Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [4025,2,Mod(1,4025)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4025, 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("4025.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 4025 = 5^{2} \cdot 7 \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 4025.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: | \(32.1397868136\) |
| Analytic rank: | \(0\) |
| Dimension: | \(5\) |
| Coefficient field: | 5.5.122821.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - 2x^{4} - 4x^{3} + 4x^{2} + 3x - 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 805) |
| 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
\( x^{5} - 2x^{4} - 4x^{3} + 4x^{2} + 3x - 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{3} - 2\nu^{2} - 3\nu + 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{4} - 2\nu^{3} - 3\nu^{2} + 2\nu + 1 \)
|
| \(\beta_{4}\) | \(=\) |
\( -\nu^{4} + 2\nu^{3} + 4\nu^{2} - 4\nu - 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{4} + \beta_{3} + 2\beta _1 + 1 \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{4} + 2\beta_{3} + \beta_{2} + 7\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 7\beta_{4} + 8\beta_{3} + 2\beta_{2} + 18\beta _1 + 2 \)
|
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 |
|
−1.79802 | 1.59027 | 1.23287 | 0 | −2.85933 | 1.00000 | 1.37931 | −0.471046 | 0 | |||||||||||||||||||||||||||||||||
| 1.2 | −0.173158 | −1.11762 | −1.97002 | 0 | 0.193524 | 1.00000 | 0.687440 | −1.75093 | 0 | ||||||||||||||||||||||||||||||||||
| 1.3 | 0.733292 | 2.28713 | −1.46228 | 0 | 1.67714 | 1.00000 | −2.53886 | 2.23098 | 0 | ||||||||||||||||||||||||||||||||||
| 1.4 | 1.78759 | −0.0742225 | 1.19548 | 0 | −0.132679 | 1.00000 | −1.43816 | −2.99449 | 0 | ||||||||||||||||||||||||||||||||||
| 1.5 | 2.45030 | 3.31444 | 4.00395 | 0 | 8.12135 | 1.00000 | 4.91027 | 7.98549 | 0 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \(1\) |
| \(7\) | \(-1\) |
| \(23\) | \(1\) |
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 | 4025.2.a.r | 5 | |
| 5.b | even | 2 | 1 | 805.2.a.k | ✓ | 5 | |
| 15.d | odd | 2 | 1 | 7245.2.a.bi | 5 | ||
| 35.c | odd | 2 | 1 | 5635.2.a.x | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 805.2.a.k | ✓ | 5 | 5.b | even | 2 | 1 | |
| 4025.2.a.r | 5 | 1.a | even | 1 | 1 | trivial | |
| 5635.2.a.x | 5 | 35.c | odd | 2 | 1 | ||
| 7245.2.a.bi | 5 | 15.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4025))\):
|
\( T_{2}^{5} - 3T_{2}^{4} - 2T_{2}^{3} + 10T_{2}^{2} - 4T_{2} - 1 \)
|
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\( T_{3}^{5} - 6T_{3}^{4} + 8T_{3}^{3} + 7T_{3}^{2} - 13T_{3} - 1 \)
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\( T_{11}^{5} + 11T_{11}^{4} + 14T_{11}^{3} - 185T_{11}^{2} - 612T_{11} - 452 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} - 3 T^{4} + \cdots - 1 \)
|
| $3$ |
\( T^{5} - 6 T^{4} + \cdots - 1 \)
|
| $5$ |
\( T^{5} \)
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| $7$ |
\( (T - 1)^{5} \)
|
| $11$ |
\( T^{5} + 11 T^{4} + \cdots - 452 \)
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| $13$ |
\( T^{5} - 7 T^{4} + \cdots - 761 \)
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| $17$ |
\( T^{5} + T^{4} + \cdots - 1444 \)
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| $19$ |
\( T^{5} - 2 T^{4} + \cdots - 452 \)
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| $23$ |
\( (T + 1)^{5} \)
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| $29$ |
\( T^{5} + 14 T^{4} + \cdots - 7448 \)
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| $31$ |
\( T^{5} + 6 T^{4} + \cdots + 2489 \)
|
| $37$ |
\( T^{5} - T^{4} + \cdots - 13628 \)
|
| $41$ |
\( T^{5} - 7 T^{4} + \cdots - 331 \)
|
| $43$ |
\( T^{5} - 12 T^{4} + \cdots - 8236 \)
|
| $47$ |
\( T^{5} - 24 T^{4} + \cdots + 289 \)
|
| $53$ |
\( T^{5} - 21 T^{4} + \cdots + 548 \)
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| $59$ |
\( T^{5} + T^{4} + \cdots + 176 \)
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| $61$ |
\( T^{5} - 7 T^{4} + \cdots - 2252 \)
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| $67$ |
\( T^{5} - 35 T^{4} + \cdots + 596 \)
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| $71$ |
\( T^{5} + 32 T^{4} + \cdots - 54361 \)
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| $73$ |
\( T^{5} - 7 T^{4} + \cdots + 4297 \)
|
| $79$ |
\( T^{5} - 18 T^{4} + \cdots - 11236 \)
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| $83$ |
\( T^{5} - T^{4} + \cdots + 10204 \)
|
| $89$ |
\( T^{5} - T^{4} + \cdots - 16204 \)
|
| $97$ |
\( T^{5} - 9 T^{4} + \cdots - 4948 \)
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