Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [160,3,Mod(31,160)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(160, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0, 0]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("160.31");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 160 = 2^{5} \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 160.b (of order \(2\), degree \(1\), minimal) |
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: | no |
| Analytic conductor: | \(4.35968422976\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(i, \sqrt{5})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 3x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{4} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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
\( x^{4} + 3x^{2} + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu^{3} + 2\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( 2\nu^{3} + 6\nu \)
|
| \(\beta_{3}\) | \(=\) |
\( 2\nu^{2} + 3 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} - \beta_1 ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} - 3 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -\beta_{2} + 3\beta_1 ) / 4 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/160\mathbb{Z}\right)^\times\).
| \(n\) | \(31\) | \(97\) | \(101\) |
| \(\chi(n)\) | \(-1\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 31.1 |
|
0 | − | 5.23607i | 0 | 2.23607 | 0 | − | 10.1803i | 0 | −18.4164 | 0 | ||||||||||||||||||||||||||||
| 31.2 | 0 | − | 0.763932i | 0 | −2.23607 | 0 | 12.1803i | 0 | 8.41641 | 0 | ||||||||||||||||||||||||||||||
| 31.3 | 0 | 0.763932i | 0 | −2.23607 | 0 | − | 12.1803i | 0 | 8.41641 | 0 | ||||||||||||||||||||||||||||||
| 31.4 | 0 | 5.23607i | 0 | 2.23607 | 0 | 10.1803i | 0 | −18.4164 | 0 | |||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 4.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 160.3.b.a | ✓ | 4 |
| 3.b | odd | 2 | 1 | 1440.3.e.b | 4 | ||
| 4.b | odd | 2 | 1 | inner | 160.3.b.a | ✓ | 4 |
| 5.b | even | 2 | 1 | 800.3.b.d | 4 | ||
| 5.c | odd | 4 | 1 | 800.3.h.c | 4 | ||
| 5.c | odd | 4 | 1 | 800.3.h.j | 4 | ||
| 8.b | even | 2 | 1 | 320.3.b.b | 4 | ||
| 8.d | odd | 2 | 1 | 320.3.b.b | 4 | ||
| 12.b | even | 2 | 1 | 1440.3.e.b | 4 | ||
| 16.e | even | 4 | 1 | 1280.3.g.a | 4 | ||
| 16.e | even | 4 | 1 | 1280.3.g.d | 4 | ||
| 16.f | odd | 4 | 1 | 1280.3.g.a | 4 | ||
| 16.f | odd | 4 | 1 | 1280.3.g.d | 4 | ||
| 20.d | odd | 2 | 1 | 800.3.b.d | 4 | ||
| 20.e | even | 4 | 1 | 800.3.h.c | 4 | ||
| 20.e | even | 4 | 1 | 800.3.h.j | 4 | ||
| 24.f | even | 2 | 1 | 2880.3.e.a | 4 | ||
| 24.h | odd | 2 | 1 | 2880.3.e.a | 4 | ||
| 40.e | odd | 2 | 1 | 1600.3.b.n | 4 | ||
| 40.f | even | 2 | 1 | 1600.3.b.n | 4 | ||
| 40.i | odd | 4 | 1 | 1600.3.h.d | 4 | ||
| 40.i | odd | 4 | 1 | 1600.3.h.m | 4 | ||
| 40.k | even | 4 | 1 | 1600.3.h.d | 4 | ||
| 40.k | even | 4 | 1 | 1600.3.h.m | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 160.3.b.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 160.3.b.a | ✓ | 4 | 4.b | odd | 2 | 1 | inner |
| 320.3.b.b | 4 | 8.b | even | 2 | 1 | ||
| 320.3.b.b | 4 | 8.d | odd | 2 | 1 | ||
| 800.3.b.d | 4 | 5.b | even | 2 | 1 | ||
| 800.3.b.d | 4 | 20.d | odd | 2 | 1 | ||
| 800.3.h.c | 4 | 5.c | odd | 4 | 1 | ||
| 800.3.h.c | 4 | 20.e | even | 4 | 1 | ||
| 800.3.h.j | 4 | 5.c | odd | 4 | 1 | ||
| 800.3.h.j | 4 | 20.e | even | 4 | 1 | ||
| 1280.3.g.a | 4 | 16.e | even | 4 | 1 | ||
| 1280.3.g.a | 4 | 16.f | odd | 4 | 1 | ||
| 1280.3.g.d | 4 | 16.e | even | 4 | 1 | ||
| 1280.3.g.d | 4 | 16.f | odd | 4 | 1 | ||
| 1440.3.e.b | 4 | 3.b | odd | 2 | 1 | ||
| 1440.3.e.b | 4 | 12.b | even | 2 | 1 | ||
| 1600.3.b.n | 4 | 40.e | odd | 2 | 1 | ||
| 1600.3.b.n | 4 | 40.f | even | 2 | 1 | ||
| 1600.3.h.d | 4 | 40.i | odd | 4 | 1 | ||
| 1600.3.h.d | 4 | 40.k | even | 4 | 1 | ||
| 1600.3.h.m | 4 | 40.i | odd | 4 | 1 | ||
| 1600.3.h.m | 4 | 40.k | even | 4 | 1 | ||
| 2880.3.e.a | 4 | 24.f | even | 2 | 1 | ||
| 2880.3.e.a | 4 | 24.h | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} + 28T_{3}^{2} + 16 \)
acting on \(S_{3}^{\mathrm{new}}(160, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} + 28T^{2} + 16 \)
|
| $5$ |
\( (T^{2} - 5)^{2} \)
|
| $7$ |
\( T^{4} + 252 T^{2} + 15376 \)
|
| $11$ |
\( T^{4} + 240T^{2} + 6400 \)
|
| $13$ |
\( (T^{2} - 32 T + 236)^{2} \)
|
| $17$ |
\( (T^{2} + 20 T + 20)^{2} \)
|
| $19$ |
\( (T^{2} + 144)^{2} \)
|
| $23$ |
\( T^{4} + 1308 T^{2} + 309136 \)
|
| $29$ |
\( (T^{2} + 36 T - 396)^{2} \)
|
| $31$ |
\( T^{4} + 752T^{2} + 256 \)
|
| $37$ |
\( (T^{2} - 56 T + 764)^{2} \)
|
| $41$ |
\( (T^{2} - 80 T + 620)^{2} \)
|
| $43$ |
\( T^{4} + 732 T^{2} + 15376 \)
|
| $47$ |
\( T^{4} + 2492 T^{2} + 1008016 \)
|
| $53$ |
\( (T^{2} - 64 T + 44)^{2} \)
|
| $59$ |
\( T^{4} + 9888 T^{2} + 8386816 \)
|
| $61$ |
\( (T^{2} - 32 T - 2164)^{2} \)
|
| $67$ |
\( T^{4} + 15260 T^{2} + 57456400 \)
|
| $71$ |
\( T^{4} + 16688 T^{2} + 26050816 \)
|
| $73$ |
\( (T^{2} + 4 T - 17996)^{2} \)
|
| $79$ |
\( T^{4} + 24768 T^{2} + 119771136 \)
|
| $83$ |
\( T^{4} + 7868 T^{2} + 2835856 \)
|
| $89$ |
\( (T + 30)^{4} \)
|
| $97$ |
\( (T^{2} - 132 T + 1476)^{2} \)
|
| show more | |
| show less |