Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1428,2,Mod(169,1428)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1428, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1428.169");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1428 = 2^{2} \cdot 3 \cdot 7 \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1428.d (of order \(2\), degree \(1\), minimal) |
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: | no |
| Analytic conductor: | \(11.4026374086\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.0.38738176.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 2x^{5} + 2x^{4} + 22x^{3} + 64x^{2} + 48x + 18 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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
\( x^{6} - 2x^{5} + 2x^{4} + 22x^{3} + 64x^{2} + 48x + 18 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -23\nu^{5} + 56\nu^{4} - 77\nu^{3} - 406\nu^{2} - 1362\nu - 585 ) / 459 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 7\nu^{5} - 37\nu^{4} + 70\nu^{3} + 77\nu^{2} + 42\nu - 567 ) / 153 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 31\nu^{5} - 142\nu^{4} + 310\nu^{3} + 341\nu^{2} + 186\nu - 1287 ) / 459 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 95\nu^{5} - 218\nu^{4} + 185\nu^{3} + 2269\nu^{2} + 4854\nu + 2097 ) / 459 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 35\nu^{5} - 83\nu^{4} + 95\nu^{3} + 793\nu^{2} + 1944\nu + 837 ) / 153 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( 2\beta_{5} - \beta_{4} + 5\beta_1 \)
|
| \(\nu^{3}\) | \(=\) |
\( 6\beta_{5} - 5\beta_{4} + 5\beta_{3} - 6\beta_{2} + 8\beta _1 - 8 \)
|
| \(\nu^{4}\) | \(=\) |
\( 21\beta_{3} - 31\beta_{2} - 56 \)
|
| \(\nu^{5}\) | \(=\) |
\( -85\beta_{5} + 64\beta_{4} + 64\beta_{3} - 85\beta_{2} - 135\beta _1 - 135 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1428\mathbb{Z}\right)^\times\).
| \(n\) | \(409\) | \(715\) | \(953\) | \(1261\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 169.1 |
|
0 | − | 1.00000i | 0 | − | 3.08060i | 0 | − | 1.00000i | 0 | −1.00000 | 0 | |||||||||||||||||||||||||||||||||
| 169.2 | 0 | − | 1.00000i | 0 | 1.27082i | 0 | − | 1.00000i | 0 | −1.00000 | 0 | |||||||||||||||||||||||||||||||||||
| 169.3 | 0 | − | 1.00000i | 0 | 2.80979i | 0 | − | 1.00000i | 0 | −1.00000 | 0 | |||||||||||||||||||||||||||||||||||
| 169.4 | 0 | 1.00000i | 0 | − | 2.80979i | 0 | 1.00000i | 0 | −1.00000 | 0 | ||||||||||||||||||||||||||||||||||||
| 169.5 | 0 | 1.00000i | 0 | − | 1.27082i | 0 | 1.00000i | 0 | −1.00000 | 0 | ||||||||||||||||||||||||||||||||||||
| 169.6 | 0 | 1.00000i | 0 | 3.08060i | 0 | 1.00000i | 0 | −1.00000 | 0 | |||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 17.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1428.2.d.b | ✓ | 6 |
| 3.b | odd | 2 | 1 | 4284.2.d.d | 6 | ||
| 17.b | even | 2 | 1 | inner | 1428.2.d.b | ✓ | 6 |
| 51.c | odd | 2 | 1 | 4284.2.d.d | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1428.2.d.b | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 1428.2.d.b | ✓ | 6 | 17.b | even | 2 | 1 | inner |
| 4284.2.d.d | 6 | 3.b | odd | 2 | 1 | ||
| 4284.2.d.d | 6 | 51.c | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{6} + 19T_{5}^{4} + 103T_{5}^{2} + 121 \)
acting on \(S_{2}^{\mathrm{new}}(1428, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( (T^{2} + 1)^{3} \)
|
| $5$ |
\( T^{6} + 19 T^{4} + \cdots + 121 \)
|
| $7$ |
\( (T^{2} + 1)^{3} \)
|
| $11$ |
\( T^{6} + 51 T^{4} + \cdots + 2601 \)
|
| $13$ |
\( (T^{3} + 5 T^{2} - T - 3)^{2} \)
|
| $17$ |
\( T^{6} - 4 T^{5} + \cdots + 4913 \)
|
| $19$ |
\( (T^{3} + 7 T^{2} + T - 27)^{2} \)
|
| $23$ |
\( (T^{2} + 9)^{3} \)
|
| $29$ |
\( T^{6} + 76 T^{4} + \cdots + 144 \)
|
| $31$ |
\( T^{6} + 52 T^{4} + \cdots + 4 \)
|
| $37$ |
\( T^{6} + 52 T^{4} + \cdots + 324 \)
|
| $41$ |
\( T^{6} + 135 T^{4} + \cdots + 9801 \)
|
| $43$ |
\( (T^{3} - 7 T^{2} - 45 T + 99)^{2} \)
|
| $47$ |
\( (T^{3} - 6 T^{2} + \cdots + 242)^{2} \)
|
| $53$ |
\( (T^{3} - 6 T^{2} + \cdots + 102)^{2} \)
|
| $59$ |
\( (T^{3} - 22 T^{2} + \cdots - 318)^{2} \)
|
| $61$ |
\( T^{6} + 180 T^{4} + \cdots + 139876 \)
|
| $67$ |
\( (T^{3} + 4 T^{2} + \cdots - 324)^{2} \)
|
| $71$ |
\( T^{6} + 344 T^{4} + \cdots + 169744 \)
|
| $73$ |
\( T^{6} + 460 T^{4} + \cdots + 2383936 \)
|
| $79$ |
\( T^{6} + 168 T^{4} + \cdots + 4356 \)
|
| $83$ |
\( (T^{3} + 16 T^{2} + \cdots - 144)^{2} \)
|
| $89$ |
\( (T^{3} + 16 T^{2} + \cdots + 16)^{2} \)
|
| $97$ |
\( T^{6} + 108 T^{4} + \cdots + 576 \)
|
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