Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [8800,2,Mod(1,8800)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8800, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("8800.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 8800 = 2^{5} \cdot 5^{2} \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 8800.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: | \(70.2683537787\) |
| Analytic rank: | \(1\) |
| Dimension: | \(5\) |
| Coefficient field: | 5.5.5060976.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - 2x^{4} - 7x^{3} + 12x^{2} - x - 2 \)
|
| 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
\( x^{5} - 2x^{4} - 7x^{3} + 12x^{2} - x - 2 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{4} - \nu^{3} - 8\nu^{2} + 4\nu + 3 \)
|
| \(\beta_{4}\) | \(=\) |
\( -\nu^{4} + 2\nu^{3} + 7\nu^{2} - 11\nu \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{4} + \beta_{3} + \beta_{2} + 7\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{4} + 2\beta_{3} + 9\beta_{2} + 3\beta _1 + 30 \)
|
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 |
|
0 | −2.56529 | 0 | 0 | 0 | −1.28048 | 0 | 3.58073 | 0 | |||||||||||||||||||||||||||||||||
| 1.2 | 0 | −0.342572 | 0 | 0 | 0 | −1.74484 | 0 | −2.88264 | 0 | ||||||||||||||||||||||||||||||||||
| 1.3 | 0 | 0.594438 | 0 | 0 | 0 | −3.46571 | 0 | −2.64664 | 0 | ||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 1.24959 | 0 | 0 | 0 | 3.00650 | 0 | −1.43851 | 0 | ||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 3.06383 | 0 | 0 | 0 | −0.515465 | 0 | 6.38707 | 0 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(5\) | \( +1 \) |
| \(11\) | \( +1 \) |
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 | 8800.2.a.bw | yes | 5 |
| 4.b | odd | 2 | 1 | 8800.2.a.bt | yes | 5 | |
| 5.b | even | 2 | 1 | 8800.2.a.bs | ✓ | 5 | |
| 20.d | odd | 2 | 1 | 8800.2.a.bx | yes | 5 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 8800.2.a.bs | ✓ | 5 | 5.b | even | 2 | 1 | |
| 8800.2.a.bt | yes | 5 | 4.b | odd | 2 | 1 | |
| 8800.2.a.bw | yes | 5 | 1.a | even | 1 | 1 | trivial |
| 8800.2.a.bx | yes | 5 | 20.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8800))\):
|
\( T_{3}^{5} - 2T_{3}^{4} - 7T_{3}^{3} + 12T_{3}^{2} - T_{3} - 2 \)
|
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\( T_{7}^{5} + 4T_{7}^{4} - 5T_{7}^{3} - 34T_{7}^{2} - 39T_{7} - 12 \)
|
|
\( T_{13}^{5} - T_{13}^{4} - 36T_{13}^{3} - 28T_{13}^{2} + 247T_{13} + 361 \)
|
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\( T_{17}^{5} - 39T_{17}^{3} - 36T_{17}^{2} + 43T_{17} + 40 \)
|
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\( T_{19}^{5} + 5T_{19}^{4} - 48T_{19}^{3} - 288T_{19}^{2} - 237T_{19} + 23 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} \)
|
| $3$ |
\( T^{5} - 2 T^{4} + \cdots - 2 \)
|
| $5$ |
\( T^{5} \)
|
| $7$ |
\( T^{5} + 4 T^{4} + \cdots - 12 \)
|
| $11$ |
\( (T + 1)^{5} \)
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| $13$ |
\( T^{5} - T^{4} + \cdots + 361 \)
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| $17$ |
\( T^{5} - 39 T^{3} + \cdots + 40 \)
|
| $19$ |
\( T^{5} + 5 T^{4} + \cdots + 23 \)
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| $23$ |
\( T^{5} - T^{4} + \cdots - 25 \)
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| $29$ |
\( T^{5} + 3 T^{4} + \cdots + 9 \)
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| $31$ |
\( T^{5} + 11 T^{4} + \cdots - 1385 \)
|
| $37$ |
\( T^{5} - 38 T^{3} + \cdots - 32 \)
|
| $41$ |
\( T^{5} - 8 T^{4} + \cdots - 8356 \)
|
| $43$ |
\( T^{5} - 7 T^{4} + \cdots + 464 \)
|
| $47$ |
\( T^{5} + 10 T^{4} + \cdots + 64 \)
|
| $53$ |
\( T^{5} - 8 T^{4} + \cdots - 15514 \)
|
| $59$ |
\( T^{5} + 6 T^{4} + \cdots + 2560 \)
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| $61$ |
\( T^{5} + 2 T^{4} + \cdots + 14890 \)
|
| $67$ |
\( T^{5} + 6 T^{4} + \cdots - 736 \)
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| $71$ |
\( T^{5} + 11 T^{4} + \cdots - 1744 \)
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| $73$ |
\( T^{5} - 89 T^{3} + \cdots - 4132 \)
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| $79$ |
\( T^{5} + 6 T^{4} + \cdots + 61760 \)
|
| $83$ |
\( T^{5} - 21 T^{4} + \cdots - 3643 \)
|
| $89$ |
\( T^{5} - 5 T^{4} + \cdots - 9991 \)
|
| $97$ |
\( T^{5} + 13 T^{4} + \cdots - 6693 \)
|
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