Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [650,6,Mod(599,650)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(650, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("650.599");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 650 = 2 \cdot 5^{2} \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 650.b (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: | \(104.249482878\) |
| 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: |
\( x^{6} - 2x^{5} + 2x^{4} - 390x^{3} + 32400x^{2} - 135000x + 281250 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{3} \) |
| Twist minimal: | no (minimal twist has level 130) |
| 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
\( x^{6} - 2x^{5} + 2x^{4} - 390x^{3} + 32400x^{2} - 135000x + 281250 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -169\nu^{5} - 12\nu^{4} - 13\nu^{3} + 2210\nu^{2} - 5407350\nu + 11548125 ) / 11529375 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 181\nu^{5} - 337\nu^{4} + 63687\nu^{3} - 66040\nu^{2} + 17388900\nu - 35983125 ) / 11529375 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -24\nu^{5} - 1667\nu^{4} - 4368\nu^{3} + 4680\nu^{2} + 18000\nu - 33136375 ) / 768625 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -14\nu^{5} + 13\nu^{4} - 2548\nu^{3} + 2730\nu^{2} + 10500\nu + 1440075 ) / 461175 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 8071\nu^{5} + 783\nu^{4} - 37583\nu^{3} - 4679090\nu^{2} + 251282400\nu - 537339375 ) / 11529375 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{4} + \beta_{2} - \beta _1 + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -5\beta_{5} - 3\beta_{2} - 242\beta_1 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -5\beta_{5} - 177\beta_{4} - 5\beta_{3} + 177\beta_{2} + 328\beta _1 + 328 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( 468\beta_{4} - 455\beta_{3} - 21077 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -65\beta_{5} - 32049\beta_{4} + 65\beta_{3} - 32049\beta_{2} - 107636\beta _1 + 107636 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/650\mathbb{Z}\right)^\times\).
| \(n\) | \(27\) | \(301\) |
| \(\chi(n)\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 599.1 |
|
− | 4.00000i | − | 9.60750i | −16.0000 | 0 | −38.4300 | 16.6832i | 64.0000i | 150.696 | 0 | ||||||||||||||||||||||||||||||||||
| 599.2 | − | 4.00000i | 3.71700i | −16.0000 | 0 | 14.8680 | − | 10.2576i | 64.0000i | 229.184 | 0 | |||||||||||||||||||||||||||||||||||
| 599.3 | − | 4.00000i | 27.8905i | −16.0000 | 0 | 111.562 | − | 240.426i | 64.0000i | −534.880 | 0 | |||||||||||||||||||||||||||||||||||
| 599.4 | 4.00000i | − | 27.8905i | −16.0000 | 0 | 111.562 | 240.426i | − | 64.0000i | −534.880 | 0 | |||||||||||||||||||||||||||||||||||
| 599.5 | 4.00000i | − | 3.71700i | −16.0000 | 0 | 14.8680 | 10.2576i | − | 64.0000i | 229.184 | 0 | |||||||||||||||||||||||||||||||||||
| 599.6 | 4.00000i | 9.60750i | −16.0000 | 0 | −38.4300 | − | 16.6832i | − | 64.0000i | 150.696 | 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 | 650.6.b.j | 6 | |
| 5.b | even | 2 | 1 | inner | 650.6.b.j | 6 | |
| 5.c | odd | 4 | 1 | 130.6.a.f | ✓ | 3 | |
| 5.c | odd | 4 | 1 | 650.6.a.j | 3 | ||
| 20.e | even | 4 | 1 | 1040.6.a.l | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 130.6.a.f | ✓ | 3 | 5.c | odd | 4 | 1 | |
| 650.6.a.j | 3 | 5.c | odd | 4 | 1 | ||
| 650.6.b.j | 6 | 1.a | even | 1 | 1 | trivial | |
| 650.6.b.j | 6 | 5.b | even | 2 | 1 | inner | |
| 1040.6.a.l | 3 | 20.e | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{6} + 884T_{3}^{4} + 83824T_{3}^{2} + 992016 \)
acting on \(S_{6}^{\mathrm{new}}(650, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 16)^{3} \)
|
| $3$ |
\( T^{6} + 884 T^{4} + \cdots + 992016 \)
|
| $5$ |
\( T^{6} \)
|
| $7$ |
\( T^{6} + \cdots + 1692828736 \)
|
| $11$ |
\( (T^{3} - 48 T^{2} + \cdots + 74436948)^{2} \)
|
| $13$ |
\( (T^{2} + 28561)^{3} \)
|
| $17$ |
\( T^{6} + \cdots + 16\!\cdots\!56 \)
|
| $19$ |
\( (T^{3} - 360 T^{2} + \cdots + 3217051244)^{2} \)
|
| $23$ |
\( T^{6} + \cdots + 10\!\cdots\!00 \)
|
| $29$ |
\( (T^{3} - 3078 T^{2} + \cdots - 20058120792)^{2} \)
|
| $31$ |
\( (T^{3} + 5388 T^{2} + \cdots - 146606313068)^{2} \)
|
| $37$ |
\( T^{6} + \cdots + 12\!\cdots\!04 \)
|
| $41$ |
\( (T^{3} + \cdots - 6827939505000)^{2} \)
|
| $43$ |
\( T^{6} + \cdots + 41\!\cdots\!00 \)
|
| $47$ |
\( T^{6} + \cdots + 18\!\cdots\!04 \)
|
| $53$ |
\( T^{6} + \cdots + 30\!\cdots\!00 \)
|
| $59$ |
\( (T^{3} + \cdots - 7052020839900)^{2} \)
|
| $61$ |
\( (T^{3} + \cdots + 14366569778248)^{2} \)
|
| $67$ |
\( T^{6} + \cdots + 24\!\cdots\!64 \)
|
| $71$ |
\( (T^{3} + \cdots - 17725732954380)^{2} \)
|
| $73$ |
\( T^{6} + \cdots + 30\!\cdots\!00 \)
|
| $79$ |
\( (T^{3} + \cdots + 22961229800288)^{2} \)
|
| $83$ |
\( T^{6} + \cdots + 71\!\cdots\!00 \)
|
| $89$ |
\( (T^{3} + \cdots - 187529271320664)^{2} \)
|
| $97$ |
\( T^{6} + \cdots + 36\!\cdots\!00 \)
|
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