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: | \(1\) |
| 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: |
\( x^{4} - x^{3} - 75x^{2} + 136x + 578 \)
|
| 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
\( x^{4} - x^{3} - 75x^{2} + 136x + 578 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -7\nu^{3} - 10\nu^{2} + 406\nu - 68 ) / 51 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 2\nu^{3} + 32\nu^{2} + 88\nu - 1139 ) / 51 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -2\nu^{3} - 32\nu^{2} + 116\nu + 1088 ) / 51 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + \beta_{2} + 1 ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -7\beta_{3} + 2\beta _1 + 152 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 34\beta_{3} + 29\beta_{2} - 16\beta _1 - 99 ) / 2 \)
|
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 | −19.4761 | 0 | 8.30352 | 0 | 0 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | 0 | 0 | −8.08710 | 0 | −28.9656 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | 0 | 0 | 1.46562 | 0 | 34.4962 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | 0 | 0 | 16.0976 | 0 | −17.8341 | 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.k | 4 | |
| 3.b | odd | 2 | 1 | 408.4.a.h | ✓ | 4 | |
| 4.b | odd | 2 | 1 | 2448.4.a.bp | 4 | ||
| 12.b | even | 2 | 1 | 816.4.a.x | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 408.4.a.h | ✓ | 4 | 3.b | odd | 2 | 1 | |
| 816.4.a.x | 4 | 12.b | even | 2 | 1 | ||
| 1224.4.a.k | 4 | 1.a | even | 1 | 1 | trivial | |
| 2448.4.a.bp | 4 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} + 10T_{5}^{3} - 303T_{5}^{2} - 2116T_{5} + 3716 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1224))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} + 10 T^{3} + \cdots + 3716 \)
|
| $7$ |
\( T^{4} + 4 T^{3} + \cdots + 147968 \)
|
| $11$ |
\( T^{4} - 34 T^{3} + \cdots + 120336 \)
|
| $13$ |
\( T^{4} - 18 T^{3} + \cdots + 263652 \)
|
| $17$ |
\( (T + 17)^{4} \)
|
| $19$ |
\( T^{4} + 86 T^{3} + \cdots + 314448 \)
|
| $23$ |
\( T^{4} - 26 T^{3} + \cdots + 60511008 \)
|
| $29$ |
\( T^{4} + 216 T^{3} + \cdots + 1348560 \)
|
| $31$ |
\( T^{4} - 20 T^{3} + \cdots + 186656896 \)
|
| $37$ |
\( T^{4} - 532 T^{3} + \cdots + 18739968 \)
|
| $41$ |
\( T^{4} + \cdots - 13225619916 \)
|
| $43$ |
\( T^{4} - 22 T^{3} + \cdots - 325658864 \)
|
| $47$ |
\( T^{4} + \cdots - 2767428096 \)
|
| $53$ |
\( T^{4} + 1164 T^{3} + \cdots + 144333360 \)
|
| $59$ |
\( T^{4} + \cdots + 52918788416 \)
|
| $61$ |
\( T^{4} + \cdots - 21310161088 \)
|
| $67$ |
\( T^{4} + \cdots + 19413025536 \)
|
| $71$ |
\( T^{4} + \cdots - 158658138624 \)
|
| $73$ |
\( T^{4} + \cdots - 4086194960 \)
|
| $79$ |
\( T^{4} + \cdots - 63606204800 \)
|
| $83$ |
\( T^{4} + \cdots - 28091792704 \)
|
| $89$ |
\( T^{4} + \cdots - 604526726592 \)
|
| $97$ |
\( T^{4} + \cdots - 52003306944 \)
|
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