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: | \(0\) |
| 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: |
\( x^{4} - x^{3} - 371904x^{2} + 29707868x + 15171428160 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{17}]\) |
| Coefficient ring index: | \( 2^{4}\cdot 3^{2} \) |
| 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
\( x^{4} - x^{3} - 371904x^{2} + 29707868x + 15171428160 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -31\nu^{3} + 6013\nu^{2} + 6987102\nu - 1801266824 ) / 2687944 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -27\nu^{3} - 4053\nu^{2} + 9940944\nu + 157173968 ) / 191996 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 48\nu^{3} + 23205\nu^{2} - 8184987\nu - 3256713026 ) / 671986 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + \beta_{2} - 6\beta _1 + 7 ) / 24 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 415\beta_{3} - 81\beta_{2} + 3558\beta _1 + 4461881 ) / 24 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 305887\beta_{3} + 209679\beta_{2} - 2743194\beta _1 - 527490439 ) / 24 \)
|
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 | −1134.12 | 512.000 | 729.000 | −1000.00 | ||||||||||||||||||||||||||||||
| 1.2 | 8.00000 | 27.0000 | 64.0000 | −125.000 | 216.000 | 154.514 | 512.000 | 729.000 | −1000.00 | |||||||||||||||||||||||||||||||
| 1.3 | 8.00000 | 27.0000 | 64.0000 | −125.000 | 216.000 | 406.601 | 512.000 | 729.000 | −1000.00 | |||||||||||||||||||||||||||||||
| 1.4 | 8.00000 | 27.0000 | 64.0000 | −125.000 | 216.000 | 1132.01 | 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.t | ✓ | 4 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 390.8.a.t | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{4} - 559T_{7}^{3} - 1222194T_{7}^{2} + 720510328T_{7} - 80657434816 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(390))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 8)^{4} \)
|
| $3$ |
\( (T - 27)^{4} \)
|
| $5$ |
\( (T + 125)^{4} \)
|
| $7$ |
\( T^{4} + \cdots - 80657434816 \)
|
| $11$ |
\( T^{4} + \cdots + 504802793071616 \)
|
| $13$ |
\( (T + 2197)^{4} \)
|
| $17$ |
\( T^{4} + \cdots + 12\!\cdots\!96 \)
|
| $19$ |
\( T^{4} + \cdots + 12\!\cdots\!64 \)
|
| $23$ |
\( T^{4} + \cdots + 29\!\cdots\!00 \)
|
| $29$ |
\( T^{4} + \cdots - 41\!\cdots\!28 \)
|
| $31$ |
\( T^{4} + \cdots - 48\!\cdots\!12 \)
|
| $37$ |
\( T^{4} + \cdots + 23\!\cdots\!08 \)
|
| $41$ |
\( T^{4} + \cdots + 88\!\cdots\!80 \)
|
| $43$ |
\( T^{4} + \cdots + 30\!\cdots\!36 \)
|
| $47$ |
\( T^{4} + \cdots - 18\!\cdots\!24 \)
|
| $53$ |
\( T^{4} + \cdots + 19\!\cdots\!80 \)
|
| $59$ |
\( T^{4} + \cdots - 28\!\cdots\!00 \)
|
| $61$ |
\( T^{4} + \cdots - 23\!\cdots\!92 \)
|
| $67$ |
\( T^{4} + \cdots + 11\!\cdots\!12 \)
|
| $71$ |
\( T^{4} + \cdots + 24\!\cdots\!76 \)
|
| $73$ |
\( T^{4} + \cdots + 11\!\cdots\!96 \)
|
| $79$ |
\( T^{4} + \cdots - 21\!\cdots\!28 \)
|
| $83$ |
\( T^{4} + \cdots - 50\!\cdots\!48 \)
|
| $89$ |
\( T^{4} + \cdots + 21\!\cdots\!12 \)
|
| $97$ |
\( T^{4} + \cdots - 24\!\cdots\!80 \)
|
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