Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [384,6,Mod(193,384)]
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, 1, 0]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("384.193");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 384 = 2^{7} \cdot 3 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 384.d (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: | \(61.5873868082\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | 8.0.110166016.2 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} + 10x^{6} + 19x^{4} + 10x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{22}\cdot 3^{12} \) |
| 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
\( x^{8} + 10x^{6} + 19x^{4} + 10x^{2} + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( -8\nu^{5} - 72\nu^{3} - 72\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -18\nu^{7} - 171\nu^{5} - 261\nu^{3} - 81\nu ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( -69\nu^{6} - 624\nu^{4} - 720\nu^{2} - 63 \)
|
| \(\beta_{4}\) | \(=\) |
\( -42\nu^{7} - 411\nu^{5} - 699\nu^{3} - 201\nu \)
|
| \(\beta_{5}\) | \(=\) |
\( 51\nu^{6} + 480\nu^{4} + 720\nu^{2} + 225 \)
|
| \(\beta_{6}\) | \(=\) |
\( 72\nu^{7} + 720\nu^{5} + 1368\nu^{3} + 792\nu \)
|
| \(\beta_{7}\) | \(=\) |
\( 72\nu^{6} + 720\nu^{4} + 1296\nu^{2} + 360 \)
|
| \(\nu\) | \(=\) |
\( ( 2\beta_{6} + 16\beta_{2} + 9\beta_1 ) / 288 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -3\beta_{7} + 11\beta_{5} + 5\beta_{3} - 1080 ) / 432 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -4\beta_{6} + 6\beta_{4} - 60\beta_{2} - 27\beta_1 ) / 108 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 10\beta_{7} - 29\beta_{5} - 11\beta_{3} + 2232 ) / 144 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 26\beta_{6} - 48\beta_{4} + 432\beta_{2} + 177\beta_1 ) / 96 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( -5\beta_{7} + 14\beta_{5} + 5\beta_{3} - 1035 ) / 9 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( -1786\beta_{6} + 3408\beta_{4} - 30288\beta_{2} - 12123\beta_1 ) / 864 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 193.1 |
|
0 | − | 9.00000i | 0 | − | 76.5014i | 0 | 56.1972 | 0 | −81.0000 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 193.2 | 0 | − | 9.00000i | 0 | − | 42.1845i | 0 | −139.463 | 0 | −81.0000 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 193.3 | 0 | − | 9.00000i | 0 | 42.1845i | 0 | 139.463 | 0 | −81.0000 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 193.4 | 0 | − | 9.00000i | 0 | 76.5014i | 0 | −56.1972 | 0 | −81.0000 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 193.5 | 0 | 9.00000i | 0 | − | 76.5014i | 0 | −56.1972 | 0 | −81.0000 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 193.6 | 0 | 9.00000i | 0 | − | 42.1845i | 0 | 139.463 | 0 | −81.0000 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 193.7 | 0 | 9.00000i | 0 | 42.1845i | 0 | −139.463 | 0 | −81.0000 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 193.8 | 0 | 9.00000i | 0 | 76.5014i | 0 | 56.1972 | 0 | −81.0000 | 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 | 384.6.d.i | ✓ | 8 |
| 4.b | odd | 2 | 1 | inner | 384.6.d.i | ✓ | 8 |
| 8.b | even | 2 | 1 | inner | 384.6.d.i | ✓ | 8 |
| 8.d | odd | 2 | 1 | inner | 384.6.d.i | ✓ | 8 |
| 16.e | even | 4 | 1 | 768.6.a.x | 4 | ||
| 16.e | even | 4 | 1 | 768.6.a.y | 4 | ||
| 16.f | odd | 4 | 1 | 768.6.a.x | 4 | ||
| 16.f | odd | 4 | 1 | 768.6.a.y | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 384.6.d.i | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 384.6.d.i | ✓ | 8 | 4.b | odd | 2 | 1 | inner |
| 384.6.d.i | ✓ | 8 | 8.b | even | 2 | 1 | inner |
| 384.6.d.i | ✓ | 8 | 8.d | odd | 2 | 1 | inner |
| 768.6.a.x | 4 | 16.e | even | 4 | 1 | ||
| 768.6.a.x | 4 | 16.f | odd | 4 | 1 | ||
| 768.6.a.y | 4 | 16.e | even | 4 | 1 | ||
| 768.6.a.y | 4 | 16.f | odd | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(384, [\chi])\):
|
\( T_{5}^{4} + 7632T_{5}^{2} + 10414656 \)
|
|
\( T_{7}^{4} - 22608T_{7}^{2} + 61425216 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
|
| $3$ |
\( (T^{2} + 81)^{4} \)
|
| $5$ |
\( (T^{4} + 7632 T^{2} + 10414656)^{2} \)
|
| $7$ |
\( (T^{4} - 22608 T^{2} + 61425216)^{2} \)
|
| $11$ |
\( (T^{4} + 300064 T^{2} + 530104576)^{2} \)
|
| $13$ |
\( (T^{4} + 990144 T^{2} + 145716765696)^{2} \)
|
| $17$ |
\( (T^{2} - 1020 T - 113148)^{4} \)
|
| $19$ |
\( (T^{4} + 813600 T^{2} + 31116960000)^{2} \)
|
| $23$ |
\( (T^{4} + \cdots + 6280695604224)^{2} \)
|
| $29$ |
\( (T^{4} + \cdots + 613390472829504)^{2} \)
|
| $31$ |
\( (T^{4} + \cdots + 812571134405184)^{2} \)
|
| $37$ |
\( (T^{4} + \cdots + 52\!\cdots\!84)^{2} \)
|
| $41$ |
\( (T^{2} - 14676 T + 50487012)^{4} \)
|
| $43$ |
\( (T^{4} + \cdots + 12\!\cdots\!36)^{2} \)
|
| $47$ |
\( (T^{4} + \cdots + 48\!\cdots\!36)^{2} \)
|
| $53$ |
\( (T^{4} + \cdots + 14\!\cdots\!76)^{2} \)
|
| $59$ |
\( (T^{4} + \cdots + 289249749934336)^{2} \)
|
| $61$ |
\( (T^{4} + \cdots + 10\!\cdots\!76)^{2} \)
|
| $67$ |
\( (T^{4} + \cdots + 10\!\cdots\!16)^{2} \)
|
| $71$ |
\( (T^{4} + \cdots + 28\!\cdots\!24)^{2} \)
|
| $73$ |
\( (T^{2} - 62204 T - 1264854524)^{4} \)
|
| $79$ |
\( (T^{4} + \cdots + 28\!\cdots\!56)^{2} \)
|
| $83$ |
\( (T^{4} + \cdots + 70\!\cdots\!00)^{2} \)
|
| $89$ |
\( (T^{2} + 11700 T - 15195788892)^{4} \)
|
| $97$ |
\( (T^{2} + 65636 T - 14557756988)^{4} \)
|
| show more | |
| show less |