Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [333,4,Mod(1,333)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(333, 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("333.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 333 = 3^{2} \cdot 37 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 333.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: | \(19.6476360319\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.21208.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 8x^{2} + 13x - 3 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 37) |
| 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
\( x^{4} - x^{3} - 8x^{2} + 13x - 3 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{3} + \nu^{2} - 6\nu + 1 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 7\nu + 5 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{3} + \beta_{2} - \beta _1 + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 7\beta _1 - 5 \)
|
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 |
|
−1.32121 | 0 | −6.25440 | −7.30237 | 0 | −30.2958 | 18.8331 | 0 | 9.64798 | ||||||||||||||||||||||||||||||
| 1.2 | −0.389055 | 0 | −7.84864 | 19.1145 | 0 | 34.7993 | 6.16599 | 0 | −7.43658 | |||||||||||||||||||||||||||||||
| 1.3 | 2.06923 | 0 | −3.71830 | 5.08678 | 0 | −27.2773 | −24.2478 | 0 | 10.5257 | |||||||||||||||||||||||||||||||
| 1.4 | 5.64104 | 0 | 23.8213 | 12.1011 | 0 | −9.22618 | 89.2487 | 0 | 68.2629 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(37\) | \(-1\) |
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 | 333.4.a.e | 4 | |
| 3.b | odd | 2 | 1 | 37.4.a.a | ✓ | 4 | |
| 12.b | even | 2 | 1 | 592.4.a.f | 4 | ||
| 15.d | odd | 2 | 1 | 925.4.a.a | 4 | ||
| 21.c | even | 2 | 1 | 1813.4.a.b | 4 | ||
| 24.f | even | 2 | 1 | 2368.4.a.g | 4 | ||
| 24.h | odd | 2 | 1 | 2368.4.a.l | 4 | ||
| 111.d | odd | 2 | 1 | 1369.4.a.c | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 37.4.a.a | ✓ | 4 | 3.b | odd | 2 | 1 | |
| 333.4.a.e | 4 | 1.a | even | 1 | 1 | trivial | |
| 592.4.a.f | 4 | 12.b | even | 2 | 1 | ||
| 925.4.a.a | 4 | 15.d | odd | 2 | 1 | ||
| 1369.4.a.c | 4 | 111.d | odd | 2 | 1 | ||
| 1813.4.a.b | 4 | 21.c | even | 2 | 1 | ||
| 2368.4.a.g | 4 | 24.f | even | 2 | 1 | ||
| 2368.4.a.l | 4 | 24.h | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} - 6T_{2}^{3} - T_{2}^{2} + 16T_{2} + 6 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(333))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - 6 T^{3} + \cdots + 6 \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} - 29 T^{3} + \cdots - 8592 \)
|
| $7$ |
\( T^{4} + 32 T^{3} + \cdots - 265324 \)
|
| $11$ |
\( T^{4} - 11 T^{3} + \cdots + 169317 \)
|
| $13$ |
\( T^{4} + 45 T^{3} + \cdots - 4630592 \)
|
| $17$ |
\( T^{4} - 56 T^{3} + \cdots - 1776 \)
|
| $19$ |
\( T^{4} + 144 T^{3} + \cdots - 115380208 \)
|
| $23$ |
\( T^{4} - 275 T^{3} + \cdots - 93600684 \)
|
| $29$ |
\( T^{4} - 29 T^{3} + \cdots - 20921868 \)
|
| $31$ |
\( T^{4} - 111 T^{3} + \cdots + 443578864 \)
|
| $37$ |
\( (T - 37)^{4} \)
|
| $41$ |
\( T^{4} - 9 T^{3} + \cdots + 239475447 \)
|
| $43$ |
\( T^{4} + 190 T^{3} + \cdots + 118152128 \)
|
| $47$ |
\( T^{4} + \cdots - 4884453612 \)
|
| $53$ |
\( T^{4} + 318 T^{3} + \cdots + 1515396 \)
|
| $59$ |
\( T^{4} + \cdots - 19839940416 \)
|
| $61$ |
\( T^{4} + \cdots - 1974966496 \)
|
| $67$ |
\( T^{4} + \cdots + 3584880464 \)
|
| $71$ |
\( T^{4} + \cdots - 21095220732 \)
|
| $73$ |
\( T^{4} + \cdots - 10092864749 \)
|
| $79$ |
\( T^{4} + \cdots - 29390715884 \)
|
| $83$ |
\( T^{4} + \cdots - 146330370096 \)
|
| $89$ |
\( T^{4} + \cdots - 212866753728 \)
|
| $97$ |
\( T^{4} + \cdots + 3402009472 \)
|
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