Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1,88,Mod(1,1)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1, 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("1.1");
S:= CuspForms(chi, 88);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1 \) |
| Weight: | \( k \) | \(=\) | \( 88 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1.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: | \(47.9333631461\) |
| Analytic rank: | \(0\) |
| Dimension: | \(7\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{7} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{7} - x^{6} + \cdots - 79\!\cdots\!56 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | multiple of \( 2^{76}\cdot 3^{35}\cdot 5^{8}\cdot 7^{4}\cdot 11^{2}\cdot 13\cdot 17\cdot 29^{2} \) |
| 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 \(1,\beta_1,\ldots,\beta_{6}\) 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
\( x^{7} - x^{6} + \cdots - 79\!\cdots\!56 \)
:
| \(\beta_{1}\) | \(=\) |
\( 72\nu - 10 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 99\!\cdots\!67 \nu^{6} + \cdots - 35\!\cdots\!04 ) / 35\!\cdots\!64 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 54\!\cdots\!43 \nu^{6} + \cdots - 91\!\cdots\!28 ) / 35\!\cdots\!64 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 82\!\cdots\!71 \nu^{6} + \cdots + 61\!\cdots\!92 ) / 22\!\cdots\!00 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 15\!\cdots\!49 \nu^{6} + \cdots + 15\!\cdots\!48 ) / 89\!\cdots\!00 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 58\!\cdots\!89 \nu^{6} + \cdots + 20\!\cdots\!28 ) / 12\!\cdots\!00 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 10 ) / 72 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} - 55029\beta_{2} - 1925402030686\beta _1 + 201756781109916158185107823 ) / 5184 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( - 9 \beta_{6} + 21600 \beta_{5} - 2115827 \beta_{4} - 569859075196 \beta_{3} + \cdots - 48\!\cdots\!61 ) / 46656 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( - 5495209853591 \beta_{6} + \cdots + 10\!\cdots\!67 ) / 419904 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 16\!\cdots\!17 \beta_{6} + \cdots - 22\!\cdots\!75 ) / 139968 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( - 65\!\cdots\!19 \beta_{6} + \cdots + 75\!\cdots\!47 ) / 419904 \)
|
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 |
|
−1.89989e13 | 2.95637e20 | 2.06214e26 | −5.49540e29 | −5.61676e33 | −2.67695e35 | −9.77905e38 | −2.35857e41 | 1.04406e43 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.28058e13 | −1.04195e21 | 9.24499e24 | 2.09154e30 | 1.33429e34 | −6.15897e36 | 1.86321e39 | 7.62399e41 | −2.67838e43 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | −2.47499e12 | −1.66110e20 | −1.48617e26 | −1.54336e30 | 4.11120e32 | 1.02283e37 | 7.50812e38 | −2.95665e41 | 3.81980e42 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | 9.09417e11 | 7.58553e20 | −1.53915e26 | 1.79518e30 | 6.89842e32 | −6.36454e36 | −2.80699e38 | 2.52145e41 | 1.63257e42 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 1.28460e13 | −4.56067e20 | 1.02776e25 | −3.44534e30 | −5.85865e33 | −1.04838e37 | −1.85580e39 | −1.15260e41 | −4.42589e43 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.68092e13 | −3.69059e20 | 1.27807e26 | 4.92178e30 | −6.20358e33 | 5.74193e36 | −4.52765e38 | −1.87054e41 | 8.27312e43 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 2.19120e13 | 9.03577e20 | 3.25393e26 | −2.93989e30 | 1.97992e34 | 7.30934e36 | 3.73929e39 | 4.93193e41 | −6.44188e43 | ||||||||||||||||||||||||||||||||||||||||
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 | 1.88.a.a | ✓ | 7 |
| 3.b | odd | 2 | 1 | 9.88.a.b | 7 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1.88.a.a | ✓ | 7 | 1.a | even | 1 | 1 | trivial |
| 9.88.a.b | 7 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace is the entire newspace \(S_{88}^{\mathrm{new}}(\Gamma_0(1))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} + \cdots + 25\!\cdots\!76 \)
|
| $3$ |
\( T^{7} + \cdots - 59\!\cdots\!52 \)
|
| $5$ |
\( T^{7} + \cdots - 15\!\cdots\!00 \)
|
| $7$ |
\( T^{7} + \cdots - 47\!\cdots\!64 \)
|
| $11$ |
\( T^{7} + \cdots + 28\!\cdots\!12 \)
|
| $13$ |
\( T^{7} + \cdots + 82\!\cdots\!48 \)
|
| $17$ |
\( T^{7} + \cdots + 88\!\cdots\!56 \)
|
| $19$ |
\( T^{7} + \cdots + 40\!\cdots\!00 \)
|
| $23$ |
\( T^{7} + \cdots + 53\!\cdots\!48 \)
|
| $29$ |
\( T^{7} + \cdots - 17\!\cdots\!00 \)
|
| $31$ |
\( T^{7} + \cdots + 90\!\cdots\!92 \)
|
| $37$ |
\( T^{7} + \cdots + 31\!\cdots\!96 \)
|
| $41$ |
\( T^{7} + \cdots - 63\!\cdots\!68 \)
|
| $43$ |
\( T^{7} + \cdots - 53\!\cdots\!52 \)
|
| $47$ |
\( T^{7} + \cdots + 53\!\cdots\!16 \)
|
| $53$ |
\( T^{7} + \cdots - 95\!\cdots\!52 \)
|
| $59$ |
\( T^{7} + \cdots + 69\!\cdots\!00 \)
|
| $61$ |
\( T^{7} + \cdots - 36\!\cdots\!88 \)
|
| $67$ |
\( T^{7} + \cdots - 49\!\cdots\!44 \)
|
| $71$ |
\( T^{7} + \cdots - 28\!\cdots\!48 \)
|
| $73$ |
\( T^{7} + \cdots + 63\!\cdots\!48 \)
|
| $79$ |
\( T^{7} + \cdots + 54\!\cdots\!00 \)
|
| $83$ |
\( T^{7} + \cdots - 16\!\cdots\!52 \)
|
| $89$ |
\( T^{7} + \cdots + 40\!\cdots\!00 \)
|
| $97$ |
\( T^{7} + \cdots + 10\!\cdots\!16 \)
|
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