Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [38,8,Mod(1,38)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(38, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("38.1");
S:= CuspForms(chi, 8);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 38 = 2 \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 38.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: | \(11.8706309684\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 9097x^{2} - 110520x + 10368000 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2\cdot 3 \) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - 9097x^{2} - 110520x + 10368000 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} + 30\nu^{2} - 7627\nu - 219060 ) / 1140 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -7\nu^{3} + 360\nu^{2} + 40279\nu - 1058940 ) / 3420 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 6\beta_{3} + 14\beta_{2} + 23\beta _1 + 4548 \)
|
| \(\nu^{3}\) | \(=\) |
\( -180\beta_{3} + 720\beta_{2} + 6937\beta _1 + 82620 \)
|
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 |
|
8.00000 | −92.4845 | 64.0000 | −426.367 | −739.876 | 875.686 | 512.000 | 6366.38 | −3410.93 | ||||||||||||||||||||||||||||||
| 1.2 | 8.00000 | −26.4051 | 64.0000 | 139.358 | −211.240 | 458.899 | 512.000 | −1489.77 | 1114.87 | |||||||||||||||||||||||||||||||
| 1.3 | 8.00000 | 51.0684 | 64.0000 | 338.533 | 408.547 | −143.677 | 512.000 | 420.979 | 2708.27 | |||||||||||||||||||||||||||||||
| 1.4 | 8.00000 | 79.8211 | 64.0000 | −330.525 | 638.569 | 1294.09 | 512.000 | 4184.42 | −2644.20 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(19\) | \(-1\) |
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 | 38.8.a.e | ✓ | 4 |
| 3.b | odd | 2 | 1 | 342.8.a.o | 4 | ||
| 4.b | odd | 2 | 1 | 304.8.a.e | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 38.8.a.e | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 304.8.a.e | 4 | 4.b | odd | 2 | 1 | ||
| 342.8.a.o | 4 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} - 12T_{3}^{3} - 9043T_{3}^{2} + 164994T_{3} + 9954648 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(38))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 8)^{4} \)
|
| $3$ |
\( T^{4} - 12 T^{3} + \cdots + 9954648 \)
|
| $5$ |
\( T^{4} + \cdots + 6648480000 \)
|
| $7$ |
\( T^{4} + \cdots - 74716579192 \)
|
| $11$ |
\( T^{4} + \cdots + 88406758195632 \)
|
| $13$ |
\( T^{4} + \cdots - 31\!\cdots\!80 \)
|
| $17$ |
\( T^{4} + \cdots + 18\!\cdots\!58 \)
|
| $19$ |
\( (T - 6859)^{4} \)
|
| $23$ |
\( T^{4} + \cdots + 28\!\cdots\!76 \)
|
| $29$ |
\( T^{4} + \cdots + 27\!\cdots\!20 \)
|
| $31$ |
\( T^{4} + \cdots + 32\!\cdots\!28 \)
|
| $37$ |
\( T^{4} + \cdots + 11\!\cdots\!28 \)
|
| $41$ |
\( T^{4} + \cdots - 43\!\cdots\!00 \)
|
| $43$ |
\( T^{4} + \cdots + 32\!\cdots\!56 \)
|
| $47$ |
\( T^{4} + \cdots - 57\!\cdots\!80 \)
|
| $53$ |
\( T^{4} + \cdots + 18\!\cdots\!48 \)
|
| $59$ |
\( T^{4} + \cdots - 33\!\cdots\!00 \)
|
| $61$ |
\( T^{4} + \cdots + 23\!\cdots\!60 \)
|
| $67$ |
\( T^{4} + \cdots + 26\!\cdots\!80 \)
|
| $71$ |
\( T^{4} + \cdots + 14\!\cdots\!04 \)
|
| $73$ |
\( T^{4} + \cdots + 13\!\cdots\!58 \)
|
| $79$ |
\( T^{4} + \cdots + 85\!\cdots\!00 \)
|
| $83$ |
\( T^{4} + \cdots + 88\!\cdots\!88 \)
|
| $89$ |
\( T^{4} + \cdots + 25\!\cdots\!80 \)
|
| $97$ |
\( T^{4} + \cdots - 28\!\cdots\!80 \)
|
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