Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2106,2,Mod(703,2106)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2106, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([4, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2106.703");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2106 = 2 \cdot 3^{4} \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2106.e (of order \(3\), degree \(2\), 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: | \(16.8164946657\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\Q(\zeta_{12})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 3 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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
| \(\beta_{1}\) | \(=\) |
\( \zeta_{12}^{2} \)
|
| \(\beta_{2}\) | \(=\) |
\( \zeta_{12}^{3} + \zeta_{12} \)
|
| \(\beta_{3}\) | \(=\) |
\( -\zeta_{12}^{3} + 2\zeta_{12} \)
|
| \(\zeta_{12}\) | \(=\) |
\( ( \beta_{3} + \beta_{2} ) / 3 \)
|
| \(\zeta_{12}^{2}\) | \(=\) |
\( \beta_1 \)
|
| \(\zeta_{12}^{3}\) | \(=\) |
\( ( -\beta_{3} + 2\beta_{2} ) / 3 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2106\mathbb{Z}\right)^\times\).
| \(n\) | \(1379\) | \(1783\) |
| \(\chi(n)\) | \(-\beta_{1}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 703.1 |
|
−0.500000 | + | 0.866025i | 0 | −0.500000 | − | 0.866025i | −0.366025 | − | 0.633975i | 0 | −2.23205 | + | 3.86603i | 1.00000 | 0 | 0.732051 | ||||||||||||||||||||||
| 703.2 | −0.500000 | + | 0.866025i | 0 | −0.500000 | − | 0.866025i | 1.36603 | + | 2.36603i | 0 | 1.23205 | − | 2.13397i | 1.00000 | 0 | −2.73205 | |||||||||||||||||||||||
| 1405.1 | −0.500000 | − | 0.866025i | 0 | −0.500000 | + | 0.866025i | −0.366025 | + | 0.633975i | 0 | −2.23205 | − | 3.86603i | 1.00000 | 0 | 0.732051 | |||||||||||||||||||||||
| 1405.2 | −0.500000 | − | 0.866025i | 0 | −0.500000 | + | 0.866025i | 1.36603 | − | 2.36603i | 0 | 1.23205 | + | 2.13397i | 1.00000 | 0 | −2.73205 | |||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 9.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 2106.2.e.bd | 4 | |
| 3.b | odd | 2 | 1 | 2106.2.e.bf | 4 | ||
| 9.c | even | 3 | 1 | 2106.2.a.n | yes | 2 | |
| 9.c | even | 3 | 1 | inner | 2106.2.e.bd | 4 | |
| 9.d | odd | 6 | 1 | 2106.2.a.k | ✓ | 2 | |
| 9.d | odd | 6 | 1 | 2106.2.e.bf | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2106.2.a.k | ✓ | 2 | 9.d | odd | 6 | 1 | |
| 2106.2.a.n | yes | 2 | 9.c | even | 3 | 1 | |
| 2106.2.e.bd | 4 | 1.a | even | 1 | 1 | trivial | |
| 2106.2.e.bd | 4 | 9.c | even | 3 | 1 | inner | |
| 2106.2.e.bf | 4 | 3.b | odd | 2 | 1 | ||
| 2106.2.e.bf | 4 | 9.d | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2106, [\chi])\):
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\( T_{5}^{4} - 2T_{5}^{3} + 6T_{5}^{2} + 4T_{5} + 4 \)
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\( T_{7}^{4} + 2T_{7}^{3} + 15T_{7}^{2} - 22T_{7} + 121 \)
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\( T_{11}^{4} + 8T_{11}^{3} + 51T_{11}^{2} + 104T_{11} + 169 \)
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\( T_{19}^{2} - 3 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + T + 1)^{2} \)
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| $3$ |
\( T^{4} \)
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| $5$ |
\( T^{4} - 2 T^{3} + \cdots + 4 \)
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| $7$ |
\( T^{4} + 2 T^{3} + \cdots + 121 \)
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| $11$ |
\( T^{4} + 8 T^{3} + \cdots + 169 \)
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| $13$ |
\( (T^{2} - T + 1)^{2} \)
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| $17$ |
\( (T^{2} - 48)^{2} \)
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| $19$ |
\( (T^{2} - 3)^{2} \)
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| $23$ |
\( T^{4} - 2 T^{3} + \cdots + 676 \)
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| $29$ |
\( T^{4} + 16 T^{3} + \cdots + 3721 \)
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| $31$ |
\( T^{4} + 8 T^{3} + \cdots + 16 \)
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| $37$ |
\( (T^{2} + 12 T + 24)^{2} \)
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| $41$ |
\( T^{4} + 6 T^{3} + \cdots + 36 \)
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| $43$ |
\( T^{4} - 4 T^{3} + \cdots + 1936 \)
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| $47$ |
\( T^{4} + 8 T^{3} + \cdots + 16 \)
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| $53$ |
\( (T^{2} - 75)^{2} \)
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| $59$ |
\( T^{4} + 4 T^{3} + \cdots + 20449 \)
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| $61$ |
\( T^{4} - 4 T^{3} + \cdots + 529 \)
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| $67$ |
\( T^{4} - 4 T^{3} + \cdots + 10816 \)
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| $71$ |
\( (T^{2} - 14 T + 37)^{2} \)
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| $73$ |
\( (T^{2} + 20 T + 88)^{2} \)
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| $79$ |
\( T^{4} + 22 T^{3} + \cdots + 8836 \)
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| $83$ |
\( T^{4} + 12 T^{3} + \cdots + 1521 \)
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| $89$ |
\( (T^{2} + 18 T + 54)^{2} \)
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| $97$ |
\( T^{4} + 22 T^{3} + \cdots + 13924 \)
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