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: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.391168.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 40x^{2} + 382 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{3}\cdot 7 \) |
| Twist minimal: | yes |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - 40x^{2} + 382 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -2\nu^{3} - \nu^{2} + 40\nu + 20 ) / 3 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -2\nu^{2} + 12\nu + 40 ) / 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 7\nu^{2} - 140 ) / 3 \)
|
| \(\nu\) | \(=\) |
\( ( 2\beta_{3} + 7\beta_{2} ) / 28 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 3\beta_{3} + 140 ) / 7 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 17\beta_{3} + 70\beta_{2} - 21\beta_1 ) / 14 \)
|
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 | −13.3405 | 0 | 0 | 0 | 9.00000 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | 3.00000 | 0 | −7.81338 | 0 | 0 | 0 | 9.00000 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | 3.00000 | 0 | 14.5121 | 0 | 0 | 0 | 9.00000 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | 3.00000 | 0 | 14.6418 | 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.be | yes | 4 |
| 4.b | odd | 2 | 1 | 2352.4.a.cn | 4 | ||
| 7.b | odd | 2 | 1 | 1176.4.a.z | ✓ | 4 | |
| 28.d | even | 2 | 1 | 2352.4.a.co | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1176.4.a.z | ✓ | 4 | 7.b | odd | 2 | 1 | |
| 1176.4.a.be | yes | 4 | 1.a | even | 1 | 1 | trivial |
| 2352.4.a.cn | 4 | 4.b | odd | 2 | 1 | ||
| 2352.4.a.co | 4 | 28.d | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1176))\):
|
\( T_{5}^{4} - 8T_{5}^{3} - 300T_{5}^{2} + 1456T_{5} + 22148 \)
|
|
\( T_{11}^{4} - 40T_{11}^{3} - 232T_{11}^{2} + 4704T_{11} + 7056 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( (T - 3)^{4} \)
|
| $5$ |
\( T^{4} - 8 T^{3} + \cdots + 22148 \)
|
| $7$ |
\( T^{4} \)
|
| $11$ |
\( T^{4} - 40 T^{3} + \cdots + 7056 \)
|
| $13$ |
\( T^{4} - 48 T^{3} + \cdots + 5508 \)
|
| $17$ |
\( T^{4} - 72 T^{3} + \cdots + 695428 \)
|
| $19$ |
\( T^{4} - 32 T^{3} + \cdots + 50872896 \)
|
| $23$ |
\( T^{4} - 8 T^{3} + \cdots - 241264 \)
|
| $29$ |
\( T^{4} - 144 T^{3} + \cdots + 193813056 \)
|
| $31$ |
\( T^{4} - 48 T^{3} + \cdots + 280696384 \)
|
| $37$ |
\( T^{4} - 48 T^{3} + \cdots + 3279104 \)
|
| $41$ |
\( T^{4} - 72 T^{3} + \cdots - 110754684 \)
|
| $43$ |
\( T^{4} + \cdots - 1048085504 \)
|
| $47$ |
\( T^{4} - 160 T^{3} + \cdots + 908321344 \)
|
| $53$ |
\( T^{4} - 536 T^{3} + \cdots + 520036624 \)
|
| $59$ |
\( T^{4} + \cdots - 10001985984 \)
|
| $61$ |
\( T^{4} + \cdots - 2069850492 \)
|
| $67$ |
\( T^{4} + \cdots - 23013868544 \)
|
| $71$ |
\( T^{4} + \cdots - 188096048496 \)
|
| $73$ |
\( T^{4} + \cdots - 21509659836 \)
|
| $79$ |
\( T^{4} + \cdots - 88431606784 \)
|
| $83$ |
\( T^{4} + \cdots + 1047073024 \)
|
| $89$ |
\( T^{4} + \cdots - 1254461582844 \)
|
| $97$ |
\( T^{4} + \cdots + 13301642052 \)
|
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