Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [200,6,Mod(1,200)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(200, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("200.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 200 = 2^{3} \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 200.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: | \(32.0767639626\) |
| Analytic rank: | \(0\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.47217.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 38x - 24 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{4}\cdot 5 \) |
| 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 \(1,\beta_1,\beta_2\) 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
\( x^{3} - x^{2} - 38x - 24 \)
:
| \(\beta_{1}\) | \(=\) |
\( -\nu^{2} + 3\nu + 25 \)
|
| \(\beta_{2}\) | \(=\) |
\( 3\nu^{2} + 31\nu - 87 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + 3\beta _1 + 12 ) / 40 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 3\beta_{2} - 31\beta _1 + 1036 ) / 40 \)
|
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 | −22.6278 | 0 | 0 | 0 | 107.252 | 0 | 269.020 | 0 | |||||||||||||||||||||||||||
| 1.2 | 0 | 2.53448 | 0 | 0 | 0 | −247.370 | 0 | −236.576 | 0 | ||||||||||||||||||||||||||||
| 1.3 | 0 | 19.0934 | 0 | 0 | 0 | 210.117 | 0 | 121.557 | 0 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(5\) | \(1\) |
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 | 200.6.a.h | ✓ | 3 |
| 4.b | odd | 2 | 1 | 400.6.a.y | 3 | ||
| 5.b | even | 2 | 1 | 200.6.a.i | yes | 3 | |
| 5.c | odd | 4 | 2 | 200.6.c.g | 6 | ||
| 20.d | odd | 2 | 1 | 400.6.a.x | 3 | ||
| 20.e | even | 4 | 2 | 400.6.c.p | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 200.6.a.h | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
| 200.6.a.i | yes | 3 | 5.b | even | 2 | 1 | |
| 200.6.c.g | 6 | 5.c | odd | 4 | 2 | ||
| 400.6.a.x | 3 | 20.d | odd | 2 | 1 | ||
| 400.6.a.y | 3 | 4.b | odd | 2 | 1 | ||
| 400.6.c.p | 6 | 20.e | even | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{3} + T_{3}^{2} - 441T_{3} + 1095 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(200))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} \)
|
| $3$ |
\( T^{3} + T^{2} + \cdots + 1095 \)
|
| $5$ |
\( T^{3} \)
|
| $7$ |
\( T^{3} - 70 T^{2} + \cdots + 5574616 \)
|
| $11$ |
\( T^{3} + 19 T^{2} + \cdots - 1542819 \)
|
| $13$ |
\( T^{3} - 196 T^{2} + \cdots - 51768768 \)
|
| $17$ |
\( T^{3} + 1223 T^{2} + \cdots - 654046475 \)
|
| $19$ |
\( T^{3} + \cdots + 7033288181 \)
|
| $23$ |
\( T^{3} + \cdots + 50437840744 \)
|
| $29$ |
\( T^{3} + \cdots - 55993024512 \)
|
| $31$ |
\( T^{3} + \cdots + 14434811400 \)
|
| $37$ |
\( T^{3} + \cdots - 435216240216 \)
|
| $41$ |
\( T^{3} + \cdots - 86972031621 \)
|
| $43$ |
\( T^{3} + \cdots - 660952050112 \)
|
| $47$ |
\( T^{3} + \cdots - 324350860864 \)
|
| $53$ |
\( T^{3} + \cdots + 22325036001624 \)
|
| $59$ |
\( T^{3} + \cdots - 8618337690624 \)
|
| $61$ |
\( T^{3} + \cdots + 28184724893400 \)
|
| $67$ |
\( T^{3} + \cdots + 1261173627437 \)
|
| $71$ |
\( T^{3} + \cdots + 31597978616640 \)
|
| $73$ |
\( T^{3} + \cdots - 160799137840701 \)
|
| $79$ |
\( T^{3} + \cdots + 26059657326840 \)
|
| $83$ |
\( T^{3} + \cdots - 289154781250911 \)
|
| $89$ |
\( T^{3} + \cdots - 190810675088559 \)
|
| $97$ |
\( T^{3} + \cdots - 166573103686456 \)
|
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