Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [425,2,Mod(1,425)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(425, 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("425.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 425 = 5^{2} \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 425.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: | \(3.39364208590\) |
| Analytic rank: | \(0\) |
| Dimension: | \(5\) |
| Coefficient field: | 5.5.1893456.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - x^{4} - 10x^{3} + 10x^{2} + 23x - 25 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| 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
\( x^{5} - x^{4} - 10x^{3} + 10x^{2} + 23x - 25 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 5\nu + 5 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - 9\nu^{2} - \nu + 16 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 5\beta _1 - 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{4} + 9\beta_{2} + \beta _1 + 20 \)
|
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 |
|
−2.60242 | −1.18219 | 4.77260 | 0 | 3.07656 | 3.53650 | −7.21549 | −1.60242 | 0 | |||||||||||||||||||||||||||||||||
| 1.2 | −1.24214 | −1.66068 | −0.457096 | 0 | 2.06279 | −4.35698 | 3.05205 | −0.242137 | 0 | ||||||||||||||||||||||||||||||||||
| 1.3 | −0.150980 | 1.96189 | −1.97720 | 0 | −0.296207 | 1.54475 | 0.600480 | 0.849020 | 0 | ||||||||||||||||||||||||||||||||||
| 1.4 | 2.19447 | 2.48887 | 2.81568 | 0 | 5.46174 | −3.05725 | 1.78998 | 3.19447 | 0 | ||||||||||||||||||||||||||||||||||
| 1.5 | 2.80107 | −2.60789 | 5.84602 | 0 | −7.30489 | 1.33298 | 10.7730 | 3.80107 | 0 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \( -1 \) |
| \(17\) | \( +1 \) |
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 | 425.2.a.j | yes | 5 |
| 3.b | odd | 2 | 1 | 3825.2.a.bl | 5 | ||
| 4.b | odd | 2 | 1 | 6800.2.a.cd | 5 | ||
| 5.b | even | 2 | 1 | 425.2.a.i | ✓ | 5 | |
| 5.c | odd | 4 | 2 | 425.2.b.f | 10 | ||
| 15.d | odd | 2 | 1 | 3825.2.a.bq | 5 | ||
| 17.b | even | 2 | 1 | 7225.2.a.y | 5 | ||
| 20.d | odd | 2 | 1 | 6800.2.a.bz | 5 | ||
| 85.c | even | 2 | 1 | 7225.2.a.x | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 425.2.a.i | ✓ | 5 | 5.b | even | 2 | 1 | |
| 425.2.a.j | yes | 5 | 1.a | even | 1 | 1 | trivial |
| 425.2.b.f | 10 | 5.c | odd | 4 | 2 | ||
| 3825.2.a.bl | 5 | 3.b | odd | 2 | 1 | ||
| 3825.2.a.bq | 5 | 15.d | odd | 2 | 1 | ||
| 6800.2.a.bz | 5 | 20.d | odd | 2 | 1 | ||
| 6800.2.a.cd | 5 | 4.b | odd | 2 | 1 | ||
| 7225.2.a.x | 5 | 85.c | even | 2 | 1 | ||
| 7225.2.a.y | 5 | 17.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(425))\):
|
\( T_{2}^{5} - T_{2}^{4} - 10T_{2}^{3} + 6T_{2}^{2} + 21T_{2} + 3 \)
|
|
\( T_{3}^{5} + T_{3}^{4} - 10T_{3}^{3} - 10T_{3}^{2} + 23T_{3} + 25 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} - T^{4} - 10 T^{3} + \cdots + 3 \)
|
| $3$ |
\( T^{5} + T^{4} + \cdots + 25 \)
|
| $5$ |
\( T^{5} \)
|
| $7$ |
\( T^{5} + T^{4} + \cdots - 97 \)
|
| $11$ |
\( T^{5} - 4 T^{4} + \cdots + 60 \)
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| $13$ |
\( T^{5} - 3 T^{4} + \cdots - 227 \)
|
| $17$ |
\( (T + 1)^{5} \)
|
| $19$ |
\( T^{5} - 6 T^{4} + \cdots + 400 \)
|
| $23$ |
\( T^{5} + 4 T^{4} + \cdots - 108 \)
|
| $29$ |
\( T^{5} - 2 T^{4} + \cdots + 240 \)
|
| $31$ |
\( T^{5} - 21 T^{4} + \cdots + 2151 \)
|
| $37$ |
\( T^{5} - 2 T^{4} + \cdots - 3824 \)
|
| $41$ |
\( T^{5} + 8 T^{4} + \cdots + 48 \)
|
| $43$ |
\( T^{5} - 4 T^{4} + \cdots - 14704 \)
|
| $47$ |
\( T^{5} - 2 T^{4} + \cdots - 480 \)
|
| $53$ |
\( T^{5} + 21 T^{4} + \cdots + 9 \)
|
| $59$ |
\( T^{5} - 12 T^{4} + \cdots - 11760 \)
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| $61$ |
\( T^{5} + 2 T^{4} + \cdots + 800 \)
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| $67$ |
\( T^{5} - 12 T^{4} + \cdots - 27008 \)
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| $71$ |
\( T^{5} - 21 T^{4} + \cdots + 11853 \)
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| $73$ |
\( T^{5} + 22 T^{4} + \cdots + 41744 \)
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| $79$ |
\( T^{5} - 41 T^{4} + \cdots + 42575 \)
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| $83$ |
\( T^{5} - 8 T^{4} + \cdots - 5952 \)
|
| $89$ |
\( T^{5} + 10 T^{4} + \cdots - 18000 \)
|
| $97$ |
\( T^{5} - 20 T^{4} + \cdots + 8000 \)
|
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