Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [200,14,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, 14, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("200.1");
S:= CuspForms(chi, 14);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 200 = 2^{3} \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 14 \) |
| 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: | \(214.461857904\) |
| Analytic rank: | \(1\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - x^{5} - 1444079x^{4} - 56528771x^{3} + 433996031186x^{2} + 67576226094880x + 2042331073104000 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{26}\cdot 3^{2}\cdot 5^{5} \) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - x^{5} - 1444079x^{4} - 56528771x^{3} + 433996031186x^{2} + 67576226094880x + 2042331073104000 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 10051 \nu^{5} - 826652 \nu^{4} - 13825541277 \nu^{3} + 694719617806 \nu^{2} + \cdots + 26\!\cdots\!00 ) / 422868478500 \)
|
| \(\beta_{3}\) | \(=\) |
\( 4\nu^{2} - 240\nu - 1925399 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 9227 \nu^{5} - 4706404 \nu^{4} - 13390185429 \nu^{3} + 4684498273862 \nu^{2} + \cdots - 78\!\cdots\!00 ) / 28191231900 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 64009 \nu^{5} + 5264468 \nu^{4} + 94641308343 \nu^{3} - 3510937070554 \nu^{2} + \cdots - 23\!\cdots\!00 ) / 140956159500 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + 120\beta _1 + 1925399 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 171\beta_{5} - 277\beta_{3} + 3267\beta_{2} + 3371321\beta _1 + 230768809 ) / 8 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -15120\beta_{5} - 28566\beta_{4} + 1049625\beta_{3} + 104490\beta_{2} + 53971012\beta _1 + 1623299092899 ) / 4 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 232730037 \beta_{5} - 4698864 \beta_{4} - 346608791 \beta_{3} + 4847651469 \beta_{2} + \cdots + 103673432807651 ) / 8 \)
|
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 | −2112.86 | 0 | 0 | 0 | 178812. | 0 | 2.86987e6 | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −1196.44 | 0 | 0 | 0 | −369015. | 0 | −162860. | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −326.091 | 0 | 0 | 0 | 321957. | 0 | −1.48799e6 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | −167.067 | 0 | 0 | 0 | −315720. | 0 | −1.56641e6 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 1383.59 | 0 | 0 | 0 | 291912. | 0 | 320006. | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 1910.87 | 0 | 0 | 0 | −353507. | 0 | 2.05709e6 | 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 | 200.14.a.g | ✓ | 6 |
| 5.b | even | 2 | 1 | 200.14.a.h | yes | 6 | |
| 5.c | odd | 4 | 2 | 200.14.c.g | 12 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 200.14.a.g | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 200.14.a.h | yes | 6 | 5.b | even | 2 | 1 | |
| 200.14.c.g | 12 | 5.c | odd | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{6} + 508 T_{3}^{5} - 5668791 T_{3}^{4} - 2404039608 T_{3}^{3} + 6578985234411 T_{3}^{2} + \cdots + 36\!\cdots\!75 \)
acting on \(S_{14}^{\mathrm{new}}(\Gamma_0(200))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( T^{6} + \cdots + 36\!\cdots\!75 \)
|
| $5$ |
\( T^{6} \)
|
| $7$ |
\( T^{6} + \cdots - 69\!\cdots\!04 \)
|
| $11$ |
\( T^{6} + \cdots - 28\!\cdots\!49 \)
|
| $13$ |
\( T^{6} + \cdots + 39\!\cdots\!84 \)
|
| $17$ |
\( T^{6} + \cdots + 23\!\cdots\!25 \)
|
| $19$ |
\( T^{6} + \cdots + 39\!\cdots\!71 \)
|
| $23$ |
\( T^{6} + \cdots - 85\!\cdots\!04 \)
|
| $29$ |
\( T^{6} + \cdots + 12\!\cdots\!84 \)
|
| $31$ |
\( T^{6} + \cdots + 13\!\cdots\!00 \)
|
| $37$ |
\( T^{6} + \cdots - 26\!\cdots\!84 \)
|
| $41$ |
\( T^{6} + \cdots - 28\!\cdots\!19 \)
|
| $43$ |
\( T^{6} + \cdots + 43\!\cdots\!24 \)
|
| $47$ |
\( T^{6} + \cdots - 78\!\cdots\!44 \)
|
| $53$ |
\( T^{6} + \cdots + 84\!\cdots\!76 \)
|
| $59$ |
\( T^{6} + \cdots + 25\!\cdots\!16 \)
|
| $61$ |
\( T^{6} + \cdots + 40\!\cdots\!00 \)
|
| $67$ |
\( T^{6} + \cdots - 34\!\cdots\!61 \)
|
| $71$ |
\( T^{6} + \cdots + 47\!\cdots\!00 \)
|
| $73$ |
\( T^{6} + \cdots + 55\!\cdots\!61 \)
|
| $79$ |
\( T^{6} + \cdots - 21\!\cdots\!00 \)
|
| $83$ |
\( T^{6} + \cdots + 49\!\cdots\!31 \)
|
| $89$ |
\( T^{6} + \cdots + 28\!\cdots\!81 \)
|
| $97$ |
\( T^{6} + \cdots + 26\!\cdots\!36 \)
|
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