Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [38,10,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, 10, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("38.1");
S:= CuspForms(chi, 10);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 38 = 2 \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 10 \) |
| 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: | \(19.5713617742\) |
| 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: |
\( x^{4} - 2x^{3} - 8016x^{2} - 155839x + 2105804 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{4}\cdot 3 \) |
| 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
\( x^{4} - 2x^{3} - 8016x^{2} - 155839x + 2105804 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 2\nu^{3} - 47\nu^{2} - 12893\nu - 62941 ) / 693 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{2} + 57\nu + 3970 ) / 21 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -2\nu^{3} + 179\nu^{2} + 22001\nu - 470801 ) / 693 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + 4\beta_{2} + \beta _1 + 14 ) / 24 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 19\beta_{3} - 92\beta_{2} + 19\beta _1 + 32026 ) / 8 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 3893\beta_{3} + 9650\beta_{2} + 8051\beta _1 + 1551688 ) / 12 \)
|
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 |
|
16.0000 | −206.847 | 256.000 | −845.315 | −3309.55 | −6607.98 | 4096.00 | 23102.6 | −13525.0 | ||||||||||||||||||||||||||||||
| 1.2 | 16.0000 | 55.3919 | 256.000 | 2263.93 | 886.270 | 5766.09 | 4096.00 | −16614.7 | 36222.9 | |||||||||||||||||||||||||||||||
| 1.3 | 16.0000 | 110.471 | 256.000 | −1298.23 | 1767.54 | 6626.40 | 4096.00 | −7479.16 | −20771.6 | |||||||||||||||||||||||||||||||
| 1.4 | 16.0000 | 266.984 | 256.000 | 745.613 | 4271.74 | −3114.51 | 4096.00 | 51597.3 | 11929.8 | |||||||||||||||||||||||||||||||
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 | 38.10.a.e | ✓ | 4 |
| 3.b | odd | 2 | 1 | 342.10.a.i | 4 | ||
| 4.b | odd | 2 | 1 | 304.10.a.d | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 38.10.a.e | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 304.10.a.d | 4 | 4.b | odd | 2 | 1 | ||
| 342.10.a.i | 4 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} - 226T_{3}^{3} - 39131T_{3}^{2} + 8791740T_{3} - 337930956 \)
acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(38))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 16)^{4} \)
|
| $3$ |
\( T^{4} - 226 T^{3} + \cdots - 337930956 \)
|
| $5$ |
\( T^{4} + \cdots + 1852446724000 \)
|
| $7$ |
\( T^{4} + \cdots + 786353549326443 \)
|
| $11$ |
\( T^{4} + \cdots + 30\!\cdots\!92 \)
|
| $13$ |
\( T^{4} + \cdots - 79\!\cdots\!24 \)
|
| $17$ |
\( T^{4} + \cdots + 16\!\cdots\!09 \)
|
| $19$ |
\( (T + 130321)^{4} \)
|
| $23$ |
\( T^{4} + \cdots - 60\!\cdots\!88 \)
|
| $29$ |
\( T^{4} + \cdots - 13\!\cdots\!00 \)
|
| $31$ |
\( T^{4} + \cdots + 16\!\cdots\!44 \)
|
| $37$ |
\( T^{4} + \cdots + 88\!\cdots\!52 \)
|
| $41$ |
\( T^{4} + \cdots - 18\!\cdots\!92 \)
|
| $43$ |
\( T^{4} + \cdots - 70\!\cdots\!72 \)
|
| $47$ |
\( T^{4} + \cdots + 11\!\cdots\!00 \)
|
| $53$ |
\( T^{4} + \cdots - 13\!\cdots\!08 \)
|
| $59$ |
\( T^{4} + \cdots - 41\!\cdots\!00 \)
|
| $61$ |
\( T^{4} + \cdots - 32\!\cdots\!92 \)
|
| $67$ |
\( T^{4} + \cdots - 45\!\cdots\!48 \)
|
| $71$ |
\( T^{4} + \cdots - 13\!\cdots\!12 \)
|
| $73$ |
\( T^{4} + \cdots + 14\!\cdots\!57 \)
|
| $79$ |
\( T^{4} + \cdots + 32\!\cdots\!00 \)
|
| $83$ |
\( T^{4} + \cdots + 94\!\cdots\!24 \)
|
| $89$ |
\( T^{4} + \cdots + 42\!\cdots\!00 \)
|
| $97$ |
\( T^{4} + \cdots - 24\!\cdots\!84 \)
|
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