Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1,116,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, 116, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1.1");
S:= CuspForms(chi, 116);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1 \) |
| Weight: | \( k \) | \(=\) | \( 116 \) |
| 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: | \(83.7504016273\) |
| Analytic rank: | \(0\) |
| Dimension: | \(9\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{9} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{9} - 2 x^{8} + \cdots + 17\!\cdots\!50 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | multiple of \( 2^{142}\cdot 3^{52}\cdot 5^{17}\cdot 7^{8}\cdot 11^{3}\cdot 13^{3}\cdot 17\cdot 19^{3}\cdot 23^{3}\cdot 29^{2} \) |
| 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 \(1,\beta_1,\ldots,\beta_{8}\) 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
\( x^{9} - 2 x^{8} + \cdots + 17\!\cdots\!50 \)
:
| \(\beta_{1}\) | \(=\) |
\( 24\nu - 5 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 18\!\cdots\!49 \nu^{8} + \cdots + 90\!\cdots\!30 ) / 40\!\cdots\!84 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 90\!\cdots\!97 \nu^{8} + \cdots - 20\!\cdots\!66 ) / 40\!\cdots\!84 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 28\!\cdots\!31 \nu^{8} + \cdots + 15\!\cdots\!50 ) / 31\!\cdots\!00 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 23\!\cdots\!11 \nu^{8} + \cdots + 13\!\cdots\!50 ) / 62\!\cdots\!00 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 38\!\cdots\!21 \nu^{8} + \cdots - 42\!\cdots\!50 ) / 62\!\cdots\!00 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 20\!\cdots\!19 \nu^{8} + \cdots - 98\!\cdots\!50 ) / 12\!\cdots\!00 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 37\!\cdots\!91 \nu^{8} + \cdots + 82\!\cdots\!50 ) / 62\!\cdots\!00 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 5 ) / 24 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} - 484553\beta_{2} - 18073717211147679\beta _1 + 53001739878028145233323179512730439 ) / 576 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( \beta_{7} + 413 \beta_{6} + 664275 \beta_{5} + 189393293777 \beta_{4} + \cdots - 95\!\cdots\!86 ) / 13824 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( - 10412963909952 \beta_{8} + \cdots + 58\!\cdots\!22 ) / 41472 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 30\!\cdots\!40 \beta_{8} + \cdots - 20\!\cdots\!41 ) / 7776 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( - 17\!\cdots\!20 \beta_{8} + \cdots + 53\!\cdots\!14 ) / 20736 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 22\!\cdots\!40 \beta_{8} + \cdots - 27\!\cdots\!58 ) / 41472 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( - 26\!\cdots\!44 \beta_{8} + \cdots + 63\!\cdots\!02 ) / 124416 \)
|
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 |
|
−3.55538e17 | −3.54906e26 | 8.48689e34 | 1.00650e40 | 1.26183e44 | 6.58681e47 | −1.54056e52 | −7.26915e54 | −3.57850e57 | |||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.38832e17 | 4.64741e27 | 1.55022e34 | −1.89028e40 | −1.10995e45 | 5.62740e48 | 6.21827e51 | 1.42033e55 | 4.51459e57 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −2.28920e17 | −5.05419e27 | 1.08661e34 | −2.40937e40 | 1.15700e45 | −2.38510e48 | 7.02151e51 | 1.81497e55 | 5.51552e57 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.48197e17 | 7.29391e26 | −1.95761e34 | 6.05511e39 | −1.08093e44 | −4.10385e48 | 9.05697e51 | −6.86309e54 | −8.97348e56 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −1.62458e15 | −2.61982e27 | −4.15357e34 | 1.47212e40 | 4.25609e42 | 6.19961e48 | 1.34960e50 | −5.31663e53 | −2.39156e55 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.17803e17 | 5.60840e26 | −2.76608e34 | −2.33935e40 | 6.60688e43 | −1.66757e47 | −8.15189e51 | −7.08056e54 | −2.75583e57 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 1.25417e17 | 4.55926e27 | −2.58089e34 | 2.05215e40 | 5.71809e44 | −2.84362e48 | −8.44651e51 | 1.33917e55 | 2.57375e57 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 2.69631e17 | −3.76423e27 | 3.11624e34 | 5.22690e39 | −1.01495e45 | −4.51498e48 | −2.79767e51 | 6.77433e54 | 1.40933e57 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 3.44079e17 | 1.64109e27 | 7.68520e34 | 3.84023e38 | 5.64664e44 | 3.35173e48 | 1.21507e52 | −4.70193e54 | 1.32134e56 | ||||||||||||||||||||||||||||||||||||||||||||||
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 | 1.116.a.a | ✓ | 9 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1.116.a.a | ✓ | 9 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace is the entire newspace \(S_{116}^{\mathrm{new}}(\Gamma_0(1))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{9} + \cdots + 64\!\cdots\!04 \)
|
| $3$ |
\( T^{9} + \cdots - 25\!\cdots\!72 \)
|
| $5$ |
\( T^{9} + \cdots + 39\!\cdots\!00 \)
|
| $7$ |
\( T^{9} + \cdots + 16\!\cdots\!04 \)
|
| $11$ |
\( T^{9} + \cdots - 17\!\cdots\!12 \)
|
| $13$ |
\( T^{9} + \cdots + 84\!\cdots\!28 \)
|
| $17$ |
\( T^{9} + \cdots - 18\!\cdots\!96 \)
|
| $19$ |
\( T^{9} + \cdots + 10\!\cdots\!00 \)
|
| $23$ |
\( T^{9} + \cdots - 11\!\cdots\!72 \)
|
| $29$ |
\( T^{9} + \cdots - 88\!\cdots\!00 \)
|
| $31$ |
\( T^{9} + \cdots + 29\!\cdots\!88 \)
|
| $37$ |
\( T^{9} + \cdots - 13\!\cdots\!96 \)
|
| $41$ |
\( T^{9} + \cdots - 12\!\cdots\!12 \)
|
| $43$ |
\( T^{9} + \cdots + 41\!\cdots\!28 \)
|
| $47$ |
\( T^{9} + \cdots - 76\!\cdots\!96 \)
|
| $53$ |
\( T^{9} + \cdots - 18\!\cdots\!72 \)
|
| $59$ |
\( T^{9} + \cdots - 81\!\cdots\!00 \)
|
| $61$ |
\( T^{9} + \cdots + 22\!\cdots\!88 \)
|
| $67$ |
\( T^{9} + \cdots + 14\!\cdots\!04 \)
|
| $71$ |
\( T^{9} + \cdots - 16\!\cdots\!12 \)
|
| $73$ |
\( T^{9} + \cdots - 94\!\cdots\!72 \)
|
| $79$ |
\( T^{9} + \cdots + 25\!\cdots\!00 \)
|
| $83$ |
\( T^{9} + \cdots - 21\!\cdots\!72 \)
|
| $89$ |
\( T^{9} + \cdots + 10\!\cdots\!00 \)
|
| $97$ |
\( T^{9} + \cdots - 84\!\cdots\!96 \)
|
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