Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [160,4,Mod(1,160)]
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([0, 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("160.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 160 = 2^{5} \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 160.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: | \(9.44030560092\) |
| Analytic rank: | \(1\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{6}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - 6 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| 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 \(\beta = 2\sqrt{6}\). 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 | −8.89898 | 0 | 5.00000 | 0 | 20.4949 | 0 | 52.1918 | 0 | ||||||||||||||||||||||||
| 1.2 | 0 | 0.898979 | 0 | 5.00000 | 0 | −28.4949 | 0 | −26.1918 | 0 | |||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(5\) | \( -1 \) |
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 | 160.4.a.c | ✓ | 2 |
| 3.b | odd | 2 | 1 | 1440.4.a.t | 2 | ||
| 4.b | odd | 2 | 1 | 160.4.a.g | yes | 2 | |
| 5.b | even | 2 | 1 | 800.4.a.s | 2 | ||
| 5.c | odd | 4 | 2 | 800.4.c.i | 4 | ||
| 8.b | even | 2 | 1 | 320.4.a.s | 2 | ||
| 8.d | odd | 2 | 1 | 320.4.a.o | 2 | ||
| 12.b | even | 2 | 1 | 1440.4.a.x | 2 | ||
| 16.e | even | 4 | 2 | 1280.4.d.x | 4 | ||
| 16.f | odd | 4 | 2 | 1280.4.d.q | 4 | ||
| 20.d | odd | 2 | 1 | 800.4.a.m | 2 | ||
| 20.e | even | 4 | 2 | 800.4.c.k | 4 | ||
| 40.e | odd | 2 | 1 | 1600.4.a.cn | 2 | ||
| 40.f | even | 2 | 1 | 1600.4.a.cd | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 160.4.a.c | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
| 160.4.a.g | yes | 2 | 4.b | odd | 2 | 1 | |
| 320.4.a.o | 2 | 8.d | odd | 2 | 1 | ||
| 320.4.a.s | 2 | 8.b | even | 2 | 1 | ||
| 800.4.a.m | 2 | 20.d | odd | 2 | 1 | ||
| 800.4.a.s | 2 | 5.b | even | 2 | 1 | ||
| 800.4.c.i | 4 | 5.c | odd | 4 | 2 | ||
| 800.4.c.k | 4 | 20.e | even | 4 | 2 | ||
| 1280.4.d.q | 4 | 16.f | odd | 4 | 2 | ||
| 1280.4.d.x | 4 | 16.e | even | 4 | 2 | ||
| 1440.4.a.t | 2 | 3.b | odd | 2 | 1 | ||
| 1440.4.a.x | 2 | 12.b | even | 2 | 1 | ||
| 1600.4.a.cd | 2 | 40.f | even | 2 | 1 | ||
| 1600.4.a.cn | 2 | 40.e | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} + 8T_{3} - 8 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(160))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} \)
|
| $3$ |
\( T^{2} + 8T - 8 \)
|
| $5$ |
\( (T - 5)^{2} \)
|
| $7$ |
\( T^{2} + 8T - 584 \)
|
| $11$ |
\( T^{2} + 64T + 160 \)
|
| $13$ |
\( T^{2} - 12T - 1500 \)
|
| $17$ |
\( T^{2} - 4T - 13820 \)
|
| $19$ |
\( T^{2} + 208T + 9280 \)
|
| $23$ |
\( T^{2} + 120T + 696 \)
|
| $29$ |
\( T^{2} + 292T + 15172 \)
|
| $31$ |
\( T^{2} + 176T + 6880 \)
|
| $37$ |
\( T^{2} + 356T + 25540 \)
|
| $41$ |
\( T^{2} - 100T - 35900 \)
|
| $43$ |
\( T^{2} - 376T + 2488 \)
|
| $47$ |
\( T^{2} - 280T + 15544 \)
|
| $53$ |
\( T^{2} - 316T + 11140 \)
|
| $59$ |
\( T^{2} + 720T + 120000 \)
|
| $61$ |
\( T^{2} + 1268 T + 377380 \)
|
| $67$ |
\( T^{2} - 744T + 85368 \)
|
| $71$ |
\( T^{2} - 48T - 1140000 \)
|
| $73$ |
\( T^{2} + 940T - 124700 \)
|
| $79$ |
\( T^{2} + 32T - 709760 \)
|
| $83$ |
\( T^{2} - 1592 T + 580600 \)
|
| $89$ |
\( T^{2} + 780T + 96804 \)
|
| $97$ |
\( T^{2} - 1220 T + 333700 \)
|
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