Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [775,4,Mod(1,775)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(775, 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("775.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 775 = 5^{2} \cdot 31 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 775.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: | \(45.7264802544\) |
| Analytic rank: | \(1\) |
| 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: | no (minimal twist has level 31) |
| 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 |
|
0.438447 | 5.12311 | −7.80776 | 0 | 2.24621 | −0.807764 | −6.93087 | −0.753789 | 0 | ||||||||||||||||||||||||
| 1.2 | 4.56155 | −3.12311 | 12.8078 | 0 | −14.2462 | 19.8078 | 21.9309 | −17.2462 | 0 | |||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \(1\) |
| \(31\) | \(-1\) |
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 | 775.4.a.d | 2 | |
| 5.b | even | 2 | 1 | 31.4.a.a | ✓ | 2 | |
| 15.d | odd | 2 | 1 | 279.4.a.d | 2 | ||
| 20.d | odd | 2 | 1 | 496.4.a.c | 2 | ||
| 35.c | odd | 2 | 1 | 1519.4.a.b | 2 | ||
| 40.e | odd | 2 | 1 | 1984.4.a.e | 2 | ||
| 40.f | even | 2 | 1 | 1984.4.a.f | 2 | ||
| 155.c | odd | 2 | 1 | 961.4.a.b | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 31.4.a.a | ✓ | 2 | 5.b | even | 2 | 1 | |
| 279.4.a.d | 2 | 15.d | odd | 2 | 1 | ||
| 496.4.a.c | 2 | 20.d | odd | 2 | 1 | ||
| 775.4.a.d | 2 | 1.a | even | 1 | 1 | trivial | |
| 961.4.a.b | 2 | 155.c | odd | 2 | 1 | ||
| 1519.4.a.b | 2 | 35.c | odd | 2 | 1 | ||
| 1984.4.a.e | 2 | 40.e | odd | 2 | 1 | ||
| 1984.4.a.f | 2 | 40.f | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(775))\):
|
\( T_{2}^{2} - 5T_{2} + 2 \)
|
|
\( T_{3}^{2} - 2T_{3} - 16 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} - 5T + 2 \)
|
| $3$ |
\( T^{2} - 2T - 16 \)
|
| $5$ |
\( T^{2} \)
|
| $7$ |
\( T^{2} - 19T - 16 \)
|
| $11$ |
\( T^{2} + 26T + 16 \)
|
| $13$ |
\( T^{2} + 64T - 676 \)
|
| $17$ |
\( T^{2} - 84T + 1492 \)
|
| $19$ |
\( T^{2} + 51T - 68 \)
|
| $23$ |
\( T^{2} + 10T - 12368 \)
|
| $29$ |
\( T^{2} + 314T + 8312 \)
|
| $31$ |
\( (T - 31)^{2} \)
|
| $37$ |
\( T^{2} + 32T - 103172 \)
|
| $41$ |
\( T^{2} + 583T + 48214 \)
|
| $43$ |
\( T^{2} + 220T - 138112 \)
|
| $47$ |
\( T^{2} + 512T + 64448 \)
|
| $53$ |
\( T^{2} - 54T - 47024 \)
|
| $59$ |
\( T^{2} + 609T + 82516 \)
|
| $61$ |
\( T^{2} - 372T + 33508 \)
|
| $67$ |
\( T^{2} + 424T - 199856 \)
|
| $71$ |
\( T^{2} + 317T - 197752 \)
|
| $73$ |
\( T^{2} - 1840 T + 781052 \)
|
| $79$ |
\( T^{2} + 1132 T + 246304 \)
|
| $83$ |
\( T^{2} + 1458 T + 459616 \)
|
| $89$ |
\( T^{2} - 418T - 165776 \)
|
| $97$ |
\( T^{2} + 1363 T + 205154 \)
|
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