Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1344,4,Mod(1,1344)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1344, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("1344.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1344 = 2^{6} \cdot 3 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1344.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: | \(79.2985670477\) |
| Analytic rank: | \(0\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.22700.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 28x + 12 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{4} \) |
| Twist minimal: | no (minimal twist has level 672) |
| 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
\( x^{3} - x^{2} - 28x + 12 \)
:
| \(\beta_{1}\) | \(=\) |
\( -\nu^{2} + 7\nu + 17 \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} + \nu - 19 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + \beta _1 + 2 ) / 8 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 7\beta_{2} - \beta _1 + 150 ) / 8 \)
|
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 | 3.00000 | 0 | −15.3946 | 0 | 7.00000 | 0 | 9.00000 | 0 | |||||||||||||||||||||||||||
| 1.2 | 0 | 3.00000 | 0 | 4.31394 | 0 | 7.00000 | 0 | 9.00000 | 0 | ||||||||||||||||||||||||||||
| 1.3 | 0 | 3.00000 | 0 | 21.0807 | 0 | 7.00000 | 0 | 9.00000 | 0 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(3\) | \( -1 \) |
| \(7\) | \( -1 \) |
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 | 1344.4.a.bv | 3 | |
| 4.b | odd | 2 | 1 | 1344.4.a.bt | 3 | ||
| 8.b | even | 2 | 1 | 672.4.a.o | ✓ | 3 | |
| 8.d | odd | 2 | 1 | 672.4.a.q | yes | 3 | |
| 24.f | even | 2 | 1 | 2016.4.a.y | 3 | ||
| 24.h | odd | 2 | 1 | 2016.4.a.z | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 672.4.a.o | ✓ | 3 | 8.b | even | 2 | 1 | |
| 672.4.a.q | yes | 3 | 8.d | odd | 2 | 1 | |
| 1344.4.a.bt | 3 | 4.b | odd | 2 | 1 | ||
| 1344.4.a.bv | 3 | 1.a | even | 1 | 1 | trivial | |
| 2016.4.a.y | 3 | 24.f | even | 2 | 1 | ||
| 2016.4.a.z | 3 | 24.h | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1344))\):
|
\( T_{5}^{3} - 10T_{5}^{2} - 300T_{5} + 1400 \)
|
|
\( T_{11}^{3} - 2840T_{11} + 45280 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} \)
|
| $3$ |
\( (T - 3)^{3} \)
|
| $5$ |
\( T^{3} - 10 T^{2} + \cdots + 1400 \)
|
| $7$ |
\( (T - 7)^{3} \)
|
| $11$ |
\( T^{3} - 2840T + 45280 \)
|
| $13$ |
\( T^{3} - 50 T^{2} + \cdots + 26920 \)
|
| $17$ |
\( T^{3} - 30 T^{2} + \cdots + 275880 \)
|
| $19$ |
\( T^{3} - 140 T^{2} + \cdots - 6080 \)
|
| $23$ |
\( T^{3} + 56 T^{2} + \cdots - 3285792 \)
|
| $29$ |
\( T^{3} + 298 T^{2} + \cdots - 5325704 \)
|
| $31$ |
\( T^{3} - 80 T^{2} + \cdots + 8700160 \)
|
| $37$ |
\( T^{3} + 10 T^{2} + \cdots - 4978760 \)
|
| $41$ |
\( T^{3} - 390 T^{2} + \cdots + 21358600 \)
|
| $43$ |
\( T^{3} - 784 T^{2} + \cdots - 12439552 \)
|
| $47$ |
\( T^{3} + 248 T^{2} + \cdots + 452096 \)
|
| $53$ |
\( T^{3} + 10 T^{2} + \cdots - 32747080 \)
|
| $59$ |
\( T^{3} - 1500 T^{2} + \cdots - 90301120 \)
|
| $61$ |
\( T^{3} - 810 T^{2} + \cdots + 276011400 \)
|
| $67$ |
\( T^{3} - 1272 T^{2} + \cdots - 71066624 \)
|
| $71$ |
\( T^{3} + 160 T^{2} + \cdots + 258709600 \)
|
| $73$ |
\( T^{3} + 1170 T^{2} + \cdots - 337981800 \)
|
| $79$ |
\( T^{3} + 840 T^{2} + \cdots - 29596160 \)
|
| $83$ |
\( T^{3} - 1564 T^{2} + \cdots - 33022272 \)
|
| $89$ |
\( T^{3} + 178 T^{2} + \cdots - 53515224 \)
|
| $97$ |
\( T^{3} + 130 T^{2} + \cdots - 400247720 \)
|
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