Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [182,2,Mod(107,182)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(182, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([2, 2]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("182.107");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 182 = 2 \cdot 7 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 182.e (of order \(3\), degree \(2\), 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: | \(1.45327731679\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Relative dimension: | \(3\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | 6.0.4740147.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} + 10x^{4} + 25x^{2} + 3 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 3 \) |
| Twist minimal: | yes |
| 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 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
\( x^{6} + 10x^{4} + 25x^{2} + 3 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} + 5\nu + 1 ) / 2 \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} + 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( -\nu^{4} - 6\nu^{2} - 3 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{5} + 8\nu^{3} - \nu^{2} + 15\nu - 3 ) / 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -\nu^{5} + \nu^{4} - 8\nu^{3} + 6\nu^{2} - 12\nu + 3 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( ( 2\beta_{5} + 2\beta_{4} + \beta_{3} + \beta_{2} ) / 3 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} - 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -10\beta_{5} - 10\beta_{4} - 5\beta_{3} - 5\beta_{2} + 6\beta _1 - 3 ) / 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( -\beta_{3} - 6\beta_{2} + 15 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 50\beta_{5} + 56\beta_{4} + 25\beta_{3} + 28\beta_{2} - 48\beta _1 + 24 ) / 3 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/182\mathbb{Z}\right)^\times\).
| \(n\) | \(15\) | \(157\) |
| \(\chi(n)\) | \(-1 + \beta_{1}\) | \(-1 + \beta_{1}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 107.1 |
|
−1.00000 | −1.43685 | + | 2.48870i | 1.00000 | −0.307774 | + | 0.533081i | 1.43685 | − | 2.48870i | −0.500000 | + | 2.59808i | −1.00000 | −2.62908 | − | 4.55369i | 0.307774 | − | 0.533081i | ||||||||||||||||||||||||
| 107.2 | −1.00000 | 0.574872 | − | 0.995707i | 1.00000 | −1.76417 | + | 3.05564i | −0.574872 | + | 0.995707i | −0.500000 | + | 2.59808i | −1.00000 | 0.839045 | + | 1.45327i | 1.76417 | − | 3.05564i | |||||||||||||||||||||||||
| 107.3 | −1.00000 | 1.36198 | − | 2.35902i | 1.00000 | 2.07195 | − | 3.58872i | −1.36198 | + | 2.35902i | −0.500000 | + | 2.59808i | −1.00000 | −2.20997 | − | 3.82778i | −2.07195 | + | 3.58872i | |||||||||||||||||||||||||
| 165.1 | −1.00000 | −1.43685 | − | 2.48870i | 1.00000 | −0.307774 | − | 0.533081i | 1.43685 | + | 2.48870i | −0.500000 | − | 2.59808i | −1.00000 | −2.62908 | + | 4.55369i | 0.307774 | + | 0.533081i | |||||||||||||||||||||||||
| 165.2 | −1.00000 | 0.574872 | + | 0.995707i | 1.00000 | −1.76417 | − | 3.05564i | −0.574872 | − | 0.995707i | −0.500000 | − | 2.59808i | −1.00000 | 0.839045 | − | 1.45327i | 1.76417 | + | 3.05564i | |||||||||||||||||||||||||
| 165.3 | −1.00000 | 1.36198 | + | 2.35902i | 1.00000 | 2.07195 | + | 3.58872i | −1.36198 | − | 2.35902i | −0.500000 | − | 2.59808i | −1.00000 | −2.20997 | + | 3.82778i | −2.07195 | − | 3.58872i | |||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 91.h | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 182.2.e.c | ✓ | 6 |
| 3.b | odd | 2 | 1 | 1638.2.m.f | 6 | ||
| 7.b | odd | 2 | 1 | 1274.2.e.q | 6 | ||
| 7.c | even | 3 | 1 | 182.2.h.c | yes | 6 | |
| 7.c | even | 3 | 1 | 1274.2.g.n | 6 | ||
| 7.d | odd | 6 | 1 | 1274.2.g.m | 6 | ||
| 7.d | odd | 6 | 1 | 1274.2.h.q | 6 | ||
| 13.c | even | 3 | 1 | 182.2.h.c | yes | 6 | |
| 21.h | odd | 6 | 1 | 1638.2.p.f | 6 | ||
| 39.i | odd | 6 | 1 | 1638.2.p.f | 6 | ||
| 91.g | even | 3 | 1 | 1274.2.g.n | 6 | ||
| 91.h | even | 3 | 1 | inner | 182.2.e.c | ✓ | 6 |
| 91.m | odd | 6 | 1 | 1274.2.g.m | 6 | ||
| 91.n | odd | 6 | 1 | 1274.2.h.q | 6 | ||
| 91.v | odd | 6 | 1 | 1274.2.e.q | 6 | ||
| 273.s | odd | 6 | 1 | 1638.2.m.f | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 182.2.e.c | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 182.2.e.c | ✓ | 6 | 91.h | even | 3 | 1 | inner |
| 182.2.h.c | yes | 6 | 7.c | even | 3 | 1 | |
| 182.2.h.c | yes | 6 | 13.c | even | 3 | 1 | |
| 1274.2.e.q | 6 | 7.b | odd | 2 | 1 | ||
| 1274.2.e.q | 6 | 91.v | odd | 6 | 1 | ||
| 1274.2.g.m | 6 | 7.d | odd | 6 | 1 | ||
| 1274.2.g.m | 6 | 91.m | odd | 6 | 1 | ||
| 1274.2.g.n | 6 | 7.c | even | 3 | 1 | ||
| 1274.2.g.n | 6 | 91.g | even | 3 | 1 | ||
| 1274.2.h.q | 6 | 7.d | odd | 6 | 1 | ||
| 1274.2.h.q | 6 | 91.n | odd | 6 | 1 | ||
| 1638.2.m.f | 6 | 3.b | odd | 2 | 1 | ||
| 1638.2.m.f | 6 | 273.s | odd | 6 | 1 | ||
| 1638.2.p.f | 6 | 21.h | odd | 6 | 1 | ||
| 1638.2.p.f | 6 | 39.i | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(182, [\chi])\):
|
\( T_{3}^{6} - T_{3}^{5} + 9T_{3}^{4} - 10T_{3}^{3} + 73T_{3}^{2} - 72T_{3} + 81 \)
|
|
\( T_{5}^{6} + 15T_{5}^{4} + 18T_{5}^{3} + 225T_{5}^{2} + 135T_{5} + 81 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 1)^{6} \)
|
| $3$ |
\( T^{6} - T^{5} + \cdots + 81 \)
|
| $5$ |
\( T^{6} + 15 T^{4} + \cdots + 81 \)
|
| $7$ |
\( (T^{2} + T + 7)^{3} \)
|
| $11$ |
\( T^{6} + 4 T^{5} + \cdots + 225 \)
|
| $13$ |
\( T^{6} + 4 T^{5} + \cdots + 2197 \)
|
| $17$ |
\( (T^{3} + T^{2} - 30 T + 9)^{2} \)
|
| $19$ |
\( (T^{2} - T + 1)^{3} \)
|
| $23$ |
\( (T^{3} + 4 T^{2} - 45 T + 27)^{2} \)
|
| $29$ |
\( T^{6} + T^{5} + \cdots + 31329 \)
|
| $31$ |
\( T^{6} - 13 T^{5} + \cdots + 961 \)
|
| $37$ |
\( (T^{3} - 2 T^{2} - 31 T + 89)^{2} \)
|
| $41$ |
\( T^{6} + 3 T^{5} + \cdots + 81 \)
|
| $43$ |
\( T^{6} + 7 T^{5} + \cdots + 1521 \)
|
| $47$ |
\( T^{6} - 13 T^{5} + \cdots + 1521 \)
|
| $53$ |
\( T^{6} - 13 T^{5} + \cdots + 1521 \)
|
| $59$ |
\( (T - 3)^{6} \)
|
| $61$ |
\( T^{6} + 11 T^{5} + \cdots + 51529 \)
|
| $67$ |
\( T^{6} + 6 T^{5} + \cdots + 361 \)
|
| $71$ |
\( T^{6} + 15 T^{5} + \cdots + 342225 \)
|
| $73$ |
\( T^{6} - 22 T^{5} + \cdots + 36481 \)
|
| $79$ |
\( T^{6} - T^{5} + \cdots + 169 \)
|
| $83$ |
\( (T^{3} + 14 T^{2} + \cdots - 225)^{2} \)
|
| $89$ |
\( (T^{3} - 17 T^{2} + 66 T + 9)^{2} \)
|
| $97$ |
\( T^{6} - 27 T^{5} + \cdots + 49729 \)
|
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