Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [8670,2,Mod(1,8670)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8670, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("8670.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 8670 = 2 \cdot 3 \cdot 5 \cdot 17^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 8670.a (trivial) |
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: | yes |
| Analytic conductor: | \(69.2302985525\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{2}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - 2 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 510) |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-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 \(\beta = 2\sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−1.00000 | 1.00000 | 1.00000 | −1.00000 | −1.00000 | −0.828427 | −1.00000 | 1.00000 | 1.00000 | ||||||||||||||||||||||||
| 1.2 | −1.00000 | 1.00000 | 1.00000 | −1.00000 | −1.00000 | 4.82843 | −1.00000 | 1.00000 | 1.00000 | |||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(3\) | \( -1 \) |
| \(5\) | \( +1 \) |
| \(17\) | \( +1 \) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 8670.2.a.bf | 2 | |
| 17.b | even | 2 | 1 | 8670.2.a.bd | 2 | ||
| 17.c | even | 4 | 2 | 510.2.c.d | ✓ | 4 | |
| 51.f | odd | 4 | 2 | 1530.2.c.f | 4 | ||
| 68.f | odd | 4 | 2 | 4080.2.h.o | 4 | ||
| 85.f | odd | 4 | 2 | 2550.2.f.o | 4 | ||
| 85.i | odd | 4 | 2 | 2550.2.f.t | 4 | ||
| 85.j | even | 4 | 2 | 2550.2.c.l | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 510.2.c.d | ✓ | 4 | 17.c | even | 4 | 2 | |
| 1530.2.c.f | 4 | 51.f | odd | 4 | 2 | ||
| 2550.2.c.l | 4 | 85.j | even | 4 | 2 | ||
| 2550.2.f.o | 4 | 85.f | odd | 4 | 2 | ||
| 2550.2.f.t | 4 | 85.i | odd | 4 | 2 | ||
| 4080.2.h.o | 4 | 68.f | odd | 4 | 2 | ||
| 8670.2.a.bd | 2 | 17.b | even | 2 | 1 | ||
| 8670.2.a.bf | 2 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8670))\):
|
\( T_{7}^{2} - 4T_{7} - 4 \)
|
|
\( T_{11}^{2} - 32 \)
|
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\( T_{13}^{2} - 4T_{13} - 4 \)
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|
\( T_{23} + 6 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 1)^{2} \)
|
| $3$ |
\( (T - 1)^{2} \)
|
| $5$ |
\( (T + 1)^{2} \)
|
| $7$ |
\( T^{2} - 4T - 4 \)
|
| $11$ |
\( T^{2} - 32 \)
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| $13$ |
\( T^{2} - 4T - 4 \)
|
| $17$ |
\( T^{2} \)
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| $19$ |
\( T^{2} \)
|
| $23$ |
\( (T + 6)^{2} \)
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| $29$ |
\( T^{2} - 4T - 28 \)
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| $31$ |
\( T^{2} + 4T - 28 \)
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| $37$ |
\( (T + 2)^{2} \)
|
| $41$ |
\( T^{2} - 8 \)
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| $43$ |
\( T^{2} - 8T - 56 \)
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| $47$ |
\( T^{2} - 32 \)
|
| $53$ |
\( T^{2} - 12T + 4 \)
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| $59$ |
\( T^{2} + 16T + 56 \)
|
| $61$ |
\( (T - 6)^{2} \)
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| $67$ |
\( T^{2} - 72 \)
|
| $71$ |
\( T^{2} + 4T - 4 \)
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| $73$ |
\( T^{2} - 16T + 56 \)
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| $79$ |
\( (T - 14)^{2} \)
|
| $83$ |
\( T^{2} - 288 \)
|
| $89$ |
\( T^{2} + 12T + 4 \)
|
| $97$ |
\( T^{2} + 8T - 56 \)
|
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