Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [124,3,Mod(61,124)]
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([0, 1]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("124.61");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 124 = 2^{2} \cdot 31 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 124.c (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: | \(3.37875527807\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.0.63368.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 11x^{2} + 8 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{3} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} + 11x^{2} + 8 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} + 3\nu ) / 2 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} + 11\nu ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{2} + 6 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} - \beta_1 ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} - 6 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -3\beta_{2} + 11\beta_1 ) / 4 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 61.1 |
|
0 | − | 4.52040i | 0 | 9.21699 | 0 | −3.21699 | 0 | −11.4340 | 0 | |||||||||||||||||||||||||||||
| 61.2 | 0 | − | 1.25141i | 0 | −0.216991 | 0 | 6.21699 | 0 | 7.43398 | 0 | ||||||||||||||||||||||||||||||
| 61.3 | 0 | 1.25141i | 0 | −0.216991 | 0 | 6.21699 | 0 | 7.43398 | 0 | |||||||||||||||||||||||||||||||
| 61.4 | 0 | 4.52040i | 0 | 9.21699 | 0 | −3.21699 | 0 | −11.4340 | 0 | |||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 31.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 124.3.c.b | ✓ | 4 |
| 3.b | odd | 2 | 1 | 1116.3.h.d | 4 | ||
| 4.b | odd | 2 | 1 | 496.3.e.e | 4 | ||
| 5.b | even | 2 | 1 | 3100.3.d.b | 4 | ||
| 5.c | odd | 4 | 2 | 3100.3.f.b | 8 | ||
| 31.b | odd | 2 | 1 | inner | 124.3.c.b | ✓ | 4 |
| 93.c | even | 2 | 1 | 1116.3.h.d | 4 | ||
| 124.d | even | 2 | 1 | 496.3.e.e | 4 | ||
| 155.c | odd | 2 | 1 | 3100.3.d.b | 4 | ||
| 155.f | even | 4 | 2 | 3100.3.f.b | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 124.3.c.b | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 124.3.c.b | ✓ | 4 | 31.b | odd | 2 | 1 | inner |
| 496.3.e.e | 4 | 4.b | odd | 2 | 1 | ||
| 496.3.e.e | 4 | 124.d | even | 2 | 1 | ||
| 1116.3.h.d | 4 | 3.b | odd | 2 | 1 | ||
| 1116.3.h.d | 4 | 93.c | even | 2 | 1 | ||
| 3100.3.d.b | 4 | 5.b | even | 2 | 1 | ||
| 3100.3.d.b | 4 | 155.c | odd | 2 | 1 | ||
| 3100.3.f.b | 8 | 5.c | odd | 4 | 2 | ||
| 3100.3.f.b | 8 | 155.f | even | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} + 22T_{3}^{2} + 32 \)
acting on \(S_{3}^{\mathrm{new}}(124, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} + 22T^{2} + 32 \)
|
| $5$ |
\( (T^{2} - 9 T - 2)^{2} \)
|
| $7$ |
\( (T^{2} - 3 T - 20)^{2} \)
|
| $11$ |
\( T^{4} + 262 T^{2} + 12800 \)
|
| $13$ |
\( T^{4} + 614 T^{2} + 89888 \)
|
| $17$ |
\( T^{4} + 1456 T^{2} + 524288 \)
|
| $19$ |
\( (T^{2} + 3 T - 198)^{2} \)
|
| $23$ |
\( T^{4} + 584 T^{2} + 15488 \)
|
| $29$ |
\( T^{4} + 2678 T^{2} + 1767200 \)
|
| $31$ |
\( T^{4} - 56 T^{3} + \cdots + 923521 \)
|
| $37$ |
\( T^{4} + 1206 T^{2} + 10368 \)
|
| $41$ |
\( (T^{2} + 53 T - 388)^{2} \)
|
| $43$ |
\( T^{4} + 22T^{2} + 32 \)
|
| $47$ |
\( (T^{2} - 50 T - 1600)^{2} \)
|
| $53$ |
\( T^{4} + 6086 T^{2} + 5068928 \)
|
| $59$ |
\( (T^{2} - T - 1802)^{2} \)
|
| $61$ |
\( T^{4} + 9238 T^{2} + 15456800 \)
|
| $67$ |
\( (T^{2} - 128 T + 3740)^{2} \)
|
| $71$ |
\( (T^{2} + 7 T - 4994)^{2} \)
|
| $73$ |
\( T^{4} + 2392 T^{2} + 61952 \)
|
| $79$ |
\( T^{4} + 22928 T^{2} + 1548800 \)
|
| $83$ |
\( T^{4} + 18422 T^{2} + 82535552 \)
|
| $89$ |
\( T^{4} + 22792 T^{2} + 126723200 \)
|
| $97$ |
\( (T^{2} - 79 T - 2200)^{2} \)
|
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