Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [57,3,Mod(11,57)]
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([3, 4]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("57.11");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 57 = 3 \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 57.h (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: | \(8\) |
| Relative dimension: | \(4\) over \(\Q(\zeta_{6})\) |
| Coefficient field: | 8.0.12745506816.5 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 8x^{6} + 55x^{4} - 72x^{2} + 81 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{4}]\) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - 8x^{6} + 55x^{4} - 72x^{2} + 81 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{7} - 203\nu ) / 165 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{6} - 148 ) / 55 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -8\nu^{6} + 55\nu^{4} - 440\nu^{2} + 576 ) / 495 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -8\nu^{7} + 55\nu^{5} - 440\nu^{3} + 81\nu ) / 495 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -23\nu^{6} + 220\nu^{4} - 1265\nu^{2} + 1656 ) / 495 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 8\nu^{7} - 55\nu^{5} + 341\nu^{3} - 81\nu ) / 297 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{6} - 4\beta_{4} + \beta_{3} + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( -3\beta_{7} - 5\beta_{5} \)
|
| \(\nu^{4}\) | \(=\) |
\( 8\beta_{6} - 23\beta_{4} \)
|
| \(\nu^{5}\) | \(=\) |
\( -24\beta_{7} - 31\beta_{5} - 24\beta_{2} - 31\beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( -55\beta_{3} - 148 \)
|
| \(\nu^{7}\) | \(=\) |
\( -165\beta_{2} - 203\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/57\mathbb{Z}\right)^\times\).
| \(n\) | \(20\) | \(40\) |
| \(\chi(n)\) | \(-1\) | \(-1 + \beta_{4}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 11.1 |
|
−2.23256 | − | 1.28897i | −2.54762 | + | 1.58418i | 1.32288 | + | 2.29129i | 3.67423 | + | 2.12132i | 7.72967 | − | 0.252975i | 8.64575 | 3.49117i | 3.98074 | − | 8.07178i | −5.46863 | − | 9.47194i | ||||||||||||||||||||||||||||
| 11.2 | −1.00781 | − | 0.581861i | 2.54762 | − | 1.58418i | −1.32288 | − | 2.29129i | −3.67423 | − | 2.12132i | −3.48930 | + | 0.114197i | 3.35425 | 7.73381i | 3.98074 | − | 8.07178i | 2.46863 | + | 4.27579i | |||||||||||||||||||||||||||||
| 11.3 | 1.00781 | + | 0.581861i | 0.0981308 | − | 2.99839i | −1.32288 | − | 2.29129i | 3.67423 | + | 2.12132i | 1.84355 | − | 2.96472i | 3.35425 | − | 7.73381i | −8.98074 | − | 0.588470i | 2.46863 | + | 4.27579i | ||||||||||||||||||||||||||||
| 11.4 | 2.23256 | + | 1.28897i | −0.0981308 | + | 2.99839i | 1.32288 | + | 2.29129i | −3.67423 | − | 2.12132i | −4.08392 | + | 6.56760i | 8.64575 | − | 3.49117i | −8.98074 | − | 0.588470i | −5.46863 | − | 9.47194i | ||||||||||||||||||||||||||||
| 26.1 | −2.23256 | + | 1.28897i | −2.54762 | − | 1.58418i | 1.32288 | − | 2.29129i | 3.67423 | − | 2.12132i | 7.72967 | + | 0.252975i | 8.64575 | − | 3.49117i | 3.98074 | + | 8.07178i | −5.46863 | + | 9.47194i | ||||||||||||||||||||||||||||
| 26.2 | −1.00781 | + | 0.581861i | 2.54762 | + | 1.58418i | −1.32288 | + | 2.29129i | −3.67423 | + | 2.12132i | −3.48930 | − | 0.114197i | 3.35425 | − | 7.73381i | 3.98074 | + | 8.07178i | 2.46863 | − | 4.27579i | ||||||||||||||||||||||||||||
| 26.3 | 1.00781 | − | 0.581861i | 0.0981308 | + | 2.99839i | −1.32288 | + | 2.29129i | 3.67423 | − | 2.12132i | 1.84355 | + | 2.96472i | 3.35425 | 7.73381i | −8.98074 | + | 0.588470i | 2.46863 | − | 4.27579i | |||||||||||||||||||||||||||||
| 26.4 | 2.23256 | − | 1.28897i | −0.0981308 | − | 2.99839i | 1.32288 | − | 2.29129i | −3.67423 | + | 2.12132i | −4.08392 | − | 6.56760i | 8.64575 | 3.49117i | −8.98074 | + | 0.588470i | −5.46863 | + | 9.47194i | |||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
| 19.c | even | 3 | 1 | inner |
| 57.h | 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.h.a | ✓ | 8 |
| 3.b | odd | 2 | 1 | inner | 57.3.h.a | ✓ | 8 |
| 19.c | even | 3 | 1 | inner | 57.3.h.a | ✓ | 8 |
| 57.h | odd | 6 | 1 | inner | 57.3.h.a | ✓ | 8 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 57.3.h.a | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 57.3.h.a | ✓ | 8 | 3.b | odd | 2 | 1 | inner |
| 57.3.h.a | ✓ | 8 | 19.c | even | 3 | 1 | inner |
| 57.3.h.a | ✓ | 8 | 57.h | odd | 6 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{8} - 8T_{2}^{6} + 55T_{2}^{4} - 72T_{2}^{2} + 81 \)
acting on \(S_{3}^{\mathrm{new}}(57, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} - 8 T^{6} + \cdots + 81 \)
|
| $3$ |
\( T^{8} + 10 T^{6} + \cdots + 6561 \)
|
| $5$ |
\( (T^{4} - 18 T^{2} + 324)^{2} \)
|
| $7$ |
\( (T^{2} - 12 T + 29)^{4} \)
|
| $11$ |
\( (T^{4} + 284 T^{2} + 12996)^{2} \)
|
| $13$ |
\( (T^{2} - 13 T + 169)^{4} \)
|
| $17$ |
\( T^{8} + \cdots + 4032758016 \)
|
| $19$ |
\( (T^{4} + 24 T^{3} + \cdots + 130321)^{2} \)
|
| $23$ |
\( T^{8} - 308 T^{6} + \cdots + 3111696 \)
|
| $29$ |
\( T^{8} + \cdots + 251265597696 \)
|
| $31$ |
\( (T^{2} - 8 T - 327)^{4} \)
|
| $37$ |
\( (T^{2} - 38 T - 1011)^{4} \)
|
| $41$ |
\( T^{8} + \cdots + 2821109907456 \)
|
| $43$ |
\( (T^{4} + 16 T^{3} + \cdots + 2283121)^{2} \)
|
| $47$ |
\( (T^{4} - 72 T^{2} + 5184)^{2} \)
|
| $53$ |
\( T^{8} + \cdots + 96947540496 \)
|
| $59$ |
\( T^{8} + \cdots + 7471182096 \)
|
| $61$ |
\( (T^{4} + 70 T^{3} + \cdots + 47089)^{2} \)
|
| $67$ |
\( (T^{4} + 64 T^{3} + \cdots + 208849)^{2} \)
|
| $71$ |
\( T^{8} + \cdots + 10\!\cdots\!96 \)
|
| $73$ |
\( (T^{4} + 42 T^{3} + \cdots + 18412681)^{2} \)
|
| $79$ |
\( (T^{4} + 24 T^{3} + \cdots + 234671761)^{2} \)
|
| $83$ |
\( (T^{4} + 22176 T^{2} + 22924944)^{2} \)
|
| $89$ |
\( (T^{4} - 1134 T^{2} + 1285956)^{2} \)
|
| $97$ |
\( (T^{4} - 68 T^{3} + \cdots + 125260864)^{2} \)
|
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