Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1,118,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, 118, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1.1");
S:= CuspForms(chi, 118);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1 \) |
| Weight: | \( k \) | \(=\) | \( 118 \) |
| 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: | \(86.6887159558\) |
| Analytic rank: | \(1\) |
| 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: |
\( x^{9} - 4 x^{8} + \cdots - 93\!\cdots\!48 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | multiple of \( 2^{151}\cdot 3^{56}\cdot 5^{18}\cdot 7^{7}\cdot 11^{4}\cdot 13^{4}\cdot 17^{2}\cdot 19\cdot 23\cdot 29^{2} \) |
| 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
\( x^{9} - 4 x^{8} + \cdots - 93\!\cdots\!48 \)
:
| \(\beta_{1}\) | \(=\) |
\( 288\nu - 128 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 15\!\cdots\!87 \nu^{8} + \cdots + 12\!\cdots\!40 ) / 21\!\cdots\!16 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 25\!\cdots\!87 \nu^{8} + \cdots - 27\!\cdots\!24 ) / 10\!\cdots\!08 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 21\!\cdots\!11 \nu^{8} + \cdots + 27\!\cdots\!96 ) / 62\!\cdots\!00 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 82\!\cdots\!23 \nu^{8} + \cdots + 54\!\cdots\!28 ) / 15\!\cdots\!00 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 83\!\cdots\!21 \nu^{8} + \cdots + 16\!\cdots\!56 ) / 62\!\cdots\!00 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 29\!\cdots\!93 \nu^{8} + \cdots + 43\!\cdots\!48 ) / 62\!\cdots\!00 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 75\!\cdots\!27 \nu^{8} + \cdots + 15\!\cdots\!72 ) / 62\!\cdots\!00 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 128 ) / 288 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + 3291802\beta_{2} - 31697913853565898\beta _1 + 241647385072183322446125411887407104 ) / 82944 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( \beta_{7} - 323 \beta_{6} - 348895 \beta_{5} - 13791262570 \beta_{4} + \cdots - 76\!\cdots\!48 ) / 23887872 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 2935952636928 \beta_{8} + \cdots + 31\!\cdots\!28 ) / 214990848 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 16\!\cdots\!80 \beta_{8} + \cdots - 25\!\cdots\!44 ) / 967458816 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 13\!\cdots\!80 \beta_{8} + \cdots + 80\!\cdots\!16 ) / 967458816 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 52\!\cdots\!80 \beta_{8} + \cdots - 49\!\cdots\!68 ) / 1934917632 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( 65\!\cdots\!76 \beta_{8} + \cdots + 32\!\cdots\!24 ) / 644972544 \)
|
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 |
|
−7.14121e17 | 1.43031e28 | 3.43815e35 | 7.63780e40 | −1.02141e46 | −4.21519e49 | −1.26872e53 | 1.38022e56 | −5.45431e58 | |||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −6.09583e17 | −4.19687e27 | 2.05438e35 | −4.17042e40 | 2.55834e45 | 1.99165e49 | −2.39470e52 | −4.89422e55 | 2.54222e58 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −3.14621e17 | −1.19854e28 | −6.71671e34 | 1.37379e41 | 3.77084e45 | −3.15157e49 | 7.34076e52 | 7.70927e55 | −4.32224e58 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.76103e17 | 5.15707e27 | −1.35141e35 | −8.41896e40 | −9.08178e44 | −3.65434e49 | 5.30590e52 | −3.99606e55 | 1.48261e58 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −1.27600e17 | 8.60094e27 | −1.49872e35 | 5.94769e40 | −1.09748e45 | 4.46500e49 | 4.03248e52 | 7.42032e54 | −7.58924e57 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 2.06254e17 | −1.02253e28 | −1.23613e35 | −6.31988e40 | −2.10900e45 | 1.11715e49 | −5.97654e52 | 3.80005e55 | −1.30350e58 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 4.44230e17 | 2.03993e27 | 3.11870e34 | 7.97097e40 | 9.06197e44 | −1.30568e49 | −5.99562e52 | −6.23946e55 | 3.54095e58 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 5.65415e17 | 1.40368e28 | 1.53541e35 | −1.09678e41 | 7.93660e45 | 1.38759e48 | −7.13120e51 | 1.30475e56 | −6.20137e58 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 7.66546e17 | −7.42736e27 | 4.21439e35 | −1.57195e40 | −5.69341e45 | 1.22056e49 | 1.95688e53 | −1.13902e55 | −1.20497e58 | ||||||||||||||||||||||||||||||||||||||||||||||
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 | 1.118.a.a | ✓ | 9 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1.118.a.a | ✓ | 9 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace is the entire newspace \(S_{118}^{\mathrm{new}}(\Gamma_0(1))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{9} + \cdots + 12\!\cdots\!88 \)
|
| $3$ |
\( T^{9} + \cdots - 69\!\cdots\!16 \)
|
| $5$ |
\( T^{9} + \cdots + 19\!\cdots\!00 \)
|
| $7$ |
\( T^{9} + \cdots - 10\!\cdots\!72 \)
|
| $11$ |
\( T^{9} + \cdots + 24\!\cdots\!08 \)
|
| $13$ |
\( T^{9} + \cdots + 11\!\cdots\!64 \)
|
| $17$ |
\( T^{9} + \cdots + 57\!\cdots\!08 \)
|
| $19$ |
\( T^{9} + \cdots - 25\!\cdots\!00 \)
|
| $23$ |
\( T^{9} + \cdots + 57\!\cdots\!44 \)
|
| $29$ |
\( T^{9} + \cdots + 29\!\cdots\!00 \)
|
| $31$ |
\( T^{9} + \cdots + 17\!\cdots\!48 \)
|
| $37$ |
\( T^{9} + \cdots - 67\!\cdots\!32 \)
|
| $41$ |
\( T^{9} + \cdots - 74\!\cdots\!32 \)
|
| $43$ |
\( T^{9} + \cdots + 63\!\cdots\!04 \)
|
| $47$ |
\( T^{9} + \cdots + 69\!\cdots\!48 \)
|
| $53$ |
\( T^{9} + \cdots - 14\!\cdots\!16 \)
|
| $59$ |
\( T^{9} + \cdots + 30\!\cdots\!00 \)
|
| $61$ |
\( T^{9} + \cdots - 42\!\cdots\!92 \)
|
| $67$ |
\( T^{9} + \cdots + 22\!\cdots\!08 \)
|
| $71$ |
\( T^{9} + \cdots - 35\!\cdots\!72 \)
|
| $73$ |
\( T^{9} + \cdots + 69\!\cdots\!44 \)
|
| $79$ |
\( T^{9} + \cdots - 12\!\cdots\!00 \)
|
| $83$ |
\( T^{9} + \cdots - 86\!\cdots\!76 \)
|
| $89$ |
\( T^{9} + \cdots + 31\!\cdots\!00 \)
|
| $97$ |
\( T^{9} + \cdots + 16\!\cdots\!48 \)
|
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