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: | \(1\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\Q(\sqrt{2}, \sqrt{-3})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 2x^{2} + 4 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| 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
\( x^{4} + 2x^{2} + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 2\beta_{2} \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{3} \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/637\mathbb{Z}\right)^\times\).
| \(n\) | \(197\) | \(248\) |
| \(\chi(n)\) | \(\beta_{2}\) | \(\beta_{2}\) |
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 |
|
−2.41421 | 0.707107 | + | 1.22474i | 3.82843 | −1.91421 | − | 3.31552i | −1.70711 | − | 2.95680i | 0 | −4.41421 | 0.500000 | − | 0.866025i | 4.62132 | + | 8.00436i | ||||||||||||||||||||
| 165.2 | 0.414214 | −0.707107 | − | 1.22474i | −1.82843 | 0.914214 | + | 1.58346i | −0.292893 | − | 0.507306i | 0 | −1.58579 | 0.500000 | − | 0.866025i | 0.378680 | + | 0.655892i | |||||||||||||||||||||
| 471.1 | −2.41421 | 0.707107 | − | 1.22474i | 3.82843 | −1.91421 | + | 3.31552i | −1.70711 | + | 2.95680i | 0 | −4.41421 | 0.500000 | + | 0.866025i | 4.62132 | − | 8.00436i | |||||||||||||||||||||
| 471.2 | 0.414214 | −0.707107 | + | 1.22474i | −1.82843 | 0.914214 | − | 1.58346i | −0.292893 | + | 0.507306i | 0 | −1.58579 | 0.500000 | + | 0.866025i | 0.378680 | − | 0.655892i | |||||||||||||||||||||
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.b | 4 | |
| 7.b | odd | 2 | 1 | 637.2.h.c | 4 | ||
| 7.c | even | 3 | 1 | 637.2.f.f | yes | 4 | |
| 7.c | even | 3 | 1 | 637.2.g.f | 4 | ||
| 7.d | odd | 6 | 1 | 637.2.f.e | ✓ | 4 | |
| 7.d | odd | 6 | 1 | 637.2.g.g | 4 | ||
| 13.c | even | 3 | 1 | 637.2.g.f | 4 | ||
| 91.g | even | 3 | 1 | 637.2.f.f | yes | 4 | |
| 91.h | even | 3 | 1 | inner | 637.2.h.b | 4 | |
| 91.h | even | 3 | 1 | 8281.2.a.p | 2 | ||
| 91.k | even | 6 | 1 | 8281.2.a.x | 2 | ||
| 91.l | odd | 6 | 1 | 8281.2.a.y | 2 | ||
| 91.m | odd | 6 | 1 | 637.2.f.e | ✓ | 4 | |
| 91.n | odd | 6 | 1 | 637.2.g.g | 4 | ||
| 91.v | odd | 6 | 1 | 637.2.h.c | 4 | ||
| 91.v | odd | 6 | 1 | 8281.2.a.o | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 637.2.f.e | ✓ | 4 | 7.d | odd | 6 | 1 | |
| 637.2.f.e | ✓ | 4 | 91.m | odd | 6 | 1 | |
| 637.2.f.f | yes | 4 | 7.c | even | 3 | 1 | |
| 637.2.f.f | yes | 4 | 91.g | even | 3 | 1 | |
| 637.2.g.f | 4 | 7.c | even | 3 | 1 | ||
| 637.2.g.f | 4 | 13.c | even | 3 | 1 | ||
| 637.2.g.g | 4 | 7.d | odd | 6 | 1 | ||
| 637.2.g.g | 4 | 91.n | odd | 6 | 1 | ||
| 637.2.h.b | 4 | 1.a | even | 1 | 1 | trivial | |
| 637.2.h.b | 4 | 91.h | even | 3 | 1 | inner | |
| 637.2.h.c | 4 | 7.b | odd | 2 | 1 | ||
| 637.2.h.c | 4 | 91.v | odd | 6 | 1 | ||
| 8281.2.a.o | 2 | 91.v | odd | 6 | 1 | ||
| 8281.2.a.p | 2 | 91.h | even | 3 | 1 | ||
| 8281.2.a.x | 2 | 91.k | even | 6 | 1 | ||
| 8281.2.a.y | 2 | 91.l | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(637, [\chi])\):
|
\( T_{2}^{2} + 2T_{2} - 1 \)
|
|
\( T_{3}^{4} + 2T_{3}^{2} + 4 \)
|
|
\( T_{5}^{4} + 2T_{5}^{3} + 11T_{5}^{2} - 14T_{5} + 49 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 2 T - 1)^{2} \)
|
| $3$ |
\( T^{4} + 2T^{2} + 4 \)
|
| $5$ |
\( T^{4} + 2 T^{3} + \cdots + 49 \)
|
| $7$ |
\( T^{4} \)
|
| $11$ |
\( T^{4} + 4 T^{3} + \cdots + 4 \)
|
| $13$ |
\( (T^{2} + 7 T + 13)^{2} \)
|
| $17$ |
\( (T^{2} + 6 T + 1)^{2} \)
|
| $19$ |
\( (T^{2} + 6 T + 36)^{2} \)
|
| $23$ |
\( (T^{2} - 2)^{2} \)
|
| $29$ |
\( T^{4} + 14 T^{3} + \cdots + 1681 \)
|
| $31$ |
\( T^{4} + 8 T^{3} + \cdots + 196 \)
|
| $37$ |
\( (T^{2} + 2 T - 71)^{2} \)
|
| $41$ |
\( T^{4} + 6 T^{3} + \cdots + 1 \)
|
| $43$ |
\( T^{4} + 4 T^{3} + \cdots + 4 \)
|
| $47$ |
\( T^{4} + 4 T^{3} + \cdots + 784 \)
|
| $53$ |
\( (T^{2} - 3 T + 9)^{2} \)
|
| $59$ |
\( (T^{2} + 12 T + 18)^{2} \)
|
| $61$ |
\( T^{4} + 14 T^{3} + \cdots + 1681 \)
|
| $67$ |
\( T^{4} + 18T^{2} + 324 \)
|
| $71$ |
\( T^{4} - 12 T^{3} + \cdots + 16 \)
|
| $73$ |
\( T^{4} - 10 T^{3} + \cdots + 49 \)
|
| $79$ |
\( T^{4} + 12 T^{3} + \cdots + 324 \)
|
| $83$ |
\( (T^{2} + 12 T - 14)^{2} \)
|
| $89$ |
\( (T^{2} - 8 T - 112)^{2} \)
|
| $97$ |
\( T^{4} - 16 T^{3} + \cdots + 3136 \)
|
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