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: | \(1\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{5}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x - 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| 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 \(\beta = \sqrt{5}\). We also show the integral \(q\)-expansion of the trace form.
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 |
|
−4.23607 | 3.00000 | 9.94427 | −10.7639 | −12.7082 | 10.6525 | −8.23607 | 9.00000 | 45.5967 | ||||||||||||||||||||||||
| 1.2 | 0.236068 | 3.00000 | −7.94427 | −15.2361 | 0.708204 | −20.6525 | −3.76393 | 9.00000 | −3.59675 | |||||||||||||||||||||||||
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.a | ✓ | 2 |
| 3.b | odd | 2 | 1 | 207.4.a.c | 2 | ||
| 4.b | odd | 2 | 1 | 1104.4.a.h | 2 | ||
| 5.b | even | 2 | 1 | 1725.4.a.n | 2 | ||
| 23.b | odd | 2 | 1 | 1587.4.a.b | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 69.4.a.a | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
| 207.4.a.c | 2 | 3.b | odd | 2 | 1 | ||
| 1104.4.a.h | 2 | 4.b | odd | 2 | 1 | ||
| 1587.4.a.b | 2 | 23.b | odd | 2 | 1 | ||
| 1725.4.a.n | 2 | 5.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} + 4T_{2} - 1 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(69))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} + 4T - 1 \)
|
| $3$ |
\( (T - 3)^{2} \)
|
| $5$ |
\( T^{2} + 26T + 164 \)
|
| $7$ |
\( T^{2} + 10T - 220 \)
|
| $11$ |
\( T^{2} + 60T + 400 \)
|
| $13$ |
\( T^{2} + 24T - 4356 \)
|
| $17$ |
\( T^{2} + 150T + 5620 \)
|
| $19$ |
\( T^{2} + 46T - 4916 \)
|
| $23$ |
\( (T + 23)^{2} \)
|
| $29$ |
\( T^{2} + 216T - 2916 \)
|
| $31$ |
\( T^{2} - 324T - 4176 \)
|
| $37$ |
\( T^{2} - 140T - 15580 \)
|
| $41$ |
\( T^{2} + 364T + 12644 \)
|
| $43$ |
\( T^{2} + 126T - 51156 \)
|
| $47$ |
\( T^{2} - 120T - 303920 \)
|
| $53$ |
\( T^{2} - 490T + 59420 \)
|
| $59$ |
\( T^{2} + 32T - 268864 \)
|
| $61$ |
\( T^{2} + 908T + 58196 \)
|
| $67$ |
\( T^{2} + 874T + 190844 \)
|
| $71$ |
\( T^{2} - 488T - 88384 \)
|
| $73$ |
\( T^{2} - 652T - 252844 \)
|
| $79$ |
\( T^{2} + 1198 T + 225956 \)
|
| $83$ |
\( T^{2} + 100T - 31120 \)
|
| $89$ |
\( T^{2} - 102T - 685604 \)
|
| $97$ |
\( T^{2} - 1624T + 32764 \)
|
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