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: | \(4\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 39x^{2} - 30x + 20 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - 39x^{2} - 30x + 20 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} - \nu^{2} - 36\nu - 4 ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} + \nu^{2} - 40\nu - 42 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} - \beta_{2} + 2\beta _1 + 19 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 38\beta _1 + 23 \)
|
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 |
|
−4.76108 | −9.00000 | −9.33211 | −97.1410 | 42.8497 | −211.220 | 196.786 | 81.0000 | 462.496 | ||||||||||||||||||||||||||||||
| 1.2 | −0.233171 | −9.00000 | −31.9456 | 18.8848 | 2.09853 | −97.3695 | 14.9102 | 81.0000 | −4.40338 | |||||||||||||||||||||||||||||||
| 1.3 | 1.42878 | −9.00000 | −29.9586 | −83.8971 | −12.8590 | 175.668 | −88.5253 | 81.0000 | −119.871 | |||||||||||||||||||||||||||||||
| 1.4 | 7.56547 | −9.00000 | 25.2363 | 40.1532 | −68.0892 | 194.921 | −51.1704 | 81.0000 | 303.778 | |||||||||||||||||||||||||||||||
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.c | ✓ | 4 |
| 3.b | odd | 2 | 1 | 207.6.a.d | 4 | ||
| 4.b | odd | 2 | 1 | 1104.6.a.n | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 69.6.a.c | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 207.6.a.d | 4 | 3.b | odd | 2 | 1 | ||
| 1104.6.a.n | 4 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} - 4T_{2}^{3} - 33T_{2}^{2} + 44T_{2} + 12 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(69))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - 4 T^{3} + \cdots + 12 \)
|
| $3$ |
\( (T + 9)^{4} \)
|
| $5$ |
\( T^{4} + 122 T^{3} + \cdots + 6179904 \)
|
| $7$ |
\( T^{4} - 62 T^{3} + \cdots + 704219920 \)
|
| $11$ |
\( T^{4} + \cdots + 3071914368 \)
|
| $13$ |
\( T^{4} + \cdots + 8279608752 \)
|
| $17$ |
\( T^{4} + \cdots + 4839573197664 \)
|
| $19$ |
\( T^{4} + \cdots - 12894399037424 \)
|
| $23$ |
\( (T - 529)^{4} \)
|
| $29$ |
\( T^{4} + \cdots - 313971545146320 \)
|
| $31$ |
\( T^{4} + \cdots + 12\!\cdots\!00 \)
|
| $37$ |
\( T^{4} + \cdots + 17\!\cdots\!12 \)
|
| $41$ |
\( T^{4} + \cdots + 29\!\cdots\!48 \)
|
| $43$ |
\( T^{4} + \cdots - 579704020143984 \)
|
| $47$ |
\( T^{4} + \cdots + 13\!\cdots\!20 \)
|
| $53$ |
\( T^{4} + \cdots - 82\!\cdots\!72 \)
|
| $59$ |
\( T^{4} + \cdots - 59\!\cdots\!60 \)
|
| $61$ |
\( T^{4} + \cdots + 95\!\cdots\!80 \)
|
| $67$ |
\( T^{4} + \cdots + 10\!\cdots\!88 \)
|
| $71$ |
\( T^{4} + \cdots - 13\!\cdots\!00 \)
|
| $73$ |
\( T^{4} + \cdots + 25\!\cdots\!40 \)
|
| $79$ |
\( T^{4} + \cdots + 23\!\cdots\!72 \)
|
| $83$ |
\( T^{4} + \cdots + 16\!\cdots\!96 \)
|
| $89$ |
\( T^{4} + \cdots + 24\!\cdots\!40 \)
|
| $97$ |
\( T^{4} + \cdots - 74\!\cdots\!00 \)
|
| show more | |
| show less |