Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2112,2,Mod(1343,2112)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2112, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0, 1, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2112.1343");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2112 = 2^{6} \cdot 3 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2112.d (of order \(2\), degree \(1\), not 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: | \(16.8644049069\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | 8.0.386672896.3 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - x^{6} - 2x^{5} + 2x^{4} - 4x^{3} - 4x^{2} + 16 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{6} \) |
| Twist minimal: | no (minimal twist has level 132) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - x^{6} - 2x^{5} + 2x^{4} - 4x^{3} - 4x^{2} + 16 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{7} - \nu^{5} + 2\nu^{4} + 2\nu^{3} - 4\nu ) / 8 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{7} + 2\nu^{6} - 3\nu^{5} + 6\nu^{3} + 8\nu^{2} - 12\nu ) / 8 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{7} + \nu^{5} + 2\nu^{4} - 2\nu^{3} + 4\nu^{2} + 4\nu ) / 4 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{7} + 2\nu^{6} - \nu^{5} - 2\nu^{3} - 4\nu^{2} - 12\nu ) / 8 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( \nu^{7} + 2\nu^{6} + 3\nu^{5} - 4\nu^{4} + 2\nu^{3} - 4\nu - 24 ) / 8 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -3\nu^{7} - 2\nu^{6} - \nu^{5} - 6\nu^{3} + 16\nu^{2} + 12\nu + 16 ) / 8 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{7} + \beta_{6} + 2\beta_{2} + 1 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -\beta_{7} - 2\beta_{5} + \beta_{3} + 2 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -\beta_{7} - \beta_{6} + 2\beta_{4} + 2\beta_{2} - 1 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -\beta_{7} + 2\beta_{6} - 2\beta_{5} + 2\beta_{4} - \beta_{3} - 2\beta _1 + 8 ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( -\beta_{7} + \beta_{6} + 4\beta_{5} + 2\beta_{4} + 2\beta_{3} - 2\beta_{2} + 4\beta _1 + 5 ) / 2 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 3\beta_{7} + 4\beta_{6} + 2\beta_{5} - 2\beta_{4} - 3\beta_{3} + 12\beta_{2} + 2\beta _1 + 6 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2112\mathbb{Z}\right)^\times\).
| \(n\) | \(133\) | \(1409\) | \(1729\) | \(2047\) |
| \(\chi(n)\) | \(1\) | \(-1\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1343.1 |
|
0 | −1.47398 | − | 0.909606i | 0 | 3.81921i | 0 | − | 2.00000i | 0 | 1.34523 | + | 2.68148i | 0 | |||||||||||||||||||||||||||||||||||||
| 1343.2 | 0 | −1.47398 | + | 0.909606i | 0 | − | 3.81921i | 0 | 2.00000i | 0 | 1.34523 | − | 2.68148i | 0 | ||||||||||||||||||||||||||||||||||||||
| 1343.3 | 0 | −0.356193 | − | 1.69503i | 0 | 1.39006i | 0 | 2.00000i | 0 | −2.74625 | + | 1.20752i | 0 | |||||||||||||||||||||||||||||||||||||||
| 1343.4 | 0 | −0.356193 | + | 1.69503i | 0 | − | 1.39006i | 0 | − | 2.00000i | 0 | −2.74625 | − | 1.20752i | 0 | |||||||||||||||||||||||||||||||||||||
| 1343.5 | 0 | 1.10238 | − | 1.33595i | 0 | 0.671897i | 0 | 2.00000i | 0 | −0.569517 | − | 2.94545i | 0 | |||||||||||||||||||||||||||||||||||||||
| 1343.6 | 0 | 1.10238 | + | 1.33595i | 0 | − | 0.671897i | 0 | − | 2.00000i | 0 | −0.569517 | + | 2.94545i | 0 | |||||||||||||||||||||||||||||||||||||
| 1343.7 | 0 | 1.72779 | − | 0.121372i | 0 | 2.24274i | 0 | − | 2.00000i | 0 | 2.97054 | − | 0.419412i | 0 | ||||||||||||||||||||||||||||||||||||||
| 1343.8 | 0 | 1.72779 | + | 0.121372i | 0 | − | 2.24274i | 0 | 2.00000i | 0 | 2.97054 | + | 0.419412i | 0 | ||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 12.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 2112.2.d.l | 8 | |
| 3.b | odd | 2 | 1 | 2112.2.d.k | 8 | ||
| 4.b | odd | 2 | 1 | 2112.2.d.k | 8 | ||
| 8.b | even | 2 | 1 | 132.2.c.c | ✓ | 8 | |
| 8.d | odd | 2 | 1 | 132.2.c.d | yes | 8 | |
| 12.b | even | 2 | 1 | inner | 2112.2.d.l | 8 | |
| 24.f | even | 2 | 1 | 132.2.c.c | ✓ | 8 | |
| 24.h | odd | 2 | 1 | 132.2.c.d | yes | 8 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 132.2.c.c | ✓ | 8 | 8.b | even | 2 | 1 | |
| 132.2.c.c | ✓ | 8 | 24.f | even | 2 | 1 | |
| 132.2.c.d | yes | 8 | 8.d | odd | 2 | 1 | |
| 132.2.c.d | yes | 8 | 24.h | odd | 2 | 1 | |
| 2112.2.d.k | 8 | 3.b | odd | 2 | 1 | ||
| 2112.2.d.k | 8 | 4.b | odd | 2 | 1 | ||
| 2112.2.d.l | 8 | 1.a | even | 1 | 1 | trivial | |
| 2112.2.d.l | 8 | 12.b | even | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2112, [\chi])\):
|
\( T_{5}^{8} + 22T_{5}^{6} + 121T_{5}^{4} + 192T_{5}^{2} + 64 \)
|
|
\( T_{7}^{2} + 4 \)
|
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\( T_{23}^{4} + 2T_{23}^{3} - 43T_{23}^{2} + 100T_{23} - 56 \)
|
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\( T_{47}^{4} + 12T_{47}^{3} + 8T_{47}^{2} - 192T_{47} - 256 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
|
| $3$ |
\( T^{8} - 2 T^{7} + \cdots + 81 \)
|
| $5$ |
\( T^{8} + 22 T^{6} + \cdots + 64 \)
|
| $7$ |
\( (T^{2} + 4)^{4} \)
|
| $11$ |
\( (T + 1)^{8} \)
|
| $13$ |
\( (T^{4} - 4 T^{3} - 20 T^{2} + \cdots - 32)^{2} \)
|
| $17$ |
\( T^{8} + 104 T^{6} + \cdots + 65536 \)
|
| $19$ |
\( T^{8} + 56 T^{6} + \cdots + 1024 \)
|
| $23$ |
\( (T^{4} + 2 T^{3} - 43 T^{2} + \cdots - 56)^{2} \)
|
| $29$ |
\( (T^{2} + 16)^{4} \)
|
| $31$ |
\( T^{8} + 54 T^{6} + \cdots + 400 \)
|
| $37$ |
\( (T^{4} - 10 T^{3} + \cdots + 116)^{2} \)
|
| $41$ |
\( T^{8} + 224 T^{6} + \cdots + 1638400 \)
|
| $43$ |
\( T^{8} + 144 T^{6} + \cdots + 215296 \)
|
| $47$ |
\( (T^{4} + 12 T^{3} + \cdots - 256)^{2} \)
|
| $53$ |
\( T^{8} + 192 T^{6} + \cdots + 65536 \)
|
| $59$ |
\( (T^{4} + 22 T^{3} + \cdots - 6640)^{2} \)
|
| $61$ |
\( (T^{4} + 8 T^{3} + \cdots + 256)^{2} \)
|
| $67$ |
\( T^{8} + 422 T^{6} + \cdots + 929296 \)
|
| $71$ |
\( (T^{4} + 10 T^{3} + \cdots - 376)^{2} \)
|
| $73$ |
\( (T^{4} + 8 T^{3} + \cdots + 2144)^{2} \)
|
| $79$ |
\( T^{8} + 384 T^{6} + \cdots + 20214016 \)
|
| $83$ |
\( (T^{4} + 20 T^{3} + \cdots - 5056)^{2} \)
|
| $89$ |
\( T^{8} + 342 T^{6} + \cdots + 3810304 \)
|
| $97$ |
\( (T^{4} + 14 T^{3} + 13 T^{2} + \cdots + 4)^{2} \)
|
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