Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [384,9,Mod(319,384)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(384, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 1, 0]))
N = Newforms(chi, 9, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("384.319");
S:= CuspForms(chi, 9);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 384 = 2^{7} \cdot 3 \) |
| Weight: | \( k \) | \(=\) | \( 9 \) |
| Character orbit: | \([\chi]\) | \(=\) | 384.b (of order \(2\), degree \(1\), 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: | \(156.433386263\) |
| 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: |
\( x^{8} - 4x^{7} + 200x^{6} - 586x^{5} + 10949x^{4} - 20926x^{3} + 78946x^{2} - 68580x + 222900 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{32}\cdot 3^{16} \) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - 4x^{7} + 200x^{6} - 586x^{5} + 10949x^{4} - 20926x^{3} + 78946x^{2} - 68580x + 222900 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -432\nu^{6} + 1296\nu^{5} - 76896\nu^{4} + 151632\nu^{3} - 3639600\nu^{2} + 3564000\nu - 12572496 ) / 403 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 422 \nu^{7} - 466693 \nu^{6} + 1475675 \nu^{5} - 68583127 \nu^{4} + 138435443 \nu^{3} + \cdots + 89182264770 ) / 5257910 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 3798 \nu^{7} + 13293 \nu^{6} - 720243 \nu^{5} + 1767375 \nu^{4} - 35892171 \nu^{3} + \cdots + 38651310 ) / 5257910 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 1288 \nu^{7} + 4508 \nu^{6} - 269884 \nu^{5} + 663440 \nu^{4} - 16358356 \nu^{3} + \cdots + 84579636 ) / 976469 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 10064 \nu^{7} - 49762 \nu^{6} + 2067878 \nu^{5} - 8127346 \nu^{4} + 120250730 \nu^{3} + \cdots - 1195854900 ) / 3307395 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 10064 \nu^{7} - 81080 \nu^{6} - 1675352 \nu^{5} - 20265368 \nu^{4} - 64119512 \nu^{3} + \cdots - 4311284880 ) / 3307395 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 3072 \nu^{7} - 10752 \nu^{6} + 609024 \nu^{5} - 1495680 \nu^{4} + 31491840 \nu^{3} - 45747456 \nu^{2} + \cdots - 33862464 ) / 2423 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{6} + 8\beta_{5} + 12\beta_{4} + 16\beta_{3} + 432 ) / 864 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 36\beta_{6} + 144\beta_{5} + 144\beta_{4} + 195\beta_{3} + 27\beta_{2} - 4\beta _1 - 497664 ) / 10368 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( - 9 \beta_{7} + 652 \beta_{6} - 4784 \beta_{5} - 10992 \beta_{4} - 18621 \beta_{3} + 27 \beta_{2} + \cdots - 499392 ) / 6912 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( - 27 \beta_{7} - 9312 \beta_{6} - 25728 \beta_{5} - 33120 \beta_{4} - 56340 \beta_{3} + \cdots + 42814656 ) / 10368 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 4275 \beta_{7} - 212796 \beta_{6} + 1199088 \beta_{5} + 2870928 \beta_{4} + 8207181 \beta_{3} + \cdots + 216571968 ) / 20736 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 540 \beta_{7} + 106610 \beta_{6} + 286520 \beta_{5} + 365772 \beta_{4} + 1039705 \beta_{3} + \cdots - 302102784 ) / 864 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 535311 \beta_{7} + 23742708 \beta_{6} - 92469072 \beta_{5} - 236122704 \beta_{4} + \cdots - 26136385344 ) / 20736 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/384\mathbb{Z}\right)^\times\).
| \(n\) | \(127\) | \(133\) | \(257\) |
| \(\chi(n)\) | \(-1\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 319.1 |
|
0 | −46.7654 | 0 | − | 721.475i | 0 | − | 1935.65i | 0 | 2187.00 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 319.2 | 0 | −46.7654 | 0 | − | 402.176i | 0 | 1935.65i | 0 | 2187.00 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 319.3 | 0 | −46.7654 | 0 | 402.176i | 0 | − | 1935.65i | 0 | 2187.00 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 319.4 | 0 | −46.7654 | 0 | 721.475i | 0 | 1935.65i | 0 | 2187.00 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 319.5 | 0 | 46.7654 | 0 | − | 721.475i | 0 | 1935.65i | 0 | 2187.00 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 319.6 | 0 | 46.7654 | 0 | − | 402.176i | 0 | − | 1935.65i | 0 | 2187.00 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 319.7 | 0 | 46.7654 | 0 | 402.176i | 0 | 1935.65i | 0 | 2187.00 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 319.8 | 0 | 46.7654 | 0 | 721.475i | 0 | − | 1935.65i | 0 | 2187.00 | 0 | ||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 4.b | odd | 2 | 1 | inner |
| 8.b | even | 2 | 1 | inner |
| 8.d | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 384.9.b.a | ✓ | 8 |
| 4.b | odd | 2 | 1 | inner | 384.9.b.a | ✓ | 8 |
| 8.b | even | 2 | 1 | inner | 384.9.b.a | ✓ | 8 |
| 8.d | odd | 2 | 1 | inner | 384.9.b.a | ✓ | 8 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 384.9.b.a | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 384.9.b.a | ✓ | 8 | 4.b | odd | 2 | 1 | inner |
| 384.9.b.a | ✓ | 8 | 8.b | even | 2 | 1 | inner |
| 384.9.b.a | ✓ | 8 | 8.d | odd | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} + 682272T_{5}^{2} + 84192825600 \)
acting on \(S_{9}^{\mathrm{new}}(384, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
|
| $3$ |
\( (T^{2} - 2187)^{4} \)
|
| $5$ |
\( (T^{4} + 682272 T^{2} + 84192825600)^{2} \)
|
| $7$ |
\( (T^{2} + 3746736)^{4} \)
|
| $11$ |
\( (T^{4} + \cdots + 84\!\cdots\!16)^{2} \)
|
| $13$ |
\( (T^{4} + \cdots + 99\!\cdots\!44)^{2} \)
|
| $17$ |
\( (T^{2} - 23260 T - 7909979324)^{4} \)
|
| $19$ |
\( (T^{4} + \cdots + 45\!\cdots\!00)^{2} \)
|
| $23$ |
\( (T^{4} + \cdots + 93\!\cdots\!84)^{2} \)
|
| $29$ |
\( (T^{4} + \cdots + 27\!\cdots\!00)^{2} \)
|
| $31$ |
\( (T^{4} + \cdots + 15\!\cdots\!00)^{2} \)
|
| $37$ |
\( (T^{4} + \cdots + 57\!\cdots\!56)^{2} \)
|
| $41$ |
\( (T^{2} + \cdots - 17633303147132)^{4} \)
|
| $43$ |
\( (T^{4} + \cdots + 30\!\cdots\!04)^{2} \)
|
| $47$ |
\( (T^{4} + \cdots + 19\!\cdots\!84)^{2} \)
|
| $53$ |
\( (T^{4} + \cdots + 14\!\cdots\!44)^{2} \)
|
| $59$ |
\( (T^{4} + \cdots + 22\!\cdots\!24)^{2} \)
|
| $61$ |
\( (T^{4} + \cdots + 66\!\cdots\!56)^{2} \)
|
| $67$ |
\( (T^{4} + \cdots + 62\!\cdots\!36)^{2} \)
|
| $71$ |
\( (T^{4} + \cdots + 13\!\cdots\!24)^{2} \)
|
| $73$ |
\( (T^{2} + \cdots + 323128538436100)^{4} \)
|
| $79$ |
\( (T^{4} + \cdots + 12\!\cdots\!00)^{2} \)
|
| $83$ |
\( (T^{4} + \cdots + 27\!\cdots\!00)^{2} \)
|
| $89$ |
\( (T^{2} + \cdots + 21\!\cdots\!00)^{4} \)
|
| $97$ |
\( (T^{2} + \cdots - 10\!\cdots\!84)^{4} \)
|
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