Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2,86,Mod(1,2)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2, base_ring=CyclotomicField(1))
chi = DirichletCharacter(H, H._module([]))
N = Newforms(chi, 86, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2.1");
S:= CuspForms(chi, 86);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2 \) |
| Weight: | \( k \) | \(=\) | \( 86 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2.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: | \(91.5099153814\) |
| 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: |
\( x^{4} - 2 x^{3} + \cdots + 54\!\cdots\!68 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{48}\cdot 3^{14}\cdot 5^{6}\cdot 7^{2}\cdot 11\cdot 17 \) |
| 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
\( x^{4} - 2 x^{3} + \cdots + 54\!\cdots\!68 \)
:
| \(\beta_{1}\) | \(=\) |
\( 3840\nu - 1920 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 6416865689600 \nu^{3} + \cdots + 55\!\cdots\!20 ) / 84\!\cdots\!87 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 58\!\cdots\!00 \nu^{3} + \cdots - 45\!\cdots\!60 ) / 84\!\cdots\!87 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 1920 ) / 3840 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 30598 \beta_{3} - 27942812599 \beta_{2} + \cdots + 20\!\cdots\!00 ) / 7372800 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( - 13\!\cdots\!37 \beta_{3} + \cdots + 78\!\cdots\!00 ) / 1887436800 \)
|
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 |
|
4.39805e12 | −2.01208e20 | 1.93428e25 | 8.07669e29 | −8.84924e32 | −1.62828e36 | 8.50706e37 | 4.56726e39 | 3.55217e42 | ||||||||||||||||||||||||||||||
| 1.2 | 4.39805e12 | −7.67993e18 | 1.93428e25 | 2.91184e29 | −3.37767e31 | 1.35377e36 | 8.50706e37 | −3.58586e40 | 1.28064e42 | |||||||||||||||||||||||||||||||
| 1.3 | 4.39805e12 | 8.09227e19 | 1.93428e25 | −7.32566e29 | 3.55902e32 | −7.15356e35 | 8.50706e37 | −2.93691e40 | −3.22186e42 | |||||||||||||||||||||||||||||||
| 1.4 | 4.39805e12 | 3.56743e20 | 1.93428e25 | 4.29102e29 | 1.56897e33 | 1.16998e35 | 8.50706e37 | 9.13482e40 | 1.88721e42 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
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 | 2.86.a.b | ✓ | 4 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2.86.a.b | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} + \cdots + 44\!\cdots\!16 \)
acting on \(S_{86}^{\mathrm{new}}(\Gamma_0(2))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 4398046511104)^{4} \)
|
| $3$ |
\( T^{4} + \cdots + 44\!\cdots\!16 \)
|
| $5$ |
\( T^{4} + \cdots - 73\!\cdots\!00 \)
|
| $7$ |
\( T^{4} + \cdots + 18\!\cdots\!96 \)
|
| $11$ |
\( T^{4} + \cdots - 15\!\cdots\!84 \)
|
| $13$ |
\( T^{4} + \cdots - 11\!\cdots\!64 \)
|
| $17$ |
\( T^{4} + \cdots - 20\!\cdots\!84 \)
|
| $19$ |
\( T^{4} + \cdots + 54\!\cdots\!00 \)
|
| $23$ |
\( T^{4} + \cdots + 56\!\cdots\!56 \)
|
| $29$ |
\( T^{4} + \cdots + 39\!\cdots\!00 \)
|
| $31$ |
\( T^{4} + \cdots - 52\!\cdots\!84 \)
|
| $37$ |
\( T^{4} + \cdots + 11\!\cdots\!56 \)
|
| $41$ |
\( T^{4} + \cdots - 35\!\cdots\!84 \)
|
| $43$ |
\( T^{4} + \cdots + 72\!\cdots\!96 \)
|
| $47$ |
\( T^{4} + \cdots + 57\!\cdots\!76 \)
|
| $53$ |
\( T^{4} + \cdots - 48\!\cdots\!84 \)
|
| $59$ |
\( T^{4} + \cdots - 14\!\cdots\!00 \)
|
| $61$ |
\( T^{4} + \cdots + 98\!\cdots\!16 \)
|
| $67$ |
\( T^{4} + \cdots + 38\!\cdots\!16 \)
|
| $71$ |
\( T^{4} + \cdots - 36\!\cdots\!84 \)
|
| $73$ |
\( T^{4} + \cdots + 36\!\cdots\!56 \)
|
| $79$ |
\( T^{4} + \cdots - 16\!\cdots\!00 \)
|
| $83$ |
\( T^{4} + \cdots - 44\!\cdots\!24 \)
|
| $89$ |
\( T^{4} + \cdots + 47\!\cdots\!00 \)
|
| $97$ |
\( T^{4} + \cdots + 13\!\cdots\!76 \)
|
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