Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [6272,2,Mod(1,6272)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6272, 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("6272.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 6272 = 2^{7} \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 6272.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: | \(50.0821721477\) |
| Analytic rank: | \(0\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.316.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 4x + 2 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 896) |
| 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 a basis \(1,\beta_1,\beta_2\) 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
\( x^{3} - x^{2} - 4x + 2 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu^{2} + \nu - 3 \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 3 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{2} + \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{2} + \beta _1 + 6 ) / 2 \)
|
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 | −2.24914 | 0 | −3.30777 | 0 | 0 | 0 | 2.05863 | 0 | |||||||||||||||||||||||||||
| 1.2 | 0 | 1.14637 | 0 | 3.83221 | 0 | 0 | 0 | −1.68585 | 0 | ||||||||||||||||||||||||||||
| 1.3 | 0 | 3.10278 | 0 | −2.52444 | 0 | 0 | 0 | 6.62721 | 0 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(7\) | \(-1\) |
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 | 6272.2.a.w | 3 | |
| 4.b | odd | 2 | 1 | 6272.2.a.u | 3 | ||
| 7.b | odd | 2 | 1 | 896.2.a.j | yes | 3 | |
| 8.b | even | 2 | 1 | 6272.2.a.v | 3 | ||
| 8.d | odd | 2 | 1 | 6272.2.a.x | 3 | ||
| 21.c | even | 2 | 1 | 8064.2.a.cb | 3 | ||
| 28.d | even | 2 | 1 | 896.2.a.l | yes | 3 | |
| 56.e | even | 2 | 1 | 896.2.a.i | ✓ | 3 | |
| 56.h | odd | 2 | 1 | 896.2.a.k | yes | 3 | |
| 84.h | odd | 2 | 1 | 8064.2.a.bu | 3 | ||
| 112.j | even | 4 | 2 | 1792.2.b.p | 6 | ||
| 112.l | odd | 4 | 2 | 1792.2.b.o | 6 | ||
| 168.e | odd | 2 | 1 | 8064.2.a.ce | 3 | ||
| 168.i | even | 2 | 1 | 8064.2.a.ch | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 896.2.a.i | ✓ | 3 | 56.e | even | 2 | 1 | |
| 896.2.a.j | yes | 3 | 7.b | odd | 2 | 1 | |
| 896.2.a.k | yes | 3 | 56.h | odd | 2 | 1 | |
| 896.2.a.l | yes | 3 | 28.d | even | 2 | 1 | |
| 1792.2.b.o | 6 | 112.l | odd | 4 | 2 | ||
| 1792.2.b.p | 6 | 112.j | even | 4 | 2 | ||
| 6272.2.a.u | 3 | 4.b | odd | 2 | 1 | ||
| 6272.2.a.v | 3 | 8.b | even | 2 | 1 | ||
| 6272.2.a.w | 3 | 1.a | even | 1 | 1 | trivial | |
| 6272.2.a.x | 3 | 8.d | odd | 2 | 1 | ||
| 8064.2.a.bu | 3 | 84.h | odd | 2 | 1 | ||
| 8064.2.a.cb | 3 | 21.c | even | 2 | 1 | ||
| 8064.2.a.ce | 3 | 168.e | odd | 2 | 1 | ||
| 8064.2.a.ch | 3 | 168.i | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6272))\):
|
\( T_{3}^{3} - 2T_{3}^{2} - 6T_{3} + 8 \)
|
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\( T_{5}^{3} + 2T_{5}^{2} - 14T_{5} - 32 \)
|
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\( T_{11}^{3} + 2T_{11}^{2} - 16T_{11} - 16 \)
|
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\( T_{13}^{3} + 6T_{13}^{2} - 22T_{13} - 136 \)
|
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\( T_{23}^{3} - 28T_{23} + 16 \)
|
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\( T_{29}^{3} - 10T_{29}^{2} + 4T_{29} + 88 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} \)
|
| $3$ |
\( T^{3} - 2 T^{2} + \cdots + 8 \)
|
| $5$ |
\( T^{3} + 2 T^{2} + \cdots - 32 \)
|
| $7$ |
\( T^{3} \)
|
| $11$ |
\( T^{3} + 2 T^{2} + \cdots - 16 \)
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| $13$ |
\( T^{3} + 6 T^{2} + \cdots - 136 \)
|
| $17$ |
\( T^{3} - 2 T^{2} + \cdots - 8 \)
|
| $19$ |
\( T^{3} - 14 T^{2} + \cdots - 64 \)
|
| $23$ |
\( T^{3} - 28T + 16 \)
|
| $29$ |
\( T^{3} - 10 T^{2} + \cdots + 88 \)
|
| $31$ |
\( T^{3} + 4 T^{2} + \cdots - 256 \)
|
| $37$ |
\( T^{3} - 2 T^{2} + \cdots - 136 \)
|
| $41$ |
\( T^{3} + 14 T^{2} + \cdots - 8 \)
|
| $43$ |
\( T^{3} + 2 T^{2} + \cdots + 304 \)
|
| $47$ |
\( T^{3} + 4 T^{2} + \cdots - 64 \)
|
| $53$ |
\( (T + 2)^{3} \)
|
| $59$ |
\( T^{3} - 10 T^{2} + \cdots + 1576 \)
|
| $61$ |
\( T^{3} + 26 T^{2} + \cdots + 496 \)
|
| $67$ |
\( T^{3} - 18 T^{2} + \cdots - 32 \)
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| $71$ |
\( T^{3} \)
|
| $73$ |
\( T^{3} - 2 T^{2} + \cdots + 8 \)
|
| $79$ |
\( T^{3} - 12 T^{2} + \cdots + 512 \)
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| $83$ |
\( T^{3} + 14 T^{2} + \cdots - 368 \)
|
| $89$ |
\( T^{3} - 2 T^{2} + \cdots + 296 \)
|
| $97$ |
\( T^{3} - 10 T^{2} + \cdots + 1816 \)
|
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