Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [170,2,Mod(169,170)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(170, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("170.169");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 170 = 2 \cdot 5 \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 170.d (of order \(2\), degree \(1\), 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: | \(1.35745683436\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\zeta_{8})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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 \(\zeta_{8}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/170\mathbb{Z}\right)^\times\).
| \(n\) | \(71\) | \(137\) |
| \(\chi(n)\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 169.1 |
|
− | 1.00000i | −2.82843 | −1.00000 | −0.707107 | + | 2.12132i | 2.82843i | 4.24264 | 1.00000i | 5.00000 | 2.12132 | + | 0.707107i | |||||||||||||||||||||||||
| 169.2 | − | 1.00000i | 2.82843 | −1.00000 | 0.707107 | − | 2.12132i | − | 2.82843i | −4.24264 | 1.00000i | 5.00000 | −2.12132 | − | 0.707107i | |||||||||||||||||||||||||
| 169.3 | 1.00000i | −2.82843 | −1.00000 | −0.707107 | − | 2.12132i | − | 2.82843i | 4.24264 | − | 1.00000i | 5.00000 | 2.12132 | − | 0.707107i | |||||||||||||||||||||||||
| 169.4 | 1.00000i | 2.82843 | −1.00000 | 0.707107 | + | 2.12132i | 2.82843i | −4.24264 | − | 1.00000i | 5.00000 | −2.12132 | + | 0.707107i | ||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.b | even | 2 | 1 | inner |
| 17.b | even | 2 | 1 | inner |
| 85.c | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 170.2.d.c | ✓ | 4 |
| 3.b | odd | 2 | 1 | 1530.2.f.h | 4 | ||
| 4.b | odd | 2 | 1 | 1360.2.o.d | 4 | ||
| 5.b | even | 2 | 1 | inner | 170.2.d.c | ✓ | 4 |
| 5.c | odd | 4 | 1 | 850.2.b.b | 2 | ||
| 5.c | odd | 4 | 1 | 850.2.b.g | 2 | ||
| 15.d | odd | 2 | 1 | 1530.2.f.h | 4 | ||
| 17.b | even | 2 | 1 | inner | 170.2.d.c | ✓ | 4 |
| 20.d | odd | 2 | 1 | 1360.2.o.d | 4 | ||
| 51.c | odd | 2 | 1 | 1530.2.f.h | 4 | ||
| 68.d | odd | 2 | 1 | 1360.2.o.d | 4 | ||
| 85.c | even | 2 | 1 | inner | 170.2.d.c | ✓ | 4 |
| 85.g | odd | 4 | 1 | 850.2.b.b | 2 | ||
| 85.g | odd | 4 | 1 | 850.2.b.g | 2 | ||
| 255.h | odd | 2 | 1 | 1530.2.f.h | 4 | ||
| 340.d | odd | 2 | 1 | 1360.2.o.d | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 170.2.d.c | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 170.2.d.c | ✓ | 4 | 5.b | even | 2 | 1 | inner |
| 170.2.d.c | ✓ | 4 | 17.b | even | 2 | 1 | inner |
| 170.2.d.c | ✓ | 4 | 85.c | even | 2 | 1 | inner |
| 850.2.b.b | 2 | 5.c | odd | 4 | 1 | ||
| 850.2.b.b | 2 | 85.g | odd | 4 | 1 | ||
| 850.2.b.g | 2 | 5.c | odd | 4 | 1 | ||
| 850.2.b.g | 2 | 85.g | odd | 4 | 1 | ||
| 1360.2.o.d | 4 | 4.b | odd | 2 | 1 | ||
| 1360.2.o.d | 4 | 20.d | odd | 2 | 1 | ||
| 1360.2.o.d | 4 | 68.d | odd | 2 | 1 | ||
| 1360.2.o.d | 4 | 340.d | odd | 2 | 1 | ||
| 1530.2.f.h | 4 | 3.b | odd | 2 | 1 | ||
| 1530.2.f.h | 4 | 15.d | odd | 2 | 1 | ||
| 1530.2.f.h | 4 | 51.c | odd | 2 | 1 | ||
| 1530.2.f.h | 4 | 255.h | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} - 8 \)
acting on \(S_{2}^{\mathrm{new}}(170, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 1)^{2} \)
|
| $3$ |
\( (T^{2} - 8)^{2} \)
|
| $5$ |
\( T^{4} + 8T^{2} + 25 \)
|
| $7$ |
\( (T^{2} - 18)^{2} \)
|
| $11$ |
\( T^{4} \)
|
| $13$ |
\( (T^{2} + 36)^{2} \)
|
| $17$ |
\( T^{4} + 2T^{2} + 289 \)
|
| $19$ |
\( (T - 2)^{4} \)
|
| $23$ |
\( (T^{2} - 2)^{2} \)
|
| $29$ |
\( (T^{2} + 18)^{2} \)
|
| $31$ |
\( (T^{2} + 18)^{2} \)
|
| $37$ |
\( (T^{2} - 18)^{2} \)
|
| $41$ |
\( (T^{2} + 72)^{2} \)
|
| $43$ |
\( (T^{2} + 36)^{2} \)
|
| $47$ |
\( T^{4} \)
|
| $53$ |
\( (T^{2} + 36)^{2} \)
|
| $59$ |
\( (T + 12)^{4} \)
|
| $61$ |
\( (T^{2} + 18)^{2} \)
|
| $67$ |
\( (T^{2} + 144)^{2} \)
|
| $71$ |
\( (T^{2} + 18)^{2} \)
|
| $73$ |
\( (T^{2} - 72)^{2} \)
|
| $79$ |
\( (T^{2} + 18)^{2} \)
|
| $83$ |
\( (T^{2} + 36)^{2} \)
|
| $89$ |
\( (T - 12)^{4} \)
|
| $97$ |
\( T^{4} \)
|
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