Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [6020,2,Mod(1,6020)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6020, 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("6020.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 6020 = 2^{2} \cdot 5 \cdot 7 \cdot 43 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 6020.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: | \(48.0699420168\) |
| Analytic rank: | \(1\) |
| 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: |
\( x^{7} - x^{6} - 11x^{5} + 7x^{4} + 35x^{3} - 9x^{2} - 26x - 7 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| 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 \(1,\beta_1,\ldots,\beta_{6}\) 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
\( x^{7} - x^{6} - 11x^{5} + 7x^{4} + 35x^{3} - 9x^{2} - 26x - 7 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{6} + 2\nu^{5} + 9\nu^{4} - 13\nu^{3} - 22\nu^{2} + 16\nu + 7 ) / 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{6} + 2\nu^{5} + 9\nu^{4} - 13\nu^{3} - 25\nu^{2} + 16\nu + 19 ) / 3 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{6} - 2\nu^{5} - 9\nu^{4} + 16\nu^{3} + 22\nu^{2} - 31\nu - 10 ) / 3 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -\nu^{6} + 2\nu^{5} + 12\nu^{4} - 19\nu^{3} - 40\nu^{2} + 40\nu + 25 ) / 3 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 4\nu^{6} - 5\nu^{5} - 42\nu^{4} + 40\nu^{3} + 121\nu^{2} - 76\nu - 61 ) / 3 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{3} + \beta_{2} + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{4} + \beta_{2} + 5\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{5} + 2\beta_{4} - 6\beta_{3} + 7\beta_{2} + 2\beta _1 + 20 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{6} + 2\beta_{5} + 8\beta_{4} - \beta_{3} + 11\beta_{2} + 28\beta _1 + 11 \)
|
| \(\nu^{6}\) | \(=\) |
\( 2\beta_{6} + 13\beta_{5} + 21\beta_{4} - 34\beta_{3} + 47\beta_{2} + 25\beta _1 + 108 \)
|
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 |
|
0 | −3.25479 | 0 | 1.00000 | 0 | 1.00000 | 0 | 7.59367 | 0 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −1.54802 | 0 | 1.00000 | 0 | 1.00000 | 0 | −0.603640 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −1.44035 | 0 | 1.00000 | 0 | 1.00000 | 0 | −0.925397 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | −0.836897 | 0 | 1.00000 | 0 | 1.00000 | 0 | −2.29960 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 0.517965 | 0 | 1.00000 | 0 | 1.00000 | 0 | −2.73171 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 1.22318 | 0 | 1.00000 | 0 | 1.00000 | 0 | −1.50384 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0 | 2.33891 | 0 | 1.00000 | 0 | 1.00000 | 0 | 2.47052 | 0 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(5\) | \(-1\) |
| \(7\) | \(-1\) |
| \(43\) | \(-1\) |
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 | 6020.2.a.d | ✓ | 7 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 6020.2.a.d | ✓ | 7 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6020))\):
|
\( T_{3}^{7} + 3T_{3}^{6} - 7T_{3}^{5} - 21T_{3}^{4} + 6T_{3}^{3} + 31T_{3}^{2} + 3T_{3} - 9 \)
|
|
\( T_{11}^{7} + 3T_{11}^{6} - 33T_{11}^{5} - 129T_{11}^{4} + 106T_{11}^{3} + 746T_{11}^{2} + 474T_{11} + 9 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} \)
|
| $3$ |
\( T^{7} + 3 T^{6} + \cdots - 9 \)
|
| $5$ |
\( (T - 1)^{7} \)
|
| $7$ |
\( (T - 1)^{7} \)
|
| $11$ |
\( T^{7} + 3 T^{6} + \cdots + 9 \)
|
| $13$ |
\( T^{7} + 10 T^{6} + \cdots + 25 \)
|
| $17$ |
\( T^{7} + 9 T^{6} + \cdots - 13 \)
|
| $19$ |
\( T^{7} + 12 T^{6} + \cdots + 3159 \)
|
| $23$ |
\( T^{7} - T^{6} + \cdots + 1049 \)
|
| $29$ |
\( T^{7} + T^{6} + \cdots - 2431 \)
|
| $31$ |
\( T^{7} + 7 T^{6} + \cdots - 7649 \)
|
| $37$ |
\( T^{7} + 10 T^{6} + \cdots + 18369 \)
|
| $41$ |
\( T^{7} + 5 T^{6} + \cdots - 40607 \)
|
| $43$ |
\( (T - 1)^{7} \)
|
| $47$ |
\( T^{7} + 9 T^{6} + \cdots + 19827 \)
|
| $53$ |
\( T^{7} + 16 T^{6} + \cdots + 11011 \)
|
| $59$ |
\( T^{7} + 16 T^{6} + \cdots - 15881 \)
|
| $61$ |
\( T^{7} + 25 T^{6} + \cdots + 646151 \)
|
| $67$ |
\( T^{7} + 13 T^{6} + \cdots + 26559 \)
|
| $71$ |
\( T^{7} + 14 T^{6} + \cdots + 11011 \)
|
| $73$ |
\( T^{7} + 38 T^{6} + \cdots + 3812081 \)
|
| $79$ |
\( T^{7} + 12 T^{6} + \cdots - 1658707 \)
|
| $83$ |
\( T^{7} + T^{6} + \cdots + 1269251 \)
|
| $89$ |
\( T^{7} + 37 T^{6} + \cdots - 10243 \)
|
| $97$ |
\( T^{7} + 6 T^{6} + \cdots - 9621 \)
|
| show more | |
| show less |