Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [3997,1,Mod(13,3997)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3997, base_ring=CyclotomicField(570))
chi = DirichletCharacter(H, H._module([285, 352]))
N = Newforms(chi, 1, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("3997.13");
S:= CuspForms(chi, 1);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 3997 = 7 \cdot 571 \) |
| Weight: | \( k \) | \(=\) | \( 1 \) |
| Character orbit: | \([\chi]\) | \(=\) | 3997.cz (of order \(570\), degree \(144\), 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: | \(1.99476285549\) |
| Analytic rank: | \(0\) |
| Dimension: | \(144\) |
| Coefficient field: | \(\Q(\zeta_{570})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{144} - x^{143} + x^{142} + x^{139} - x^{138} + x^{137} - x^{129} + x^{128} - x^{127} + x^{125} + \cdots + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{4}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Projective image: | \(D_{285}\) |
| Projective field: | Galois closure of \(\mathbb{Q}[x]/(x^{285} - \cdots)\) |
$q$-expansion
comment: q-expansion
sage: f.q_expansion() # note that sage often uses an isomorphic number field
gp: mfcoefs(f, 20)
The \(q\)-expansion and trace form are shown below.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3997\mathbb{Z}\right)^\times\).
| \(n\) | \(1716\) | \(2285\) |
| \(\chi(n)\) | \(\zeta_{570}^{176}\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 13.1 |
|
0.341150 | − | 1.74650i | 0 | −2.00737 | − | 0.815325i | 0 | 0 | −0.461341 | + | 0.887223i | −1.13548 | + | 1.73798i | −0.739446 | − | 0.673216i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 34.1 | 1.66575 | − | 0.0919017i | 0 | 1.77233 | − | 0.196161i | 0 | 0 | −0.724425 | − | 0.689353i | 1.28870 | − | 0.215046i | −0.421786 | − | 0.906696i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 83.1 | 0.967953 | − | 1.17532i | 0 | −0.252734 | − | 1.29386i | 0 | 0 | 0.652586 | + | 0.757715i | −0.426247 | − | 0.230673i | −0.256156 | + | 0.966635i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 97.1 | 1.62671 | + | 1.12794i | 0 | 1.02331 | + | 2.73313i | 0 | 0 | 0.490424 | + | 0.871484i | −0.932234 | + | 3.68131i | −0.997024 | − | 0.0770854i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 139.1 | 1.15042 | − | 1.11915i | 0 | 0.0434243 | − | 1.57535i | 0 | 0 | 0.828009 | + | 0.560715i | −0.626073 | − | 0.680097i | −0.480787 | + | 0.876837i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 174.1 | −0.298517 | − | 0.362469i | 0 | 0.149439 | − | 0.765045i | 0 | 0 | −0.518970 | − | 0.854793i | −0.734890 | + | 0.397702i | −0.360939 | + | 0.932589i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 202.1 | −0.315447 | − | 1.00854i | 0 | −0.0958675 | + | 0.0664730i | 0 | 0 | 0.746821 | + | 0.665025i | −0.736619 | − | 0.573334i | 0.618553 | + | 0.785743i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 223.1 | −1.08088 | + | 0.439016i | 0 | 0.258785 | − | 0.251750i | 0 | 0 | −0.574329 | + | 0.818625i | 0.299439 | − | 0.682651i | 0.0935596 | − | 0.995614i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 237.1 | −0.579560 | + | 1.85296i | 0 | −2.27578 | − | 1.57799i | 0 | 0 | −0.995083 | + | 0.0990455i | 2.71079 | − | 2.10989i | 0.938430 | + | 0.345471i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 244.1 | 1.75911 | + | 0.714490i | 0 | 1.86720 | + | 1.81644i | 0 | 0 | −0.956036 | + | 0.293250i | 1.22410 | + | 2.79067i | 0.509516 | − | 0.860461i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 279.1 | 1.98014 | − | 0.219162i | 0 | 2.89711 | − | 0.649259i | 0 | 0 | −0.627176 | − | 0.778877i | 3.71011 | − | 1.27368i | 0.528360 | + | 0.849020i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 293.1 | −0.874198 | − | 1.72746i | 0 | −1.62767 | + | 2.21452i | 0 | 0 | 0.180881 | − | 0.983505i | 3.33875 | + | 0.557139i | 0.224056 | − | 0.974576i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 321.1 | 1.92132 | + | 0.430577i | 0 | 2.60171 | + | 1.22778i | 0 | 0 | −0.213300 | − | 0.976987i | 2.91626 | + | 2.26982i | −0.441671 | − | 0.897177i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 342.1 | −1.70487 | + | 0.188695i | 0 | 1.89518 | − | 0.424719i | 0 | 0 | −0.934564 | + | 0.355794i | −1.52855 | + | 0.524751i | 0.0715891 | − | 0.997434i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 482.1 | −0.660728 | − | 1.76473i | 0 | −1.92359 | + | 1.67525i | 0 | 0 | 0.652586 | + | 0.757715i | 2.57009 | + | 1.39086i | −0.709053 | − | 0.705155i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 531.1 | −1.26766 | − | 1.53924i | 0 | −0.570575 | + | 2.92103i | 0 | 0 | 0.922290 | + | 0.386499i | 3.46574 | − | 1.87557i | 0.775409 | + | 0.631460i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 538.1 | 1.17577 | − | 1.59970i | 0 | −0.878074 | − | 2.80736i | 0 | 0 | 0.0495838 | − | 0.998770i | −3.64560 | − | 1.25154i | 0.984487 | − | 0.175457i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 552.1 | −0.238657 | − | 1.72107i | 0 | −1.94285 | + | 0.549385i | 0 | 0 | 0.601081 | − | 0.799188i | 0.711243 | + | 1.62147i | −0.319482 | + | 0.947592i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 559.1 | −0.485567 | + | 0.959507i | 0 | −0.0926427 | − | 0.126045i | 0 | 0 | 0.431754 | − | 0.901991i | −0.894782 | + | 0.149313i | −0.857640 | + | 0.514250i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 580.1 | 1.76667 | − | 0.499567i | 0 | 2.01965 | − | 1.24147i | 0 | 0 | 0.828009 | − | 0.560715i | 1.70440 | − | 1.85147i | 0.999757 | + | 0.0220445i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| See next 80 embeddings (of 144 total) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-7}) \) |
| 571.o | even | 285 | 1 | inner |
| 3997.cz | odd | 570 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 3997.1.cz.a | ✓ | 144 |
| 7.b | odd | 2 | 1 | CM | 3997.1.cz.a | ✓ | 144 |
| 571.o | even | 285 | 1 | inner | 3997.1.cz.a | ✓ | 144 |
| 3997.cz | odd | 570 | 1 | inner | 3997.1.cz.a | ✓ | 144 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 3997.1.cz.a | ✓ | 144 | 1.a | even | 1 | 1 | trivial |
| 3997.1.cz.a | ✓ | 144 | 7.b | odd | 2 | 1 | CM |
| 3997.1.cz.a | ✓ | 144 | 571.o | even | 285 | 1 | inner |
| 3997.1.cz.a | ✓ | 144 | 3997.cz | odd | 570 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(3997, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{144} - T^{143} + \cdots + 1 \)
|
| $3$ |
\( T^{144} \)
|
| $5$ |
\( T^{144} \)
|
| $7$ |
\( (T^{72} - T^{71} + T^{67} + \cdots + 1)^{2} \)
|
| $11$ |
\( T^{144} - 6 T^{143} + \cdots + 1 \)
|
| $13$ |
\( T^{144} \)
|
| $17$ |
\( T^{144} \)
|
| $19$ |
\( T^{144} \)
|
| $23$ |
\( T^{144} - 3 T^{143} + \cdots + 1 \)
|
| $29$ |
\( (T^{36} + T^{35} - T^{33} + \cdots + 1)^{4} \)
|
| $31$ |
\( T^{144} \)
|
| $37$ |
\( T^{144} + 2 T^{143} + \cdots + 1 \)
|
| $41$ |
\( T^{144} \)
|
| $43$ |
\( T^{144} + 9 T^{143} + \cdots + 1 \)
|
| $47$ |
\( T^{144} \)
|
| $53$ |
\( T^{144} - 6 T^{143} + \cdots + 1 \)
|
| $59$ |
\( T^{144} \)
|
| $61$ |
\( T^{144} \)
|
| $67$ |
\( T^{144} - 20 T^{143} + \cdots + 1 \)
|
| $71$ |
\( T^{144} - 3 T^{143} + \cdots + 1 \)
|
| $73$ |
\( T^{144} \)
|
| $79$ |
\( T^{144} - 6 T^{143} + \cdots + 1 \)
|
| $83$ |
\( T^{144} \)
|
| $89$ |
\( T^{144} \)
|
| $97$ |
\( T^{144} \)
|
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