Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [440,2,Mod(219,440)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(440, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 1, 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("440.219");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 440 = 2^{3} \cdot 5 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 440.c (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: | \(3.51341768894\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | 8.0.599695360000.19 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 3x^{4} + 16 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{U}(1)[D_{2}]$ |
$q$-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 basis \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - 3x^{4} + 16 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - 1 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{6} - 3\nu^{2} ) / 4 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( \nu^{7} + 4\nu^{5} - 3\nu^{3} - 4\nu ) / 8 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -\nu^{7} + 3\nu^{3} ) / 4 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{4} + 1 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{7} + 2\beta_{6} + \beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( 4\beta_{5} + 3\beta_{2} \)
|
| \(\nu^{7}\) | \(=\) |
\( -4\beta_{7} + 3\beta_{3} \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/440\mathbb{Z}\right)^\times\).
| \(n\) | \(111\) | \(177\) | \(221\) | \(321\) |
| \(\chi(n)\) | \(-1\) | \(-1\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 219.1 |
|
−1.35246 | − | 0.413333i | 0 | 1.65831 | + | 1.11803i | − | 2.23607i | 0 | − | 5.22173i | −1.78069 | − | 2.19753i | 3.00000 | −0.924240 | + | 3.02420i | ||||||||||||||||||||||||||||||||
| 219.2 | −1.35246 | + | 0.413333i | 0 | 1.65831 | − | 1.11803i | 2.23607i | 0 | 5.22173i | −1.78069 | + | 2.19753i | 3.00000 | −0.924240 | − | 3.02420i | |||||||||||||||||||||||||||||||||||
| 219.3 | −0.413333 | − | 1.35246i | 0 | −1.65831 | + | 1.11803i | − | 2.23607i | 0 | 0.856447i | 2.19753 | + | 1.78069i | 3.00000 | −3.02420 | + | 0.924240i | ||||||||||||||||||||||||||||||||||
| 219.4 | −0.413333 | + | 1.35246i | 0 | −1.65831 | − | 1.11803i | 2.23607i | 0 | − | 0.856447i | 2.19753 | − | 1.78069i | 3.00000 | −3.02420 | − | 0.924240i | ||||||||||||||||||||||||||||||||||
| 219.5 | 0.413333 | − | 1.35246i | 0 | −1.65831 | − | 1.11803i | 2.23607i | 0 | 0.856447i | −2.19753 | + | 1.78069i | 3.00000 | 3.02420 | + | 0.924240i | |||||||||||||||||||||||||||||||||||
| 219.6 | 0.413333 | + | 1.35246i | 0 | −1.65831 | + | 1.11803i | − | 2.23607i | 0 | − | 0.856447i | −2.19753 | − | 1.78069i | 3.00000 | 3.02420 | − | 0.924240i | |||||||||||||||||||||||||||||||||
| 219.7 | 1.35246 | − | 0.413333i | 0 | 1.65831 | − | 1.11803i | 2.23607i | 0 | − | 5.22173i | 1.78069 | − | 2.19753i | 3.00000 | 0.924240 | + | 3.02420i | ||||||||||||||||||||||||||||||||||
| 219.8 | 1.35246 | + | 0.413333i | 0 | 1.65831 | + | 1.11803i | − | 2.23607i | 0 | 5.22173i | 1.78069 | + | 2.19753i | 3.00000 | 0.924240 | − | 3.02420i | ||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 55.d | odd | 2 | 1 | CM by \(\Q(\sqrt{-55}) \) |
| 5.b | even | 2 | 1 | inner |
| 8.d | odd | 2 | 1 | inner |
| 11.b | odd | 2 | 1 | inner |
| 40.e | odd | 2 | 1 | inner |
| 88.g | even | 2 | 1 | inner |
| 440.c | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 440.2.c.b | ✓ | 8 |
| 4.b | odd | 2 | 1 | 1760.2.c.b | 8 | ||
| 5.b | even | 2 | 1 | inner | 440.2.c.b | ✓ | 8 |
| 8.b | even | 2 | 1 | 1760.2.c.b | 8 | ||
| 8.d | odd | 2 | 1 | inner | 440.2.c.b | ✓ | 8 |
| 11.b | odd | 2 | 1 | inner | 440.2.c.b | ✓ | 8 |
| 20.d | odd | 2 | 1 | 1760.2.c.b | 8 | ||
| 40.e | odd | 2 | 1 | inner | 440.2.c.b | ✓ | 8 |
| 40.f | even | 2 | 1 | 1760.2.c.b | 8 | ||
| 44.c | even | 2 | 1 | 1760.2.c.b | 8 | ||
| 55.d | odd | 2 | 1 | CM | 440.2.c.b | ✓ | 8 |
| 88.b | odd | 2 | 1 | 1760.2.c.b | 8 | ||
| 88.g | even | 2 | 1 | inner | 440.2.c.b | ✓ | 8 |
| 220.g | even | 2 | 1 | 1760.2.c.b | 8 | ||
| 440.c | even | 2 | 1 | inner | 440.2.c.b | ✓ | 8 |
| 440.o | odd | 2 | 1 | 1760.2.c.b | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 440.2.c.b | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 440.2.c.b | ✓ | 8 | 5.b | even | 2 | 1 | inner |
| 440.2.c.b | ✓ | 8 | 8.d | odd | 2 | 1 | inner |
| 440.2.c.b | ✓ | 8 | 11.b | odd | 2 | 1 | inner |
| 440.2.c.b | ✓ | 8 | 40.e | odd | 2 | 1 | inner |
| 440.2.c.b | ✓ | 8 | 55.d | odd | 2 | 1 | CM |
| 440.2.c.b | ✓ | 8 | 88.g | even | 2 | 1 | inner |
| 440.2.c.b | ✓ | 8 | 440.c | even | 2 | 1 | inner |
| 1760.2.c.b | 8 | 4.b | odd | 2 | 1 | ||
| 1760.2.c.b | 8 | 8.b | even | 2 | 1 | ||
| 1760.2.c.b | 8 | 20.d | odd | 2 | 1 | ||
| 1760.2.c.b | 8 | 40.f | even | 2 | 1 | ||
| 1760.2.c.b | 8 | 44.c | even | 2 | 1 | ||
| 1760.2.c.b | 8 | 88.b | odd | 2 | 1 | ||
| 1760.2.c.b | 8 | 220.g | even | 2 | 1 | ||
| 1760.2.c.b | 8 | 440.o | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(440, [\chi])\):
|
\( T_{3} \)
|
|
\( T_{7}^{4} + 28T_{7}^{2} + 20 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} - 3T^{4} + 16 \)
|
| $3$ |
\( T^{8} \)
|
| $5$ |
\( (T^{2} + 5)^{4} \)
|
| $7$ |
\( (T^{4} + 28 T^{2} + 20)^{2} \)
|
| $11$ |
\( (T^{2} - 11)^{4} \)
|
| $13$ |
\( (T^{4} + 52 T^{2} + 500)^{2} \)
|
| $17$ |
\( (T^{4} - 68 T^{2} + 980)^{2} \)
|
| $19$ |
\( T^{8} \)
|
| $23$ |
\( T^{8} \)
|
| $29$ |
\( T^{8} \)
|
| $31$ |
\( (T^{2} + 80)^{4} \)
|
| $37$ |
\( T^{8} \)
|
| $41$ |
\( T^{8} \)
|
| $43$ |
\( (T^{4} - 172 T^{2} + 7220)^{2} \)
|
| $47$ |
\( T^{8} \)
|
| $53$ |
\( T^{8} \)
|
| $59$ |
\( (T + 4)^{8} \)
|
| $61$ |
\( T^{8} \)
|
| $67$ |
\( T^{8} \)
|
| $71$ |
\( (T^{2} + 220)^{4} \)
|
| $73$ |
\( (T^{4} - 292 T^{2} + 20)^{2} \)
|
| $79$ |
\( T^{8} \)
|
| $83$ |
\( (T^{4} - 332 T^{2} + 27380)^{2} \)
|
| $89$ |
\( (T^{2} - 176)^{4} \)
|
| $97$ |
\( T^{8} \)
|
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