Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [57,2,Mod(7,57)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(57, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 2]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("57.7");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 57 = 3 \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 57.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: | \(0.455147291521\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Relative dimension: | \(3\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | 6.0.954288.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - x^{5} - 2x^{4} + 3x^{3} - 6x^{2} - 9x + 27 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| 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
\( x^{6} - x^{5} - 2x^{4} + 3x^{3} - 6x^{2} - 9x + 27 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{5} + 4\nu^{4} - \nu^{3} + 9\nu^{2} - 21\nu - 9 ) / 27 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{5} + 4\nu^{4} - \nu^{3} - 18\nu^{2} + 33\nu - 9 ) / 27 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -2\nu^{5} - \nu^{4} - 2\nu^{3} + 12\nu + 36 ) / 27 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -2\nu^{5} - \nu^{4} + 7\nu^{3} + 9\nu^{2} + 12\nu + 9 ) / 27 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 4\nu^{5} + 2\nu^{4} - 5\nu^{3} + 18\nu^{2} + 3\nu - 72 ) / 27 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} - \beta _1 + 1 ) / 3 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 2\beta_{5} + 2\beta_{4} + 2\beta_{3} - \beta_{2} + \beta _1 + 2 ) / 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -2\beta_{5} + 7\beta_{4} - 11\beta_{3} + \beta_{2} - \beta _1 + 7 ) / 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 2\beta_{5} + 2\beta_{4} - 7\beta_{3} + 8\beta_{2} + 10\beta _1 + 20 ) / 3 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 7\beta_{5} - 2\beta_{4} - 20\beta_{3} + \beta_{2} - 10\beta _1 + 43 ) / 3 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/57\mathbb{Z}\right)^\times\).
| \(n\) | \(20\) | \(40\) |
| \(\chi(n)\) | \(1\) | \(-1 + \beta_{3}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 7.1 |
|
−1.04307 | − | 1.80664i | −0.500000 | − | 0.866025i | −1.17597 | + | 2.03684i | −0.675970 | − | 1.17081i | −1.04307 | + | 1.80664i | 0.351939 | 0.734191 | −0.500000 | + | 0.866025i | −1.41016 | + | 2.44247i | ||||||||||||||||||||||
| 7.2 | 0.285997 | + | 0.495361i | −0.500000 | − | 0.866025i | 0.836412 | − | 1.44871i | 1.33641 | + | 2.31473i | 0.285997 | − | 0.495361i | −3.67282 | 2.10083 | −0.500000 | + | 0.866025i | −0.764419 | + | 1.32401i | |||||||||||||||||||||||
| 7.3 | 1.25707 | + | 2.17731i | −0.500000 | − | 0.866025i | −2.16044 | + | 3.74200i | −1.66044 | − | 2.87597i | 1.25707 | − | 2.17731i | 2.32088 | −5.83502 | −0.500000 | + | 0.866025i | 4.17458 | − | 7.23058i | |||||||||||||||||||||||
| 49.1 | −1.04307 | + | 1.80664i | −0.500000 | + | 0.866025i | −1.17597 | − | 2.03684i | −0.675970 | + | 1.17081i | −1.04307 | − | 1.80664i | 0.351939 | 0.734191 | −0.500000 | − | 0.866025i | −1.41016 | − | 2.44247i | |||||||||||||||||||||||
| 49.2 | 0.285997 | − | 0.495361i | −0.500000 | + | 0.866025i | 0.836412 | + | 1.44871i | 1.33641 | − | 2.31473i | 0.285997 | + | 0.495361i | −3.67282 | 2.10083 | −0.500000 | − | 0.866025i | −0.764419 | − | 1.32401i | |||||||||||||||||||||||
| 49.3 | 1.25707 | − | 2.17731i | −0.500000 | + | 0.866025i | −2.16044 | − | 3.74200i | −1.66044 | + | 2.87597i | 1.25707 | + | 2.17731i | 2.32088 | −5.83502 | −0.500000 | − | 0.866025i | 4.17458 | + | 7.23058i | |||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 19.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 57.2.e.b | ✓ | 6 |
| 3.b | odd | 2 | 1 | 171.2.f.b | 6 | ||
| 4.b | odd | 2 | 1 | 912.2.q.l | 6 | ||
| 12.b | even | 2 | 1 | 2736.2.s.z | 6 | ||
| 19.c | even | 3 | 1 | inner | 57.2.e.b | ✓ | 6 |
| 19.c | even | 3 | 1 | 1083.2.a.l | 3 | ||
| 19.d | odd | 6 | 1 | 1083.2.a.o | 3 | ||
| 57.f | even | 6 | 1 | 3249.2.a.t | 3 | ||
| 57.h | odd | 6 | 1 | 171.2.f.b | 6 | ||
| 57.h | odd | 6 | 1 | 3249.2.a.y | 3 | ||
| 76.g | odd | 6 | 1 | 912.2.q.l | 6 | ||
| 228.m | even | 6 | 1 | 2736.2.s.z | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 57.2.e.b | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 57.2.e.b | ✓ | 6 | 19.c | even | 3 | 1 | inner |
| 171.2.f.b | 6 | 3.b | odd | 2 | 1 | ||
| 171.2.f.b | 6 | 57.h | odd | 6 | 1 | ||
| 912.2.q.l | 6 | 4.b | odd | 2 | 1 | ||
| 912.2.q.l | 6 | 76.g | odd | 6 | 1 | ||
| 1083.2.a.l | 3 | 19.c | even | 3 | 1 | ||
| 1083.2.a.o | 3 | 19.d | odd | 6 | 1 | ||
| 2736.2.s.z | 6 | 12.b | even | 2 | 1 | ||
| 2736.2.s.z | 6 | 228.m | even | 6 | 1 | ||
| 3249.2.a.t | 3 | 57.f | even | 6 | 1 | ||
| 3249.2.a.y | 3 | 57.h | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} - T_{2}^{5} + 6T_{2}^{4} - T_{2}^{3} + 28T_{2}^{2} - 15T_{2} + 9 \)
acting on \(S_{2}^{\mathrm{new}}(57, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} - T^{5} + 6 T^{4} + \cdots + 9 \)
|
| $3$ |
\( (T^{2} + T + 1)^{3} \)
|
| $5$ |
\( T^{6} + 2 T^{5} + \cdots + 144 \)
|
| $7$ |
\( (T^{3} + T^{2} - 9 T + 3)^{2} \)
|
| $11$ |
\( (T^{3} - 24 T - 36)^{2} \)
|
| $13$ |
\( T^{6} - T^{5} + 22 T^{4} + \cdots + 9 \)
|
| $17$ |
\( T^{6} \)
|
| $19$ |
\( T^{6} - 4 T^{5} + \cdots + 6859 \)
|
| $23$ |
\( T^{6} + 14 T^{5} + \cdots + 24336 \)
|
| $29$ |
\( T^{6} + 4 T^{5} + \cdots + 9216 \)
|
| $31$ |
\( (T^{3} - 15 T^{2} + \cdots + 31)^{2} \)
|
| $37$ |
\( (T + 1)^{6} \)
|
| $41$ |
\( T^{6} - 4 T^{5} + \cdots + 9216 \)
|
| $43$ |
\( T^{6} - 3 T^{5} + \cdots + 169 \)
|
| $47$ |
\( (T^{2} - 6 T + 36)^{3} \)
|
| $53$ |
\( T^{6} - 6 T^{5} + \cdots + 104976 \)
|
| $59$ |
\( T^{6} + 24 T^{4} + \cdots + 1296 \)
|
| $61$ |
\( T^{6} + 13 T^{5} + \cdots + 5329 \)
|
| $67$ |
\( T^{6} + 9 T^{5} + \cdots + 292681 \)
|
| $71$ |
\( T^{6} - 18 T^{5} + \cdots + 419904 \)
|
| $73$ |
\( T^{6} + 19 T^{5} + \cdots + 961 \)
|
| $79$ |
\( T^{6} + 11 T^{5} + \cdots + 29241 \)
|
| $83$ |
\( (T^{3} + 4 T^{2} + \cdots - 192)^{2} \)
|
| $89$ |
\( T^{6} - 16 T^{5} + \cdots + 11664 \)
|
| $97$ |
\( T^{6} - 2 T^{5} + \cdots + 2096704 \)
|
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