Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [192,9,Mod(65,192)]
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([0, 0, 1]))
N = Newforms(chi, 9, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("192.65");
S:= CuspForms(chi, 9);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 192 = 2^{6} \cdot 3 \) |
| Weight: | \( k \) | \(=\) | \( 9 \) |
| Character orbit: | \([\chi]\) | \(=\) | 192.e (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: |
\( x^{8} - 4x^{7} - 78x^{6} + 144x^{5} + 2079x^{4} + 936x^{3} - 658x^{2} + 2884x + 30633 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{42}\cdot 3^{10} \) |
| Twist minimal: | no (minimal twist has level 24) |
| 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
\( x^{8} - 4x^{7} - 78x^{6} + 144x^{5} + 2079x^{4} + 936x^{3} - 658x^{2} + 2884x + 30633 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( - 14134 \nu^{7} - 759906 \nu^{6} + 8175192 \nu^{5} + 32119854 \nu^{4} - 356737260 \nu^{3} + \cdots - 10631678772 ) / 6908733 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 90977 \nu^{7} + 306225 \nu^{6} + 8157405 \nu^{5} - 15590670 \nu^{4} - 246555357 \nu^{3} + \cdots + 77404449 ) / 41452398 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 1348973 \nu^{7} - 7217253 \nu^{6} - 97646277 \nu^{5} + 350373210 \nu^{4} + 2225360013 \nu^{3} + \cdots + 7288059399 ) / 41452398 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 2071745 \nu^{7} + 76404537 \nu^{6} - 163202991 \nu^{5} - 5405297226 \nu^{4} + \cdots - 270823955163 ) / 41452398 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 1067294 \nu^{7} - 14014314 \nu^{6} - 49856268 \nu^{5} + 883003110 \nu^{4} + 1518931968 \nu^{3} + \cdots - 41475425052 ) / 20726199 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 580555 \nu^{7} - 2572731 \nu^{6} - 41619615 \nu^{5} + 91789914 \nu^{4} + 1034741919 \nu^{3} + \cdots + 2039309301 ) / 6908733 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 17611 \nu^{7} + 68427 \nu^{6} + 1422267 \nu^{5} - 2154414 \nu^{4} - 41574963 \nu^{3} + \cdots - 10880913 ) / 170586 \)
|
| \(\nu\) | \(=\) |
\( ( 16\beta_{7} + 23\beta_{6} + 16\beta_{5} + 4\beta_{4} + 4\beta_{3} + 494\beta_{2} - 24\beta _1 + 13824 ) / 27648 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 14\beta_{7} + 5\beta_{6} + 10\beta_{5} + 4\beta_{4} - 2\beta_{3} - 370\beta_{2} + 18\beta _1 + 74304 ) / 3456 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 296 \beta_{7} + 583 \beta_{6} + 296 \beta_{5} + 56 \beta_{4} - 1024 \beta_{3} - 914 \beta_{2} + \cdots + 654336 ) / 9216 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 3056 \beta_{7} + 2423 \beta_{6} + 1192 \beta_{5} + 358 \beta_{4} - 2186 \beta_{3} - 64282 \beta_{2} + \cdots + 5871744 ) / 6912 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 21170 \beta_{7} + 36652 \beta_{6} + 9650 \beta_{5} + 1715 \beta_{4} - 60295 \beta_{3} + \cdots + 29137536 ) / 6912 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 150456 \beta_{7} + 168071 \beta_{6} + 32536 \beta_{5} + 9064 \beta_{4} - 202784 \beta_{3} + \cdots + 130065408 ) / 4608 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 6424576 \beta_{7} + 10532849 \beta_{6} + 1181248 \beta_{5} + 172084 \beta_{4} - 15833948 \beta_{3} + \cdots + 4132200960 ) / 27648 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 65.1 |
|
0 | −76.4245 | − | 26.8384i | 0 | 295.589i | 0 | 843.136 | 0 | 5120.40 | + | 4102.23i | 0 | ||||||||||||||||||||||||||||||||||||||
| 65.2 | 0 | −76.4245 | + | 26.8384i | 0 | − | 295.589i | 0 | 843.136 | 0 | 5120.40 | − | 4102.23i | 0 | ||||||||||||||||||||||||||||||||||||||
| 65.3 | 0 | −4.95410 | − | 80.8484i | 0 | 404.296i | 0 | 262.245 | 0 | −6511.91 | + | 801.062i | 0 | |||||||||||||||||||||||||||||||||||||||
| 65.4 | 0 | −4.95410 | + | 80.8484i | 0 | − | 404.296i | 0 | 262.245 | 0 | −6511.91 | − | 801.062i | 0 | ||||||||||||||||||||||||||||||||||||||
| 65.5 | 0 | 28.6420 | − | 75.7670i | 0 | − | 868.404i | 0 | −3909.88 | 0 | −4920.27 | − | 4340.23i | 0 | ||||||||||||||||||||||||||||||||||||||
| 65.6 | 0 | 28.6420 | + | 75.7670i | 0 | 868.404i | 0 | −3909.88 | 0 | −4920.27 | + | 4340.23i | 0 | |||||||||||||||||||||||||||||||||||||||
| 65.7 | 0 | 80.7366 | − | 6.52720i | 0 | 920.542i | 0 | 2012.50 | 0 | 6475.79 | − | 1053.97i | 0 | |||||||||||||||||||||||||||||||||||||||
| 65.8 | 0 | 80.7366 | + | 6.52720i | 0 | − | 920.542i | 0 | 2012.50 | 0 | 6475.79 | + | 1053.97i | 0 | ||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.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.e.j | 8 | |
| 3.b | odd | 2 | 1 | inner | 192.9.e.j | 8 | |
| 4.b | odd | 2 | 1 | 192.9.e.i | 8 | ||
| 8.b | even | 2 | 1 | 48.9.e.e | 8 | ||
| 8.d | odd | 2 | 1 | 24.9.e.a | ✓ | 8 | |
| 12.b | even | 2 | 1 | 192.9.e.i | 8 | ||
| 24.f | even | 2 | 1 | 24.9.e.a | ✓ | 8 | |
| 24.h | odd | 2 | 1 | 48.9.e.e | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 24.9.e.a | ✓ | 8 | 8.d | odd | 2 | 1 | |
| 24.9.e.a | ✓ | 8 | 24.f | even | 2 | 1 | |
| 48.9.e.e | 8 | 8.b | even | 2 | 1 | ||
| 48.9.e.e | 8 | 24.h | odd | 2 | 1 | ||
| 192.9.e.i | 8 | 4.b | odd | 2 | 1 | ||
| 192.9.e.i | 8 | 12.b | even | 2 | 1 | ||
| 192.9.e.j | 8 | 1.a | even | 1 | 1 | trivial | |
| 192.9.e.j | 8 | 3.b | odd | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{9}^{\mathrm{new}}(192, [\chi])\):
|
\( T_{5}^{8} + 1852352T_{5}^{6} + 1055033650176T_{5}^{4} + 183162750181376000T_{5}^{2} + 9126569679992258560000 \)
|
|
\( T_{7}^{4} + 792T_{7}^{3} - 9744840T_{7}^{2} + 9117353952T_{7} - 1739819024496 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
|
| $3$ |
\( T^{8} + \cdots + 18\!\cdots\!41 \)
|
| $5$ |
\( T^{8} + \cdots + 91\!\cdots\!00 \)
|
| $7$ |
\( (T^{4} + \cdots - 1739819024496)^{2} \)
|
| $11$ |
\( T^{8} + \cdots + 54\!\cdots\!24 \)
|
| $13$ |
\( (T^{4} + \cdots + 50\!\cdots\!00)^{2} \)
|
| $17$ |
\( T^{8} + \cdots + 68\!\cdots\!36 \)
|
| $19$ |
\( (T^{4} + \cdots + 14\!\cdots\!56)^{2} \)
|
| $23$ |
\( T^{8} + \cdots + 10\!\cdots\!56 \)
|
| $29$ |
\( T^{8} + \cdots + 20\!\cdots\!76 \)
|
| $31$ |
\( (T^{4} + \cdots - 24\!\cdots\!88)^{2} \)
|
| $37$ |
\( (T^{4} + \cdots + 13\!\cdots\!76)^{2} \)
|
| $41$ |
\( T^{8} + \cdots + 71\!\cdots\!76 \)
|
| $43$ |
\( (T^{4} + \cdots + 40\!\cdots\!44)^{2} \)
|
| $47$ |
\( T^{8} + \cdots + 14\!\cdots\!56 \)
|
| $53$ |
\( T^{8} + \cdots + 12\!\cdots\!44 \)
|
| $59$ |
\( T^{8} + \cdots + 36\!\cdots\!44 \)
|
| $61$ |
\( (T^{4} + \cdots - 77\!\cdots\!96)^{2} \)
|
| $67$ |
\( (T^{4} + \cdots + 23\!\cdots\!36)^{2} \)
|
| $71$ |
\( T^{8} + \cdots + 11\!\cdots\!96 \)
|
| $73$ |
\( (T^{4} + \cdots + 58\!\cdots\!04)^{2} \)
|
| $79$ |
\( (T^{4} + \cdots - 61\!\cdots\!00)^{2} \)
|
| $83$ |
\( T^{8} + \cdots + 98\!\cdots\!96 \)
|
| $89$ |
\( T^{8} + \cdots + 25\!\cdots\!36 \)
|
| $97$ |
\( (T^{4} + \cdots - 90\!\cdots\!04)^{2} \)
|
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