Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [69,6,Mod(1,69)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(69, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("69.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 69 = 3 \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 69.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: | \(11.0664835671\) |
| 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: |
\( x^{5} - 113x^{3} - 257x^{2} + 1404x + 2197 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{5} \) |
| 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
\( x^{5} - 113x^{3} - 257x^{2} + 1404x + 2197 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 7\nu^{4} - 13\nu^{3} - 700\nu^{2} - 499\nu + 5941 ) / 468 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{4} + 13\nu^{3} - 56\nu^{2} - 419\nu + 4511 ) / 117 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 23\nu^{4} - 221\nu^{3} - 1676\nu^{2} + 11821\nu + 18005 ) / 468 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{4} - 4\beta_{3} + \beta_{2} + 6\beta _1 + 180 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -7\beta_{4} - 7\beta_{3} + 19\beta_{2} + 86\beta _1 + 298 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -63\beta_{4} - 213\beta_{3} + 219\beta_{2} + 531\beta _1 + 7856 ) / 2 \)
|
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 |
|
−9.32729 | 9.00000 | 54.9984 | 99.7124 | −83.9456 | 125.983 | −214.513 | 81.0000 | −930.047 | |||||||||||||||||||||||||||||||||
| 1.2 | −3.54286 | 9.00000 | −19.4482 | −92.1306 | −31.8857 | −8.98894 | 182.273 | 81.0000 | 326.406 | ||||||||||||||||||||||||||||||||||
| 1.3 | 2.24792 | 9.00000 | −26.9469 | 53.3906 | 20.2313 | 89.8688 | −132.508 | 81.0000 | 120.018 | ||||||||||||||||||||||||||||||||||
| 1.4 | 9.27315 | 9.00000 | 53.9914 | −37.4928 | 83.4584 | 154.850 | 203.929 | 81.0000 | −347.676 | ||||||||||||||||||||||||||||||||||
| 1.5 | 9.34908 | 9.00000 | 55.4053 | 70.5203 | 84.1417 | −89.7132 | 218.818 | 81.0000 | 659.300 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(23\) | \(1\) |
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 | 69.6.a.e | ✓ | 5 |
| 3.b | odd | 2 | 1 | 207.6.a.f | 5 | ||
| 4.b | odd | 2 | 1 | 1104.6.a.r | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 69.6.a.e | ✓ | 5 | 1.a | even | 1 | 1 | trivial |
| 207.6.a.f | 5 | 3.b | odd | 2 | 1 | ||
| 1104.6.a.r | 5 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{5} - 8T_{2}^{4} - 107T_{2}^{3} + 770T_{2}^{2} + 1740T_{2} - 6440 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(69))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} - 8 T^{4} + \cdots - 6440 \)
|
| $3$ |
\( (T - 9)^{5} \)
|
| $5$ |
\( T^{5} + \cdots - 1296820224 \)
|
| $7$ |
\( T^{5} + \cdots - 1413831904 \)
|
| $11$ |
\( T^{5} + \cdots + 5578745996288 \)
|
| $13$ |
\( T^{5} + \cdots - 2474410088160 \)
|
| $17$ |
\( T^{5} + \cdots + 13327498013440 \)
|
| $19$ |
\( T^{5} + \cdots + 30\!\cdots\!60 \)
|
| $23$ |
\( (T + 529)^{5} \)
|
| $29$ |
\( T^{5} + \cdots + 32\!\cdots\!40 \)
|
| $31$ |
\( T^{5} + \cdots + 16\!\cdots\!48 \)
|
| $37$ |
\( T^{5} + \cdots + 84\!\cdots\!80 \)
|
| $41$ |
\( T^{5} + \cdots - 22\!\cdots\!12 \)
|
| $43$ |
\( T^{5} + \cdots - 23\!\cdots\!72 \)
|
| $47$ |
\( T^{5} + \cdots + 94\!\cdots\!64 \)
|
| $53$ |
\( T^{5} + \cdots + 77\!\cdots\!36 \)
|
| $59$ |
\( T^{5} + \cdots - 10\!\cdots\!88 \)
|
| $61$ |
\( T^{5} + \cdots + 11\!\cdots\!60 \)
|
| $67$ |
\( T^{5} + \cdots - 11\!\cdots\!24 \)
|
| $71$ |
\( T^{5} + \cdots - 42\!\cdots\!28 \)
|
| $73$ |
\( T^{5} + \cdots + 27\!\cdots\!68 \)
|
| $79$ |
\( T^{5} + \cdots - 39\!\cdots\!64 \)
|
| $83$ |
\( T^{5} + \cdots - 29\!\cdots\!24 \)
|
| $89$ |
\( T^{5} + \cdots + 28\!\cdots\!76 \)
|
| $97$ |
\( T^{5} + \cdots + 55\!\cdots\!56 \)
|
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