Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2116,4,Mod(1,2116)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2116, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("2116.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2116 = 2^{2} \cdot 23^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2116.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: | \(124.848041572\) |
| Analytic rank: | \(0\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.1229.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 7x + 6 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 92) |
| 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 \(1,\beta_1,\beta_2\) 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
\( x^{3} - x^{2} - 7x + 6 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu^{2} + \nu - 5 \)
|
| \(\beta_{2}\) | \(=\) |
\( -\nu^{2} + \nu + 5 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{2} + \beta _1 + 10 ) / 2 \)
|
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 | −6.13366 | 0 | −14.8525 | 0 | 19.8565 | 0 | 10.6218 | 0 | |||||||||||||||||||||||||||
| 1.2 | 0 | −1.18060 | 0 | 8.78065 | 0 | −9.15411 | 0 | −25.6062 | 0 | ||||||||||||||||||||||||||||
| 1.3 | 0 | 3.31427 | 0 | 16.0718 | 0 | 35.2976 | 0 | −16.0156 | 0 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(23\) | \( -1 \) |
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 | 2116.4.a.a | 3 | |
| 23.b | odd | 2 | 1 | 92.4.a.a | ✓ | 3 | |
| 69.c | even | 2 | 1 | 828.4.a.f | 3 | ||
| 92.b | even | 2 | 1 | 368.4.a.k | 3 | ||
| 115.c | odd | 2 | 1 | 2300.4.a.b | 3 | ||
| 115.e | even | 4 | 2 | 2300.4.c.b | 6 | ||
| 184.e | odd | 2 | 1 | 1472.4.a.w | 3 | ||
| 184.h | even | 2 | 1 | 1472.4.a.p | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 92.4.a.a | ✓ | 3 | 23.b | odd | 2 | 1 | |
| 368.4.a.k | 3 | 92.b | even | 2 | 1 | ||
| 828.4.a.f | 3 | 69.c | even | 2 | 1 | ||
| 1472.4.a.p | 3 | 184.h | even | 2 | 1 | ||
| 1472.4.a.w | 3 | 184.e | odd | 2 | 1 | ||
| 2116.4.a.a | 3 | 1.a | even | 1 | 1 | trivial | |
| 2300.4.a.b | 3 | 115.c | odd | 2 | 1 | ||
| 2300.4.c.b | 6 | 115.e | even | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2116))\):
|
\( T_{3}^{3} + 4T_{3}^{2} - 17T_{3} - 24 \)
|
|
\( T_{5}^{3} - 10T_{5}^{2} - 228T_{5} + 2096 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} \)
|
| $3$ |
\( T^{3} + 4 T^{2} + \cdots - 24 \)
|
| $5$ |
\( T^{3} - 10 T^{2} + \cdots + 2096 \)
|
| $7$ |
\( T^{3} - 46 T^{2} + \cdots + 6416 \)
|
| $11$ |
\( T^{3} - 64 T^{2} + \cdots + 88184 \)
|
| $13$ |
\( T^{3} + 44 T^{2} + \cdots - 3334 \)
|
| $17$ |
\( T^{3} - 88 T^{2} + \cdots + 1617496 \)
|
| $19$ |
\( T^{3} - 94 T^{2} + \cdots + 184984 \)
|
| $23$ |
\( T^{3} \)
|
| $29$ |
\( T^{3} - 308 T^{2} + \cdots - 1008698 \)
|
| $31$ |
\( T^{3} + 140 T^{2} + \cdots - 1074768 \)
|
| $37$ |
\( T^{3} + 26 T^{2} + \cdots + 4672784 \)
|
| $41$ |
\( T^{3} - 584 T^{2} + \cdots + 21405186 \)
|
| $43$ |
\( T^{3} - 478 T^{2} + \cdots + 14441984 \)
|
| $47$ |
\( T^{3} + 28 T^{2} + \cdots - 25906224 \)
|
| $53$ |
\( T^{3} + 356 T^{2} + \cdots - 63184 \)
|
| $59$ |
\( T^{3} - 144 T^{2} + \cdots - 15495744 \)
|
| $61$ |
\( T^{3} - 1052 T^{2} + \cdots + 55378432 \)
|
| $67$ |
\( T^{3} - 2008 T^{2} + \cdots - 228170232 \)
|
| $71$ |
\( T^{3} - 360 T^{2} + \cdots + 7542816 \)
|
| $73$ |
\( T^{3} + 252 T^{2} + \cdots + 34560066 \)
|
| $79$ |
\( T^{3} - 720 T^{2} + \cdots + 932658192 \)
|
| $83$ |
\( T^{3} - 1404 T^{2} + \cdots + 464610136 \)
|
| $89$ |
\( T^{3} + 534 T^{2} + \cdots - 77599776 \)
|
| $97$ |
\( T^{3} + 736 T^{2} + \cdots - 67270584 \)
|
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