Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [69,4,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, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("69.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 69 = 3 \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| 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: | \(4.07113179040\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.2009704.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 27x^{2} - 6x + 112 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2 \) |
| 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
\( x^{4} - 27x^{2} - 6x + 112 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} + \nu^{2} - 18\nu - 20 ) / 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{3} + 3\nu^{2} + 18\nu - 36 ) / 4 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 14 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{3} + 3\beta_{2} + 18\beta _1 + 6 \)
|
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 |
|
−3.84414 | 3.00000 | 6.77740 | 8.70815 | −11.5324 | −26.5728 | 4.69983 | 9.00000 | −33.4753 | ||||||||||||||||||||||||||||||
| 1.2 | −1.09731 | 3.00000 | −6.79592 | 3.76407 | −3.29192 | 27.3318 | 16.2356 | 9.00000 | −4.13033 | |||||||||||||||||||||||||||||||
| 1.3 | 3.45983 | 3.00000 | 3.97043 | 7.89053 | 10.3795 | 4.57766 | −13.9416 | 9.00000 | 27.2999 | |||||||||||||||||||||||||||||||
| 1.4 | 5.48161 | 3.00000 | 22.0481 | −16.3627 | 16.4448 | −19.3367 | 77.0061 | 9.00000 | −89.6942 | |||||||||||||||||||||||||||||||
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 | 69.4.a.d | ✓ | 4 |
| 3.b | odd | 2 | 1 | 207.4.a.d | 4 | ||
| 4.b | odd | 2 | 1 | 1104.4.a.t | 4 | ||
| 5.b | even | 2 | 1 | 1725.4.a.p | 4 | ||
| 23.b | odd | 2 | 1 | 1587.4.a.g | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 69.4.a.d | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 207.4.a.d | 4 | 3.b | odd | 2 | 1 | ||
| 1104.4.a.t | 4 | 4.b | odd | 2 | 1 | ||
| 1587.4.a.g | 4 | 23.b | odd | 2 | 1 | ||
| 1725.4.a.p | 4 | 5.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} - 4T_{2}^{3} - 21T_{2}^{2} + 56T_{2} + 80 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(69))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - 4 T^{3} + \cdots + 80 \)
|
| $3$ |
\( (T - 3)^{4} \)
|
| $5$ |
\( T^{4} - 4 T^{3} + \cdots - 4232 \)
|
| $7$ |
\( T^{4} + 14 T^{3} + \cdots + 64288 \)
|
| $11$ |
\( T^{4} - 70 T^{3} + \cdots + 73984 \)
|
| $13$ |
\( T^{4} + 12 T^{3} + \cdots + 277632 \)
|
| $17$ |
\( T^{4} - 178 T^{3} + \cdots - 19990000 \)
|
| $19$ |
\( T^{4} - 96 T^{3} + \cdots + 26764064 \)
|
| $23$ |
\( (T - 23)^{4} \)
|
| $29$ |
\( T^{4} + 24 T^{3} + \cdots + 77040 \)
|
| $31$ |
\( T^{4} + 400 T^{3} + \cdots + 214004736 \)
|
| $37$ |
\( T^{4} + 358 T^{3} + \cdots - 89676928 \)
|
| $41$ |
\( T^{4} + \cdots - 1851640112 \)
|
| $43$ |
\( T^{4} + 196 T^{3} + \cdots - 66620160 \)
|
| $47$ |
\( T^{4} + \cdots - 1761689600 \)
|
| $53$ |
\( T^{4} + \cdots + 2541177224 \)
|
| $59$ |
\( T^{4} - 1236 T^{3} + \cdots - 895185728 \)
|
| $61$ |
\( T^{4} + \cdots + 25271105312 \)
|
| $67$ |
\( T^{4} + \cdots - 6404665408 \)
|
| $71$ |
\( T^{4} + \cdots + 93159882752 \)
|
| $73$ |
\( T^{4} + \cdots - 205893068432 \)
|
| $79$ |
\( T^{4} - 254 T^{3} + \cdots - 199673280 \)
|
| $83$ |
\( T^{4} + \cdots - 9163210624 \)
|
| $89$ |
\( T^{4} + \cdots + 477698056688 \)
|
| $97$ |
\( T^{4} + \cdots - 759203302672 \)
|
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