Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [637,2,Mod(165,637)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(637, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([4, 4]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("637.165");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 637 = 7^{2} \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 637.h (of order \(3\), degree \(2\), not 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: | \(5.08647060876\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\Q(\sqrt{-3}, \sqrt{5})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} + 2x^{2} + x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 91) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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
\( x^{4} - x^{3} + 2x^{2} + x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} + 1 ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{3} + 2\nu^{2} - 2\nu - 1 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + \beta_{2} + \beta_1 \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{2} - 1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/637\mathbb{Z}\right)^\times\).
| \(n\) | \(197\) | \(248\) |
| \(\chi(n)\) | \(\beta_{3}\) | \(\beta_{3}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 165.1 |
|
0.381966 | 0.190983 | + | 0.330792i | −1.85410 | −0.190983 | − | 0.330792i | 0.0729490 | + | 0.126351i | 0 | −1.47214 | 1.42705 | − | 2.47172i | −0.0729490 | − | 0.126351i | ||||||||||||||||||||
| 165.2 | 2.61803 | 1.30902 | + | 2.26728i | 4.85410 | −1.30902 | − | 2.26728i | 3.42705 | + | 5.93583i | 0 | 7.47214 | −1.92705 | + | 3.33775i | −3.42705 | − | 5.93583i | |||||||||||||||||||||
| 471.1 | 0.381966 | 0.190983 | − | 0.330792i | −1.85410 | −0.190983 | + | 0.330792i | 0.0729490 | − | 0.126351i | 0 | −1.47214 | 1.42705 | + | 2.47172i | −0.0729490 | + | 0.126351i | |||||||||||||||||||||
| 471.2 | 2.61803 | 1.30902 | − | 2.26728i | 4.85410 | −1.30902 | + | 2.26728i | 3.42705 | − | 5.93583i | 0 | 7.47214 | −1.92705 | − | 3.33775i | −3.42705 | + | 5.93583i | |||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 91.h | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 637.2.h.g | 4 | |
| 7.b | odd | 2 | 1 | 637.2.h.f | 4 | ||
| 7.c | even | 3 | 1 | 91.2.f.a | ✓ | 4 | |
| 7.c | even | 3 | 1 | 637.2.g.b | 4 | ||
| 7.d | odd | 6 | 1 | 637.2.f.c | 4 | ||
| 7.d | odd | 6 | 1 | 637.2.g.c | 4 | ||
| 13.c | even | 3 | 1 | 637.2.g.b | 4 | ||
| 21.h | odd | 6 | 1 | 819.2.o.c | 4 | ||
| 28.g | odd | 6 | 1 | 1456.2.s.h | 4 | ||
| 91.g | even | 3 | 1 | 91.2.f.a | ✓ | 4 | |
| 91.h | even | 3 | 1 | inner | 637.2.h.g | 4 | |
| 91.h | even | 3 | 1 | 1183.2.a.g | 2 | ||
| 91.k | even | 6 | 1 | 1183.2.a.c | 2 | ||
| 91.l | odd | 6 | 1 | 8281.2.a.n | 2 | ||
| 91.m | odd | 6 | 1 | 637.2.f.c | 4 | ||
| 91.n | odd | 6 | 1 | 637.2.g.c | 4 | ||
| 91.v | odd | 6 | 1 | 637.2.h.f | 4 | ||
| 91.v | odd | 6 | 1 | 8281.2.a.bb | 2 | ||
| 91.x | odd | 12 | 2 | 1183.2.c.c | 4 | ||
| 273.bm | odd | 6 | 1 | 819.2.o.c | 4 | ||
| 364.q | odd | 6 | 1 | 1456.2.s.h | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 91.2.f.a | ✓ | 4 | 7.c | even | 3 | 1 | |
| 91.2.f.a | ✓ | 4 | 91.g | even | 3 | 1 | |
| 637.2.f.c | 4 | 7.d | odd | 6 | 1 | ||
| 637.2.f.c | 4 | 91.m | odd | 6 | 1 | ||
| 637.2.g.b | 4 | 7.c | even | 3 | 1 | ||
| 637.2.g.b | 4 | 13.c | even | 3 | 1 | ||
| 637.2.g.c | 4 | 7.d | odd | 6 | 1 | ||
| 637.2.g.c | 4 | 91.n | odd | 6 | 1 | ||
| 637.2.h.f | 4 | 7.b | odd | 2 | 1 | ||
| 637.2.h.f | 4 | 91.v | odd | 6 | 1 | ||
| 637.2.h.g | 4 | 1.a | even | 1 | 1 | trivial | |
| 637.2.h.g | 4 | 91.h | even | 3 | 1 | inner | |
| 819.2.o.c | 4 | 21.h | odd | 6 | 1 | ||
| 819.2.o.c | 4 | 273.bm | odd | 6 | 1 | ||
| 1183.2.a.c | 2 | 91.k | even | 6 | 1 | ||
| 1183.2.a.g | 2 | 91.h | even | 3 | 1 | ||
| 1183.2.c.c | 4 | 91.x | odd | 12 | 2 | ||
| 1456.2.s.h | 4 | 28.g | odd | 6 | 1 | ||
| 1456.2.s.h | 4 | 364.q | odd | 6 | 1 | ||
| 8281.2.a.n | 2 | 91.l | odd | 6 | 1 | ||
| 8281.2.a.bb | 2 | 91.v | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(637, [\chi])\):
|
\( T_{2}^{2} - 3T_{2} + 1 \)
|
|
\( T_{3}^{4} - 3T_{3}^{3} + 8T_{3}^{2} - 3T_{3} + 1 \)
|
|
\( T_{5}^{4} + 3T_{5}^{3} + 8T_{5}^{2} + 3T_{5} + 1 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} - 3 T + 1)^{2} \)
|
| $3$ |
\( T^{4} - 3 T^{3} + \cdots + 1 \)
|
| $5$ |
\( T^{4} + 3 T^{3} + \cdots + 1 \)
|
| $7$ |
\( T^{4} \)
|
| $11$ |
\( T^{4} - 3 T^{3} + \cdots + 81 \)
|
| $13$ |
\( (T^{2} + 5 T + 13)^{2} \)
|
| $17$ |
\( (T^{2} - 6 T - 11)^{2} \)
|
| $19$ |
\( T^{4} + 3 T^{3} + \cdots + 81 \)
|
| $23$ |
\( (T^{2} - 20)^{2} \)
|
| $29$ |
\( T^{4} + 3 T^{3} + \cdots + 841 \)
|
| $31$ |
\( T^{4} + 4 T^{3} + \cdots + 1681 \)
|
| $37$ |
\( (T - 4)^{4} \)
|
| $41$ |
\( T^{4} + 6 T^{3} + \cdots + 16 \)
|
| $43$ |
\( T^{4} + 5 T^{3} + \cdots + 9025 \)
|
| $47$ |
\( T^{4} + 5T^{2} + 25 \)
|
| $53$ |
\( T^{4} + 12 T^{3} + \cdots + 961 \)
|
| $59$ |
\( (T^{2} - 5)^{2} \)
|
| $61$ |
\( (T^{2} - 6 T + 36)^{2} \)
|
| $67$ |
\( T^{4} - 12 T^{3} + \cdots + 81 \)
|
| $71$ |
\( T^{4} - 6 T^{3} + \cdots + 13456 \)
|
| $73$ |
\( (T^{2} - 2 T + 4)^{2} \)
|
| $79$ |
\( (T^{2} + 4 T + 16)^{2} \)
|
| $83$ |
\( (T^{2} - 45)^{2} \)
|
| $89$ |
\( (T^{2} - 21 T + 79)^{2} \)
|
| $97$ |
\( T^{4} + 31 T^{3} + \cdots + 52441 \)
|
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