Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [4518,2,Mod(1,4518)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4518, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("4518.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 4518 = 2 \cdot 3^{2} \cdot 251 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 4518.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: | \(36.0764116332\) |
| Analytic rank: | \(1\) |
| Dimension: | \(5\) |
| Coefficient field: | 5.5.138917.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - 6x^{3} - 2x^{2} + 3x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 502) |
| 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
\( x^{5} - 6x^{3} - 2x^{2} + 3x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{4} - 6\nu^{2} - 2\nu + 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( -\nu^{4} + \nu^{3} + 5\nu^{2} - 3\nu - 1 \)
|
| \(\beta_{4}\) | \(=\) |
\( -\nu^{4} + \nu^{3} + 6\nu^{2} - 3\nu - 4 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{4} - \beta_{3} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{4} + \beta_{2} + 5\beta _1 + 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( 6\beta_{4} - 6\beta_{3} + \beta_{2} + 2\beta _1 + 16 \)
|
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 | 0 | 1.00000 | −3.81582 | 0 | 2.11419 | 1.00000 | 0 | −3.81582 | |||||||||||||||||||||||||||||||||
| 1.2 | 1.00000 | 0 | 1.00000 | −3.18773 | 0 | −2.51166 | 1.00000 | 0 | −3.18773 | ||||||||||||||||||||||||||||||||||
| 1.3 | 1.00000 | 0 | 1.00000 | −1.24476 | 0 | 2.42122 | 1.00000 | 0 | −1.24476 | ||||||||||||||||||||||||||||||||||
| 1.4 | 1.00000 | 0 | 1.00000 | −0.781072 | 0 | 2.79232 | 1.00000 | 0 | −0.781072 | ||||||||||||||||||||||||||||||||||
| 1.5 | 1.00000 | 0 | 1.00000 | 2.02938 | 0 | −3.81607 | 1.00000 | 0 | 2.02938 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(3\) | \(-1\) |
| \(251\) | \(1\) |
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 | 4518.2.a.t | 5 | |
| 3.b | odd | 2 | 1 | 502.2.a.d | ✓ | 5 | |
| 12.b | even | 2 | 1 | 4016.2.a.d | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 502.2.a.d | ✓ | 5 | 3.b | odd | 2 | 1 | |
| 4016.2.a.d | 5 | 12.b | even | 2 | 1 | ||
| 4518.2.a.t | 5 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4518))\):
|
\( T_{5}^{5} + 7T_{5}^{4} + 9T_{5}^{3} - 24T_{5}^{2} - 52T_{5} - 24 \)
|
|
\( T_{7}^{5} - T_{7}^{4} - 19T_{7}^{3} + 28T_{7}^{2} + 80T_{7} - 137 \)
|
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\( T_{11}^{5} + 7T_{11}^{4} + 13T_{11}^{3} - 10T_{11} + 1 \)
|
|
\( T_{13}^{5} - 4T_{13}^{4} - 23T_{13}^{3} + 138T_{13}^{2} - 200T_{13} + 72 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1)^{5} \)
|
| $3$ |
\( T^{5} \)
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| $5$ |
\( T^{5} + 7 T^{4} + \cdots - 24 \)
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| $7$ |
\( T^{5} - T^{4} + \cdots - 137 \)
|
| $11$ |
\( T^{5} + 7 T^{4} + \cdots + 1 \)
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| $13$ |
\( T^{5} - 4 T^{4} + \cdots + 72 \)
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| $17$ |
\( T^{5} + 6 T^{4} + \cdots + 531 \)
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| $19$ |
\( T^{5} + 10 T^{4} + \cdots - 129 \)
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| $23$ |
\( T^{5} + 13 T^{4} + \cdots + 4703 \)
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| $29$ |
\( T^{5} + 4 T^{4} + \cdots + 123 \)
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| $31$ |
\( T^{5} + 13 T^{4} + \cdots - 1948 \)
|
| $37$ |
\( T^{5} - 16 T^{4} + \cdots - 10025 \)
|
| $41$ |
\( T^{5} + 6 T^{4} + \cdots - 2733 \)
|
| $43$ |
\( T^{5} + 8 T^{4} + \cdots + 16633 \)
|
| $47$ |
\( T^{5} + 29 T^{4} + \cdots + 1448 \)
|
| $53$ |
\( T^{5} + 25 T^{4} + \cdots - 2316 \)
|
| $59$ |
\( T^{5} + 11 T^{4} + \cdots + 64 \)
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| $61$ |
\( T^{5} - 11 T^{4} + \cdots - 107 \)
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| $67$ |
\( T^{5} + 6 T^{4} + \cdots + 138664 \)
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| $71$ |
\( T^{5} + 15 T^{4} + \cdots - 536 \)
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| $73$ |
\( T^{5} + 9 T^{4} + \cdots - 28608 \)
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| $79$ |
\( T^{5} + 29 T^{4} + \cdots - 247763 \)
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| $83$ |
\( T^{5} - 9 T^{4} + \cdots - 216 \)
|
| $89$ |
\( T^{5} - 9 T^{4} + \cdots + 1107 \)
|
| $97$ |
\( T^{5} + 6 T^{4} + \cdots - 8 \)
|
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