Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [93,2,Mod(25,93)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(93, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 2]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("93.25");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 93 = 3 \cdot 31 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 93.e (of order \(3\), degree \(2\), 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: | \(0.742608738798\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Relative dimension: | \(3\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | 6.0.591408.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - x^{5} + 4x^{4} + x^{3} + 10x^{2} - 3x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - x^{5} + 4x^{4} + x^{3} + 10x^{2} - 3x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{5} + 4\nu^{4} - 16\nu^{3} + 10\nu^{2} - 3\nu + 12 ) / 37 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -4\nu^{5} + 16\nu^{4} - 27\nu^{3} + 40\nu^{2} - 12\nu + 85 ) / 37 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 12\nu^{5} - 11\nu^{4} + 44\nu^{3} + 28\nu^{2} + 110\nu + 4 ) / 37 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -25\nu^{5} + 26\nu^{4} - 104\nu^{3} - 9\nu^{2} - 260\nu + 78 ) / 37 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{5} + 2\beta_{4} - \beta_{2} + \beta _1 - 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} - 4\beta_{2} - 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -4\beta_{5} - 7\beta_{4} + 4\beta_{3} - 6\beta_1 \)
|
| \(\nu^{5}\) | \(=\) |
\( -6\beta_{5} - 8\beta_{4} + 17\beta_{2} - 17\beta _1 + 8 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/93\mathbb{Z}\right)^\times\).
| \(n\) | \(32\) | \(34\) |
| \(\chi(n)\) | \(1\) | \(-1 + \beta_{4}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 25.1 |
|
−1.67513 | −0.500000 | − | 0.866025i | 0.806063 | −1.64363 | + | 2.84685i | 0.837565 | + | 1.45071i | 1.07816 | + | 1.86743i | 2.00000 | −0.500000 | + | 0.866025i | 2.75329 | − | 4.76884i | ||||||||||||||||||||||||
| 25.2 | −0.539189 | −0.500000 | − | 0.866025i | −1.70928 | 1.43968 | − | 2.49360i | 0.269594 | + | 0.466951i | −1.31545 | − | 2.27842i | 2.00000 | −0.500000 | + | 0.866025i | −0.776260 | + | 1.34452i | |||||||||||||||||||||||||
| 25.3 | 2.21432 | −0.500000 | − | 0.866025i | 2.90321 | −1.79605 | + | 3.11085i | −1.10716 | − | 1.91766i | −1.76271 | − | 3.05311i | 2.00000 | −0.500000 | + | 0.866025i | −3.97703 | + | 6.88842i | |||||||||||||||||||||||||
| 67.1 | −1.67513 | −0.500000 | + | 0.866025i | 0.806063 | −1.64363 | − | 2.84685i | 0.837565 | − | 1.45071i | 1.07816 | − | 1.86743i | 2.00000 | −0.500000 | − | 0.866025i | 2.75329 | + | 4.76884i | |||||||||||||||||||||||||
| 67.2 | −0.539189 | −0.500000 | + | 0.866025i | −1.70928 | 1.43968 | + | 2.49360i | 0.269594 | − | 0.466951i | −1.31545 | + | 2.27842i | 2.00000 | −0.500000 | − | 0.866025i | −0.776260 | − | 1.34452i | |||||||||||||||||||||||||
| 67.3 | 2.21432 | −0.500000 | + | 0.866025i | 2.90321 | −1.79605 | − | 3.11085i | −1.10716 | + | 1.91766i | −1.76271 | + | 3.05311i | 2.00000 | −0.500000 | − | 0.866025i | −3.97703 | − | 6.88842i | |||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 31.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 93.2.e.b | ✓ | 6 |
| 3.b | odd | 2 | 1 | 279.2.h.d | 6 | ||
| 4.b | odd | 2 | 1 | 1488.2.q.m | 6 | ||
| 31.c | even | 3 | 1 | inner | 93.2.e.b | ✓ | 6 |
| 31.c | even | 3 | 1 | 2883.2.a.h | 3 | ||
| 31.e | odd | 6 | 1 | 2883.2.a.g | 3 | ||
| 93.g | even | 6 | 1 | 8649.2.a.n | 3 | ||
| 93.h | odd | 6 | 1 | 279.2.h.d | 6 | ||
| 93.h | odd | 6 | 1 | 8649.2.a.o | 3 | ||
| 124.i | odd | 6 | 1 | 1488.2.q.m | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 93.2.e.b | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 93.2.e.b | ✓ | 6 | 31.c | even | 3 | 1 | inner |
| 279.2.h.d | 6 | 3.b | odd | 2 | 1 | ||
| 279.2.h.d | 6 | 93.h | odd | 6 | 1 | ||
| 1488.2.q.m | 6 | 4.b | odd | 2 | 1 | ||
| 1488.2.q.m | 6 | 124.i | odd | 6 | 1 | ||
| 2883.2.a.g | 3 | 31.e | odd | 6 | 1 | ||
| 2883.2.a.h | 3 | 31.c | even | 3 | 1 | ||
| 8649.2.a.n | 3 | 93.g | even | 6 | 1 | ||
| 8649.2.a.o | 3 | 93.h | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{3} - 4T_{2} - 2 \)
acting on \(S_{2}^{\mathrm{new}}(93, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{3} - 4 T - 2)^{2} \)
|
| $3$ |
\( (T^{2} + T + 1)^{3} \)
|
| $5$ |
\( T^{6} + 4 T^{5} + \cdots + 1156 \)
|
| $7$ |
\( T^{6} + 4 T^{5} + \cdots + 400 \)
|
| $11$ |
\( T^{6} - 2 T^{5} + \cdots + 2500 \)
|
| $13$ |
\( T^{6} + 7 T^{5} + \cdots + 1 \)
|
| $17$ |
\( T^{6} - 2 T^{5} + \cdots + 10816 \)
|
| $19$ |
\( T^{6} - 5 T^{5} + \cdots + 25 \)
|
| $23$ |
\( (T^{3} - 12 T^{2} + \cdots + 16)^{2} \)
|
| $29$ |
\( (T^{3} - 4 T^{2} - 32 T - 2)^{2} \)
|
| $31$ |
\( T^{6} - 27 T^{4} + \cdots + 29791 \)
|
| $37$ |
\( T^{6} - 3 T^{5} + \cdots + 134689 \)
|
| $41$ |
\( T^{6} - 4 T^{5} + \cdots + 1444 \)
|
| $43$ |
\( T^{6} - T^{5} + \cdots + 105625 \)
|
| $47$ |
\( (T^{3} + 6 T^{2} - 88 T + 58)^{2} \)
|
| $53$ |
\( T^{6} - 10 T^{5} + \cdots + 2500 \)
|
| $59$ |
\( T^{6} - 16 T^{5} + \cdots + 6400 \)
|
| $61$ |
\( (T^{3} - 10 T^{2} + \cdots - 20)^{2} \)
|
| $67$ |
\( T^{6} + 24 T^{5} + \cdots + 21904 \)
|
| $71$ |
\( T^{6} + 20 T^{5} + \cdots + 73984 \)
|
| $73$ |
\( T^{6} + 3 T^{5} + \cdots + 10609 \)
|
| $79$ |
\( T^{6} - 4 T^{5} + \cdots + 400 \)
|
| $83$ |
\( T^{6} - 16 T^{5} + \cdots + 3844 \)
|
| $89$ |
\( (T^{3} - 6 T^{2} + \cdots - 270)^{2} \)
|
| $97$ |
\( (T^{3} + 17 T^{2} + \cdots + 67)^{2} \)
|
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