Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1,120,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, 120, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1.1");
S:= CuspForms(chi, 120);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1 \) |
| Weight: | \( k \) | \(=\) | \( 120 \) |
| 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: | \(89.6776908760\) |
| Analytic rank: | \(0\) |
| Dimension: | \(10\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{10} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{10} - 5 x^{9} + \cdots + 23\!\cdots\!68 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | multiple of \( 2^{171}\cdot 3^{61}\cdot 5^{22}\cdot 7^{9}\cdot 11^{6}\cdot 13^{3}\cdot 17^{4} \) |
| 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 \(1,\beta_1,\ldots,\beta_{9}\) 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
\( x^{10} - 5 x^{9} + \cdots + 23\!\cdots\!68 \)
:
| \(\beta_{1}\) | \(=\) |
\( 24\nu - 12 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 48\!\cdots\!81 \nu^{9} + \cdots + 20\!\cdots\!04 ) / 48\!\cdots\!44 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 34\!\cdots\!41 \nu^{9} + \cdots - 72\!\cdots\!56 ) / 80\!\cdots\!24 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 47\!\cdots\!59 \nu^{9} + \cdots + 26\!\cdots\!72 ) / 37\!\cdots\!00 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 20\!\cdots\!87 \nu^{9} + \cdots + 49\!\cdots\!96 ) / 43\!\cdots\!00 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 70\!\cdots\!99 \nu^{9} + \cdots + 14\!\cdots\!08 ) / 15\!\cdots\!00 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 76\!\cdots\!23 \nu^{9} + \cdots + 69\!\cdots\!84 ) / 12\!\cdots\!00 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 16\!\cdots\!21 \nu^{9} + \cdots + 12\!\cdots\!32 ) / 10\!\cdots\!00 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 49\!\cdots\!01 \nu^{9} + \cdots + 15\!\cdots\!08 ) / 37\!\cdots\!00 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 12 ) / 24 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} - 4314966\beta_{2} + 216035480204442455\beta _1 + 1083421825434706698581133759471027360 ) / 576 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( \beta_{7} - 194 \beta_{6} + 4418884 \beta_{5} + 285635101119 \beta_{4} + \cdots + 23\!\cdots\!04 ) / 13824 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( - 27276333880 \beta_{9} + 39762557716528 \beta_{8} + \cdots + 24\!\cdots\!12 ) / 41472 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 30\!\cdots\!20 \beta_{9} + \cdots + 71\!\cdots\!24 ) / 62208 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( - 18\!\cdots\!60 \beta_{9} + \cdots + 13\!\cdots\!60 ) / 62208 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 48\!\cdots\!40 \beta_{9} + \cdots + 80\!\cdots\!76 ) / 124416 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( - 80\!\cdots\!00 \beta_{9} + \cdots + 10\!\cdots\!56 ) / 124416 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( 70\!\cdots\!80 \beta_{9} + \cdots + 10\!\cdots\!52 ) / 31104 \)
|
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 |
|
−1.59803e18 | −3.81224e28 | 1.88909e36 | 5.86174e40 | 6.09207e46 | −2.05200e50 | −1.95675e54 | 8.54313e56 | −9.36724e58 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.16148e18 | 2.32093e28 | 6.84414e35 | −4.71757e41 | −2.69571e46 | −1.44318e49 | −2.29977e52 | −6.03321e55 | 5.47935e59 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −9.92540e17 | 9.16621e27 | 3.20523e35 | 6.42021e41 | −9.09783e45 | 1.00631e50 | 3.41525e53 | −5.14984e56 | −6.37232e59 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −4.25584e17 | −2.69872e28 | −4.83493e35 | −2.34399e41 | 1.14853e46 | 5.11468e49 | 4.88615e53 | 1.29303e56 | 9.97565e58 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 9.66294e16 | 4.03037e28 | −6.55277e35 | −8.26042e40 | 3.89453e45 | −7.51518e49 | −1.27540e53 | 1.02539e57 | −7.98200e57 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 3.14287e17 | −2.85616e27 | −5.65837e35 | 3.01555e41 | −8.97656e44 | −2.21350e50 | −3.86715e53 | −5.90846e56 | 9.47749e58 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 8.58138e17 | 9.54518e27 | 7.17860e34 | −1.94208e41 | 8.19107e45 | 3.78566e50 | −5.08728e53 | −5.07893e56 | −1.66657e59 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 1.04139e18 | −4.57629e28 | 4.19888e35 | 5.48255e41 | −4.76572e46 | 1.41978e50 | −2.54857e53 | 1.49524e57 | 5.70949e59 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 1.25784e18 | −1.09661e28 | 9.17555e35 | −6.18260e41 | −1.37937e46 | −3.09107e50 | 3.18161e53 | −4.78748e56 | −7.77674e59 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 1.52927e18 | 3.29001e28 | 1.67406e36 | 6.57038e41 | 5.03133e46 | −6.63364e49 | 1.54372e54 | 4.83416e56 | 1.00479e60 | |||||||||||||||||||||||||||||||||||||||||||||||||
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 | 1.120.a.a | ✓ | 10 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1.120.a.a | ✓ | 10 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace is the entire newspace \(S_{120}^{\mathrm{new}}(\Gamma_0(1))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{10} + \cdots + 40\!\cdots\!24 \)
|
| $3$ |
\( T^{10} + \cdots - 39\!\cdots\!24 \)
|
| $5$ |
\( T^{10} + \cdots - 44\!\cdots\!00 \)
|
| $7$ |
\( T^{10} + \cdots + 27\!\cdots\!24 \)
|
| $11$ |
\( T^{10} + \cdots - 38\!\cdots\!76 \)
|
| $13$ |
\( T^{10} + \cdots - 12\!\cdots\!24 \)
|
| $17$ |
\( T^{10} + \cdots + 11\!\cdots\!24 \)
|
| $19$ |
\( T^{10} + \cdots - 96\!\cdots\!00 \)
|
| $23$ |
\( T^{10} + \cdots + 19\!\cdots\!76 \)
|
| $29$ |
\( T^{10} + \cdots + 60\!\cdots\!00 \)
|
| $31$ |
\( T^{10} + \cdots + 37\!\cdots\!24 \)
|
| $37$ |
\( T^{10} + \cdots - 53\!\cdots\!76 \)
|
| $41$ |
\( T^{10} + \cdots + 85\!\cdots\!24 \)
|
| $43$ |
\( T^{10} + \cdots + 23\!\cdots\!76 \)
|
| $47$ |
\( T^{10} + \cdots + 13\!\cdots\!24 \)
|
| $53$ |
\( T^{10} + \cdots - 15\!\cdots\!24 \)
|
| $59$ |
\( T^{10} + \cdots + 42\!\cdots\!00 \)
|
| $61$ |
\( T^{10} + \cdots - 32\!\cdots\!76 \)
|
| $67$ |
\( T^{10} + \cdots + 45\!\cdots\!24 \)
|
| $71$ |
\( T^{10} + \cdots + 86\!\cdots\!24 \)
|
| $73$ |
\( T^{10} + \cdots - 10\!\cdots\!24 \)
|
| $79$ |
\( T^{10} + \cdots + 51\!\cdots\!00 \)
|
| $83$ |
\( T^{10} + \cdots + 18\!\cdots\!76 \)
|
| $89$ |
\( T^{10} + \cdots - 15\!\cdots\!00 \)
|
| $97$ |
\( T^{10} + \cdots + 86\!\cdots\!24 \)
|
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