Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1380,4,Mod(1,1380)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1380, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("1380.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1380 = 2^{2} \cdot 3 \cdot 5 \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1380.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: | \(81.4226358079\) |
| 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: |
\( x^{5} - 2x^{4} - 176x^{3} + 306x^{2} + 3519x - 7104 \)
|
| 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
\( x^{5} - 2x^{4} - 176x^{3} + 306x^{2} + 3519x - 7104 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 5\nu^{4} + 127\nu^{3} - 861\nu^{2} - 19095\nu + 15984 ) / 4944 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -29\nu^{4} + 5\nu^{3} + 4005\nu^{2} - 1725\nu - 4704 ) / 4944 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -25\nu^{4} - 17\nu^{3} + 4305\nu^{2} + 5865\nu - 67560 ) / 2472 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 25\nu^{4} + 17\nu^{3} - 4305\nu^{2} - 921\nu + 65088 ) / 2472 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{4} + \beta_{3} + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -5\beta_{4} + \beta_{3} - 10\beta_{2} + 2\beta _1 + 143 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 145\beta_{4} + 153\beta_{3} + 80\beta _1 + 105 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -725\beta_{4} + 105\beta_{3} - 1722\beta_{2} + 290\beta _1 + 19383 ) / 2 \)
|
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 | 3.00000 | 0 | 5.00000 | 0 | −32.6481 | 0 | 9.00000 | 0 | |||||||||||||||||||||||||||||||||
| 1.2 | 0 | 3.00000 | 0 | 5.00000 | 0 | −9.69768 | 0 | 9.00000 | 0 | ||||||||||||||||||||||||||||||||||
| 1.3 | 0 | 3.00000 | 0 | 5.00000 | 0 | −2.81836 | 0 | 9.00000 | 0 | ||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 3.00000 | 0 | 5.00000 | 0 | 1.87635 | 0 | 9.00000 | 0 | ||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 3.00000 | 0 | 5.00000 | 0 | 20.2878 | 0 | 9.00000 | 0 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(3\) | \(-1\) |
| \(5\) | \(-1\) |
| \(23\) | \(1\) |
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 | 1380.4.a.h | ✓ | 5 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1380.4.a.h | ✓ | 5 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{5} + 23T_{7}^{4} - 527T_{7}^{3} - 7051T_{7}^{2} - 3182T_{7} + 33968 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1380))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} \)
|
| $3$ |
\( (T - 3)^{5} \)
|
| $5$ |
\( (T - 5)^{5} \)
|
| $7$ |
\( T^{5} + 23 T^{4} + \cdots + 33968 \)
|
| $11$ |
\( T^{5} + 42 T^{4} + \cdots - 2339040 \)
|
| $13$ |
\( T^{5} + 80 T^{4} + \cdots + 5552 \)
|
| $17$ |
\( T^{5} + 81 T^{4} + \cdots + 38403648 \)
|
| $19$ |
\( T^{5} + 128 T^{4} + \cdots - 993453248 \)
|
| $23$ |
\( (T + 23)^{5} \)
|
| $29$ |
\( T^{5} + \cdots + 5443847628 \)
|
| $31$ |
\( T^{5} + \cdots + 14890703872 \)
|
| $37$ |
\( T^{5} + \cdots + 372360139936 \)
|
| $41$ |
\( T^{5} + \cdots - 4168498709988 \)
|
| $43$ |
\( T^{5} + 596 T^{4} + \cdots + 304456704 \)
|
| $47$ |
\( T^{5} + \cdots + 212305155072 \)
|
| $53$ |
\( T^{5} + \cdots - 157836488076 \)
|
| $59$ |
\( T^{5} + \cdots - 31975671211008 \)
|
| $61$ |
\( T^{5} + \cdots + 491988370560 \)
|
| $67$ |
\( T^{5} + \cdots + 5271316396288 \)
|
| $71$ |
\( T^{5} + \cdots + 13639030783488 \)
|
| $73$ |
\( T^{5} + \cdots + 24603032783824 \)
|
| $79$ |
\( T^{5} + \cdots + 761082085376 \)
|
| $83$ |
\( T^{5} + \cdots + 30371586154200 \)
|
| $89$ |
\( T^{5} + \cdots + 218886100830720 \)
|
| $97$ |
\( T^{5} + \cdots + 30050842904192 \)
|
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