Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [160,3,Mod(33,160)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(160, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([0, 0, 3]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("160.33");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 160 = 2^{5} \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 160.p (of order \(4\), degree \(2\), 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: | \(6\) |
| Relative dimension: | \(3\) over \(\Q(i)\) |
| Coefficient field: | 6.0.3534400.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 2x^{5} - 3x^{4} + 16x^{3} + x^{2} - 12x + 40 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{9}]\) |
| Coefficient ring index: | \( 2^{4} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 2x^{5} - 3x^{4} + 16x^{3} + x^{2} - 12x + 40 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 7\nu^{5} + 100\nu^{4} - 681\nu^{3} + 1100\nu^{2} + 757\nu - 3140 ) / 890 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 17\nu^{5} - 75\nu^{4} - \nu^{3} + 510\nu^{2} + 122\nu - 760 ) / 445 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 9\nu^{5} - 24\nu^{4} - 11\nu^{3} + 92\nu^{2} + 7\nu + 6 ) / 178 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -61\nu^{5} + 400\nu^{4} - 677\nu^{3} - 940\nu^{2} + 3829\nu - 1880 ) / 890 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 39\nu^{5} - 15\nu^{4} - 107\nu^{3} + 280\nu^{2} + 594\nu + 560 ) / 445 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{4} + 2\beta_{2} - \beta _1 + 2 ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 2\beta_{5} - \beta_{4} - 8\beta_{3} + 4\beta_{2} + \beta _1 + 6 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 4\beta_{5} - \beta_{4} - 11\beta_{3} + 4\beta_{2} - 2\beta _1 - 7 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 28\beta_{5} - 9\beta_{4} - 60\beta_{3} - 2\beta_{2} + 5\beta _1 - 38 ) / 4 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 64\beta_{5} - 17\beta_{4} - 26\beta_{3} - 38\beta_{2} - \beta _1 - 184 ) / 4 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/160\mathbb{Z}\right)^\times\).
| \(n\) | \(31\) | \(97\) | \(101\) |
| \(\chi(n)\) | \(1\) | \(\beta_{3}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 33.1 |
|
0 | −2.29240 | − | 2.29240i | 0 | 1.25511 | − | 4.83991i | 0 | −4.80261 | + | 4.80261i | 0 | 1.51021i | 0 | ||||||||||||||||||||||||||||||
| 33.2 | 0 | −0.602705 | − | 0.602705i | 0 | −3.63675 | + | 3.43134i | 0 | 6.67079 | − | 6.67079i | 0 | − | 8.27349i | 0 | ||||||||||||||||||||||||||||||
| 33.3 | 0 | 2.89511 | + | 2.89511i | 0 | 4.38164 | + | 2.40857i | 0 | −5.86818 | + | 5.86818i | 0 | 7.76328i | 0 | |||||||||||||||||||||||||||||||
| 97.1 | 0 | −2.29240 | + | 2.29240i | 0 | 1.25511 | + | 4.83991i | 0 | −4.80261 | − | 4.80261i | 0 | − | 1.51021i | 0 | ||||||||||||||||||||||||||||||
| 97.2 | 0 | −0.602705 | + | 0.602705i | 0 | −3.63675 | − | 3.43134i | 0 | 6.67079 | + | 6.67079i | 0 | 8.27349i | 0 | |||||||||||||||||||||||||||||||
| 97.3 | 0 | 2.89511 | − | 2.89511i | 0 | 4.38164 | − | 2.40857i | 0 | −5.86818 | − | 5.86818i | 0 | − | 7.76328i | 0 | ||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.c | odd | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 160.3.p.e | ✓ | 6 |
| 4.b | odd | 2 | 1 | 160.3.p.f | yes | 6 | |
| 5.b | even | 2 | 1 | 800.3.p.j | 6 | ||
| 5.c | odd | 4 | 1 | inner | 160.3.p.e | ✓ | 6 |
| 5.c | odd | 4 | 1 | 800.3.p.j | 6 | ||
| 8.b | even | 2 | 1 | 320.3.p.m | 6 | ||
| 8.d | odd | 2 | 1 | 320.3.p.n | 6 | ||
| 20.d | odd | 2 | 1 | 800.3.p.i | 6 | ||
| 20.e | even | 4 | 1 | 160.3.p.f | yes | 6 | |
| 20.e | even | 4 | 1 | 800.3.p.i | 6 | ||
| 40.i | odd | 4 | 1 | 320.3.p.m | 6 | ||
| 40.k | even | 4 | 1 | 320.3.p.n | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 160.3.p.e | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 160.3.p.e | ✓ | 6 | 5.c | odd | 4 | 1 | inner |
| 160.3.p.f | yes | 6 | 4.b | odd | 2 | 1 | |
| 160.3.p.f | yes | 6 | 20.e | even | 4 | 1 | |
| 320.3.p.m | 6 | 8.b | even | 2 | 1 | ||
| 320.3.p.m | 6 | 40.i | odd | 4 | 1 | ||
| 320.3.p.n | 6 | 8.d | odd | 2 | 1 | ||
| 320.3.p.n | 6 | 40.k | even | 4 | 1 | ||
| 800.3.p.i | 6 | 20.d | odd | 2 | 1 | ||
| 800.3.p.i | 6 | 20.e | even | 4 | 1 | ||
| 800.3.p.j | 6 | 5.b | even | 2 | 1 | ||
| 800.3.p.j | 6 | 5.c | odd | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(160, [\chi])\):
|
\( T_{3}^{6} + 16T_{3}^{3} + 196T_{3}^{2} + 224T_{3} + 128 \)
|
|
\( T_{13}^{6} + 14T_{13}^{5} + 98T_{13}^{4} - 4000T_{13}^{3} + 62500T_{13}^{2} - 125000T_{13} + 125000 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( T^{6} + 16 T^{3} + \cdots + 128 \)
|
| $5$ |
\( T^{6} - 4 T^{5} + \cdots + 15625 \)
|
| $7$ |
\( T^{6} + 8 T^{5} + \cdots + 282752 \)
|
| $11$ |
\( (T^{3} - 16 T^{2} + \cdots + 1600)^{2} \)
|
| $13$ |
\( T^{6} + 14 T^{5} + \cdots + 125000 \)
|
| $17$ |
\( T^{6} + 14 T^{5} + \cdots + 1445000 \)
|
| $19$ |
\( T^{6} + 1216 T^{4} + \cdots + 40960000 \)
|
| $23$ |
\( T^{6} + 56 T^{5} + \cdots + 282752 \)
|
| $29$ |
\( T^{6} + 1824 T^{4} + \cdots + 46022656 \)
|
| $31$ |
\( (T^{3} - 104 T^{2} + \cdots - 34400)^{2} \)
|
| $37$ |
\( T^{6} + \cdots + 2332445000 \)
|
| $41$ |
\( (T^{3} - 60 T^{2} + \cdots + 1600)^{2} \)
|
| $43$ |
\( T^{6} + \cdots + 2069560448 \)
|
| $47$ |
\( T^{6} + \cdots + 24436414592 \)
|
| $53$ |
\( T^{6} + 122 T^{5} + \cdots + 85805000 \)
|
| $59$ |
\( T^{6} + \cdots + 25600000000 \)
|
| $61$ |
\( (T^{3} + 108 T^{2} + \cdots + 18080)^{2} \)
|
| $67$ |
\( T^{6} + \cdots + 7869603968 \)
|
| $71$ |
\( (T^{3} - 168 T^{2} + \cdots + 628000)^{2} \)
|
| $73$ |
\( T^{6} + \cdots + 2775125000 \)
|
| $79$ |
\( T^{6} + \cdots + 10485760000 \)
|
| $83$ |
\( T^{6} + \cdots + 34995996800 \)
|
| $89$ |
\( T^{6} + 7552 T^{4} + \cdots + 4194304 \)
|
| $97$ |
\( T^{6} + \cdots + 2269515125000 \)
|
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