Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [192,9,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, 9, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("192.127");
S:= CuspForms(chi, 9);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 192 = 2^{6} \cdot 3 \) |
| Weight: | \( k \) | \(=\) | \( 9 \) |
| 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: | \(78.2166931317\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 342x^{6} + 43937x^{4} - 2512824x^{2} + 53993104 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{40}\cdot 3^{8} \) |
| Twist minimal: | no (minimal twist has level 96) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - 342x^{6} + 43937x^{4} - 2512824x^{2} + 53993104 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -27\nu^{7} + 9234\nu^{5} - 987903\nu^{3} + 33920532\nu ) / 58784 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 348\nu^{7} - 89624\nu^{5} + 7765724\nu^{3} - 226222192\nu ) / 123079 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 871 \nu^{7} + 235136 \nu^{6} + 297882 \nu^{5} - 123211264 \nu^{4} - 31869019 \nu^{3} + \cdots - 607942246912 ) / 3938528 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 27 \nu^{7} - 352704 \nu^{6} - 23930 \nu^{5} + 90292224 \nu^{4} + 3471527 \nu^{3} + \cdots + 225427939968 ) / 984632 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 1044 \nu^{7} + 44088 \nu^{6} + 268872 \nu^{5} - 11286528 \nu^{4} - 23297172 \nu^{3} + \cdots - 27168260064 ) / 123079 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 14562 \nu^{7} - 20207 \nu^{6} + 3716348 \nu^{5} + 5172992 \nu^{4} - 315278826 \nu^{3} + \cdots + 12452119196 ) / 492316 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 14562 \nu^{7} + 773377 \nu^{6} + 3716348 \nu^{5} - 197984512 \nu^{4} - 315278826 \nu^{3} + \cdots - 476576561956 ) / 492316 \)
|
| \(\nu\) | \(=\) |
\( ( -2\beta_{7} + 2\beta_{6} + 9\beta_{5} + 27\beta_{2} ) / 3456 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -32\beta_{7} + 32\beta_{6} - 144\beta_{4} - 9\beta_{2} - 64\beta _1 + 1181952 ) / 13824 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -53\beta_{7} + 197\beta_{6} + 255\beta_{5} + 2313\beta_{2} + 256\beta_1 ) / 1152 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -5440\beta_{7} + 5440\beta_{6} - 24624\beta_{4} - 864\beta_{3} - 1539\beta_{2} - 10528\beta _1 + 100535040 ) / 13824 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -12515\beta_{7} + 85523\beta_{6} + 64683\beta_{5} + 993357\beta_{2} + 218880\beta_1 ) / 3456 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( - 228960 \beta_{7} + 228960 \beta_{6} - 1055472 \beta_{4} - 73728 \beta_{3} - 65967 \beta_{2} + \cdots + 2834839296 ) / 4608 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( -975111\beta_{7} + 10137399\beta_{6} + 5437845\beta_{5} + 119757555\beta_{2} + 39232256\beta_1 ) / 3456 \)
|
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 | − | 46.7654i | 0 | −455.790 | 0 | − | 1689.76i | 0 | −2187.00 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 127.2 | 0 | − | 46.7654i | 0 | −261.868 | 0 | 1340.75i | 0 | −2187.00 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 127.3 | 0 | − | 46.7654i | 0 | 611.370 | 0 | 4427.25i | 0 | −2187.00 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 127.4 | 0 | − | 46.7654i | 0 | 722.288 | 0 | − | 891.264i | 0 | −2187.00 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 127.5 | 0 | 46.7654i | 0 | −455.790 | 0 | 1689.76i | 0 | −2187.00 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 127.6 | 0 | 46.7654i | 0 | −261.868 | 0 | − | 1340.75i | 0 | −2187.00 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 127.7 | 0 | 46.7654i | 0 | 611.370 | 0 | − | 4427.25i | 0 | −2187.00 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 127.8 | 0 | 46.7654i | 0 | 722.288 | 0 | 891.264i | 0 | −2187.00 | 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 | 192.9.g.f | 8 | |
| 4.b | odd | 2 | 1 | inner | 192.9.g.f | 8 | |
| 8.b | even | 2 | 1 | 96.9.g.a | ✓ | 8 | |
| 8.d | odd | 2 | 1 | 96.9.g.a | ✓ | 8 | |
| 24.f | even | 2 | 1 | 288.9.g.f | 8 | ||
| 24.h | odd | 2 | 1 | 288.9.g.f | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 96.9.g.a | ✓ | 8 | 8.b | even | 2 | 1 | |
| 96.9.g.a | ✓ | 8 | 8.d | odd | 2 | 1 | |
| 192.9.g.f | 8 | 1.a | even | 1 | 1 | trivial | |
| 192.9.g.f | 8 | 4.b | odd | 2 | 1 | inner | |
| 288.9.g.f | 8 | 24.f | even | 2 | 1 | ||
| 288.9.g.f | 8 | 24.h | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} - 616T_{5}^{3} - 396168T_{5}^{2} + 157725920T_{5} + 52706093200 \)
acting on \(S_{9}^{\mathrm{new}}(192, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
|
| $3$ |
\( (T^{2} + 2187)^{4} \)
|
| $5$ |
\( (T^{4} - 616 T^{3} + \cdots + 52706093200)^{2} \)
|
| $7$ |
\( T^{8} + \cdots + 79\!\cdots\!84 \)
|
| $11$ |
\( T^{8} + \cdots + 65\!\cdots\!84 \)
|
| $13$ |
\( (T^{4} + \cdots - 81\!\cdots\!44)^{2} \)
|
| $17$ |
\( (T^{4} + \cdots + 28\!\cdots\!68)^{2} \)
|
| $19$ |
\( T^{8} + \cdots + 10\!\cdots\!84 \)
|
| $23$ |
\( T^{8} + \cdots + 21\!\cdots\!76 \)
|
| $29$ |
\( (T^{4} + \cdots - 40\!\cdots\!92)^{2} \)
|
| $31$ |
\( T^{8} + \cdots + 13\!\cdots\!76 \)
|
| $37$ |
\( (T^{4} + \cdots + 30\!\cdots\!84)^{2} \)
|
| $41$ |
\( (T^{4} + \cdots - 37\!\cdots\!12)^{2} \)
|
| $43$ |
\( T^{8} + \cdots + 70\!\cdots\!00 \)
|
| $47$ |
\( T^{8} + \cdots + 26\!\cdots\!16 \)
|
| $53$ |
\( (T^{4} + \cdots + 34\!\cdots\!72)^{2} \)
|
| $59$ |
\( T^{8} + \cdots + 31\!\cdots\!64 \)
|
| $61$ |
\( (T^{4} + \cdots + 19\!\cdots\!00)^{2} \)
|
| $67$ |
\( T^{8} + \cdots + 41\!\cdots\!96 \)
|
| $71$ |
\( T^{8} + \cdots + 13\!\cdots\!16 \)
|
| $73$ |
\( (T^{4} + \cdots + 60\!\cdots\!52)^{2} \)
|
| $79$ |
\( T^{8} + \cdots + 16\!\cdots\!24 \)
|
| $83$ |
\( T^{8} + \cdots + 54\!\cdots\!16 \)
|
| $89$ |
\( (T^{4} + \cdots - 54\!\cdots\!04)^{2} \)
|
| $97$ |
\( (T^{4} + \cdots - 66\!\cdots\!12)^{2} \)
|
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