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: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 7121x^{2} - 128406x + 3057138 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{18}\cdot 3^{2}\cdot 5^{2}\cdot 17 \) |
| Twist minimal: | no (minimal twist has level 40) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - x^{3} - 7121x^{2} - 128406x + 3057138 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 116\nu^{3} - 4424\nu^{2} - 844984\nu + 4173126 ) / 11355 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -2032\nu^{3} + 378208\nu^{2} - 1887712\nu - 1139688552 ) / 2271 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 75332\nu^{3} - 1369448\nu^{2} - 385106968\nu - 2684620098 ) / 3785 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} - 3\beta_{2} - 2211\beta _1 + 16320 ) / 65280 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 111\beta_{3} + 653\beta_{2} - 159061\beta _1 + 464891520 ) / 130560 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 553\beta_{3} - 553\beta_{2} - 749923\beta _1 + 390318720 ) / 3840 \)
|
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 | −2181.88 | 0 | 0 | 0 | 131313. | 0 | 3.16628e6 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | −931.336 | 0 | 0 | 0 | −500587. | 0 | −726936. | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | 341.481 | 0 | 0 | 0 | 519291. | 0 | −1.47771e6 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | 1835.74 | 0 | 0 | 0 | −262440. | 0 | 1.77560e6 | 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.f | 4 | |
| 5.b | even | 2 | 1 | 40.14.a.d | ✓ | 4 | |
| 5.c | odd | 4 | 2 | 200.14.c.f | 8 | ||
| 20.d | odd | 2 | 1 | 80.14.a.k | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 40.14.a.d | ✓ | 4 | 5.b | even | 2 | 1 | |
| 80.14.a.k | 4 | 20.d | odd | 2 | 1 | ||
| 200.14.a.f | 4 | 1.a | even | 1 | 1 | trivial | |
| 200.14.c.f | 8 | 5.c | odd | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} + 936T_{3}^{3} - 4119216T_{3}^{2} - 2472664320T_{3} + 1273839436800 \)
acting on \(S_{14}^{\mathrm{new}}(\Gamma_0(200))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} + \cdots + 1273839436800 \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( T^{4} + \cdots + 89\!\cdots\!64 \)
|
| $11$ |
\( T^{4} + \cdots + 18\!\cdots\!96 \)
|
| $13$ |
\( T^{4} + \cdots + 16\!\cdots\!64 \)
|
| $17$ |
\( T^{4} + \cdots + 13\!\cdots\!00 \)
|
| $19$ |
\( T^{4} + \cdots + 62\!\cdots\!96 \)
|
| $23$ |
\( T^{4} + \cdots - 25\!\cdots\!96 \)
|
| $29$ |
\( T^{4} + \cdots + 19\!\cdots\!96 \)
|
| $31$ |
\( T^{4} + \cdots + 18\!\cdots\!00 \)
|
| $37$ |
\( T^{4} + \cdots + 60\!\cdots\!76 \)
|
| $41$ |
\( T^{4} + \cdots - 15\!\cdots\!56 \)
|
| $43$ |
\( T^{4} + \cdots + 20\!\cdots\!24 \)
|
| $47$ |
\( T^{4} + \cdots - 53\!\cdots\!24 \)
|
| $53$ |
\( T^{4} + \cdots - 33\!\cdots\!84 \)
|
| $59$ |
\( T^{4} + \cdots + 11\!\cdots\!24 \)
|
| $61$ |
\( T^{4} + \cdots - 57\!\cdots\!00 \)
|
| $67$ |
\( T^{4} + \cdots - 65\!\cdots\!96 \)
|
| $71$ |
\( T^{4} + \cdots + 67\!\cdots\!00 \)
|
| $73$ |
\( T^{4} + \cdots - 94\!\cdots\!16 \)
|
| $79$ |
\( T^{4} + \cdots - 48\!\cdots\!00 \)
|
| $83$ |
\( T^{4} + \cdots - 80\!\cdots\!36 \)
|
| $89$ |
\( T^{4} + \cdots - 17\!\cdots\!84 \)
|
| $97$ |
\( T^{4} + \cdots - 34\!\cdots\!24 \)
|
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