Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [200,6,Mod(49,200)]
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, 1]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("200.49");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 200 = 2^{3} \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 200.c (of order \(2\), degree \(1\), not minimal) |
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: | no |
| Analytic conductor: | \(32.0767639626\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(i, \sqrt{241})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 121x^{2} + 3600 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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
\( x^{4} + 121x^{2} + 3600 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} + 61\nu ) / 60 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} + 181\nu ) / 60 \)
|
| \(\beta_{3}\) | \(=\) |
\( 2\nu^{2} + 121 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} - \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} - 121 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -61\beta_{2} + 181\beta_1 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/200\mathbb{Z}\right)^\times\).
| \(n\) | \(101\) | \(151\) | \(177\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 49.1 |
|
0 | − | 19.5242i | 0 | 0 | 0 | − | 35.0483i | 0 | −138.193 | 0 | ||||||||||||||||||||||||||||
| 49.2 | 0 | − | 11.5242i | 0 | 0 | 0 | − | 27.0483i | 0 | 110.193 | 0 | |||||||||||||||||||||||||||||
| 49.3 | 0 | 11.5242i | 0 | 0 | 0 | 27.0483i | 0 | 110.193 | 0 | |||||||||||||||||||||||||||||||
| 49.4 | 0 | 19.5242i | 0 | 0 | 0 | 35.0483i | 0 | −138.193 | 0 | |||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 200.6.c.f | 4 | |
| 4.b | odd | 2 | 1 | 400.6.c.o | 4 | ||
| 5.b | even | 2 | 1 | inner | 200.6.c.f | 4 | |
| 5.c | odd | 4 | 1 | 200.6.a.e | ✓ | 2 | |
| 5.c | odd | 4 | 1 | 200.6.a.f | yes | 2 | |
| 20.d | odd | 2 | 1 | 400.6.c.o | 4 | ||
| 20.e | even | 4 | 1 | 400.6.a.r | 2 | ||
| 20.e | even | 4 | 1 | 400.6.a.u | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 200.6.a.e | ✓ | 2 | 5.c | odd | 4 | 1 | |
| 200.6.a.f | yes | 2 | 5.c | odd | 4 | 1 | |
| 200.6.c.f | 4 | 1.a | even | 1 | 1 | trivial | |
| 200.6.c.f | 4 | 5.b | even | 2 | 1 | inner | |
| 400.6.a.r | 2 | 20.e | even | 4 | 1 | ||
| 400.6.a.u | 2 | 20.e | even | 4 | 1 | ||
| 400.6.c.o | 4 | 4.b | odd | 2 | 1 | ||
| 400.6.c.o | 4 | 20.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} + 514T_{3}^{2} + 50625 \)
acting on \(S_{6}^{\mathrm{new}}(200, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} + 514 T^{2} + 50625 \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( T^{4} + 1960 T^{2} + 898704 \)
|
| $11$ |
\( (T^{2} + 200 T - 96281)^{2} \)
|
| $13$ |
\( T^{4} + \cdots + 318150146304 \)
|
| $17$ |
\( T^{4} + \cdots + 1795640625 \)
|
| $19$ |
\( (T^{2} - 840 T - 662521)^{2} \)
|
| $23$ |
\( T^{4} + \cdots + 1855011312144 \)
|
| $29$ |
\( (T^{2} - 4680 T + 196736)^{2} \)
|
| $31$ |
\( (T^{2} + 5008 T - 23931140)^{2} \)
|
| $37$ |
\( T^{4} + \cdots + 302523267094416 \)
|
| $41$ |
\( (T^{2} + 5334 T - 204026247)^{2} \)
|
| $43$ |
\( T^{4} + \cdots + 77\!\cdots\!84 \)
|
| $47$ |
\( (T^{2} + 169937296)^{2} \)
|
| $53$ |
\( T^{4} + \cdots + 26\!\cdots\!96 \)
|
| $59$ |
\( (T^{2} - 81776 T + 1630122048)^{2} \)
|
| $61$ |
\( (T^{2} + 46932 T + 550186580)^{2} \)
|
| $67$ |
\( T^{4} + \cdots + 13\!\cdots\!29 \)
|
| $71$ |
\( (T^{2} - 7448 T - 14650800)^{2} \)
|
| $73$ |
\( T^{4} + \cdots + 87\!\cdots\!29 \)
|
| $79$ |
\( (T^{2} - 108104 T + 662040300)^{2} \)
|
| $83$ |
\( T^{4} + \cdots + 11\!\cdots\!29 \)
|
| $89$ |
\( (T^{2} + 70990 T - 4823083791)^{2} \)
|
| $97$ |
\( T^{4} + \cdots + 12\!\cdots\!76 \)
|
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