Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1050,3,Mod(451,1050)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1050, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 0, 1]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1050.451");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1050.p (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: | \(28.6104277578\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(4\) over \(\Q(\zeta_{6})\) |
| Coefficient field: | 8.0.151613669376.6 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 12x^{6} + 95x^{4} - 588x^{2} + 2401 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| 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
\( x^{8} - 12x^{6} + 95x^{4} - 588x^{2} + 2401 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -11\nu^{7} + 475\nu^{5} - 1045\nu^{3} + 6468\nu ) / 32585 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -12\nu^{6} + 95\nu^{4} - 1140\nu^{2} + 7056 ) / 4655 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -23\nu^{6} + 570\nu^{4} - 2185\nu^{2} + 13524 ) / 4655 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -\nu^{6} + 18 ) / 95 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -12\nu^{7} + 95\nu^{5} - 209\nu^{3} + 2401\nu ) / 6517 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( \nu^{7} - 12\nu^{5} + 95\nu^{3} - 588\nu ) / 343 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{5} + \beta_{4} - 6\beta_{3} + 6 \)
|
| \(\nu^{3}\) | \(=\) |
\( 5\beta_{7} + 7\beta_{6} + 5\beta_{2} + 5\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 12\beta_{4} - 23\beta_{3} \)
|
| \(\nu^{5}\) | \(=\) |
\( 11\beta_{7} + 95\beta_{2} \)
|
| \(\nu^{6}\) | \(=\) |
\( -95\beta_{5} + 18 \)
|
| \(\nu^{7}\) | \(=\) |
\( -665\beta_{6} + 665\beta_{2} + 113\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1050\mathbb{Z}\right)^\times\).
| \(n\) | \(127\) | \(451\) | \(701\) |
| \(\chi(n)\) | \(1\) | \(\beta_{3}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 451.1 |
|
−0.707107 | − | 1.22474i | −1.50000 | − | 0.866025i | −1.00000 | + | 1.73205i | 0 | 2.44949i | −0.440173 | − | 6.98615i | 2.82843 | 1.50000 | + | 2.59808i | 0 | ||||||||||||||||||||||||||||||||
| 451.2 | −0.707107 | − | 1.22474i | −1.50000 | − | 0.866025i | −1.00000 | + | 1.73205i | 0 | 2.44949i | 3.97571 | + | 5.76140i | 2.82843 | 1.50000 | + | 2.59808i | 0 | |||||||||||||||||||||||||||||||||
| 451.3 | 0.707107 | + | 1.22474i | −1.50000 | − | 0.866025i | −1.00000 | + | 1.73205i | 0 | − | 2.44949i | −3.97571 | − | 5.76140i | −2.82843 | 1.50000 | + | 2.59808i | 0 | ||||||||||||||||||||||||||||||||
| 451.4 | 0.707107 | + | 1.22474i | −1.50000 | − | 0.866025i | −1.00000 | + | 1.73205i | 0 | − | 2.44949i | 0.440173 | + | 6.98615i | −2.82843 | 1.50000 | + | 2.59808i | 0 | ||||||||||||||||||||||||||||||||
| 901.1 | −0.707107 | + | 1.22474i | −1.50000 | + | 0.866025i | −1.00000 | − | 1.73205i | 0 | − | 2.44949i | −0.440173 | + | 6.98615i | 2.82843 | 1.50000 | − | 2.59808i | 0 | ||||||||||||||||||||||||||||||||
| 901.2 | −0.707107 | + | 1.22474i | −1.50000 | + | 0.866025i | −1.00000 | − | 1.73205i | 0 | − | 2.44949i | 3.97571 | − | 5.76140i | 2.82843 | 1.50000 | − | 2.59808i | 0 | ||||||||||||||||||||||||||||||||
| 901.3 | 0.707107 | − | 1.22474i | −1.50000 | + | 0.866025i | −1.00000 | − | 1.73205i | 0 | 2.44949i | −3.97571 | + | 5.76140i | −2.82843 | 1.50000 | − | 2.59808i | 0 | |||||||||||||||||||||||||||||||||
| 901.4 | 0.707107 | − | 1.22474i | −1.50000 | + | 0.866025i | −1.00000 | − | 1.73205i | 0 | 2.44949i | 0.440173 | − | 6.98615i | −2.82843 | 1.50000 | − | 2.59808i | 0 | |||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.d | odd | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1050.3.p.c | ✓ | 8 |
| 5.b | even | 2 | 1 | 1050.3.p.d | yes | 8 | |
| 5.c | odd | 4 | 2 | 1050.3.q.b | 16 | ||
| 7.d | odd | 6 | 1 | inner | 1050.3.p.c | ✓ | 8 |
| 35.i | odd | 6 | 1 | 1050.3.p.d | yes | 8 | |
| 35.k | even | 12 | 2 | 1050.3.q.b | 16 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1050.3.p.c | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 1050.3.p.c | ✓ | 8 | 7.d | odd | 6 | 1 | inner |
| 1050.3.p.d | yes | 8 | 5.b | even | 2 | 1 | |
| 1050.3.p.d | yes | 8 | 35.i | odd | 6 | 1 | |
| 1050.3.q.b | 16 | 5.c | odd | 4 | 2 | ||
| 1050.3.q.b | 16 | 35.k | even | 12 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1050, [\chi])\):
|
\( T_{11}^{8} + 4T_{11}^{7} + 58T_{11}^{6} + 16T_{11}^{5} + 1954T_{11}^{4} + 2440T_{11}^{3} + 15940T_{11}^{2} - 16376T_{11} + 31684 \)
|
|
\( T_{17}^{8} + 24 T_{17}^{7} - 506 T_{17}^{6} - 16752 T_{17}^{5} + 368850 T_{17}^{4} + 4615176 T_{17}^{3} + \cdots + 4284749764 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{4} + 2 T^{2} + 4)^{2} \)
|
| $3$ |
\( (T^{2} + 3 T + 3)^{4} \)
|
| $5$ |
\( T^{8} \)
|
| $7$ |
\( T^{8} + 132 T^{6} + \cdots + 5764801 \)
|
| $11$ |
\( T^{8} + 4 T^{7} + \cdots + 31684 \)
|
| $13$ |
\( T^{8} + 540 T^{6} + \cdots + 3500641 \)
|
| $17$ |
\( T^{8} + \cdots + 4284749764 \)
|
| $19$ |
\( T^{8} - 72 T^{7} + \cdots + 648364369 \)
|
| $23$ |
\( T^{8} - 60 T^{7} + \cdots + 258309184 \)
|
| $29$ |
\( (T^{4} + 12 T^{3} + \cdots - 109496)^{2} \)
|
| $31$ |
\( T^{8} - 96 T^{7} + \cdots + 641507584 \)
|
| $37$ |
\( T^{8} + \cdots + 749278940881 \)
|
| $41$ |
\( T^{8} + \cdots + 16384512004 \)
|
| $43$ |
\( (T^{4} + 56 T^{3} + \cdots - 6447728)^{2} \)
|
| $47$ |
\( T^{8} + \cdots + 1293619615876 \)
|
| $53$ |
\( T^{8} + \cdots + 157223352196 \)
|
| $59$ |
\( T^{8} + \cdots + 14372454374404 \)
|
| $61$ |
\( T^{8} + \cdots + 16806974336161 \)
|
| $67$ |
\( T^{8} + \cdots + 122387325921 \)
|
| $71$ |
\( (T^{4} + 32 T^{3} + \cdots + 27042304)^{2} \)
|
| $73$ |
\( T^{8} + \cdots + 19\!\cdots\!61 \)
|
| $79$ |
\( T^{8} + \cdots + 334870712077729 \)
|
| $83$ |
\( T^{8} + \cdots + 29997547204 \)
|
| $89$ |
\( T^{8} + \cdots + 9916238788036 \)
|
| $97$ |
\( T^{8} + \cdots + 2745758363089 \)
|
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