Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2,80,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, 80, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2.1");
S:= CuspForms(chi, 80);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2 \) |
| Weight: | \( k \) | \(=\) | \( 80 \) |
| 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: | \(79.0474097443\) |
| Analytic rank: | \(1\) |
| Dimension: | \(3\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{3} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 5157021731103247543589585180x + 141562397820564875200991893221092433132672 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{23}\cdot 3^{9}\cdot 5^{4}\cdot 7\cdot 11 \) |
| 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 \(1,\beta_1,\beta_2\) 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
\( x^{3} - x^{2} - 5157021731103247543589585180x + 141562397820564875200991893221092433132672 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 288\nu^{2} + 10566252004493664\nu - 990148172371827050453201852544 ) / 16696982899 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 788207616\nu^{2} + 36596022868804296296448\nu - 2709869202888734729468024946465988608 ) / 16696982899 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} - 2736832\beta _1 + 153280512000 ) / 459841536000 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( - 36688375015603 \beta_{2} + \cdots + 15\!\cdots\!00 ) / 459841536000 \)
|
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 |
|
5.49756e11 | −8.21312e18 | 3.02231e23 | −6.30299e27 | −4.51521e30 | 3.82944e33 | 1.66153e35 | 1.81857e37 | −3.46510e39 | |||||||||||||||||||||||||||
| 1.2 | 5.49756e11 | −3.97815e18 | 3.02231e23 | 4.89479e27 | −2.18701e30 | −4.89031e32 | 1.66153e35 | −3.34439e37 | 2.69094e39 | ||||||||||||||||||||||||||||
| 1.3 | 5.49756e11 | 7.60629e18 | 3.02231e23 | −2.65106e27 | 4.18160e30 | −1.85699e33 | 1.66153e35 | 8.58611e36 | −1.45744e39 | ||||||||||||||||||||||||||||
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 | 2.80.a.a | ✓ | 3 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2.80.a.a | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{3} + \cdots - 24\!\cdots\!72 \)
acting on \(S_{80}^{\mathrm{new}}(\Gamma_0(2))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 549755813888)^{3} \)
|
| $3$ |
\( T^{3} + \cdots - 24\!\cdots\!72 \)
|
| $5$ |
\( T^{3} + \cdots - 81\!\cdots\!00 \)
|
| $7$ |
\( T^{3} + \cdots - 34\!\cdots\!64 \)
|
| $11$ |
\( T^{3} + \cdots - 57\!\cdots\!48 \)
|
| $13$ |
\( T^{3} + \cdots + 65\!\cdots\!68 \)
|
| $17$ |
\( T^{3} + \cdots + 81\!\cdots\!16 \)
|
| $19$ |
\( T^{3} + \cdots - 11\!\cdots\!00 \)
|
| $23$ |
\( T^{3} + \cdots - 12\!\cdots\!12 \)
|
| $29$ |
\( T^{3} + \cdots + 30\!\cdots\!00 \)
|
| $31$ |
\( T^{3} + \cdots - 58\!\cdots\!68 \)
|
| $37$ |
\( T^{3} + \cdots - 32\!\cdots\!44 \)
|
| $41$ |
\( T^{3} + \cdots - 50\!\cdots\!08 \)
|
| $43$ |
\( T^{3} + \cdots - 67\!\cdots\!32 \)
|
| $47$ |
\( T^{3} + \cdots - 27\!\cdots\!84 \)
|
| $53$ |
\( T^{3} + \cdots - 54\!\cdots\!12 \)
|
| $59$ |
\( T^{3} + \cdots - 11\!\cdots\!00 \)
|
| $61$ |
\( T^{3} + \cdots - 48\!\cdots\!28 \)
|
| $67$ |
\( T^{3} + \cdots + 18\!\cdots\!96 \)
|
| $71$ |
\( T^{3} + \cdots - 75\!\cdots\!08 \)
|
| $73$ |
\( T^{3} + \cdots - 42\!\cdots\!52 \)
|
| $79$ |
\( T^{3} + \cdots - 45\!\cdots\!00 \)
|
| $83$ |
\( T^{3} + \cdots + 55\!\cdots\!08 \)
|
| $89$ |
\( T^{3} + \cdots + 72\!\cdots\!00 \)
|
| $97$ |
\( T^{3} + \cdots - 95\!\cdots\!04 \)
|
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