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: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{-3}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| 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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/169\mathbb{Z}\right)^\times\).
| \(n\) | \(2\) |
| \(\chi(n)\) | \(-\zeta_{6}\) |
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 |
|
1.50000 | − | 2.59808i | 0.500000 | − | 0.866025i | −0.500000 | − | 0.866025i | 9.00000 | −1.50000 | − | 2.59808i | 7.50000 | + | 12.9904i | 21.0000 | 13.0000 | + | 22.5167i | 13.5000 | − | 23.3827i | ||||||||||
| 146.1 | 1.50000 | + | 2.59808i | 0.500000 | + | 0.866025i | −0.500000 | + | 0.866025i | 9.00000 | −1.50000 | + | 2.59808i | 7.50000 | − | 12.9904i | 21.0000 | 13.0000 | − | 22.5167i | 13.5000 | + | 23.3827i | |||||||||||
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.c | 2 | |
| 13.b | even | 2 | 1 | 169.4.c.b | 2 | ||
| 13.c | even | 3 | 1 | 169.4.a.b | 1 | ||
| 13.c | even | 3 | 1 | inner | 169.4.c.c | 2 | |
| 13.d | odd | 4 | 2 | 169.4.e.d | 4 | ||
| 13.e | even | 6 | 1 | 169.4.a.c | 1 | ||
| 13.e | even | 6 | 1 | 169.4.c.b | 2 | ||
| 13.f | odd | 12 | 2 | 13.4.b.a | ✓ | 2 | |
| 13.f | odd | 12 | 2 | 169.4.e.d | 4 | ||
| 39.h | odd | 6 | 1 | 1521.4.a.d | 1 | ||
| 39.i | odd | 6 | 1 | 1521.4.a.i | 1 | ||
| 39.k | even | 12 | 2 | 117.4.b.a | 2 | ||
| 52.l | even | 12 | 2 | 208.4.f.b | 2 | ||
| 65.o | even | 12 | 2 | 325.4.d.a | 2 | ||
| 65.s | odd | 12 | 2 | 325.4.c.b | 2 | ||
| 65.t | even | 12 | 2 | 325.4.d.b | 2 | ||
| 104.u | even | 12 | 2 | 832.4.f.c | 2 | ||
| 104.x | odd | 12 | 2 | 832.4.f.e | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 13.4.b.a | ✓ | 2 | 13.f | odd | 12 | 2 | |
| 117.4.b.a | 2 | 39.k | even | 12 | 2 | ||
| 169.4.a.b | 1 | 13.c | even | 3 | 1 | ||
| 169.4.a.c | 1 | 13.e | even | 6 | 1 | ||
| 169.4.c.b | 2 | 13.b | even | 2 | 1 | ||
| 169.4.c.b | 2 | 13.e | even | 6 | 1 | ||
| 169.4.c.c | 2 | 1.a | even | 1 | 1 | trivial | |
| 169.4.c.c | 2 | 13.c | even | 3 | 1 | inner | |
| 169.4.e.d | 4 | 13.d | odd | 4 | 2 | ||
| 169.4.e.d | 4 | 13.f | odd | 12 | 2 | ||
| 208.4.f.b | 2 | 52.l | even | 12 | 2 | ||
| 325.4.c.b | 2 | 65.s | odd | 12 | 2 | ||
| 325.4.d.a | 2 | 65.o | even | 12 | 2 | ||
| 325.4.d.b | 2 | 65.t | even | 12 | 2 | ||
| 832.4.f.c | 2 | 104.u | even | 12 | 2 | ||
| 832.4.f.e | 2 | 104.x | odd | 12 | 2 | ||
| 1521.4.a.d | 1 | 39.h | odd | 6 | 1 | ||
| 1521.4.a.i | 1 | 39.i | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} - 3T_{2} + 9 \)
acting on \(S_{4}^{\mathrm{new}}(169, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} - 3T + 9 \)
|
| $3$ |
\( T^{2} - T + 1 \)
|
| $5$ |
\( (T - 9)^{2} \)
|
| $7$ |
\( T^{2} - 15T + 225 \)
|
| $11$ |
\( T^{2} + 48T + 2304 \)
|
| $13$ |
\( T^{2} \)
|
| $17$ |
\( T^{2} + 45T + 2025 \)
|
| $19$ |
\( T^{2} - 6T + 36 \)
|
| $23$ |
\( T^{2} - 162T + 26244 \)
|
| $29$ |
\( T^{2} - 144T + 20736 \)
|
| $31$ |
\( (T + 264)^{2} \)
|
| $37$ |
\( T^{2} - 303T + 91809 \)
|
| $41$ |
\( T^{2} + 192T + 36864 \)
|
| $43$ |
\( T^{2} + 97T + 9409 \)
|
| $47$ |
\( (T + 111)^{2} \)
|
| $53$ |
\( (T + 414)^{2} \)
|
| $59$ |
\( T^{2} - 522T + 272484 \)
|
| $61$ |
\( T^{2} + 376T + 141376 \)
|
| $67$ |
\( T^{2} + 36T + 1296 \)
|
| $71$ |
\( T^{2} - 357T + 127449 \)
|
| $73$ |
\( (T - 1098)^{2} \)
|
| $79$ |
\( (T + 830)^{2} \)
|
| $83$ |
\( (T - 438)^{2} \)
|
| $89$ |
\( T^{2} + 438T + 191844 \)
|
| $97$ |
\( T^{2} + 852T + 725904 \)
|
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