Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [8037,2,Mod(1,8037)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8037, 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("8037.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 8037 = 3^{2} \cdot 19 \cdot 47 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 8037.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: | \(64.1757681045\) |
| 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: |
\( x^{7} - 3x^{6} - 4x^{5} + 14x^{4} + 4x^{3} - 17x^{2} - x + 5 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 2679) |
| 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
\( x^{7} - 3x^{6} - 4x^{5} + 14x^{4} + 4x^{3} - 17x^{2} - x + 5 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 2\nu^{2} - 2\nu + 3 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - 2\nu^{3} - 3\nu^{2} + 4\nu + 1 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{5} - 3\nu^{4} - 3\nu^{3} + 11\nu^{2} + \nu - 7 \)
|
| \(\beta_{6}\) | \(=\) |
\( \nu^{6} - 3\nu^{5} - 3\nu^{4} + 11\nu^{3} + \nu^{2} - 7\nu + 1 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 2\beta_{2} + 4\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{4} + 2\beta_{3} + 7\beta_{2} + 7\beta _1 + 7 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{5} + 3\beta_{4} + 9\beta_{3} + 16\beta_{2} + 21\beta _1 + 9 \)
|
| \(\nu^{6}\) | \(=\) |
\( \beta_{6} + 3\beta_{5} + 12\beta_{4} + 22\beta_{3} + 46\beta_{2} + 46\beta _1 + 34 \)
|
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 |
|
−1.61407 | 0 | 0.605217 | 1.82857 | 0 | 3.04786 | 2.25128 | 0 | −2.95144 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | −0.949093 | 0 | −1.09922 | 1.64272 | 0 | 0.492594 | 2.94145 | 0 | −1.55910 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | −0.209428 | 0 | −1.95614 | −0.161157 | 0 | −2.90787 | 0.828525 | 0 | 0.0337507 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0.333763 | 0 | −1.88860 | 3.79815 | 0 | 0.575782 | −1.29787 | 0 | 1.26768 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 1.65583 | 0 | 0.741759 | 3.75472 | 0 | 1.84065 | −2.08343 | 0 | 6.21716 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 2.11003 | 0 | 2.45224 | −2.30708 | 0 | −2.96487 | 0.954250 | 0 | −4.86802 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 2.67297 | 0 | 5.14475 | 1.44408 | 0 | 2.91586 | 8.40580 | 0 | 3.85997 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(19\) | \(-1\) |
| \(47\) | \(-1\) |
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 | 8037.2.a.l | 7 | |
| 3.b | odd | 2 | 1 | 2679.2.a.j | ✓ | 7 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2679.2.a.j | ✓ | 7 | 3.b | odd | 2 | 1 | |
| 8037.2.a.l | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8037))\):
|
\( T_{2}^{7} - 4T_{2}^{6} - T_{2}^{5} + 16T_{2}^{4} - 5T_{2}^{3} - 15T_{2}^{2} + 2T_{2} + 1 \)
|
|
\( T_{5}^{7} - 10T_{5}^{6} + 29T_{5}^{5} + 7T_{5}^{4} - 164T_{5}^{3} + 251T_{5}^{2} - 98T_{5} - 23 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} - 4 T^{6} + \cdots + 1 \)
|
| $3$ |
\( T^{7} \)
|
| $5$ |
\( T^{7} - 10 T^{6} + \cdots - 23 \)
|
| $7$ |
\( T^{7} - 3 T^{6} + \cdots - 40 \)
|
| $11$ |
\( T^{7} + T^{6} + \cdots + 11 \)
|
| $13$ |
\( T^{7} + 4 T^{6} + \cdots + 343 \)
|
| $17$ |
\( T^{7} - 16 T^{6} + \cdots + 11 \)
|
| $19$ |
\( (T - 1)^{7} \)
|
| $23$ |
\( T^{7} - 3 T^{6} + \cdots + 3475 \)
|
| $29$ |
\( T^{7} - 22 T^{6} + \cdots - 2719 \)
|
| $31$ |
\( T^{7} + 5 T^{6} + \cdots + 11987 \)
|
| $37$ |
\( T^{7} - T^{6} + \cdots + 805 \)
|
| $41$ |
\( T^{7} - 2 T^{6} + \cdots + 5393 \)
|
| $43$ |
\( T^{7} + T^{6} + \cdots - 1193 \)
|
| $47$ |
\( (T - 1)^{7} \)
|
| $53$ |
\( T^{7} - 15 T^{6} + \cdots - 28393 \)
|
| $59$ |
\( T^{7} - 21 T^{6} + \cdots - 20071 \)
|
| $61$ |
\( T^{7} + T^{6} + \cdots - 3937 \)
|
| $67$ |
\( T^{7} + 4 T^{6} + \cdots - 212551 \)
|
| $71$ |
\( T^{7} - 36 T^{6} + \cdots - 415163 \)
|
| $73$ |
\( T^{7} - 2 T^{6} + \cdots + 465125 \)
|
| $79$ |
\( T^{7} + 17 T^{6} + \cdots - 14408 \)
|
| $83$ |
\( T^{7} - 16 T^{6} + \cdots - 266923 \)
|
| $89$ |
\( T^{7} - 53 T^{6} + \cdots + 45973 \)
|
| $97$ |
\( T^{7} + 19 T^{6} + \cdots + 289 \)
|
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