Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [384,5,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, 5, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("384.319");
S:= CuspForms(chi, 5);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 384 = 2^{7} \cdot 3 \) |
| Weight: | \( k \) | \(=\) | \( 5 \) |
| 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: | \(39.6940658242\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | 8.0.49787136.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} + 3x^{6} + 5x^{4} + 12x^{2} + 16 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{28}\cdot 3^{4} \) |
| 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
\( x^{8} + 3x^{6} + 5x^{4} + 12x^{2} + 16 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 24\nu^{6} + 108 ) / 5 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 3\nu^{7} - 3\nu^{5} + 3\nu^{3} + 24\nu ) / 8 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -3\nu^{7} - 5\nu^{5} + 5\nu^{3} - 16\nu ) / 5 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 7\nu^{7} - 24\nu^{6} + 25\nu^{5} - 40\nu^{4} + 55\nu^{3} - 120\nu^{2} + 184\nu - 208 ) / 10 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 7\nu^{7} + 48\nu^{6} + 25\nu^{5} + 80\nu^{4} + 55\nu^{3} + 240\nu^{2} + 184\nu + 416 ) / 10 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -\nu^{7} - 128\nu^{6} + \nu^{5} - 384\nu^{4} - \nu^{3} - 128\nu^{2} - 8\nu - 768 ) / 8 \)
|
| \(\beta_{7}\) | \(=\) |
\( -36\nu^{7} - 60\nu^{5} - 132\nu^{3} - 192\nu \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{7} + 8\beta_{5} + 16\beta_{4} - 12\beta_{3} + 32\beta_{2} ) / 384 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 3\beta_{6} + 12\beta_{5} - 12\beta_{4} + \beta_{2} - 8\beta _1 - 288 ) / 384 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -\beta_{7} + 60\beta_{3} ) / 192 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -9\beta_{6} - 4\beta_{5} + 4\beta_{4} - 3\beta_{2} - 24\beta _1 - 96 ) / 384 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -\beta_{7} + 8\beta_{5} + 16\beta_{4} - 132\beta_{3} - 352\beta_{2} ) / 384 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 5\beta _1 - 108 ) / 24 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( -7\beta_{7} - 56\beta_{5} - 112\beta_{4} - 156\beta_{3} + 416\beta_{2} ) / 384 \)
|
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 | −5.19615 | 0 | − | 39.7490i | 0 | − | 46.0431i | 0 | 27.0000 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 319.2 | 0 | −5.19615 | 0 | − | 23.7490i | 0 | 9.38251i | 0 | 27.0000 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 319.3 | 0 | −5.19615 | 0 | 23.7490i | 0 | − | 9.38251i | 0 | 27.0000 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 319.4 | 0 | −5.19615 | 0 | 39.7490i | 0 | 46.0431i | 0 | 27.0000 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 319.5 | 0 | 5.19615 | 0 | − | 39.7490i | 0 | 46.0431i | 0 | 27.0000 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 319.6 | 0 | 5.19615 | 0 | − | 23.7490i | 0 | − | 9.38251i | 0 | 27.0000 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 319.7 | 0 | 5.19615 | 0 | 23.7490i | 0 | 9.38251i | 0 | 27.0000 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 319.8 | 0 | 5.19615 | 0 | 39.7490i | 0 | − | 46.0431i | 0 | 27.0000 | 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 | 384.5.b.c | ✓ | 8 |
| 3.b | odd | 2 | 1 | 1152.5.b.k | 8 | ||
| 4.b | odd | 2 | 1 | inner | 384.5.b.c | ✓ | 8 |
| 8.b | even | 2 | 1 | inner | 384.5.b.c | ✓ | 8 |
| 8.d | odd | 2 | 1 | inner | 384.5.b.c | ✓ | 8 |
| 12.b | even | 2 | 1 | 1152.5.b.k | 8 | ||
| 16.e | even | 4 | 1 | 768.5.g.c | 4 | ||
| 16.e | even | 4 | 1 | 768.5.g.g | 4 | ||
| 16.f | odd | 4 | 1 | 768.5.g.c | 4 | ||
| 16.f | odd | 4 | 1 | 768.5.g.g | 4 | ||
| 24.f | even | 2 | 1 | 1152.5.b.k | 8 | ||
| 24.h | odd | 2 | 1 | 1152.5.b.k | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 384.5.b.c | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 384.5.b.c | ✓ | 8 | 4.b | odd | 2 | 1 | inner |
| 384.5.b.c | ✓ | 8 | 8.b | even | 2 | 1 | inner |
| 384.5.b.c | ✓ | 8 | 8.d | odd | 2 | 1 | inner |
| 768.5.g.c | 4 | 16.e | even | 4 | 1 | ||
| 768.5.g.c | 4 | 16.f | odd | 4 | 1 | ||
| 768.5.g.g | 4 | 16.e | even | 4 | 1 | ||
| 768.5.g.g | 4 | 16.f | odd | 4 | 1 | ||
| 1152.5.b.k | 8 | 3.b | odd | 2 | 1 | ||
| 1152.5.b.k | 8 | 12.b | even | 2 | 1 | ||
| 1152.5.b.k | 8 | 24.f | even | 2 | 1 | ||
| 1152.5.b.k | 8 | 24.h | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} + 2144T_{5}^{2} + 891136 \)
acting on \(S_{5}^{\mathrm{new}}(384, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
|
| $3$ |
\( (T^{2} - 27)^{4} \)
|
| $5$ |
\( (T^{4} + 2144 T^{2} + 891136)^{2} \)
|
| $7$ |
\( (T^{4} + 2208 T^{2} + 186624)^{2} \)
|
| $11$ |
\( (T^{4} - 45408 T^{2} + 412252416)^{2} \)
|
| $13$ |
\( (T^{4} + 36864 T^{2} + 107495424)^{2} \)
|
| $17$ |
\( (T^{2} - 348 T - 34236)^{4} \)
|
| $19$ |
\( (T^{4} - 491616 T^{2} + 19955517696)^{2} \)
|
| $23$ |
\( (T^{4} + 825984 T^{2} + 36790308864)^{2} \)
|
| $29$ |
\( (T^{4} + 1032032 T^{2} + 247876528384)^{2} \)
|
| $31$ |
\( (T^{4} + 1389216 T^{2} + 189245880576)^{2} \)
|
| $37$ |
\( (T^{4} + 862848 T^{2} + 140607000576)^{2} \)
|
| $41$ |
\( (T^{2} + 252 T - 5209596)^{4} \)
|
| $43$ |
\( (T^{4} + \cdots + 1176686901504)^{2} \)
|
| $47$ |
\( (T^{4} + \cdots + 54724012314624)^{2} \)
|
| $53$ |
\( (T^{4} + 1263200 T^{2} + 394886560000)^{2} \)
|
| $59$ |
\( (T^{2} - 1714608)^{4} \)
|
| $61$ |
\( (T^{4} + \cdots + 14729384300544)^{2} \)
|
| $67$ |
\( (T^{4} + \cdots + 4145361152256)^{2} \)
|
| $71$ |
\( (T^{4} + \cdots + 424196534145024)^{2} \)
|
| $73$ |
\( (T^{2} + 2564 T + 611332)^{4} \)
|
| $79$ |
\( (T^{4} + \cdots + 37\!\cdots\!44)^{2} \)
|
| $83$ |
\( (T^{4} + \cdots + 470880972843264)^{2} \)
|
| $89$ |
\( (T^{2} + 7908 T + 9182916)^{4} \)
|
| $97$ |
\( (T^{2} + 11300 T + 25471300)^{4} \)
|
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