Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [5390,2,Mod(1,5390)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5390, 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("5390.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 5390 = 2 \cdot 5 \cdot 7^{2} \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 5390.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: | \(43.0393666895\) |
| Analytic rank: | \(0\) |
| Dimension: | \(5\) |
| Coefficient field: | 5.5.12760689.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - 12x^{3} - 3x^{2} + 31x + 7 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 770) |
| 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
\( x^{5} - 12x^{3} - 3x^{2} + 31x + 7 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 5 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{4} - \nu^{3} - 9\nu^{2} + 4\nu + 13 ) / 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{4} + 3\nu^{3} + 7\nu^{2} - 18\nu - 7 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 5 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{4} + \beta_{3} + \beta_{2} + 7\beta _1 + 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{4} + 3\beta_{3} + 10\beta_{2} + 3\beta _1 + 34 \)
|
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 | −3.07519 | 1.00000 | 1.00000 | −3.07519 | 0 | 1.00000 | 6.45677 | 1.00000 | |||||||||||||||||||||||||||||||||
| 1.2 | 1.00000 | −1.84492 | 1.00000 | 1.00000 | −1.84492 | 0 | 1.00000 | 0.403739 | 1.00000 | ||||||||||||||||||||||||||||||||||
| 1.3 | 1.00000 | 0.225302 | 1.00000 | 1.00000 | 0.225302 | 0 | 1.00000 | −2.94924 | 1.00000 | ||||||||||||||||||||||||||||||||||
| 1.4 | 1.00000 | 2.16285 | 1.00000 | 1.00000 | 2.16285 | 0 | 1.00000 | 1.67790 | 1.00000 | ||||||||||||||||||||||||||||||||||
| 1.5 | 1.00000 | 2.53196 | 1.00000 | 1.00000 | 2.53196 | 0 | 1.00000 | 3.41082 | 1.00000 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(5\) | \(-1\) |
| \(7\) | \(1\) |
| \(11\) | \(-1\) |
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 | 5390.2.a.ci | 5 | |
| 7.b | odd | 2 | 1 | 5390.2.a.ch | 5 | ||
| 7.c | even | 3 | 2 | 770.2.i.m | ✓ | 10 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 770.2.i.m | ✓ | 10 | 7.c | even | 3 | 2 | |
| 5390.2.a.ch | 5 | 7.b | odd | 2 | 1 | ||
| 5390.2.a.ci | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5390))\):
|
\( T_{3}^{5} - 12T_{3}^{3} + 3T_{3}^{2} + 31T_{3} - 7 \)
|
|
\( T_{13}^{5} + T_{13}^{4} - 28T_{13}^{3} - 20T_{13}^{2} + 168T_{13} + 144 \)
|
|
\( T_{17}^{5} - 43T_{17}^{3} + 75T_{17}^{2} + 172T_{17} - 224 \)
|
|
\( T_{19}^{5} - 4T_{19}^{4} - 21T_{19}^{3} + 75T_{19}^{2} + 90T_{19} - 304 \)
|
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\( T_{31}^{5} - 6T_{31}^{4} - 13T_{31}^{3} + 67T_{31}^{2} + 90T_{31} + 24 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1)^{5} \)
|
| $3$ |
\( T^{5} - 12 T^{3} + \cdots - 7 \)
|
| $5$ |
\( (T - 1)^{5} \)
|
| $7$ |
\( T^{5} \)
|
| $11$ |
\( (T - 1)^{5} \)
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| $13$ |
\( T^{5} + T^{4} + \cdots + 144 \)
|
| $17$ |
\( T^{5} - 43 T^{3} + \cdots - 224 \)
|
| $19$ |
\( T^{5} - 4 T^{4} + \cdots - 304 \)
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| $23$ |
\( T^{5} - 7 T^{4} + \cdots + 48 \)
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| $29$ |
\( T^{5} - 4 T^{4} + \cdots - 317 \)
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| $31$ |
\( T^{5} - 6 T^{4} + \cdots + 24 \)
|
| $37$ |
\( T^{5} - 16 T^{4} + \cdots - 10368 \)
|
| $41$ |
\( T^{5} + T^{4} + \cdots - 3584 \)
|
| $43$ |
\( T^{5} - 13 T^{4} + \cdots - 16 \)
|
| $47$ |
\( T^{5} - 8 T^{4} + \cdots - 9728 \)
|
| $53$ |
\( T^{5} - 4 T^{4} + \cdots - 37244 \)
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| $59$ |
\( T^{5} - 9 T^{4} + \cdots - 17024 \)
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| $61$ |
\( T^{5} - 2 T^{4} + \cdots - 5033 \)
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| $67$ |
\( T^{5} - 5 T^{4} + \cdots - 649 \)
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| $71$ |
\( T^{5} - 28 T^{4} + \cdots + 12544 \)
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| $73$ |
\( T^{5} + T^{4} + \cdots + 5416 \)
|
| $79$ |
\( T^{5} - 23 T^{4} + \cdots - 79952 \)
|
| $83$ |
\( T^{5} + 23 T^{4} + \cdots + 12672 \)
|
| $89$ |
\( T^{5} + 9 T^{4} + \cdots + 13302 \)
|
| $97$ |
\( T^{5} + 13 T^{4} + \cdots + 120192 \)
|
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