Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [961,2,Mod(1,961)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(961, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("961.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 961 = 31^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 961.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: | \(7.67362363425\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\sqrt{2}, \sqrt{5})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 6x^{2} + 4 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| 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 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
\( x^{4} - 6x^{2} + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} - 2 ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} - 4\nu ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 2\beta_{2} + 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{3} + 4\beta_1 \)
|
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 |
|
0.381966 | −2.28825 | −1.85410 | −2.23607 | −0.874032 | −1.00000 | −1.47214 | 2.23607 | −0.854102 | ||||||||||||||||||||||||||||||
| 1.2 | 0.381966 | 2.28825 | −1.85410 | −2.23607 | 0.874032 | −1.00000 | −1.47214 | 2.23607 | −0.854102 | |||||||||||||||||||||||||||||||
| 1.3 | 2.61803 | −0.874032 | 4.85410 | 2.23607 | −2.28825 | −1.00000 | 7.47214 | −2.23607 | 5.85410 | |||||||||||||||||||||||||||||||
| 1.4 | 2.61803 | 0.874032 | 4.85410 | 2.23607 | 2.28825 | −1.00000 | 7.47214 | −2.23607 | 5.85410 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(31\) | \( -1 \) |
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 31.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 961.2.a.h | ✓ | 4 |
| 3.b | odd | 2 | 1 | 8649.2.a.r | 4 | ||
| 31.b | odd | 2 | 1 | inner | 961.2.a.h | ✓ | 4 |
| 31.c | even | 3 | 2 | 961.2.c.h | 8 | ||
| 31.d | even | 5 | 2 | 961.2.d.h | 8 | ||
| 31.d | even | 5 | 2 | 961.2.d.j | 8 | ||
| 31.e | odd | 6 | 2 | 961.2.c.h | 8 | ||
| 31.f | odd | 10 | 2 | 961.2.d.h | 8 | ||
| 31.f | odd | 10 | 2 | 961.2.d.j | 8 | ||
| 31.g | even | 15 | 4 | 961.2.g.i | 16 | ||
| 31.g | even | 15 | 4 | 961.2.g.p | 16 | ||
| 31.h | odd | 30 | 4 | 961.2.g.i | 16 | ||
| 31.h | odd | 30 | 4 | 961.2.g.p | 16 | ||
| 93.c | even | 2 | 1 | 8649.2.a.r | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 961.2.a.h | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 961.2.a.h | ✓ | 4 | 31.b | odd | 2 | 1 | inner |
| 961.2.c.h | 8 | 31.c | even | 3 | 2 | ||
| 961.2.c.h | 8 | 31.e | odd | 6 | 2 | ||
| 961.2.d.h | 8 | 31.d | even | 5 | 2 | ||
| 961.2.d.h | 8 | 31.f | odd | 10 | 2 | ||
| 961.2.d.j | 8 | 31.d | even | 5 | 2 | ||
| 961.2.d.j | 8 | 31.f | odd | 10 | 2 | ||
| 961.2.g.i | 16 | 31.g | even | 15 | 4 | ||
| 961.2.g.i | 16 | 31.h | odd | 30 | 4 | ||
| 961.2.g.p | 16 | 31.g | even | 15 | 4 | ||
| 961.2.g.p | 16 | 31.h | odd | 30 | 4 | ||
| 8649.2.a.r | 4 | 3.b | odd | 2 | 1 | ||
| 8649.2.a.r | 4 | 93.c | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(961))\):
|
\( T_{2}^{2} - 3T_{2} + 1 \)
|
|
\( T_{3}^{4} - 6T_{3}^{2} + 4 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} - 3 T + 1)^{2} \)
|
| $3$ |
\( T^{4} - 6T^{2} + 4 \)
|
| $5$ |
\( (T^{2} - 5)^{2} \)
|
| $7$ |
\( (T + 1)^{4} \)
|
| $11$ |
\( (T^{2} - 18)^{2} \)
|
| $13$ |
\( T^{4} - 54T^{2} + 324 \)
|
| $17$ |
\( T^{4} - 14T^{2} + 4 \)
|
| $19$ |
\( (T - 1)^{4} \)
|
| $23$ |
\( T^{4} - 54T^{2} + 324 \)
|
| $29$ |
\( T^{4} - 14T^{2} + 4 \)
|
| $31$ |
\( T^{4} \)
|
| $37$ |
\( (T^{2} - 18)^{2} \)
|
| $41$ |
\( (T^{2} - 6 T - 11)^{2} \)
|
| $43$ |
\( T^{4} - 94T^{2} + 4 \)
|
| $47$ |
\( (T^{2} + 6 T - 36)^{2} \)
|
| $53$ |
\( T^{4} - 216T^{2} + 5184 \)
|
| $59$ |
\( (T^{2} - 6 T - 71)^{2} \)
|
| $61$ |
\( T^{4} - 214T^{2} + 3844 \)
|
| $67$ |
\( (T - 6)^{4} \)
|
| $71$ |
\( (T^{2} - 6 T - 11)^{2} \)
|
| $73$ |
\( (T^{2} - 18)^{2} \)
|
| $79$ |
\( T^{4} - 126T^{2} + 324 \)
|
| $83$ |
\( (T^{2} - 10)^{2} \)
|
| $89$ |
\( T^{4} - 270T^{2} + 8100 \)
|
| $97$ |
\( (T + 7)^{4} \)
|
| show more | |
| show less |