Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1666,2,Mod(883,1666)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1666, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("1666.883");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1666 = 2 \cdot 7^{2} \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1666.b (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: | \(13.3030769767\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | 8.0.10070523904.3 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} + 16x^{6} + 38x^{4} + 16x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{17}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} + 16x^{6} + 38x^{4} + 16x^{2} + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{6} + 17\nu^{4} + 55\nu^{2} + 35 ) / 12 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{7} + 17\nu^{5} + 55\nu^{3} + 71\nu ) / 12 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{7} - 15\nu^{5} - 23\nu^{3} + 7\nu ) / 4 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -2\nu^{6} - 31\nu^{4} - 62\nu^{2} - 13 ) / 6 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{6} + 16\nu^{4} + 37\nu^{2} + 8 ) / 2 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -11\nu^{7} - 175\nu^{5} - 401\nu^{3} - 133\nu ) / 12 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -5\nu^{7} - 79\nu^{5} - 175\nu^{3} - 53\nu ) / 4 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{7} - \beta_{6} - \beta_{3} + \beta_{2} ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{5} - \beta_{4} + 2\beta _1 - 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -15\beta_{7} + 17\beta_{6} + 11\beta_{3} - 5\beta_{2} ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( 16\beta_{5} + 18\beta_{4} - 24\beta _1 + 45 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 201\beta_{7} - 233\beta_{6} - 133\beta_{3} + 53\beta_{2} ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( -217\beta_{5} - 251\beta_{4} + 310\beta _1 - 580 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( -2663\beta_{7} + 3097\beta_{6} + 1727\beta_{3} - 673\beta_{2} ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1666\mathbb{Z}\right)^\times\).
| \(n\) | \(785\) | \(885\) |
| \(\chi(n)\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 883.1 |
|
1.00000 | − | 2.37608i | 1.00000 | 1.53436i | − | 2.37608i | 0 | 1.00000 | −2.64575 | 1.53436i | ||||||||||||||||||||||||||||||||||||||||
| 883.2 | 1.00000 | − | 2.37608i | 1.00000 | 1.53436i | − | 2.37608i | 0 | 1.00000 | −2.64575 | 1.53436i | |||||||||||||||||||||||||||||||||||||||||
| 883.3 | 1.00000 | − | 0.595188i | 1.00000 | − | 2.76510i | − | 0.595188i | 0 | 1.00000 | 2.64575 | − | 2.76510i | |||||||||||||||||||||||||||||||||||||||
| 883.4 | 1.00000 | − | 0.595188i | 1.00000 | − | 2.76510i | − | 0.595188i | 0 | 1.00000 | 2.64575 | − | 2.76510i | |||||||||||||||||||||||||||||||||||||||
| 883.5 | 1.00000 | 0.595188i | 1.00000 | 2.76510i | 0.595188i | 0 | 1.00000 | 2.64575 | 2.76510i | |||||||||||||||||||||||||||||||||||||||||||
| 883.6 | 1.00000 | 0.595188i | 1.00000 | 2.76510i | 0.595188i | 0 | 1.00000 | 2.64575 | 2.76510i | |||||||||||||||||||||||||||||||||||||||||||
| 883.7 | 1.00000 | 2.37608i | 1.00000 | − | 1.53436i | 2.37608i | 0 | 1.00000 | −2.64575 | − | 1.53436i | |||||||||||||||||||||||||||||||||||||||||
| 883.8 | 1.00000 | 2.37608i | 1.00000 | − | 1.53436i | 2.37608i | 0 | 1.00000 | −2.64575 | − | 1.53436i | |||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.b | odd | 2 | 1 | inner |
| 17.b | even | 2 | 1 | inner |
| 119.d | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1666.2.b.o | ✓ | 8 |
| 7.b | odd | 2 | 1 | inner | 1666.2.b.o | ✓ | 8 |
| 17.b | even | 2 | 1 | inner | 1666.2.b.o | ✓ | 8 |
| 119.d | odd | 2 | 1 | inner | 1666.2.b.o | ✓ | 8 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1666.2.b.o | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 1666.2.b.o | ✓ | 8 | 7.b | odd | 2 | 1 | inner |
| 1666.2.b.o | ✓ | 8 | 17.b | even | 2 | 1 | inner |
| 1666.2.b.o | ✓ | 8 | 119.d | odd | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1666, [\chi])\):
|
\( T_{3}^{4} + 6T_{3}^{2} + 2 \)
|
|
\( T_{5}^{4} + 10T_{5}^{2} + 18 \)
|
|
\( T_{11}^{4} + 10T_{11}^{2} + 18 \)
|
|
\( T_{13}^{2} - 14 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1)^{8} \)
|
| $3$ |
\( (T^{4} + 6 T^{2} + 2)^{2} \)
|
| $5$ |
\( (T^{4} + 10 T^{2} + 18)^{2} \)
|
| $7$ |
\( T^{8} \)
|
| $11$ |
\( (T^{4} + 10 T^{2} + 18)^{2} \)
|
| $13$ |
\( (T^{2} - 14)^{4} \)
|
| $17$ |
\( T^{8} + 36 T^{6} + \cdots + 83521 \)
|
| $19$ |
\( (T^{4} - 44 T^{2} + 36)^{2} \)
|
| $23$ |
\( (T^{4} + 12 T^{2} + 8)^{2} \)
|
| $29$ |
\( (T^{4} + 42 T^{2} + 98)^{2} \)
|
| $31$ |
\( (T^{4} + 76 T^{2} + 72)^{2} \)
|
| $37$ |
\( (T^{4} + 54 T^{2} + 162)^{2} \)
|
| $41$ |
\( (T^{4} + 20 T^{2} + 72)^{2} \)
|
| $43$ |
\( (T^{2} - 4 T - 24)^{4} \)
|
| $47$ |
\( (T^{4} - 44 T^{2} + 36)^{2} \)
|
| $53$ |
\( (T^{2} + 4 T - 24)^{4} \)
|
| $59$ |
\( (T^{4} - 116 T^{2} + 2916)^{2} \)
|
| $61$ |
\( (T^{4} + 10 T^{2} + 18)^{2} \)
|
| $67$ |
\( (T^{2} + 4 T - 108)^{4} \)
|
| $71$ |
\( (T^{4} + 168 T^{2} + 1568)^{2} \)
|
| $73$ |
\( (T^{4} + 160 T^{2} + 4608)^{2} \)
|
| $79$ |
\( (T^{4} + 320 T^{2} + 18432)^{2} \)
|
| $83$ |
\( (T^{2} - 126)^{4} \)
|
| $89$ |
\( (T^{4} - 212 T^{2} + 36)^{2} \)
|
| $97$ |
\( (T^{4} + 76 T^{2} + 72)^{2} \)
|
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