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: | \(\Q(\zeta_{12})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 3 \) |
| 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
| \(\beta_{1}\) | \(=\) |
\( \zeta_{12}^{2} \)
|
| \(\beta_{2}\) | \(=\) |
\( \zeta_{12}^{3} + \zeta_{12} \)
|
| \(\beta_{3}\) | \(=\) |
\( -\zeta_{12}^{3} + 2\zeta_{12} \)
|
| \(\zeta_{12}\) | \(=\) |
\( ( \beta_{3} + \beta_{2} ) / 3 \)
|
| \(\zeta_{12}^{2}\) | \(=\) |
\( \beta_1 \)
|
| \(\zeta_{12}^{3}\) | \(=\) |
\( ( -\beta_{3} + 2\beta_{2} ) / 3 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/169\mathbb{Z}\right)^\times\).
| \(n\) | \(2\) |
| \(\chi(n)\) | \(-1 + \beta_{1}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 22.1 |
|
−1.73205 | + | 3.00000i | 3.50000 | − | 6.06218i | −2.00000 | − | 3.46410i | 13.8564 | 12.1244 | + | 21.0000i | 11.2583 | + | 19.5000i | −13.8564 | −11.0000 | − | 19.0526i | −24.0000 | + | 41.5692i | ||||||||||||||||
| 22.2 | 1.73205 | − | 3.00000i | 3.50000 | − | 6.06218i | −2.00000 | − | 3.46410i | −13.8564 | −12.1244 | − | 21.0000i | −11.2583 | − | 19.5000i | 13.8564 | −11.0000 | − | 19.0526i | −24.0000 | + | 41.5692i | |||||||||||||||||
| 146.1 | −1.73205 | − | 3.00000i | 3.50000 | + | 6.06218i | −2.00000 | + | 3.46410i | 13.8564 | 12.1244 | − | 21.0000i | 11.2583 | − | 19.5000i | −13.8564 | −11.0000 | + | 19.0526i | −24.0000 | − | 41.5692i | |||||||||||||||||
| 146.2 | 1.73205 | + | 3.00000i | 3.50000 | + | 6.06218i | −2.00000 | + | 3.46410i | −13.8564 | −12.1244 | + | 21.0000i | −11.2583 | + | 19.5000i | 13.8564 | −11.0000 | + | 19.0526i | −24.0000 | − | 41.5692i | |||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 13.b | even | 2 | 1 | inner |
| 13.c | even | 3 | 1 | inner |
| 13.e | even | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 169.4.c.i | 4 | |
| 13.b | even | 2 | 1 | inner | 169.4.c.i | 4 | |
| 13.c | even | 3 | 1 | 169.4.a.h | 2 | ||
| 13.c | even | 3 | 1 | inner | 169.4.c.i | 4 | |
| 13.d | odd | 4 | 1 | 13.4.e.a | ✓ | 2 | |
| 13.d | odd | 4 | 1 | 169.4.e.b | 2 | ||
| 13.e | even | 6 | 1 | 169.4.a.h | 2 | ||
| 13.e | even | 6 | 1 | inner | 169.4.c.i | 4 | |
| 13.f | odd | 12 | 1 | 13.4.e.a | ✓ | 2 | |
| 13.f | odd | 12 | 2 | 169.4.b.b | 2 | ||
| 13.f | odd | 12 | 1 | 169.4.e.b | 2 | ||
| 39.f | even | 4 | 1 | 117.4.q.c | 2 | ||
| 39.h | odd | 6 | 1 | 1521.4.a.q | 2 | ||
| 39.i | odd | 6 | 1 | 1521.4.a.q | 2 | ||
| 39.k | even | 12 | 1 | 117.4.q.c | 2 | ||
| 52.f | even | 4 | 1 | 208.4.w.a | 2 | ||
| 52.l | even | 12 | 1 | 208.4.w.a | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 13.4.e.a | ✓ | 2 | 13.d | odd | 4 | 1 | |
| 13.4.e.a | ✓ | 2 | 13.f | odd | 12 | 1 | |
| 117.4.q.c | 2 | 39.f | even | 4 | 1 | ||
| 117.4.q.c | 2 | 39.k | even | 12 | 1 | ||
| 169.4.a.h | 2 | 13.c | even | 3 | 1 | ||
| 169.4.a.h | 2 | 13.e | even | 6 | 1 | ||
| 169.4.b.b | 2 | 13.f | odd | 12 | 2 | ||
| 169.4.c.i | 4 | 1.a | even | 1 | 1 | trivial | |
| 169.4.c.i | 4 | 13.b | even | 2 | 1 | inner | |
| 169.4.c.i | 4 | 13.c | even | 3 | 1 | inner | |
| 169.4.c.i | 4 | 13.e | even | 6 | 1 | inner | |
| 169.4.e.b | 2 | 13.d | odd | 4 | 1 | ||
| 169.4.e.b | 2 | 13.f | odd | 12 | 1 | ||
| 208.4.w.a | 2 | 52.f | even | 4 | 1 | ||
| 208.4.w.a | 2 | 52.l | even | 12 | 1 | ||
| 1521.4.a.q | 2 | 39.h | odd | 6 | 1 | ||
| 1521.4.a.q | 2 | 39.i | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} + 12T_{2}^{2} + 144 \)
acting on \(S_{4}^{\mathrm{new}}(169, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + 12T^{2} + 144 \)
|
| $3$ |
\( (T^{2} - 7 T + 49)^{2} \)
|
| $5$ |
\( (T^{2} - 192)^{2} \)
|
| $7$ |
\( T^{4} + 507 T^{2} + 257049 \)
|
| $11$ |
\( T^{4} + 507 T^{2} + 257049 \)
|
| $13$ |
\( T^{4} \)
|
| $17$ |
\( (T^{2} - 27 T + 729)^{2} \)
|
| $19$ |
\( T^{4} + 7803 T^{2} + 60886809 \)
|
| $23$ |
\( (T^{2} - 57 T + 3249)^{2} \)
|
| $29$ |
\( (T^{2} - 69 T + 4761)^{2} \)
|
| $31$ |
\( (T^{2} - 5292)^{2} \)
|
| $37$ |
\( T^{4} + 1587 T^{2} + 2518569 \)
|
| $41$ |
\( T^{4} + \cdots + 23897140569 \)
|
| $43$ |
\( (T^{2} + 85 T + 7225)^{2} \)
|
| $47$ |
\( (T^{2} - 117612)^{2} \)
|
| $53$ |
\( (T - 426)^{4} \)
|
| $59$ |
\( T^{4} + 363 T^{2} + 131769 \)
|
| $61$ |
\( (T^{2} - 17 T + 289)^{2} \)
|
| $67$ |
\( T^{4} + 27075 T^{2} + 733055625 \)
|
| $71$ |
\( T^{4} + \cdots + 116081259849 \)
|
| $73$ |
\( (T^{2} - 1009200)^{2} \)
|
| $79$ |
\( (T + 1244)^{4} \)
|
| $83$ |
\( (T^{2} - 181548)^{2} \)
|
| $89$ |
\( T^{4} + \cdots + 8833556169 \)
|
| $97$ |
\( T^{4} + \cdots + 2325951361449 \)
|
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