Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2156,1,Mod(197,2156)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2156, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 1]))
N = Newforms(chi, 1, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2156.197");
S:= CuspForms(chi, 1);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2156 = 2^{2} \cdot 7^{2} \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 1 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2156.h (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: | yes |
| Analytic conductor: | \(1.07598416724\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\zeta_{24})^+\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 4x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Projective image: | \(D_{12}\) |
| Projective field: | Galois closure of \(\mathbb{Q}[x]/(x^{12} - \cdots)\) |
$q$-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 \(\nu = \zeta_{24} + \zeta_{24}^{-1}\):
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 4\nu \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 4\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2156\mathbb{Z}\right)^\times\).
| \(n\) | \(981\) | \(1079\) | \(1277\) |
| \(\chi(n)\) | \(-1\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 197.1 |
|
0 | −1.93185 | 0 | −0.517638 | 0 | 0 | 0 | 2.73205 | 0 | ||||||||||||||||||||||||||||||
| 197.2 | 0 | −0.517638 | 0 | −1.93185 | 0 | 0 | 0 | −0.732051 | 0 | |||||||||||||||||||||||||||||||
| 197.3 | 0 | 0.517638 | 0 | 1.93185 | 0 | 0 | 0 | −0.732051 | 0 | |||||||||||||||||||||||||||||||
| 197.4 | 0 | 1.93185 | 0 | 0.517638 | 0 | 0 | 0 | 2.73205 | 0 | |||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 11.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-11}) \) |
| 7.b | odd | 2 | 1 | inner |
| 77.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 2156.1.h.c | ✓ | 4 |
| 7.b | odd | 2 | 1 | inner | 2156.1.h.c | ✓ | 4 |
| 7.c | even | 3 | 2 | 2156.1.k.d | 8 | ||
| 7.d | odd | 6 | 2 | 2156.1.k.d | 8 | ||
| 11.b | odd | 2 | 1 | CM | 2156.1.h.c | ✓ | 4 |
| 77.b | even | 2 | 1 | inner | 2156.1.h.c | ✓ | 4 |
| 77.h | odd | 6 | 2 | 2156.1.k.d | 8 | ||
| 77.i | even | 6 | 2 | 2156.1.k.d | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2156.1.h.c | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 2156.1.h.c | ✓ | 4 | 7.b | odd | 2 | 1 | inner |
| 2156.1.h.c | ✓ | 4 | 11.b | odd | 2 | 1 | CM |
| 2156.1.h.c | ✓ | 4 | 77.b | even | 2 | 1 | inner |
| 2156.1.k.d | 8 | 7.c | even | 3 | 2 | ||
| 2156.1.k.d | 8 | 7.d | odd | 6 | 2 | ||
| 2156.1.k.d | 8 | 77.h | odd | 6 | 2 | ||
| 2156.1.k.d | 8 | 77.i | even | 6 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} - 4T_{3}^{2} + 1 \)
acting on \(S_{1}^{\mathrm{new}}(2156, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} - 4T^{2} + 1 \)
|
| $5$ |
\( T^{4} - 4T^{2} + 1 \)
|
| $7$ |
\( T^{4} \)
|
| $11$ |
\( (T + 1)^{4} \)
|
| $13$ |
\( T^{4} \)
|
| $17$ |
\( T^{4} \)
|
| $19$ |
\( T^{4} \)
|
| $23$ |
\( (T^{2} - 3)^{2} \)
|
| $29$ |
\( T^{4} \)
|
| $31$ |
\( T^{4} - 4T^{2} + 1 \)
|
| $37$ |
\( (T - 1)^{4} \)
|
| $41$ |
\( T^{4} \)
|
| $43$ |
\( T^{4} \)
|
| $47$ |
\( (T^{2} - 2)^{2} \)
|
| $53$ |
\( T^{4} \)
|
| $59$ |
\( T^{4} - 4T^{2} + 1 \)
|
| $61$ |
\( T^{4} \)
|
| $67$ |
\( (T^{2} - 3)^{2} \)
|
| $71$ |
\( (T^{2} - 3)^{2} \)
|
| $73$ |
\( T^{4} \)
|
| $79$ |
\( T^{4} \)
|
| $83$ |
\( T^{4} \)
|
| $89$ |
\( T^{4} - 4T^{2} + 1 \)
|
| $97$ |
\( T^{4} - 4T^{2} + 1 \)
|
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