Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [4928,2,Mod(1,4928)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4928, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("4928.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 4928 = 2^{6} \cdot 7 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 4928.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: | \(39.3502781161\) |
| Analytic rank: | \(0\) |
| Dimension: | \(5\) |
| Coefficient field: | 5.5.2042356.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - 13x^{3} - 8x^{2} + 24x + 4 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | no (minimal twist has level 2464) |
| 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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) 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
\( x^{5} - 13x^{3} - 8x^{2} + 24x + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{4} - 3\nu^{3} - 6\nu^{2} + 14\nu - 2 ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{4} - \nu^{3} - 10\nu^{2} + 8 ) / 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{4} + 3\nu^{3} + 8\nu^{2} - 16\nu - 8 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 2\beta_{4} + 2\beta_{2} + \beta _1 + 10 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 4\beta_{4} + 2\beta_{3} + 2\beta_{2} + 9\beta _1 + 10 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 24\beta_{4} + 6\beta_{3} + 22\beta_{2} + 19\beta _1 + 94 ) / 2 \)
|
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 | −3.19289 | 0 | −3.87387 | 0 | −1.00000 | 0 | 7.19452 | 0 | |||||||||||||||||||||||||||||||||
| 1.2 | 0 | −2.79184 | 0 | 2.72505 | 0 | −1.00000 | 0 | 4.79436 | 0 | ||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −0.660872 | 0 | −1.73253 | 0 | −1.00000 | 0 | −2.56325 | 0 | ||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 0.421158 | 0 | 3.36074 | 0 | −1.00000 | 0 | −2.82263 | 0 | ||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 3.22444 | 0 | 0.520614 | 0 | −1.00000 | 0 | 7.39700 | 0 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(7\) | \(1\) |
| \(11\) | \(1\) |
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 | 4928.2.a.ck | 5 | |
| 4.b | odd | 2 | 1 | 4928.2.a.cl | 5 | ||
| 8.b | even | 2 | 1 | 2464.2.a.z | yes | 5 | |
| 8.d | odd | 2 | 1 | 2464.2.a.y | ✓ | 5 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2464.2.a.y | ✓ | 5 | 8.d | odd | 2 | 1 | |
| 2464.2.a.z | yes | 5 | 8.b | even | 2 | 1 | |
| 4928.2.a.ck | 5 | 1.a | even | 1 | 1 | trivial | |
| 4928.2.a.cl | 5 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4928))\):
|
\( T_{3}^{5} + 3T_{3}^{4} - 10T_{3}^{3} - 32T_{3}^{2} - 4T_{3} + 8 \)
|
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\( T_{5}^{5} - T_{5}^{4} - 18T_{5}^{3} + 20T_{5}^{2} + 56T_{5} - 32 \)
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\( T_{13}^{5} - 4T_{13}^{4} - 36T_{13}^{3} + 60T_{13}^{2} + 440T_{13} + 416 \)
|
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\( T_{17}^{5} - 16T_{17}^{4} + 80T_{17}^{3} - 92T_{17}^{2} - 216T_{17} + 256 \)
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\( T_{19}^{5} - 52T_{19}^{3} + 16T_{19}^{2} + 320T_{19} - 64 \)
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\( T_{23}^{5} + 11T_{23}^{4} + 8T_{23}^{3} - 144T_{23}^{2} - 96T_{23} + 512 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} \)
|
| $3$ |
\( T^{5} + 3 T^{4} + \cdots + 8 \)
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| $5$ |
\( T^{5} - T^{4} + \cdots - 32 \)
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| $7$ |
\( (T + 1)^{5} \)
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| $11$ |
\( (T + 1)^{5} \)
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| $13$ |
\( T^{5} - 4 T^{4} + \cdots + 416 \)
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| $17$ |
\( T^{5} - 16 T^{4} + \cdots + 256 \)
|
| $19$ |
\( T^{5} - 52 T^{3} + \cdots - 64 \)
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| $23$ |
\( T^{5} + 11 T^{4} + \cdots + 512 \)
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| $29$ |
\( T^{5} - 4 T^{4} + \cdots - 256 \)
|
| $31$ |
\( T^{5} - 5 T^{4} + \cdots - 256 \)
|
| $37$ |
\( T^{5} - T^{4} + \cdots - 1136 \)
|
| $41$ |
\( T^{5} - 18 T^{4} + \cdots + 1168 \)
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| $43$ |
\( T^{5} + 18 T^{4} + \cdots - 9344 \)
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| $47$ |
\( T^{5} - 12 T^{4} + \cdots - 59648 \)
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| $53$ |
\( T^{5} - 2 T^{4} + \cdots + 29728 \)
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| $59$ |
\( T^{5} + 5 T^{4} + \cdots - 6584 \)
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| $61$ |
\( T^{5} + 10 T^{4} + \cdots - 4864 \)
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| $67$ |
\( T^{5} - 13 T^{4} + \cdots - 3328 \)
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| $71$ |
\( T^{5} + 9 T^{4} + \cdots - 256 \)
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| $73$ |
\( T^{5} - 22 T^{4} + \cdots - 16 \)
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| $79$ |
\( T^{5} - 6 T^{4} + \cdots + 17152 \)
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| $83$ |
\( T^{5} - 228 T^{3} + \cdots + 5696 \)
|
| $89$ |
\( T^{5} - 15 T^{4} + \cdots + 21136 \)
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| $97$ |
\( T^{5} - 21 T^{4} + \cdots - 60464 \)
|
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