Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [966,4,Mod(1,966)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(966, 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("966.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 966 = 2 \cdot 3 \cdot 7 \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 966.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: | \(56.9958450655\) |
| Analytic rank: | \(1\) |
| Dimension: | \(5\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{5} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - 319x^{3} - 666x^{2} + 23460x + 101568 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| 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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) 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
\( x^{5} - 319x^{3} - 666x^{2} + 23460x + 101568 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 4\nu^{4} - 23\nu^{3} - 1023\nu^{2} + 3500\nu + 51980 ) / 644 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -4\nu^{4} + 23\nu^{3} + 1345\nu^{2} - 4466\nu - 93196 ) / 644 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -5\nu^{4} + 69\nu^{3} + 1480\nu^{2} - 11942\nu - 106996 ) / 644 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 2\beta_{3} + 2\beta_{2} + 3\beta _1 + 128 \)
|
| \(\nu^{3}\) | \(=\) |
\( 16\beta_{4} - 10\beta_{3} + 10\beta_{2} + 173\beta _1 + 404 \)
|
| \(\nu^{4}\) | \(=\) |
\( 92\beta_{4} + 454\beta_{3} + 730\beta_{2} + 887\beta _1 + 22064 \)
|
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 |
|
−2.00000 | −3.00000 | 4.00000 | −20.1660 | 6.00000 | 7.00000 | −8.00000 | 9.00000 | 40.3320 | |||||||||||||||||||||||||||||||||
| 1.2 | −2.00000 | −3.00000 | 4.00000 | −15.3686 | 6.00000 | 7.00000 | −8.00000 | 9.00000 | 30.7372 | ||||||||||||||||||||||||||||||||||
| 1.3 | −2.00000 | −3.00000 | 4.00000 | 1.26807 | 6.00000 | 7.00000 | −8.00000 | 9.00000 | −2.53613 | ||||||||||||||||||||||||||||||||||
| 1.4 | −2.00000 | −3.00000 | 4.00000 | 8.95143 | 6.00000 | 7.00000 | −8.00000 | 9.00000 | −17.9029 | ||||||||||||||||||||||||||||||||||
| 1.5 | −2.00000 | −3.00000 | 4.00000 | 10.3151 | 6.00000 | 7.00000 | −8.00000 | 9.00000 | −20.6302 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(3\) | \(1\) |
| \(7\) | \(-1\) |
| \(23\) | \(1\) |
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 | 966.4.a.l | ✓ | 5 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 966.4.a.l | ✓ | 5 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{5} + 15T_{5}^{4} - 303T_{5}^{3} - 2332T_{5}^{2} + 32028T_{5} - 36288 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(966))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 2)^{5} \)
|
| $3$ |
\( (T + 3)^{5} \)
|
| $5$ |
\( T^{5} + 15 T^{4} + \cdots - 36288 \)
|
| $7$ |
\( (T - 7)^{5} \)
|
| $11$ |
\( T^{5} + 19 T^{4} + \cdots + 62634096 \)
|
| $13$ |
\( T^{5} + 19 T^{4} + \cdots - 4287000 \)
|
| $17$ |
\( T^{5} + 42 T^{4} + \cdots + 117524736 \)
|
| $19$ |
\( T^{5} + \cdots + 14087905104 \)
|
| $23$ |
\( (T + 23)^{5} \)
|
| $29$ |
\( T^{5} + \cdots - 1129841688 \)
|
| $31$ |
\( T^{5} + \cdots + 126870808576 \)
|
| $37$ |
\( T^{5} + \cdots - 378516115880 \)
|
| $41$ |
\( T^{5} + \cdots - 589161509700 \)
|
| $43$ |
\( T^{5} + \cdots - 421735977792 \)
|
| $47$ |
\( T^{5} + \cdots - 1783648512 \)
|
| $53$ |
\( T^{5} + \cdots - 4637528572932 \)
|
| $59$ |
\( T^{5} + \cdots + 257121804240 \)
|
| $61$ |
\( T^{5} + \cdots - 170741533284 \)
|
| $67$ |
\( T^{5} + \cdots + 9598713835584 \)
|
| $71$ |
\( T^{5} + \cdots - 7396227885312 \)
|
| $73$ |
\( T^{5} + \cdots - 1792003628016 \)
|
| $79$ |
\( T^{5} + \cdots - 1271307835392 \)
|
| $83$ |
\( T^{5} + \cdots + 9587344906272 \)
|
| $89$ |
\( T^{5} + \cdots - 2368861030872 \)
|
| $97$ |
\( T^{5} + \cdots + 25515582030104 \)
|
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