Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [169,4,Mod(22,169)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(169, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([4]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("169.22");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 169 = 13^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 169.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: | \(9.97132279097\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\Q(\sqrt{-3}, \sqrt{17})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} + 5x^{2} + 4x + 16 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{9}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 13) |
| 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
\( x^{4} - x^{3} + 5x^{2} + 4x + 16 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{3} + 5\nu^{2} - 5\nu + 16 ) / 20 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} + 4 ) / 5 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + 4\beta_{2} + \beta _1 - 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( 5\beta_{3} - 4 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/169\mathbb{Z}\right)^\times\).
| \(n\) | \(2\) |
| \(\chi(n)\) | \(-1 + \beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 22.1 |
|
−1.28078 | + | 2.21837i | 1.84233 | − | 3.19101i | 0.719224 | + | 1.24573i | 0.561553 | 4.71922 | + | 8.17394i | −9.08854 | − | 15.7418i | −24.1771 | 6.71165 | + | 11.6249i | −0.719224 | + | 1.24573i | ||||||||||||||||
| 22.2 | 0.780776 | − | 1.35234i | −4.34233 | + | 7.52113i | 2.78078 | + | 4.81645i | −3.56155 | 6.78078 | + | 11.7446i | 13.5885 | + | 23.5360i | 21.1771 | −24.2116 | − | 41.9358i | −2.78078 | + | 4.81645i | |||||||||||||||||
| 146.1 | −1.28078 | − | 2.21837i | 1.84233 | + | 3.19101i | 0.719224 | − | 1.24573i | 0.561553 | 4.71922 | − | 8.17394i | −9.08854 | + | 15.7418i | −24.1771 | 6.71165 | − | 11.6249i | −0.719224 | − | 1.24573i | |||||||||||||||||
| 146.2 | 0.780776 | + | 1.35234i | −4.34233 | − | 7.52113i | 2.78078 | − | 4.81645i | −3.56155 | 6.78078 | − | 11.7446i | 13.5885 | − | 23.5360i | 21.1771 | −24.2116 | + | 41.9358i | −2.78078 | − | 4.81645i | |||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 13.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 169.4.c.g | 4 | |
| 13.b | even | 2 | 1 | 169.4.c.j | 4 | ||
| 13.c | even | 3 | 1 | 13.4.a.b | ✓ | 2 | |
| 13.c | even | 3 | 1 | inner | 169.4.c.g | 4 | |
| 13.d | odd | 4 | 2 | 169.4.e.f | 8 | ||
| 13.e | even | 6 | 1 | 169.4.a.g | 2 | ||
| 13.e | even | 6 | 1 | 169.4.c.j | 4 | ||
| 13.f | odd | 12 | 2 | 169.4.b.f | 4 | ||
| 13.f | odd | 12 | 2 | 169.4.e.f | 8 | ||
| 39.h | odd | 6 | 1 | 1521.4.a.r | 2 | ||
| 39.i | odd | 6 | 1 | 117.4.a.d | 2 | ||
| 52.j | odd | 6 | 1 | 208.4.a.h | 2 | ||
| 65.n | even | 6 | 1 | 325.4.a.f | 2 | ||
| 65.q | odd | 12 | 2 | 325.4.b.e | 4 | ||
| 91.n | odd | 6 | 1 | 637.4.a.b | 2 | ||
| 104.n | odd | 6 | 1 | 832.4.a.z | 2 | ||
| 104.r | even | 6 | 1 | 832.4.a.s | 2 | ||
| 143.k | odd | 6 | 1 | 1573.4.a.b | 2 | ||
| 156.p | even | 6 | 1 | 1872.4.a.bb | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 13.4.a.b | ✓ | 2 | 13.c | even | 3 | 1 | |
| 117.4.a.d | 2 | 39.i | odd | 6 | 1 | ||
| 169.4.a.g | 2 | 13.e | even | 6 | 1 | ||
| 169.4.b.f | 4 | 13.f | odd | 12 | 2 | ||
| 169.4.c.g | 4 | 1.a | even | 1 | 1 | trivial | |
| 169.4.c.g | 4 | 13.c | even | 3 | 1 | inner | |
| 169.4.c.j | 4 | 13.b | even | 2 | 1 | ||
| 169.4.c.j | 4 | 13.e | even | 6 | 1 | ||
| 169.4.e.f | 8 | 13.d | odd | 4 | 2 | ||
| 169.4.e.f | 8 | 13.f | odd | 12 | 2 | ||
| 208.4.a.h | 2 | 52.j | odd | 6 | 1 | ||
| 325.4.a.f | 2 | 65.n | even | 6 | 1 | ||
| 325.4.b.e | 4 | 65.q | odd | 12 | 2 | ||
| 637.4.a.b | 2 | 91.n | odd | 6 | 1 | ||
| 832.4.a.s | 2 | 104.r | even | 6 | 1 | ||
| 832.4.a.z | 2 | 104.n | odd | 6 | 1 | ||
| 1521.4.a.r | 2 | 39.h | odd | 6 | 1 | ||
| 1573.4.a.b | 2 | 143.k | odd | 6 | 1 | ||
| 1872.4.a.bb | 2 | 156.p | even | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} + T_{2}^{3} + 5T_{2}^{2} - 4T_{2} + 16 \)
acting on \(S_{4}^{\mathrm{new}}(169, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + T^{3} + \cdots + 16 \)
|
| $3$ |
\( T^{4} + 5 T^{3} + \cdots + 1024 \)
|
| $5$ |
\( (T^{2} + 3 T - 2)^{2} \)
|
| $7$ |
\( T^{4} - 9 T^{3} + \cdots + 244036 \)
|
| $11$ |
\( T^{4} + 80 T^{3} + \cdots + 976144 \)
|
| $13$ |
\( T^{4} \)
|
| $17$ |
\( T^{4} + 19 T^{3} + \cdots + 1295044 \)
|
| $19$ |
\( T^{4} - 84 T^{3} + \cdots + 6697744 \)
|
| $23$ |
\( T^{4} + 196 T^{3} + \cdots + 80856064 \)
|
| $29$ |
\( T^{4} + \cdots + 1496451856 \)
|
| $31$ |
\( (T^{2} + 86 T - 3064)^{2} \)
|
| $37$ |
\( T^{4} + 209 T^{3} + \cdots + 116942596 \)
|
| $41$ |
\( T^{4} - 230 T^{3} + \cdots + 124724224 \)
|
| $43$ |
\( T^{4} + \cdots + 4397811856 \)
|
| $47$ |
\( (T^{2} - 435 T - 14918)^{2} \)
|
| $53$ |
\( (T^{2} + 118 T - 344)^{2} \)
|
| $59$ |
\( T^{4} - 368 T^{3} + \cdots + 991746064 \)
|
| $61$ |
\( T^{4} + \cdots + 15981005056 \)
|
| $67$ |
\( T^{4} + \cdots + 51799939216 \)
|
| $71$ |
\( T^{4} + \cdots + 49503580036 \)
|
| $73$ |
\( (T^{2} - 456 T - 235316)^{2} \)
|
| $79$ |
\( (T^{2} + 1008 T + 247216)^{2} \)
|
| $83$ |
\( (T^{2} - 1958 T + 817664)^{2} \)
|
| $89$ |
\( T^{4} + \cdots + 260316284944 \)
|
| $97$ |
\( T^{4} + \cdots + 776999938576 \)
|
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