Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2,84,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, 84, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2.1");
S:= CuspForms(chi, 84);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2 \) |
| Weight: | \( k \) | \(=\) | \( 84 \) |
| 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: | \(87.2544256533\) |
| Analytic rank: | \(0\) |
| 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: |
\( x^{3} - x^{2} - 287609867501924274375802127400x - 41230865304567060522794640394926417995512500 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{24}\cdot 3^{7}\cdot 5^{3}\cdot 7\cdot 11\cdot 17 \) |
| 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 \(1,\beta_1,\beta_2\) 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
\( x^{3} - x^{2} - 287609867501924274375802127400x - 41230865304567060522794640394926417995512500 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -18\nu^{2} + 8140017811031118\nu + 3451318410023088579170355185100 ) / 52252679285 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 10166454\nu^{2} - 58893267023738651754\nu - 1949314991936272011652225019633331300 ) / 52252679285 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + 564803\beta _1 + 28952985600 ) / 86858956800 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 452223211723951 \beta_{2} + \cdots + 16\!\cdots\!00 ) / 86858956800 \)
|
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 |
|
−2.19902e12 | −1.03294e20 | 4.83570e24 | −4.29409e28 | 2.27146e32 | −1.14250e35 | −1.06338e37 | 6.67886e39 | 9.44280e40 | |||||||||||||||||||||||||||
| 1.2 | −2.19902e12 | −7.32556e17 | 4.83570e24 | −1.23719e29 | 1.61091e30 | 6.61589e34 | −1.06338e37 | −3.99030e39 | 2.72062e41 | ||||||||||||||||||||||||||||
| 1.3 | −2.19902e12 | 2.32406e18 | 4.83570e24 | 1.63642e29 | −5.11067e30 | −8.77702e33 | −1.06338e37 | −3.98544e39 | −3.59853e41 | ||||||||||||||||||||||||||||
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 | 2.84.a.a | ✓ | 3 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2.84.a.a | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{3} + \cdots - 17\!\cdots\!12 \)
acting on \(S_{84}^{\mathrm{new}}(\Gamma_0(2))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 2199023255552)^{3} \)
|
| $3$ |
\( T^{3} + \cdots - 17\!\cdots\!12 \)
|
| $5$ |
\( T^{3} + \cdots - 86\!\cdots\!00 \)
|
| $7$ |
\( T^{3} + \cdots - 66\!\cdots\!64 \)
|
| $11$ |
\( T^{3} + \cdots - 13\!\cdots\!28 \)
|
| $13$ |
\( T^{3} + \cdots - 49\!\cdots\!12 \)
|
| $17$ |
\( T^{3} + \cdots + 25\!\cdots\!96 \)
|
| $19$ |
\( T^{3} + \cdots - 32\!\cdots\!00 \)
|
| $23$ |
\( T^{3} + \cdots + 19\!\cdots\!68 \)
|
| $29$ |
\( T^{3} + \cdots + 19\!\cdots\!00 \)
|
| $31$ |
\( T^{3} + \cdots - 28\!\cdots\!08 \)
|
| $37$ |
\( T^{3} + \cdots - 16\!\cdots\!04 \)
|
| $41$ |
\( T^{3} + \cdots - 27\!\cdots\!28 \)
|
| $43$ |
\( T^{3} + \cdots + 17\!\cdots\!68 \)
|
| $47$ |
\( T^{3} + \cdots - 69\!\cdots\!64 \)
|
| $53$ |
\( T^{3} + \cdots + 12\!\cdots\!48 \)
|
| $59$ |
\( T^{3} + \cdots + 11\!\cdots\!00 \)
|
| $61$ |
\( T^{3} + \cdots + 23\!\cdots\!92 \)
|
| $67$ |
\( T^{3} + \cdots - 28\!\cdots\!24 \)
|
| $71$ |
\( T^{3} + \cdots + 74\!\cdots\!32 \)
|
| $73$ |
\( T^{3} + \cdots - 48\!\cdots\!72 \)
|
| $79$ |
\( T^{3} + \cdots + 40\!\cdots\!00 \)
|
| $83$ |
\( T^{3} + \cdots + 10\!\cdots\!48 \)
|
| $89$ |
\( T^{3} + \cdots + 16\!\cdots\!00 \)
|
| $97$ |
\( T^{3} + \cdots - 51\!\cdots\!84 \)
|
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