Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1280,2,Mod(1,1280)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1280, 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("1280.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1280 = 2^{8} \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1280.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: | \(10.2208514587\) |
| Analytic rank: | \(1\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{3}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - 3 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 40) |
| 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 \(\beta = \sqrt{3}\). 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 | −2.73205 | 0 | −1.00000 | 0 | −0.732051 | 0 | 4.46410 | 0 | ||||||||||||||||||||||||
| 1.2 | 0 | 0.732051 | 0 | −1.00000 | 0 | 2.73205 | 0 | −2.46410 | 0 | |||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(5\) | \( +1 \) |
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 | 1280.2.a.a | 2 | |
| 4.b | odd | 2 | 1 | 1280.2.a.n | 2 | ||
| 5.b | even | 2 | 1 | 6400.2.a.ce | 2 | ||
| 8.b | even | 2 | 1 | 1280.2.a.o | 2 | ||
| 8.d | odd | 2 | 1 | 1280.2.a.d | 2 | ||
| 16.e | even | 4 | 2 | 40.2.d.a | ✓ | 4 | |
| 16.f | odd | 4 | 2 | 160.2.d.a | 4 | ||
| 20.d | odd | 2 | 1 | 6400.2.a.be | 2 | ||
| 40.e | odd | 2 | 1 | 6400.2.a.cj | 2 | ||
| 40.f | even | 2 | 1 | 6400.2.a.z | 2 | ||
| 48.i | odd | 4 | 2 | 360.2.k.e | 4 | ||
| 48.k | even | 4 | 2 | 1440.2.k.e | 4 | ||
| 80.i | odd | 4 | 2 | 200.2.f.c | 4 | ||
| 80.j | even | 4 | 2 | 800.2.f.e | 4 | ||
| 80.k | odd | 4 | 2 | 800.2.d.e | 4 | ||
| 80.q | even | 4 | 2 | 200.2.d.f | 4 | ||
| 80.s | even | 4 | 2 | 800.2.f.c | 4 | ||
| 80.t | odd | 4 | 2 | 200.2.f.e | 4 | ||
| 240.t | even | 4 | 2 | 7200.2.k.j | 4 | ||
| 240.z | odd | 4 | 2 | 7200.2.d.o | 4 | ||
| 240.bb | even | 4 | 2 | 1800.2.d.p | 4 | ||
| 240.bd | odd | 4 | 2 | 7200.2.d.n | 4 | ||
| 240.bf | even | 4 | 2 | 1800.2.d.l | 4 | ||
| 240.bm | odd | 4 | 2 | 1800.2.k.j | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 40.2.d.a | ✓ | 4 | 16.e | even | 4 | 2 | |
| 160.2.d.a | 4 | 16.f | odd | 4 | 2 | ||
| 200.2.d.f | 4 | 80.q | even | 4 | 2 | ||
| 200.2.f.c | 4 | 80.i | odd | 4 | 2 | ||
| 200.2.f.e | 4 | 80.t | odd | 4 | 2 | ||
| 360.2.k.e | 4 | 48.i | odd | 4 | 2 | ||
| 800.2.d.e | 4 | 80.k | odd | 4 | 2 | ||
| 800.2.f.c | 4 | 80.s | even | 4 | 2 | ||
| 800.2.f.e | 4 | 80.j | even | 4 | 2 | ||
| 1280.2.a.a | 2 | 1.a | even | 1 | 1 | trivial | |
| 1280.2.a.d | 2 | 8.d | odd | 2 | 1 | ||
| 1280.2.a.n | 2 | 4.b | odd | 2 | 1 | ||
| 1280.2.a.o | 2 | 8.b | even | 2 | 1 | ||
| 1440.2.k.e | 4 | 48.k | even | 4 | 2 | ||
| 1800.2.d.l | 4 | 240.bf | even | 4 | 2 | ||
| 1800.2.d.p | 4 | 240.bb | even | 4 | 2 | ||
| 1800.2.k.j | 4 | 240.bm | odd | 4 | 2 | ||
| 6400.2.a.z | 2 | 40.f | even | 2 | 1 | ||
| 6400.2.a.be | 2 | 20.d | odd | 2 | 1 | ||
| 6400.2.a.ce | 2 | 5.b | even | 2 | 1 | ||
| 6400.2.a.cj | 2 | 40.e | odd | 2 | 1 | ||
| 7200.2.d.n | 4 | 240.bd | odd | 4 | 2 | ||
| 7200.2.d.o | 4 | 240.z | odd | 4 | 2 | ||
| 7200.2.k.j | 4 | 240.t | even | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1280))\):
|
\( T_{3}^{2} + 2T_{3} - 2 \)
|
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\( T_{7}^{2} - 2T_{7} - 2 \)
|
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\( T_{11} + 2 \)
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\( T_{13}^{2} - 12 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} \)
|
| $3$ |
\( T^{2} + 2T - 2 \)
|
| $5$ |
\( (T + 1)^{2} \)
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| $7$ |
\( T^{2} - 2T - 2 \)
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| $11$ |
\( (T + 2)^{2} \)
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| $13$ |
\( T^{2} - 12 \)
|
| $17$ |
\( T^{2} - 12 \)
|
| $19$ |
\( T^{2} + 8T + 4 \)
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| $23$ |
\( T^{2} - 2T - 26 \)
|
| $29$ |
\( T^{2} - 48 \)
|
| $31$ |
\( T^{2} + 4T - 8 \)
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| $37$ |
\( (T + 2)^{2} \)
|
| $41$ |
\( T^{2} - 4T - 8 \)
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| $43$ |
\( T^{2} + 14T + 46 \)
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| $47$ |
\( T^{2} - 10T + 22 \)
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| $53$ |
\( T^{2} - 16T + 52 \)
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| $59$ |
\( T^{2} + 8T + 4 \)
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| $61$ |
\( T^{2} - 4T - 44 \)
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| $67$ |
\( T^{2} + 18T + 78 \)
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| $71$ |
\( T^{2} + 4T - 8 \)
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| $73$ |
\( T^{2} + 8T + 4 \)
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| $79$ |
\( T^{2} + 16T + 16 \)
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| $83$ |
\( T^{2} + 6T + 6 \)
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| $89$ |
\( T^{2} + 4T - 44 \)
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| $97$ |
\( T^{2} + 8T - 92 \)
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