Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [42,5,Mod(13,42)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(42, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1]))
N = Newforms(chi, 5, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("42.13");
S:= CuspForms(chi, 5);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 42 = 2 \cdot 3 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 5 \) |
| Character orbit: | \([\chi]\) | \(=\) | 42.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: | \(4.34153844952\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\sqrt{2}, \sqrt{-3})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 2x^{2} + 4 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{4} \) |
| 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 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
\( x^{4} + 2x^{2} + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu^{3} \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} + 1 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} + 4\nu \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} - \beta_1 ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} - 1 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/42\mathbb{Z}\right)^\times\).
| \(n\) | \(29\) | \(31\) |
| \(\chi(n)\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 13.1 |
|
−2.82843 | − | 5.19615i | 8.00000 | 2.86976i | 14.6969i | −28.9411 | − | 39.5400i | −22.6274 | −27.0000 | − | 8.11689i | ||||||||||||||||||||||||||
| 13.2 | −2.82843 | 5.19615i | 8.00000 | − | 2.86976i | − | 14.6969i | −28.9411 | + | 39.5400i | −22.6274 | −27.0000 | 8.11689i | |||||||||||||||||||||||||||
| 13.3 | 2.82843 | − | 5.19615i | 8.00000 | − | 16.7262i | − | 14.6969i | 38.9411 | − | 29.7420i | 22.6274 | −27.0000 | − | 47.3087i | |||||||||||||||||||||||||
| 13.4 | 2.82843 | 5.19615i | 8.00000 | 16.7262i | 14.6969i | 38.9411 | + | 29.7420i | 22.6274 | −27.0000 | 47.3087i | |||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 42.5.c.a | ✓ | 4 |
| 3.b | odd | 2 | 1 | 126.5.c.b | 4 | ||
| 4.b | odd | 2 | 1 | 336.5.f.a | 4 | ||
| 5.b | even | 2 | 1 | 1050.5.f.a | 4 | ||
| 5.c | odd | 4 | 2 | 1050.5.h.a | 8 | ||
| 7.b | odd | 2 | 1 | inner | 42.5.c.a | ✓ | 4 |
| 7.c | even | 3 | 1 | 294.5.g.a | 4 | ||
| 7.c | even | 3 | 1 | 294.5.g.b | 4 | ||
| 7.d | odd | 6 | 1 | 294.5.g.a | 4 | ||
| 7.d | odd | 6 | 1 | 294.5.g.b | 4 | ||
| 12.b | even | 2 | 1 | 1008.5.f.f | 4 | ||
| 21.c | even | 2 | 1 | 126.5.c.b | 4 | ||
| 28.d | even | 2 | 1 | 336.5.f.a | 4 | ||
| 35.c | odd | 2 | 1 | 1050.5.f.a | 4 | ||
| 35.f | even | 4 | 2 | 1050.5.h.a | 8 | ||
| 84.h | odd | 2 | 1 | 1008.5.f.f | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 42.5.c.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 42.5.c.a | ✓ | 4 | 7.b | odd | 2 | 1 | inner |
| 126.5.c.b | 4 | 3.b | odd | 2 | 1 | ||
| 126.5.c.b | 4 | 21.c | even | 2 | 1 | ||
| 294.5.g.a | 4 | 7.c | even | 3 | 1 | ||
| 294.5.g.a | 4 | 7.d | odd | 6 | 1 | ||
| 294.5.g.b | 4 | 7.c | even | 3 | 1 | ||
| 294.5.g.b | 4 | 7.d | odd | 6 | 1 | ||
| 336.5.f.a | 4 | 4.b | odd | 2 | 1 | ||
| 336.5.f.a | 4 | 28.d | even | 2 | 1 | ||
| 1008.5.f.f | 4 | 12.b | even | 2 | 1 | ||
| 1008.5.f.f | 4 | 84.h | odd | 2 | 1 | ||
| 1050.5.f.a | 4 | 5.b | even | 2 | 1 | ||
| 1050.5.f.a | 4 | 35.c | odd | 2 | 1 | ||
| 1050.5.h.a | 8 | 5.c | odd | 4 | 2 | ||
| 1050.5.h.a | 8 | 35.f | even | 4 | 2 | ||
Hecke kernels
This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(42, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} - 8)^{2} \)
|
| $3$ |
\( (T^{2} + 27)^{2} \)
|
| $5$ |
\( T^{4} + 288T^{2} + 2304 \)
|
| $7$ |
\( T^{4} - 20 T^{3} + \cdots + 5764801 \)
|
| $11$ |
\( (T^{2} + 204 T - 3708)^{2} \)
|
| $13$ |
\( T^{4} + 103968 T^{2} + 300259584 \)
|
| $17$ |
\( T^{4} + 43200 T^{2} + 282240000 \)
|
| $19$ |
\( T^{4} + \cdots + 11903682816 \)
|
| $23$ |
\( (T^{2} + 300 T - 70812)^{2} \)
|
| $29$ |
\( (T^{2} + 924 T - 100188)^{2} \)
|
| $31$ |
\( T^{4} + \cdots + 3537890141184 \)
|
| $37$ |
\( (T^{2} - 2084 T + 826564)^{2} \)
|
| $41$ |
\( T^{4} + \cdots + 25205540742144 \)
|
| $43$ |
\( (T^{2} - 308 T - 1553372)^{2} \)
|
| $47$ |
\( T^{4} + \cdots + 25374512185344 \)
|
| $53$ |
\( (T^{2} - 4644 T + 3939876)^{2} \)
|
| $59$ |
\( T^{4} + \cdots + 997521767610624 \)
|
| $61$ |
\( T^{4} + \cdots + 363850436618496 \)
|
| $67$ |
\( (T^{2} + 7244 T + 2055076)^{2} \)
|
| $71$ |
\( (T^{2} + 8076 T + 16079652)^{2} \)
|
| $73$ |
\( T^{4} + \cdots + 18235222953984 \)
|
| $79$ |
\( (T^{2} + 7148 T - 2197916)^{2} \)
|
| $83$ |
\( T^{4} + \cdots + 28\!\cdots\!96 \)
|
| $89$ |
\( T^{4} + \cdots + 36\!\cdots\!36 \)
|
| $97$ |
\( T^{4} + \cdots + 425280742096896 \)
|
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