Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [384,7,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, 7, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("384.319");
S:= CuspForms(chi, 7);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 384 = 2^{7} \cdot 3 \) |
| Weight: | \( k \) | \(=\) | \( 7 \) |
| 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: | \(88.3407681100\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | 8.0.1731891456.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 9x^{6} + 65x^{4} - 144x^{2} + 256 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{32}\cdot 3^{12} \) |
| 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
\( x^{8} - 9x^{6} + 65x^{4} - 144x^{2} + 256 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 9\nu^{7} - 225\nu^{5} + 1305\nu^{3} - 4896\nu ) / 320 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -81\nu^{6} + 585\nu^{4} - 5265\nu^{2} + 6984 ) / 130 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 9\nu^{7} - 65\nu^{5} + 377\nu^{3} - 256\nu ) / 208 \)
|
| \(\beta_{4}\) | \(=\) |
\( 6\nu^{6} - 54\nu^{4} + 294\nu^{2} - 432 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -3456\nu^{6} - 513216 ) / 65 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 51\nu^{7} - 507\nu^{5} + 3939\nu^{3} - 15456\nu ) / 13 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 243\nu^{7} - 1755\nu^{5} + 13635\nu^{3} - 6912\nu ) / 20 \)
|
| \(\nu\) | \(=\) |
\( ( 4\beta_{7} - 9\beta_{6} - 432\beta_{3} + 192\beta_1 ) / 6912 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{5} - 36\beta_{4} - 432\beta_{2} + 15552 ) / 6912 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 5\beta_{7} - 1404\beta_{3} ) / 864 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -3\beta_{5} - 108\beta_{4} - 784\beta_{2} - 28224 ) / 2304 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 116\beta_{7} + 261\beta_{6} - 43632\beta_{3} - 19392\beta_1 ) / 6912 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( -65\beta_{5} - 513216 ) / 3456 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( -724\beta_{7} + 1629\beta_{6} + 302832\beta_{3} - 134592\beta_1 ) / 6912 \)
|
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 | −15.5885 | 0 | − | 20.0000i | 0 | − | 529.850i | 0 | 243.000 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 319.2 | 0 | −15.5885 | 0 | − | 20.0000i | 0 | 155.727i | 0 | 243.000 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 319.3 | 0 | −15.5885 | 0 | 20.0000i | 0 | − | 155.727i | 0 | 243.000 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 319.4 | 0 | −15.5885 | 0 | 20.0000i | 0 | 529.850i | 0 | 243.000 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 319.5 | 0 | 15.5885 | 0 | − | 20.0000i | 0 | − | 155.727i | 0 | 243.000 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 319.6 | 0 | 15.5885 | 0 | − | 20.0000i | 0 | 529.850i | 0 | 243.000 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 319.7 | 0 | 15.5885 | 0 | 20.0000i | 0 | − | 529.850i | 0 | 243.000 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 319.8 | 0 | 15.5885 | 0 | 20.0000i | 0 | 155.727i | 0 | 243.000 | 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 | 384.7.b.d | ✓ | 8 |
| 4.b | odd | 2 | 1 | inner | 384.7.b.d | ✓ | 8 |
| 8.b | even | 2 | 1 | inner | 384.7.b.d | ✓ | 8 |
| 8.d | odd | 2 | 1 | inner | 384.7.b.d | ✓ | 8 |
| 16.e | even | 4 | 1 | 768.7.g.c | 4 | ||
| 16.e | even | 4 | 1 | 768.7.g.e | 4 | ||
| 16.f | odd | 4 | 1 | 768.7.g.c | 4 | ||
| 16.f | odd | 4 | 1 | 768.7.g.e | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 384.7.b.d | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 384.7.b.d | ✓ | 8 | 4.b | odd | 2 | 1 | inner |
| 384.7.b.d | ✓ | 8 | 8.b | even | 2 | 1 | inner |
| 384.7.b.d | ✓ | 8 | 8.d | odd | 2 | 1 | inner |
| 768.7.g.c | 4 | 16.e | even | 4 | 1 | ||
| 768.7.g.c | 4 | 16.f | odd | 4 | 1 | ||
| 768.7.g.e | 4 | 16.e | even | 4 | 1 | ||
| 768.7.g.e | 4 | 16.f | odd | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{2} + 400 \)
acting on \(S_{7}^{\mathrm{new}}(384, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
|
| $3$ |
\( (T^{2} - 243)^{4} \)
|
| $5$ |
\( (T^{2} + 400)^{4} \)
|
| $7$ |
\( (T^{4} + 304992 T^{2} + 6808230144)^{2} \)
|
| $11$ |
\( (T^{4} + \cdots + 1010555709696)^{2} \)
|
| $13$ |
\( (T^{4} + \cdots + 1207818584064)^{2} \)
|
| $17$ |
\( (T^{2} - 1116 T - 50450364)^{4} \)
|
| $19$ |
\( (T^{4} + \cdots + 285122947021056)^{2} \)
|
| $23$ |
\( (T^{4} + \cdots + 30\!\cdots\!04)^{2} \)
|
| $29$ |
\( (T^{4} + \cdots + 75\!\cdots\!44)^{2} \)
|
| $31$ |
\( (T^{4} + \cdots + 21\!\cdots\!16)^{2} \)
|
| $37$ |
\( (T^{4} + \cdots + 74\!\cdots\!16)^{2} \)
|
| $41$ |
\( (T^{2} + 24732 T - 303937596)^{4} \)
|
| $43$ |
\( (T^{4} + \cdots + 62\!\cdots\!24)^{2} \)
|
| $47$ |
\( (T^{4} + \cdots + 96\!\cdots\!24)^{2} \)
|
| $53$ |
\( (T^{4} + \cdots + 46\!\cdots\!00)^{2} \)
|
| $59$ |
\( (T^{4} + \cdots + 48\!\cdots\!24)^{2} \)
|
| $61$ |
\( (T^{4} + \cdots + 15\!\cdots\!84)^{2} \)
|
| $67$ |
\( (T^{4} + \cdots + 44\!\cdots\!96)^{2} \)
|
| $71$ |
\( (T^{4} + \cdots + 27\!\cdots\!24)^{2} \)
|
| $73$ |
\( (T^{2} - 19228 T - 147928769852)^{4} \)
|
| $79$ |
\( (T^{4} + \cdots + 88\!\cdots\!44)^{2} \)
|
| $83$ |
\( (T^{4} + \cdots + 37\!\cdots\!84)^{2} \)
|
| $89$ |
\( (T^{2} - 1659708 T + 683581488516)^{4} \)
|
| $97$ |
\( (T^{2} + 1232420 T - 31455232700)^{4} \)
|
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