Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [49,12,Mod(18,49)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(49, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([4]))
N = Newforms(chi, 12, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("49.18");
S:= CuspForms(chi, 12);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 49 = 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 12 \) |
| Character orbit: | \([\chi]\) | \(=\) | 49.c (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: | \(37.6488158474\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\Q(\sqrt{-3}, \sqrt{-1123})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 280x^{2} - 281x + 78961 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{9}]\) |
| Coefficient ring index: | \( 2^{2}\cdot 3 \) |
| Twist minimal: | no (minimal twist has level 7) |
| 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
\( x^{4} - x^{3} - 280x^{2} - 281x + 78961 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} + 280\nu^{2} - 280\nu - 78961 ) / 78680 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -2\nu^{3} + 2\nu^{2} + 1122\nu + 281 ) / 281 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 563\nu^{3} + 280\nu^{2} + 157080\nu - 315563 ) / 78680 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + \beta_{2} - 3\beta_1 ) / 6 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{3} + 2\beta_{2} + 1683\beta _1 + 1683 ) / 6 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 280\beta_{3} - 140\beta_{2} + 1263 ) / 3 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/49\mathbb{Z}\right)^\times\).
| \(n\) | \(3\) |
| \(\chi(n)\) | \(-1 - \beta_{1}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 18.1 |
|
−15.5215 | − | 26.8841i | −144.129 | + | 249.639i | 542.163 | − | 939.054i | −3084.78 | − | 5343.00i | 8948.43 | 0 | −97237.1 | 47027.0 | + | 81453.2i | −95761.2 | + | 165863.i | ||||||||||||||||||
| 18.2 | 42.5215 | + | 73.6495i | 204.129 | − | 353.562i | −2592.16 | + | 4489.76i | −3665.22 | − | 6348.34i | 34719.6 | 0 | −266723. | 5235.99 | + | 9069.00i | 311701. | − | 539882.i | |||||||||||||||||||
| 30.1 | −15.5215 | + | 26.8841i | −144.129 | − | 249.639i | 542.163 | + | 939.054i | −3084.78 | + | 5343.00i | 8948.43 | 0 | −97237.1 | 47027.0 | − | 81453.2i | −95761.2 | − | 165863.i | |||||||||||||||||||
| 30.2 | 42.5215 | − | 73.6495i | 204.129 | + | 353.562i | −2592.16 | − | 4489.76i | −3665.22 | + | 6348.34i | 34719.6 | 0 | −266723. | 5235.99 | − | 9069.00i | 311701. | + | 539882.i | |||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 49.12.c.e | 4 | |
| 7.b | odd | 2 | 1 | 49.12.c.d | 4 | ||
| 7.c | even | 3 | 1 | 49.12.a.c | 2 | ||
| 7.c | even | 3 | 1 | inner | 49.12.c.e | 4 | |
| 7.d | odd | 6 | 1 | 7.12.a.a | ✓ | 2 | |
| 7.d | odd | 6 | 1 | 49.12.c.d | 4 | ||
| 21.g | even | 6 | 1 | 63.12.a.c | 2 | ||
| 28.f | even | 6 | 1 | 112.12.a.d | 2 | ||
| 35.i | odd | 6 | 1 | 175.12.a.a | 2 | ||
| 35.k | even | 12 | 2 | 175.12.b.a | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 7.12.a.a | ✓ | 2 | 7.d | odd | 6 | 1 | |
| 49.12.a.c | 2 | 7.c | even | 3 | 1 | ||
| 49.12.c.d | 4 | 7.b | odd | 2 | 1 | ||
| 49.12.c.d | 4 | 7.d | odd | 6 | 1 | ||
| 49.12.c.e | 4 | 1.a | even | 1 | 1 | trivial | |
| 49.12.c.e | 4 | 7.c | even | 3 | 1 | inner | |
| 63.12.a.c | 2 | 21.g | even | 6 | 1 | ||
| 112.12.a.d | 2 | 28.f | even | 6 | 1 | ||
| 175.12.a.a | 2 | 35.i | odd | 6 | 1 | ||
| 175.12.b.a | 4 | 35.k | even | 12 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{12}^{\mathrm{new}}(49, [\chi])\):
|
\( T_{2}^{4} - 54T_{2}^{3} + 5556T_{2}^{2} + 142560T_{2} + 6969600 \)
|
|
\( T_{3}^{4} - 120T_{3}^{3} + 132084T_{3}^{2} + 14122080T_{3} + 13849523856 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - 54 T^{3} + \cdots + 6969600 \)
|
| $3$ |
\( T^{4} + \cdots + 13849523856 \)
|
| $5$ |
\( T^{4} + \cdots + 20\!\cdots\!00 \)
|
| $7$ |
\( T^{4} \)
|
| $11$ |
\( T^{4} + \cdots + 68\!\cdots\!96 \)
|
| $13$ |
\( (T^{2} + \cdots - 3004246048160)^{2} \)
|
| $17$ |
\( T^{4} + \cdots + 18\!\cdots\!76 \)
|
| $19$ |
\( T^{4} + \cdots + 36\!\cdots\!00 \)
|
| $23$ |
\( T^{4} + \cdots + 11\!\cdots\!96 \)
|
| $29$ |
\( (T^{2} + \cdots - 11\!\cdots\!40)^{2} \)
|
| $31$ |
\( T^{4} + \cdots + 39\!\cdots\!64 \)
|
| $37$ |
\( T^{4} + \cdots + 15\!\cdots\!96 \)
|
| $41$ |
\( (T^{2} + \cdots + 28\!\cdots\!16)^{2} \)
|
| $43$ |
\( (T^{2} + \cdots + 54\!\cdots\!20)^{2} \)
|
| $47$ |
\( T^{4} + \cdots + 23\!\cdots\!24 \)
|
| $53$ |
\( T^{4} + \cdots + 96\!\cdots\!44 \)
|
| $59$ |
\( T^{4} + \cdots + 11\!\cdots\!00 \)
|
| $61$ |
\( T^{4} + \cdots + 20\!\cdots\!36 \)
|
| $67$ |
\( T^{4} + \cdots + 13\!\cdots\!04 \)
|
| $71$ |
\( (T^{2} + \cdots + 13\!\cdots\!60)^{2} \)
|
| $73$ |
\( T^{4} + \cdots + 11\!\cdots\!64 \)
|
| $79$ |
\( T^{4} + \cdots + 40\!\cdots\!00 \)
|
| $83$ |
\( (T^{2} + \cdots - 37\!\cdots\!88)^{2} \)
|
| $89$ |
\( T^{4} + \cdots + 42\!\cdots\!00 \)
|
| $97$ |
\( (T^{2} + \cdots + 90\!\cdots\!08)^{2} \)
|
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