Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2,88,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, 88, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2.1");
S:= CuspForms(chi, 88);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2 \) |
| Weight: | \( k \) | \(=\) | \( 88 \) |
| 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: | \(95.8667262922\) |
| 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: |
\( x^{3} - 11473904362186221800312196301729x - 156905659743614387346100645850205598702591560 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{23}\cdot 3^{9}\cdot 5^{3}\cdot 7\cdot 11\cdot 29 \) |
| 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
\( x^{3} - 11473904362186221800312196301729x - 156905659743614387346100645850205598702591560 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 77760\nu^{2} + 116872286813505034560\nu - 594807202135733738128184256281631360 ) / 926145375996031 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 68041581120 \nu^{2} + \cdots + 52\!\cdots\!20 ) / 926145375996031 \)
|
| \(\nu\) | \(=\) |
\( ( 3\beta_{2} + 2625061\beta_1 ) / 889026969600 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( - 22211633904531 \beta_{2} + \cdots + 33\!\cdots\!00 ) / 4379443200 \)
|
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 |
|
8.79609e12 | −8.60735e20 | 7.73713e25 | −5.73526e28 | −7.57111e33 | 1.36940e36 | 6.80565e38 | 4.17607e41 | −5.04479e41 | |||||||||||||||||||||||||||
| 1.2 | 8.79609e12 | 1.94339e18 | 7.73713e25 | −1.48628e30 | 1.70942e31 | 3.06439e35 | 6.80565e38 | −3.23254e41 | −1.30735e43 | ||||||||||||||||||||||||||||
| 1.3 | 8.79609e12 | 5.36528e20 | 7.73713e25 | 2.66105e30 | 4.71935e33 | −4.57229e36 | 6.80565e38 | −3.53951e40 | 2.34068e43 | ||||||||||||||||||||||||||||
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 | 2.88.a.a | ✓ | 3 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2.88.a.a | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{3} + \cdots + 89\!\cdots\!48 \)
acting on \(S_{88}^{\mathrm{new}}(\Gamma_0(2))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 8796093022208)^{3} \)
|
| $3$ |
\( T^{3} + \cdots + 89\!\cdots\!48 \)
|
| $5$ |
\( T^{3} + \cdots - 22\!\cdots\!00 \)
|
| $7$ |
\( T^{3} + \cdots + 19\!\cdots\!36 \)
|
| $11$ |
\( T^{3} + \cdots - 22\!\cdots\!08 \)
|
| $13$ |
\( T^{3} + \cdots + 67\!\cdots\!08 \)
|
| $17$ |
\( T^{3} + \cdots + 91\!\cdots\!76 \)
|
| $19$ |
\( T^{3} + \cdots - 54\!\cdots\!00 \)
|
| $23$ |
\( T^{3} + \cdots - 28\!\cdots\!52 \)
|
| $29$ |
\( T^{3} + \cdots + 58\!\cdots\!00 \)
|
| $31$ |
\( T^{3} + \cdots - 26\!\cdots\!48 \)
|
| $37$ |
\( T^{3} + \cdots + 47\!\cdots\!36 \)
|
| $41$ |
\( T^{3} + \cdots + 10\!\cdots\!52 \)
|
| $43$ |
\( T^{3} + \cdots - 34\!\cdots\!32 \)
|
| $47$ |
\( T^{3} + \cdots - 85\!\cdots\!44 \)
|
| $53$ |
\( T^{3} + \cdots - 12\!\cdots\!92 \)
|
| $59$ |
\( T^{3} + \cdots - 12\!\cdots\!00 \)
|
| $61$ |
\( T^{3} + \cdots + 40\!\cdots\!12 \)
|
| $67$ |
\( T^{3} + \cdots + 39\!\cdots\!56 \)
|
| $71$ |
\( T^{3} + \cdots - 10\!\cdots\!28 \)
|
| $73$ |
\( T^{3} + \cdots - 14\!\cdots\!92 \)
|
| $79$ |
\( T^{3} + \cdots + 14\!\cdots\!00 \)
|
| $83$ |
\( T^{3} + \cdots - 27\!\cdots\!12 \)
|
| $89$ |
\( T^{3} + \cdots - 54\!\cdots\!00 \)
|
| $97$ |
\( T^{3} + \cdots + 22\!\cdots\!36 \)
|
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