Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1232,4,Mod(1,1232)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1232, 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("1232.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1232 = 2^{4} \cdot 7 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1232.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: | \(72.6903531271\) |
| Analytic rank: | \(0\) |
| Dimension: | \(5\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{5} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - 2x^{4} - 105x^{3} + 92x^{2} + 1858x - 2576 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{3} \) |
| Twist minimal: | no (minimal twist has level 616) |
| 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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) 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
\( x^{5} - 2x^{4} - 105x^{3} + 92x^{2} + 1858x - 2576 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 5\nu^{4} - 13\nu^{3} - 363\nu^{2} + 61\nu + 978 ) / 257 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 4\nu^{4} + 41\nu^{3} - 496\nu^{2} - 3395\nu + 7156 ) / 257 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -10\nu^{4} + 26\nu^{3} + 983\nu^{2} - 636\nu - 12493 ) / 257 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{4} + 2\beta_{2} + 2\beta _1 + 41 \)
|
| \(\nu^{3}\) | \(=\) |
\( 4\beta_{4} + 5\beta_{3} + 4\beta_{2} + 75\beta _1 + 40 \)
|
| \(\nu^{4}\) | \(=\) |
\( 83\beta_{4} + 13\beta_{3} + 207\beta_{2} + 328\beta _1 + 2885 \)
|
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 | −9.98555 | 0 | −20.3895 | 0 | 7.00000 | 0 | 72.7112 | 0 | |||||||||||||||||||||||||||||||||
| 1.2 | 0 | −4.06945 | 0 | 10.6234 | 0 | 7.00000 | 0 | −10.4396 | 0 | ||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −1.45755 | 0 | −4.53944 | 0 | 7.00000 | 0 | −24.8755 | 0 | ||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 5.28837 | 0 | 17.5419 | 0 | 7.00000 | 0 | 0.966854 | 0 | ||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 8.22418 | 0 | −17.2362 | 0 | 7.00000 | 0 | 40.6371 | 0 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(7\) | \(-1\) |
| \(11\) | \(-1\) |
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 | 1232.4.a.z | 5 | |
| 4.b | odd | 2 | 1 | 616.4.a.g | ✓ | 5 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 616.4.a.g | ✓ | 5 | 4.b | odd | 2 | 1 | |
| 1232.4.a.z | 5 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1232))\):
|
\( T_{3}^{5} + 2T_{3}^{4} - 105T_{3}^{3} - 92T_{3}^{2} + 1858T_{3} + 2576 \)
|
|
\( T_{5}^{5} + 14T_{5}^{4} - 479T_{5}^{3} - 5256T_{5}^{2} + 52388T_{5} + 297296 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} \)
|
| $3$ |
\( T^{5} + 2 T^{4} + \cdots + 2576 \)
|
| $5$ |
\( T^{5} + 14 T^{4} + \cdots + 297296 \)
|
| $7$ |
\( (T - 7)^{5} \)
|
| $11$ |
\( (T - 11)^{5} \)
|
| $13$ |
\( T^{5} - 194 T^{4} + \cdots + 165725120 \)
|
| $17$ |
\( T^{5} + \cdots - 1652853792 \)
|
| $19$ |
\( T^{5} + \cdots - 1871657024 \)
|
| $23$ |
\( T^{5} + \cdots + 1356850592 \)
|
| $29$ |
\( T^{5} + \cdots + 558300459168 \)
|
| $31$ |
\( T^{5} + \cdots - 7177964236 \)
|
| $37$ |
\( T^{5} + \cdots - 575625945376 \)
|
| $41$ |
\( T^{5} + \cdots - 260317168544 \)
|
| $43$ |
\( T^{5} + \cdots - 37111868928 \)
|
| $47$ |
\( T^{5} + \cdots + 8257603328 \)
|
| $53$ |
\( T^{5} + \cdots - 118934413056 \)
|
| $59$ |
\( T^{5} + \cdots - 50642124297976 \)
|
| $61$ |
\( T^{5} + \cdots + 865511238448 \)
|
| $67$ |
\( T^{5} + \cdots + 74164434252000 \)
|
| $71$ |
\( T^{5} + \cdots + 254347095519296 \)
|
| $73$ |
\( T^{5} + \cdots - 6750753157408 \)
|
| $79$ |
\( T^{5} + \cdots - 4640907655424 \)
|
| $83$ |
\( T^{5} + \cdots + 90576457428736 \)
|
| $89$ |
\( T^{5} + \cdots - 29444789628360 \)
|
| $97$ |
\( T^{5} + \cdots + 11\!\cdots\!88 \)
|
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