Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [4,40,Mod(1,4)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 40, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("4.1");
S:= CuspForms(chi, 40);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 4 = 2^{2} \) |
| Weight: | \( k \) | \(=\) | \( 40 \) |
| Character orbit: | \([\chi]\) | \(=\) | 4.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: | \(38.5358205559\) |
| Analytic rank: | \(1\) |
| Dimension: | \(3\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{3} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 3400790375346x + 2307443739169452696 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{22}\cdot 3^{3}\cdot 5\cdot 7\cdot 13 \) |
| 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 \(1,\beta_1,\beta_2\) 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
\( x^{3} - x^{2} - 3400790375346x + 2307443739169452696 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 128\nu^{2} + 216555520\nu - 290200850881408 ) / 53785 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -4405376\nu^{2} - 3844622295040\nu + 9987841481929073536 ) / 53785 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + 34417\beta _1 + 22364160 ) / 67092480 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -5287\beta_{2} - 93862849\beta _1 + 475348875504432384 ) / 209664 \)
|
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 | −3.66534e9 | 0 | −7.71630e13 | 0 | 2.50960e16 | 0 | 9.38218e18 | 0 | |||||||||||||||||||||||||||
| 1.2 | 0 | −4.61337e8 | 0 | 3.28780e13 | 0 | −1.06405e16 | 0 | −3.83972e18 | 0 | ||||||||||||||||||||||||||||
| 1.3 | 0 | 2.85669e9 | 0 | −3.67409e13 | 0 | 1.18298e16 | 0 | 4.10814e18 | 0 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
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 | 4.40.a.a | ✓ | 3 |
| 4.b | odd | 2 | 1 | 16.40.a.d | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 4.40.a.a | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
| 16.40.a.d | 3 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace is the entire newspace \(S_{40}^{\mathrm{new}}(\Gamma_0(4))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} \)
|
| $3$ |
\( T^{3} + \cdots - 48\!\cdots\!76 \)
|
| $5$ |
\( T^{3} + \cdots - 93\!\cdots\!00 \)
|
| $7$ |
\( T^{3} + \cdots + 31\!\cdots\!84 \)
|
| $11$ |
\( T^{3} + \cdots - 13\!\cdots\!00 \)
|
| $13$ |
\( T^{3} + \cdots + 86\!\cdots\!56 \)
|
| $17$ |
\( T^{3} + \cdots - 54\!\cdots\!08 \)
|
| $19$ |
\( T^{3} + \cdots - 22\!\cdots\!44 \)
|
| $23$ |
\( T^{3} + \cdots + 11\!\cdots\!32 \)
|
| $29$ |
\( T^{3} + \cdots - 67\!\cdots\!24 \)
|
| $31$ |
\( T^{3} + \cdots + 23\!\cdots\!04 \)
|
| $37$ |
\( T^{3} + \cdots - 27\!\cdots\!76 \)
|
| $41$ |
\( T^{3} + \cdots - 28\!\cdots\!76 \)
|
| $43$ |
\( T^{3} + \cdots - 90\!\cdots\!00 \)
|
| $47$ |
\( T^{3} + \cdots + 30\!\cdots\!92 \)
|
| $53$ |
\( T^{3} + \cdots + 20\!\cdots\!68 \)
|
| $59$ |
\( T^{3} + \cdots + 13\!\cdots\!48 \)
|
| $61$ |
\( T^{3} + \cdots - 23\!\cdots\!28 \)
|
| $67$ |
\( T^{3} + \cdots - 53\!\cdots\!56 \)
|
| $71$ |
\( T^{3} + \cdots - 13\!\cdots\!12 \)
|
| $73$ |
\( T^{3} + \cdots - 66\!\cdots\!04 \)
|
| $79$ |
\( T^{3} + \cdots + 13\!\cdots\!72 \)
|
| $83$ |
\( T^{3} + \cdots + 14\!\cdots\!32 \)
|
| $89$ |
\( T^{3} + \cdots + 29\!\cdots\!24 \)
|
| $97$ |
\( T^{3} + \cdots - 17\!\cdots\!16 \)
|
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