Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1224,4,Mod(1,1224)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1224, 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("1224.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1224 = 2^{3} \cdot 3^{2} \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1224.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: | \(72.2183378470\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 80x^{2} - 135x - 25 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{4} \) |
| Twist minimal: | no (minimal twist has level 408) |
| 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
\( x^{4} - x^{3} - 80x^{2} - 135x - 25 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} - \nu^{2} - 75\nu - 105 ) / 5 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{3} + \nu^{2} + 95\nu + 100 ) / 5 \)
|
| \(\beta_{3}\) | \(=\) |
\( -\nu^{3} + 3\nu^{2} + 73\nu + 22 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + \beta _1 + 1 ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 2\beta_{3} + \beta_{2} + 11\beta _1 + 167 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( \beta_{3} + 38\beta_{2} + 53\beta _1 + 331 ) / 2 \)
|
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 | 0 | 0 | −16.6735 | 0 | 2.61724 | 0 | 0 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | 0 | 0 | −1.24200 | 0 | 18.7513 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | 0 | 0 | 2.07927 | 0 | 35.4573 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | 0 | 0 | 17.8362 | 0 | −24.8258 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(3\) | \(-1\) |
| \(17\) | \(-1\) |
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 | 1224.4.a.m | 4 | |
| 3.b | odd | 2 | 1 | 408.4.a.i | ✓ | 4 | |
| 4.b | odd | 2 | 1 | 2448.4.a.br | 4 | ||
| 12.b | even | 2 | 1 | 816.4.a.w | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 408.4.a.i | ✓ | 4 | 3.b | odd | 2 | 1 | |
| 816.4.a.w | 4 | 12.b | even | 2 | 1 | ||
| 1224.4.a.m | 4 | 1.a | even | 1 | 1 | trivial | |
| 2448.4.a.br | 4 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} - 2T_{5}^{3} - 299T_{5}^{2} + 252T_{5} + 768 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1224))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} - 2 T^{3} + \cdots + 768 \)
|
| $7$ |
\( T^{4} - 32 T^{3} + \cdots - 43200 \)
|
| $11$ |
\( T^{4} + 58 T^{3} + \cdots - 3418624 \)
|
| $13$ |
\( T^{4} - 66 T^{3} + \cdots - 1564236 \)
|
| $17$ |
\( (T - 17)^{4} \)
|
| $19$ |
\( T^{4} - 154 T^{3} + \cdots - 41254672 \)
|
| $23$ |
\( T^{4} + 126 T^{3} + \cdots - 44860684 \)
|
| $29$ |
\( T^{4} + 136 T^{3} + \cdots + 10624512 \)
|
| $31$ |
\( T^{4} - 264 T^{3} + \cdots - 112222000 \)
|
| $37$ |
\( T^{4} + \cdots - 1163461120 \)
|
| $41$ |
\( T^{4} + 298 T^{3} + \cdots + 29521524 \)
|
| $43$ |
\( T^{4} - 574 T^{3} + \cdots + 58628944 \)
|
| $47$ |
\( T^{4} + \cdots + 7956696512 \)
|
| $53$ |
\( T^{4} + \cdots + 65249465968 \)
|
| $59$ |
\( T^{4} + \cdots - 2693294400 \)
|
| $61$ |
\( T^{4} + \cdots - 85995296256 \)
|
| $67$ |
\( T^{4} - 400 T^{3} + \cdots + 381102336 \)
|
| $71$ |
\( T^{4} + \cdots - 10932225600 \)
|
| $73$ |
\( T^{4} + \cdots + 10864990960 \)
|
| $79$ |
\( T^{4} + \cdots + 25531153936 \)
|
| $83$ |
\( T^{4} + \cdots - 21640044224 \)
|
| $89$ |
\( T^{4} + \cdots + 15073311040 \)
|
| $97$ |
\( T^{4} + \cdots - 2563246629312 \)
|
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