Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [3724,2,Mod(1,3724)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3724, 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("3724.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 3724 = 2^{2} \cdot 7^{2} \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 3724.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: | \(29.7362897127\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 4x^{7} - 4x^{6} + 24x^{5} + x^{4} - 40x^{3} + 6x^{2} + 16x - 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - 4x^{7} - 4x^{6} + 24x^{5} + x^{4} - 40x^{3} + 6x^{2} + 16x - 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -2\nu^{7} + 2\nu^{6} + 27\nu^{5} - 19\nu^{4} - 85\nu^{3} + 46\nu^{2} + 48\nu - 31 ) / 13 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 2\nu^{7} - 2\nu^{6} - 27\nu^{5} + 19\nu^{4} + 85\nu^{3} - 33\nu^{2} - 61\nu - 8 ) / 13 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{7} + \nu^{6} + 20\nu^{5} - 29\nu^{4} - 75\nu^{3} + 101\nu^{2} + 63\nu - 61 ) / 13 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -2\nu^{7} + 2\nu^{6} + 27\nu^{5} - 6\nu^{4} - 124\nu^{3} - 6\nu^{2} + 165\nu + 8 ) / 13 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -5\nu^{7} + 18\nu^{6} + 22\nu^{5} - 93\nu^{4} - 24\nu^{3} + 115\nu^{2} + 3\nu - 19 ) / 13 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -6\nu^{7} + 19\nu^{6} + 42\nu^{5} - 122\nu^{4} - 86\nu^{3} + 177\nu^{2} + 40\nu - 15 ) / 13 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + \beta_{2} + \beta _1 + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{7} - \beta_{6} - \beta_{4} + 3\beta_{3} + 3\beta_{2} + 5\beta _1 + 4 \)
|
| \(\nu^{4}\) | \(=\) |
\( 3\beta_{7} - 3\beta_{6} + \beta_{5} - 3\beta_{4} + 13\beta_{3} + 12\beta_{2} + 10\beta _1 + 21 \)
|
| \(\nu^{5}\) | \(=\) |
\( 14\beta_{7} - 14\beta_{6} + 3\beta_{5} - 12\beta_{4} + 42\beta_{3} + 38\beta_{2} + 37\beta _1 + 54 \)
|
| \(\nu^{6}\) | \(=\) |
\( 45\beta_{7} - 44\beta_{6} + 14\beta_{5} - 38\beta_{4} + 149\beta_{3} + 129\beta_{2} + 101\beta _1 + 200 \)
|
| \(\nu^{7}\) | \(=\) |
\( 163\beta_{7} - 162\beta_{6} + 45\beta_{5} - 129\beta_{4} + 488\beta_{3} + 417\beta_{2} + 340\beta _1 + 613 \)
|
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 | −1.80685 | 0 | 0.516282 | 0 | 0 | 0 | 0.264703 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −1.39347 | 0 | 3.30592 | 0 | 0 | 0 | −1.05823 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −0.674986 | 0 | −1.82398 | 0 | 0 | 0 | −2.54439 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 0.0616581 | 0 | −2.14505 | 0 | 0 | 0 | −2.99620 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 0.923026 | 0 | −0.765692 | 0 | 0 | 0 | −2.14802 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 1.47289 | 0 | −3.38647 | 0 | 0 | 0 | −0.830597 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0 | 2.14459 | 0 | 3.48761 | 0 | 0 | 0 | 1.59929 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0 | 3.27314 | 0 | 0.811390 | 0 | 0 | 0 | 7.71345 | 0 | |||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(7\) | \(1\) |
| \(19\) | \(1\) |
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 | 3724.2.a.p | yes | 8 |
| 7.b | odd | 2 | 1 | 3724.2.a.o | ✓ | 8 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 3724.2.a.o | ✓ | 8 | 7.b | odd | 2 | 1 | |
| 3724.2.a.p | yes | 8 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{8} - 4T_{3}^{7} - 4T_{3}^{6} + 24T_{3}^{5} + T_{3}^{4} - 40T_{3}^{3} + 6T_{3}^{2} + 16T_{3} - 1 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3724))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
|
| $3$ |
\( T^{8} - 4 T^{7} + \cdots - 1 \)
|
| $5$ |
\( T^{8} - 22 T^{6} + \cdots + 49 \)
|
| $7$ |
\( T^{8} \)
|
| $11$ |
\( T^{8} + 8 T^{7} + \cdots - 31 \)
|
| $13$ |
\( T^{8} - 38 T^{6} + \cdots - 1 \)
|
| $17$ |
\( T^{8} - 8 T^{7} + \cdots + 6529 \)
|
| $19$ |
\( (T + 1)^{8} \)
|
| $23$ |
\( T^{8} - 4 T^{7} + \cdots - 36823 \)
|
| $29$ |
\( T^{8} + 4 T^{7} + \cdots - 4841 \)
|
| $31$ |
\( T^{8} - 24 T^{7} + \cdots - 833 \)
|
| $37$ |
\( T^{8} - 12 T^{7} + \cdots + 28799 \)
|
| $41$ |
\( T^{8} - 20 T^{7} + \cdots + 11999 \)
|
| $43$ |
\( T^{8} - 12 T^{7} + \cdots + 29968 \)
|
| $47$ |
\( T^{8} - 24 T^{7} + \cdots + 104057 \)
|
| $53$ |
\( T^{8} - 4 T^{7} + \cdots - 34537 \)
|
| $59$ |
\( T^{8} - 52 T^{7} + \cdots - 235841 \)
|
| $61$ |
\( T^{8} + 16 T^{7} + \cdots - 208103 \)
|
| $67$ |
\( T^{8} - 8 T^{7} + \cdots - 4196801 \)
|
| $71$ |
\( T^{8} - 16 T^{7} + \cdots + 136759 \)
|
| $73$ |
\( T^{8} - 24 T^{7} + \cdots + 59809 \)
|
| $79$ |
\( T^{8} + 20 T^{7} + \cdots - 65552 \)
|
| $83$ |
\( T^{8} - 48 T^{7} + \cdots + 1858841 \)
|
| $89$ |
\( T^{8} - 12 T^{7} + \cdots - 671504 \)
|
| $97$ |
\( T^{8} - 20 T^{7} + \cdots - 2367793 \)
|
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