Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [124,6,Mod(123,124)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(124, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 1]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("124.123");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 124 = 2^{2} \cdot 31 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 124.d (of order \(2\), degree \(1\), 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: | \(19.8875936568\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.0.21717639.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - x^{3} + 8 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{U}(1)[D_{2}]$ |
$q$-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
\( x^{6} - x^{3} + 8 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{5} + 2\nu^{4} - \nu^{2} - 2\nu ) / 4 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -\nu^{5} + 2\nu^{4} + \nu^{2} + 2\nu ) / 4 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{5} + \beta_{4} \)
|
| \(\nu^{5}\) | \(=\) |
\( -2\beta_{5} + 2\beta_{4} + \beta_{2} + \beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/124\mathbb{Z}\right)^\times\).
| \(n\) | \(63\) | \(65\) |
| \(\chi(n)\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 123.1 |
|
−3.86012 | − | 4.13515i | 0 | −2.19890 | + | 31.9244i | 16.6271 | 0 | 247.491i | 140.500 | − | 114.139i | −243.000 | −64.1825 | − | 68.7554i | ||||||||||||||||||||||||||||
| 123.2 | −3.86012 | + | 4.13515i | 0 | −2.19890 | − | 31.9244i | 16.6271 | 0 | − | 247.491i | 140.500 | + | 114.139i | −243.000 | −64.1825 | + | 68.7554i | ||||||||||||||||||||||||||||
| 123.3 | −1.65108 | − | 5.41054i | 0 | −26.5479 | + | 17.8665i | −104.061 | 0 | 56.7963i | 140.500 | + | 114.139i | −243.000 | 171.814 | + | 563.028i | |||||||||||||||||||||||||||||
| 123.4 | −1.65108 | + | 5.41054i | 0 | −26.5479 | − | 17.8665i | −104.061 | 0 | − | 56.7963i | 140.500 | − | 114.139i | −243.000 | 171.814 | − | 563.028i | ||||||||||||||||||||||||||||
| 123.5 | 5.51120 | − | 1.27539i | 0 | 28.7468 | − | 14.0579i | 87.4343 | 0 | − | 190.695i | 140.500 | − | 114.139i | −243.000 | 481.869 | − | 111.513i | ||||||||||||||||||||||||||||
| 123.6 | 5.51120 | + | 1.27539i | 0 | 28.7468 | + | 14.0579i | 87.4343 | 0 | 190.695i | 140.500 | + | 114.139i | −243.000 | 481.869 | + | 111.513i | |||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 31.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-31}) \) |
| 4.b | odd | 2 | 1 | inner |
| 124.d | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 124.6.d.a | ✓ | 6 |
| 4.b | odd | 2 | 1 | inner | 124.6.d.a | ✓ | 6 |
| 31.b | odd | 2 | 1 | CM | 124.6.d.a | ✓ | 6 |
| 124.d | even | 2 | 1 | inner | 124.6.d.a | ✓ | 6 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 124.6.d.a | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 124.6.d.a | ✓ | 6 | 4.b | odd | 2 | 1 | inner |
| 124.6.d.a | ✓ | 6 | 31.b | odd | 2 | 1 | CM |
| 124.6.d.a | ✓ | 6 | 124.d | even | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3} \)
acting on \(S_{6}^{\mathrm{new}}(124, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} - 281 T^{3} + 32768 \)
|
| $3$ |
\( T^{6} \)
|
| $5$ |
\( (T^{3} - 9375 T + 151282)^{2} \)
|
| $7$ |
\( T^{6} + \cdots + 7185139587036 \)
|
| $11$ |
\( T^{6} \)
|
| $13$ |
\( T^{6} \)
|
| $17$ |
\( T^{6} \)
|
| $19$ |
\( T^{6} + \cdots + 59\!\cdots\!00 \)
|
| $23$ |
\( T^{6} \)
|
| $29$ |
\( T^{6} \)
|
| $31$ |
\( (T^{2} + 28629151)^{3} \)
|
| $37$ |
\( T^{6} \)
|
| $41$ |
\( (T^{3} - 347568603 T + 859266041398)^{2} \)
|
| $43$ |
\( T^{6} \)
|
| $47$ |
\( (T^{2} + 916727164)^{3} \)
|
| $53$ |
\( T^{6} \)
|
| $59$ |
\( T^{6} + \cdots + 59\!\cdots\!96 \)
|
| $61$ |
\( T^{6} \)
|
| $67$ |
\( (T^{2} + 1715039164)^{3} \)
|
| $71$ |
\( T^{6} + \cdots + 12\!\cdots\!04 \)
|
| $73$ |
\( T^{6} \)
|
| $79$ |
\( T^{6} \)
|
| $83$ |
\( T^{6} \)
|
| $89$ |
\( T^{6} \)
|
| $97$ |
\( (T^{3} + \cdots - 14\!\cdots\!06)^{2} \)
|
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