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: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} + \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} + 77x^{4} + 1396x^{2} + 576 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{8}\cdot 5^{2} \) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} + 77x^{4} + 1396x^{2} + 576 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{5} + 101\nu^{3} + 2332\nu ) / 1488 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 3\nu^{4} + 179\nu^{2} + 1478 ) / 62 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -49\nu^{5} - 3461\nu^{3} - 53260\nu ) / 1488 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 53\nu^{5} + 3865\nu^{3} + 82428\nu ) / 496 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -\nu^{4} - 33\nu^{2} + 198 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{4} + 3\beta_{3} - 12\beta_1 ) / 40 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 3\beta_{5} + 31\beta_{2} - 1036 ) / 40 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -41\beta_{4} - 83\beta_{3} + 2452\beta_1 ) / 40 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -179\beta_{5} - 1023\beta_{2} + 42108 ) / 40 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 1809\beta_{4} + 1387\beta_{3} - 160148\beta_1 ) / 40 \)
|
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 | − | 22.6278i | 0 | 0 | 0 | − | 107.252i | 0 | −269.020 | 0 | ||||||||||||||||||||||||||||||||||
| 49.2 | 0 | − | 19.0934i | 0 | 0 | 0 | 210.117i | 0 | −121.557 | 0 | ||||||||||||||||||||||||||||||||||||
| 49.3 | 0 | − | 2.53448i | 0 | 0 | 0 | − | 247.370i | 0 | 236.576 | 0 | |||||||||||||||||||||||||||||||||||
| 49.4 | 0 | 2.53448i | 0 | 0 | 0 | 247.370i | 0 | 236.576 | 0 | |||||||||||||||||||||||||||||||||||||
| 49.5 | 0 | 19.0934i | 0 | 0 | 0 | − | 210.117i | 0 | −121.557 | 0 | ||||||||||||||||||||||||||||||||||||
| 49.6 | 0 | 22.6278i | 0 | 0 | 0 | 107.252i | 0 | −269.020 | 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.g | 6 | |
| 4.b | odd | 2 | 1 | 400.6.c.p | 6 | ||
| 5.b | even | 2 | 1 | inner | 200.6.c.g | 6 | |
| 5.c | odd | 4 | 1 | 200.6.a.h | ✓ | 3 | |
| 5.c | odd | 4 | 1 | 200.6.a.i | yes | 3 | |
| 20.d | odd | 2 | 1 | 400.6.c.p | 6 | ||
| 20.e | even | 4 | 1 | 400.6.a.x | 3 | ||
| 20.e | even | 4 | 1 | 400.6.a.y | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 200.6.a.h | ✓ | 3 | 5.c | odd | 4 | 1 | |
| 200.6.a.i | yes | 3 | 5.c | odd | 4 | 1 | |
| 200.6.c.g | 6 | 1.a | even | 1 | 1 | trivial | |
| 200.6.c.g | 6 | 5.b | even | 2 | 1 | inner | |
| 400.6.a.x | 3 | 20.e | even | 4 | 1 | ||
| 400.6.a.y | 3 | 20.e | even | 4 | 1 | ||
| 400.6.c.p | 6 | 4.b | odd | 2 | 1 | ||
| 400.6.c.p | 6 | 20.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{6} + 883T_{3}^{4} + 192291T_{3}^{2} + 1199025 \)
acting on \(S_{6}^{\mathrm{new}}(200, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( T^{6} + 883 T^{4} + \cdots + 1199025 \)
|
| $5$ |
\( T^{6} \)
|
| $7$ |
\( T^{6} + \cdots + 31076343547456 \)
|
| $11$ |
\( (T^{3} + 19 T^{2} + \cdots - 1542819)^{2} \)
|
| $13$ |
\( T^{6} + \cdots + 26\!\cdots\!24 \)
|
| $17$ |
\( T^{6} + \cdots + 42\!\cdots\!25 \)
|
| $19$ |
\( (T^{3} + 1221 T^{2} + \cdots - 7033288181)^{2} \)
|
| $23$ |
\( T^{6} + \cdots + 25\!\cdots\!36 \)
|
| $29$ |
\( (T^{3} + 11912 T^{2} + \cdots + 55993024512)^{2} \)
|
| $31$ |
\( (T^{3} - 7442 T^{2} + \cdots + 14434811400)^{2} \)
|
| $37$ |
\( T^{6} + \cdots + 18\!\cdots\!56 \)
|
| $41$ |
\( (T^{3} - 3223 T^{2} + \cdots - 86972031621)^{2} \)
|
| $43$ |
\( T^{6} + \cdots + 43\!\cdots\!44 \)
|
| $47$ |
\( T^{6} + \cdots + 10\!\cdots\!96 \)
|
| $53$ |
\( T^{6} + \cdots + 49\!\cdots\!76 \)
|
| $59$ |
\( (T^{3} + \cdots + 8618337690624)^{2} \)
|
| $61$ |
\( (T^{3} + \cdots + 28184724893400)^{2} \)
|
| $67$ |
\( T^{6} + \cdots + 15\!\cdots\!69 \)
|
| $71$ |
\( (T^{3} + \cdots + 31597978616640)^{2} \)
|
| $73$ |
\( T^{6} + \cdots + 25\!\cdots\!01 \)
|
| $79$ |
\( (T^{3} + \cdots - 26059657326840)^{2} \)
|
| $83$ |
\( T^{6} + \cdots + 83\!\cdots\!21 \)
|
| $89$ |
\( (T^{3} + \cdots + 190810675088559)^{2} \)
|
| $97$ |
\( T^{6} + \cdots + 27\!\cdots\!36 \)
|
| show more | |
| show less |