Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1050,2,Mod(299,1050)]
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([3, 3, 5]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1050.299");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1050.u (of order \(6\), 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: | \(8.38429221223\) |
| Analytic rank: | \(1\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{6})\) |
| 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: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 42) |
| 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 primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1050\mathbb{Z}\right)^\times\).
| \(n\) | \(127\) | \(451\) | \(701\) |
| \(\chi(n)\) | \(-1\) | \(\zeta_{12}^{2}\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 299.1 |
|
−0.500000 | + | 0.866025i | −1.50000 | + | 0.866025i | −0.500000 | − | 0.866025i | 0 | − | 1.73205i | −0.866025 | + | 2.50000i | 1.00000 | 1.50000 | − | 2.59808i | 0 | |||||||||||||||||||
| 299.2 | −0.500000 | + | 0.866025i | −1.50000 | + | 0.866025i | −0.500000 | − | 0.866025i | 0 | − | 1.73205i | 0.866025 | − | 2.50000i | 1.00000 | 1.50000 | − | 2.59808i | 0 | ||||||||||||||||||||
| 899.1 | −0.500000 | − | 0.866025i | −1.50000 | − | 0.866025i | −0.500000 | + | 0.866025i | 0 | 1.73205i | −0.866025 | − | 2.50000i | 1.00000 | 1.50000 | + | 2.59808i | 0 | |||||||||||||||||||||
| 899.2 | −0.500000 | − | 0.866025i | −1.50000 | − | 0.866025i | −0.500000 | + | 0.866025i | 0 | 1.73205i | 0.866025 | + | 2.50000i | 1.00000 | 1.50000 | + | 2.59808i | 0 | |||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.d | odd | 6 | 1 | inner |
| 15.d | odd | 2 | 1 | inner |
| 105.p | even | 6 | 1 | inner |
Twists
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1050, [\chi])\):
|
\( T_{11}^{4} - 9T_{11}^{2} + 81 \)
|
|
\( T_{13}^{2} - 12 \)
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\( T_{17}^{2} + 6T_{17} + 12 \)
|
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\( T_{23}^{2} + 6T_{23} + 36 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + T + 1)^{2} \)
|
| $3$ |
\( (T^{2} + 3 T + 3)^{2} \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( T^{4} + 11T^{2} + 49 \)
|
| $11$ |
\( T^{4} - 9T^{2} + 81 \)
|
| $13$ |
\( (T^{2} - 12)^{2} \)
|
| $17$ |
\( (T^{2} + 6 T + 12)^{2} \)
|
| $19$ |
\( (T^{2} + 6 T + 12)^{2} \)
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| $23$ |
\( (T^{2} + 6 T + 36)^{2} \)
|
| $29$ |
\( (T^{2} + 9)^{2} \)
|
| $31$ |
\( (T^{2} + 3 T + 3)^{2} \)
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| $37$ |
\( T^{4} - 4T^{2} + 16 \)
|
| $41$ |
\( (T^{2} - 48)^{2} \)
|
| $43$ |
\( (T^{2} + 64)^{2} \)
|
| $47$ |
\( (T^{2} + 12 T + 48)^{2} \)
|
| $53$ |
\( (T^{2} + 9 T + 81)^{2} \)
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| $59$ |
\( T^{4} + 3T^{2} + 9 \)
|
| $61$ |
\( T^{4} \)
|
| $67$ |
\( T^{4} - 4T^{2} + 16 \)
|
| $71$ |
\( (T^{2} + 144)^{2} \)
|
| $73$ |
\( T^{4} + 48T^{2} + 2304 \)
|
| $79$ |
\( (T^{2} + T + 1)^{2} \)
|
| $83$ |
\( (T^{2} + 75)^{2} \)
|
| $89$ |
\( T^{4} + 108 T^{2} + 11664 \)
|
| $97$ |
\( (T^{2} - 27)^{2} \)
|
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