Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1176,4,Mod(1,1176)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1176, 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("1176.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1176 = 2^{3} \cdot 3 \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1176.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: | \(69.3862461668\) |
| Analytic rank: | \(1\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.58461.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 65x - 126 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | no (minimal twist has level 168) |
| 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
\( x^{3} - x^{2} - 65x - 126 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{2} + \nu - 45 ) / 3 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -2\nu^{2} + 10\nu + 84 ) / 3 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + 2\beta _1 + 2 ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{2} + 10\beta _1 + 178 ) / 4 \)
|
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 | −3.00000 | 0 | −10.1566 | 0 | 0 | 0 | 9.00000 | 0 | |||||||||||||||||||||||||||
| 1.2 | 0 | −3.00000 | 0 | −0.239289 | 0 | 0 | 0 | 9.00000 | 0 | ||||||||||||||||||||||||||||
| 1.3 | 0 | −3.00000 | 0 | 21.3959 | 0 | 0 | 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 | 1176.4.a.x | 3 | |
| 4.b | odd | 2 | 1 | 2352.4.a.cj | 3 | ||
| 7.b | odd | 2 | 1 | 1176.4.a.y | 3 | ||
| 7.d | odd | 6 | 2 | 168.4.q.e | ✓ | 6 | |
| 21.g | even | 6 | 2 | 504.4.s.g | 6 | ||
| 28.d | even | 2 | 1 | 2352.4.a.ch | 3 | ||
| 28.f | even | 6 | 2 | 336.4.q.l | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 168.4.q.e | ✓ | 6 | 7.d | odd | 6 | 2 | |
| 336.4.q.l | 6 | 28.f | even | 6 | 2 | ||
| 504.4.s.g | 6 | 21.g | even | 6 | 2 | ||
| 1176.4.a.x | 3 | 1.a | even | 1 | 1 | trivial | |
| 1176.4.a.y | 3 | 7.b | odd | 2 | 1 | ||
| 2352.4.a.ch | 3 | 28.d | even | 2 | 1 | ||
| 2352.4.a.cj | 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1176))\):
|
\( T_{5}^{3} - 11T_{5}^{2} - 220T_{5} - 52 \)
|
|
\( T_{11}^{3} + 19T_{11}^{2} - 624T_{11} + 3276 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} \)
|
| $3$ |
\( (T + 3)^{3} \)
|
| $5$ |
\( T^{3} - 11 T^{2} + \cdots - 52 \)
|
| $7$ |
\( T^{3} \)
|
| $11$ |
\( T^{3} + 19 T^{2} + \cdots + 3276 \)
|
| $13$ |
\( T^{3} - 22 T^{2} + \cdots - 26976 \)
|
| $17$ |
\( T^{3} - 104 T^{2} + \cdots + 4032 \)
|
| $19$ |
\( T^{3} + 202 T^{2} + \cdots - 2225968 \)
|
| $23$ |
\( T^{3} + 280 T^{2} + \cdots - 1532736 \)
|
| $29$ |
\( T^{3} + 73 T^{2} + \cdots + 970992 \)
|
| $31$ |
\( T^{3} - 131 T^{2} + \cdots + 4156607 \)
|
| $37$ |
\( T^{3} + 326 T^{2} + \cdots - 17796 \)
|
| $41$ |
\( T^{3} - 516 T^{2} + \cdots + 15002144 \)
|
| $43$ |
\( T^{3} - 36 T^{2} + \cdots + 7204222 \)
|
| $47$ |
\( T^{3} - 126 T^{2} + \cdots - 11682152 \)
|
| $53$ |
\( T^{3} + 385 T^{2} + \cdots + 178128 \)
|
| $59$ |
\( T^{3} + 285 T^{2} + \cdots - 1793232 \)
|
| $61$ |
\( T^{3} - 34 T^{2} + \cdots - 22240152 \)
|
| $67$ |
\( T^{3} + 100 T^{2} + \cdots - 27246966 \)
|
| $71$ |
\( T^{3} - 34 T^{2} + \cdots + 207049704 \)
|
| $73$ |
\( T^{3} - 108 T^{2} + \cdots + 409658074 \)
|
| $79$ |
\( T^{3} + 2463 T^{2} + \cdots + 492780761 \)
|
| $83$ |
\( T^{3} - 115 T^{2} + \cdots - 111709668 \)
|
| $89$ |
\( T^{3} - 110 T^{2} + \cdots + 347278464 \)
|
| $97$ |
\( T^{3} - 2941 T^{2} + \cdots - 866695284 \)
|
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