Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [43,3,Mod(42,43)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(43, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("43.42");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 43 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 43.b (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: | \(1.17166513675\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} + \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} + 20x^{4} + 121x^{2} + 214 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| 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 basis \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} + 20x^{4} + 121x^{2} + 214 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{5} - 11\nu^{3} - 18\nu ) / 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{4} - 11\nu^{2} - 22 ) / 4 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{4} + 15\nu^{2} + 46 ) / 4 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{5} + 15\nu^{3} + 50\nu ) / 4 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{4} + \beta_{3} - 6 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{5} + \beta_{2} - 8\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -11\beta_{4} - 15\beta_{3} + 44 \)
|
| \(\nu^{5}\) | \(=\) |
\( -11\beta_{5} - 15\beta_{2} + 70\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/43\mathbb{Z}\right)^\times\).
| \(n\) | \(3\) |
| \(\chi(n)\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 42.1 |
|
− | 3.18991i | 0.724539i | −6.17554 | − | 7.66434i | 2.31122 | 10.1297i | 6.93980i | 8.47504 | −24.4486 | ||||||||||||||||||||||||||||||||||
| 42.2 | − | 2.58315i | − | 3.59415i | −2.67267 | 7.02284i | −9.28424 | − | 0.845536i | − | 3.42869i | −3.91793 | 18.1411 | |||||||||||||||||||||||||||||||||
| 42.3 | − | 1.77533i | 5.61757i | 0.848217 | 2.98959i | 9.97302 | − | 6.83184i | − | 8.60717i | −22.5571 | 5.30751 | ||||||||||||||||||||||||||||||||||
| 42.4 | 1.77533i | − | 5.61757i | 0.848217 | − | 2.98959i | 9.97302 | 6.83184i | 8.60717i | −22.5571 | 5.30751 | |||||||||||||||||||||||||||||||||||
| 42.5 | 2.58315i | 3.59415i | −2.67267 | − | 7.02284i | −9.28424 | 0.845536i | 3.42869i | −3.91793 | 18.1411 | ||||||||||||||||||||||||||||||||||||
| 42.6 | 3.18991i | − | 0.724539i | −6.17554 | 7.66434i | 2.31122 | − | 10.1297i | − | 6.93980i | 8.47504 | −24.4486 | ||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 43.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 43.3.b.b | ✓ | 6 |
| 3.b | odd | 2 | 1 | 387.3.b.c | 6 | ||
| 4.b | odd | 2 | 1 | 688.3.b.e | 6 | ||
| 43.b | odd | 2 | 1 | inner | 43.3.b.b | ✓ | 6 |
| 129.d | even | 2 | 1 | 387.3.b.c | 6 | ||
| 172.d | even | 2 | 1 | 688.3.b.e | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 43.3.b.b | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 43.3.b.b | ✓ | 6 | 43.b | odd | 2 | 1 | inner |
| 387.3.b.c | 6 | 3.b | odd | 2 | 1 | ||
| 387.3.b.c | 6 | 129.d | even | 2 | 1 | ||
| 688.3.b.e | 6 | 4.b | odd | 2 | 1 | ||
| 688.3.b.e | 6 | 172.d | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} + 20T_{2}^{4} + 121T_{2}^{2} + 214 \)
acting on \(S_{3}^{\mathrm{new}}(43, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + 20 T^{4} + \cdots + 214 \)
|
| $3$ |
\( T^{6} + 45 T^{4} + \cdots + 214 \)
|
| $5$ |
\( T^{6} + 117 T^{4} + \cdots + 25894 \)
|
| $7$ |
\( T^{6} + 150 T^{4} + \cdots + 3424 \)
|
| $11$ |
\( (T^{3} - 19 T^{2} + \cdots + 244)^{2} \)
|
| $13$ |
\( (T^{3} - 15 T^{2} + \cdots + 3500)^{2} \)
|
| $17$ |
\( (T^{3} + 10 T^{2} + \cdots - 125)^{2} \)
|
| $19$ |
\( T^{6} + 1475 T^{4} + \cdots + 53500000 \)
|
| $23$ |
\( (T^{3} + 40 T^{2} + \cdots - 14125)^{2} \)
|
| $29$ |
\( T^{6} + 3425 T^{4} + \cdots + 163843750 \)
|
| $31$ |
\( (T^{3} + 56 T^{2} + \cdots - 11041)^{2} \)
|
| $37$ |
\( T^{6} + 2955 T^{4} + \cdots + 316432384 \)
|
| $41$ |
\( (T^{3} + 86 T^{2} + \cdots + 2989)^{2} \)
|
| $43$ |
\( T^{6} + \cdots + 6321363049 \)
|
| $47$ |
\( (T^{3} - 15 T^{2} + \cdots - 13500)^{2} \)
|
| $53$ |
\( (T^{3} + 55 T^{2} + \cdots - 140500)^{2} \)
|
| $59$ |
\( (T^{3} + 6 T^{2} + \cdots + 105424)^{2} \)
|
| $61$ |
\( T^{6} + \cdots + 150281500000 \)
|
| $67$ |
\( (T^{3} + 35 T^{2} + \cdots + 36500)^{2} \)
|
| $71$ |
\( T^{6} + \cdots + 13696000000 \)
|
| $73$ |
\( T^{6} + 7370 T^{4} + \cdots + 601120864 \)
|
| $79$ |
\( (T^{3} - 89 T^{2} + \cdots - 11236)^{2} \)
|
| $83$ |
\( (T^{3} - 5 T^{2} + \cdots - 59500)^{2} \)
|
| $89$ |
\( T^{6} + 8650 T^{4} + \cdots + 53500000 \)
|
| $97$ |
\( (T^{3} + 190 T^{2} + \cdots - 1046125)^{2} \)
|
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