Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [57,3,Mod(31,57)]
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, 5]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("57.31");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 57 = 3 \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 57.g (of order \(6\), 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: | \(1.55313750685\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Relative dimension: | \(3\) over \(\Q(\zeta_{6})\) |
| Coefficient field: | 6.0.92607408.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 3x^{5} + 20x^{4} - 35x^{3} + 94x^{2} - 77x + 43 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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
\( x^{6} - 3x^{5} + 20x^{4} - 35x^{3} + 94x^{2} - 77x + 43 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu + 6 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -2\nu^{5} + 5\nu^{4} - 26\nu^{3} + 34\nu^{2} - 56\nu + 11 ) / 23 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{5} + 9\nu^{4} - 10\nu^{3} + 98\nu^{2} - 87\nu + 52 ) / 23 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{3} - 2\nu^{2} + 9\nu - 7 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 - 6 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{5} + 2\beta_{2} - 7\beta _1 - 5 \)
|
| \(\nu^{4}\) | \(=\) |
\( 2\beta_{5} + 2\beta_{4} + \beta_{3} - 6\beta_{2} - 14\beta _1 + 45 \)
|
| \(\nu^{5}\) | \(=\) |
\( -8\beta_{5} + 5\beta_{4} - 9\beta_{3} - 24\beta_{2} + 45\beta _1 + 81 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 31.1 |
|
−1.78805 | + | 1.03233i | −1.50000 | + | 0.866025i | 0.131406 | − | 0.227602i | −3.20750 | − | 5.55555i | 1.78805 | − | 3.09699i | −2.26281 | − | 7.71601i | 1.50000 | − | 2.59808i | 11.4703 | + | 6.62239i | |||||||||||||||||||||
| 31.2 | 0.204011 | − | 0.117786i | −1.50000 | + | 0.866025i | −1.97225 | + | 3.41604i | 2.88028 | + | 4.98878i | −0.204011 | + | 0.353358i | 1.94451 | 1.87150i | 1.50000 | − | 2.59808i | 1.17522 | + | 0.678513i | |||||||||||||||||||||||
| 31.3 | 3.08403 | − | 1.78057i | −1.50000 | + | 0.866025i | 4.34085 | − | 7.51857i | 2.32722 | + | 4.03087i | −3.08403 | + | 5.34170i | −10.6817 | − | 16.6722i | 1.50000 | − | 2.59808i | 14.3545 | + | 8.28756i | ||||||||||||||||||||||
| 46.1 | −1.78805 | − | 1.03233i | −1.50000 | − | 0.866025i | 0.131406 | + | 0.227602i | −3.20750 | + | 5.55555i | 1.78805 | + | 3.09699i | −2.26281 | 7.71601i | 1.50000 | + | 2.59808i | 11.4703 | − | 6.62239i | |||||||||||||||||||||||
| 46.2 | 0.204011 | + | 0.117786i | −1.50000 | − | 0.866025i | −1.97225 | − | 3.41604i | 2.88028 | − | 4.98878i | −0.204011 | − | 0.353358i | 1.94451 | − | 1.87150i | 1.50000 | + | 2.59808i | 1.17522 | − | 0.678513i | ||||||||||||||||||||||
| 46.3 | 3.08403 | + | 1.78057i | −1.50000 | − | 0.866025i | 4.34085 | + | 7.51857i | 2.32722 | − | 4.03087i | −3.08403 | − | 5.34170i | −10.6817 | 16.6722i | 1.50000 | + | 2.59808i | 14.3545 | − | 8.28756i | |||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 19.d | odd | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 57.3.g.b | ✓ | 6 |
| 3.b | odd | 2 | 1 | 171.3.p.c | 6 | ||
| 4.b | odd | 2 | 1 | 912.3.be.f | 6 | ||
| 19.d | odd | 6 | 1 | inner | 57.3.g.b | ✓ | 6 |
| 57.f | even | 6 | 1 | 171.3.p.c | 6 | ||
| 76.f | even | 6 | 1 | 912.3.be.f | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 57.3.g.b | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 57.3.g.b | ✓ | 6 | 19.d | odd | 6 | 1 | inner |
| 171.3.p.c | 6 | 3.b | odd | 2 | 1 | ||
| 171.3.p.c | 6 | 57.f | even | 6 | 1 | ||
| 912.3.be.f | 6 | 4.b | odd | 2 | 1 | ||
| 912.3.be.f | 6 | 76.f | even | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} - 3T_{2}^{5} - 4T_{2}^{4} + 21T_{2}^{3} + 46T_{2}^{2} - 21T_{2} + 3 \)
acting on \(S_{3}^{\mathrm{new}}(57, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} - 3 T^{5} + \cdots + 3 \)
|
| $3$ |
\( (T^{2} + 3 T + 3)^{3} \)
|
| $5$ |
\( T^{6} - 4 T^{5} + \cdots + 29584 \)
|
| $7$ |
\( (T^{3} + 11 T^{2} + \cdots - 47)^{2} \)
|
| $11$ |
\( (T^{3} + 18 T^{2} + \cdots - 1084)^{2} \)
|
| $13$ |
\( T^{6} + 3 T^{5} + \cdots + 2883 \)
|
| $17$ |
\( T^{6} - 38 T^{5} + \cdots + 52186176 \)
|
| $19$ |
\( T^{6} + 10 T^{5} + \cdots + 47045881 \)
|
| $23$ |
\( T^{6} - 54 T^{5} + \cdots + 21049744 \)
|
| $29$ |
\( T^{6} + 102 T^{5} + \cdots + 487228608 \)
|
| $31$ |
\( T^{6} + 2345 T^{4} + \cdots + 112326483 \)
|
| $37$ |
\( T^{6} + \cdots + 3652564347 \)
|
| $41$ |
\( T^{6} - 96 T^{5} + \cdots + 1354752 \)
|
| $43$ |
\( T^{6} + \cdots + 1477402969 \)
|
| $47$ |
\( T^{6} + 50 T^{5} + \cdots + 5798464 \)
|
| $53$ |
\( T^{6} + \cdots + 6327041328 \)
|
| $59$ |
\( T^{6} - 1048 T^{4} + \cdots + 25509168 \)
|
| $61$ |
\( T^{6} + \cdots + 1215986641 \)
|
| $67$ |
\( T^{6} + \cdots + 11787475467 \)
|
| $71$ |
\( T^{6} + \cdots + 149905029888 \)
|
| $73$ |
\( T^{6} + \cdots + 434693631969 \)
|
| $79$ |
\( T^{6} + \cdots + 32898206883 \)
|
| $83$ |
\( (T^{3} + 174 T^{2} + \cdots - 1174072)^{2} \)
|
| $89$ |
\( T^{6} + \cdots + 591631797168 \)
|
| $97$ |
\( T^{6} + 228 T^{5} + \cdots + 68659968 \)
|
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