896 \)
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| $79$ |
\( T^{6} - 6 T^{5} + \cdots + 296448 \)
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| $83$ |
\( T^{6} - 24 T^{5} + \cdots + 1083392 \)
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| $89$ |
\( T^{6} + 14 T^{5} + \cdots - 354176 \)
|
| $97$ |
\( T^{6} + 2 T^{5} + \cdots - 2528 \)
|
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