Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [390,8,Mod(1,390)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(390, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("390.1");
S:= CuspForms(chi, 8);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 390 = 2 \cdot 3 \cdot 5 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 390.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: | \(121.830159939\) |
| Analytic rank: | \(1\) |
| 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: |
\( x^{4} - x^{3} - 1426336x^{2} + 179014540x + 344930456016 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{3}\cdot 3 \) |
| 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
\( x^{4} - x^{3} - 1426336x^{2} + 179014540x + 344930456016 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -155\nu^{3} - 93121\nu^{2} + 135065684\nu + 45727312868 ) / 20930572 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 155\nu^{3} + 93121\nu^{2} - 9482252\nu - 45769174012 ) / 20930572 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -9\nu^{3} - 72925\nu^{2} + 6762236\nu + 50810000019 ) / 5232643 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + \beta _1 + 2 ) / 6 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -465\beta_{3} - 16\beta_{2} + 92\beta _1 + 4279261 ) / 6 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 93121\beta_{3} + 293668\beta_{2} + 1968\beta _1 - 266354957 ) / 2 \)
|
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 | −27.0000 | 64.0000 | −125.000 | 216.000 | −1423.48 | −512.000 | 729.000 | 1000.00 | ||||||||||||||||||||||||||||||
| 1.2 | −8.00000 | −27.0000 | 64.0000 | −125.000 | 216.000 | 123.164 | −512.000 | 729.000 | 1000.00 | |||||||||||||||||||||||||||||||
| 1.3 | −8.00000 | −27.0000 | 64.0000 | −125.000 | 216.000 | 1017.26 | −512.000 | 729.000 | 1000.00 | |||||||||||||||||||||||||||||||
| 1.4 | −8.00000 | −27.0000 | 64.0000 | −125.000 | 216.000 | 1284.06 | −512.000 | 729.000 | 1000.00 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(3\) | \(1\) |
| \(5\) | \(1\) |
| \(13\) | \(-1\) |
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 | 390.8.a.o | ✓ | 4 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 390.8.a.o | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{4} - 1001T_{7}^{3} - 1861554T_{7}^{2} + 2101981392T_{7} - 229009281696 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(390))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 8)^{4} \)
|
| $3$ |
\( (T + 27)^{4} \)
|
| $5$ |
\( (T + 125)^{4} \)
|
| $7$ |
\( T^{4} + \cdots - 229009281696 \)
|
| $11$ |
\( T^{4} + \cdots + 207950782605696 \)
|
| $13$ |
\( (T - 2197)^{4} \)
|
| $17$ |
\( T^{4} + \cdots - 27\!\cdots\!04 \)
|
| $19$ |
\( T^{4} + \cdots + 90\!\cdots\!44 \)
|
| $23$ |
\( T^{4} + \cdots + 30\!\cdots\!00 \)
|
| $29$ |
\( T^{4} + \cdots + 15\!\cdots\!72 \)
|
| $31$ |
\( T^{4} + \cdots - 15\!\cdots\!52 \)
|
| $37$ |
\( T^{4} + \cdots - 31\!\cdots\!12 \)
|
| $41$ |
\( T^{4} + \cdots - 10\!\cdots\!80 \)
|
| $43$ |
\( T^{4} + \cdots + 97\!\cdots\!56 \)
|
| $47$ |
\( T^{4} + \cdots - 85\!\cdots\!24 \)
|
| $53$ |
\( T^{4} + \cdots + 10\!\cdots\!60 \)
|
| $59$ |
\( T^{4} + \cdots - 92\!\cdots\!80 \)
|
| $61$ |
\( T^{4} + \cdots - 42\!\cdots\!92 \)
|
| $67$ |
\( T^{4} + \cdots - 34\!\cdots\!28 \)
|
| $71$ |
\( T^{4} + \cdots - 11\!\cdots\!64 \)
|
| $73$ |
\( T^{4} + \cdots + 25\!\cdots\!96 \)
|
| $79$ |
\( T^{4} + \cdots - 26\!\cdots\!08 \)
|
| $83$ |
\( T^{4} + \cdots - 12\!\cdots\!28 \)
|
| $89$ |
\( T^{4} + \cdots + 96\!\cdots\!52 \)
|
| $97$ |
\( T^{4} + \cdots + 13\!\cdots\!40 \)
|
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