Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [65,4,Mod(1,65)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(65, 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("65.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 65 = 5 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 65.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: | \(3.83512415037\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{17}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x - 4 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| 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 \(\beta = \frac{1}{2}(1 + \sqrt{17})\). 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 |
|
−1.56155 | −8.24621 | −5.56155 | −5.00000 | 12.8769 | 33.1231 | 21.1771 | 41.0000 | 7.80776 | ||||||||||||||||||||||||
| 1.2 | 2.56155 | 8.24621 | −1.43845 | −5.00000 | 21.1231 | 24.8769 | −24.1771 | 41.0000 | −12.8078 | |||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \( +1 \) |
| \(13\) | \( +1 \) |
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 | 65.4.a.c | ✓ | 2 |
| 3.b | odd | 2 | 1 | 585.4.a.h | 2 | ||
| 4.b | odd | 2 | 1 | 1040.4.a.k | 2 | ||
| 5.b | even | 2 | 1 | 325.4.a.g | 2 | ||
| 5.c | odd | 4 | 2 | 325.4.b.f | 4 | ||
| 13.b | even | 2 | 1 | 845.4.a.d | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 65.4.a.c | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
| 325.4.a.g | 2 | 5.b | even | 2 | 1 | ||
| 325.4.b.f | 4 | 5.c | odd | 4 | 2 | ||
| 585.4.a.h | 2 | 3.b | odd | 2 | 1 | ||
| 845.4.a.d | 2 | 13.b | even | 2 | 1 | ||
| 1040.4.a.k | 2 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} - T_{2} - 4 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(65))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} - T - 4 \)
|
| $3$ |
\( T^{2} - 68 \)
|
| $5$ |
\( (T + 5)^{2} \)
|
| $7$ |
\( T^{2} - 58T + 824 \)
|
| $11$ |
\( T^{2} + 86T + 1832 \)
|
| $13$ |
\( (T + 13)^{2} \)
|
| $17$ |
\( T^{2} + 28T - 4156 \)
|
| $19$ |
\( T^{2} - 166T + 4832 \)
|
| $23$ |
\( T^{2} - 60T - 16508 \)
|
| $29$ |
\( T^{2} - 120T + 268 \)
|
| $31$ |
\( T^{2} + 78T - 65952 \)
|
| $37$ |
\( T^{2} - 360T + 26892 \)
|
| $41$ |
\( T^{2} + 72T - 64052 \)
|
| $43$ |
\( T^{2} + 44T - 6316 \)
|
| $47$ |
\( T^{2} - 362T + 16424 \)
|
| $53$ |
\( T^{2} + 396T + 36 \)
|
| $59$ |
\( T^{2} - 18T - 128592 \)
|
| $61$ |
\( (T - 442)^{2} \)
|
| $67$ |
\( T^{2} - 1322 T + 249496 \)
|
| $71$ |
\( T^{2} - 634T - 318544 \)
|
| $73$ |
\( T^{2} + 60T - 478908 \)
|
| $79$ |
\( T^{2} + 180T + 7488 \)
|
| $83$ |
\( T^{2} - 714T - 74528 \)
|
| $89$ |
\( T^{2} + 852T + 120276 \)
|
| $97$ |
\( T^{2} - 32T - 2970052 \)
|
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