Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [124,2,Mod(5,124)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(124, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 4]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("124.5");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 124 = 2^{2} \cdot 31 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 124.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: | \(0.990144985064\) |
| 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: | \(\Z[a_1, \ldots, a_{5}]\) |
| 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 \(\left(\mathbb{Z}/124\mathbb{Z}\right)^\times\).
| \(n\) | \(63\) | \(65\) |
| \(\chi(n)\) | \(1\) | \(-1 + \beta_{3}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 5.1 |
|
0 | −1.04307 | + | 1.80664i | 0 | 0.175970 | + | 0.304788i | 0 | −1.71903 | + | 2.97746i | 0 | −0.675970 | − | 1.17081i | 0 | ||||||||||||||||||||||||||||
| 5.2 | 0 | 0.285997 | − | 0.495361i | 0 | −1.83641 | − | 3.18076i | 0 | 1.62241 | − | 2.81009i | 0 | 1.33641 | + | 2.31473i | 0 | |||||||||||||||||||||||||||||
| 5.3 | 0 | 1.25707 | − | 2.17731i | 0 | 1.16044 | + | 2.00994i | 0 | −0.403374 | + | 0.698664i | 0 | −1.66044 | − | 2.87597i | 0 | |||||||||||||||||||||||||||||
| 25.1 | 0 | −1.04307 | − | 1.80664i | 0 | 0.175970 | − | 0.304788i | 0 | −1.71903 | − | 2.97746i | 0 | −0.675970 | + | 1.17081i | 0 | |||||||||||||||||||||||||||||
| 25.2 | 0 | 0.285997 | + | 0.495361i | 0 | −1.83641 | + | 3.18076i | 0 | 1.62241 | + | 2.81009i | 0 | 1.33641 | − | 2.31473i | 0 | |||||||||||||||||||||||||||||
| 25.3 | 0 | 1.25707 | + | 2.17731i | 0 | 1.16044 | − | 2.00994i | 0 | −0.403374 | − | 0.698664i | 0 | −1.66044 | + | 2.87597i | 0 | |||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 31.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 124.2.e.a | ✓ | 6 |
| 3.b | odd | 2 | 1 | 1116.2.i.f | 6 | ||
| 4.b | odd | 2 | 1 | 496.2.i.i | 6 | ||
| 5.b | even | 2 | 1 | 3100.2.i.a | 6 | ||
| 5.c | odd | 4 | 2 | 3100.2.y.a | 12 | ||
| 31.c | even | 3 | 1 | inner | 124.2.e.a | ✓ | 6 |
| 31.c | even | 3 | 1 | 3844.2.a.i | 3 | ||
| 31.e | odd | 6 | 1 | 3844.2.a.j | 3 | ||
| 93.h | odd | 6 | 1 | 1116.2.i.f | 6 | ||
| 124.i | odd | 6 | 1 | 496.2.i.i | 6 | ||
| 155.j | even | 6 | 1 | 3100.2.i.a | 6 | ||
| 155.o | odd | 12 | 2 | 3100.2.y.a | 12 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 124.2.e.a | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 124.2.e.a | ✓ | 6 | 31.c | even | 3 | 1 | inner |
| 496.2.i.i | 6 | 4.b | odd | 2 | 1 | ||
| 496.2.i.i | 6 | 124.i | odd | 6 | 1 | ||
| 1116.2.i.f | 6 | 3.b | odd | 2 | 1 | ||
| 1116.2.i.f | 6 | 93.h | odd | 6 | 1 | ||
| 3100.2.i.a | 6 | 5.b | even | 2 | 1 | ||
| 3100.2.i.a | 6 | 155.j | even | 6 | 1 | ||
| 3100.2.y.a | 12 | 5.c | odd | 4 | 2 | ||
| 3100.2.y.a | 12 | 155.o | odd | 12 | 2 | ||
| 3844.2.a.i | 3 | 31.c | even | 3 | 1 | ||
| 3844.2.a.j | 3 | 31.e | odd | 6 | 1 | ||
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(124, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( T^{6} - T^{5} + 6 T^{4} + \cdots + 9 \)
|
| $5$ |
\( T^{6} + T^{5} + 10 T^{4} + \cdots + 9 \)
|
| $7$ |
\( T^{6} + T^{5} + \cdots + 81 \)
|
| $11$ |
\( T^{6} + 3 T^{5} + \cdots + 81 \)
|
| $13$ |
\( (T^{2} - T + 1)^{3} \)
|
| $17$ |
\( T^{6} + 9 T^{5} + \cdots + 6561 \)
|
| $19$ |
\( T^{6} - T^{5} + \cdots + 4489 \)
|
| $23$ |
\( (T^{3} - 8 T^{2} + \cdots + 144)^{2} \)
|
| $29$ |
\( (T^{3} - 2 T^{2} + \cdots + 108)^{2} \)
|
| $31$ |
\( T^{6} - 14 T^{5} + \cdots + 29791 \)
|
| $37$ |
\( T^{6} + 17 T^{5} + \cdots + 12769 \)
|
| $41$ |
\( T^{6} + 9 T^{5} + \cdots + 81 \)
|
| $43$ |
\( T^{6} + 3 T^{5} + \cdots + 169 \)
|
| $47$ |
\( (T^{3} - 4 T^{2} + \cdots + 864)^{2} \)
|
| $53$ |
\( T^{6} + 9 T^{5} + \cdots + 301401 \)
|
| $59$ |
\( T^{6} - 3 T^{5} + \cdots + 81 \)
|
| $61$ |
\( (T^{3} - 22 T^{2} + \cdots - 204)^{2} \)
|
| $67$ |
\( T^{6} + 19 T^{5} + \cdots + 31329 \)
|
| $71$ |
\( T^{6} + 5 T^{5} + \cdots + 9 \)
|
| $73$ |
\( T^{6} + 5 T^{5} + \cdots + 160801 \)
|
| $79$ |
\( T^{6} + 29 T^{5} + \cdots + 638401 \)
|
| $83$ |
\( T^{6} + 41 T^{5} + \cdots + 6155361 \)
|
| $89$ |
\( (T^{3} - 6 T^{2} + \cdots + 324)^{2} \)
|
| $97$ |
\( (T^{3} + 18 T^{2} + \cdots - 452)^{2} \)
|
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