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: | \(4\) |
| Coefficient field: | \(\Q(i, \sqrt{61})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 31x^{2} + 225 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{6} \) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} + 31x^{2} + 225 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} + 16\nu ) / 15 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 4\nu^{3} + 184\nu ) / 15 \)
|
| \(\beta_{3}\) | \(=\) |
\( 8\nu^{2} + 124 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} - 4\beta_1 ) / 8 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} - 124 ) / 8 \)
|
| \(\nu^{3}\) | \(=\) |
\( -2\beta_{2} + 23\beta_1 \)
|
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 | − | 51.2410i | 0 | 164.205 | 0 | −81.0000 | 0 | ||||||||||||||||||||||||||||
| 193.2 | 0 | − | 9.00000i | 0 | 11.2410i | 0 | −148.205 | 0 | −81.0000 | 0 | ||||||||||||||||||||||||||||||
| 193.3 | 0 | 9.00000i | 0 | − | 11.2410i | 0 | −148.205 | 0 | −81.0000 | 0 | ||||||||||||||||||||||||||||||
| 193.4 | 0 | 9.00000i | 0 | 51.2410i | 0 | 164.205 | 0 | −81.0000 | 0 | |||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 8.b | even | 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.h | yes | 4 |
| 4.b | odd | 2 | 1 | 384.6.d.g | ✓ | 4 | |
| 8.b | even | 2 | 1 | inner | 384.6.d.h | yes | 4 |
| 8.d | odd | 2 | 1 | 384.6.d.g | ✓ | 4 | |
| 16.e | even | 4 | 1 | 768.6.a.q | 2 | ||
| 16.e | even | 4 | 1 | 768.6.a.r | 2 | ||
| 16.f | odd | 4 | 1 | 768.6.a.m | 2 | ||
| 16.f | odd | 4 | 1 | 768.6.a.v | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 384.6.d.g | ✓ | 4 | 4.b | odd | 2 | 1 | |
| 384.6.d.g | ✓ | 4 | 8.d | odd | 2 | 1 | |
| 384.6.d.h | yes | 4 | 1.a | even | 1 | 1 | trivial |
| 384.6.d.h | yes | 4 | 8.b | even | 2 | 1 | inner |
| 768.6.a.m | 2 | 16.f | odd | 4 | 1 | ||
| 768.6.a.q | 2 | 16.e | even | 4 | 1 | ||
| 768.6.a.r | 2 | 16.e | even | 4 | 1 | ||
| 768.6.a.v | 2 | 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 \(S_{6}^{\mathrm{new}}(384, [\chi])\):
|
\( T_{5}^{4} + 2752T_{5}^{2} + 331776 \)
|
|
\( T_{7}^{2} - 16T_{7} - 24336 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( (T^{2} + 81)^{2} \)
|
| $5$ |
\( T^{4} + 2752 T^{2} + 331776 \)
|
| $7$ |
\( (T^{2} - 16 T - 24336)^{2} \)
|
| $11$ |
\( T^{4} + 90400 T^{2} + 195105024 \)
|
| $13$ |
\( T^{4} + \cdots + 32267655424 \)
|
| $17$ |
\( (T^{2} - 924 T - 2425660)^{2} \)
|
| $19$ |
\( T^{4} + \cdots + 480748089600 \)
|
| $23$ |
\( (T^{2} - 3584 T + 2738880)^{2} \)
|
| $29$ |
\( T^{4} + \cdots + 219728206925824 \)
|
| $31$ |
\( (T^{2} - 3536 T - 19127952)^{2} \)
|
| $37$ |
\( T^{4} + \cdots + 567838398009600 \)
|
| $41$ |
\( (T^{2} - 2516 T - 164087580)^{2} \)
|
| $43$ |
\( T^{4} + \cdots + 61\!\cdots\!84 \)
|
| $47$ |
\( (T^{2} - 4576 T - 81437760)^{2} \)
|
| $53$ |
\( T^{4} + \cdots + 686591217664 \)
|
| $59$ |
\( T^{4} + \cdots + 14\!\cdots\!00 \)
|
| $61$ |
\( T^{4} + \cdots + 56\!\cdots\!56 \)
|
| $67$ |
\( T^{4} + \cdots + 20\!\cdots\!64 \)
|
| $71$ |
\( (T^{2} - 75520 T + 1179861696)^{2} \)
|
| $73$ |
\( (T^{2} - 17100 T - 2681559900)^{2} \)
|
| $79$ |
\( (T^{2} - 100496 T + 1974396528)^{2} \)
|
| $83$ |
\( T^{4} + \cdots + 13\!\cdots\!64 \)
|
| $89$ |
\( (T^{2} + 90356 T + 2033493540)^{2} \)
|
| $97$ |
\( (T^{2} + 99908 T - 5288174460)^{2} \)
|
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