Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1450,2,Mod(1,1450)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1450, 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("1450.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1450 = 2 \cdot 5^{2} \cdot 29 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1450.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: | \(11.5783082931\) |
| Analytic rank: | \(1\) |
| Dimension: | \(5\) |
| Coefficient field: | 5.5.3661564.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - 2x^{4} - 10x^{3} + 3x^{2} + 20x + 8 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 290) |
| 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
\( x^{5} - 2x^{4} - 10x^{3} + 3x^{2} + 20x + 8 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{4} - 4\nu^{3} - 2\nu^{2} + 9\nu - 2 ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{4} - 4\nu^{3} - 4\nu^{2} + 13\nu + 6 ) / 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - 3\nu^{3} - 7\nu^{2} + 9\nu + 11 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{3} + \beta_{2} + 2\beta _1 + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{4} - 5\beta_{3} + 3\beta_{2} + 10\beta _1 + 7 \)
|
| \(\nu^{4}\) | \(=\) |
\( 4\beta_{4} - 22\beta_{3} + 16\beta_{2} + 35\beta _1 + 38 \)
|
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 | −2.81278 | 1.00000 | 0 | 2.81278 | −0.647892 | −1.00000 | 4.91176 | 0 | |||||||||||||||||||||||||||||||||
| 1.2 | −1.00000 | −2.35864 | 1.00000 | 0 | 2.35864 | −4.21797 | −1.00000 | 2.56319 | 0 | ||||||||||||||||||||||||||||||||||
| 1.3 | −1.00000 | −1.48580 | 1.00000 | 0 | 1.48580 | 4.37538 | −1.00000 | −0.792385 | 0 | ||||||||||||||||||||||||||||||||||
| 1.4 | −1.00000 | 0.681929 | 1.00000 | 0 | −0.681929 | −0.936197 | −1.00000 | −2.53497 | 0 | ||||||||||||||||||||||||||||||||||
| 1.5 | −1.00000 | 2.97530 | 1.00000 | 0 | −2.97530 | −3.57331 | −1.00000 | 5.85241 | 0 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(5\) | \(-1\) |
| \(29\) | \(-1\) |
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 | 1450.2.a.t | 5 | |
| 5.b | even | 2 | 1 | 1450.2.a.u | 5 | ||
| 5.c | odd | 4 | 2 | 290.2.b.b | ✓ | 10 | |
| 15.e | even | 4 | 2 | 2610.2.e.i | 10 | ||
| 20.e | even | 4 | 2 | 2320.2.d.h | 10 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 290.2.b.b | ✓ | 10 | 5.c | odd | 4 | 2 | |
| 1450.2.a.t | 5 | 1.a | even | 1 | 1 | trivial | |
| 1450.2.a.u | 5 | 5.b | even | 2 | 1 | ||
| 2320.2.d.h | 10 | 20.e | even | 4 | 2 | ||
| 2610.2.e.i | 10 | 15.e | even | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1450))\):
|
\( T_{3}^{5} + 3T_{3}^{4} - 8T_{3}^{3} - 29T_{3}^{2} - 7T_{3} + 20 \)
|
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\( T_{7}^{5} + 5T_{7}^{4} - 13T_{7}^{3} - 94T_{7}^{2} - 116T_{7} - 40 \)
|
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\( T_{13}^{5} + 7T_{13}^{4} - 2T_{13}^{3} - 25T_{13}^{2} + T_{13} + 8 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 1)^{5} \)
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| $3$ |
\( T^{5} + 3 T^{4} + \cdots + 20 \)
|
| $5$ |
\( T^{5} \)
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| $7$ |
\( T^{5} + 5 T^{4} + \cdots - 40 \)
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| $11$ |
\( T^{5} - 27 T^{3} + \cdots - 16 \)
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| $13$ |
\( T^{5} + 7 T^{4} + \cdots + 8 \)
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| $17$ |
\( T^{5} + 9 T^{4} + \cdots + 1712 \)
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| $19$ |
\( T^{5} + 4 T^{4} + \cdots + 640 \)
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| $23$ |
\( T^{5} + 13 T^{4} + \cdots - 1616 \)
|
| $29$ |
\( (T - 1)^{5} \)
|
| $31$ |
\( T^{5} - 5 T^{4} + \cdots + 184 \)
|
| $37$ |
\( T^{5} - 6 T^{4} + \cdots + 512 \)
|
| $41$ |
\( T^{5} + 4 T^{4} + \cdots + 608 \)
|
| $43$ |
\( T^{5} + T^{4} + \cdots + 664 \)
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| $47$ |
\( T^{5} + 14 T^{4} + \cdots - 64 \)
|
| $53$ |
\( T^{5} + 5 T^{4} + \cdots - 4 \)
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| $59$ |
\( T^{5} + 9 T^{4} + \cdots + 4000 \)
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| $61$ |
\( T^{5} - 5 T^{4} + \cdots - 9808 \)
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| $67$ |
\( T^{5} + 2 T^{4} + \cdots + 1024 \)
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| $71$ |
\( T^{5} + 6 T^{4} + \cdots + 62464 \)
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| $73$ |
\( T^{5} + 9 T^{4} + \cdots + 54848 \)
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| $79$ |
\( T^{5} + 17 T^{4} + \cdots - 3020 \)
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| $83$ |
\( T^{5} + 30 T^{4} + \cdots - 58304 \)
|
| $89$ |
\( T^{5} - 10 T^{4} + \cdots + 160 \)
|
| $97$ |
\( T^{5} - 15 T^{4} + \cdots + 57896 \)
|
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