Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1,64,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, 64, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1.1");
S:= CuspForms(chi, 64);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1 \) |
| Weight: | \( k \) | \(=\) | \( 64 \) |
| 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: | \(25.1360966918\) |
| Analytic rank: | \(0\) |
| Dimension: | \(5\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{5} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - 2 x^{4} + \cdots - 35\!\cdots\!34 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{37}\cdot 3^{17}\cdot 5^{3}\cdot 7^{2}\cdot 13 \) |
| 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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) 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
\( x^{5} - 2 x^{4} + \cdots - 35\!\cdots\!34 \)
:
| \(\beta_{1}\) | \(=\) |
\( 72\nu - 29 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 98361 \nu^{4} + 1875869661375 \nu^{3} + \cdots + 22\!\cdots\!18 ) / 24\!\cdots\!16 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 65803509 \nu^{4} + \cdots - 41\!\cdots\!66 ) / 24\!\cdots\!16 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 2665776547737 \nu^{4} + \cdots + 58\!\cdots\!78 ) / 60\!\cdots\!40 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 29 ) / 72 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + 669\beta_{2} - 12164792\beta _1 + 10567661868921261571 ) / 5184 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( - 18215 \beta_{4} + 9690671 \beta_{3} + 203947988361 \beta_{2} + \cdots - 20\!\cdots\!17 ) / 5832 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 2779066165000 \beta_{4} + \cdots + 35\!\cdots\!37 ) / 46656 \)
|
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 |
|
−4.68363e9 | −3.45751e14 | 1.27130e19 | −3.68153e21 | 1.61937e24 | 6.76929e26 | −1.63440e28 | −1.02502e30 | 1.72429e31 | |||||||||||||||||||||||||||||||||
| 1.2 | −1.14986e9 | 2.06926e15 | −7.90120e18 | −9.56939e21 | −2.37936e24 | −1.15073e26 | 1.96908e28 | 3.13728e30 | 1.10034e31 | ||||||||||||||||||||||||||||||||||
| 1.3 | −9.25761e8 | −5.88048e14 | −8.36634e18 | 1.60235e22 | 5.44392e23 | −7.44490e26 | 1.62839e28 | −7.98761e29 | −1.48340e31 | ||||||||||||||||||||||||||||||||||
| 1.4 | 2.54406e9 | −9.84533e14 | −2.75111e18 | −1.69176e22 | −2.50471e24 | 1.59356e26 | −3.04638e28 | −1.75256e29 | −4.30395e31 | ||||||||||||||||||||||||||||||||||
| 1.5 | 4.72250e9 | 8.02316e14 | 1.30786e19 | 1.36438e22 | 3.78894e24 | 4.00095e26 | 1.82063e28 | −5.00850e29 | 6.44329e31 | ||||||||||||||||||||||||||||||||||
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 | 1.64.a.a | ✓ | 5 |
| 3.b | odd | 2 | 1 | 9.64.a.c | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1.64.a.a | ✓ | 5 | 1.a | even | 1 | 1 | trivial |
| 9.64.a.c | 5 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace is the entire newspace \(S_{64}^{\mathrm{new}}(\Gamma_0(1))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} + \cdots + 59\!\cdots\!24 \)
|
| $3$ |
\( T^{5} + \cdots + 33\!\cdots\!68 \)
|
| $5$ |
\( T^{5} + \cdots + 13\!\cdots\!00 \)
|
| $7$ |
\( T^{5} + \cdots - 36\!\cdots\!76 \)
|
| $11$ |
\( T^{5} + \cdots + 11\!\cdots\!68 \)
|
| $13$ |
\( T^{5} + \cdots - 46\!\cdots\!32 \)
|
| $17$ |
\( T^{5} + \cdots + 31\!\cdots\!24 \)
|
| $19$ |
\( T^{5} + \cdots + 28\!\cdots\!00 \)
|
| $23$ |
\( T^{5} + \cdots + 19\!\cdots\!68 \)
|
| $29$ |
\( T^{5} + \cdots - 48\!\cdots\!00 \)
|
| $31$ |
\( T^{5} + \cdots - 48\!\cdots\!32 \)
|
| $37$ |
\( T^{5} + \cdots + 71\!\cdots\!24 \)
|
| $41$ |
\( T^{5} + \cdots + 72\!\cdots\!68 \)
|
| $43$ |
\( T^{5} + \cdots - 13\!\cdots\!32 \)
|
| $47$ |
\( T^{5} + \cdots + 18\!\cdots\!24 \)
|
| $53$ |
\( T^{5} + \cdots - 42\!\cdots\!32 \)
|
| $59$ |
\( T^{5} + \cdots - 31\!\cdots\!00 \)
|
| $61$ |
\( T^{5} + \cdots - 11\!\cdots\!32 \)
|
| $67$ |
\( T^{5} + \cdots + 17\!\cdots\!24 \)
|
| $71$ |
\( T^{5} + \cdots + 24\!\cdots\!68 \)
|
| $73$ |
\( T^{5} + \cdots - 26\!\cdots\!32 \)
|
| $79$ |
\( T^{5} + \cdots - 37\!\cdots\!00 \)
|
| $83$ |
\( T^{5} + \cdots - 15\!\cdots\!32 \)
|
| $89$ |
\( T^{5} + \cdots + 42\!\cdots\!00 \)
|
| $97$ |
\( T^{5} + \cdots - 82\!\cdots\!76 \)
|
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