Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [3856,2,Mod(1,3856)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3856, 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("3856.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 3856 = 2^{4} \cdot 241 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 3856.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: | \(30.7903150194\) |
| Analytic rank: | \(0\) |
| Dimension: | \(7\) |
| Coefficient field: | 7.7.31056073.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{7} - 3x^{6} - 3x^{5} + 11x^{4} + x^{3} - 9x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 241) |
| 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
\( x^{7} - 3x^{6} - 3x^{5} + 11x^{4} + x^{3} - 9x^{2} + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{5} - 2\nu^{4} - 4\nu^{3} + 5\nu^{2} + 2\nu - 1 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{5} - 2\nu^{4} - 4\nu^{3} + 4\nu^{2} + 3\nu + 1 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{6} - 3\nu^{5} - 2\nu^{4} + 9\nu^{3} - 3\nu^{2} - 4\nu + 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{6} - 2\nu^{5} - 5\nu^{4} + 6\nu^{3} + 7\nu^{2} - 3\nu - 2 \)
|
| \(\beta_{6}\) | \(=\) |
\( -\nu^{6} + 3\nu^{5} + 3\nu^{4} - 11\nu^{3} - \nu^{2} + 8\nu \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{3} + \beta_{2} + \beta _1 + 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{6} - \beta_{5} - \beta_{3} + 2\beta_{2} + 4\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -\beta_{6} - 2\beta_{5} + \beta_{4} - 6\beta_{3} + 8\beta_{2} + 8\beta _1 + 8 \)
|
| \(\nu^{5}\) | \(=\) |
\( -6\beta_{6} - 8\beta_{5} + 2\beta_{4} - 11\beta_{3} + 20\beta_{2} + 25\beta _1 + 11 \)
|
| \(\nu^{6}\) | \(=\) |
\( -11\beta_{6} - 19\beta_{5} + 9\beta_{4} - 39\beta_{3} + 61\beta_{2} + 62\beta _1 + 44 \)
|
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 | −2.33806 | 0 | −3.89634 | 0 | 3.68231 | 0 | 2.46654 | 0 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −0.980039 | 0 | −1.69135 | 0 | 1.30586 | 0 | −2.03952 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | 0.186202 | 0 | −2.25110 | 0 | −3.52970 | 0 | −2.96533 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 0.494846 | 0 | −1.23324 | 0 | −1.36627 | 0 | −2.75513 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 0.815004 | 0 | 0.961999 | 0 | 4.61392 | 0 | −2.33577 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 2.37146 | 0 | −2.63180 | 0 | 2.01025 | 0 | 2.62382 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0 | 2.45059 | 0 | 2.74184 | 0 | 0.283608 | 0 | 3.00540 | 0 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(241\) | \(1\) |
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 | 3856.2.a.j | 7 | |
| 4.b | odd | 2 | 1 | 241.2.a.a | ✓ | 7 | |
| 12.b | even | 2 | 1 | 2169.2.a.e | 7 | ||
| 20.d | odd | 2 | 1 | 6025.2.a.f | 7 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 241.2.a.a | ✓ | 7 | 4.b | odd | 2 | 1 | |
| 2169.2.a.e | 7 | 12.b | even | 2 | 1 | ||
| 3856.2.a.j | 7 | 1.a | even | 1 | 1 | trivial | |
| 6025.2.a.f | 7 | 20.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3856))\):
|
\( T_{3}^{7} - 3T_{3}^{6} - 5T_{3}^{5} + 19T_{3}^{4} - 4T_{3}^{3} - 14T_{3}^{2} + 8T_{3} - 1 \)
|
|
\( T_{5}^{7} + 8T_{5}^{6} + 12T_{5}^{5} - 50T_{5}^{4} - 165T_{5}^{3} - 93T_{5}^{2} + 137T_{5} + 127 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} \)
|
| $3$ |
\( T^{7} - 3 T^{6} + \cdots - 1 \)
|
| $5$ |
\( T^{7} + 8 T^{6} + \cdots + 127 \)
|
| $7$ |
\( T^{7} - 7 T^{6} + \cdots - 61 \)
|
| $11$ |
\( T^{7} - 18 T^{6} + \cdots - 1069 \)
|
| $13$ |
\( T^{7} + T^{6} - 48 T^{5} + \cdots - 1 \)
|
| $17$ |
\( T^{7} + 2 T^{6} + \cdots - 1039 \)
|
| $19$ |
\( T^{7} - 6 T^{6} + \cdots + 5983 \)
|
| $23$ |
\( T^{7} - 22 T^{6} + \cdots - 1369 \)
|
| $29$ |
\( T^{7} + 16 T^{6} + \cdots - 10769 \)
|
| $31$ |
\( T^{7} - 18 T^{6} + \cdots + 617 \)
|
| $37$ |
\( T^{7} - 8 T^{6} + \cdots - 78167 \)
|
| $41$ |
\( T^{7} + 15 T^{6} + \cdots + 101009 \)
|
| $43$ |
\( T^{7} + 14 T^{6} + \cdots - 296569 \)
|
| $47$ |
\( T^{7} - 10 T^{6} + \cdots + 7793 \)
|
| $53$ |
\( T^{7} - 15 T^{6} + \cdots - 230663 \)
|
| $59$ |
\( T^{7} - 18 T^{6} + \cdots + 2076763 \)
|
| $61$ |
\( T^{7} - 4 T^{6} + \cdots + 23149 \)
|
| $67$ |
\( T^{7} + 18 T^{6} + \cdots - 2288147 \)
|
| $71$ |
\( T^{7} - 50 T^{6} + \cdots + 255937 \)
|
| $73$ |
\( T^{7} - 378 T^{5} + \cdots + 11879 \)
|
| $79$ |
\( T^{7} - 15 T^{6} + \cdots - 52709 \)
|
| $83$ |
\( T^{7} - 24 T^{6} + \cdots - 4333 \)
|
| $89$ |
\( T^{7} + 13 T^{6} + \cdots - 89477 \)
|
| $97$ |
\( T^{7} - T^{6} + \cdots + 40121 \)
|
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