Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1216,4,Mod(1,1216)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1216, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("1216.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1216 = 2^{6} \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1216.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: | \(71.7463225670\) |
| Analytic rank: | \(1\) |
| 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: |
\( x^{5} - x^{4} - 20x^{3} + 24x^{2} + 2x - 4 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{3} \) |
| Twist minimal: | no (minimal twist has level 608) |
| 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
\( x^{5} - x^{4} - 20x^{3} + 24x^{2} + 2x - 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{4} + \nu^{3} + 18\nu^{2} - 24\nu + 14 ) / 2 \)
|
| \(\beta_{2}\) | \(=\) |
\( -\nu^{4} + 20\nu^{2} - 2\nu - 8 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{4} - 20\nu^{2} + 6\nu + 7 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - \nu^{3} - 20\nu^{2} + 24\nu + 2 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + \beta_{2} + 1 ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{4} - 2\beta _1 + 16 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -2\beta_{4} + 11\beta_{3} + 9\beta_{2} - 1 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -20\beta_{4} - \beta_{3} - 3\beta_{2} - 40\beta _1 + 303 ) / 2 \)
|
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 | −9.59869 | 0 | −0.0570048 | 0 | −24.6071 | 0 | 65.1348 | 0 | |||||||||||||||||||||||||||||||||
| 1.2 | 0 | −6.71806 | 0 | −17.1669 | 0 | 31.2505 | 0 | 18.1323 | 0 | ||||||||||||||||||||||||||||||||||
| 1.3 | 0 | 2.66179 | 0 | 7.98961 | 0 | 27.0741 | 0 | −19.9149 | 0 | ||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 6.60208 | 0 | −1.58045 | 0 | −19.8743 | 0 | 16.5875 | 0 | ||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 10.0529 | 0 | −16.1852 | 0 | 6.15678 | 0 | 74.0603 | 0 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(19\) | \(-1\) |
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 | 1216.4.a.bc | 5 | |
| 4.b | odd | 2 | 1 | 1216.4.a.z | 5 | ||
| 8.b | even | 2 | 1 | 608.4.a.f | ✓ | 5 | |
| 8.d | odd | 2 | 1 | 608.4.a.g | yes | 5 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 608.4.a.f | ✓ | 5 | 8.b | even | 2 | 1 | |
| 608.4.a.g | yes | 5 | 8.d | odd | 2 | 1 | |
| 1216.4.a.z | 5 | 4.b | odd | 2 | 1 | ||
| 1216.4.a.bc | 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1216))\):
|
\( T_{3}^{5} - 3T_{3}^{4} - 140T_{3}^{3} + 384T_{3}^{2} + 4256T_{3} - 11392 \)
|
|
\( T_{5}^{5} + 27T_{5}^{4} + 53T_{5}^{3} - 2199T_{5}^{2} - 3634T_{5} - 200 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} \)
|
| $3$ |
\( T^{5} - 3 T^{4} - 140 T^{3} + \cdots - 11392 \)
|
| $5$ |
\( T^{5} + 27 T^{4} + 53 T^{3} + \cdots - 200 \)
|
| $7$ |
\( T^{5} - 20 T^{4} - 1174 T^{3} + \cdots - 2547508 \)
|
| $11$ |
\( T^{5} + 67 T^{4} - 1415 T^{3} + \cdots + 42711800 \)
|
| $13$ |
\( T^{5} + 191 T^{4} + \cdots - 38142080 \)
|
| $17$ |
\( T^{5} + 52 T^{4} - 6732 T^{3} + \cdots - 9857070 \)
|
| $19$ |
\( (T - 19)^{5} \)
|
| $23$ |
\( T^{5} - 35 T^{4} + \cdots - 5659603968 \)
|
| $29$ |
\( T^{5} - 33 T^{4} + \cdots + 1528050752 \)
|
| $31$ |
\( T^{5} + 694 T^{4} + \cdots + 1546708480 \)
|
| $37$ |
\( T^{5} + 108 T^{4} + \cdots + 264959360 \)
|
| $41$ |
\( T^{5} - 54 T^{4} + \cdots + 397858568960 \)
|
| $43$ |
\( T^{5} - 427 T^{4} + \cdots - 331567864112 \)
|
| $47$ |
\( T^{5} - 79 T^{4} + \cdots - 905506740224 \)
|
| $53$ |
\( T^{5} + 1025 T^{4} + \cdots - 210846255280 \)
|
| $59$ |
\( T^{5} - 649 T^{4} + \cdots - 34775624292320 \)
|
| $61$ |
\( T^{5} + 1413 T^{4} + \cdots - 29987676575940 \)
|
| $67$ |
\( T^{5} + 915 T^{4} + \cdots + 24348069855968 \)
|
| $71$ |
\( T^{5} - 1444 T^{4} + \cdots + 35176103924480 \)
|
| $73$ |
\( T^{5} - 2682 T^{4} + \cdots + 71728412586510 \)
|
| $79$ |
\( T^{5} + 836 T^{4} + \cdots - 31064499767680 \)
|
| $83$ |
\( T^{5} - 502 T^{4} + \cdots - 8567040270336 \)
|
| $89$ |
\( T^{5} - 996 T^{4} + \cdots - 13823629765888 \)
|
| $97$ |
\( T^{5} + \cdots - 461524239069440 \)
|
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