Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [392,4,Mod(1,392)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(392, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("392.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 392 = 2^{3} \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 392.a (trivial) |
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: | yes |
| Analytic conductor: | \(23.1287487223\) |
| Analytic rank: | \(1\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.1929.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 10x + 13 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{3} \) |
| Twist minimal: | no (minimal twist has level 56) |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-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 \(1,\beta_1,\beta_2\) 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
\( x^{3} - x^{2} - 10x + 13 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu - 1 \)
|
| \(\beta_{2}\) | \(=\) |
\( 4\nu^{2} + 2\nu - 29 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{2} - \beta _1 + 28 ) / 4 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | |||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
0 | −9.54289 | 0 | −15.8094 | 0 | 0 | 0 | 64.0667 | 0 | |||||||||||||||||||||||||||
| 1.2 | 0 | −0.261560 | 0 | 19.5009 | 0 | 0 | 0 | −26.9316 | 0 | ||||||||||||||||||||||||||||
| 1.3 | 0 | 2.80445 | 0 | −6.69159 | 0 | 0 | 0 | −19.1351 | 0 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(7\) | \(1\) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 392.4.a.i | 3 | |
| 4.b | odd | 2 | 1 | 784.4.a.be | 3 | ||
| 7.b | odd | 2 | 1 | 392.4.a.l | 3 | ||
| 7.c | even | 3 | 2 | 56.4.i.b | ✓ | 6 | |
| 7.d | odd | 6 | 2 | 392.4.i.m | 6 | ||
| 21.h | odd | 6 | 2 | 504.4.s.h | 6 | ||
| 28.d | even | 2 | 1 | 784.4.a.bb | 3 | ||
| 28.g | odd | 6 | 2 | 112.4.i.e | 6 | ||
| 56.k | odd | 6 | 2 | 448.4.i.m | 6 | ||
| 56.p | even | 6 | 2 | 448.4.i.j | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 56.4.i.b | ✓ | 6 | 7.c | even | 3 | 2 | |
| 112.4.i.e | 6 | 28.g | odd | 6 | 2 | ||
| 392.4.a.i | 3 | 1.a | even | 1 | 1 | trivial | |
| 392.4.a.l | 3 | 7.b | odd | 2 | 1 | ||
| 392.4.i.m | 6 | 7.d | odd | 6 | 2 | ||
| 448.4.i.j | 6 | 56.p | even | 6 | 2 | ||
| 448.4.i.m | 6 | 56.k | odd | 6 | 2 | ||
| 504.4.s.h | 6 | 21.h | odd | 6 | 2 | ||
| 784.4.a.bb | 3 | 28.d | even | 2 | 1 | ||
| 784.4.a.be | 3 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(392))\):
|
\( T_{3}^{3} + 7T_{3}^{2} - 25T_{3} - 7 \)
|
|
\( T_{5}^{3} + 3T_{5}^{2} - 333T_{5} - 2063 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} \)
|
| $3$ |
\( T^{3} + 7 T^{2} + \cdots - 7 \)
|
| $5$ |
\( T^{3} + 3 T^{2} + \cdots - 2063 \)
|
| $7$ |
\( T^{3} \)
|
| $11$ |
\( T^{3} + 3 T^{2} + \cdots + 38637 \)
|
| $13$ |
\( T^{3} + 26 T^{2} + \cdots + 26328 \)
|
| $17$ |
\( T^{3} + 31 T^{2} + \cdots - 125571 \)
|
| $19$ |
\( T^{3} + 89 T^{2} + \cdots - 22529 \)
|
| $23$ |
\( T^{3} + 201 T^{2} + \cdots + 85687 \)
|
| $29$ |
\( T^{3} - 190 T^{2} + \cdots + 48504 \)
|
| $31$ |
\( T^{3} + 339 T^{2} + \cdots - 2554243 \)
|
| $37$ |
\( T^{3} + 535 T^{2} + \cdots - 885387 \)
|
| $41$ |
\( T^{3} - 58 T^{2} + \cdots + 3860392 \)
|
| $43$ |
\( T^{3} - 268 T^{2} + \cdots + 24343488 \)
|
| $47$ |
\( T^{3} + 205 T^{2} + \cdots - 11237437 \)
|
| $53$ |
\( T^{3} - 757 T^{2} + \cdots + 74662809 \)
|
| $59$ |
\( T^{3} + 1799 T^{2} + \cdots + 196666617 \)
|
| $61$ |
\( T^{3} - 625 T^{2} + \cdots + 16537453 \)
|
| $67$ |
\( T^{3} + 495 T^{2} + \cdots - 112177071 \)
|
| $71$ |
\( T^{3} - 640 T^{2} + \cdots - 7291392 \)
|
| $73$ |
\( T^{3} + 443 T^{2} + \cdots + 100339897 \)
|
| $79$ |
\( T^{3} - 79 T^{2} + \cdots + 147181471 \)
|
| $83$ |
\( T^{3} + 2372 T^{2} + \cdots + 266787264 \)
|
| $89$ |
\( T^{3} - 821 T^{2} + \cdots + 96776649 \)
|
| $97$ |
\( T^{3} + 342 T^{2} + \cdots + 217321448 \)
|
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