Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1932,4,Mod(1,1932)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1932, 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("1932.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1932 = 2^{2} \cdot 3 \cdot 7 \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1932.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: | \(113.991690131\) |
| Analytic rank: | \(1\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 2x^{5} - 284x^{4} - 618x^{3} + 12170x^{2} + 20060x - 81607 \)
|
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 2x^{5} - 284x^{4} - 618x^{3} + 12170x^{2} + 20060x - 81607 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{5} - 15\nu^{4} + 5\nu^{3} + 727\nu^{2} - 23601\nu - 7485 ) / 4606 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{2} - 3\nu - 92 ) / 7 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -3\nu^{5} + 45\nu^{4} + 643\nu^{3} - 7445\nu^{2} - 35135\nu + 168531 ) / 4606 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 7\nu^{5} - 11\nu^{4} - 1845\nu^{3} - 6097\nu^{2} + 49865\nu + 152337 ) / 4606 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 7\beta_{3} + 3\beta _1 + 92 \)
|
| \(\nu^{3}\) | \(=\) |
\( 7\beta_{4} + 56\beta_{3} + 21\beta_{2} + 185\beta _1 + 514 \)
|
| \(\nu^{4}\) | \(=\) |
\( 49\beta_{5} + 140\beta_{4} + 1953\beta_{3} + 77\beta_{2} + 1769\beta _1 + 19050 \)
|
| \(\nu^{5}\) | \(=\) |
\( 735\beta_{5} + 2065\beta_{4} + 23926\beta_{3} + 5656\beta_{2} + 47030\beta _1 + 223781 \)
|
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 | −10.5878 | 0 | 7.00000 | 0 | 9.00000 | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −3.00000 | 0 | −7.32765 | 0 | 7.00000 | 0 | 9.00000 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −3.00000 | 0 | −2.86445 | 0 | 7.00000 | 0 | 9.00000 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | −3.00000 | 0 | 3.04385 | 0 | 7.00000 | 0 | 9.00000 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | −3.00000 | 0 | 7.05594 | 0 | 7.00000 | 0 | 9.00000 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | −3.00000 | 0 | 18.6801 | 0 | 7.00000 | 0 | 9.00000 | 0 | |||||||||||||||||||||||||||||||||||||
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 | 1932.4.a.f | ✓ | 6 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1932.4.a.f | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{6} - 8T_{5}^{5} - 259T_{5}^{4} + 478T_{5}^{3} + 12355T_{5}^{2} - 5014T_{5} - 89160 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1932))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( (T + 3)^{6} \)
|
| $5$ |
\( T^{6} - 8 T^{5} + \cdots - 89160 \)
|
| $7$ |
\( (T - 7)^{6} \)
|
| $11$ |
\( T^{6} + 63 T^{5} + \cdots + 2490900 \)
|
| $13$ |
\( T^{6} + 28 T^{5} + \cdots - 857872140 \)
|
| $17$ |
\( T^{6} + \cdots - 17023178160 \)
|
| $19$ |
\( T^{6} + \cdots + 8639531364 \)
|
| $23$ |
\( (T - 23)^{6} \)
|
| $29$ |
\( T^{6} + \cdots + 122200153220 \)
|
| $31$ |
\( T^{6} + \cdots - 973404374048 \)
|
| $37$ |
\( T^{6} + \cdots + 24647613477580 \)
|
| $41$ |
\( T^{6} + \cdots - 69205198102510 \)
|
| $43$ |
\( T^{6} + \cdots + 7431566951520 \)
|
| $47$ |
\( T^{6} + \cdots - 72970150468608 \)
|
| $53$ |
\( T^{6} + \cdots + 192376437961410 \)
|
| $59$ |
\( T^{6} + \cdots - 37\!\cdots\!20 \)
|
| $61$ |
\( T^{6} + \cdots + 56\!\cdots\!90 \)
|
| $67$ |
\( T^{6} + \cdots + 28\!\cdots\!12 \)
|
| $71$ |
\( T^{6} + \cdots + 17390635293088 \)
|
| $73$ |
\( T^{6} + \cdots - 35\!\cdots\!44 \)
|
| $79$ |
\( T^{6} + \cdots + 710227476919200 \)
|
| $83$ |
\( T^{6} + \cdots + 44\!\cdots\!76 \)
|
| $89$ |
\( T^{6} + \cdots - 29\!\cdots\!96 \)
|
| $97$ |
\( T^{6} + \cdots + 13\!\cdots\!60 \)
|
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