Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [768,4,Mod(1,768)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(768, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("768.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 768 = 2^{8} \cdot 3 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 768.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: | \(45.3134668844\) |
| 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: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 384) |
| 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 |
|
0 | −3.00000 | 0 | −2.82843 | 0 | −14.1421 | 0 | 9.00000 | 0 | ||||||||||||||||||||||||
| 1.2 | 0 | −3.00000 | 0 | 2.82843 | 0 | 14.1421 | 0 | 9.00000 | 0 | |||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(3\) | \(1\) |
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 8.d | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 768.4.a.h | 2 | |
| 3.b | odd | 2 | 1 | 2304.4.a.bb | 2 | ||
| 4.b | odd | 2 | 1 | 768.4.a.m | 2 | ||
| 8.b | even | 2 | 1 | 768.4.a.m | 2 | ||
| 8.d | odd | 2 | 1 | inner | 768.4.a.h | 2 | |
| 12.b | even | 2 | 1 | 2304.4.a.bh | 2 | ||
| 16.e | even | 4 | 2 | 384.4.d.d | ✓ | 4 | |
| 16.f | odd | 4 | 2 | 384.4.d.d | ✓ | 4 | |
| 24.f | even | 2 | 1 | 2304.4.a.bb | 2 | ||
| 24.h | odd | 2 | 1 | 2304.4.a.bh | 2 | ||
| 48.i | odd | 4 | 2 | 1152.4.d.n | 4 | ||
| 48.k | even | 4 | 2 | 1152.4.d.n | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 384.4.d.d | ✓ | 4 | 16.e | even | 4 | 2 | |
| 384.4.d.d | ✓ | 4 | 16.f | odd | 4 | 2 | |
| 768.4.a.h | 2 | 1.a | even | 1 | 1 | trivial | |
| 768.4.a.h | 2 | 8.d | odd | 2 | 1 | inner | |
| 768.4.a.m | 2 | 4.b | odd | 2 | 1 | ||
| 768.4.a.m | 2 | 8.b | even | 2 | 1 | ||
| 1152.4.d.n | 4 | 48.i | odd | 4 | 2 | ||
| 1152.4.d.n | 4 | 48.k | even | 4 | 2 | ||
| 2304.4.a.bb | 2 | 3.b | odd | 2 | 1 | ||
| 2304.4.a.bb | 2 | 24.f | even | 2 | 1 | ||
| 2304.4.a.bh | 2 | 12.b | even | 2 | 1 | ||
| 2304.4.a.bh | 2 | 24.h | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(768))\):
|
\( T_{5}^{2} - 8 \)
|
|
\( T_{7}^{2} - 200 \)
|
|
\( T_{11} - 20 \)
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|
\( T_{19} - 52 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} \)
|
| $3$ |
\( (T + 3)^{2} \)
|
| $5$ |
\( T^{2} - 8 \)
|
| $7$ |
\( T^{2} - 200 \)
|
| $11$ |
\( (T - 20)^{2} \)
|
| $13$ |
\( T^{2} - 1568 \)
|
| $17$ |
\( (T + 34)^{2} \)
|
| $19$ |
\( (T - 52)^{2} \)
|
| $23$ |
\( T^{2} - 3872 \)
|
| $29$ |
\( T^{2} - 40328 \)
|
| $31$ |
\( T^{2} - 12168 \)
|
| $37$ |
\( T^{2} - 73728 \)
|
| $41$ |
\( (T + 26)^{2} \)
|
| $43$ |
\( (T - 252)^{2} \)
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| $47$ |
\( T^{2} - 119072 \)
|
| $53$ |
\( T^{2} - 464648 \)
|
| $59$ |
\( (T - 364)^{2} \)
|
| $61$ |
\( T^{2} - 540800 \)
|
| $67$ |
\( (T - 628)^{2} \)
|
| $71$ |
\( T^{2} - 111392 \)
|
| $73$ |
\( (T - 338)^{2} \)
|
| $79$ |
\( T^{2} - 622728 \)
|
| $83$ |
\( (T - 1036)^{2} \)
|
| $89$ |
\( (T - 234)^{2} \)
|
| $97$ |
\( (T + 178)^{2} \)
|
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