Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [4970,2,Mod(1,4970)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4970, 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("4970.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 4970 = 2 \cdot 5 \cdot 7 \cdot 71 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 4970.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: | \(39.6856498046\) |
| 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: |
\( x^{7} - x^{6} - 11x^{5} + 9x^{4} + 35x^{3} - 24x^{2} - 33x + 22 \)
|
| 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 \(1,\beta_1,\ldots,\beta_{6}\) 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
\( x^{7} - x^{6} - 11x^{5} + 9x^{4} + 35x^{3} - 24x^{2} - 33x + 22 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{6} - 2\nu^{5} - 8\nu^{4} + 15\nu^{3} + 11\nu^{2} - 21\nu + 6 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{6} - 2\nu^{5} - 8\nu^{4} + 16\nu^{3} + 10\nu^{2} - 26\nu + 9 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{6} - 3\nu^{5} - 7\nu^{4} + 24\nu^{3} + 4\nu^{2} - 38\nu + 16 \)
|
| \(\beta_{5}\) | \(=\) |
\( 2\nu^{6} - 4\nu^{5} - 16\nu^{4} + 31\nu^{3} + 22\nu^{2} - 48\nu + 12 \)
|
| \(\beta_{6}\) | \(=\) |
\( 3\nu^{6} - 8\nu^{5} - 21\nu^{4} + 64\nu^{3} + 10\nu^{2} - 103\nu + 52 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{5} - \beta_{3} - \beta_{2} + \beta _1 + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{5} - 2\beta_{2} + 6\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{6} + 8\beta_{5} - 2\beta_{4} - 9\beta_{3} - 8\beta_{2} + 9\beta _1 + 13 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{6} + 10\beta_{5} - 3\beta_{4} - 2\beta_{3} - 18\beta_{2} + 39\beta _1 + 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( 10\beta_{6} + 58\beta_{5} - 22\beta_{4} - 65\beta_{3} - 58\beta_{2} + 70\beta _1 + 69 \)
|
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 | −2.42002 | 1.00000 | −1.00000 | −2.42002 | −1.00000 | 1.00000 | 2.85650 | −1.00000 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | 1.00000 | −1.51489 | 1.00000 | −1.00000 | −1.51489 | −1.00000 | 1.00000 | −0.705105 | −1.00000 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | 1.00000 | −1.49310 | 1.00000 | −1.00000 | −1.49310 | −1.00000 | 1.00000 | −0.770660 | −1.00000 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | 1.00000 | 0.666862 | 1.00000 | −1.00000 | 0.666862 | −1.00000 | 1.00000 | −2.55529 | −1.00000 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 1.00000 | 1.18572 | 1.00000 | −1.00000 | 1.18572 | −1.00000 | 1.00000 | −1.59407 | −1.00000 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.00000 | 1.89954 | 1.00000 | −1.00000 | 1.89954 | −1.00000 | 1.00000 | 0.608257 | −1.00000 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 1.00000 | 2.67589 | 1.00000 | −1.00000 | 2.67589 | −1.00000 | 1.00000 | 4.16038 | −1.00000 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(5\) | \(1\) |
| \(7\) | \(1\) |
| \(71\) | \(-1\) |
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 | 4970.2.a.x | ✓ | 7 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 4970.2.a.x | ✓ | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4970))\):
|
\( T_{3}^{7} - T_{3}^{6} - 11T_{3}^{5} + 9T_{3}^{4} + 35T_{3}^{3} - 24T_{3}^{2} - 33T_{3} + 22 \)
|
|
\( T_{11}^{7} + 9T_{11}^{6} + 16T_{11}^{5} - 40T_{11}^{4} - 128T_{11}^{3} - 92T_{11}^{2} - 21T_{11} - 1 \)
|
|
\( T_{13}^{7} - 44T_{13}^{5} + 4T_{13}^{4} + 316T_{13}^{3} + 25T_{13}^{2} - 487T_{13} - 131 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1)^{7} \)
|
| $3$ |
\( T^{7} - T^{6} + \cdots + 22 \)
|
| $5$ |
\( (T + 1)^{7} \)
|
| $7$ |
\( (T + 1)^{7} \)
|
| $11$ |
\( T^{7} + 9 T^{6} + \cdots - 1 \)
|
| $13$ |
\( T^{7} - 44 T^{5} + \cdots - 131 \)
|
| $17$ |
\( T^{7} + 5 T^{6} + \cdots - 40064 \)
|
| $19$ |
\( T^{7} - 2 T^{6} + \cdots + 44 \)
|
| $23$ |
\( T^{7} + 8 T^{6} + \cdots + 2453 \)
|
| $29$ |
\( T^{7} + 19 T^{6} + \cdots - 86036 \)
|
| $31$ |
\( T^{7} + 6 T^{6} + \cdots + 1532 \)
|
| $37$ |
\( T^{7} + 10 T^{6} + \cdots - 41519 \)
|
| $41$ |
\( T^{7} + 19 T^{6} + \cdots - 848 \)
|
| $43$ |
\( T^{7} + 22 T^{6} + \cdots + 36904 \)
|
| $47$ |
\( T^{7} - 2 T^{6} + \cdots + 16552 \)
|
| $53$ |
\( T^{7} + 3 T^{6} + \cdots - 94816 \)
|
| $59$ |
\( T^{7} - 13 T^{6} + \cdots + 4192 \)
|
| $61$ |
\( T^{7} + 4 T^{6} + \cdots + 590972 \)
|
| $67$ |
\( T^{7} + 23 T^{6} + \cdots - 569312 \)
|
| $71$ |
\( (T - 1)^{7} \)
|
| $73$ |
\( T^{7} - 283 T^{5} + \cdots - 18592 \)
|
| $79$ |
\( T^{7} + 40 T^{6} + \cdots + 4524908 \)
|
| $83$ |
\( T^{7} - 24 T^{6} + \cdots + 61102 \)
|
| $89$ |
\( T^{7} + 4 T^{6} + \cdots - 209824 \)
|
| $97$ |
\( T^{7} + 35 T^{6} + \cdots - 181796 \)
|
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