Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [5415,2,Mod(1,5415)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5415, 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("5415.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 5415 = 3 \cdot 5 \cdot 19^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 5415.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: | \(43.2389926945\) |
| Analytic rank: | \(0\) |
| Dimension: | \(5\) |
| Coefficient field: | 5.5.8797896.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - x^{4} - 8x^{3} + 5x^{2} + 13x - 4 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 285) |
| 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
\( x^{5} - x^{4} - 8x^{3} + 5x^{2} + 13x - 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 5\nu + 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - \nu^{3} - 6\nu^{2} + 3\nu + 4 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 5\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{4} + \beta_{3} + 7\beta_{2} + 2\beta _1 + 15 \)
|
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 |
|
−2.69159 | −1.00000 | 5.24466 | 1.00000 | 2.69159 | −0.797044 | −8.73329 | 1.00000 | −2.69159 | |||||||||||||||||||||||||||||||||
| 1.2 | −1.64661 | −1.00000 | 0.711327 | 1.00000 | 1.64661 | 4.47988 | 2.12194 | 1.00000 | −1.64661 | ||||||||||||||||||||||||||||||||||
| 1.3 | −0.290697 | −1.00000 | −1.91550 | 1.00000 | 0.290697 | −0.486575 | 1.13822 | 1.00000 | −0.290697 | ||||||||||||||||||||||||||||||||||
| 1.4 | 1.38140 | −1.00000 | −0.0917248 | 1.00000 | −1.38140 | −4.36264 | −2.88952 | 1.00000 | 1.38140 | ||||||||||||||||||||||||||||||||||
| 1.5 | 2.24750 | −1.00000 | 3.05123 | 1.00000 | −2.24750 | 3.16638 | 2.36264 | 1.00000 | 2.24750 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( +1 \) |
| \(5\) | \( -1 \) |
| \(19\) | \( +1 \) |
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 | 5415.2.a.y | 5 | |
| 19.b | odd | 2 | 1 | 5415.2.a.z | 5 | ||
| 19.c | even | 3 | 2 | 285.2.i.f | ✓ | 10 | |
| 57.h | odd | 6 | 2 | 855.2.k.i | 10 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 285.2.i.f | ✓ | 10 | 19.c | even | 3 | 2 | |
| 855.2.k.i | 10 | 57.h | odd | 6 | 2 | ||
| 5415.2.a.y | 5 | 1.a | even | 1 | 1 | trivial | |
| 5415.2.a.z | 5 | 19.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5415))\):
|
\( T_{2}^{5} + T_{2}^{4} - 8T_{2}^{3} - 5T_{2}^{2} + 13T_{2} + 4 \)
|
|
\( T_{7}^{5} - 2T_{7}^{4} - 23T_{7}^{3} + 36T_{7}^{2} + 72T_{7} + 24 \)
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\( T_{11}^{5} - 5T_{11}^{4} - 32T_{11}^{3} + 148T_{11}^{2} + 256T_{11} - 992 \)
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\( T_{13}^{5} + 8T_{13}^{4} + T_{13}^{3} - 70T_{13}^{2} - 44T_{13} + 128 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} + T^{4} - 8 T^{3} + \cdots + 4 \)
|
| $3$ |
\( (T + 1)^{5} \)
|
| $5$ |
\( (T - 1)^{5} \)
|
| $7$ |
\( T^{5} - 2 T^{4} + \cdots + 24 \)
|
| $11$ |
\( T^{5} - 5 T^{4} + \cdots - 992 \)
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| $13$ |
\( T^{5} + 8 T^{4} + \cdots + 128 \)
|
| $17$ |
\( T^{5} - 10 T^{4} + \cdots - 5024 \)
|
| $19$ |
\( T^{5} \)
|
| $23$ |
\( T^{5} + 2 T^{4} + \cdots + 384 \)
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| $29$ |
\( T^{5} + 7 T^{4} + \cdots + 3008 \)
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| $31$ |
\( T^{5} + 9 T^{4} + \cdots + 117 \)
|
| $37$ |
\( T^{5} - 6 T^{4} + \cdots - 9024 \)
|
| $41$ |
\( T^{5} - 12 T^{4} + \cdots - 192 \)
|
| $43$ |
\( T^{5} + 8 T^{4} + \cdots + 128 \)
|
| $47$ |
\( T^{5} - 6 T^{4} + \cdots + 2136 \)
|
| $53$ |
\( T^{5} - 8 T^{4} + \cdots - 4328 \)
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| $59$ |
\( T^{5} + T^{4} + \cdots - 4544 \)
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| $61$ |
\( T^{5} - 7 T^{4} + \cdots - 24316 \)
|
| $67$ |
\( T^{5} + 14 T^{4} + \cdots + 8376 \)
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| $71$ |
\( T^{5} + 27 T^{4} + \cdots - 6096 \)
|
| $73$ |
\( T^{5} - 26 T^{4} + \cdots + 46208 \)
|
| $79$ |
\( T^{5} - 23 T^{4} + \cdots + 109248 \)
|
| $83$ |
\( T^{5} + 12 T^{4} + \cdots - 2304 \)
|
| $89$ |
\( T^{5} + 9 T^{4} + \cdots + 11232 \)
|
| $97$ |
\( (T - 2)^{5} \)
|
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