Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1005,2,Mod(1,1005)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1005, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("1005.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1005 = 3 \cdot 5 \cdot 67 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1005.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: | \(8.02496540314\) |
| Analytic rank: | \(0\) |
| Dimension: | \(7\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{7} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{7} - 3x^{6} - 6x^{5} + 15x^{4} + 14x^{3} - 15x^{2} - 6x + 4 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 2 \) |
| 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 \(1,\beta_1,\ldots,\beta_{6}\) 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
\( x^{7} - 3x^{6} - 6x^{5} + 15x^{4} + 14x^{3} - 15x^{2} - 6x + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 2\nu^{2} - 3\nu + 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{5} - 3\nu^{4} - 4\nu^{3} + 11\nu^{2} + 5\nu - 5 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{6} - 3\nu^{5} - 5\nu^{4} + 12\nu^{3} + 10\nu^{2} - 4\nu - 1 \)
|
| \(\beta_{6}\) | \(=\) |
\( -\nu^{6} + 2\nu^{5} + 9\nu^{4} - 10\nu^{3} - 25\nu^{2} + 3\nu + 8 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 2\beta_{2} + 5\beta _1 + 4 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{6} + \beta_{5} + \beta_{4} + 2\beta_{3} + 8\beta_{2} + 10\beta _1 + 18 \)
|
| \(\nu^{5}\) | \(=\) |
\( 3\beta_{6} + 3\beta_{5} + 4\beta_{4} + 10\beta_{3} + 21\beta_{2} + 34\beta _1 + 42 \)
|
| \(\nu^{6}\) | \(=\) |
\( 14\beta_{6} + 15\beta_{5} + 17\beta_{4} + 28\beta_{3} + 69\beta_{2} + 86\beta _1 + 139 \)
|
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 |
|
−2.05670 | 1.00000 | 2.23001 | 1.00000 | −2.05670 | −1.79080 | −0.473069 | 1.00000 | −2.05670 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.42013 | 1.00000 | 0.0167760 | 1.00000 | −1.42013 | 3.66225 | 2.81644 | 1.00000 | −1.42013 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0.236891 | 1.00000 | −1.94388 | 1.00000 | 0.236891 | −0.885216 | −0.934271 | 1.00000 | 0.236891 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0.546929 | 1.00000 | −1.70087 | 1.00000 | 0.546929 | 3.60369 | −2.02411 | 1.00000 | 0.546929 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 1.71932 | 1.00000 | 0.956075 | 1.00000 | 1.71932 | −0.918902 | −1.79484 | 1.00000 | 1.71932 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 2.29566 | 1.00000 | 3.27007 | 1.00000 | 2.29566 | 2.26654 | 2.91565 | 1.00000 | 2.29566 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 2.67802 | 1.00000 | 5.17182 | 1.00000 | 2.67802 | −2.93755 | 8.49420 | 1.00000 | 2.67802 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(5\) | \(-1\) |
| \(67\) | \(-1\) |
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 | 1005.2.a.i | ✓ | 7 |
| 3.b | odd | 2 | 1 | 3015.2.a.l | 7 | ||
| 5.b | even | 2 | 1 | 5025.2.a.bb | 7 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1005.2.a.i | ✓ | 7 | 1.a | even | 1 | 1 | trivial |
| 3015.2.a.l | 7 | 3.b | odd | 2 | 1 | ||
| 5025.2.a.bb | 7 | 5.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{7} - 4T_{2}^{6} - 3T_{2}^{5} + 25T_{2}^{4} - 11T_{2}^{3} - 33T_{2}^{2} + 25T_{2} - 4 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1005))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} - 4 T^{6} + \cdots - 4 \)
|
| $3$ |
\( (T - 1)^{7} \)
|
| $5$ |
\( (T - 1)^{7} \)
|
| $7$ |
\( T^{7} - 3 T^{6} + \cdots - 128 \)
|
| $11$ |
\( T^{7} - 5 T^{6} + \cdots - 32 \)
|
| $13$ |
\( T^{7} + T^{6} - 17 T^{5} + \cdots + 8 \)
|
| $17$ |
\( T^{7} - 11 T^{6} + \cdots - 4 \)
|
| $19$ |
\( T^{7} - 8 T^{6} + \cdots + 17648 \)
|
| $23$ |
\( T^{7} - 11 T^{6} + \cdots + 256 \)
|
| $29$ |
\( T^{7} - 50 T^{5} + \cdots - 28 \)
|
| $31$ |
\( T^{7} + 3 T^{6} + \cdots + 112 \)
|
| $37$ |
\( T^{7} + 5 T^{6} + \cdots - 652 \)
|
| $41$ |
\( T^{7} - T^{6} + \cdots + 56 \)
|
| $43$ |
\( T^{7} + 3 T^{6} + \cdots - 386848 \)
|
| $47$ |
\( T^{7} - 10 T^{6} + \cdots - 5648 \)
|
| $53$ |
\( T^{7} - 3 T^{6} + \cdots + 6176 \)
|
| $59$ |
\( T^{7} + 4 T^{6} + \cdots - 85076 \)
|
| $61$ |
\( T^{7} + 9 T^{6} + \cdots - 494672 \)
|
| $67$ |
\( (T - 1)^{7} \)
|
| $71$ |
\( T^{7} - 6 T^{6} + \cdots + 23728 \)
|
| $73$ |
\( T^{7} + 9 T^{6} + \cdots + 574076 \)
|
| $79$ |
\( T^{7} + 11 T^{6} + \cdots - 7027024 \)
|
| $83$ |
\( T^{7} - 30 T^{6} + \cdots - 87104 \)
|
| $89$ |
\( T^{7} - 13 T^{6} + \cdots + 812644 \)
|
| $97$ |
\( T^{7} + 7 T^{6} + \cdots - 220744 \)
|
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