Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2016,4,Mod(1,2016)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2016, 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("2016.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2016 = 2^{5} \cdot 3^{2} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2016.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: | \(118.947850572\) |
| Analytic rank: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.777569.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 41x^{2} + 58x + 244 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{6}\cdot 5 \) |
| 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
\( x^{4} - x^{3} - 41x^{2} + 58x + 244 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{3} - 5\nu^{2} + 39\nu + 78 ) / 7 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -4\nu^{3} + 8\nu^{2} + 128\nu - 248 ) / 7 \)
|
| \(\beta_{3}\) | \(=\) |
\( 3\nu^{3} + 11\nu^{2} - 73\nu - 173 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + \beta_{2} + 17\beta _1 + 19 ) / 40 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + 11\beta_{2} - 23\beta _1 + 819 ) / 40 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 17\beta_{3} - 8\beta_{2} + 249\beta _1 - 117 ) / 20 \)
|
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 | −18.9067 | 0 | 7.00000 | 0 | 0 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | 0 | 0 | −3.25902 | 0 | 7.00000 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | 0 | 0 | 3.09003 | 0 | 7.00000 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | 0 | 0 | 9.07567 | 0 | 7.00000 | 0 | 0 | 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 | 2016.4.a.bb | yes | 4 |
| 3.b | odd | 2 | 1 | 2016.4.a.bd | yes | 4 | |
| 4.b | odd | 2 | 1 | 2016.4.a.ba | ✓ | 4 | |
| 12.b | even | 2 | 1 | 2016.4.a.bc | yes | 4 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2016.4.a.ba | ✓ | 4 | 4.b | odd | 2 | 1 | |
| 2016.4.a.bb | yes | 4 | 1.a | even | 1 | 1 | trivial |
| 2016.4.a.bc | yes | 4 | 12.b | even | 2 | 1 | |
| 2016.4.a.bd | yes | 4 | 3.b | odd | 2 | 1 | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2016))\):
|
\( T_{5}^{4} + 10T_{5}^{3} - 180T_{5}^{2} - 128T_{5} + 1728 \)
|
|
\( T_{11}^{4} + 22T_{11}^{3} - 2564T_{11}^{2} - 100320T_{11} - 944896 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} + 10 T^{3} + \cdots + 1728 \)
|
| $7$ |
\( (T - 7)^{4} \)
|
| $11$ |
\( T^{4} + 22 T^{3} + \cdots - 944896 \)
|
| $13$ |
\( T^{4} - 20 T^{3} + \cdots + 463024 \)
|
| $17$ |
\( T^{4} - 10 T^{3} + \cdots + 19008768 \)
|
| $19$ |
\( T^{4} - 20 T^{3} + \cdots + 20377600 \)
|
| $23$ |
\( T^{4} + 158 T^{3} + \cdots - 7181568 \)
|
| $29$ |
\( T^{4} + 124 T^{3} + \cdots + 361751296 \)
|
| $31$ |
\( T^{4} - 44 T^{3} + \cdots - 75875328 \)
|
| $37$ |
\( T^{4} - 268 T^{3} + \cdots + 600975792 \)
|
| $41$ |
\( T^{4} + 298 T^{3} + \cdots + 320102656 \)
|
| $43$ |
\( T^{4} + 104 T^{3} + \cdots + 168902656 \)
|
| $47$ |
\( T^{4} + 364 T^{3} + \cdots - 345165824 \)
|
| $53$ |
\( T^{4} + \cdots - 2573288448 \)
|
| $59$ |
\( T^{4} + \cdots + 1157222400 \)
|
| $61$ |
\( T^{4} + \cdots + 9226101936 \)
|
| $67$ |
\( T^{4} + \cdots + 65770078208 \)
|
| $71$ |
\( T^{4} + \cdots + 3702938368 \)
|
| $73$ |
\( T^{4} + \cdots + 55107116496 \)
|
| $79$ |
\( T^{4} + \cdots - 36323446784 \)
|
| $83$ |
\( T^{4} + \cdots - 38098501632 \)
|
| $89$ |
\( T^{4} + \cdots - 8257579776 \)
|
| $97$ |
\( T^{4} + \cdots + 318527314128 \)
|
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