Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1,80,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, 80, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1.1");
S:= CuspForms(chi, 80);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1 \) |
| Weight: | \( k \) | \(=\) | \( 80 \) |
| 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: | \(39.5237048722\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 3 x^{5} + \cdots - 76\!\cdots\!88 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | multiple of \( 2^{54}\cdot 3^{24}\cdot 5^{6}\cdot 7^{3}\cdot 11\cdot 13^{2} \) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 3 x^{5} + \cdots - 76\!\cdots\!88 \)
:
| \(\beta_{1}\) | \(=\) |
\( 24\nu - 12 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 597229958995063 \nu^{5} + \cdots + 70\!\cdots\!24 ) / 13\!\cdots\!68 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 26\!\cdots\!35 \nu^{5} + \cdots - 56\!\cdots\!04 ) / 68\!\cdots\!04 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 23\!\cdots\!29 \nu^{5} + \cdots + 68\!\cdots\!76 ) / 35\!\cdots\!40 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 53\!\cdots\!83 \nu^{5} + \cdots - 85\!\cdots\!08 ) / 82\!\cdots\!80 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 12 ) / 24 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} - 8640\beta_{2} + 109101065932\beta _1 + 862745281906774673011680 ) / 576 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( - 369 \beta_{5} - 10406371 \beta_{4} - 15040375037 \beta_{3} + \cdots + 11\!\cdots\!28 ) / 1728 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( - 126301704456581 \beta_{5} + \cdots + 56\!\cdots\!08 ) / 1728 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 28\!\cdots\!01 \beta_{5} + \cdots + 60\!\cdots\!64 ) / 288 \)
|
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 |
|
−1.23625e12 | −2.24416e18 | 9.23858e23 | −5.14707e27 | 2.77435e30 | 4.93307e32 | −3.94854e35 | −4.42334e37 | 6.36308e39 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −8.42585e11 | 1.18388e19 | 1.05486e23 | 5.37236e27 | −9.97521e30 | −2.88361e33 | 4.20430e35 | 9.08882e37 | −4.52667e39 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −4.68329e11 | −9.12223e18 | −3.85131e23 | 5.54574e27 | 4.27221e30 | 1.91401e33 | 4.63456e35 | 3.39455e37 | −2.59723e39 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 2.26554e11 | 2.57159e18 | −5.53136e23 | −2.99197e27 | 5.82605e29 | −6.95219e31 | −2.62259e35 | −4.26565e37 | −6.77842e38 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 1.07427e12 | −9.45275e18 | 5.49591e23 | −5.27758e26 | −1.01548e31 | −3.45278e33 | −5.89470e34 | 4.00848e37 | −5.66955e38 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 1.23026e12 | 8.35089e18 | 9.09069e23 | 3.84167e27 | 1.02737e31 | 3.79445e33 | 3.74744e35 | 2.04677e37 | 4.72625e39 | |||||||||||||||||||||||||||||||||||||
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 | 1.80.a.a | ✓ | 6 |
| 3.b | odd | 2 | 1 | 9.80.a.b | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1.80.a.a | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 9.80.a.b | 6 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace is the entire newspace \(S_{80}^{\mathrm{new}}(\Gamma_0(1))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + \cdots - 14\!\cdots\!56 \)
|
| $3$ |
\( T^{6} + \cdots - 49\!\cdots\!64 \)
|
| $5$ |
\( T^{6} + \cdots - 93\!\cdots\!00 \)
|
| $7$ |
\( T^{6} + \cdots - 24\!\cdots\!56 \)
|
| $11$ |
\( T^{6} + \cdots + 60\!\cdots\!44 \)
|
| $13$ |
\( T^{6} + \cdots + 13\!\cdots\!16 \)
|
| $17$ |
\( T^{6} + \cdots + 31\!\cdots\!44 \)
|
| $19$ |
\( T^{6} + \cdots - 13\!\cdots\!00 \)
|
| $23$ |
\( T^{6} + \cdots + 34\!\cdots\!96 \)
|
| $29$ |
\( T^{6} + \cdots - 48\!\cdots\!00 \)
|
| $31$ |
\( T^{6} + \cdots - 31\!\cdots\!96 \)
|
| $37$ |
\( T^{6} + \cdots + 25\!\cdots\!44 \)
|
| $41$ |
\( T^{6} + \cdots - 29\!\cdots\!16 \)
|
| $43$ |
\( T^{6} + \cdots + 20\!\cdots\!56 \)
|
| $47$ |
\( T^{6} + \cdots - 10\!\cdots\!56 \)
|
| $53$ |
\( T^{6} + \cdots + 38\!\cdots\!36 \)
|
| $59$ |
\( T^{6} + \cdots + 31\!\cdots\!00 \)
|
| $61$ |
\( T^{6} + \cdots - 70\!\cdots\!56 \)
|
| $67$ |
\( T^{6} + \cdots - 14\!\cdots\!56 \)
|
| $71$ |
\( T^{6} + \cdots - 22\!\cdots\!76 \)
|
| $73$ |
\( T^{6} + \cdots + 11\!\cdots\!96 \)
|
| $79$ |
\( T^{6} + \cdots + 49\!\cdots\!00 \)
|
| $83$ |
\( T^{6} + \cdots + 46\!\cdots\!76 \)
|
| $89$ |
\( T^{6} + \cdots - 11\!\cdots\!00 \)
|
| $97$ |
\( T^{6} + \cdots - 98\!\cdots\!56 \)
|
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