Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [56,3,Mod(43,56)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(56, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 1, 0]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("56.43");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 56 = 2^{3} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 56.g (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: | \(1.52588948042\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\sqrt{2}, \sqrt{-7})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 6x^{2} + 16 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2 \) |
| 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
\( x^{4} + 6x^{2} + 16 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} + 2\nu ) / 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{2} + 3 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} - 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( 4\beta_{2} - 2\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/56\mathbb{Z}\right)^\times\).
| \(n\) | \(15\) | \(17\) | \(29\) |
| \(\chi(n)\) | \(-1\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 43.1 |
|
−0.707107 | − | 1.87083i | 0.585786 | −3.00000 | + | 2.64575i | − | 9.03316i | −0.414214 | − | 1.09591i | − | 2.64575i | 7.07107 | + | 3.74166i | −8.65685 | −16.8995 | + | 6.38741i | ||||||||||||||||||
| 43.2 | −0.707107 | + | 1.87083i | 0.585786 | −3.00000 | − | 2.64575i | 9.03316i | −0.414214 | + | 1.09591i | 2.64575i | 7.07107 | − | 3.74166i | −8.65685 | −16.8995 | − | 6.38741i | |||||||||||||||||||||
| 43.3 | 0.707107 | − | 1.87083i | 3.41421 | −3.00000 | − | 2.64575i | 1.54985i | 2.41421 | − | 6.38741i | 2.64575i | −7.07107 | + | 3.74166i | 2.65685 | 2.89949 | + | 1.09591i | |||||||||||||||||||||
| 43.4 | 0.707107 | + | 1.87083i | 3.41421 | −3.00000 | + | 2.64575i | − | 1.54985i | 2.41421 | + | 6.38741i | − | 2.64575i | −7.07107 | − | 3.74166i | 2.65685 | 2.89949 | − | 1.09591i | |||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 8.d | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 56.3.g.a | ✓ | 4 |
| 3.b | odd | 2 | 1 | 504.3.g.a | 4 | ||
| 4.b | odd | 2 | 1 | 224.3.g.a | 4 | ||
| 7.b | odd | 2 | 1 | 392.3.g.h | 4 | ||
| 7.c | even | 3 | 2 | 392.3.k.i | 8 | ||
| 7.d | odd | 6 | 2 | 392.3.k.j | 8 | ||
| 8.b | even | 2 | 1 | 224.3.g.a | 4 | ||
| 8.d | odd | 2 | 1 | inner | 56.3.g.a | ✓ | 4 |
| 12.b | even | 2 | 1 | 2016.3.g.a | 4 | ||
| 16.e | even | 4 | 2 | 1792.3.d.g | 8 | ||
| 16.f | odd | 4 | 2 | 1792.3.d.g | 8 | ||
| 24.f | even | 2 | 1 | 504.3.g.a | 4 | ||
| 24.h | odd | 2 | 1 | 2016.3.g.a | 4 | ||
| 28.d | even | 2 | 1 | 1568.3.g.h | 4 | ||
| 56.e | even | 2 | 1 | 392.3.g.h | 4 | ||
| 56.h | odd | 2 | 1 | 1568.3.g.h | 4 | ||
| 56.k | odd | 6 | 2 | 392.3.k.i | 8 | ||
| 56.m | even | 6 | 2 | 392.3.k.j | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 56.3.g.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 56.3.g.a | ✓ | 4 | 8.d | odd | 2 | 1 | inner |
| 224.3.g.a | 4 | 4.b | odd | 2 | 1 | ||
| 224.3.g.a | 4 | 8.b | even | 2 | 1 | ||
| 392.3.g.h | 4 | 7.b | odd | 2 | 1 | ||
| 392.3.g.h | 4 | 56.e | even | 2 | 1 | ||
| 392.3.k.i | 8 | 7.c | even | 3 | 2 | ||
| 392.3.k.i | 8 | 56.k | odd | 6 | 2 | ||
| 392.3.k.j | 8 | 7.d | odd | 6 | 2 | ||
| 392.3.k.j | 8 | 56.m | even | 6 | 2 | ||
| 504.3.g.a | 4 | 3.b | odd | 2 | 1 | ||
| 504.3.g.a | 4 | 24.f | even | 2 | 1 | ||
| 1568.3.g.h | 4 | 28.d | even | 2 | 1 | ||
| 1568.3.g.h | 4 | 56.h | odd | 2 | 1 | ||
| 1792.3.d.g | 8 | 16.e | even | 4 | 2 | ||
| 1792.3.d.g | 8 | 16.f | odd | 4 | 2 | ||
| 2016.3.g.a | 4 | 12.b | even | 2 | 1 | ||
| 2016.3.g.a | 4 | 24.h | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} - 4T_{3} + 2 \)
acting on \(S_{3}^{\mathrm{new}}(56, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + 6T^{2} + 16 \)
|
| $3$ |
\( (T^{2} - 4 T + 2)^{2} \)
|
| $5$ |
\( T^{4} + 84T^{2} + 196 \)
|
| $7$ |
\( (T^{2} + 7)^{2} \)
|
| $11$ |
\( (T^{2} - 8 T - 56)^{2} \)
|
| $13$ |
\( T^{4} + 84T^{2} + 196 \)
|
| $17$ |
\( (T^{2} - 36 T + 292)^{2} \)
|
| $19$ |
\( (T^{2} - 4 T - 718)^{2} \)
|
| $23$ |
\( T^{4} + 1848 T^{2} + 753424 \)
|
| $29$ |
\( (T^{2} + 504)^{2} \)
|
| $31$ |
\( T^{4} + 2464 T^{2} + 614656 \)
|
| $37$ |
\( T^{4} + 3696 T^{2} + 906304 \)
|
| $41$ |
\( (T^{2} + 20 T - 188)^{2} \)
|
| $43$ |
\( (T^{2} + 40 T + 392)^{2} \)
|
| $47$ |
\( T^{4} + 1344 T^{2} + 50176 \)
|
| $53$ |
\( T^{4} + 9632 T^{2} + 614656 \)
|
| $59$ |
\( (T^{2} - 92 T + 1874)^{2} \)
|
| $61$ |
\( T^{4} + 1652 T^{2} + 329476 \)
|
| $67$ |
\( (T^{2} + 112 T + 2624)^{2} \)
|
| $71$ |
\( T^{4} + 10752 T^{2} + 3211264 \)
|
| $73$ |
\( (T^{2} - 116 T + 3236)^{2} \)
|
| $79$ |
\( T^{4} + 8064 T^{2} + 9834496 \)
|
| $83$ |
\( (T^{2} - 44 T + 146)^{2} \)
|
| $89$ |
\( (T^{2} - 156 T + 4932)^{2} \)
|
| $97$ |
\( (T^{2} + 68 T - 15772)^{2} \)
|
| show more | |
| show less |