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: | \(\Q(\sqrt{2}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - 2 \)
|
| 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 \(\beta = 2\sqrt{2}\). 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 |
|
−3.82843 | −3.00000 | 6.65685 | 15.3137 | 11.4853 | −16.8284 | 5.14214 | 9.00000 | −58.6274 | ||||||||||||||||||||||||
| 1.2 | 1.82843 | −3.00000 | −4.65685 | −7.31371 | −5.48528 | −11.1716 | −23.1421 | 9.00000 | −13.3726 | |||||||||||||||||||||||||
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.b | ✓ | 2 |
| 3.b | odd | 2 | 1 | 207.4.a.b | 2 | ||
| 4.b | odd | 2 | 1 | 1104.4.a.q | 2 | ||
| 5.b | even | 2 | 1 | 1725.4.a.m | 2 | ||
| 23.b | odd | 2 | 1 | 1587.4.a.c | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 69.4.a.b | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
| 207.4.a.b | 2 | 3.b | odd | 2 | 1 | ||
| 1104.4.a.q | 2 | 4.b | odd | 2 | 1 | ||
| 1587.4.a.c | 2 | 23.b | odd | 2 | 1 | ||
| 1725.4.a.m | 2 | 5.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} + 2T_{2} - 7 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(69))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} + 2T - 7 \)
|
| $3$ |
\( (T + 3)^{2} \)
|
| $5$ |
\( T^{2} - 8T - 112 \)
|
| $7$ |
\( T^{2} + 28T + 188 \)
|
| $11$ |
\( T^{2} + 72T + 904 \)
|
| $13$ |
\( T^{2} + 60T + 612 \)
|
| $17$ |
\( T^{2} - 96T + 2104 \)
|
| $19$ |
\( T^{2} + 148T + 868 \)
|
| $23$ |
\( (T - 23)^{2} \)
|
| $29$ |
\( T^{2} + 204T - 24444 \)
|
| $31$ |
\( T^{2} + 168T - 16272 \)
|
| $37$ |
\( T^{2} - 116T - 11428 \)
|
| $41$ |
\( T^{2} - 4T - 5404 \)
|
| $43$ |
\( T^{2} + 420T + 41508 \)
|
| $47$ |
\( T^{2} + 48T - 61376 \)
|
| $53$ |
\( T^{2} - 32T - 544 \)
|
| $59$ |
\( T^{2} + 40T - 199312 \)
|
| $61$ |
\( T^{2} + 764T + 126716 \)
|
| $67$ |
\( T^{2} + 988T + 73508 \)
|
| $71$ |
\( T^{2} + 224T - 641824 \)
|
| $73$ |
\( T^{2} - 820T + 143012 \)
|
| $79$ |
\( T^{2} - 1772 T + 757148 \)
|
| $83$ |
\( T^{2} - 1480 T + 191432 \)
|
| $89$ |
\( T^{2} - 744T - 365624 \)
|
| $97$ |
\( T^{2} + 260 T - 1646588 \)
|
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