Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [192,7,Mod(127,192)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(192, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0, 0]))
N = Newforms(chi, 7, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("192.127");
S:= CuspForms(chi, 7);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 192 = 2^{6} \cdot 3 \) |
| Weight: | \( k \) | \(=\) | \( 7 \) |
| Character orbit: | \([\chi]\) | \(=\) | 192.g (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: | \(44.1703840550\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.0.50898483.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} + 8x^{4} - 10x^{3} + 64x^{2} - 40x + 25 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{27}\cdot 3^{5} \) |
| Twist minimal: | no (minimal twist has level 12) |
| 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
\( x^{6} + 8x^{4} - 10x^{3} + 64x^{2} - 40x + 25 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 9\nu^{5} + 72\nu^{3} - 45\nu^{2} + 576\nu - 180 ) / 20 \)
|
| \(\beta_{2}\) | \(=\) |
\( -2\nu^{4} - 10\nu^{3} - 16\nu^{2} + 10\nu - 35 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 259\nu^{5} + 120\nu^{4} + 2112\nu^{3} - 2255\nu^{2} + 16616\nu - 5380 ) / 20 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 257\nu^{5} + 120\nu^{4} + 1776\nu^{3} - 2245\nu^{2} + 11368\nu - 3740 ) / 10 \)
|
| \(\beta_{5}\) | \(=\) |
\( 64\nu^{4} + 32\nu^{3} + 512\nu^{2} - 320\nu + 2571 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{5} - 9\beta_{4} + 18\beta_{3} + 32\beta_{2} - 4\beta _1 - 11 ) / 4608 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 5\beta_{5} - 3\beta_{4} - 42\beta_{3} + 16\beta_{2} + 1380\beta _1 - 12295 ) / 4608 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -\beta_{5} - 32\beta_{2} + 1451 ) / 288 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 45\beta_{5} - 21\beta_{4} + 426\beta_{3} + 288\beta_{2} - 11060\beta _1 - 98415 ) / 4608 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 89\beta_{5} + 561\beta_{4} - 1362\beta_{3} + 2128\beta_{2} + 17396\beta _1 - 154339 ) / 4608 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/192\mathbb{Z}\right)^\times\).
| \(n\) | \(65\) | \(127\) | \(133\) |
| \(\chi(n)\) | \(1\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 127.1 |
|
0 | − | 15.5885i | 0 | −172.232 | 0 | 545.623i | 0 | −243.000 | 0 | |||||||||||||||||||||||||||||||||||
| 127.2 | 0 | − | 15.5885i | 0 | −18.1171 | 0 | − | 321.465i | 0 | −243.000 | 0 | |||||||||||||||||||||||||||||||||||
| 127.3 | 0 | − | 15.5885i | 0 | 212.349 | 0 | 87.6116i | 0 | −243.000 | 0 | ||||||||||||||||||||||||||||||||||||
| 127.4 | 0 | 15.5885i | 0 | −172.232 | 0 | − | 545.623i | 0 | −243.000 | 0 | ||||||||||||||||||||||||||||||||||||
| 127.5 | 0 | 15.5885i | 0 | −18.1171 | 0 | 321.465i | 0 | −243.000 | 0 | |||||||||||||||||||||||||||||||||||||
| 127.6 | 0 | 15.5885i | 0 | 212.349 | 0 | − | 87.6116i | 0 | −243.000 | 0 | ||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 4.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 192.7.g.e | 6 | |
| 3.b | odd | 2 | 1 | 576.7.g.p | 6 | ||
| 4.b | odd | 2 | 1 | inner | 192.7.g.e | 6 | |
| 8.b | even | 2 | 1 | 12.7.d.a | ✓ | 6 | |
| 8.d | odd | 2 | 1 | 12.7.d.a | ✓ | 6 | |
| 12.b | even | 2 | 1 | 576.7.g.p | 6 | ||
| 16.e | even | 4 | 2 | 768.7.b.h | 12 | ||
| 16.f | odd | 4 | 2 | 768.7.b.h | 12 | ||
| 24.f | even | 2 | 1 | 36.7.d.e | 6 | ||
| 24.h | odd | 2 | 1 | 36.7.d.e | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 12.7.d.a | ✓ | 6 | 8.b | even | 2 | 1 | |
| 12.7.d.a | ✓ | 6 | 8.d | odd | 2 | 1 | |
| 36.7.d.e | 6 | 24.f | even | 2 | 1 | ||
| 36.7.d.e | 6 | 24.h | odd | 2 | 1 | ||
| 192.7.g.e | 6 | 1.a | even | 1 | 1 | trivial | |
| 192.7.g.e | 6 | 4.b | odd | 2 | 1 | inner | |
| 576.7.g.p | 6 | 3.b | odd | 2 | 1 | ||
| 576.7.g.p | 6 | 12.b | even | 2 | 1 | ||
| 768.7.b.h | 12 | 16.e | even | 4 | 2 | ||
| 768.7.b.h | 12 | 16.f | odd | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{3} - 22T_{5}^{2} - 37300T_{5} - 662600 \)
acting on \(S_{7}^{\mathrm{new}}(192, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( (T^{2} + 243)^{3} \)
|
| $5$ |
\( (T^{3} - 22 T^{2} + \cdots - 662600)^{2} \)
|
| $7$ |
\( T^{6} + \cdots + 236143965745152 \)
|
| $11$ |
\( T^{6} + \cdots + 36\!\cdots\!68 \)
|
| $13$ |
\( (T^{3} - 1674 T^{2} + \cdots + 94471112)^{2} \)
|
| $17$ |
\( (T^{3} - 6110 T^{2} + \cdots + 2355174680)^{2} \)
|
| $19$ |
\( T^{6} + \cdots + 19\!\cdots\!48 \)
|
| $23$ |
\( T^{6} + \cdots + 26\!\cdots\!92 \)
|
| $29$ |
\( (T^{3} - 42430 T^{2} + \cdots - 391390375016)^{2} \)
|
| $31$ |
\( T^{6} + \cdots + 15\!\cdots\!28 \)
|
| $37$ |
\( (T^{3} + \cdots + 154889066690024)^{2} \)
|
| $41$ |
\( (T^{3} + \cdots - 1472139052456)^{2} \)
|
| $43$ |
\( T^{6} + \cdots + 13\!\cdots\!68 \)
|
| $47$ |
\( T^{6} + \cdots + 54\!\cdots\!00 \)
|
| $53$ |
\( (T^{3} + \cdots + 201035875990744)^{2} \)
|
| $59$ |
\( T^{6} + \cdots + 17\!\cdots\!28 \)
|
| $61$ |
\( (T^{3} + \cdots + 59\!\cdots\!84)^{2} \)
|
| $67$ |
\( T^{6} + \cdots + 16\!\cdots\!12 \)
|
| $71$ |
\( T^{6} + \cdots + 40\!\cdots\!00 \)
|
| $73$ |
\( (T^{3} + \cdots - 54\!\cdots\!28)^{2} \)
|
| $79$ |
\( T^{6} + \cdots + 12\!\cdots\!72 \)
|
| $83$ |
\( T^{6} + \cdots + 55\!\cdots\!52 \)
|
| $89$ |
\( (T^{3} + \cdots + 49\!\cdots\!24)^{2} \)
|
| $97$ |
\( (T^{3} + \cdots - 65\!\cdots\!16)^{2} \)
|
| show more | |
| show less |