Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [384,3,Mod(257,384)]
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([0, 0, 1]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("384.257");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 384 = 2^{7} \cdot 3 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 384.e (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: | \(10.4632421514\) |
| 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: |
\( x^{8} + 18x^{6} + 99x^{4} + 170x^{2} + 81 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{10}\cdot 3 \) |
| 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
\( x^{8} + 18x^{6} + 99x^{4} + 170x^{2} + 81 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{6} + 16\nu^{4} + 70\nu^{2} - 6\nu + 63 ) / 6 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 2\nu^{7} + 33\nu^{5} + 159\nu^{3} + 193\nu ) / 9 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{4} + 11\nu^{2} + 18 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 2\nu^{7} + 36\nu^{5} + 180\nu^{3} + 178\nu ) / 9 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -\nu^{7} - 16\nu^{5} - 3\nu^{4} - 70\nu^{3} - 27\nu^{2} - 63\nu - 27 ) / 3 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( \nu^{7} + 16\nu^{5} - 3\nu^{4} + 70\nu^{3} - 27\nu^{2} + 57\nu - 27 ) / 3 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( \nu^{6} + 16\nu^{4} + 70\nu^{2} + 6\nu + 61 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{7} - 3\beta _1 + 1 ) / 6 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{7} + 3\beta_{6} + 3\beta_{5} + 6\beta_{3} - 3\beta _1 - 53 ) / 12 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -13\beta_{7} - 3\beta_{6} + 3\beta_{5} - 3\beta_{4} + 12\beta_{2} + 39\beta _1 - 13 ) / 12 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -11\beta_{7} - 33\beta_{6} - 33\beta_{5} - 54\beta_{3} + 33\beta _1 + 367 ) / 12 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 101\beta_{7} + 21\beta_{6} - 21\beta_{5} + 57\beta_{4} - 120\beta_{2} - 303\beta _1 + 101 ) / 12 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 59\beta_{7} + 159\beta_{6} + 159\beta_{5} + 222\beta_{3} - 141\beta _1 - 1453 ) / 6 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( -413\beta_{7} - 54\beta_{6} + 54\beta_{5} - 351\beta_{4} + 540\beta_{2} + 1239\beta _1 - 413 ) / 6 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 257.1 |
|
0 | −2.69031 | − | 1.32750i | 0 | 0.640013i | 0 | −2.72077 | 0 | 5.47550 | + | 7.14275i | 0 | ||||||||||||||||||||||||||||||||||||||
| 257.2 | 0 | −2.69031 | + | 1.32750i | 0 | − | 0.640013i | 0 | −2.72077 | 0 | 5.47550 | − | 7.14275i | 0 | ||||||||||||||||||||||||||||||||||||||
| 257.3 | 0 | 0.246559 | − | 2.98985i | 0 | − | 6.63641i | 0 | −0.578158 | 0 | −8.87842 | − | 1.47435i | 0 | ||||||||||||||||||||||||||||||||||||||
| 257.4 | 0 | 0.246559 | + | 2.98985i | 0 | 6.63641i | 0 | −0.578158 | 0 | −8.87842 | + | 1.47435i | 0 | |||||||||||||||||||||||||||||||||||||||
| 257.5 | 0 | 1.57844 | − | 2.55118i | 0 | 1.31534i | 0 | 10.2329 | 0 | −4.01705 | − | 8.05378i | 0 | |||||||||||||||||||||||||||||||||||||||
| 257.6 | 0 | 1.57844 | + | 2.55118i | 0 | − | 1.31534i | 0 | 10.2329 | 0 | −4.01705 | + | 8.05378i | 0 | ||||||||||||||||||||||||||||||||||||||
| 257.7 | 0 | 2.86531 | − | 0.888828i | 0 | 8.59176i | 0 | −10.9340 | 0 | 7.41997 | − | 5.09353i | 0 | |||||||||||||||||||||||||||||||||||||||
| 257.8 | 0 | 2.86531 | + | 0.888828i | 0 | − | 8.59176i | 0 | −10.9340 | 0 | 7.41997 | + | 5.09353i | 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 | 384.3.e.c | yes | 8 |
| 3.b | odd | 2 | 1 | inner | 384.3.e.c | yes | 8 |
| 4.b | odd | 2 | 1 | 384.3.e.b | yes | 8 | |
| 8.b | even | 2 | 1 | 384.3.e.a | ✓ | 8 | |
| 8.d | odd | 2 | 1 | 384.3.e.d | yes | 8 | |
| 12.b | even | 2 | 1 | 384.3.e.b | yes | 8 | |
| 16.e | even | 4 | 2 | 768.3.h.h | 16 | ||
| 16.f | odd | 4 | 2 | 768.3.h.g | 16 | ||
| 24.f | even | 2 | 1 | 384.3.e.d | yes | 8 | |
| 24.h | odd | 2 | 1 | 384.3.e.a | ✓ | 8 | |
| 48.i | odd | 4 | 2 | 768.3.h.h | 16 | ||
| 48.k | even | 4 | 2 | 768.3.h.g | 16 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 384.3.e.a | ✓ | 8 | 8.b | even | 2 | 1 | |
| 384.3.e.a | ✓ | 8 | 24.h | odd | 2 | 1 | |
| 384.3.e.b | yes | 8 | 4.b | odd | 2 | 1 | |
| 384.3.e.b | yes | 8 | 12.b | even | 2 | 1 | |
| 384.3.e.c | yes | 8 | 1.a | even | 1 | 1 | trivial |
| 384.3.e.c | yes | 8 | 3.b | odd | 2 | 1 | inner |
| 384.3.e.d | yes | 8 | 8.d | odd | 2 | 1 | |
| 384.3.e.d | yes | 8 | 24.f | even | 2 | 1 | |
| 768.3.h.g | 16 | 16.f | odd | 4 | 2 | ||
| 768.3.h.g | 16 | 48.k | even | 4 | 2 | ||
| 768.3.h.h | 16 | 16.e | even | 4 | 2 | ||
| 768.3.h.h | 16 | 48.i | odd | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(384, [\chi])\):
|
\( T_{7}^{4} + 4T_{7}^{3} - 108T_{7}^{2} - 368T_{7} - 176 \)
|
|
\( T_{13}^{4} - 360T_{13}^{2} - 256T_{13} + 7824 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
|
| $3$ |
\( T^{8} - 4 T^{7} + \cdots + 6561 \)
|
| $5$ |
\( T^{8} + 120 T^{6} + \cdots + 2304 \)
|
| $7$ |
\( (T^{4} + 4 T^{3} + \cdots - 176)^{2} \)
|
| $11$ |
\( T^{8} + 488 T^{6} + \cdots + 20214016 \)
|
| $13$ |
\( (T^{4} - 360 T^{2} + \cdots + 7824)^{2} \)
|
| $17$ |
\( T^{8} + 1248 T^{6} + \cdots + 991494144 \)
|
| $19$ |
\( (T^{4} + 12 T^{3} + \cdots - 98352)^{2} \)
|
| $23$ |
\( T^{8} + \cdots + 48132849664 \)
|
| $29$ |
\( T^{8} + \cdots + 78767790336 \)
|
| $31$ |
\( (T^{4} - 28 T^{3} + \cdots + 50256)^{2} \)
|
| $37$ |
\( (T^{4} + 16 T^{3} + \cdots - 488432)^{2} \)
|
| $41$ |
\( T^{8} + \cdots + 3916979306496 \)
|
| $43$ |
\( (T^{4} - 68 T^{3} + \cdots - 917424)^{2} \)
|
| $47$ |
\( T^{8} + \cdots + 424144797696 \)
|
| $53$ |
\( T^{8} + \cdots + 870911869798656 \)
|
| $59$ |
\( T^{8} + \cdots + 12745356964096 \)
|
| $61$ |
\( (T^{4} - 80 T^{3} + \cdots - 4584688)^{2} \)
|
| $67$ |
\( (T^{4} + 140 T^{3} + \cdots - 6784816)^{2} \)
|
| $71$ |
\( T^{8} + \cdots + 1706597351424 \)
|
| $73$ |
\( (T^{4} + 40 T^{3} + \cdots - 899312)^{2} \)
|
| $79$ |
\( (T^{4} - 204 T^{3} + \cdots - 38762928)^{2} \)
|
| $83$ |
\( T^{8} + \cdots + 56638749360384 \)
|
| $89$ |
\( T^{8} + \cdots + 92\!\cdots\!56 \)
|
| $97$ |
\( (T^{4} - 48 T^{3} + \cdots + 8416272)^{2} \)
|
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