Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [169,3,Mod(70,169)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(169, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([3]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("169.70");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 169 = 13^{2} \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 169.d (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: | \(4.60491646769\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(i)\) |
| Coefficient field: | \(\Q(i, \sqrt{10})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 25 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{4}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 13) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} + 25 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} ) / 5 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} ) / 5 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 5\beta_{2} \)
|
| \(\nu^{3}\) | \(=\) |
\( 5\beta_{3} \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/169\mathbb{Z}\right)^\times\).
| \(n\) | \(2\) |
| \(\chi(n)\) | \(\beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 70.1 |
|
−0.581139 | + | 0.581139i | −4.16228 | 3.32456i | −3.58114 | + | 3.58114i | 2.41886 | − | 2.41886i | 4.58114 | + | 4.58114i | −4.25658 | − | 4.25658i | 8.32456 | − | 4.16228i | |||||||||||||||||||
| 70.2 | 2.58114 | − | 2.58114i | 2.16228 | − | 9.32456i | −0.418861 | + | 0.418861i | 5.58114 | − | 5.58114i | 1.41886 | + | 1.41886i | −13.7434 | − | 13.7434i | −4.32456 | 2.16228i | ||||||||||||||||||||
| 99.1 | −0.581139 | − | 0.581139i | −4.16228 | − | 3.32456i | −3.58114 | − | 3.58114i | 2.41886 | + | 2.41886i | 4.58114 | − | 4.58114i | −4.25658 | + | 4.25658i | 8.32456 | 4.16228i | ||||||||||||||||||||
| 99.2 | 2.58114 | + | 2.58114i | 2.16228 | 9.32456i | −0.418861 | − | 0.418861i | 5.58114 | + | 5.58114i | 1.41886 | − | 1.41886i | −13.7434 | + | 13.7434i | −4.32456 | − | 2.16228i | ||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 13.d | odd | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 169.3.d.d | 4 | |
| 13.b | even | 2 | 1 | 13.3.d.a | ✓ | 4 | |
| 13.c | even | 3 | 2 | 169.3.f.d | 8 | ||
| 13.d | odd | 4 | 1 | 13.3.d.a | ✓ | 4 | |
| 13.d | odd | 4 | 1 | inner | 169.3.d.d | 4 | |
| 13.e | even | 6 | 2 | 169.3.f.f | 8 | ||
| 13.f | odd | 12 | 2 | 169.3.f.d | 8 | ||
| 13.f | odd | 12 | 2 | 169.3.f.f | 8 | ||
| 39.d | odd | 2 | 1 | 117.3.j.a | 4 | ||
| 39.f | even | 4 | 1 | 117.3.j.a | 4 | ||
| 52.b | odd | 2 | 1 | 208.3.t.c | 4 | ||
| 52.f | even | 4 | 1 | 208.3.t.c | 4 | ||
| 65.d | even | 2 | 1 | 325.3.j.a | 4 | ||
| 65.f | even | 4 | 1 | 325.3.g.a | 4 | ||
| 65.g | odd | 4 | 1 | 325.3.j.a | 4 | ||
| 65.h | odd | 4 | 1 | 325.3.g.a | 4 | ||
| 65.h | odd | 4 | 1 | 325.3.g.b | 4 | ||
| 65.k | even | 4 | 1 | 325.3.g.b | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 13.3.d.a | ✓ | 4 | 13.b | even | 2 | 1 | |
| 13.3.d.a | ✓ | 4 | 13.d | odd | 4 | 1 | |
| 117.3.j.a | 4 | 39.d | odd | 2 | 1 | ||
| 117.3.j.a | 4 | 39.f | even | 4 | 1 | ||
| 169.3.d.d | 4 | 1.a | even | 1 | 1 | trivial | |
| 169.3.d.d | 4 | 13.d | odd | 4 | 1 | inner | |
| 169.3.f.d | 8 | 13.c | even | 3 | 2 | ||
| 169.3.f.d | 8 | 13.f | odd | 12 | 2 | ||
| 169.3.f.f | 8 | 13.e | even | 6 | 2 | ||
| 169.3.f.f | 8 | 13.f | odd | 12 | 2 | ||
| 208.3.t.c | 4 | 52.b | odd | 2 | 1 | ||
| 208.3.t.c | 4 | 52.f | even | 4 | 1 | ||
| 325.3.g.a | 4 | 65.f | even | 4 | 1 | ||
| 325.3.g.a | 4 | 65.h | odd | 4 | 1 | ||
| 325.3.g.b | 4 | 65.h | odd | 4 | 1 | ||
| 325.3.g.b | 4 | 65.k | even | 4 | 1 | ||
| 325.3.j.a | 4 | 65.d | even | 2 | 1 | ||
| 325.3.j.a | 4 | 65.g | odd | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} - 4T_{2}^{3} + 8T_{2}^{2} + 12T_{2} + 9 \)
acting on \(S_{3}^{\mathrm{new}}(169, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - 4 T^{3} + \cdots + 9 \)
|
| $3$ |
\( (T^{2} + 2 T - 9)^{2} \)
|
| $5$ |
\( T^{4} + 8 T^{3} + \cdots + 9 \)
|
| $7$ |
\( T^{4} - 12 T^{3} + \cdots + 169 \)
|
| $11$ |
\( T^{4} - 4 T^{3} + \cdots + 6084 \)
|
| $13$ |
\( T^{4} \)
|
| $17$ |
\( T^{4} + 738 T^{2} + 123201 \)
|
| $19$ |
\( T^{4} + 400 \)
|
| $23$ |
\( T^{4} + 828 T^{2} + 54756 \)
|
| $29$ |
\( (T^{2} - 20 T - 150)^{2} \)
|
| $31$ |
\( T^{4} + 40 T^{3} + \cdots + 400 \)
|
| $37$ |
\( T^{4} + 40 T^{3} + \cdots + 42025 \)
|
| $41$ |
\( T^{4} + 32 T^{3} + \cdots + 11664 \)
|
| $43$ |
\( T^{4} + 1062 T^{2} + 123201 \)
|
| $47$ |
\( T^{4} - 4 T^{3} + \cdots + 6985449 \)
|
| $53$ |
\( (T^{2} + 40 T + 150)^{2} \)
|
| $59$ |
\( T^{4} + 56 T^{3} + \cdots + 12446784 \)
|
| $61$ |
\( (T^{2} + 148 T + 5436)^{2} \)
|
| $67$ |
\( T^{4} - 84 T^{3} + \cdots + 40170244 \)
|
| $71$ |
\( T^{4} + 284 T^{3} + \cdots + 85322169 \)
|
| $73$ |
\( T^{4} + 4000000 \)
|
| $79$ |
\( (T^{2} - 32 T - 954)^{2} \)
|
| $83$ |
\( T^{4} - 52 T^{3} + \cdots + 2762244 \)
|
| $89$ |
\( T^{4} + 200 T^{3} + \cdots + 2624400 \)
|
| $97$ |
\( T^{4} - 68 T^{3} + \cdots + 449524804 \)
|
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