Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2368,4,Mod(1,2368)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2368, 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("2368.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2368 = 2^{6} \cdot 37 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2368.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: | \(139.716522894\) |
| Analytic rank: | \(1\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.15629.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - 26x - 45 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 74) |
| 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
\( x^{3} - 26x - 45 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3\nu - 17 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3\beta _1 + 17 \)
|
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 | −6.12805 | 0 | −18.7464 | 0 | −21.9178 | 0 | 10.5531 | 0 | |||||||||||||||||||||||||||
| 1.2 | 0 | −5.42375 | 0 | 1.04699 | 0 | 25.1006 | 0 | 2.41712 | 0 | ||||||||||||||||||||||||||||
| 1.3 | 0 | 7.55181 | 0 | 10.6994 | 0 | 3.81714 | 0 | 30.0298 | 0 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(37\) | \(-1\) |
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 | 2368.4.a.e | 3 | |
| 4.b | odd | 2 | 1 | 2368.4.a.f | 3 | ||
| 8.b | even | 2 | 1 | 74.4.a.c | ✓ | 3 | |
| 8.d | odd | 2 | 1 | 592.4.a.c | 3 | ||
| 24.h | odd | 2 | 1 | 666.4.a.n | 3 | ||
| 40.f | even | 2 | 1 | 1850.4.a.i | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 74.4.a.c | ✓ | 3 | 8.b | even | 2 | 1 | |
| 592.4.a.c | 3 | 8.d | odd | 2 | 1 | ||
| 666.4.a.n | 3 | 24.h | odd | 2 | 1 | ||
| 1850.4.a.i | 3 | 40.f | even | 2 | 1 | ||
| 2368.4.a.e | 3 | 1.a | even | 1 | 1 | trivial | |
| 2368.4.a.f | 3 | 4.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 \(S_{4}^{\mathrm{new}}(\Gamma_0(2368))\):
|
\( T_{3}^{3} + 4T_{3}^{2} - 54T_{3} - 251 \)
|
|
\( T_{5}^{3} + 7T_{5}^{2} - 209T_{5} + 210 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} \)
|
| $3$ |
\( T^{3} + 4 T^{2} + \cdots - 251 \)
|
| $5$ |
\( T^{3} + 7 T^{2} + \cdots + 210 \)
|
| $7$ |
\( T^{3} - 7 T^{2} + \cdots + 2100 \)
|
| $11$ |
\( T^{3} + 74 T^{2} + \cdots - 60751 \)
|
| $13$ |
\( T^{3} + 107 T^{2} + \cdots - 64466 \)
|
| $17$ |
\( T^{3} - 24 T^{2} + \cdots - 61536 \)
|
| $19$ |
\( T^{3} + 228 T^{2} + \cdots - 10800 \)
|
| $23$ |
\( T^{3} + 149 T^{2} + \cdots - 691166 \)
|
| $29$ |
\( T^{3} - 325 T^{2} + \cdots - 637750 \)
|
| $31$ |
\( T^{3} - 23 T^{2} + \cdots + 1309500 \)
|
| $37$ |
\( (T - 37)^{3} \)
|
| $41$ |
\( T^{3} + 464 T^{2} + \cdots - 19358953 \)
|
| $43$ |
\( T^{3} + 588 T^{2} + \cdots - 42631272 \)
|
| $47$ |
\( T^{3} - 155 T^{2} + \cdots + 23334804 \)
|
| $53$ |
\( T^{3} - 579 T^{2} + \cdots + 20188 \)
|
| $59$ |
\( T^{3} + 1258 T^{2} + \cdots - 208191120 \)
|
| $61$ |
\( T^{3} + 291 T^{2} + \cdots + 25288348 \)
|
| $67$ |
\( T^{3} - 127 T^{2} + \cdots + 33004112 \)
|
| $71$ |
\( T^{3} + 201 T^{2} + \cdots + 147028116 \)
|
| $73$ |
\( T^{3} - 42 T^{2} + \cdots - 14055069 \)
|
| $79$ |
\( T^{3} - 413 T^{2} + \cdots + 122660820 \)
|
| $83$ |
\( T^{3} - 1037 T^{2} + \cdots + 539472 \)
|
| $89$ |
\( T^{3} - 660 T^{2} + \cdots + 986707200 \)
|
| $97$ |
\( T^{3} - 1384 T^{2} + \cdots - 15830176 \)
|
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