Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [105,3,Mod(11,105)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(105, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([3, 0, 4]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("105.11");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 105 = 3 \cdot 5 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 105.t (of order \(6\), 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: | \(2.86104277578\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(4\) over \(\Q(\zeta_{6})\) |
| Coefficient field: | 8.0.3317760000.8 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} + 4x^{6} + 7x^{4} + 36x^{2} + 81 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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
\( x^{8} + 4x^{6} + 7x^{4} + 36x^{2} + 81 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{6} + 14\nu^{4} - 7\nu^{2} - 36 ) / 63 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 4\nu^{6} + 7\nu^{4} + 28\nu^{2} + 144 ) / 63 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -4\nu^{7} - 7\nu^{5} + 35\nu^{3} - 81\nu ) / 189 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -5\nu^{7} + 7\nu^{5} - 35\nu^{3} - 180\nu ) / 189 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -8\nu^{6} - 14\nu^{4} + 7\nu^{2} - 162 ) / 63 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -\nu^{7} - 4\nu^{5} - 7\nu^{3} - 36\nu ) / 27 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{6} + 2\beta_{3} - 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{7} - \beta_{5} + 3\beta_{4} - \beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{3} + 4\beta_{2} \)
|
| \(\nu^{5}\) | \(=\) |
\( -5\beta_{7} + 7\beta_{5} \)
|
| \(\nu^{6}\) | \(=\) |
\( -7\beta_{6} - 7\beta_{2} - 22 \)
|
| \(\nu^{7}\) | \(=\) |
\( -21\beta_{5} - 21\beta_{4} - 29\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/105\mathbb{Z}\right)^\times\).
| \(n\) | \(22\) | \(31\) | \(71\) |
| \(\chi(n)\) | \(1\) | \(-\beta_{3}\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 11.1 |
|
−3.16124 | + | 1.82514i | −2.61469 | + | 1.47085i | 4.66228 | − | 8.07530i | 1.93649 | − | 1.11803i | 5.58114 | − | 9.42188i | 7.00000 | 19.4361i | 4.67319 | − | 7.69164i | −4.08114 | + | 7.06874i | ||||||||||||||||||||||||||||
| 11.2 | −0.711747 | + | 0.410927i | −2.25829 | − | 1.97487i | −1.66228 | + | 2.87915i | 1.93649 | − | 1.11803i | 2.41886 | + | 0.477612i | 7.00000 | − | 6.01972i | 1.19979 | + | 8.91967i | −0.918861 | + | 1.59151i | ||||||||||||||||||||||||||||
| 11.3 | 0.711747 | − | 0.410927i | 2.83943 | + | 0.968306i | −1.66228 | + | 2.87915i | −1.93649 | + | 1.11803i | 2.41886 | − | 0.477612i | 7.00000 | 6.01972i | 7.12477 | + | 5.49888i | −0.918861 | + | 1.59151i | |||||||||||||||||||||||||||||
| 11.4 | 3.16124 | − | 1.82514i | 0.0335498 | + | 2.99981i | 4.66228 | − | 8.07530i | −1.93649 | + | 1.11803i | 5.58114 | + | 9.42188i | 7.00000 | − | 19.4361i | −8.99775 | + | 0.201286i | −4.08114 | + | 7.06874i | ||||||||||||||||||||||||||||
| 86.1 | −3.16124 | − | 1.82514i | −2.61469 | − | 1.47085i | 4.66228 | + | 8.07530i | 1.93649 | + | 1.11803i | 5.58114 | + | 9.42188i | 7.00000 | − | 19.4361i | 4.67319 | + | 7.69164i | −4.08114 | − | 7.06874i | ||||||||||||||||||||||||||||
| 86.2 | −0.711747 | − | 0.410927i | −2.25829 | + | 1.97487i | −1.66228 | − | 2.87915i | 1.93649 | + | 1.11803i | 2.41886 | − | 0.477612i | 7.00000 | 6.01972i | 1.19979 | − | 8.91967i | −0.918861 | − | 1.59151i | |||||||||||||||||||||||||||||
| 86.3 | 0.711747 | + | 0.410927i | 2.83943 | − | 0.968306i | −1.66228 | − | 2.87915i | −1.93649 | − | 1.11803i | 2.41886 | + | 0.477612i | 7.00000 | − | 6.01972i | 7.12477 | − | 5.49888i | −0.918861 | − | 1.59151i | ||||||||||||||||||||||||||||
| 86.4 | 3.16124 | + | 1.82514i | 0.0335498 | − | 2.99981i | 4.66228 | + | 8.07530i | −1.93649 | − | 1.11803i | 5.58114 | − | 9.42188i | 7.00000 | 19.4361i | −8.99775 | − | 0.201286i | −4.08114 | − | 7.06874i | |||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
| 7.c | even | 3 | 1 | inner |
| 21.h | odd | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 105.3.t.a | ✓ | 8 |
| 3.b | odd | 2 | 1 | inner | 105.3.t.a | ✓ | 8 |
| 7.c | even | 3 | 1 | inner | 105.3.t.a | ✓ | 8 |
| 21.h | odd | 6 | 1 | inner | 105.3.t.a | ✓ | 8 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 105.3.t.a | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 105.3.t.a | ✓ | 8 | 3.b | odd | 2 | 1 | inner |
| 105.3.t.a | ✓ | 8 | 7.c | even | 3 | 1 | inner |
| 105.3.t.a | ✓ | 8 | 21.h | odd | 6 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{8} - 14T_{2}^{6} + 187T_{2}^{4} - 126T_{2}^{2} + 81 \)
acting on \(S_{3}^{\mathrm{new}}(105, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} - 14 T^{6} + \cdots + 81 \)
|
| $3$ |
\( T^{8} + 4 T^{7} + \cdots + 6561 \)
|
| $5$ |
\( (T^{4} - 5 T^{2} + 25)^{2} \)
|
| $7$ |
\( (T - 7)^{8} \)
|
| $11$ |
\( T^{8} - 126 T^{6} + \cdots + 531441 \)
|
| $13$ |
\( (T^{2} - 2 T - 249)^{4} \)
|
| $17$ |
\( T^{8} + \cdots + 10226063376 \)
|
| $19$ |
\( (T^{4} - 18 T^{3} + \cdots + 167281)^{2} \)
|
| $23$ |
\( T^{8} + \cdots + 66074188401 \)
|
| $29$ |
\( (T^{4} + 740 T^{2} + 72900)^{2} \)
|
| $31$ |
\( (T^{4} - 12 T^{3} + \cdots + 206116)^{2} \)
|
| $37$ |
\( (T^{4} + 6 T^{3} + \cdots + 2825761)^{2} \)
|
| $41$ |
\( (T^{2} + 405)^{4} \)
|
| $43$ |
\( (T^{2} - 12 T - 214)^{4} \)
|
| $47$ |
\( T^{8} + \cdots + 66074188401 \)
|
| $53$ |
\( T^{8} + \cdots + 3154956561 \)
|
| $59$ |
\( T^{8} + \cdots + 20\!\cdots\!96 \)
|
| $61$ |
\( (T^{4} - 132 T^{3} + \cdots + 5740816)^{2} \)
|
| $67$ |
\( (T^{4} - 144 T^{3} + \cdots + 25240576)^{2} \)
|
| $71$ |
\( (T^{4} + 2340 T^{2} + 72900)^{2} \)
|
| $73$ |
\( (T^{4} + 144 T^{3} + \cdots + 22033636)^{2} \)
|
| $79$ |
\( (T^{4} + 172 T^{3} + \cdots + 53377636)^{2} \)
|
| $83$ |
\( (T^{4} + 13160 T^{2} + 32400)^{2} \)
|
| $89$ |
\( T^{8} - 2736 T^{6} + \cdots + 26873856 \)
|
| $97$ |
\( (T^{2} + 112 T + 2886)^{4} \)
|
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