Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [124,2,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, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("124.123");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 124 = 2^{2} \cdot 31 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| 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: | \(0.990144985064\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\sqrt{6}, \sqrt{-7})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 2x^{3} - 7x^{2} + 8x + 58 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| 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
\( x^{4} - 2x^{3} - 7x^{2} + 8x + 58 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -2\nu^{3} + 3\nu^{2} - 16 ) / 31 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} + 14\nu^{2} - 31\nu - 54 ) / 31 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -2\nu^{3} + 3\nu^{2} + 31\nu - 16 ) / 31 \)
|
| \(\nu\) | \(=\) |
\( \beta_{3} - \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 2\beta_{3} + 2\beta_{2} - \beta _1 + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( 3\beta_{3} + 3\beta_{2} - 17\beta _1 - 2 \)
|
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 |
|
0.500000 | − | 1.32288i | −2.44949 | −1.50000 | − | 1.32288i | −2.00000 | −1.22474 | + | 3.24037i | 0 | −2.50000 | + | 1.32288i | 3.00000 | −1.00000 | + | 2.64575i | ||||||||||||||||||||
| 123.2 | 0.500000 | − | 1.32288i | 2.44949 | −1.50000 | − | 1.32288i | −2.00000 | 1.22474 | − | 3.24037i | 0 | −2.50000 | + | 1.32288i | 3.00000 | −1.00000 | + | 2.64575i | |||||||||||||||||||||
| 123.3 | 0.500000 | + | 1.32288i | −2.44949 | −1.50000 | + | 1.32288i | −2.00000 | −1.22474 | − | 3.24037i | 0 | −2.50000 | − | 1.32288i | 3.00000 | −1.00000 | − | 2.64575i | |||||||||||||||||||||
| 123.4 | 0.500000 | + | 1.32288i | 2.44949 | −1.50000 | + | 1.32288i | −2.00000 | 1.22474 | + | 3.24037i | 0 | −2.50000 | − | 1.32288i | 3.00000 | −1.00000 | − | 2.64575i | |||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 4.b | odd | 2 | 1 | inner |
| 31.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 | 124.2.d.b | ✓ | 4 |
| 3.b | odd | 2 | 1 | 1116.2.g.b | 4 | ||
| 4.b | odd | 2 | 1 | inner | 124.2.d.b | ✓ | 4 |
| 8.b | even | 2 | 1 | 1984.2.h.e | 4 | ||
| 8.d | odd | 2 | 1 | 1984.2.h.e | 4 | ||
| 12.b | even | 2 | 1 | 1116.2.g.b | 4 | ||
| 31.b | odd | 2 | 1 | inner | 124.2.d.b | ✓ | 4 |
| 93.c | even | 2 | 1 | 1116.2.g.b | 4 | ||
| 124.d | even | 2 | 1 | inner | 124.2.d.b | ✓ | 4 |
| 248.b | even | 2 | 1 | 1984.2.h.e | 4 | ||
| 248.g | odd | 2 | 1 | 1984.2.h.e | 4 | ||
| 372.b | odd | 2 | 1 | 1116.2.g.b | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 124.2.d.b | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 124.2.d.b | ✓ | 4 | 4.b | odd | 2 | 1 | inner |
| 124.2.d.b | ✓ | 4 | 31.b | odd | 2 | 1 | inner |
| 124.2.d.b | ✓ | 4 | 124.d | even | 2 | 1 | inner |
| 1116.2.g.b | 4 | 3.b | odd | 2 | 1 | ||
| 1116.2.g.b | 4 | 12.b | even | 2 | 1 | ||
| 1116.2.g.b | 4 | 93.c | even | 2 | 1 | ||
| 1116.2.g.b | 4 | 372.b | odd | 2 | 1 | ||
| 1984.2.h.e | 4 | 8.b | even | 2 | 1 | ||
| 1984.2.h.e | 4 | 8.d | odd | 2 | 1 | ||
| 1984.2.h.e | 4 | 248.b | even | 2 | 1 | ||
| 1984.2.h.e | 4 | 248.g | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(124, [\chi])\):
|
\( T_{3}^{2} - 6 \)
|
|
\( T_{5} + 2 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} - T + 2)^{2} \)
|
| $3$ |
\( (T^{2} - 6)^{2} \)
|
| $5$ |
\( (T + 2)^{4} \)
|
| $7$ |
\( T^{4} \)
|
| $11$ |
\( (T^{2} - 6)^{2} \)
|
| $13$ |
\( (T^{2} + 42)^{2} \)
|
| $17$ |
\( T^{4} \)
|
| $19$ |
\( (T^{2} + 28)^{2} \)
|
| $23$ |
\( (T^{2} - 24)^{2} \)
|
| $29$ |
\( (T^{2} + 42)^{2} \)
|
| $31$ |
\( T^{4} - 34T^{2} + 961 \)
|
| $37$ |
\( (T^{2} + 42)^{2} \)
|
| $41$ |
\( (T + 8)^{4} \)
|
| $43$ |
\( (T^{2} - 6)^{2} \)
|
| $47$ |
\( (T^{2} + 112)^{2} \)
|
| $53$ |
\( (T^{2} + 42)^{2} \)
|
| $59$ |
\( (T^{2} + 28)^{2} \)
|
| $61$ |
\( (T^{2} + 42)^{2} \)
|
| $67$ |
\( (T^{2} + 28)^{2} \)
|
| $71$ |
\( (T^{2} + 28)^{2} \)
|
| $73$ |
\( (T^{2} + 168)^{2} \)
|
| $79$ |
\( (T^{2} - 96)^{2} \)
|
| $83$ |
\( (T^{2} - 150)^{2} \)
|
| $89$ |
\( (T^{2} + 168)^{2} \)
|
| $97$ |
\( (T - 4)^{4} \)
|
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